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if p iq is one of the roots of quadratic equation prove that the other root is p iq - Mathematics - TopperLearning.com | us8i0zii
if p +iq is one of the roots of quadratic equation prove that the other root is p- iq
Asked by aishwarya ghogare | 26th Aug, 2010, 09:59: PM
\mathrm{The}\quad \mathrm{roots}\quad \mathrm{of}\quad a\quad \mathrm{quadratic}\quad \mathrm{equation}\quad {\mathrm{ax}}^{2}\quad +\quad \mathrm{bx}\quad +c\quad =0\quad \mathrm{are}
\frac{-b+\sqrt{{b}^{2}-4\mathrm{ac}}}{2a}\quad \mathrm{and}\quad \quad \frac{-b-\sqrt{{b}^{2}-4\mathrm{ac}}}{2a}
\mathrm{These}\quad \mathrm{roots}\quad \mathrm{are}\quad \mathrm{complex}\quad \mathrm{if}\quad {b}^{2}-4\mathrm{ac}\quad <\quad 0
\mathrm{Hence}\quad \sqrt{{b}^{2}-4\mathrm{ac}}\quad \mathrm{is}\quad \mathrm{the}\quad \mathrm{imaginary}\quad \mathrm{part}\quad \mathrm{of}\quad \mathrm{the}\quad \mathrm{root}.
\mathrm{Therefore}\quad \mathrm{the}\quad \mathrm{complex}\quad \mathrm{roots}\quad \mathrm{are}\quad \mathrm{always}\quad \mathrm{conjugates}\quad \mathrm{of}\quad \mathrm{each}\quad \mathrm{other}.
\mathrm{So}\quad \mathrm{if}\quad \mathrm{one}\quad \mathrm{root}\quad \mathrm{is}\quad p+\mathrm{iq}\quad \mathrm{then}\quad \mathrm{the}\quad \mathrm{other}\quad \mathrm{is}\quad p-\mathrm{iq}\quad ,\quad \mathrm{its}\quad \mathrm{complex}\quad \mathrm{conjugate}.
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Evaluate the integrals \int_0^{\frac{\pi}{4}}\cos^2(4t-\frac{\pi}{4})dtZS
{\int }_{0}^{\frac{\pi }{4}}{\mathrm{cos}}^{2}\left(4t-\frac{\pi }{4}\right)dt
Given integral is
{\int }_{0}^{\frac{\pi }{4}}{\mathrm{cos}}^{2}\left(4t-\frac{\pi }{4}\right)dt
u=4t-\frac{\pi }{4}
u=4t-\frac{\pi }{4}
\frac{du}{dt}=4
dt=\frac{du}{4}
u=4t-\frac{\pi }{4}
dt=\frac{du}{4}
{\int }_{0}^{\frac{\pi }{4}}{\mathrm{cos}}^{2}\left(4t-\frac{\pi }{4}\right)dt
=\frac{1}{8}{\left[u+\frac{\mathrm{sin}\left(2u\right)}{2}\right]}_{-\frac{\pi }{4}}^{\frac{3\pi }{4}}
=\frac{1}{8}\left[\frac{3\pi }{4}+\frac{\pi }{4}+\frac{1}{2}\left(\mathrm{sin}\frac{3\pi }{2}-\mathrm{sin}\left(-\frac{\pi }{2}\right)\right)\right]
=\frac{\pi }{8}
\left({x}^{2}+2xy-4{y}^{2}\right)dx-\left({x}^{2}-8xy-4{y}^{2}\right)dy=0
x={t}^{2},y=2t,0\le t\le 5
\int {x}^{3}\sqrt{49-{x}^{2}}dx
{\int }_{\frac{1}{8}}^{1}{x}^{-\frac{1}{3}}{\left(1-{x}^{\frac{2}{3}}\right)}^{\frac{3}{2}}dx
Integrals Evaluate the following integrals. Include absolute values only when needed.
\int \frac{{e}^{2x}}{4+{e}^{2x}}dx
a\left(t\right)=2i+2tk,v\left(0\right)=3i-j,r\left(0\right)=j+k
\int {e}^{3x}\mathrm{cos}2xdx
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SAT Data - Tables | Brilliant Math & Science Wiki
SAT Data - Tables
To successfully solve problems with data tables on the SAT, you need to know how to interpret:
SAT Tips for Data - Tables
\begin{array}{|c|c|c|} \hline \text{Number of Eggs in a Carton} & \text{Price}\\ \hline 6 & \$3.99\\ \hline 12 & \$5.80\\ \hline 18 & \$6.30\\ \hline \end{array}
Which of the following is the closest approximation of the price of 1 egg when buying a carton of 12?
\ \ 0.40
\ \ 0.50
\ \ 0.60
\ \ 0.70
\ \ 5.80
According to the table, a carton of 12 eggs costs $5.80. So, 1 egg costs
\frac{\$5.80}{12}=\$0.48.
The best approximation is $0.50.
This is approximately how much 1 egg costs when buying a carton of 18 eggs.
This is approximately how much 1 egg costs when buying a carton of 6 eggs.
This answer is just offered to confuse you.
This is how much a 12-egg carton costs. You must divide this by 12 in order to obtain the price of one egg in a 12-egg carton.
\begin{array}{|c|c|c|} \hline \text{Number of Eggs in a Carton} & \text{Price}\\ \hline 1 & \$1.00\\ \hline 6 & \$3.99\\ \hline 12 & \$5.80\\ \hline 18 & \$6.30\\ \hline \end{array}
What would be the least amount of money needed to purchase exactly 100 eggs?
\ \ \$31.50
\ \ \$35.00
\ \ \$39.49
\ \ \$40.80
\ \ \$41.50
1 egg in a 6-egg carton costs $3.99/6=$0.67, 1 egg in a 12-egg carton costs $5.80/12=$0.48, and 1 egg in an 18-egg carton costs $6.20/18= $0.35. Since the price per egg is lowest when purchasing 18-egg cartons, we should ideally buy as many of those as possible to reduce the cost.
18 doesn't divide 100 exactly. So we cannot purchase 100 eggs only with 18-egg cartons. The maximum 18-egg cartons we can purchase is 5:
5 cartons of 18, 1 carton of 6, and 4 single eggs
=5 \cdot \$6.30+1\cdot \$3.99 + 4\cdot \$1 = \$39.49.
Clearly, the only other combination involving 5 cartons of 18 eggs will be more expensive:
5 cartons of 18 and 10 single eggs
=5\cdot \$6.30 + 10\cdot \$1 = \$41.50.
What happens if we purchase four 18-egg cartons?
4 cartons of 18, 2 cartons of 12, and 3 single eggs
=4 \cdot \$6.30 + 2 \cdot \$5.80 + 3\cdot \$1 = \$39.80.
And, substituting 6-egg cartons for the 12-egg cartons in this last combination will only increase the price since the price per egg in a 6-egg carton is higher than that in a 12-egg carton.
What about combinations with three 18-egg cartons? Let's say we could buy three 18-egg cartons, and let's also imagine we were able to buy all the remaining
100-3\cdot 18 = 46
eggs at the price we would pay for an egg when buying 12-egg cartons, $0.48. Note that if we were able to do this, this would be the cheapest price involving three 18-egg cartons.
Then we would have to pay
3\cdot \$6.30 + 46 \cdot \$0.48 = \$40.98 > \$39.49.
By analogous reasoning, we reject all other combinations involving three, two, one, or zero 18-egg cartons.
This is the price of five 18-egg cartons, or 90 eggs each at $0.35. But, we need 100 eggs, not 90.
If we were to buy the hundred eggs at the cheapest rate of
\$0.35
(1 egg in an 18-egg carton costs $0.35), we would get this wrong answer. The eggs are only sold as singles, in which case the price of one egg is
\$1.00
or in cartons of 6, 12, or 18.
This is the combination of 4 cartons of 18, 2 cartons of 12, and 4 single eggs
=4 \cdot \$6.30 + 2 \cdot \$5.80 + 4\cdot \$1.00 = \$40.80
This is the combination of 5 cartons of 18 eggs and 10 single eggs
=5\cdot \$6.30 + 10\cdot \$1.00 = \$41.50.
\begin{array}{|c|c|c|c|c|} \hline & \text{High School} & \text{Middle School} & \text{ Total}\\ \hline \text{Boys} & a & 2b & X\\ \hline \text{Girls} & 2a & b & Y\\ \hline \text{Total} & M & N & T\\ \hline \end{array}
In the table above, each letter represents the number of students in that category. Which of the following equals
M-N?
\ \ X
\ \ Y
\ \ 3(Y-X)
\ \ \frac{T}{3}
\ \ X+Y
Cite as: SAT Data - Tables. Brilliant.org. Retrieved from https://brilliant.org/wiki/sat-data-tables/
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Evaluate the integral. \int \tan^{5}xdx
Holly Guerrero 2021-12-31 Answered
\int {\mathrm{tan}}^{5}xdx
{\mathrm{tan}}^{5}x
into simpler powers
{\mathrm{tan}}^{3}x
{\mathrm{tan}}^{2}x
and change
{\mathrm{tan}}^{2}x
{\mathrm{sec}}^{2}x-1
\int {\mathrm{tan}}^{5}xdx=\int {\mathrm{tan}}^{3}x\left({\mathrm{tan}}^{2}xdx\right)
=\int {\mathrm{tan}}^{3}x\left({\mathrm{sec}}^{2}x-1\right)dx
=\int \left({\mathrm{tan}}^{3}x{\mathrm{sec}}^{2}x-{\mathrm{tan}}^{3}x\right)dx
use the integral theorem and integrate both the of
\int \left({\mathrm{tan}}^{3}x{\mathrm{sec}}^{2}x-{\mathrm{tan}}^{3}x\right)dx
terms with respect to dx and simplify to obtain the value of the integral.
\int \left({\mathrm{tan}}^{3}x{\mathrm{sec}}^{2}x-{\mathrm{tan}}^{3}x\right)dx=\int {\mathrm{tan}}^{3}x{\mathrm{sec}}^{2}xdx-\int {\mathrm{tan}}^{3}xdx
=\int \left({\mathrm{tan}}^{3}x{\mathrm{sec}}^{2}x\right)dx-\int {\mathrm{tan}}^{2}xdx\left(\mathrm{tan}x\right)dx
=\int {\mathrm{tan}}^{3}x{\mathrm{sec}}^{2}xdx-\int \left({\mathrm{sec}}^{2}x-1\right)\mathrm{tan}xdx
=\int {\mathrm{tan}}^{3}x{\mathrm{sec}}^{2}xdx-\int \mathrm{tan}x{\mathrm{sec}}^{2}xdx+\int \mathrm{tan}xdx
=\frac{1}{4}{\mathrm{tan}}^{4}x-\frac{1}{2}{\mathrm{tan}}^{2}x+\mathrm{ln}\mathrm{sec}x+C
\int {\mathrm{tan}\left(x\right)}^{5}dx
\frac{1}{4}×{\mathrm{tan}\left(x\right)}^{4}-\int {\mathrm{tan}\left(x\right)}^{3}dx
\frac{1}{4}×{\mathrm{tan}\left(x\right)}^{4}-\left(\frac{1}{2}×{\mathrm{tan}\left(x\right)}^{2}-\int \mathrm{tan}\left(x\right)dx\right)
\frac{1}{4}×{\mathrm{tan}\left(x\right)}^{4}-\left(\frac{1}{2}×{\mathrm{tan}\left(x\right)}^{2}-\int \frac{\mathrm{sin}\left(x\right)}{\mathrm{cos}\left(x\right)}dx\right)
\frac{1}{4}×{\mathrm{tan}\left(x\right)}^{4}-\left(\frac{1}{2}×{\mathrm{tan}\left(x\right)}^{2}-\int -\frac{1}{t}dt\right)
Use properties of integrals
\frac{1}{4}×{\mathrm{tan}\left(x\right)}^{4}-\left(\frac{1}{2}×{\mathrm{tan}\left(x\right)}^{2}+\int \frac{1}{t}dt\right)
\frac{1}{4}×{\mathrm{tan}\left(x\right)}^{4}-\left(\frac{1}{2}×{\mathrm{tan}\left(x\right)}^{2}+\mathrm{ln}\left(|t|\right)\right)
\frac{1}{4}×{\mathrm{tan}\left(x\right)}^{4}-\left(\frac{1}{2}×{\mathrm{tan}\left(x\right)}^{2}+\mathrm{ln}\left(|\mathrm{cos}\left(x\right)|\right)\right)
\frac{{\mathrm{tan}\left(x\right)}^{4}}{4}-\frac{{\mathrm{tan}\left(x\right)}^{2}}{2}-\mathrm{ln}\left(|\mathrm{cos}\left(x\right)|\right)
\frac{{\mathrm{tan}\left(x\right)}^{4}}{4}-\frac{{\mathrm{tan}\left(x\right)}^{2}}{2}-\mathrm{ln}\left(|\mathrm{cos}\left(x\right)|\right)+C
\begin{array}{}{\mathrm{tan}}^{5}\left(x\right)dx\\ =\int \left({\mathrm{tan}}^{2}\left(x\right){\right)}^{2}\mathrm{tan}\left(x\right)dx\\ =\int \left({\mathrm{sec}}^{2}\left(x\right)-1{\right)}^{2}\mathrm{tan}\left(x\right)dx\\ =\int \frac{\left({u}^{2}-1{\right)}^{2}}{u}du\\ =\int \left({u}^{3}-2u+\frac{1}{u}\right)du\\ =\int {u}^{3}du-2\int udu+\int \frac{1}{u}du\\ \int {u}^{3}du\\ =\frac{{u}^{4}}{4}\\ \int udu\\ =\frac{{u}^{2}}{2}\\ \int \frac{1}{u}du\\ =\mathrm{ln}\left(u\right)\\ \int {u}^{3}du-2\int udu+\int \frac{1}{u}du\\ =\mathrm{ln}\left(u\right)+\frac{{u}^{4}}{4}-{u}^{2}\\ =\mathrm{ln}\left(\mathrm{sec}\left(x\right)\right)+\frac{{\mathrm{sec}}^{4}\left(x\right)}{4}-{\mathrm{sec}}^{2}\left(x\right)\\ \int {\mathrm{tan}}^{5}\left(x\right)dx\\ =\mathrm{ln}\left(|\mathrm{sec}\left(x\right)|\right)+\frac{{\mathrm{sec}}^{4}\left(x\right)}{4}-{\mathrm{sec}}^{2}\left(x\right)+C\end{array}
{\int }_{0}^{\pi }{\left(1-\mathrm{cos}2x\right)}^{\frac{3}{2}}dx
{\int }_{0}^{1}\frac{2\mathrm{ln}\left(x\right)}{\sqrt{x}}dx
\frac{dy}{dx}=2x
Use the methods introduced evaluate the following integrals.
{\int }_{-2}^{1}\frac{3}{{x}^{2}+4x+13}dx
Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.
\int \frac{dx}{\sqrt{9{x}^{2}-100}},x>\frac{10}{3}
\frac{dy}{dx}=\frac{{x}^{3}}{{y}^{2}}
\int \frac{{e}^{-x}}{1+{e}^{-x}}dx
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3Blue1Brown - Limits and the definition of derivatives
Chapter 8Limits and the definition of derivatives
"Calculus is all about making curvy things look straight" Anonymous Professor
The last several lessons have been about the idea of a derivative, and before moving on to integrals, I want to take some time to talk about limits.
To be honest, the idea of a limit is not really anything new. If you know what the word "approach" means you pretty much already know what a limit is, you could say the rest is a matter of assigning fancy notation to the intuitive idea of one value that just gets closer to another. But there are a number of reasons to dive deeper into this topic.
For one thing it’s worth showing is how the way I’ve been describing derivatives so far lines up with the the formal definition of a derivative as it’s typically presented in most courses and textbooks. I want to give you a little confidence that thinking of terms like
dx
df
as concrete non-zero nudges is not just some trick for building intuition; it’s actually backed up by the formal definition of a derivative in all of its rigor.
Let’s take a look at the formal definition of the derivative. As a reminder, when you have some function
f(x)
, to think about the derivative at a particular input, maybe
x=2
, you start by imagining nudging that input by some tiny
dx
, and looking at the resulting change to the output,
df
df/dx
, which can nicely be thought of as the rise-over-run slope between the starting point on the graph and the nudged point, is almost what the derivative is. The actual derivative is whatever this ratio approaches as
dx
0
Just to spell out what is meant here, that nudge to the output "
df
" is is the difference between
f
at the starting-input plus
dx
f
at the starting-input; the change to the output caused by the nudge
dx
To express that you want to find what this ratio approaches as
dx
0
, you write "lim", for limit, with "
dx
0
" below it.
Now, you’ll almost never see terms with a lowercase
d
dx
, inside a limit like this. Instead the standard is to use a different variable, like
\Delta x
(delta
x
), or commonly "
h
" for some reason.
The way I like to think of it is that terms with this lowercase
d
in the typical derivative expression have built into them the idea of a limit, the idea that
dx
is supposed to eventually approach
0
So in a sense this lefthand side "
df/dx
", the ratio we’ve been thinking about in the past lessons, is just shorthand for what the righthand side spells out in more detail, writing out exactly what we mean by
df
, and writing out this limit process explicitly. And that righthand side is the formal definition of a derivative, as you’d commonly see it in any calculus textbook.
The formal derivative definition. Notice that nothing about this definition references the idea of an "infinitely small" change.
No infinitely small rant
Now, if you’ll pardon me for a small rant here, I want to emphasize that nothing about this righthand side references the paradoxical idea of an "infinitely small" change. The point of limits is to avoid that.
h
is the exact same thing as the "
dx
" I’ve been referencing throughout the series. It’s a nudge to the input of
with some non-zero, finitely small size, like
0.001
, that we’re analyzing for arbitrarily small choices of
h
. In fact, the reason people introduce a new variable name into this formal definition, rather than just using
dx
, is to be clear that these changes to the input are ordinary numbers that have nothing to do with infinitesimals.
Rather than interpret
dx
as an "infinitely small change", as others like to do, I think you can and should interpret
dx
as a concrete, finitely small nudge, just so long as you remember to ask what happens as it approaches
0
. This method helps build a stronger intuition for where the rules of calculus come from and, more than that, it’s not just some trick for building intuitions. Everything I’ve been saying about derivatives with this concrete-finitely-small-nudge philosophy is a translation of the formal definition of derivatives.
Long story short, the big fuss about limits is that they let us avoid talking about infinitely small changes by instead asking what happens as the size of some change to our variable approaches
0
So why do we teach the limit notation and how does it help us formalize what it means for one value to approach another?
Maybe you are convinced that, on an intuitive level, the language and tools we have invented so far give us what we need to solve these types of problems. When thinking about small nudges to a function's input, as long as we imagine that nudge approaching zero, things usually work out. In the same way, we can zoom in on a point on a curve and convince ourselves that if we zoom in far enough the curve really starts to look straight.
Zoom in on the exponential function
e^x
x=1
and the rate of change (slope) starts to look a whole lot like
1
But, you may still have doubts and questions, as certainly some of the creators of calculus did. For example, just how far do we have to zoom in on a point before a function starts to look like its tangent line at that point?
If you are applying calculus, your primary concern is probably the accuracy of the approximation and you might play a cat and mouse game to choose a small enough
dx
so that the approximation is is accurate enough to calculate a useful result. A mathematician, on the other hand, is playing a different game; they are interested in whether they can rig the cat and mouse game of choosing and smaller values and getting a more accurate result so that they will always win.
Even more than this, mathematicians care about ensuring all their statements have a precise and airtight meaning, which does not rest on visual intuition. As sensible as it might seem to say that this ratio approaches a certain value, and to draw a line through the graph whose slope approaches that of a tangent line, those seeking rigor insist on something better. In the next chapter, we’ll take a look the epsilon-delta definition of limits, which finally puts this idea of "approaching" onto a firm foundation.
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Estimate and compare liquidation costs across stocks - MATLAB liquidityFactor - MathWorks Australia
Determine Liquidity Factor for Stocks
Estimate and compare liquidation costs across stocks
lf = liquidityFactor(k,trade)
lf = liquidityFactor(k,trade) returns the ratio of liquidation costs due to liquidity demand by stock for an equal investment value, or liquidity factor. liquidityFactor uses the Kissell Research Group (KRG) transaction cost analysis object k and trade data trade.
Determine liquidity factor lf for each stock using the Kissell Research Group transaction cost analysis object k. Display the first three liquidity factor values.
lf = liquidityFactor(k,TradeData);
lf(1:3)
lf returns the ratios for stock comparison due to liquidity demands.
Example: trade = table({'XYZ'},100.00,860000,0.27,'VariableNames',{'Symbol' 'Price' 'ADV' 'Volatility'})
Example: trade = struct('Symbol','XYZ','Price',100.00,'ADV',860000,'Volatility',0.27)
lf — Liquidity factor
Liquidity factor, returned as a vector. The vector values are ratios that compare the liquidation costs due to liquidity demands across stocks in trade for the dollar value and execution strategy.
The Liquidity Factor (LF) is a stock-specific measure of price sensitivity to investment dollars.
LF provides investors with a fair and consistent comparison of expected liquidation costs across stocks. LF incorporates stock-specific information to determine its sensitivity to order flow and investment dollars. The LF metric shows the ratio of liquidation costs due to liquidity demand by stock for an equal investment value in each stock. Market impact relies on the order size or shares traded which vary from order to order. LF provides an apples-to-apples comparison across financial instruments. Consider a stock I that has an LF = 0.10 and a stock II that has an LF = 0.20. Stock II is twice as expensive to transact for an equal dollar value. An investor buys or sells $1 million dollars of stock in stock I and stock II utilizing the same execution strategy. The cost of stock II is twice as large as stock I. The LF metric incorporates stock liquidity, volatility, and price to determine the LF trading cost parameter.
The LF model is
\text{LF}={a}_{1}\cdot {\left(\frac{1}{ADV}\right)}^{{a}_{2}}\cdot {\sigma }^{{a}_{3}}\cdot {\left(\frac{1}{Price}\right)}^{{a}_{2}}\cdot Pric{e}^{{a}_{5}}.
\sigma
is price volatility. ADV is the average daily volume of the stock. Price is the current stock price in local currency.
{a}_{1}
{a}_{2}
{a}_{3}
{a}_{5}
{a}_{1}
{a}_{2}
{a}_{3}
{a}_{5}
Price shape
You can expand the LF model to include a stock-specific factor such as market capitalization, beta, P/E ratio, and Debt/Equity ratio. In this case,
{X}_{k}
denotes the stock-specific factor and
{a}_{k}
denotes the corresponding shape parameter. For details about implementing an expanded LF model, contact the Kissell Research Group.
krg | iStar | marketImpact | priceAppreciation | timingRisk
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3Blue1Brown - Trig Derivatives through geometry
Chapter 4Trig Derivatives through geometry
Let's try to reason through what the derivatives of the functions sine and cosine should be. For background, you should be comfortable with how to think about both of these functions using the unit circle; that is, the circle with radius
1
For example, how would you interpret the value
\sin(0.8)
\theta = 0.8
is understood to be in radians? You might imagine walking around a circle with a radius of
1
, starting from the rightmost point, until you’ve traversed the distance
0.8
in arc length. This is the same thing as saying you've traversed an angle of
0.8
radians. Then
\sin(\theta)
is your height above the
x
-axis at this point.
As theta increases, and you walk around the circle, your height bobs up and down and up and down. So the graph of
\sin(\theta)
\theta
, which plots this height as a function of arc length, is a wave pattern. This is the quintessential wave pattern.
Just from looking at this graph, we can get a feel from the shape of the derivative function. The slope at
0
is something positive, then as
\sin(\theta)
approaches its peak, the slope goes down to
0
. Then the slope is negative for a little while before coming back up to
0
\sin(\theta)
graph levels out. If you’re familiar with the graphs of trig functions, you might guess that this derivative graph should be exactly
\cos(\theta)
, whose graph is just a shifted-back copy of the sine graph.
But all this tells us is that the peaks and valleys of the derivative graph seem to line up with the graph of cosine. How could we know that this derivative actually is the cosine of theta, and not just some new function that happens looks similar to it. As with the previous examples of this video, a more exact understanding of the derivative requires looking at what the function itself represents, rather than the graph of the function.
Think back to the walk around the unit circle, having traversed an arc length of
\theta
\sin(\theta)
is the height of this point. Consider a slight nudge of
d
-theta along the circumference of the circle; a tiny step in your walk around the unit circle. How much does this change
\sin(\theta)
? How much does that step change your height above the x-axis? This is best observed by zooming in on the point where you are on the circle.
Zoomed in close enough the circle basically looks like a straight line in this neighborhood. Consider the right triangle pictured below, where the hypotenuse represents a straight-line approximation of the nudge
d \theta
along the circumference, and this left side represents the change in height; the resulting tiny nudge to
\sin(\theta)
This tiny triangle is actually similar to this larger triangle with the defining angle theta, and whose hypotenuse is the radius of the circle with length
1
. Specifically, the angle between its height
d(\sin(\theta))
and its hypotenuse
d\theta
is precisely equals to
\theta
Similar triangle explanation
Think about what the derivative of sine is supposed to mean. It’s the ratio between that
d\left(\sin(\theta)\right)
, the tiny change to the output of sine, divided by
d \theta
, the tiny change to the input of the function. From the picture, that’s the ratio between the length of the side adjacent to this little right-triangle divided by the hypotenuse. Well, let’s see, adjacent divided by hypotenuse; that’s exactly what
\cos(\theta)
means!
Notice, by considering the slope of the graph, we can get a quick intuitive feel for the rough shape that the derivative of
\sin(\theta)
should have, which is enough to make an educated guess. But to more to understand why this derivative is precisely
\cos(\theta)
, we had to begin our line of reasoning with the defining features of
\sin(\theta)
For those of you who enjoy pausing and pondering, take a moment to find a similar line of reasoning which explains what the derivative of
\cos(\theta)
should be.
In the next lesson we'll figure out the derivatives of functions that combine simple functions like these, either as sums, products, or functions compositions. Similar to this lesson, we’ll try to understand each rule geometrically, in a way that makes it intuitively reasonable and memorable.
Power Rule through geometry
Visualizing the chain rule and product rule
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Richardson Cascadic Multigrid Method for 2D Poisson Equation Based on a Fourth Order Compact Scheme
Li Ming, Li Chen-Liang, "Richardson Cascadic Multigrid Method for 2D Poisson Equation Based on a Fourth Order Compact Scheme", Journal of Applied Mathematics, vol. 2014, Article ID 490540, 6 pages, 2014. https://doi.org/10.1155/2014/490540
Li Ming1 and Li Chen-Liang2
2School of Mathematics and Computing Science, Guilin University of Electronics Technology, Guilin, Guangxi 541004, China
Based on a fourth order compact difference scheme, a Richardson cascadic multigrid (RCMG) method for 2D Poisson equation is proposed, in which the an initial value on the each grid level is given by the Richardson extrapolation technique (Wang and Zhang (2009)) and a cubic interpolation operator. The numerical experiments show that the new method is of higher accuracy and less computation time.
Poisson equation is a partial differential equation (PDE) with broad applications in theoretical physics, mechanical engineering and other fields, such as groundwater flow [1, 2], fluid pressure prediction [3], electromagnetics [4], semiconductor modeling [5], and electrical power network modeling [6].
We consider the following two-dimensional (2D) Poisson equation: where is a rectangular domain or union of rectangular domains with Dirichlet boundary . The solution and the forcing function are assumed to be sufficiently smooth.
Multigrid (MG) method is one of the most effective algorithms to solve the large scale problem. In 1996, cascadic multigrid (CMG) method proposed by Bornemann and Deuflhard [7] and then analyzed by Shi et al. (see [8–11]) and Shaidurov (see [12]). In the recent years, there have been several theoretical analyses and the applications of these methods for the plate bending problems (see [13]), the parabolic problems (see [10]), the nonlinear problems (see [14, 15]), and the Stokes problems (see [16]). In order to improve the efficiency of the CMG, some new extrapolation formulas and extrapolation cascadic multigrid (EXCMG) methods are proposed by Chen et al. (see [17–20]). These new methods can provide a better initial value for smoothing operator on the refined grid level to accelerate their convergence rate.
Based on the Richardson extrapolation technique, Wang and Zhang [21] presented a multiscale multigrid algorithm. Numerical experiments show that the new method is of higher accuracy solution and higher efficiency.
In this paper, in order to develop a more efficient CMG method, we use the Richardson extrapolation technique presented in [21] and a new extrapolation formula; a new Richardson extrapolation cascadic multigrid (RCMG) method for 2D Poisson equation is proposed.
The sections are arranged as follows: the fourth order compact difference scheme and Richardson extrapolation technique are given in Section 2. Chen’s new extrapolation formula and EXCMG method are introduced in Section 3. In Section 4, we present the RCMG method. In Section 5, the numerical experiments show the effectiveness of the new method.
2. Fourth Order Compact Difference Scheme and Richardson Extrapolation Technique
For convenience, we consider the rectangular domain . We discretize with uniform mesh sizes and in the and coordinate directions. The mesh points are with and , and , . Let's denote the mesh aspect ratio , and be the solution at the grid point , we can rewrite the fourth order compact difference scheme of (1) into the following form [22]: The coefficients in (2) are If the domain is subdivided into a sequence of grids (or ), with step length (namely, , by using the fourth order compact difference scheme (see (2)), a series of linear equations of the model problem (1) are given as follows
Assume the fourth order accurate solutions and on the grid and the grid are given, respectively (Figure 1). In 2009, Wang and Zhang [21] applied the Richardson extrapolation (where to get a sixth order accurate solution on .
Four types of points on grid.
The above extrapolation operator is rewritten as the following iterative operator .
Algorithm 1. Consider .
Step 2. Update every (even, even) grid point on by Richardson extrapolation formula (see (5)); then use direct interpolation to get . Consider Step 3. Update every (odd, odd) grid point on . From (2), for each (odd, odd) point , the updated solution is Here, represents the right-hand side part of (2).
Step 4. Update every (odd, even) grid point on . From (2), for each (odd, even) grid point, the updated value is
Step 5. Update every (even, odd) grid point on . From (2), the idea is similar to the (odd, even) grid point. Let .
Step 6. If or , stop. Else, let and return to Step 3.
3. New Extrapolation Formula and EXCMG Method
Based on an asymptotic expansion of finite element method, a new extrapolation formula and an extrapolation cascadic multigrid (EXCMG) method are proposed by Chen et al. (see [17–20]). The numerical experiments show that the EXCMG method is of high accuracy and efficiency. Now we rewrite the new extrapolation formula as follows.
Let us denote the above new extrapolation formula by operator
Now let , on , denote the exact solutions, the EXCMG method is as following:
Algorithm 2 (EXCMG). For , consider the following
Step 1. Extrapolate by using the new extrapolation formula (see (10))
Step 2. Compute the initial value on by using quadratic interpolation operator .
Step 3. Smooth times to get the iterative solution on by using some classical iterative operator .
Step 4. Return to Step 1 if , until you get the final iterative solution on the finest grid .
4. Richardson Cascadic Multigrid Method
One of the main tasks in cascadic multigrid method is constructing a suitable interpolation. Based on a new extrapolation-interpolation formula, Chen [17–20] proposed the following extrapolation cascadic multigrid (EXCMG) method, in which the new extrapolation and quadratic interpolation are used to provide a better initial value on refined grid.
In this section, we use RET operator and a cubic interpolation to interpolate the initial guess on the refined grid . Then a classical iterative operator (such as conjugate gradient method) is used as a smoothing operator to compute the high accuracy solution on the fine grid . Similar to the standard CMG method, we propose the following Richardson cascadic multigrid (RCMG) method.
Algorithm 3 [RCMG]
Step 1. Exactly solve the equation on coarsest grid .
Step 2. Run Algorithm 1; we have
Step 3. Use a cubic interpolation operator to have the initial value on the gird level .
Step 4. Smoothing times by using the classical iterative operator , on the level . Set ;
Step 5. Return to Step 2, if .
The difference between RCMG method and EXCMG method is that
5. Numerical Experiment and Comparison
Numerical experiments are conducted to solve a 2D Poisson equation (1) on the unit square domain .
Example 4. The exact solution ; the forcing function
We use the conjugate gradient (CG) method as a smoothing iterative operator in EXCMG method and RCMG method. In EXCMG method, the number of iterations on each grid level has to increase from finer to coarser grids; in this paper let . And in RCMG, we set the number of iteration (Step 2) and (Step 4) be . We set of RET in the RCMG method (on Step 2).
5.1. Comparison of the Initial Errors
Assume that the exact solutions of the difference equation on grids and are given. We compare EXCMG method with RCMG method for the initial error on grid .
From Figure 2, the accuracy of the initial error on the next grid of RCMG method is higher than EXCMG method. Namely, a better initial value on the fine grid can be got by using RCMG method. Based on the results of the literature [17–20], the RCMG method can obtain good convergence rate.
Example 4, grid , initial error of EXCMG ((a) scale ) and RCMG ((b) scale ).
5.2. Comparison between EXCMG Method and RCMG Method
Let denote the maximum absolute error between the computed solution and the exact solution on the finest grid points. The “” denotes the computing time (unit: second) of EXCMG method and RCMG method.
From Figures 3 and 4 and Tables 1 and 2, we see that, under the same conditions, the RCMG method can obtain higher computational precision and spend less computing time than EXCMG method.
EXCMG RCMG
Numerical results of EXCMG and RCMG for Example 4.
Comparison of the maximum error and cpu time for Example 4 with , taking step lengths , , , , and , respectively.
In this paper, based on a fourth order compact scheme, we present a Richardson cascadic multigrid method for 2D Poisson problem by using Richardson technique presented by [21]. The numerical results show that RCMG method has higher computational accuracy and higher efficiency.
This work is supported by the National Natural Science Foundation of China (Grant no. 11161014), the National Natural Science Foundation of Yunnan Province (Grant no. 2012FD054), and Scientific Research Starting Foundation for Master or Ph.D. of Honghe University (Grant no. XJ1S0925).
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Find derivatives for the functions. Assume a, b, c, and
Find derivatives for the functions. Assume a, b, c, and k are constants. s(t)=6 t^{-2}+3 t^{3}-4 t^{1 /. 2}
{s}^{\prime }\left(t\right){=}^{1}{\left(6{t}^{-2}\right)}^{\prime }+{\left(3{t}^{3}\right)}^{\prime }-{\left(4{t}^{\frac{1}{2}}\right)}^{\prime }
Find derivative of given function:
{s}^{\prime }\left(t\right){=}^{1}{\left(6{t}^{-2}\right)}^{\prime }+{\left(3{t}^{3}\right)}^{\prime }-{\left(4{t}^{\frac{1}{2}}\right)}^{\prime }
{=}^{2}6\stackrel{˙}{-2{t}^{3}}+3\stackrel{˙}{3}{t}^{2}-4\stackrel{˙}{\frac{1}{2}}{t}^{-\frac{1}{2}}
=-12{t}^{-3}+9{t}^{2}-2{t}^{-\frac{1}{2}}
(1) Derivative of sum
(2) Power rule
y=5{x}^{3}+7{x}^{2}-3x+1
\frac{\left(\mathrm{arctan}\frac{1}{2}+\mathrm{arctan}\frac{1}{3}\right)}{\left(\mathrm{arcot}\frac{1}{2}+\mathrm{arcot}\frac{1}{3}\right)}
{\int }_{-\infty }^{\infty }{e}^{x/2}sech\left(x\right)dx
without Residue Calculus
Find the derivatives of the functions. \(\displaystyle{f{{\left({x}\right)}}}={\ln{
{e}^{2{x}^{2}-x+1/x}
development of
\sum _{m=-\left(M-1\right)}^{M-1}{e}^{-imh\xi }
\sum _{m=-\left(M-1\right)}^{M-1}{e}^{-imh\xi }=\frac{\mathrm{sin}\left(M-\frac{1}{2}\right)h\xi }{\mathrm{sin}\frac{1}{2}h\xi }
Can you show the process of the development?
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Spherical law of cosines - Wikipedia
In spherical trigonometry, the law of cosines (also called the cosine rule for sides[1]) is a theorem relating the sides and angles of spherical triangles, analogous to the ordinary law of cosines from plane trigonometry.
Given a unit sphere, a "spherical triangle" on the surface of the sphere is defined by the great circles connecting three points u, v, and w on the sphere (shown at right). If the lengths of these three sides are a (from u to v), b (from u to w), and c (from v to w), and the angle of the corner opposite c is C, then the (first) spherical law of cosines states:[2][1]
{\displaystyle \cos c=\cos a\cos b+\sin a\sin b\cos C\,}
Since this is a unit sphere, the lengths a, b, and c are simply equal to the angles (in radians) subtended by those sides from the center of the sphere. (For a non-unit sphere, the lengths are the subtended angles times the radius, and the formula still holds if a, b and c are reinterpreted as the subtended angles). As a special case, for C = π/2, then cos C = 0, and one obtains the spherical analogue of the Pythagorean theorem:
{\displaystyle \cos c=\cos a\cos b\,}
If the law of cosines is used to solve for c, the necessity of inverting the cosine magnifies rounding errors when c is small. In this case, the alternative formulation of the law of haversines is preferable.[3]
A variation on the law of cosines, the second spherical law of cosines,[4] (also called the cosine rule for angles[1]) states:
{\displaystyle \cos C=-\cos A\cos B+\sin A\sin B\cos c\,}
where A and B are the angles of the corners opposite to sides a and b, respectively. It can be obtained from consideration of a spherical triangle dual to the given one.
3 Planar limit: small angles
Let u, v, and w denote the unit vectors from the center of the sphere to those corners of the triangle. The angles and distances do not change if the coordinate system is rotated, so we can rotate the coordinate system so that
{\displaystyle \mathbf {u} }
is at the north pole and
{\displaystyle \mathbf {v} }
is somewhere on the prime meridian (longitude of 0). With this rotation, the spherical coordinates for
{\displaystyle \mathbf {v} }
{\displaystyle (r,\theta ,\phi )=(1,a,0)}
, where θ is the angle measured from the north pole not from the equator, and the spherical coordinates for
{\displaystyle \mathbf {w} }
{\displaystyle (r,\theta ,\phi )=(1,b,C)}
. The Cartesian coordinates for
{\displaystyle \mathbf {v} }
{\displaystyle (x,y,z)=(\sin a,0,\cos a)}
and the Cartesian coordinates for
{\displaystyle \mathbf {w} }
{\displaystyle (x,y,z)=(\sin b\cos C,\sin b\sin C,\cos b)}
{\displaystyle \cos c}
is the dot product of the two Cartesian vectors, which is
{\displaystyle \sin a\sin b\cos C+\cos a\cos b}
Let u, v, and w denote the unit vectors from the center of the sphere to those corners of the triangle. We have u · u = 1, v · w = cos c, u · v = cos a, and u · w = cos b. The vectors u × v and u × w have lengths sin a and sin b respectively and the angle between them is C, so
sin a sin b cos C = (u × v) · (u × w) = (u · u)(v · w) − (u · v)(u · w) = cos c − cos a cos b,
using cross products, dot products, and the Binet–Cauchy identity (p × q) · (r × s) = (p · r)(q · s) − (p · s)(q · r).
The first and second spherical laws of cosines can be rearranged to put the sides (a, b, c) and angles (A, B, C) on opposite sides of the equations:
{\displaystyle {\begin{aligned}\cos C&={\frac {\cos c-\cos a\cos b}{\sin a\sin b}}\\\\\cos c&={\frac {\cos C+\cos A\cos B}{\sin A\sin B}}\\\end{aligned}}}
Planar limit: small angles[edit]
For small spherical triangles, i.e. for small a, b, and c, the spherical law of cosines is approximately the same as the ordinary planar law of cosines,
{\displaystyle c^{2}\approx a^{2}+b^{2}-2ab\cos C\,.}
To prove this, we will use the small-angle approximation obtained from the Maclaurin series for the cosine and sine functions:
{\displaystyle \cos a=1-{\frac {a^{2}}{2}}+O\left(a^{4}\right),\,\sin a=a+O\left(a^{3}\right)}
Substituting these expressions into the spherical law of cosines nets:
{\displaystyle 1-{\frac {c^{2}}{2}}+O\left(c^{4}\right)=1-{\frac {a^{2}}{2}}-{\frac {b^{2}}{2}}+{\frac {a^{2}b^{2}}{4}}+O\left(a^{4}\right)+O\left(b^{4}\right)+\cos(C)\left(ab+O\left(a^{3}b\right)+O\left(ab^{3}\right)+O\left(a^{3}b^{3}\right)\right)}
or after simplifying:
{\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos C+O\left(c^{4}\right)+O\left(a^{4}\right)+O\left(b^{4}\right)+O\left(a^{2}b^{2}\right)+O\left(a^{3}b\right)+O\left(ab^{3}\right)+O\left(a^{3}b^{3}\right).}
The big O terms for a and b are dominated by O(a4) + O(b4) as a and b get small, so we can write this last expression as:
{\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos C+O\left(a^{4}\right)+O\left(b^{4}\right)+O\left(c^{4}\right).}
^ a b c W. Gellert, S. Gottwald, M. Hellwich, H. Kästner, and H. Küstner, The VNR Concise Encyclopedia of Mathematics, 2nd ed., ch. 12 (Van Nostrand Reinhold: New York, 1989).
^ Romuald Ireneus 'Scibor-Marchocki, Spherical trigonometry, Elementary-Geometry Trigonometry web page (1997).
^ R. W. Sinnott, "Virtues of the Haversine", Sky and Telescope 68 (2), 159 (1984).
^ Reiman, István (1999). Geometria és határterületei. Szalay Könyvkiadó és Kereskedőház Kft. p. 83.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Spherical_law_of_cosines&oldid=1019368890"
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Answer the questions using the table below: P(A) = ?
Answer the questions using the table below: P(A) = ? P (A/B) = ? P (X) = ?
P\left(A\right)=?
P\left(\frac{A}{B}\right)=?
P\left(X\right)=?
P\left(\frac{A}{X}\right)=?
Find total:
Find P(A) using the following table.
\begin{array}{|ccc|}\hline \text{Grade}& \text{Frequency}& \text{Relative frequency}\\ A& 5\\ B& 11\\ C& 16\\ D& 5\\ F& 1\\ \hline\end{array}
A survey was designed to study how business operations vary according to their size. Companies were classified as small, medium, or large.
\le
Describe in your own words the process you use to write marginal relative frequencies for data given in a two-way table.
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Risk Management - IndexZoo
Risk Management: The parameters are set upon the inception of the Bear token, and changes as market Fluctuates and as the Bear manages risks.
Liquidation Safety Margin
Margin Call Safety Margin
Liquidation Price: The indexed price that will liquidates the Bear's margin trading account once reached. The formula for perpetual futures liquidation price are given by DEX, and are generally the same across exchanges.
Liquidation Price = (AverageEntryPrice * Leverage) / (Leverage - 1 + (MaintenanceMargin * Leverage)
Margin Call Price: This is the price where all our cash are loss to the float P/L, although we are not liquidated yet, we do now want to get there and pay the fine.
MarginCallPrice = AverageEntryPrice + CashLeft / (NetExposure/AverageEntryPrice)
Liquidation Safety Margin: Percent move needed for Liquidation.
LiquidationMargin = (LiquidationPrice - IndexedPrice) / IndexedPrice
When this drops below 30%, meaning a 30% price move against us will liquidate us, we deleverage to maintain 30% safe margin.
When we are deleveraged, and the safe margin is above 30%, vice versa. We re-leverge to our target exposure.
Margin Call Safety Margin: This is the price that will trigger margin call when reached, although it won't get us liquidated, it means effectively cash left is zero after the float loss. Therefore, we enforce a mandatory deleveraging when safety margin is dropped below 15%, to the target of resetting the margin back to 15%.
MarginCallMargin = (MarginCallPrice - IndexedPrice) / IndexedPrice
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Continuous Probability Distributions - Uniform Distribution | Brilliant Math & Science Wiki
Sameer Kailasa contributed
A real-valued continuous random variable
X
is uniformly distributed if the probability that
X
lands in an interval is proportional to the length of that interval.
S
be a finite set. A uniformly distributed random variable
X
S
should be equally likely to land at any element of
S
x \in S
P(X = x) = 1/|S|
|S|
S
S
is infinite, say, a subinterval of
\mathbb{R}
1/|S| = 1/\infty = 0
, so defining
X
by giving the probabilities that
X
equals a certain element of
S
will not work. Instead, one defines
X
by assigning probabilities to subsets of
S
. These probabilities are assigned by weighting subsets based on their measure. For example, when
S = [0,1]
[a,b] \subset [0,1]
, one has the probability
P(X \in [a,b]) = b-a
The uniform distribution on
[a,b]
\mathcal{U}[a,b]
, and has PDF
p(x) = \left\{ \begin{array}{lr} 1/(b-a) & : x \in [a,b]\\ 0 & : x \notin [a,b] \end{array} \right.
Integrating this function, one observes the cumulative density function for
\mathcal{U}[a,b]
f(x) = \int_{-\infty}^{x} p(y) \, dy = \left\{ \begin{array}{lr} 0 & : x \in (-\infty, a) \\ (x-a)/(b-a) & : x \in [a,b]\\ 1 & : x \in (b, \infty) \end{array} \right.
Consider the following example computation, using this information:
X_1, X_2, \cdots, X_n
be independent, and identically distributed
\mathcal{U}[0,1]
. Compute the probability distribution function for the random variable
\text{max}(X_1, X_2, \cdots, X_n)
X_i
p(x) = 1
x\in [0,1]
p(x) = 0
elsewhere. Let
q(x)
denote the CDF for
Y:= \text{max}(X_1, X_2, \cdots, X_n)
g(x) = P(Y\le x) = P(X_i \le x \, : \, 1\le i \le n) = P(X_1 \le x)^n = \left(\int_{-\infty}^{x} p(y) \, dy \right)^n
= \left\{ \begin{array}{lr} 0 & : x \in (-\infty, 0) \\ x^n & : x \in [0,1]\\ 1 & : x \in (1, \infty) \end{array} \right.
Differentiating this gives the PDF
q(x) = \left\{ \begin{array}{lr} n x^{n-1} & : x \in [0,1]\\ 0 & : x \notin [0,1] \end{array} \right.
Cite as: Continuous Probability Distributions - Uniform Distribution. Brilliant.org. Retrieved from https://brilliant.org/wiki/continuous-probability-distributions-uniform/
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Find the value of \int_{-\pi}^{\pi} (4 \arctan(e^x)-\pi)dx I've tried to show
{\int }_{-\pi }^{\pi }\left(4\mathrm{arctan}\left({e}^{x}\right)-\pi \right)dx
f\left(x\right)=4\mathrm{arctan}\left({e}^{x}\right)-\pi
\mathrm{arctan}u+\mathrm{arctan}\frac{1}{u}=\frac{\pi }{2}
(∗):
-f\left(-x\right)=-4{\mathrm{arctan}e}^{-x}+\pi =-4\left(\frac{\pi }{2}-{\mathrm{arctan}e}^{x}\right)+\pi =4{\mathrm{arctan}e}^{x}-\pi =f\left(x\right)
(∗) can be proved geometrically:
\mathrm{arctan}u
is the angle with opposite side u and adjacent side 1, and
\mathrm{arctan}\frac{1}{u}
is the other acute angle in the right triangle. Their sum must thus be
\frac{\pi }{2}
. Alternatively, differentiate the left-hand side and show it is constant, then choose a convenient value for u since any u works, say u=1.
autormtak0w
I={\int }_{-\pi }^{0}4\mathrm{arctan}\left({e}^{x}\right)-\pi dx+{\int }_{0}^{\pi }4\mathrm{arctan}\left({e}^{x}\right)-\pi dx
Upon making the substitution u=−x, we find that
I=-{\int }_{\pi }^{0}4\mathrm{arctan}\left({e}^{-u}\right)-\pi du+{\int }_{0}^{\pi }4\mathrm{arctan}\left({e}^{x}\right)-\pi dx
={\int }_{0}^{\pi }4\mathrm{arctan}\left({e}^{-x}\right)-\pi dx+{\int }_{0}^{\pi }4\mathrm{arctan}\left({e}^{x}\right)-\pi dx
={\int }_{0}^{\pi }4\left(\mathrm{arctan}\left({e}^{x}\right)+\mathrm{arctan}\left({e}^{-x}\right)\right)-2\pi dx
\mathrm{arctan}\left(u+\mathrm{arctan}\left(\frac{1}{u}\right)\right)=\left\{\begin{array}{cc}\frac{\pi }{2}& \text{ }\phantom{\rule{1em}{0ex}}\text{if}\phantom{\rule{1em}{0ex}}\text{ }u>0\\ -\frac{\pi }{2}& \text{ }\phantom{\rule{1em}{0ex}}\text{if}\phantom{\rule{1em}{0ex}}\text{ }u<0\end{array}
x\in \left[0,\pi \right]
u={e}^{x}>0
\mathrm{arctan}\left(u\right)+\mathrm{arctan}\left(\frac{1}{u}\right)=\frac{\pi }{2}
, meaning that the integrand is the zero function. Hence, I=0.
The answer expands on my comment, which remarked that the integrand of the given integral is twice the Gudermannian function, gd, which appears most famously in the equation governing the Mercator projection in cartography.
Differentiating the integrand gives
\frac{d}{dx}\left[4\mathrm{arctan}\left({e}^{x}\right)-\pi \right]=4\cdot \frac{1}{1+\left({e}^{x}{\right)}^{2}}\cdot {e}^{x}=\frac{4}{{e}^{x}+{e}^{-x}}=2\mathrm{sec}hx
In particular, this derivative is even. Since evaluating the integrand at x=0 gives
4\mathrm{arctan}{e}^{0}-\pi =0
the integrand is odd; since the integral is taken over an interval symmetric around 0, by symmetry
{\int }_{-a}^{a}\left[4\mathrm{arctan}\left({e}^{x}\right)-\pi \right]dx=0
for any a: In particular, the occurrence of
\pi
in the limits of the integral is something of a red herring.
\mathrm{sin}x+\mathrm{sin}y=a\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\mathrm{cos}x+\mathrm{cos}y=b
\mathrm{tan}\left(x-\frac{y}{2}\right)
\frac{1+\mathrm{tan}v}{1-\mathrm{tan}v}=\frac{\mathrm{cot}v+1}{\mathrm{cot}v-1}
\frac{1}{\mathrm{csc}\theta -\mathrm{cos}\theta }=\frac{1+\mathrm{cos}\theta }{\mathrm{sin}\theta }
\mathrm{cos}x=-\frac{12}{13}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\mathrm{csc}x<0
\mathrm{cot}\left(2x\right)
Find the value of differentiated function
f\left(x\right)=\mathrm{cos}\left[{\mathrm{cot}}^{-1}\left(\frac{\mathrm{cos}x}{\sqrt{1-\mathrm{cos}2x}}\right)\right]
\frac{\pi }{4}<x<\frac{\pi }{2}
,then what is the value of
\frac{d\left(f\left(x\right)\right)}{d\left(\mathrm{cot}\left(x\right)\right)}
\left(\mathrm{cos}x/1+\mathrm{sin}x\right)+\left(1+\mathrm{sin}x/\mathrm{cos}x\right)
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Linear Momentum - Maple Help
Home : Support : Online Help : Math Apps : Natural Sciences : Physics : Linear Momentum
The linear momentum of an object, denoted
\stackrel{→}{p }
, is defined as the product of its mass
m
\stackrel{→}{v}
\stackrel{→}{p }= m\stackrel{→}{v}
As a result, a massive object moving quickly has a greater linear momentum than a small object moving slowly.
Note that linear momentum is a vector quantity, as it is the product of a scalar (mass) with a vector (velocity).
The following demonstration shows how the momentum varies with different values of mass and velocity. Click anywhere on the plot to set the ball moving in that direction. Then, adjust the velocity and mass sliders to see the effect that they have on the ball's momentum.
\mathrm{Mass}:
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Grignard reaction - Simple English Wikipedia, the free encyclopedia
A solution of a carbonyl compound is added to a Grignard reagent. (See gallery below)
The Grignard reaction (pronounced /ɡriɲar/) is an organometallic chemical reaction in which alkyl- or aryl-magnesium halides (Grignard reagents) attack electrophilic carbon atoms that are present within polar bonds (for example, in a carbonyl group as in the example shown below). Grignard reagents act as nucleophiles. The Grignard reaction produces a carbon–carbon bond. It alters hybridization about the reaction center.[1] The Grignard reaction is an important tool in the formation of carbon–carbon bonds.[2][3] It also can form carbon–phosphorus, carbon–tin, carbon–silicon, carbon–boron and other carbon–heteroatom bonds.
It is a nucleophilic organometallic addition reaction. The high pKa value of the alkyl component (pKa = ~45) makes the reaction irreversible. Grignard reactions are not ionic. The Grignard reagent exists as an organometallic cluster (in ether).
The disadvantage of Grignard reagents is that they readily react with protic solvents (such as water), or with functional groups with acidic protons, such as alcohols and amines. Atmospheric humidity can alter the yield of making a Grignard reagent from magnesium turnings and an alkyl halide. One of many methods used to exclude water from the reaction atmosphere is to flame-dry the reaction vessel to evaporate all moisture, which is then sealed to prevent moisture from returning. Chemists then use ultrasound to activate the surface of the magnesium so that it consumes any water present. This can allow Grignard reagents to form with less sensitivity to water being present.[4]
Another disadvantage of Grignard reagents is that they do not readily form carbon–carbon bonds by reacting with alkyl halides by an SN2 mechanism.
François Auguste Victor Grignard discovered Grignard reactions and reagents. They are named after this French chemist (University of Nancy, France) who was awarded the 1912 Nobel Prize in Chemistry for this work.
2 Making a Grignard reagent
3 Reactions of Grignard reagents
3.1 Reactions with carbonyl compounds
3.2 Reactions with other electrophiles
3.3 Formation of bonds to B, Si, P, Sn
3.4 Carbon–carbon coupling reactions
4 Grignard degradation
Reaction mechanism[change | change source]
The addition of the Grignard reagent to a carbonyl typically proceeds through a six-membered ring transition state.[5]
However, with steric hindered Grignard reagents, the reaction may proceed by single-electron transfer.
Grignard reactions will not work if water is present; water causes the reagent to rapidly decompose. So, most Grignard reactions occur in solvents such as anhydrous diethyl ether or tetrahydrofuran (THF), because the oxygen in these solvents stabilizes the magnesium reagent. The reagent may also react with oxygen present in the atmosphere. This will insert an oxygen atom between the carbon base and the magnesium halide group. Usually, this side-reaction may be limited by the volatile solvent vapors displacing air above the reaction mixture. However, chemists may perform the reactions in nitrogen or argon atmospheres. In small scale reactions, the solvent vapors do not have enough space to protect the magnesium from oxygen.
Making a Grignard reagent[change | change source]
Grignard reagents are formed by the action of an alkyl or aryl halide on magnesium metal.[6] The reaction is conducted by adding the organic halide to a suspension of magnesium in an ether, which provides ligands required to stabilize the organomagnesium compound. Typical solvents are diethyl ether and tetrahydrofuran. Oxygen and protic solvents such as water or alcohols are not compatible with Grignard reagents. The reaction proceeds through single electron transfer.[7][8]
X− + Mg•+ → XMg•
R• + XMg• → RMgX
Grignard reactions often start slowly. First, there is an induction period during which reactive magnesium becomes exposed to the organic reagents. After this induction period, the reactions can be highly exothermic. Alkyl and aryl bromides and iodides are common substrates. Chlorides are also used, but fluorides are generally unreactive, except with specially activated magnesium, such as Rieke magnesium.
Many Grignard reagents, such as methylmagnesium chloride, phenylmagnesium bromide, and allylmagnesium bromide are available commercially in tetrahydrofuran or diethyl ether solutions.
Using the Schlenk equilibrium, Grignard reagents form varying amounts of diorganomagnesium compounds (R = organic group, X = halide):
Initiation[change | change source]
Many methods have been developed to initiate Grignard reactions that are slow to start. These methods weaken the layer of MgO that covers the magnesium. They expose the magnesium to the organic halide to start the reaction that makes the Grignard reagent.
Mechanical methods include crushing of the Mg pieces in situ, rapid stirring, or using ultrasound (sonication) of the suspension. Iodine, methyl iodide, and 1,2-dibromoethane are commonly employed activating agents. Chemists use 1,2-dibromoethane because its action can be monitored by the observation of bubbles of ethylene. Also, the side-products are innocuous:
The addition of a small amount of mercuric chloride will amalgamate the surface of the metal, allowing it to react.
Grignard reagents are produced in industry for use in place, or for sale. As with at bench-scale, the main problem is that of initiation. A portion of a previous batch of Grignard reagent is often used as the initiator. Grignard reactions are exothermic; this exothermicity must be considered when a reaction is scaled-up from laboratory to production plant.[9]
Reactions of Grignard reagents[change | change source]
Reactions with carbonyl compounds[change | change source]
Grignard reagents will react with a variety of carbonyl derivatives.[10]
The most common application is for alkylation of aldehydes and ketones, as in this example:[11]
Note that the acetal function (a masked carbonyl) does not react.
Such reactions usually involve a water-based (aqueous) acidic workup, though this is rarely shown in reaction schemes. In cases where the Grignard reagent is adding to a prochiral aldehyde or ketone, the Felkin-Anh model or Cram's Rule can usually predict which stereoisomer will form.
Reactions with other electrophiles[change | change source]
In addition, Grignard reagents will react with electrophiles.
Another example is making salicylaldehyde (not shown above). First, bromoethane reacts with Mg in ether. Second, phenol in THF converts the phenol into Ar-OMgBr. Third, benzene is added in the presence of paraformaldehyde powder and triethylamine. Fourth, the mixture is distilled to remove the solvents. Next, 10% HCl is added. Salicylaldehyde will be the major product as long as everything is very dry and under inert conditions. The reaction works also with iodoethane instead of bromoethane.[12][13][14]
Formation of bonds to B, Si, P, Sn[change | change source]
The Grignard reagent is very useful for forming carbon–heteroatom bonds.
Carbon–carbon coupling reactions[change | change source]
A Grignard reagent can also be involved in coupling reactions. For example, nonylmagnesium bromide reacts with methyl p-chlorobenzoate to give p-nonylbenzoic acid, in the presence of Tris(acetylacetonato)iron(III), often symbolized as Fe(acac)3, after workup with NaOH to hydrolyze the ester, shown as follows. Without the Fe(acac)3, the Grignard reagent would attack the ester group over the aryl halide.[15]
For the coupling of aryl halides with aryl Grignards, nickel chloride in tetrahydrofuran (THF) is also a good catalyst. Additionally, an effective catalyst for the couplings of alkyl halides is dilithium tetrachlorocuprate (Li2CuCl4), prepared by mixing lithium chloride (LiCl) and copper(II) chloride (CuCl2) in THF. The Kumada-Corriu coupling gives access to [substituted] styrenes.
Oxidation[change | change source]
{\displaystyle {\begin{array}{l}{\mathsf {R{-}MgX}}\quad +\quad {\mathsf {O2}}\quad \longrightarrow \quad {\color {Red}{\mathsf {R^{\bullet }+O_{2}^{\bullet {-}}}}}\quad +\quad {\mathsf {MgX^{+}}}\longrightarrow &{\mathsf {R{-}O{-}O{-}MgX}}&+\quad {\mathsf {H_{3}O^{+}}}&\longrightarrow \quad {\mathsf {R{-}O{-}O{-}H}}&+\quad {\mathsf {HO{-}MgX+H^{+}}}\\&\quad \ \ {\Bigg \downarrow }{\mathsf {R{-}MgX}}\\&{\mathsf {R{-}O{-}MgX}}&+\quad {\mathsf {H_{3}O^{+}}}&\longrightarrow \quad {\mathsf {R{-}O{-}H}}&+\quad {\mathsf {HO{-}MgX+H^{+}}}\end{array}}}
A reaction of Grignards with oxygen in presence of an alkene makes an ethylene extended alcohol. These are useful in synthesizing larger compounds.[16] This modification requires aryl or vinyl Grignard reagents. Adding just the Grignard and the alkene does not result in a reaction, showing that the presence of oxygen is essential. The only drawback is the requirement of at least two equivalents of Grignard reagent in the reaction. This can addressed by using a dual Grignard system with a cheap reducing Grignard reagent such as n-butylmagnesium bromide.
Nucleophilic aliphatic substitution[change | change source]
Elimination[change | change source]
Grignard degradation[change | change source]
Grignard degradation[17][18] at one time was a tool in structure identification (elucidation) in which a Grignard RMgBr formed from a heteroaryl bromide HetBr reacts with water to Het-H (bromine replaced by a hydrogen atom) and MgBrOH. This hydrolysis method allows the determination of the number of halogen atoms in an organic compound. In modern usage, Grignard degradation is used in the chemical analysis of certain triacylglycerols.[19]
Industrial use[change | change source]
An example of the Grignard reaction is a key step in the industrial production of Tamoxifen.[20] (Tamoxifen is currently used for the treatment of estrogen receptor positive breast cancer in women.):[21]
Magnesium turnings placed on a flask.
Covered with THF and a small piece of iodine added.
A solution of alkyl bromide was added while heating.
After completion of the addition, the mixture was heated for a while.
Formation of the Grignard reagent had completed. A small amount of magnesium still remained in the flask.
The Grignard reagent thus prepared was cooled to 0°C before the addition of carbonyl compound. The solution became cloudy since the Grignard reagent precipitated out.
A solution of carbonyl compound was added to the Grignard reagent.
The solution was warmed to room temperature. The reaction was complete.
↑ Shirley, D.A. (1954), "The Synthesis of Ketones from Acid Halides and Organometallic Compounds of Magnesium, Zinc, and Cadmium", Org. React, 8: 28–58
↑ Huryn, D. M. (1991), "Carbanions of Alkali and Alkaline Earth Cations: (Ii) Selectivity of Carbonyl Addition Reactions", Comp. Org. Syn, 1: 49–75, doi:10.1016/B978-0-08-052349-1.00002-0, ISBN 9780080523491
↑ Smith, David H. (1999), "Grignard Reactions in "Wet" Ether", Journal of Chemical Education, 76 (10): 1427, Bibcode:1999JChEd..76.1427S, doi:10.1021/ed076p1427
↑ Maruyama, K.; Katagiri, T. (1989), "Mechanism of the Grignard reaction", J. Phys. Org. Chem, 2 (3): 205–213, doi:10.1002/poc.610020303
↑ Lai Yee Hing (1981), "Grignard Reagents from Chemically Activated Magnesium", Synthesis, 1981 (8): 585–604, doi:10.1055/s-1981-29537
↑ Philip E. Rakita (1996). "5. Safe Handling Practices of Industrial Scale Grignard Ragents" (Google Books excerpt). In Gary S. Silverman, Philip E. Rakita (ed.). Handbook of Grignard reagents. CRC Press. pp. 79–88. ISBN 0824795458.
↑ Henry Gilman and R. H. Kirby (1941). "Butyric acid, α-methyl". Org. Synth. Coll. Vol. 1: 361.
↑ Haugan, Jarle André; Songe, Pål; Rømming, Christian; Rise, Frode; Hartshorn, Michael P.; Merchán, Manuela; Robinson, Ward T.; Roos, Björn O.; Vallance, Claire (1997), "Total Synthesis of C31-Methyl Ketone Apocarotenoids 2: The First Total Synthesis of (3R)-Triophaxanthin." (PDF), Acta Chimica Scandinavica, 51: 1096–1103, doi:10.3891/acta.chem.scand.51-1096, archived from the original (PDF) on 2011-08-11, retrieved 2009-11-26
↑ Wang, R. X et al. Synth. Commun. 1994, 24, 1757-1760.
↑ C. Ji ; Peters, D. G. Tetrahedron Letters, Volume 42, Issue 35, 27 August 2001, Pages 6065-6067 http://www.sciencedirect.com/science/article/pii/S0040403901011789
↑ A. Fürstner; A. Leitner; G. Seidel (2004), "4-Nonylbenzoic Acid", Org. Synth., 81: 33–42
↑ Nobe, Youhei; Arayama, Kyohei; Urabe, Hirokazu (2005-12-01). "Air-Assisted Addition of Grignard Reagents to Olefins. A Simple Protocol for a Three-Component Coupling Process Yielding Alcohols". Journal of the American Chemical Society. 127 (51): 18006–18007. doi:10.1021/ja055732b. ISSN 0002-7863.
↑ Steinkopf, Wilhelm; Jacob, Hans; Penz, Herbert (1934), "Studien in der Thiophenreihe. XXVI. Isomere Bromthiophene und die Konstitution der Thiophendisulfonsäuren", Justus Liebig S Annalen der Chemie, 512: 136–164, doi:10.1002/jlac.19345120113
↑ Steinkopf, Wilhelm; V. Petersdorff, Hans-JüRgen (1940), "Studien in der Thiophenreihe. LI. Atophanartige Derivate des Dithienyls und Diphenyls", Justus Liebig S Annalen der Chemie, 543: 119–128, doi:10.1002/jlac.19405430110
↑ Myher JJ, Kuksis A (February 1979), "Stereospecific analysis of triacylglycerols via racemic phosphatidylcholines and phospholipase C", Can. J. Biochem., 57 (2): 117–24, doi:10.1139/o79-015, PMID 455112
↑ Jordan VC (1993), "Fourteenth Gaddum Memorial Lecture. A current view of tamoxifen for the treatment and prevention of breast cancer", Br J Pharmacol, 110 (2): 507–17, doi:10.1111/j.1476-5381.1993.tb13840.x, PMC 2175926, PMID 8242225
Wikimedia Commons has media related to Grignard reactions.
Rakita, Philip E; Gary S. Silverman, eds. (1996), Handbook of Grignard reagents, New York, N.Y: Marcel Dekker, ISBN 0-8247-9545-8
Grignard knowledge: Alkyl coupling chemistry with inexpensive transition metals by Larry J. Westrum, Fine Chemistry November/December 2002, pp. 10–13 [1] Archived 2016-10-10 at the Wayback Machine
Electron counting, 18-Electron rule, polyhedral skeletal electron pair theory, isolobal principle, π backbonding, hapticity, d electron count
Oxidative addition / Reductive elimination, β-hydride elimination, transmetalation, carbometalation
Gilman reagents, Grignard reagents, cyclopentadienyl complexes, metallocenes, sandwich compounds, transition metal carbene complexes
Monsanto process, Zigler-Natta process, Shell higher olefin process, olefin metathesis
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Grignard_reaction&oldid=8149742"
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Compute conic-sector index of linear system - MATLAB getSectorIndex - MathWorks í•œêµ
{â«}_{0}^{T}y{\left(t\right)}^{T}Q\text{â}y\left(t\right)dt<0,
for all T ≥ 0.
{â«}_{0}^{T}{\left(\begin{array}{c}y\left(t\right)\\ u\left(t\right)\end{array}\right)}^{T}Q\text{â}\left(\begin{array}{c}y\left(t\right)\\ u\left(t\right)\end{array}\right)dt<0,
for all T ≥ 0. To do so, use getSectorIndex with H = [G;I], where I = eyes(nu), and nu is the number of inputs of G.
G\left(s\right)=\left(s+2\right)/\left(s+1\right)
S=\left\{\left(y,u\right):0.1{u}^{2}<uy<10{u}^{2}\right\}.
Q=\left[\begin{array}{cc}1& -\left(a+b\right)/2\\ -\left(a+b\right)/2& ab\end{array}\right];\phantom{\rule{1em}{0ex}}a=0.1,\phantom{\rule{0.2777777777777778em}{0ex}}b=10.
y\left(t\right)=G\phantom{\rule{0.1em}{0ex}}u\left(t\right)
0.1{â«}_{0}^{T}u{\left(t\right)}^{2}<\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}{â«}_{0}^{T}u\left(t\right)y\left(t\right)dt<\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}10{â«}_{0}^{T}u{\left(t\right)}^{2}dt.
Compute the R-index and its frequency with a relative accuracy of 0.0001%. Also, specify fband = [1,10] to compute the index in the frequency interval [–10,–1] ∪ [1,10].
Now compute the index in the frequency interval [–5,–1] ∪ [1,5]. To do so, specify fband = [1,5].
An LTI model, for frequency-dependent sector geometry. Q satisfies Q(s)’ = Q(–s). In other words, Q(s) evaluates to a Hermitian matrix at each frequency.
Frequency interval for calculating the sector index, specified as an array of the form [fmin,fmax] with 0 ≤ fmin < fmax. When you provide fband, getSectorIndex restricts to the specified frequency interval the inequalities that define the index.
For models with complex coefficients, getSectorIndex computes the index in the range [–fmax,–fmin]∪[fmin,fmax]. As a result, the function can return indices at a negative frequency.
Q={W}_{1}{W}_{1}^{T}â{W}_{2}{W}_{2}^{T},\text{â}{W}_{1}^{T}{W}_{2}=0.
{â«}_{0}^{T}y{\left(t\right)}^{T}\left({W}_{1}{W}_{1}^{T}â{R}^{2}{W}_{2}{W}_{2}^{T}\right)\text{â}y\left(t\right)dt<0,
for all T ≥ 0. Varying R is equivalent to adjusting the slant angle of the cone specified by Q until the cone fits tightly around the output trajectories of H. The cone base-to-height ratio is proportional to R.
Q={W}_{1}{W}_{1}^{T}â{W}_{2}{W}_{2}^{T},\text{â}{W}_{1}^{T}{W}_{2}=0.
Q\left(j\mathrm{Ï}\right)=Z{\left(j\mathrm{Ï}\right)}^{H}\left({W}_{1}{W}_{1}^{T}â{W}_{2}{W}_{2}^{T}\right)Z\left(j\mathrm{Ï}\right).
Directional sector index of the system H for the sector specified by Q in the direction dQ, returned as a scalar value, or an array if H is an array. The directional index is the largest Ï„ which satisfies:
{â«}_{0}^{T}y{\left(t\right)}^{T}\left(Q+\mathrm{Ï}dQ\right)\text{â}y\left(t\right)dt<0,
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Find edges of objects in grayscale pixel stream - Simulink - MathWorks Nordic
Gv,Gh
Output the binary image
Output the gradient components
Source of threshold value
Gradient Data Type
Find edges of objects in grayscale pixel stream
Vision HDL Toolbox / Analysis & Enhancement
The Edge Detector block finds the edges in a grayscale pixel stream by using the Sobel, Prewitt, or Roberts method. The block convolves the input pixels with derivative approximation matrices to find the gradient of pixel magnitude along two orthogonal directions. It then compares the sum of the squares of the gradients to the square of a configurable threshold to determine if the gradients represent an edge.
By default, the block returns a binary image as a stream of pixel values. A pixel value of 1 indicates that the pixel is an edge. You can disable the edge output. You can also enable output of the gradient values in the two orthogonal directions at each pixel.
This block uses a streaming pixel interface with a bus for frame control signals. This interface enables the block to operate independently of image size and format. The pixel, Edge, and gradient ports on this block support single pixel streaming or multipixel streaming. Single pixel streaming accepts and returns a single pixel value each clock cycle. Multipixel streaming accepts and returns a vector of M pixels per clock cycle to support high-frame-rate or high-resolution formats. The M value corresponds to the Number of pixels parameter of the Frame To Pixels block. Along with the pixel, the block accepts and returns a pixelcontrol bus containing five control signals. The control signals indicate the validity of each pixel and their location in the frame. For multipixel streaming, one set of control signals applies to all pixels in the vector. To convert a frame (pixel matrix) into a serial pixel stream and control signals, use the Frame To Pixels block. For a full description of the interface, see Streaming Pixel Interface.
This block supports single pixel streaming or multipixel streaming. For single pixel streaming, specify a single input pixel as a scalar intensity value. For multipixel streaming, specify a vector of four or eight pixel intensity values. For details of how to set up your model for multipixel streaming, see Filter Multipixel Video Streams.
Th — Threshold value
Threshold value that defines an edge, specified as a scalar. The block compares the square of this value to the sum of the squares of the gradients.
Data Types: single | double | int | uint | fixed point
Edge — Boolean pixel value, indicating whether pixel is an edge
For single pixel streaming, Edge is a Boolean scalar. For multipixel streaming, Edge is a vector of M-by-1 Boolean values. Each pixel value indicates whether the pixel is an edge.
Gv,Gh — Vertical and horizontal gradient
Vertical and horizontal gradient values calculated over the kernel centered at a pixel location.
For single pixel streaming, the block returns Gv and Gh as scalar values. For multipixel streaming, the block returns Gv and Gh as vectors of M-by-1 values.
These ports are visible when you set Method to Sobel or Prewitt.
G45,G135 — Orthogonal gradient
Orthogonal gradient values calculated over the kernel centered at a pixel location.
For single pixel streaming, the block returns G45 and G135 as scalar values. For multipixel streaming, the block returns G45 and G135 as vectors of M-by-1 values.
These ports are visible when you set Method to Roberts.
Method — Edge detection algorithm
Sobel (default) | Prewitt | Roberts
When you select Sobel or Prewitt, the block calculates horizontal and vertical gradients, Gv and Gh. When you select Roberts, the block calculates orthogonal gradients, G45 and G135. For details of each method, see Algorithms.
If you select Prewitt, the full-precision internal data type is large due to the 1/6 coefficient. Consider selecting Output the gradient components, so that you can customize the data type to a smaller size.
Output the binary image — Enable edge output port
When this parameter is selected, the block returns a stream of binary pixels representing the edges detected in the input frame.
You must select at least one of Output the binary image and Output the gradient components.
Output the gradient components — Enable gradient output ports
When this parameter is selected, the block returns a stream of values representing the gradients calculated in the two orthogonal directions at each pixel. When you set Method to Sobel or Prewitt, the output ports Gv and Gh appear on the block. When you set Method to Roberts, the output ports G45 and G135 appear on the block.
Source of threshold value — Source for gradient threshold that indicates an edge
You can set the threshold from an input port or from the dialog box. The default value is Property. Selecting Input port enables the Th port.
Threshold value — Gradient threshold value that indicates an edge
The block compares the square of this value to the sum of the squares of the gradients. The block casts this value to the data type of the gradients.
This option is visible when you set Source of threshold value to Property.
Line buffer size — Size of the line memory buffer
The block allocates (N – 1)-by-Line buffer size memory locations to store the pixels, where N is the number of lines in the differential approximation matrix. If you set Method to Sobel or Prewitt, then N is 3. If you set Method to Roberts, then is 2.
Symmetric (default) | None
Symmetric — Set the value of the padding pixels to mirror the edge of the image. This option prevents edges from being detected at the boundaries of the active frame.
Rounding mode — Rounding method for internal fixed-point calculations
Specify a rounding method for internal fixed-point calculations.
Gradient Data Type — Data type for gradient output ports
Inherit via internal rule (default) | data type expression
Data type for the two gradient output ports. By default, the block automatically chooses full-precision data types.
To enable this parameter, on the Main tab, select Output the gradient components.
The Edge Detector block provides three methods for detecting edges in an input image. The methods use different derivative approximation matrices to find two orthogonal gradients. The Sobel and Prewitt methods calculate the gradient in horizontal and vertical directions. The Roberts method calculates the gradients at 45 degrees and 135 degrees. The block uses the same matrices as the Edge Detection block in Computer Vision Toolbox™.
When you use multipixel streaming, the block uses a single line memory and implements one filter for each of the M input pixels, in parallel. This increase in hardware resources is a trade off for increasing throughput compared to single-pixel streaming.
\frac{1}{8}\left[\begin{array}{ccc}1& 0& -1\\ 2& 0& -2\\ 1& 0& -1\end{array}\right]
\frac{1}{8}\left[\begin{array}{ccc}1& 2& 1\\ 0& 0& 0\\ -1& -2& -1\end{array}\right]
\frac{1}{6}\left[\begin{array}{ccc}1& 0& -1\\ 1& 0& -1\\ 1& 0& -1\end{array}\right]
\frac{1}{6}\left[\begin{array}{ccc}1& 1& 1\\ 0& 0& 0\\ -1& -1& -1\end{array}\right]
\frac{1}{2}\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]
\frac{1}{2}\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]
The Prewitt coefficients require extra bits of precision because they are not powers of two. The block uses 16 bits to represent the Prewitt coefficients. For 8-bit input, the default size of the full-precision gradients is 27 bits. When using the Prewitt method, a good practice is to reduce the word length used for the gradient calculation. Select the Output the gradient components check box, and then on the Data Types tab, specify a smaller word length using Gradient Data Type.
The block convolves the neighborhood of the input pixel with the derivative matrices, D1 and D2. It then compares the sum of the squares of the gradients to the square of the threshold. Computing the square of the threshold avoids constructing a square root circuit. The block casts the gradients to the type you specified on the Data Types tab. The type conversion on the square of the threshold matches the type of the sum of the squares of the gradients.
Edge Detection (Computer Vision Toolbox) | Frame To Pixels
visionhdl.EdgeDetector
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Unisoft Hypervector Spaces
Abdullah M. Al-Roqi, "Unisoft Hypervector Spaces", Journal of Applied Mathematics, vol. 2014, Article ID 418702, 6 pages, 2014. https://doi.org/10.1155/2014/418702
Abdullah M. Al-Roqi1
The notion of unisoft subfields, unisoft algebras over unisoft subfields, and unisoft hypervector spaces are introduced, and their properties and characterizations are considered. In connection with linear transformations, unisoft hypervector spaces are discussed.
The hyperstructure theory was introduced by Marty [1] at the 8th Congress of Scandinavian Mathematicians in 1934. As a generalization of fuzzy vector spaces, the fuzzy hypervector spaces are studied by Ameri and Dehghan (see [2, 3]). Molodtsov [4] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. Molodtsov pointed out to several directions for the applications of soft sets. At present, works on the soft set theory are progressing rapidly. Maji et al. [5] described the application of soft set theory to a decision-making problem. Maji et al. [6] also studied several operations on the theory of soft sets. Chen et al. [7] presented a new definition of soft set parametrization reduction and compared this definition to the related concept of attributes reduction in rough set theory. The algebraic structure of set theories dealing with uncertainties has been studied by some authors. Çağman et al. [8] introduced fuzzy parameterized (FP) soft sets and their related properties. They proposed a decision-making method based on FP soft set theory and provided an example which shows that the method can be successfully applied to the problems that contain uncertainties. Feng [9] considered the application of soft rough approximations in multicriteria group decision-making problems. Aktaş and Çağman [10] studied the basic concepts of soft set theory and compared soft sets to fuzzy and rough sets, providing examples to clarify their differences. They also discussed the notion of soft groups. After that, many algebraic properties of soft sets are studied (see [11–21]).
In this paper, we introduce the notion of unisoft subfields, unisoft algebras over unisoft subfields, and unisoft hypervector spaces. We study their properties and characterizations. In connection with linear transformations, we discuss unisoft hypervector spaces.
A soft set theory introduced by Molodtsov [4] and Çağman and Enginoğlu [22] provided new definitions and various results on soft set theory.
Definition 1 (see [4, 22]) . A soft set over is defined to be the set of ordered pairs where such that if .
A map is called a hyperoperation or join operation, where is the set of all nonempty subsets of . The join operation is extended to subsets of in natural way, so that is given by
The notations and are used for and , respectively. Generally, the singleton is identified by its element .
Definition 2 (see [23]) . Let be a field and be an abelian group. A hypervector space over is defined to be the quadruplet , where “” is a mapping such that for all and the following conditions hold: (H1), (H2), (H3), (H4), (H5).
A hypervector space over a field is said to be strongly left distributive (see [2]) if it satisfies the following condition:
3. Unisoft Algebras over a Unisoft Field
In what follows let be a field unless otherwise specified.
Definition 3. A soft set over is called a unisoft subfield of if the following conditions are satisfied: (1), (2), (3), (4).
Proposition 4. If is a unisoft subfield of , then (1), (2), (3).
Proof. (1) For all , we have
(2) Let be such that . Then
(3) It follows from (1).
It is easy to show that the following theorem holds.
Theorem 5. A soft set over is a unisoft subfield of if and only if the nonempty -exclusive set of is a subfield of for all .
Definition 6. Let be an algebra over and let be a unisoft subfield of . A soft set is called an unisoft algebra over if it satisfies the following conditions: (1), (2), (3), (4).
Proposition 7. Let be an algebra over and let be a unisoft subfield of . If is a unisoft algebra over , then for all .
Proof. For any , we have .
We provide a characterization of a unisoft algebra over .
Theorem 8. For any algebra over , let be a unisoft subfield of . Then a soft set is a unisoft algebra over if and only if it satisfies (3) and (4) of Definition 6 and
Proof. Assume that is a unisoft algebra over . Using (1) and (2) of Definition 6, we have for all and .
Conversely, suppose that satisfies (3) and (4) of Definition 6 and (8). Then
By using Definition 6(3) and Proposition 4(3), we obtain for all . Thus for all and . Therefore is a unisoft algebra over .
For any sets and , let be a function and and be soft sets over .
(1) The soft set where , is called the unisoft preimage of under .
(2) The soft set where is called the unisoft image of under .
Theorem 9. Let and be algebras over . For any algebraic homomorphism ,(1)if is a unisoft algebra over , then the unisoft preimage of under is also a unisoft algebra over .(2)If is a unisoft algebra over , then the unisoft image of under is also a unisoft algebra over .
Proof. (1) For any and , we have and . Therefore, by Theorem 8, is a unisoft algebra over .
(2) Let . If or , then
Assume that and . Then , and so
For all , if at least one of and is empty, then the inclusion is clear. Assume that and . Then
Since for all , it follows that for all . Therefore is a unisoft algebra over .
4. Unisoft Hypervector Spaces
Definition 10. Let be a hypervector space over and a unisoft subfield of . A soft set over is called a unisoft hypervector space of related to if the following assertions are valid: (1), (2), (3), (4) where is the zero of .
Proposition 11. Let be a hypervector space over and a unisoft subfield of . If is a unisoft hypervector space of related to , then (1), (2), (3).
Proof. It is an immediate consequence of Definition 10 and Proposition 4.
Proposition 12. Let be a hypervector space over . If is a unisoft hypervector space of related to a unisoft subfield of , then
Proof. Let . Since by (H5), we have . Using Definition 10(3) we have
Hence for all .
Theorem 13. Assume that a hypervector space over is strongly left distributive. Let be a unisoft subfield of . Then a soft set over is a unisoft hypervector space of related to if and only if the following conditions are true: (1), (2)for all and all .
Proof. Assume that is a unisoft hypervector space of related to . The second condition follows from Proposition 11(2) and Definition 10(4). Let and . Then
Conversely suppose the conditions (1) and (2) are true. For all , we have
Since is a unisoft subfield of , we have and . Note that for all . It follows that for all . Let and . Then
Clearly, . Therefore is a unisoft hypervector space of related to .
Theorem 14. Let be a hypervector space over and a unisoft subfield of . If a soft set over is a unisoft hypervector space of related to , then the nonempty -exclusive set of is a subhypervector space of over the field for all .
Proof. Let . Then and . It follows that
Hence . Note that is a subfield of (see Theorem 5). Let and . Then and so which shows that . Therefore is a hypervector space over the field for all .
Let and be hypervector spaces over . A mapping is called linear transformation (see [3]) if it satisfies(i), (ii).
Theorem 15. Let and be hypervector spaces over and let be a unisoft subfield of . For any linear transformation , if is a unisoft hypervector space of related to , then is a unisoft hypervector space of related to .
Proof. Let and . Since is a linear transformation, we have
Obviously, for all . It follows from Theorem 13 that is a unisoft hypervector space of related to .
Theorem 16. Let and be hypervector spaces over and let be a unisoft subfield of . For any linear transformation , if is a unisoft hypervector space of related to , then is a unisoft hypervector space of related to .
Proof. Let and . If at least one of and is empty, then the inclusion is clear. Assume that and are nonempty. Then there exist such that and . Thus, since is linear. Hence, . Then
Obviously, for all . Therefore is a unisoft hypervector space of related to by Theorem 13.
The author would like to express his sincere thanks to the anonymous referees for their valuable suggestions and comments that improved this paper.
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Copyright © 2014 Abdullah M. Al-Roqi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Parametric Modeling - MATLAB & Simulink - MathWorks España
Available Parametric Modeling Functions
Time-Domain Based Modeling
Prony's Method (ARMA Modeling)
Steiglitz-McBride Method (ARMA Modeling)
Frequency-Domain Based Modeling
Parametric modeling techniques find the parameters for a mathematical model describing a signal, system, or process. These techniques use known information about the system to determine the model. Applications for parametric modeling include speech and music synthesis, data compression, high-resolution spectral estimation, communications, manufacturing, and simulation.
The toolbox parametric modeling functions operate with the rational transfer function model. Given appropriate information about an unknown system (impulse or frequency response data, or input and output sequences), these functions find the coefficients of a linear system that models the system.
One important application of the parametric modeling functions is in the design of filters that have a prescribed time or frequency response.
Here is a summary of the parametric modeling functions in this toolbox.
Generate all-pole filter coefficients that model an input data sequence using the Levinson-Durbin algorithm.
Generate all-pole filter coefficients that model an input data sequence by minimizing the forward prediction error.
Generate all-pole filter coefficients that model an input data sequence by minimizing the forward and backward prediction errors.
Generate all-pole filter coefficients that model an input data sequence using an estimate of the autocorrelation function.
lpc, levinson
Linear Predictive Coding. Generate all-pole recursive filter whose impulse response matches a given sequence.
Generate IIR filter whose impulse response matches a given sequence.
Find IIR filter whose output, given a specified input sequence, matches a given output sequence.
invfreqz, invfreqs
Generate digital or analog filter coefficients given complex frequency response data.
The lpc, prony, and stmcb functions find the coefficients of a digital rational transfer function that approximates a given time-domain impulse response. The algorithms differ in complexity and accuracy of the resulting model.
Linear prediction modeling assumes that each output sample of a signal, x(k), is a linear combination of the past n outputs (that is, it can be linearly predicted from these outputs), and that the coefficients are constant from sample to sample:
x\left(k\right)=-a\left(2\right)x\left(k-1\right)-a\left(3\right)x\left(k-2\right)-\cdots -a\left(n+1\right)x\left(k-n\right).
An nth-order all-pole model of a signal x is
a = lpc(x,n)
To illustrate lpc, create a sample signal that is the impulse response of an all-pole filter with additive white noise:
x = impz(1,[1 0.1 0.1 0.1 0.1],10) + randn(10,1)/10;
The coefficients for a fourth-order all-pole filter that models the system are
a = lpc(x,4)
lpc first calls xcorr to find a biased estimate of the correlation function of x, and then uses the Levinson-Durbin recursion, implemented in the levinson function, to find the model coefficients a. The Levinson-Durbin recursion is a fast algorithm for solving a system of symmetric Toeplitz linear equations. lpc's entire algorithm for n = 4 is
r = xcorr(x);
r(1:length(x)-1) = []; % Remove corr. at negative lags
a = levinson(r,4)
You could form the linear prediction coefficients with other assumptions by passing a different correlation estimate to levinson, such as the biased correlation estimate:
r = xcorr(x,'biased');
The prony function models a signal using a specified number of poles and zeros. Given a sequence x and numerator and denominator orders n and m, respectively, the statement
[b,a] = prony(x,n,m)
finds the numerator and denominator coefficients of an IIR filter whose impulse response approximates the sequence x.
The prony function implements the method described in [4] Parks and Burrus. This method uses a variation of the covariance method of AR modeling to find the denominator coefficients a, and then finds the numerator coefficients b for which the resulting filter's impulse response matches exactly the first n + 1 samples of x. The filter is not necessarily stable, but it can potentially recover the coefficients exactly if the data sequence is truly an autoregressive moving-average (ARMA) process of the correct order.
The functions prony and stmcb (described next) are more accurately described as ARX models in system identification terminology. ARMA modeling assumes noise only at the inputs, while ARX assumes an external input. prony and stmcb know the input signal: it is an impulse for prony and is arbitrary for stmcb.
A model for the test sequence x (from the earlier lpc example) using a third-order IIR filter is
[b,a] = prony(x,3,3)
The impz command shows how well this filter's impulse response matches the original sequence:
[x impz(b,a,10)]
Notice that the first four samples match exactly. For an example of exact recovery, recover the coefficients of a Butterworth filter from its impulse response:
h = impz(b,a,26);
[bb,aa] = prony(h,4,4);
Try this example; you'll see that bb and aa match the original filter coefficients to within a tolerance of 10-13.
The stmcb function determines the coefficients for the system b(z)/a(z) given an approximate impulse response x, as well as the desired number of zeros and poles. This function identifies an unknown system based on both input and output sequences that describe the system's behavior, or just the impulse response of the system. In its default mode, stmcb works like prony.
[b,a] = stmcb(x,3,3)
stmcb also finds systems that match given input and output sequences:
y = filter(1,[1 1],x); % Create an output signal.
[b,a] = stmcb(y,x,0,1)
In this example, stmcb correctly identifies the system used to create y from x.
The Steiglitz-McBride method is a fast iterative algorithm that solves for the numerator and denominator coefficients simultaneously in an attempt to minimize the signal error between the filter output and the given output signal. This algorithm usually converges rapidly, but might not converge if the model order is too large. As for prony, stmcb's resulting filter is not necessarily stable due to its exact modeling approach.
stmcb provides control over several important algorithmic parameters; modify these parameters if you are having trouble modeling the data. To change the number of iterations from the default of five and provide an initial estimate for the denominator coefficients:
n = 10; % Number of iterations
a = lpc(x,3); % Initial estimates for denominator
[b,a] = stmcb(x,3,3,n,a);
The function uses an all-pole model created with prony as an initial estimate when you do not provide one of your own.
To compare the functions lpc, prony, and stmcb, compute the signal error in each case:
a1 = lpc(x,3);
[b2,a2] = prony(x,3,3);
[b3,a3] = stmcb(x,3,3);
[x-impz(1,a1,10) x-impz(b2,a2,10) x-impz(b3,a3,10)]
In comparing modeling capabilities for a given order IIR model, the last result shows that for this example, stmcb performs best, followed by prony, then lpc. This relative performance is typical of the modeling functions.
The invfreqs and invfreqz functions implement the inverse operations of freqs and freqz; they find an analog or digital transfer function of a specified order that matches a given complex frequency response. Though the following examples demonstrate invfreqz, the discussion also applies to invfreqs.
To recover the original filter coefficients from the frequency response of a simple digital filter:
[b,a] = butter(4,0.4) % Design Butterworth lowpass
[h,w] = freqz(b,a,64); % Compute frequency response
[b4,a4] = invfreqz(h,w,4,4) % Model: n = 4, m = 4
The vector of frequencies w has the units in rad/sample, and the frequencies need not be equally spaced. invfreqz finds a filter of any order to fit the frequency data; a third-order example is
[b4,a4] = invfreqz(h,w,3,3) % Find third-order IIR
Both invfreqs and invfreqz design filters with real coefficients; for a data point at positive frequency f, the functions fit the frequency response at both f and -f.
By default invfreqz uses an equation error method to identify the best model from the data. This finds b and a in
by creating a system of linear equations and solving them with the MATLAB® \ operator. Here A(w(k)) and B(w(k)) are the Fourier transforms of the polynomials a and b respectively at the frequency w(k), and n is the number of frequency points (the length of h and w). wt(k) weights the error relative to the error at different frequencies. The syntax
invfreqz(h,w,n,m,wt)
includes a weighting vector. In this mode, the filter resulting from invfreqz is not guaranteed to be stable.
invfreqz provides a superior ("output-error") algorithm that solves the direct problem of minimizing the weighted sum of the squared error between the actual frequency response points and the desired response
To use this algorithm, specify a parameter for the iteration count after the weight vector parameter:
wt = ones(size(w)); % Create unit weighting vector
[b30,a30] = invfreqz(h,w,3,3,wt,30) % 30 iterations
The resulting filter is always stable.
Graphically compare the results of the first and second algorithms to the original Butterworth filter with FVTool (and select the Magnitude and Phase Responses):
fvtool(b,a,b4,a4,b30,a30)
To verify the superiority of the fit numerically, type
sum(abs(h-freqz(b4,a4,w)).^2) % Total error, algorithm 1
sum(abs(h-freqz(b30,a30,w)).^2) % Total error, algorithm 2
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Brinell Hardness Number Calculator - BHN
Brinell hardness test — What is Brinell hardness number?
How to calculate Brinell hardness number
Example: Using the brinell hardness number calculator
Applications and similar tests
The Brinell hardness number calculator will return the BHN or Brinell hardness number, which helps estimate the indentation hardness for a material sample. It is not to be confused with water hardness. This hardness number is estimated after performing an indentation experiment or a test known as the Brinell hardness test.
Swedish engineer Johan August Brinell proposed this procedure for indentation hardness in the year 1900. Since then, the Brinell hardness test has become a standardized test in metallurgy and material science to estimate the hardness more so because of its non-destructive nature.
The basic idea behind this test is the calculation of pressure based on the force applied and the indentation area. Read on to understand the Brinell hardness scale and conduct the Brinell hardness test.
The Brinell hardness test is a non-destructive test performed by using a small ball-shaped indentor that is pressed onto the material specimen with a standard load value. The depth and the diameter of the indentor are then recorded, and you use them to calculate the Brinell hardness number.
Brinell hardness test with a ball indenter.
The indentor is a ball made of steel or tungsten carbide having a diameter, D. Consider the ball indentor pressed at a load or force value of P newtons. The equation gives the Brinell hardness number:
\scriptsize \text{HBW} = \frac{\text{Test force}}{\text{Surface area of indentation}}
\scriptsize \text{HBW} = 0.102\frac{2P}{\pi D \left ( D - \sqrt{D^2 - d^2}\right)}
P – Test force (in Newtons); and
d – Diameter of indentation (in mm).
You can find the details of the Brinell hardness test in the international standards ISO 6506-1 for metallic testing.
To calculate Brinell hardness number:
Enter the load applied, P.
Insert the diameter of the indentor, D.
Fill in the diameter of indentation, d.
The calculator will determine the surface area, A, and the Brinell hardness number (HBW).
Determine the Brinell hardness number for a material specimen having an indentation diameter of 3 mm. The test is conducted using an indentor with a 10 mm diameter and a test force of 294.2 N.
Enter the load applied, P = 294.2 N.
Insert the diameter of the indentor, D = 10 mm.
Fill in the diameter of indentation, d = 3 mm.
The Brinell hardness number (HBW) is then calculated:
\qquad \scriptsize \begin{align*} \text{HBW} &= 0.102\frac{2 \times 294.2}{10\pi \left( 10 - \sqrt{10^2 - 3^2}\right)} \\ &= 4.1475 \text{ MPa} \end{align*}
You can represent the Brinell hardness for a material specimen as per the ISO 6506-1. For example:
600 HBW 1 / 30 / 20
600 – Brinell hardness value;
HBW – Hardness symbol;
1 – Ball diameter in mm;
30 – Applied test force (in kgf); and
20 – Duration time of the test force.
The following table contains the Brinell hardness numbers for some materials.
AW-6060 aluminum
The Brinell hardness scale is often connected with the ultimate tensile strength (UTS) of materials or alloys, but it is not applicable for all ranges of Brinell hardness number. You can also lookup Meyer's index or law, which is an empirical relation for converting the Brinell hardness number to ultimate tensile strength (UTS). Still, it is limited to a series of indentor diameters.
Some of the applications of this test include testing hardness for rocks, inspecting welded joints, characterization of surface coatings and thin films, and material behavior. Researchers and engineers also use it to develop material modeling frameworks for crystalline materials. Some variations of this test are broadly categorized as micro or nano indentations; they are primarily used but not limited to thin surface layers like films and coatings.
Some other indentation tests similar to the Brinell hardness test are Vickers hardness and Rockwell hardness, which are applicable for a wide range of categories of metals and alloys. In addition to this, some material-specific tests also exist, such as the Janka hardness test for wood, Knoop hardness test for ceramics, or Shore and Barcol hardness tests for polymers and composites, respectively.
The Brinell hardness test is a non-destructive indentation test that is used to measure the hardness of a material. It is performed using a small indentor that is impressed upon the material surface, after which the diameter of indentation is measured to estimate the Brinell hardness number.
How do I perform the Brinell hardness test?
As per the testing standard ISO 6506-1:
Begin the test by bringing the indentor in contact with the test surface.
Apply the force perpendicular to the test surface, but without any shock, or vibration, until the applied force is attained as per the test standard.
Maintain the test force for 14 seconds. It could be longer for specific materials.
Remove the load and measure the diameter of indentation in two perpendicular directions.
Use the formula to obtain the Brinell hardness number.
What is the Brinell hardness number for copper?
The Brinell hardness number for mild steel is 35 HB. This value implies that during the Brinell hardness test, the user will observe an indentation of diameter about 1.35 mm when a load of 500 N is applied on the material specimen using an indentor of diameter having 10 mm.
What is the Brinell hardness number for aluminum?
The Brinell hardness number for aluminum is 15 HB. This value implies that during the Brinell hardness test, the user will observe an indentation of diameter about 2.05 mm when a load of 500 N is applied on the material specimen using an indentor of diameter having 10 mm.
Applied load (P)
Diameter of indenter (D)
Diameter of indentation (d)
Surface area of indentation (A)
Brinell hardness number (HBW)
Bernoulli equation calculator helps you determine the speed and pressure, as well as the flow rate, of an incompressible fluid.
Enter the orbital period calculator, where you can calculate the orbital period of a binary system, a satellite around the Earth and much more while learning about the universe and the laws that rule it.
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Circle - New World Encyclopedia
Previous (Circadian rhythm)
Next (Circulatory system)
This article is about the shape and mathematical concept of circle. For other uses of the term, see Circle (disambiguation).
In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a given point, the center. The length of the circle is called its circumference, and any continuous portion of the circle is called an arc.
A circle is a simple closed curve that divides the plane into an interior and exterior. The interior of the circle is called a disk.
1.2 Sagitta properties
4 An alternative definition of a circle
5 Calculating the parameters of a circle
5.3 Plane unit normal
5.4 Parametric Equation
Mathematically, a circle can be understood in several other ways as well. For instance, it is a special case of an ellipse in which the two foci coincide (that is, they are the same point). Alternatively, a circle can be thought of as the conic section attained when a right circular cone is intersected with a plane perpendicular to the axis of the cone.
All circles have similar properties. Some of these are noted below.
For any circle, the area enclosed and the square of its radius are in a fixed proportion, equal to the mathematical constant π.
For any circle, the circumference and radius are in a fixed proportion, equal to 2π.
The circle is the shape with the highest area for a given length of perimeter.
The circle is a highly symmetrical shape. Every line through the center forms a line of reflection symmetry. In addition, there is rotational symmetry around the center for every angle. The symmetry group is called the orthogonal group O(2,R), and the group of rotations alone is called the circle group T.
A line segment that connects one point of a circle to another is called a chord. The diameter is a chord that runs through the center of the circle.
The diameter is longest chord of the circle.
Chords equidistant from the center of a circle are equal in length. Conversely, chords that are equal in length are equidistant from the center.
A line drawn through the center of a circle perpendicular to a chord bisects the chord. Alternatively, one can state that a line drawn through the center of a circle bisecting a chord is perpendicular to the chord. This line is called the perpendicular bisector of the chord. Thus, one could also state that the perpendicular bisector of a chord passes through the center of the circle.
If two angles are inscribed on the same chord and on opposite sides of the chord, then they are supplemental.
An inscribed angle subtended by a diameter is a right angle.
Sagitta properties
The sagitta is a line segment drawn perpendicular to a chord, between the midpoint of that chord and the circumference of the circle.
Given the length of a chord, y, and the length x of the sagitta, the Pythagorean theorem can be used to calculate the radius of the unique circle which will fit around the 2 lines :
{\displaystyle r={\frac {y^{2}}{8x}}+{\frac {x}{2}}}
The line drawn perpendicular to the end point of a radius is a tangent to the circle.
A line drawn perpendicular to a tangent at the point of contact with a circle passes through the center of the circle.
Tangents drawn from a point outside the circle are equal in length.
Two tangents can always be drawn from a point outside of the circle.
Secant-secant theorem
The chord theorem states that if two chords, CD and EF, intersect at G, then
{\displaystyle CG\times DG=EG\times FG}
. (Chord theorem)
If a tangent from an external point D meets the circle at C and a secant from the external point D meets the circle at G and E respectively, then
{\displaystyle DC^{2}=DG\times DE}
. (tangent-secant theorem)
If two secants, DG and DE, also cut the circle at H and F respectively, then
{\displaystyle DH\times DG=DF\times DE}
. (Corollary of the tangent-secant theorem)
The angle between a tangent and chord is equal to the subtended angle on the opposite side of the chord. (Tangent chord property)
If the angle subtended by the chord at the center is 90 degrees then l = √(2) × r, where l is the length of the chord and r is the radius of the circle.
If two secants are inscribed in the circle as shown at right, then the measurement of angle A is equal to one half the difference of the measurements of the enclosed arcs (DE and BC). This is the secant-secant theorem.
In an x-y coordinate system, the circle with center (a, b) and radius r is the set of all points (x, y) such that
{\displaystyle \left(x-a\right)^{2}+\left(y-b\right)^{2}=r^{2}.}
If the circle is centered at the origin (0, 0), then this formula can be simplified to
{\displaystyle x^{2}+y^{2}=r^{2}\!\ }
and its tangent will be
{\displaystyle xx_{1}+yy_{1}=r^{2}\!\ }
{\displaystyle x_{1}}
{\displaystyle y_{1}}
are the coordinates of the common point.
When expressed in parametric equations, (x, y) can be written using the trigonometric functions sine and cosine as
{\displaystyle x=a+r\,\cos t,\,\!}
{\displaystyle y=b+r\,\sin t\,\!}
where t is a parametric variable, understood as the angle the ray to (x, y) makes with the x-axis.
In homogeneous coordinates each conic section with equation of a circle is
{\displaystyle ax^{2}+ay^{2}+2b_{1}xz+2b_{2}yz+cz^{2}=0.}
It can be proven that a conic section is a circle if and only if the point I(1,i,0) and J(1,-i,0) lie on the conic section. These points are called the circular points at infinity.
In polar coordinates the equation of a circle is
{\displaystyle r^{2}-2rr_{0}\cos(\theta -\varphi )+r_{0}^{2}=a^{2}.\,}
In the complex plane, a circle with a center at c and radius r has the equation
{\displaystyle |z-c|^{2}=r^{2}}
{\displaystyle |z-c|^{2}=z{\overline {z}}-{\overline {c}}z-c{\overline {z}}+c{\overline {c}}}
, the slightly generalized equation
{\displaystyle pz{\overline {z}}+gz+{\overline {gz}}=q}
for real p, q and complex g is sometimes called a generalized circle. It is important to note that not all generalized circles are actually circles.
The slope of a circle at a point (x, y) can be expressed with the following formula, assuming the center is at the origin and (x, y) is on the circle:
{\displaystyle y'=-{\frac {x}{y}}.}
More generally, the slope at a point (x, y) on the circle
{\displaystyle (x-a)^{2}+(y-b)^{2}=r^{2}}
, (i.e., the circle centered at [a, b] with radius r units), is given by
{\displaystyle y'={\frac {a-x}{y-b}},}
{\displaystyle y\neq b}
, of course.
The area enclosed by a circle is
{\displaystyle A=r^{2}\cdot \pi ={\frac {d^{2}\cdot \pi }{4}}\approx 0{.}7854\cdot d^{2},}
that is, approximately 79 percent of the circumscribed square.
Length of a circle's circumference is
{\displaystyle c=\pi d=2\pi \cdot r.}
Alternate formula for circumference:
Given that the ratio circumference c to the Area A is
{\displaystyle {\frac {c}{A}}={\frac {2\pi r}{\pi r^{2}}}.}
The r and the π can be canceled, leaving
{\displaystyle {\frac {c}{A}}={\frac {2}{r}}.}
Therefore solving for c:
{\displaystyle c={\frac {2A}{r}}}
So the circumference is equal to 2 times the area, divided by the radius. This can be used to calculate the circumference when a value for π cannot be computed.
The diameter of a circle is
{\displaystyle d=2r=2\cdot {\sqrt {\frac {A}{\pi }}}\approx 1{.}1284\cdot {\sqrt {A}}.}
An inscribed angle
{\displaystyle \psi }
is exactly half of the corresponding central angle
{\displaystyle \theta }
(see Figure). Hence, all inscribed angles that subtend the same arc have the same value (cf. the blue and green angles
{\displaystyle \psi }
in the Figure). Angles inscribed on the arc are supplementary. In particular, every inscribed angle that subtends a diameter is a right angle.
An alternative definition of a circle
{\displaystyle {\frac {d_{1}}{d_{2}}}={\textrm {constant}}}
Apollonius' definition of a circle
Apollonius of Perga showed that a circle may also be defined as the set of points having a constant ratio of distances to two foci, A and B.
The proof is as follows. A line segment PC bisects the interior angle APB, since the segments are similar:
{\displaystyle {\frac {AP}{BP}}={\frac {AC}{BC}}}
Analogously, a line segment PD bisects the corresponding exterior angle. Since the interior and exterior angles sum to
{\displaystyle 180^{\circ }}
, the angle CPD is exactly
{\displaystyle 90^{\circ }}
, i.e., a right angle. The set of points P that form a right angle with a given line segment CD form a circle, of which CD is the diameter.
As a point of clarification, note that C and D are determined by A, B, and the desired ratio (i.e. A and B are not arbitrary points lying on an extension of the diameter of an existing circle).
Calculating the parameters of a circle
Given three non-collinear points lying on the circle
{\displaystyle \mathrm {P_{1}} ={\begin{bmatrix}x_{1}\\y_{1}\\z_{1}\end{bmatrix}},\mathrm {P_{2}} ={\begin{bmatrix}x_{2}\\y_{2}\\z_{2}\end{bmatrix}},\mathrm {P_{3}} ={\begin{bmatrix}x_{3}\\y_{3}\\z_{3}\end{bmatrix}}}
{\displaystyle \mathrm {r} ={\frac {\left|P_{1}-P_{2}\right|\left|P_{2}-P_{3}\right|\left|P_{3}-P_{1}\right|}{2\left|\left(P_{1}-P_{2}\right)\times \left(P_{2}-P_{3}\right)\right|}}}
The center of the circle is given by
{\displaystyle \mathrm {P_{c}} =\alpha \,P_{1}+\beta \,P_{2}+\gamma \,P_{3}}
{\displaystyle \alpha ={\frac {\left|P_{2}-P_{3}\right|^{2}\left(P_{1}-P_{2}\right)\cdot \left(P_{1}-P_{3}\right)}{2\left|\left(P_{1}-P_{2}\right)\times \left(P_{2}-P_{3}\right)\right|^{2}}}}
{\displaystyle \beta ={\frac {\left|P_{1}-P_{3}\right|^{2}\left(P_{2}-P_{1}\right)\cdot \left(P_{2}-P_{3}\right)}{2\left|\left(P_{1}-P_{2}\right)\times \left(P_{2}-P_{3}\right)\right|^{2}}}}
{\displaystyle \gamma ={\frac {\left|P_{1}-P_{2}\right|^{2}\left(P_{3}-P_{1}\right)\cdot \left(P_{3}-P_{2}\right)}{2\left|\left(P_{1}-P_{2}\right)\times \left(P_{2}-P_{3}\right)\right|^{2}}}}
Plane unit normal
A unit normal of the plane containing the circle is given by
{\displaystyle {\hat {n}}={\frac {\left(P_{2}-P_{1}\right)\times \left(P_{3}-P_{1}\right)}{\left|\left(P_{2}-P_{1}\right)\times \left(P_{3}-P_{1}\right)\right|}}}
Given the radius,
{\displaystyle \mathrm {r} }
, center,
{\displaystyle \mathrm {P_{c}} }
, a point on the circle,
{\displaystyle \mathrm {P_{0}} }
and a unit normal of the plane containing the circle,
{\displaystyle {\hat {n}}}
, the parametric equation of the circle starting from the point
{\displaystyle \mathrm {P_{0}} }
and proceeding counterclockwise is given by the following equation:
{\displaystyle \mathrm {R} \left(s\right)=\mathrm {P_{c}} +\cos \left({\frac {\mathrm {s} }{\mathrm {r} }}\right)\left(P_{0}-P_{c}\right)+\sin \left({\frac {\mathrm {s} }{\mathrm {r} }}\right)\left[{\hat {n}}\times \left(P_{0}-P_{c}\right)\right]}
Altshiller-Court, Nathan. 2007. College Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle. New York, NY: Dover Publications. ISBN 0486458059.
Pedoe, Dan. 1997. Circles: A Mathematical View. Washington, DC: The Mathematical Association of America. ISBN 0883855186.
Interactive Java applets – for the properties of and elementary constructions involving circles.
Interactive Standard Form Equation of Circle – Click and drag points to see standard form equation in action.
What Is Circle? – at cut-the-knot.
History of "Circle"
Retrieved from https://www.newworldencyclopedia.org/p/index.php?title=Circle&oldid=1003427
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Shock Index Calculator | Shock Index formula
How to calculate shock index
Shock index score interpretation
Prehospital shock index formula
The shock index calculator - an example
Modified shock index formula
What is age shock index?
The shock index calculator quickly assesses just two parameters - heart rate and systolic blood pressure - to give you a valuable predictive measure. Don't forget to hit the advanced mode button and read on to find out:
How the prehospital shock index formula was introduced;
How to perform a shock index interpretation after you calculate it; and
The shock index formula is:
shock index = heart rate [beats/minute] / systolic blood pressure [mmHg]
The normal shock index should be smaller than 0.7. Analyzing the formula, you can clearly see the bigger the shock index is, the worse the patient's condition. It means that the disproportion between heart rate and systolic blood pressure is growing, which indicates increasing circulatory failure.
It is an old formula, first presented over 50 years ago - in 1967. Its utility still remains, however, and the index is widely used, mainly in emergency medicine.
The normal shock index value is more than 0.5 and less than 0.7. The general rule is that the greater the shock index, the worse state the patient is in. It means either the heart is beating too fast and/or the blood pressure is dropping too much, both of which are signs of uncompensated shock.
One piece of research showed an almost three times higher mortality rate in patients with a shock index over > 0.9, compared to patients within the normal shock index. In the same paper, the absolute change of the shock index during observation in the hospital was also a prognostic sign - a 0.3 or greater difference in the shock index correlated with an almost five times higher risk of death than a change of less than 0.3.
But mortality is not all there is. A shock index over 0.9 is linked to an increased risk of massive transfusion as well. Another study also found a correlation between a shock index greater than one, hyperlactatemia, and 28-day mortality.
Shock index was first proposed by two Swiss doctors, Allgöwer and Burri, in 1967. Therefore, you might also find the formula under the name of Allgöwer's index, or Allgöwer's shock index, as well.
It has been used mainly in emergency medicine to assess the severity of shock. The index is quick to find and can help you estimate the risk of fatal outcomes, especially:
Increased early mortality;
Need of massive transfusion;
An occult shock; and
The normal shock index ranges from 0.5 to 0.7. The bigger the index, the higher is the risk of all the outcomes mentioned above.
How do you calculate the shock index for this patient?
A male patient, aged 63, was admitted to the emergency room at 6 AM. He presented with chest pain, shortness of breath; he was pale and sweaty. He has a history of hypertension, prediabetes, and he underwent a left knee replacement two years ago. His body temperature was 36.9°C (98.4°F), his blood pressure was 165/109 mmHg, heart rate was 104 beats/minute. ECG showed manifestations of myocardial ischemia.
The formula for shock index is:
shock index = heart rate [beats/minute]/systolic blood pressure [mmHg]
From the history, we have two important parameters:
heart rate equal to 104 beats/minute; and
systolic blood pressure equal to 165 mmHg.
shock index = 104 / 165 ≈ 0.63
The patient's shock index is 0.63, which is within the normal range.
Some clinicians prefer to use the modified shock index formula (MSI formula) instead of the 'regular' one. The difference lies in the pressure we're taking into account - in the modified formula, we're using mean arterial pressure (MAP) instead of systolic blood pressure.
modified shock index = heart rate [beats/minute] / MAP [mmHg]
You can calculate MSI in our shock index calculator - hit the advanced mode to expand the calculator panel.
To count MAP on your own, use the formula:
MAP = \frac{2 × DBP + SBP}{3}
MAP - Mean arterial pressure;
DBP - Diastolic blood pressure; and
SBP - Systolic blood pressure.
Work on standardizing the modified shock index is still being done; however, a value of 1 is considered too high. You interpret this measure the same as the standard shock index - the bigger the value, the worse the prognosis.
Elevated MSI increases the risk of:
ICU (Intensive Care Unit) admission;
Sepsis and hyperlactatemia;
Short- and long-term mortality; and
Age shock index is the shock index (SI) multiplied by the patient's age in years. It is a modified shock index and can identify patients at high risk of death due to acute myocardial infarction (AMI). It gives similar results to the GRACE score; however, it's much quicker to calculate bedside.
shock index = age [years] × heart rate [beats/minute] / systolic blood pressure [mmHg]
You can also calculate the age SI using our shock index calculator - again, hit the advanced mode to expand the calculator panel.
In one papers, there's a suggestion to consider any value above 50 as an elevated age shock index.
A compensated shock is a state where the body is still able to compensate shock (meaning relative or total fluid loss) via different mechanisms (circulatory centralization, release of specific hormones). Patients in a compensated shock won't be visibly hypotensive.
What is volemia?
Hypovolemia is a state of pathologically low fluids in your circulatory system. It can be due to bleeding or extreme dehydration. On the other hand, hypervolemia means too much fluid - e.g., due to renal or heart failure.
When we talking about volemia, we never talk about the term itself - we always refer to either hypovolemia or hypervolemia.
Does shock lower heart rate?
Generally, no. Shock will most probably increase your heart rate. This is because your body tries to compensate for relatively too little fluids, blood, and oxygen in the body - so it tries to speed up the blood circulation.
But this doesn't always mean that shock is always present with tachycardia (fast heart beating). If a patient is in shock because of conduction abnormalities, and their heartbeat is abnormally slow or not at all (asystole), you won't observe such a relationship.
What are the most common causes of shock?
The most common causes of shock include:
Severe allergic reaction (or anaphylaxis);
Major blood loss (due to an injury);
Heart failure (especially acute heart failure, e.g., due to myocardial infarction);
Severe, significant burns; and
Shock Index (SI)
The happiness calculator will help you measure your subjective happiness according to Subjective Happiness Scale (SHS).
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Poise and stokes viscosity units
How to convert poise to stokes?
How to use the poise-stokes converter?
This Poise-Stokes converter will help you easily convert dynamic viscosity to kinematic viscosity and vice versa. In this stokes to poise calculator, you will learn:
The difference between dynamic and kinematic viscosity;
The relationship between these two types of viscosities; and
How to convert poise to stokes, stokes to poise, or either to other viscosity units.
Keep on reading to start learning!
Poise and stokes are units of measure used to quantify viscosity. Poise is a unit of measurement used particularly for dynamic viscosity, while stokes is for kinematic viscosity. Viscosity, which describes a fluid's consistency or "thickness," comes in these two types for some distinct reasons.
The dynamic viscosity tells us how much force is required for a fluid to move at a particular speed. When formulating the mixture of, let's say, a paste in a tube, we want the paste to have a specific dynamic viscosity. That way, it won't be either too hard to squeeze the paste out of the tube or too runny that a lot of paste comes out even with a little squeezing pressure.
On the other hand, we use kinematic viscosity to describe the speed of the fluid due to an applied force. One particular use of kinematic viscosity is for fuels. By determining the viscosity of fuels in terms of kinematic viscosity, we get to model the speed fuel droplets will be sprayed out of an injection nozzle due to applied pressure.
Now that we know the difference between the two types of viscosities, let's go back to the measurement units. A poise (P), named after Jean Léonard Marie Poiseuille, who also derived the Poiseuille's law equation, has a value equivalent to 0.1 pascals-second (Pa⋅s). We can also equate 1 poise in terms of other units at values shown in this table:
1 poise (P)
100 centipoise (cP)
100 millipascal seconds (mPa⋅s)
0.1 pascal seconds (Pa⋅s)
0.00209 slugs per foot second (slug/(ft⋅s))
0.00209 pound force second per square foot (lbf⋅s/ft²)
0.06720 pounds per foot second (lb/(ft⋅s))
1 dyne second per square centimeter (dyn⋅s/cm²)
1 grams per centimeter second (g/(cm⋅s))
0.1 kilogram per meter second (kg/(m⋅s))
0.0000145 reyn (reyn)
For kinematic viscosity, we measure it in terms of stokes. Stokes is named after Sir George Gabriel Stokes, who also derived Stokes' law. One stokes (St) is equivalent to 1 square centimeter per second (cm²/s). Here, we also have the other units which have a kinematic viscosity equal to 1 stoke:
1 stoke (St)
100 centistokes (cSt)
100 square millimeters per second (mm²/s)
1 square centimeter per second (cm²/s)
0.00010 square meters per second (m²/s)
0.15500 square inches per second (in²/s)
0.00108 square feet per second (ft²/s)
💡 For practical reasons, we usually use centipoise and centistokes as units for dynamic and kinematic viscosities, respectively.
We may want to express the viscosity values from one type to another in some cases. Converting from poise to stokes, and even from stokes to poise, is quite an easy thing to do. However, to do that, we need the density of the fluid in question. Once we know that, we can then use the following formula for conversion:
ν = η/ρ
ν - Kinematic viscosity in stokes (St);
η - Dynamic viscosity in poise (P); and
ρ - Density in grams per cubic centimeter (g/cm³).
We may also see this formula expressed in the form:
ν_{T} = η_{T} / ρ_{T}
In this form, the subscripted T denotes that we can only observe these parameters in a particular fluid at a specific temperature. That indicates that the fluid's density, hence its viscosity too, changes depending on the fluid's temperature. For the simplicity, we'll be using the form without the subscripted T.
Using our conversion formula, we can derive the formula to convert stokes to poise. We only need to multiply both sides of the equation by the density, ρ, to get:
η = ρ \times ν
Now that we know how to convert poise to stokes, and vice versa, why don't we consider an example? 🙂
Let's say we want to convert the dynamic viscosity of water at 20ºC. At that temperature, water's density, ρ, is approximately 0.9982 g/cm³, and its dynamic viscosity, η, is around 0.010016 poise. Substituting these values in our formula, we have:
ν = 0.010016 poise / 0.9982 g/cm³
ν = 0.010034 stokes (St) or 1.0034 centistokes (cSt)
You can learn more about water viscosity and how temperature affects it by checking out our water viscosity calculator.
⚠️ When converting between the two types of viscosities, make sure you divide the dynamic viscosity in poise by the density in grams per cubic centimeters to obtain the kinematic viscosity in stokes. If these parameters are in different units, convert them first to these units to avoid miscalculations.
Using the poise-stokes calculator makes converting viscosity values a breeze. Here are the steps you can follow when using the poise-stokes converter:
Enter the density of the fluid you're interested in.
Input your known viscosity.
If you enter a value for the dynamic viscosity, you'll instantly see its equivalent kinematic viscosity, and vice versa. You can also convert dynamic and kinematic viscosity values to their other viscosity units using this converter. Simply type in the value of your known viscosity, and then choose another unit from the drop-down menu.
What unit is used to measure viscosity?
We measure a fluid's viscosity in terms of poise for dynamic viscosity and stokes for kinematic viscosity. We can convert poise to stokes (or dynamic viscosities to kinematic viscosities) by dividing the value in poise by the fluid's density in grams per cubic centimeter.
How many stokes is in a poise?
Stokes is directly proportional to poise, with the fluid's density being the constant. By dividing a value of dynamic viscosity in poise by a density in grams per cubic centimeter, we get the kinematic viscosity value in stokes. If the fluid density is equal to 1 g/cm³, then the amount of kinematic viscosity in stokes will be the same as the dynamic viscosity in poise.
How do I convert stokes to poise?
To convert stokes to poise, you only have to multiply the value in stokes by the fluid density in units of grams per cubic centimeter. If you have a density value in terms of kilograms per cubic meter, you must first divide it by 1000 to get the density in grams per cubic centimeter before multiplying it by the viscosity in stokes.
Kinematic viscosity is a type of viscosity that tells us something about the speed of the fluid due to an applied force. This is a different type of viscosity than dynamic viscosity, which describes the amount of force required to move a particular fluid at a certain speed.
How much poise does 0.025 stokes diesel have?
To convert the kinematic viscosity of a particular mix of diesel from 0.025 stokes to poise:
We first determine the density of diesel. Diesel has a density of 0.9 g/cm³.
We then multiply the kinematic viscosity by this density. Therefore, we have 0.025 stokes × 0.9 g/cm³ to obtain 0.0225 poise as its equivalent dynamic viscosity.
Dynamic viscosity (η)
Use this tool to calculate the force of a pneumatic cylinder.
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Function of a real variable — Wikipedia Republished // WIKI 2
{\displaystyle X}
{\displaystyle \mathbb {B} }
{\displaystyle \mathbb {B} }
{\displaystyle X}
{\displaystyle \mathbb {B} ^{n}}
{\displaystyle X}
{\displaystyle X}
{\displaystyle \mathbb {Z} }
{\displaystyle \mathbb {Z} }
{\displaystyle X}
{\displaystyle X}
{\displaystyle \mathbb {R} }
{\displaystyle \mathbb {R} }
{\displaystyle X}
{\displaystyle \mathbb {R} ^{n}}
{\displaystyle X}
{\displaystyle X}
{\displaystyle \mathbb {C} }
{\displaystyle \mathbb {C} }
{\displaystyle X}
{\displaystyle \mathbb {C} ^{n}}
{\displaystyle X}
In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers
{\displaystyle \mathbb {R} }
, or a subset of
{\displaystyle \mathbb {R} }
that contains an interval of positive length. Most real functions that are considered and studied are differentiable in some interval. The most widely considered such functions are the real functions, which are the real-valued functions of a real variable, that is, the functions of a real variable whose codomain is the set of real numbers.
Nevertheless, the codomain of a function of a real variable may be any set. However, it is often assumed to have a structure of
{\displaystyle \mathbb {R} }
-vector space over the reals. That is, the codomain may be a Euclidean space, a coordinate vector, the set of matrices of real numbers of a given size, or an
{\displaystyle \mathbb {R} }
-algebra, such as the complex numbers or the quaternions. The structure
{\displaystyle \mathbb {R} }
-vector space of the codomain induces a structure of
{\displaystyle \mathbb {R} }
-vector space on the functions. If the codomain has a structure of
{\displaystyle \mathbb {R} }
-algebra, the same is true for the functions.
The image of a function of a real variable is a curve in the codomain. In this context, a function that defines curve is called a parametric equation of the curve.
When the codomain of a function of a real variable is a finite-dimensional vector space, the function may be viewed as a sequence of real functions. This is often used in applications.
Real Analysis - Function Of Several Variable |Mathematical Science | CSIR UGC NET | Gajendra Purohit
UPSC Mathematics Optional | Calculus | Lecture 10 - Functions of Real Variable Part 1
Engineering Mathematics - Functions of Single Variable - Limit, Continuity, and Differentiability
Functions | Calculus | Engineering Mathematics | GATE/ESE 2021 Exam Preparation | Rohit Sinha
1 Real function
1.1 Basic examples
2 General definition
2.1 Image
2.2 Domain
2.3 Algebraic structure
2.4 Continuity and limit
3 Calculus
3.1 Theorems
4 Implicit functions
5 One-dimensional space curves in ℝn
5.1 Formulation
5.2 Tangent line to curve
5.3 Normal plane to curve
5.4 Relation to kinematics
6 Matrix valued functions
7 Banach and Hilbert spaces and quantum mechanics
8 Complex-valued function of a real variable
9 Cardinality of sets of functions of a real variable
The graph of a real function
A real function is a function from a subset of
{\displaystyle \mathbb {R} }
{\displaystyle \mathbb {R} ,}
{\displaystyle \mathbb {R} }
denotes as usual the set of real numbers. That is, the domain of a real function is a subset
{\displaystyle \mathbb {R} }
, and its codomain is
{\displaystyle \mathbb {R} .}
It is generally assumed that the domain contains an interval of positive length.
All polynomial functions, including constant functions and linear functions
The Heaviside step function is defined everywhere, but not continuous at zero.
The absolute value is defined and continuous everywhere, and is differentiable everywhere, except for zero.
The cubic root is defined and continuous everywhere, and is differentiable everywhere, except for zero.
A rational function is a quotient of two polynomial functions, and is not defined at the zeros of the denominator.
The tangent function is not defined for
{\displaystyle {\frac {\pi }{2}}+k\pi ,}
where k is any integer.
The logarithm function is defined only for positive values of the variable.
The square root is defined only for nonnegative values of the variable, and not differentiable at 0 (it is differentiable for all positive values of the variable).
A real-valued function of a real variable is a function that takes as input a real number, commonly represented by the variable x, for producing another real number, the value of the function, commonly denoted f(x). For simplicity, in this article a real-valued function of a real variable will be simply called a function. To avoid any ambiguity, the other types of functions that may occur will be explicitly specified.
{\displaystyle f:X\to \mathbb {R} }
A simple example of a function in one variable could be:
{\displaystyle f:X\to \mathbb {R} }
{\displaystyle X=\{x\in \mathbb {R} \,:\,x\geq 0\}}
{\displaystyle f(x)={\sqrt {x}}}
which is the square root of x.
Main article: Image (mathematics)
The image of a function
{\displaystyle f(x)}
is the set of all values of f when the variable x runs in the whole domain of f. For a continuous (see below for a definition) real-valued function with a connected domain, the image is either an interval or a single value. In the latter case, the function is a constant function.
Conversely, it is sometimes possible to enlarge naturally the domain of a given function, for example by continuity or by analytic continuation. This means that it is not worthy to explicitly define the domain of a function of a real variable.
For every real number r, the constant function
{\displaystyle (x)\mapsto r}
, is everywhere defined.
For every real number r and every function f, the function
{\displaystyle rf:(x)\mapsto rf(x)}
has the same domain as f (or is everywhere defined if r = 0).
If f and g are two functions of respective domains X and Y such that X∩Y contains an open subset of ℝ, then
{\displaystyle f+g:(x)\mapsto f(x)+g(x)}
{\displaystyle f\,g:(x)\mapsto f(x)\,g(x)}
are functions that have a domain containing X∩Y.
It follows that the functions of n variables that are everywhere defined and the functions of n variables that are defined in some neighbourhood of a given point both form commutative algebras over the reals (ℝ-algebras).
One may similarly define
{\displaystyle 1/f:(x)\mapsto 1/f(x),}
which is a function only if the set of the points (x) in the domain of f such that f(x) ≠ 0 contains an open subset of ℝ. This constraint implies that the above two algebras are not fields.
Limit of a real function of a real variable.
Until the second part of 19th century, only continuous functions were considered by mathematicians. At that time, the notion of continuity was elaborated for the functions of one or several real variables a rather long time before the formal definition of a topological space and a continuous map between topological spaces. As continuous functions of a real variable are ubiquitous in mathematics, it is worth defining this notion without reference to the general notion of continuous maps between topological space.
For defining the continuity, it is useful to consider the distance function of ℝ, which is an everywhere defined function of 2 real variables:
{\displaystyle d(x,y)=|x-y|}
A function f is continuous at a point
{\displaystyle a}
which is interior to its domain, if, for every positive real number ε, there is a positive real number φ such that
{\displaystyle |f(x)-f(a)|<\varepsilon }
{\displaystyle x}
{\displaystyle d(x,a)<\varphi .}
In other words, φ may be chosen small enough for having the image by f of the interval of radius φ centered at
{\displaystyle a}
contained in the interval of length 2ε centered at
{\displaystyle f(a).}
A function is continuous if it is continuous at every point of its domain.
The limit of a real-valued function of a real variable is as follows.[1] Let a be a point in topological closure of the domain X of the function f. The function, f has a limit L when x tends toward a, denoted
{\displaystyle L=\lim _{x\to a}f(x),}
{\displaystyle |f(x)-L|<\varepsilon }
for all x in the domain such that
{\displaystyle d(x,a)<\delta .}
{\displaystyle f(a)=\lim _{x\to a}f(x).}
One can collect a number of functions each of a real variable, say
{\displaystyle y_{1}=f_{1}(x)\,,\quad y_{2}=f_{2}(x)\,,\ldots ,y_{n}=f_{n}(x)}
into a vector parametrized by x:
{\displaystyle \mathbf {y} =(y_{1},y_{2},\ldots ,y_{n})=[f_{1}(x),f_{2}(x),\ldots ,f_{n}(x)]}
{\displaystyle {\frac {d\mathbf {y} }{dx}}=\left({\frac {dy_{1}}{dx}},{\frac {dy_{2}}{dx}},\ldots ,{\frac {dy_{n}}{dx}}\right)}
One can also perform line integrals along a space curve parametrized by x, with position vector r = r(x), by integrating with respect to the variable x:
{\displaystyle \int _{a}^{b}\mathbf {y} (x)\cdot d\mathbf {r} =\int _{a}^{b}\mathbf {y} (x)\cdot {\frac {d\mathbf {r} (x)}{dx}}dx}
where · is the dot product, and x = a and x = b are the start and endpoints of the curve.
With the definitions of integration and derivatives, key theorems can be formulated, including the fundamental theorem of calculus integration by parts, and Taylor's theorem. Evaluating a mixture of integrals and derivatives can be done by using theorem differentiation under the integral sign.
A real-valued implicit function of a real variable is not written in the form "y = f(x)". Instead, the mapping is from the space ℝ2 to the zero element in ℝ (just the ordinary zero 0):
{\displaystyle \phi :\mathbb {R} ^{2}\to \{0\}}
{\displaystyle \phi (x,y)=0}
{\displaystyle y=f(x)}
{\displaystyle \phi (x,y)=y-f(x)=0}
One-dimensional space curves in ℝn
Space curve in 3d. The position vector r is parametrized by a scalar t. At r = a the red line is the tangent to the curve, and the blue plane is normal to the curve.
{\displaystyle {\begin{aligned}r_{1}:\mathbb {R} \rightarrow \mathbb {R} &\quad r_{2}:\mathbb {R} \rightarrow \mathbb {R} &\cdots &\quad r_{n}:\mathbb {R} \rightarrow \mathbb {R} \\r_{1}=r_{1}(t)&\quad r_{2}=r_{2}(t)&\cdots &\quad r_{n}=r_{n}(t)\\\end{aligned}}}
{\displaystyle \mathbf {r} :\mathbb {R} \rightarrow \mathbb {R} ^{n}\,,\quad \mathbf {r} =\mathbf {r} (t)}
then the parametrized n-tuple,
{\displaystyle \mathbf {r} (t)=[r_{1}(t),r_{2}(t),\ldots ,r_{n}(t)]}
describes a one-dimensional space curve.
Tangent line to curve
At a point r(t = c) = a = (a1, a2, ..., an) for some constant t = c, the equations of the one-dimensional tangent line to the curve at that point are given in terms of the ordinary derivatives of r1(t), r2(t), ..., rn(t), and r with respect to t:
{\displaystyle {\frac {r_{1}(t)-a_{1}}{dr_{1}(t)/dt}}={\frac {r_{2}(t)-a_{2}}{dr_{2}(t)/dt}}=\cdots ={\frac {r_{n}(t)-a_{n}}{dr_{n}(t)/dt}}}
Normal plane to curve
{\displaystyle (p_{1}-a_{1}){\frac {dr_{1}(t)}{dt}}+(p_{2}-a_{2}){\frac {dr_{2}(t)}{dt}}+\cdots +(p_{n}-a_{n}){\frac {dr_{n}(t)}{dt}}=0}
or in terms of the dot product:
{\displaystyle (\mathbf {p} -\mathbf {a} )\cdot {\frac {d\mathbf {r} (t)}{dt}}=0}
Relation to kinematics
The physical and geometric interpretation of dr(t)/dt is the "velocity" of a point-like particle moving along the path r(t), treating r as the spatial position vector coordinates parametrized by time t, and is a vector tangent to the space curve for all t in the instantaneous direction of motion. At t = c, the space curve has a tangent vector dr(t)/dt|t = c, and the hyperplane normal to the space curve at t = c is also normal to the tangent at t = c. Any vector in this plane (p − a) must be normal to dr(t)/dt|t = c.
Similarly, d2r(t)/dt2 is the "acceleration" of the particle, and is a vector normal to the curve directed along the radius of curvature.
Matrix valued functions
A matrix can also be a function of a single variable. For example, the rotation matrix in 2d:
{\displaystyle R(\theta )={\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}}}
is a matrix valued function of rotation angle of about the origin. Similarly, in special relativity, the Lorentz transformation matrix for a pure boost (without rotations):
{\displaystyle \Lambda (\beta )={\begin{bmatrix}{\frac {1}{\sqrt {1-\beta ^{2}}}}&-{\frac {\beta }{\sqrt {1-\beta ^{2}}}}&0&0\\-{\frac {\beta }{\sqrt {1-\beta ^{2}}}}&{\frac {1}{\sqrt {1-\beta ^{2}}}}&0&0\\0&0&1&0\\0&0&0&1\\\end{bmatrix}}}
is a function of the boost parameter β = v/c, in which v is the relative velocity between the frames of reference (a continuous variable), and c is the speed of light, a constant.
Banach and Hilbert spaces and quantum mechanics
Generalizing the previous section, the output of a function of a real variable can also lie in a Banach space or a Hilbert space. In these spaces, division and multiplication and limits are all defined, so notions such as derivative and integral still apply. This occurs especially often in quantum mechanics, where one takes the derivative of a ket or an operator. This occurs, for instance, in the general time-dependent Schrödinger equation:
{\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi ={\hat {H}}\Psi }
Complex-valued function of a real variable
If f(x) is such a complex valued function, it may be decomposed as
f(x) = g(x) + ih(x),
Cardinality of sets of functions of a real variable
The cardinality of the set of real-valued functions of a real variable,
{\displaystyle \mathbb {R} ^{\mathbb {R} }=\{f:\mathbb {R} \to \mathbb {R} \}}
{\displaystyle \beth _{2}=2^{\mathfrak {c}}}
, which is strictly larger than the cardinality of the continuum (i.e., set of all real numbers). This fact is easily verified by cardinal arithmetic:
{\displaystyle \mathrm {card} (\mathbb {R} ^{\mathbb {R} })=\mathrm {card} (\mathbb {R} )^{\mathrm {card} (\mathbb {R} )}={\mathfrak {c}}^{\mathfrak {c}}=(2^{\aleph _{0}})^{\mathfrak {c}}=2^{\aleph _{0}\cdot {\mathfrak {c}}}=2^{\mathfrak {c}}.}
{\displaystyle X}
{\displaystyle 2\leq \mathrm {card} (X)\leq {\mathfrak {c}}}
, then the cardinality of the set
{\displaystyle X^{\mathbb {R} }=\{f:\mathbb {R} \to X\}}
{\displaystyle 2^{\mathfrak {c}}}
{\displaystyle 2^{\mathfrak {c}}=\mathrm {card} (2^{\mathbb {R} })\leq \mathrm {card} (X^{\mathbb {R} })\leq \mathrm {card} (\mathbb {R} ^{\mathbb {R} })=2^{\mathfrak {c}}.}
However, the set of continuous functions
{\displaystyle C^{0}(\mathbb {R} )=\{f:\mathbb {R} \to \mathbb {R} :f\ \mathrm {continuous} \}}
has a strictly smaller cardinality, the cardinality of the continuum,
{\displaystyle {\mathfrak {c}}}
. This follows from the fact that a continuous function is completely determined by its value on a dense subset of its domain.[2] Thus, the cardinality of the set of continuous real-valued functions on the reals is no greater than the cardinality of the set of real-valued functions of a rational variable. By cardinal arithmetic:
{\displaystyle \mathrm {card} (C^{0}(\mathbb {R} ))\leq \mathrm {card} (\mathbb {R} ^{\mathbb {Q} })=(2^{\aleph _{0}})^{\aleph _{0}}=2^{\aleph _{0}\cdot \aleph _{0}}=2^{\aleph _{0}}={\mathfrak {c}}.}
On the other hand, since there is a clear bijection between
{\displaystyle \mathbb {R} }
and the set of constant functions
{\displaystyle \{f:\mathbb {R} \to \mathbb {R} :f(x)\equiv x_{0}\}}
, which forms a subset of
{\displaystyle C^{0}(\mathbb {R} )}
{\displaystyle \mathrm {card} (C^{0}(\mathbb {R} ))\geq {\mathfrak {c}}}
must also hold. Hence,
{\displaystyle \mathrm {card} (C^{0}(\mathbb {R} ))={\mathfrak {c}}}
Function of several complex variables
^ R. Courant. Differential and Integral Calculus. Vol. 2. Wiley Classics Library. pp. 46–47. ISBN 0-471-60840-8.
^ Rudin, W. (1976). Principles of Mathematical Analysis. New York: McGraw-Hill. pp. 98–99. ISBN 0-07-054235X.
F. Ayres, E. Mendelson (2009). Calculus. Schaum's outline series (5th ed.). McGraw Hill. ISBN 978-0-07-150861-2.
R. Wrede, M. R. Spiegel (2010). Advanced calculus. Schaum's outline series (3rd ed.). McGraw Hill. ISBN 978-0-07-162366-7.
N. Bourbaki (2004). Functions of a Real Variable: Elementary Theory. Springer. ISBN 354-065-340-6.
L. A. Talman (2007) Differentiability for Multivariable Functions
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(Redirected from Mathematical Logic)
10 Foundations of mathematics
Subfields and scope[edit]
Set theory and paradoxes[edit]
Symbolic logic[edit]
Beginnings of the other branches[edit]
Formal logical systems [edit]
Other classical logics[edit]
The most well studied infinitary logic is
{\displaystyle L_{\omega _{1},\omega }}
. In this logic, quantifiers may only be nested to finite depths, as in first-order logic, but formulas may have finite or countably infinite conjunctions and disjunctions within them. Thus, for example, it is possible to say that an object is a whole number using a formula of
{\displaystyle L_{\omega _{1},\omega }}
{\displaystyle (x=0)\lor (x=1)\lor (x=2)\lor \cdots .}
Nonclassical and modal logic[edit]
Algebraic logic[edit]
Model theory[edit]
Algorithmically unsolvable problems[edit]
Proof theory and constructive mathematics[edit]
Connections with computer science[edit]
Foundations of mathematics[edit]
Undergraduate texts[edit]
Graduate texts[edit]
Research papers, monographs, texts, and surveys[edit]
Classical papers, texts, and collections[edit]
Retrieved from "https://en.wikipedia.org/w/index.php?title=Mathematical_logic&oldid=1088839951"
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Matrix Multiplication Practice Problems Online | Brilliant
Compute the following matrix multiplication:
\begin{pmatrix} 2 & 2 \\ 1 & 3 \end{pmatrix} \times \begin{pmatrix} 2 & 4 \\ 1 & 3 \end{pmatrix}.
\begin{pmatrix} 6 & 14 \\ 5 & 13 \end{pmatrix}
22
\begin{pmatrix} 8 & 16 \\ 5 & 11 \end{pmatrix}
\begin{pmatrix} 4 & 8 \\ 1 & 9 \end{pmatrix}
At an amusement park, tickets for the Pirate ship ride are $
5.00
per adult and $
9.00
per child. One sunny day,
558
adults and
880
children purchase tickets for this ride. The following day is rainy and only
219
406
children purchase tickets for this ride. By representing the number of rides in a matrix, calculate the total ticket sales (in dollars) for the ride over the two days.
\begin{pmatrix} 2 & 2 & 3 \\ 2 & 2 & 1 \end{pmatrix} \times \begin{pmatrix} 2 & 0 \\ 1 & 2 \\ 3 & 1 \end{pmatrix}.
\begin{pmatrix} 15 & 7 \\ 9 & 5 \end{pmatrix}
\begin{pmatrix} 4 & 0 & 0 \\ 2 & 4 & 0 \\ 0 & 0 & 0 \end{pmatrix}
\begin{pmatrix} 4 & 4 & 6 \\ 6 & 6 & 5 \\ 8 & 8 & 10 \end{pmatrix}
\begin{pmatrix} 4 & 0 \\ 2 & 4 \end{pmatrix}
x+ y
which satisfies the following equation:
\begin{pmatrix} 7 & 1 \\ 3 & 2 \end{pmatrix} \times \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 2 \\ -51 \end{pmatrix}.
-25
-28
-31
31
3 \times 3
A
A \times \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 5 \\ 5 \\ 6 \end{pmatrix}, A \times \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 5 \\ 3 \\ 3 \end{pmatrix}, A \times \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 6 \\ 4 \\ 3 \end{pmatrix}.
\begin{pmatrix} 1 & 3 & 2 \\ 2 & 0 & 1 \\ 3 & 3 & 0 \end{pmatrix}
\begin{pmatrix} 3 & 2 & 3 \\ 3 & 2 & 1 \\ 3 & 3 & 0 \end{pmatrix}
\begin{pmatrix} 3 & 3 & 3 \\ 2 & 2 & 3 \\ 3 & 1 & 0 \end{pmatrix}
\begin{pmatrix} 3 & 3 & 2 \\ 2 & 1 & 0 \\ 0 & 1 & 3 \end{pmatrix}
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$IDLE staking - Idle Finance
Governance > $IDLE staking
Staking allows users to lock their $IDLE for a flexible period (up to 4 years) in return for a series of benefits. The contracts for IDLE staking are based on the Curve VotingEscrow contracts, decided via community vote.
sktIDLE holders have access to:
Buybacks distribution
Gauges voting
Liquidity providers rewards boost
Depending on the lock time, a specific amount of stkIDLE is generated from the $IDLE locked. The lock scale is as follows
1 $IDLE
0.25 stkIDLE (25%)
1 stkIDLE (100%)
stkIDLE linearly decreases st from the lockup date to the end date. Only at the end of the lock time, it is possible to withdraw the starting locked $IDLE tokens. The general formula to compute the stkIDLE balance at any point in time is:
Q_{\text{stkIDLE}} = \frac{\text{lock time remaining in seconds}}{\text{max lock time in seconds}} \times Q_{\text{IDLE lock}}
Users can increase their $stkIDLE by either staking more $IDLE into their existing lock or increasing their lock end date, or both. The maximum lockup duration is 4 years.
Example: 5000 IDLE locked for 3 years and 2 months (with 30 days/month) would give:
Q_{\text{stkIDLE}} = \frac{(3\times12 + 2)_{m} \times30_{d}\times24_{h}\times3600_s}{4\times365\times24\times3600} \times 5000
Q_{\text{stkIDLE}} = \frac{1140}{1460} \times 12 = 3904.1095
As described above, the quantity of stkIDLE decreases constantly in proportion to the reduction in lock time (lock expiration), down to 0 at expiration. For example, two months after the initial lock, the stkIDLE holder from the example above would have a stkIDLE balance of:
Q_{\text{stkIDLE}} = \frac{1080}{1460} \times 12 = 3698.6301
$stkIDLE
The staking contract is implemented as a non-standard ERC-20 token, which is non-transferable and can only be created by staking $IDLE by callingcreate_lock(uint256 _value, utin256 _unlock_time).
By default, smart contracts cannot participate in staking, as this would allow for trivial which could be implemented to circumvent the non-transferable nature of $stkIDLE.
However, Smart contracts can be whitelisted via a governance proposal as described here.
The $IDLE voting power deposited in the staking contract is delegated to a community multisig, which is used to vote on governance proposals based on the snapshot polls for stkIDLE token holders.
The community multisig is located at: 0xb08696efcf019a6128ed96067b55dd7d0ab23ce4
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The Measure of Time - Wikisource, the free online library
In French: Poincaré, Henri (1898), “La mesure du temps”, Revue de métaphysique et de morale 6: 1-13
English translation: Poincaré, Henri (1913), “The Measure of Time”, The Foundations of Science (The Value of Science), New York: Science Press, pp. 222-234, <http://www.archive.org/details/foundationsscie01poingoog>
Henri Poincaré246510The Measure of Time1898George Bruce Halsted
To measure time they use the pendulum and they suppose by definition that all the beats of this pendulum are of equal duration. But this is only a first approximation; the temperature, the resistance of the air, the barometric pressure, make the pace of the pendulum vary. If we could escape these sources of error, we should obtain a much closer approximation, but it would still be only an approximation. New causes, hitherto neglected, electric, magnetic or others, would introduce minute perturbations.
Let me explain myself. I suppose that at a certain place in the world the phenomenon
{\displaystyle \alpha }
happens, causing as consequence at the end of a certain time the effect
{\displaystyle \alpha '}
. At another place in the world very far away from the first, happens the phenomenon
{\displaystyle \beta }
, which causes as consequence the effect
{\displaystyle \beta '}
. The phenomena
{\displaystyle \alpha }
{\displaystyle \beta }
are simultaneous, as are also the effects
{\displaystyle \alpha '}
{\displaystyle \beta '}
Later, the phenomenon
{\displaystyle \alpha }
is reproduced under approximately the same conditions as before, and simultaneously the phenomenon
{\displaystyle \beta }
is also reproduced at a very distant place in the world and almost under the same circumstances. The effects
{\displaystyle \alpha '}
{\displaystyle \beta '}
also take place. Let us suppose that the effect
{\displaystyle \alpha '}
happens perceptibly before the effect
{\displaystyle \beta '}
If experience made us witness such a sight, our postulate would be contradicted. For experience would tell us that the first duration
{\displaystyle \alpha \alpha '}
is equal to the first duration
{\displaystyle \beta \beta '}
and that the second duration
{\displaystyle \alpha \alpha '}
is less than the second duration
{\displaystyle \beta \beta '}
. On the other hand, our postulate would require that the two durations
{\displaystyle \alpha \alpha '}
should be equal to each other, as likewise the two durations
{\displaystyle \beta \beta '}
. The equality and the inequality deduced from experience would be incompatible with the two equalities deduced from the postulate.
Under these conditions, it is clear that the causes which have produced a certain effect will never be reproduced except approximately. Then we should modify our postulate and our definition. Instead of saying: 'The same causes take the same time to produce the same effects,' we should say : 'Causes almost identical take almost the same time to produce almost the same effects.'
If now it be supposed that another way of measuring time is adopted, the experiments on which Newton's law is founded would none the less have the same meaning. Only the enunciation of the law would be different, because it would be translated into another language; it would evidently be much less simple. So that the definition implicitly adopted by the astronomers may be summed up thus: Time should be so defined that the equations of mechanics may be as simple as possible. In other words, there is not one way of measuring time more true than another; that which is generally adopted is only more convenient. Of two watches, we have no right to say that the one goes true, the other wrong; we can only say that it is advantageous to conform to the indications of the first.
We should first ask ourselves how one could have had the idea of putting into the same frame so many worlds impenetrable to one another. We should like to represent to ourselves the external universe, and only by so doing could we feel that we understood it. We know we never can attain this representation: our weakness is too great. But at least we desire the ability to conceive an infinite intelligence for which this representation could be possible, a sort of great consciousness which should see all, and which should classify all in its time, as we classify, in our time, the little we see.
It has also been said that two facts should be regarded as simultaneous when the order of their succession may be inverted at will. It is evident that this definition would not suit two physical facts which happen far from one another, and that, in what concerns them, we no longer even understand what this reversibility would be; besides, succession itself must first be defined.
I execute a voluntary act
{\displaystyle A}
and I feel afterward a sensation
{\displaystyle D}
, which I regard as a consequence of the act
{\displaystyle A}
; on the other hand, for whatever reason, I infer that this consequence is not immediate, but that outside my consciousness two facts
{\displaystyle B}
{\displaystyle C}
, which I have not witnessed, have happened, and in such a way that
{\displaystyle B}
is the effect of
{\displaystyle A}
{\displaystyle C}
{\displaystyle B}
{\displaystyle D}
{\displaystyle C}
But why? If I think I have reason to regard the four facts
{\displaystyle A,B,C,D,}
as bound to one another by a causal connection, why range them in the causal order
{\displaystyle ABCD}
, and at the same time in the chronologic order
{\displaystyle ABCD}
, rather than in any other order?
I clearly see that in the act
{\displaystyle A}
I have the feeling of having been active, while in undergoing the sensation
{\displaystyle D}
I have that of having been passive. This is why I regard
{\displaystyle A}
as the initial cause and
{\displaystyle D}
as the ultimate effect; this is why I put
{\displaystyle A}
at the beginning of the chain and
{\displaystyle D}
at the end; but why put
{\displaystyle B}
{\displaystyle C}
{\displaystyle C}
{\displaystyle B}
If this question is put, the reply ordinarily is: we know that it is
{\displaystyle B}
which is the cause of
{\displaystyle C}
because we always see
{\displaystyle B}
happen before
{\displaystyle C}
. These two phenomena, when witnessed, happen in a certain order; when analogous phenomena happen without witness, there is no reason to invert this order.
Doubtless, but take care; we never know directly the physical phenomena
{\displaystyle B}
{\displaystyle C}
. What we know are sensations
{\displaystyle B'}
{\displaystyle C'}
produced respectively by
{\displaystyle B}
{\displaystyle C}
. Our consciousness tells us immediately that
{\displaystyle B'}
{\displaystyle C'}
and we suppose that
{\displaystyle B}
{\displaystyle C}
succeed one another in the same order.
Not to lose ourselves in this infinite complexity, let us make a simpler hypothesis. Consider three stars, for example, the sun, Jupiter and Saturn; but, for greater simplicity, regard them as reduced to material points and isolated from the rest of the world. The positions and the velocities of three bodies at a given instant suffice to determine their positions and velocities at the following instant, and consequently at any instant. Their positions at the instan{\displaystyle t}
determine their positions at the instant
{\displaystyle t+h}
as well as their positions at the instant
{\displaystyle t-h}
Even more; the position of Jupiter at the instan{\displaystyle t}
, together with that of Saturn at the instant
{\displaystyle t+a}
, determines the position of Jupiter at any instant and that of Saturn at any instant
The aggregate of positions occupied by Jupiter at the instant
{\displaystyle t+e}
and Saturn at the instant
{\displaystyle t+a+e}
is bound to the aggregate of positions occupied by Jupiter at the instan{\displaystyle t}
{\displaystyle t+a}
, by laws as precise as that of Newton, though more complicated. Then why not regard one of these aggregates as the cause of the other, which would lead to considering as simultaneous the instan{\displaystyle t}
of Jupiter and the instant
{\displaystyle t+a}
of Saturn?
He has begun by supposing that light has a constant velocity, and in particular that its velocity is the same in all directions. That is a postulate without which no measurement of this velocity could be attempted. This postulate could never be verified directly by experiment; it might be contradicted by it if the results of different measurements were not concordant. We should think ourselves fortunate that this contradiction has not happened and that the slight discordances which may happen can be readily explained.
Or else finally they use the telegraph. It is clear first that the reception of the signal at Berlin, for instance, is after the sending of this same signal from Paris. This is the rule of cause and effect analyzed above. But how much after? In general, the duration of the transmission is neglected and the two events are regarded as simultaneous. But, to be rigorous, a little correction would still have to be made by a complicated calculation; in practise it is not made, because it would be well within the errors of observation; its theoretic necessity is none the less from our point of view, which is that of a rigorous definition. From this discussion, I wish to emphasize two things: (1) The rules applied are exceedingly various. (2) It is difficult to separate the qualitative problem of simultaneity from the quantitative problem of the measurement of time; no matter whether a chronometer is used, or whether account must be taken of a velocity of transmission, as that of light, because such a velocity could not be measured without measuring a time.
↑ Etude sur les diverses grandeurs, Paris, Gauthier-Villars, 1897.
Retrieved from "https://en.wikisource.org/w/index.php?title=The_Measure_of_Time&oldid=10776727"
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3Blue1Brown - How secure is 256 bit security?
Text adaptation by River Way
What is 256 bit security?
If a protocol is described as having "
n
-bit security", it means an attacker would have to run some computation
2^n
times to break the system.
For example, the last post touched on the function SHA256, which takes in an input of arbitrary length, and produces an output which is always 256 bits long. It's believed that this is a crytographic hash function, which means that it's computationally infeasible to reverse it. More specifically, if you want to find an input producing a particular output, there is no better method than to repeatedly guess and check.
These cryptographic hash functions are used, for example, to store passwords without revealing what those passwords are. If your database stores usernames and the associated hash of the corresponding password, then when a user enters their password, you can check if it’s correct by computing its hash and comparing it with what is stored. But anyone looking at the database should have no way to reverse engineer what the passwords are.
A core theme of cryptography is to find tasks which are easy in one direction but hard in another, so it should make sense that these one-way hash functions are a very common building block in all sorts of security protocols, including not just password storage, but digital signatures, proofs of work, and many more. So how hard is this guess-and-check process, exactly? The outputs of this hash function behave like a random sequence of numbers, so it’s helpful to think of rolling a die.
How many times, on average, do you need to roll a six-sided die until you get a particular number, say a
1
3
6
9
12
For SHA256, there are not six possible outputs, there are
2^{256}
. So on average it will take
2^{256}
guesses to find an input with a particular hash
. But how difficult is this, exactly? A number like
2^{256}
is so far removed from anything we ever deal with that it can be hard to appreciate its size. Nevertheless, let’s give it a try.
Appreciating Large Powers of Two
The human mind is best at breaking down concepts into smaller pieces to analyze.
2^{256}
2^{32}
multiplied by itself eight times.
2^{32}
is about 4 billion, which is a number we can at least start to think about, so all we need to do is appreciate what multiplying 4 billion times itself eight successive times feels like.
4 Billion Hashes Per Second
The GPU on your computer can let you run parallel computations incredibly quickly. If you specially program a GPU to run a cryptographic hash function over and over, a really good one can do just under a billion hashes per second.
Let’s say you cram a computer with extra GPUs so that it can guess and check 4 billion times per second. So our first 4 billion is the number of hashes per second per computer.
4 Billion Computers
Now picture 4 billion of these GPU-packed computers. For comparison, even though Google does not make the number of servers it runs public, estimates have it somewhere in the single-digit millions.
In reality, most of those servers are much less powerful than our imagined GPU-packed machine, but if Google replaced all its millions of servers with machines like this, 4 billion machines would mean about 1,000 copies of this souped-up Google, which I’ll call one KiloGoogle++.
4 Billion KiloGoogles
There are about 7.9 billion people on Earth, so imagine giving a little over half the people on Earth their own personal KiloGoogle.
If you were the president of this KiloGoogle filled Earth, how many guesses could you order to be checked per second?
2^{32}
2^{64}
2^{96}
2^{128}
Now imagine 4 billion copies of this Earth. The Milky Way has between 100 and 400 billion stars, so this would be akin to 1% of all stars in the galaxy having a copy of Earth, where half the population on each has their own personal KiloGoogle.
4 Billion Milky Ways
Now picture 4 billion copies of the Milky Way, and call this your GigaGalactic Super Computer, running
(2^{32})^5 = 2^{160}
guesses per second.
4 billion seconds is about 126 years, which is still a reasonable amount of time to want to keep something hidden. Maybe you’re hiding a digital treasure for your great-great-great grandchildren!
4 Billion Seconds Squared
4 billion times 126 years is about 507 billion years. This is roughly 37 times the age of the universe. The Earth will have been swallowed up by the sun a long time before this, hopefully nobody cares about your password anymore.
1 in 4 Billion Success Rate
So even if you were to have your GPU-packed KiloGoogle-per-person multi-planetary GigaGalactic Super Computer guessing numbers for 37 times the age of the universe, it would still only have a 1 in 4 billion chance of finding a correct guess.
If you had your very own KiloGoogle, how many years would it take you to match a GigaGalactic Super Computer with its success rate of 1 in 4 billion?
4 \times 10^{40}
5 \times 10^{50}
6 \times 10^{60}
7 \times 10^{70}
Modern Bitcoin Hashing
In 2021, all the miners put together make guesses at a rate of about 100 billion billion (
10^{20}\approx 2^{66}
) hashes per second. Which is four times more than what was described as a KiloGoogle.
This is not because there are actually billions of GPU packed machines out there, but because miners use something about ten thousand times better than a GPU: Application-Specific Integrated Circuits.
These are pieces of hardware specifically designed for bitcoin mining and nothing else. It turns out that there are a lot of efficiency gains to be had when you throw out the need for general computation, and design your integrated circuits for a single, specific task.
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Soundness — Wikipedia Republished // WIKI 2
In logic, more precisely in deductive reasoning, an argument is sound if it is both valid in form and its premises are true.[1] Soundness also has a related meaning in mathematical logic, wherein logical systems are sound if and only if every formula that can be proved in the system is logically valid with respect to the semantics of the system.
[Natural Deductive Logic] Soundness Preliminaries || Lecture 12
IntroToLogic - An Introduction to Symbolic Logic
2 Use in mathematical logic
2.1 Logical systems
2.1.1 Soundness
2.1.2 Strong soundness
2.1.3 Arithmetic soundness
2.2 Relation to completeness
In deductive reasoning, a sound argument is an argument that is valid and all of its premises are true (and as a consequence its conclusion is true as well). An argument is valid if, assuming its premises are true, the conclusion must be true. An example of a sound argument is the following well-known syllogism:
(premises)
However, an argument can be valid without being sound. For example:
This argument is valid as the conclusion must be true assuming the premises are true. However, the first premise is false. Not all birds can fly (for example, penguins). For an argument to be sound, the argument must be valid and its premises must be true.[2]
In mathematical logic, a logical system has the soundness property if and only if every formula that can be proved in the system is logically valid with respect to the semantics of the system. In most cases, this comes down to its rules having the property of preserving truth.[3] The converse of soundness is known as completeness.
A logical system with syntactic entailment
{\displaystyle \vdash }
and semantic entailment
{\displaystyle \models }
is sound if for any sequence
{\displaystyle A_{1},A_{2},...,A_{n}}
of sentences in its language, if
{\displaystyle A_{1},A_{2},...,A_{n}\vdash C}
{\displaystyle A_{1},A_{2},...,A_{n}\models C}
. In other words, a system is sound when all of its theorems are tautologies.
Most proofs of soundness are trivial.[citation needed] For example, in an axiomatic system, proof of soundness amounts to verifying the validity of the axioms and that the rules of inference preserve validity (or the weaker property, truth). If the system allows Hilbert-style deduction, it requires only verifying the validity of the axioms and one rule of inference, namely modus ponens. (and sometimes substitution)
Soundness of a deductive system is the property that any sentence that is provable in that deductive system is also true on all interpretations or structures of the semantic theory for the language upon which that theory is based. In symbols, where S is the deductive system, L the language together with its semantic theory, and P a sentence of L: if ⊢S P, then also ⊨L P.
Strong soundness of a deductive system is the property that any sentence P of the language upon which the deductive system is based that is derivable from a set Γ of sentences of that language is also a logical consequence of that set, in the sense that any model that makes all members of Γ true will also make P true. In symbols where Γ is a set of sentences of L: if Γ ⊢S P, then also Γ ⊨L P. Notice that in the statement of strong soundness, when Γ is empty, we have the statement of weak soundness.
If T is a theory whose objects of discourse can be interpreted as natural numbers, we say T is arithmetically sound if all theorems of T are actually true about the standard mathematical integers. For further information, see ω-consistent theory.
The converse of the soundness property is the semantic completeness property. A deductive system with a semantic theory is strongly complete if every sentence P that is a semantic consequence of a set of sentences Γ can be derived in the deduction system from that set. In symbols: whenever Γ ⊨ P, then also Γ ⊢ P. Completeness of first-order logic was first explicitly established by Gödel, though some of the main results were contained in earlier work of Skolem.
Gödel's first incompleteness theorem shows that for languages sufficient for doing a certain amount of arithmetic, there can be no consistent and effective deductive system that is complete with respect to the intended interpretation of the symbolism of that language. Thus, not all sound deductive systems are complete in this special sense of completeness, in which the class of models (up to isomorphism) is restricted to the intended one. The original completeness proof applies to all classical models, not some special proper subclass of intended ones.
Soundness (interactive proof)
^ Smith, Peter (2010). "Types of proof system" (PDF). p. 5.
^ Gensler, Harry J., 1945- (January 6, 2017). Introduction to logic (Third ed.). New York. ISBN 978-1-138-91058-4. OCLC 957680480. {{cite book}}: CS1 maint: multiple names: authors list (link)
^ Mindus, Patricia (2009-09-18). A Real Mind: The Life and Work of Axel Hägerström. Springer Science & Business Media. ISBN 978-90-481-2895-2.
Copi, Irving (1979), Symbolic Logic (5th ed.), Macmillan Publishing Co., ISBN 0-02-324880-7
Boolos, Burgess, Jeffrey. Computability and Logic, 4th Ed, Cambridge, 2002.
Wiktionary has definitions related to Soundness
Validity and Soundness in the Internet Encyclopedia of Philosophy.
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Working With Logarithms | Boundless Algebra | Course Hero
Logarithms of Products
A useful property of logarithms states that the logarithm of a product of two quantities is the sum of the logarithms of the two factors. In symbols,
\log_b(xy)=\log_b(x)+\log_b(y).
Relate the product rule for logarithms to the rules for operating with exponents, and use this rule to rewrite logarithms of products
The logarithm of a product is the sum of the logarithms of the factors.
The product rule does not apply when the base of the two logarithms are different.
x^3
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of
1000
10
3
10^3=1000.
x=b^y
y
is the logarithm base
b
x
, written:
y=\log_b(x)
\log_{10}(1000)=3
It is useful to think of logarithms as inverses of exponentials. So, for example:
\displaystyle \log_b(b^z)=z
\displaystyle b^{\log_b(z)}=z
Logarithms were introduced by John Napier in the early 17th century as a means to simplify calculations. Logarithms were rapidly adopted by navigators, scientists, engineers, and others to perform computations more easily by using slide rules and logarithm tables. Tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition, because of the fact that the logarithm of a product is the sum of the logarithms of the factors:
\displaystyle log_b(xy) = log_b(x) + log_b(y)
We can see that this rule is true by writing the logarithms in terms of exponentials.
\log_b(x)=v
\log_b(y)=w.
Writing these equations as exponentials:
\displaystyle b^v=x
\displaystyle b^w=y.
\displaystyle \begin{align} xy&=b^vb^w\\ &=b^{v+w} \end{align}
Taking the logarithm base
b
of both sides of this last equation yields:
\displaystyle \begin{align} \log_b(xy)&=\log_b(b^{v+w})\\ &=v+w\\ &=\log_b(x) + \log_b(y) \end{align}
This is a very useful property of logarithms, because it can sometimes simplify more complex expressions. For example:
\displaystyle \log_{10}(10^x\cdot 100^{x^3+1})=\log_{10} (10^x)+\log_{10}(100^{x^3+1})
100
10^2
\displaystyle \begin{align} x+\log_{10}(10^{2(x^3+1)}) &= x+2(x^3+1)\\ &=2x^3+x+2 \end{align}
Logarithms of Powers
The logarithm of the
p\text{th}
power of a quantity is
p
times the logarithm of the quantity. In symbols,
\log_b(x^p)=p\log_b(x).
Relate the power rule for logarithms to the rules for operating with exponents, and use this rule to rewrite logarithms of powers
An exponent,
p
, signifies that a number is being multiplied by itself
p
number of times. Because the logarithm of a product is the sum of the logarithms of the factors, the logarithm of a number,
x
, to an exponent,
p
, is the same as the logarithm of
x
p
times, so it is equal to
p\log_b(x).
logarithm: The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number.
x^3
We have already seen that the logarithm of a product is the sum of the logarithms of the factors:
\displaystyle \log _b \left( {xy} \right) = \log _b \left( x \right) + \log _b \left( y \right)
If we apply this rule repeatedly we can devise another rule for simplifying expressions of the form
\log_b x^p
x^p
x \cdot x \cdot x \cdots x
p
x
\displaystyle \begin{align} \log_b(x^p) &= \log_b (x \cdot x \cdots x) \\ &= \log_b x + \log_b x + \cdots +\log_b x \\ &= p\log_b x \end{align}
p
x
p
summands by the product rule formula.
\log_3(3^x\cdot 9x^{100})
First expand the log:
\displaystyle \log_3(3^x\cdot 9x^{100}) =\log_3 (3^x) + \log_3 9 + \log_3(x^{100})
Next use the product and power rule to simplify:
\displaystyle \log_3 (3^x) + \log_3 9 + \log_3 (x^{100})= x+2+100\log_3 x
2^{(x+1)}=10^3
x
Start by taking the logarithm with base
2
\displaystyle \begin{align} \log_2 (2^{(x+1)}) &= \log_2 (10^3)\\ x+1&=3\log_2(10)\\ x&=3\log_2(10)-1 \end{align}
Therefore a solution would be
x=3\log_2(10) -1.
Logarithms of Quotients
The logarithm of the ratio of two quantities is the difference of the logarithms of the quantities. In symbols,
\log_b\left( \frac{x}{y}\right) = \log_bx - \log_by.
Relate the quotient rule for logarithms to the rules for operating with exponents, and use this rule to rewrite logarithms of quotients
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number.
The logarithm of the ratio or quotient of two numbers is the difference of the logarithms.
x^3
\displaystyle \log_b(xy) = \log_bx + \log_by
Similarly, the logarithm of the ratio of two quantities is the difference of the logarithms:
\displaystyle \log_b\left( \frac{x}{y}\right) = \log_bx - log_by.
We can show that this is true by the following example:
u=\log_b x
v=\log_b y
b^u=x
b^v=y.
\displaystyle \begin{align} \log_b\left(\frac{x}{y}\right)&=\log_b\left({b^u \over b^v}\right)\\ &= \log_b(b^{u-v}) \\ &=u-v\\ &= \log_b x - \log_b y \end{align}
Another way to show that this rule is true, is to apply both the power and product rules and the fact that dividing by
is the same is multiplying by
y^{-1}.
\displaystyle \begin{align} \log_b\left(\frac{x}{y}\right)&=\log_b(x\cdot y^{-1})\\ & = \log_bx + \log_b(y^{-1})\\& = \log_bx -\log_by \end{align}
Example: write the expression
\log_2\left({x^4y^9 \over z^{100}}\right)
in a simpler way
By applying the product, power, and quotient rules, you could write this expression as:
\displaystyle \log_2(x^4)+\log_2(y^9)-\log_2(z^{100}) = 4\log_2x+9\log_2y-100\log_2z.
Changing Logarithmic Bases
A logarithm written in one base can be converted to an equal quantity written in a different base.
Use the change of base formula to convert logarithms to different bases
The base of a logarithm can be changed by expressing it as the quotient of two logarithms with a common base.
Changing a logarithm's base to
10
makes it much simpler to evaluate; it can be done on a calculator.
Most common scientific calculators have a key for computing logarithms with base
10
, but do not have keys for other bases. So, if you needed to get an approximation to a number like
\log_4(9)
it can be difficult to do so. One could easily guess that it is between
1
2
9
4^1
4^2
, but it is difficult to get an accurate approximation. Fortunately, there is a change of base formula that can help.
The change of base formula for logarithms is:
\displaystyle \log_a(x)=\frac{\log_b(x)}{\log_b(a)}
Thus, for example, we could calculate that
\log_4(9)=\frac{\log_{10}(9)}{\log_{10}(4)}
which could be computed on almost any handheld calculator.
To see why the formula is true, give
\log_a(x)
a name like
z
\displaystyle z=\log_a(x)
a^z=x
Now take the logarithm with base
b
of both sides, yielding:
\displaystyle \log_b a^z = \log_bx
Using the power rule gives:
\displaystyle z \cdot \log_ba = \log_b x
\log_ba
\displaystyle z={\log_b x \over \log_ba}.
\log_a x ={\log_b x \over \log_b a}.
\log_5(10^{x^2+1})
might be easier to graph on a graphing calculator or other device if it were written in base
10
instead of base 5. The change-of-base formula can be applied to it:
\displaystyle \log_5(10^{x^2+1}) = {\log_{10}(10^{x^2+1}) \over \log_{10}5}
{x^2+1 \over \log_{10} 5}.
Logarithmic function. Provided by: Wikipedia. Located at: http://en.wikipedia.org/wiki/Logarithmic_function%23Product.2C_quotient.2C_power.2C_and_root. License: CC BY-SA: Attribution-ShareAlike
Rory Adams (Free High School Science Texts Project), Mark Horner, and Heather Williams, Logarithms. September 17, 2013. Provided by: OpenStax CNX. Located at: http://cnx.org/content/m31883/latest/. License: CC BY: Attribution
Algebra/Logarithms. Provided by: Wikibooks. Located at: http://en.wikibooks.org/wiki/Algebra/Logarithms. License: CC BY-SA: Attribution-ShareAlike
Kenny Felder, Logarithm Concepts -- Properties of Logarithms. September 18, 2013. Provided by: OpenStax CNX. Located at: http://cnx.org/content/m18239/latest/. License: CC BY: Attribution
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Differential Subordination for Certian Subclasses of -Valent Functions Assoicated with Generalized Linear Operator
R. M. El-Ashwah, M. K. Aouf, S. M. El-Deeb, "Differential Subordination for Certian Subclasses of -Valent Functions Assoicated with Generalized Linear Operator", Journal of Mathematics, vol. 2013, Article ID 692045, 8 pages, 2013. https://doi.org/10.1155/2013/692045
R. M. El-Ashwah ,1 M. K. Aouf,2 and S. M. El-Deeb1
1Department of Mathematics, Faculty of Science at Damietta, University of Mansoura, New Damietta 34517, Egypt
2Department of Mathematics, Faculty of Science, University of Mansoura, Mansoura 33516, Egypt
By making use of the differential subordination analytic functions, we investigate inclusion relationships among certain classes of analytic and -valent functions defined by generalized linear operator.
Let denote the class of functions of the form which are analytic and valent in the open unit disc
For functions , given by (1) and given by the Hadamard product (or convolution) of and is defined by
For , we say that the function is subordinate to denoted by , if there exists a Schwarz function , that is, with and , such that for all . It is well known that, if the function is univalent in , then is equivalent to and (see [1, 2]).
Now, we introduce the linear multiplier operator by It is easily verified from (4) that By specializing the parameters , , , , and , we obtain the following operators:(i) (see [3]);(ii) (see [4, 5]); (iii) (see [6, 7]); (iv) (see [8]); (v) (see [9, 10]);(vi) (see [11]); (vii) (see [12]); (viii) (see [13]); (ix) (see [14]).
By using the multiplier operator , we define the following classes of functions.
Definition 1. For fixed parameters and , with and , we say that the function is in the class if it satisfies the following subordination condition: We note that(i) was studied by Cho et al. [15]; (ii) was studied by Aouf et al. [16]; (iii) was studied by Patel [17].
For complex numbers , , and , the Gaussian hypergeometric function is defined by where and . The series (8) converges absolutely for , hence it represents an analytic function in (see [18, Chapter 14]).
If , from the fact that , we deduce that the image is symmetric with respect to the real axis, and that maps the unit disc onto the disc . If , the function maps the unit disc onto the half plan , hence we obtain the following.
Remark 2. The function is in the class if and only if
Definition 3. The function is in the class if it satisfies the following the inequality: where ; from (9) and (10) it follows, respectively, that We note that when and , the class was studied by Aouf et al. [16].
Let us consider the first-order differential subordination
A univalent function is called its dominant, if for all analytic functions that satisfy this differential subordination. A dominant is called the best dominant, if for all dominants . For the general theory of the first-order differential subordination and its applications, we refer the reader to [1, 2].
The object of the present paper is to obtain several inclusion relationships and other interesting properties of functions belonging to the subclasses and by using the theory of differential subordination.
To establish our main results, we will require the following lemmas.
Lemma 4 (see [19]). Let , and let be a convex function with
If is analytic in , with , then
Lemma 5 (see [20]). Let , and consider the integral operator defined by where the powers are the principal ones.
If then the order of starlikeness of the class , that is, the largest number such that , is given by the number , where
Moreover, if , where and with , then where
Lemma 6 (see [21]). Let be analytic in with and for , and let with , .
(i) Let and satisfy either or . If satisfies then and this is the best dominant.
(ii) Let be such that , and if satisfies then and this is the best dominant.
2. Inclusion Relationships
Unless otherwise mentioned, we assume throughout this paper that and the power is the principal one.
Theorem 7. Let (1)Supposing that for all , then .(2)Moreover, if we suppose in addition that then where the bound is the best possible.
Proof. Let , and put the function is analytic in , with and . Differentiating (29) logarithmically with respect to , we have then, using (5) in (30), we obtain By differentiating both sides of (31) logarithmically with respect to and multiplying by , we have Combining (32) together with , we obtain that the function satisfies the Briot-Bouquet differential subordination as follows:
Now we will use Lemma 4 for the special case and . Since is a convex function in , a simple computation shows that whenever (25) holds, we have ; that is, . If in addition, we suppose that the inequality (26) holds, then all the assumptions of Lemma 5 are verified for the above values of , , and . Then it follows the inclusion , where the bound given by (28) is the best possible.
From Theorem 7, according to the definitions (7) and (11), we deduce the next inclusions.
Corollary 8. Let , such that (25) holds.(1)Supposing that for all , then (2)If we suppose in addition that (26) holds, then where is given by (28). As a consequence of the last inclusion, one has .
For the special case , Theorem 7 reduces to the following.
Corollary 9. Let .(1)Supposing that for all , then(2)If we suppose in addition that then where the bound is the best possible.
Theorem 10. Let , where and , then for , where
Proof. Since , the function given by is analytic in with and . Using (5) in (42) and taking the logarithmic differentiation in the resulting equation, we obtain If we denote , then and and substituting in (43) we obtain hence By using the well-known results [22] together with the inequality (45), we get Since the right hand side term of the inequality (47) is nonnegative whenever is given by (41), using the fact that the real part of an analytic function is harmonic, we deduce that for .
For a function , let the integral operator defined by Saitoh [23] and Saitoh et al. [24] From (4) and (48), we have
We now prove the next result.
Theorem 11. Let and (i)Supposing that for all , then (ii)Moreover, if we suppose in addition that then where the bound is the best possible.
Proof. Let , and suppose that for all . Let then is analytic in , with and . Taking the logarithmic differentiation in (55), we have Now, by using (49) in (56), we obtain By differentiating in both sides of (57) logarithmical with respect to and multiplying by , we have Since , from (58), we obtain that the function satisfies the Briot-Bouquet differential subordination
Now we will use Lemma 4 for the special case and ; we have , that is, . If we suppose in addition that the inequality (52) holds, then all the assumptions of the Lemma 5 are satisfied for , , and , hence it follows the inclusion , and the bound given by (54) is the best possible.
Taking in Theorem 11, we obtain the next corollary.
Corollary 12. Let and .(1)Supposing that for all , then(2)If we suppose in addition that then where the bound is the best possible.
Theorem 13. Let , and let with and . Suppose that If with for all , then implies where is the best dominant.
Proof. Let us put then is analytic in , with and for all . By differentiating both sides of (68) logarithmical with respect to and using (5), we have Now the assertions of Theorem 13 follows by using Lemma 6 for the special case .
Putting and , in Theorem 13, we obtain the following corollary.
Corollary 14. Assume that satisfies either If with for all , then implies and is the best dominant.
3. Properties Involving the Multiplier Operator
Theorem 15. If , then for all , with , and , the next subordination holds
Proof. If , from (7) it follows that Moreover, the function defined by (74) and the function given by are convex in . By combining a general subordination theorem [25, Theorem 4] with (74), we get For every analytic function in with , we have and thus, from (76) and (77), we deduce This last subordination implies and by simplification, we get the assertion of Theorem 15.
Corollary 16. If , then for the next inequalities hold All of the estimates asserted here are sharp.
Proof. Taking and in (73) and using the definition of subordination, we obtain where is analytic function in with and for . According to the well-known Schwarz's theorem, we have for all .(i)If , then we find from (83) that (ii)If , we can easily obtain This proves the inequality (80) for . Similarly, we can prove the other inequalities in (80) and (81). Now, for and , we observe from (83) that and for , (82) is a direct consequence of (83).
It is easy to see that all of the estimates in Corollary 16 are sharp, being attained by the function defined by
Remark 17. (i) Putting and in our results, we obtain the results obtained by Aouf et al. [16].
(ii) By specializing the parameters , and , we obtain various results for different operators defined in Section 1.
The authors thank the referees for their valuable suggestions which led to improvement of this paper.
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Ceramic with room temperature stable cubic crystal structure
Yttria-stabilized zirconia (YSZ) crystal structure
Pure zirconium dioxide undergoes a phase transformation from monoclinic (stable at room temperature) to tetragonal (at about 1173 °C) and then to cubic (at about 2370 °C), according to the scheme:
monoclinic (1173 °C)
{\displaystyle \leftrightarrow }
{\displaystyle \leftrightarrow }
cubic (2690 °C)
{\displaystyle \leftrightarrow }
Obtaining stable sintered zirconia ceramic products is difficult because of the large volume change accompanying the transition from tetragonal to monoclinic (about 5%). Stabilization of the cubic polymorph of zirconia over wider range of temperatures is accomplished by substitution of some of the Zr4+ ions (ionic radius of 0.82 Å, too small for ideal lattice of fluorite characteristic for the cubic zirconia) in the crystal lattice with slightly larger ions, e.g., those of Y3+ (ionic radius of 0.96 Å). The resulting doped zirconia materials are termed stabilized zirconias.[1]
Materials related to YSZ include calcia-, magnesia-, ceria- or alumina-stabilized zirconias, or partially stabilized zirconias (PSZ). Hafnia stabilized Zirconia is also known[citation needed].
Although 8–9 mol% YSZ is known to not be completely stabilized in the pure cubic YSZ phase up to temperatures above 1000 °C.[2]
Commonly used abbreviations in conjunction with yttria-stabilized zirconia are:
Partly stabilized zirconia ZrO2:
PSZ – Partially Stabilized Zirconia
TZP – Tetragonal Zirconia Polycrystal
4YSZ: with 4 mol-% Y2O3 partially Stabilized ZrO2, Yttria Stabilized Zirconia
Fully stabilized zirconias ZrO2:
FSZ – Fully Stabilized Zirconia
CSZ – Cubic Stabilized Zirconia
8YSZ – with 8 mol% Y2O3 Fully Stabilized ZrO2
8YDZ – 8–9 mol% Y2O3-doped ZrO2: the material is not completely stabilized and decomposes at high application temperatures, see next paragraphs[2][3][4])
The thermal expansion coefficients depends on the modification of zirconia as follows:
Monoclinic: 7·10−6/K[5]
Tetragonal: 12·10−6/K[5]
Y2O3 stabilized: 10,5·10−6/K[5]
Ionic conductivity of YSZ and its degradation
By the addition of yttria to pure zirconia (e.g., fully stabilized YSZ) Y3+ ions replace Zr4+ on the cationic sublattice. Thereby, oxygen vacancies are generated due to charge neutrality:[6]
{\displaystyle {\text{Y}}_{2}{\text{O}}_{3}\rightarrow 2{\text{Y}}_{\text{Zr}}^{'}+3{\text{O}}_{\text{O}}^{\text{x}}+{\text{V}}_{\text{O}}^{\bullet \bullet }{\text{ with }}[{\text{V}}_{\text{O}}^{\bullet \bullet }]={\frac {1}{2}}[{\text{Y}}_{\text{Zr}}^{'}],}
meaning two Y3+ ions generate one vacancy on the anionic sublattice. This facilitates moderate conductivity of yttrium stabilized zirconia for O2− ions (and thus electrical conductivity) at elevated and high temperature. This ability to conduct O2− ions makes yttria-stabilized zirconia well suited for application as solid electrolyte in solid oxide fuel cells.
For low dopant concentrations, the ionic conductivity of the stabilized zirconias increases with increasing Y2O3 content. It has a maximum around 8–9 mol% almost independent of the temperature (800–1200 °C).[1][2] Unfortunately, 8-9 mol% YSZ (8YSZ, 8YDZ) also turned out to be situated in the 2-phase field (c+t) of the YSZ phase diagram at these temperatures, which causes the material's decomposition into Y-enriched and depleted regions on the nm-scale and, consequently, the electrical degradation during operation.[3] The microstructural and chemical changes on the nm-scale are accompanied by the drastic decrease of the oxygen-ion conductivity of 8YSZ (degradation of 8YSZ) of about 40% at 950 °C within 2500 hrs.[4] Traces of impurities like Ni, dissolved in the 8YSZ, e.g., due to fuel-cell fabrication, can have a severe impact on the decomposition rate (acceleration of inherent decomposition of the 8YSZ by orders of magnitude) such that the degradation of conductivity even becomes problematic at low operation temperatures in the range of 500–700 °C.[7]
Nowadays, more complex ceramics like co-doped Zirconia (e.g., with Scandia, ...) are in use as solid electrolytes.
Multiple metal-free dental crowns
YSZ has a number of applications:
For its hardness and chemical inertness (e.g., tooth crowns).
As a refractory (e.g., in jet engines).
As a thermal barrier coating in gas turbines
As an electroceramic due to its ion-conducting properties (e.g., to determine oxygen content in exhaust gases, to measure pH in high-temperature water, in fuel cells).
Used in the production of a solid oxide fuel cell (SOFC). YSZ is used as the solid electrolyte, which enables oxygen ion conduction while blocking electronic conduction. In order to achieve sufficient ion conduction, an SOFC with a YSZ electrolyte must be operated at high temperatures (800 °C-1000 °C).[8] While it is advantageous that YSZ retains mechanical robustness at those temperatures, the high temperature necessary is often a disadvantage of SOFCs. The high density of YSZ is also necessary in order to physically separate the gaseous fuel from oxygen, or else the electrochemical system would produce no electrical power.[9][10]
For its hardness and optical properties in monocrystal form (see "cubic zirconia"), it is used as jewelry.
As a material for non-metallic knife blades, produced by Boker and Kyocera companies.
In water-based pastes for do-it-yourself ceramics and cements. These contain microscopic YSZ milled fibers or sub-micrometer particles, often with potassium silicate and zirconium acetate binders (at mildly acidic pH). The cementation occurs on removal of water. The resulting ceramic material is suitable for very high temperature applications.
YSZ doped with rare-earth materials can act as a thermographic phosphor and a luminescent material.[11]
Historically used for glowing rods in Nernst lamps.
As a high precision alignment sleeve for optical fiber connector ferrules.[12]
Cubic zirconia – The cubic crystalline form of zirconium dioxide
Sintering – Process of forming and bonding material by heat or pressure
Superconducting wire – Wires exhibiting zero resistance
^ a b H. Yanagida, K. Koumoto, M. Miyayama, "The Chemistry of Ceramics", John Wiley & Sons, 1996. ISBN 0 471 95627 9.
^ a b c Butz, Benjamin (2011). Yttria-doped zirconia as solid electrolyte for fuel-cell applications : Fundamental aspects. Südwestdt. Verl. für Hochschulschr. ISBN 978-3-8381-1775-1.
^ a b Butz, B.; Schneider, R.; Gerthsen, D.; Schowalter, M.; Rosenauer, A. (1 October 2009). "Decomposition of 8.5 mol.% Y2O3-doped zirconia and its contribution to the degradation of ionic conductivity". Acta Materialia. 57 (18): 5480–5490. doi:10.1016/j.actamat.2009.07.045.
^ a b Butz, B.; Kruse, P.; Störmer, H.; Gerthsen, D.; Müller, A.; Weber, A.; Ivers-Tiffée, E. (1 December 2006). "Correlation between microstructure and degradation in conductivity for cubic Y2O3-doped ZrO2". Solid State Ionics. 177 (37–38): 3275–3284. doi:10.1016/j.ssi.2006.09.003.
^ a b c Matweb: CeramTec 848 Zirconia (ZrO2) & Zirconium Oxide, Zirconia, ZrO2
^ Hund, F (1951). "Anomale Mischkristalle im System ZrO2–Y2O3. Kristallbau der Nernst-Stifte". Zeitschrift für Elektrochemie und Angewandte Physikalische Chemie. 55: 363–366.
^ Butz, B.; Lefarth, A.; Störmer, H.; Utz, A.; Ivers-Tiffée, E.; Gerthsen, D. (25 April 2012). "Accelerated degradation of 8.5 mol% Y2O3-doped zirconia by dissolved Ni". Solid State Ionics. 214: 37–44. doi:10.1016/j.ssi.2012.02.023.
^ Song, B.; Ruiz-Trejo, E.; Brandon, N.P. (August 2018). "Enhanced mechanical stability of Ni-YSZ scaffold demonstrated by nanoindentation and Electrochemical Impedance Spectroscopy". Journal of Power Sources. 395: 205–211. Bibcode:2018JPS...395..205S. doi:10.1016/j.jpowsour.2018.05.075.
^ Minh, N.Q. (1993). "Ceramic Fuel-Cells". Journal of the American Ceramic Society. 76 (3): 563–588. doi:10.1111/j.1151-2916.1993.tb03645.x.
^ De Guire, Eileen (2003). Solid Oxide Fuel Cells (Report). CSA.
^ American Ceramic Society (29 May 2009). Progress in Thermal Barrier Coatings. John Wiley and Sons. pp. 139–. ISBN 978-0-470-40838-4. Retrieved 23 October 2011.
^ "DIAMOND SA | Fiber Optic Interconnect Solutions".
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Static and Dynamic Components of the Redshift
1Macronix Research Corporation, Ottawa, Canada
2National Research Council, Ottawa, Canada (Retired)
We analyse the possibility that the observed cosmological redshift may be cumulatively due to the expansion of the universe and the tired light phenomenon. Since the source of both the redshifts is the same, they both independently relate to the same proper distance of the light source. Using this approach we have developed a hybrid model combining the Einstein de Sitter model and the tired light model that yields a slightly better fit to Supernovae Ia redshift data using one parameter than the standard ΛCDM model with two parameters. We have shown that the ratio of tired light component to the Einstein de Sitter component of redshift has evolved from 2.5 in the past, corresponding to redshift 1000, to its present value of 1.5. The hybrid model yields Hubble constant
{H}_{0}=69.11\left(\pm 0.53\right)\text{km}\cdot {\text{s}}^{-1}\cdot {\text{Mpc}}^{-1}
and the deceleration parameter
{q}_{0}=-0.4
. The component of Hubble constant responsible for expansion of the universe is 40% of
{H}_{0}
and for the tired light is 60% of
{H}_{0}
. Consequently, the critical density is only 16% of its currently accepted value; a lot less dark matter is needed to make up the critical density. In addition, the best data fit yields the cosmological constant density parameter
{\Omega }_{\Lambda }=0
. The tired light effect may thus be considered equivalent to the cosmological constant in the hybrid model.
d={R}_{0}\mathrm{ln}\left(1+z\right)
{R}_{0}=c/{H}_{0}
{H}_{0}
d={R}_{0}{\int }_{0}^{z}\left[\text{d}u/\sqrt{\left({\Omega }_{m,0}{\left(1+u\right)}^{3}+{\Omega }_{\Lambda ,0}\right)}\right]
{\Omega }_{m,0}
{\Omega }_{\Lambda ,0}
{\Omega }_{m,0}+{\Omega }_{\Lambda ,0}=1
{d}_{L}
{d}_{L}=\sqrt{L/4\text{π}f}
f=L/\left(4\text{π}{d}^{2}\right)
{z}_{X}
{z}_{M}
{\lambda }_{e}
{\lambda }_{X}
{\lambda }_{X}=\left(1+{z}_{X}\right){\lambda }_{e}
{\lambda }_{0}=\left(1+{z}_{M}\right){\lambda }_{X}=\left(1+{z}_{M}\right)\left(1+{z}_{X}\right){\lambda }_{e}
{\lambda }_{0}=\left(1+z\right){\lambda }_{e}
1+z=\left(1+{z}_{X}\right)\left(1+{z}_{M}\right)
a\left(t\right)
{a}_{X}\left(t\right)=1/\left(1+{z}_{X}\right)
\propto 1/\left(1+{z}_{X}\right)
\propto {a}_{X}\left(t\right)=1/\left(1+{z}_{X}\right)
{f}_{X}=f/{\left(1+{z}_{X}\right)}^{2}
\propto 1/\left(1+{z}_{M}\right)
{f}_{M}={f}_{X}/\left(1+{z}_{M}\right)
\propto 1/\sqrt{1+z}
{f}_{B}=L/\left[4\text{π}{d}^{2}{\left(1+{z}_{X}\right)}^{2}\left(1+{z}_{M}\right)\sqrt{1+z}\right]
\begin{array}{c}{d}_{L}=d\left(1+{z}_{X}\right){\left(1+{z}_{M}\right)}^{\frac{1}{2}}{\left(1+z\right)}^{\frac{1}{4}}\\ =d{\left(1+{z}_{X}\right)}^{\frac{1}{2}}{\left(1+z\right)}^{\frac{1}{2}}{\left(1+z\right)}^{\frac{1}{4}}\end{array}
{d}_{P}\left(t\right)
{z}_{X}
{z}_{M}
{q}_{0}
z\ll 1
{d}_{M}={R}_{M}\mathrm{ln}\left(1+{z}_{M}\right)
{R}_{M}=c/{H}_{M}
{H}_{M}
{\Omega }_{m,0}=0.3
{\Omega }_{\Lambda ,0}=0.7
{\Omega }_{m,0}=1
{d}_{X}=2{R}_{X}\left(1-\frac{1}{\sqrt{1+{z}_{X}}}\right)
{R}_{X}=c/{H}_{X}
{H}_{X}
z={z}_{X}={z}_{M}⇒0
d={R}_{0}z
{d}_{M}\left({t}_{0}\right)={R}_{M}{z}_{M}\left(1-\frac{1}{2}{z}_{M}+\cdots \right)
{d}_{X}\left({t}_{0}\right)={R}_{X}{z}_{X}\left(1-\frac{3}{4}{z}_{X}-\cdots \right)
{R}_{0}z={R}_{M}{z}_{M}={R}_{X}{z}_{X}
z={z}_{X}+{z}_{M}
z=\left(1-w\right)z+wz
1-w
{q}_{0}
z⇒0
{d}_{P}\left({t}_{0}\right)={R}_{0}z\left[1-\frac{1}{2}\left(1+{q}_{0}\right)z\right]
\left(1+{q}_{0}\right)z/2={z}_{M}/2=wz/2
{q}_{0}=w-1
\left(1+{q}_{0}\right)z/2=3{z}_{X}/4=3\left(1-w\right)z/4
{q}_{0}=\left(1-3w\right)/2
{q}_{0}
w=0.6
{q}_{0}=-0.4
{\Omega }_{m,0}\equiv b
{\Omega }_{\Lambda ,0}\equiv 1-b
u=0
\begin{array}{c}{d}_{X}\left({t}_{0}\right)={R}_{X}{\int }_{0}^{{z}_{X}}\left[\text{d}u\left(1-\frac{3}{2}bu+\frac{3}{8}b\left(9b-4\right){u}^{2}-\cdots \right)\right]\\ =\frac{{R}_{X}}{8}{z}_{X}\left(8-6b{z}_{X}+\cdots \right)\\ ={R}_{0}z\left(1-\frac{3}{4}b\left(1-w\right)z+\cdots \right)\end{array}
1+{q}_{0}=3b\left(1-w\right)/2
1+{q}_{0}=w
w=1/\left(1+\frac{2}{3b}\right)
{q}_{0}=-1/\left(1+\frac{3}{2}b\right)
b=1
w=0.6
{q}_{0}=-0.4
b=0.3
w=0.31
{q}_{0}=-0.69
{H}_{X}=0.4{H}_{0}
{H}_{M}=0.6{H}_{0}
{H}_{X}=0.69{H}_{0}
{H}_{M}=0.31{H}_{0}
{z}_{X},{z}_{M}
z\ll 0
{d}_{M}
{d}_{X}
{R}_{M}\mathrm{ln}\left(1+{z}_{M}\right)=2{R}_{X}\left(1-\frac{1}{\sqrt{1+{z}_{X}}}\right)
\frac{{R}_{0}z}{{z}_{M}}\mathrm{ln}\left[\frac{1+z}{1+{z}_{X}}\right]=\frac{2{R}_{0}z}{{z}_{X}}\left(1-\frac{1}{\sqrt{1+{z}_{X}}}\right)
\mathrm{ln}\left(\frac{y}{x}\right)=\frac{2\left(y-x\right)}{x\left(x-1\right)}\left(1-\frac{1}{\sqrt{x}}\right)
y=1+z
x=1+{z}_{X}
{z}_{X}
{z}_{M}
1+{z}_{M}=\left(1+z\right)/\left(1+{z}_{X}\right)
{d}_{L}
\mu
{d}_{L}
\mu =5\mathrm{log}\left({d}_{L}\right)+25
\mu =5\mathrm{log}\left(\frac{2{R}_{0}\left(y-1\right)}{x-1}\left(1-\frac{1}{\sqrt{x}}\right)\sqrt{xy}\right)+1.25\mathrm{log}\left(y\right)+25
{R}_{M}\mathrm{ln}\left(1+{z}_{M}\right)={R}_{X}{\int }_{0}^{{z}_{X}}\left[\text{d}u/\sqrt{\left({\Omega }_{m,0}{\left(1+u\right)}^{3}+1-{\Omega }_{m,0}\right)}\right]
{z}_{X},{z}_{M}
\begin{array}{c}\mu =5\mathrm{log}\left(\frac{{R}_{0}\left(y-1\right)}{x-1}{\int }_{0}^{x-1}\left[\text{d}u/\sqrt{\left({\Omega }_{m,0}{\left(1+u\right)}^{3}+1-{\Omega }_{m,0}\right)}\right]\sqrt{xy}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+1.25\mathrm{log}\left(y\right)+25\end{array}
0.015\le z\le 1.414
{H}_{0}
{\Omega }_{m}
{H}_{0}
{H}_{0}=70.52
{H}_{0}
{H}_{0},{\Omega }_{m}
{a}_{X}
1/\left(1+z\right)
{a}_{X}
1/\left(1+z\right)
{z}_{X}
{z}_{M}
{a}_{X}
1/\left(1+z\right)
{a}_{X}\left(z\right)={\left(1+z\right)}^{-0.42}
{z}_{M}/{z}_{X}
H{\left(t\right)}^{2}\equiv {\left(\frac{\stackrel{˙}{a}}{a}\right)}^{2}=\frac{8\text{π}G}{3{c}^{2}}\epsilon +\frac{\Lambda }{3}
a
\stackrel{˙}{a}
cz\equiv v\left(t\right)=H\left(t\right)d\left(t\right)
c{z}_{X}\equiv v\left(t\right)={H}_{X}\left(t\right)d\left(t\right)
{a}_{X}\left[\equiv 1/\left(1+{z}_{X}\right)\right]
{H}_{X}{\left(t\right)}^{2}\equiv {\left(\frac{{\stackrel{˙}{a}}_{X}}{{a}_{X}}\right)}^{2}=\frac{8\text{π}G}{3{c}^{2}}\epsilon
a\left(t\right)
1/\left(1+z\right)
{a}_{X}\left(z\right)
{a}_{X}
1/\left(1+z\right)
\frac{\Lambda }{3}=H{\left(t\right)}^{2}-{H}_{X}{\left(t\right)}^{2}
t={t}_{0}
\Lambda =3{H}_{0}^{2}\left(1-{0.4}^{2}\right)=2.52{H}_{0}^{2}=1.22\times {10}^{-35}\text{\hspace{0.17em}}{\text{s}}^{-2}
{H}_{0}=68\text{\hspace{0.17em}}\text{km}\cdot {\text{s}}^{-1}\cdot {\text{Mpc}}^{-1}=2.2\times {10}^{-18}\text{\hspace{0.17em}}{\text{s}}^{-1}
{c}^{2}
\Lambda =1.36\times {10}^{-52}\text{\hspace{0.17em}}{\text{m}}^{-2}
{\Omega }_{\Lambda }=0.84
{H}_{X}=0.4{H}_{0}
{\left(1+z\right)}^{-1}
{\left(1+{z}_{X}\right)}^{-1}
{\left(1+z\right)}^{-1}
{H}_{0}=69.11\left(\pm 0.53\right)
{q}_{0}=-0.4
=1.2\times {10}^{-35}\text{\hspace{0.17em}}{\text{s}}^{-2}
\left(1.4\times {10}^{-52}\text{\hspace{0.17em}}{\text{m}}^{-2}\right)
{\Omega }_{\Lambda }=0.84
1/\left(1+z\right)
0.4{H}_{0}
0.6{H}_{0}
Gupta, R.P. (2018) Static and Dynamic Components of the Redshift. International Journal of Astronomy and Astrophysics, 8, 219-229. https://doi.org/10.4236/ijaa.2018.83016
1. Penzias, A.A. and Wilson, R.W. (1965) A Measurement of Excess Antenna Temperature at 4080 Mc/s. The Astrophysical Journal, 142, 419-421. https://doi.org/10.1086/148307
2. Lineweaver, C.H. and Barbosa, D. (1998) Cosmic Microwave Background Observations: Implications for Hubble’s Constant and the Spectral Parameters N and Q in Cold Dark Matter Critical Density Universes. Astronomy & Astrophysics, 329, 799-808.
3. Blanchard, A., Douspis, M., Rowan-Robinson, M. and Sarkar, S. (2003) An Alternative to the Cosmological “Concordance Model”. Astronomy & Astrophysics, 412, 35-44. https://doi.org/10.1051/0004-6361:20031425
4. López-Corredoira, M. (2017) Test and Problems of the Standard Model in Cosmology. Foundations of Physics, 47, 711-768. https://doi.org/10.1007/s10701-017-0073-8
5. Orlov, V.V. and Raikov, A.A. (2016) Cosmological Tests and Evolution of Extragalactic Objects. Astronomy Reports, 60, 477-485. https://doi.org/10.1134/S1063772916030112
6. Gupta, R.P. (2018) Mass of the Universe and the Redshift. International Journal of Astronomy and Astrophysics, 8, 68-78. https://doi.org/10.4236/ijaa.2018.81005
7. Poisson, E. (2004) The Motion of Point Particles in Curved Spacetime. Living Reviews in Relativity, 7, 6. https://doi.org/10.12942/lrr-2004-6
8. Fischer, E. (2007) Redshift from Gravitational Back Reaction. arXivastro-ph/0703791
9. Peebles, P.J.E. (1993) Principles of Physical Cosmology. Princeton, NJ.
10. Ryden, B. (2017) Introduction to Cosmology. Cambridge, UK.
11. Amanullah, R., et al. (2010) Spectra and Hubble Space Telescope Light Curves of Six Type Ia Supernovae at 0.511 < z < 1.12 and the UNION2 Compilation. The Astrophysical Journal, 716, 712-738. https://doi.org/10.1088/0004-637X/716/1/712
12. Blondin, S., et al. (2008) Time Dilation in Type Ia Supernova Spectra at High Redshift. The Astrophysical Journal, 682, 724-736. https://doi.org/10.1086/589568
13. Crawford, D.F. (2017) A Problem with the Analysis of Type Ia Supernovae. Open Astronomy, 26, 111-119. https://doi.org/10.1515/astro-2017-0013
14. Hawkins, M.R.S. (2010) On Time Dilation in Quasar Light Curves. Monthly Notices of the Royal Astronomical Society, 405, 1940-1946. https://doi.org/10.1111/j.1365-2966.2010.16581.x
15. Chang, H.-Y. (2001) Fourier Analysis of Gamma-Ray Burst Light Curves: Searching for a Direct Signature of Cosmological Time Dilation. The Astrophysical Journal, 557, L85-L88. https://doi.org/10.1086/323331
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Generate orthogonal variable spreading factor (OVSF) code from set of orthogonal codes - Simulink - MathWorks América Latina
Generate orthogonal variable spreading factor (OVSF) code from set of orthogonal codes
The OVSF Code Generator block generates an OVSF code from a set of orthogonal codes. OVSF codes were first introduced for 3G communication systems. OVSF codes are primarily used to preserve orthogonality between different channels in a communication system.
OVSF codes are defined as the rows of an N-by-N matrix, CN, which is defined recursively as follows. First, define C1 = [1]. Next, assume that CN is defined and let CN(k) denote the kth row of CN. Define C2N by
{C}_{2N}=\left[\begin{array}{cc}{C}_{N}\left(0\right)& {C}_{N}\left(0\right)\\ {C}_{N}\left(0\right)& -{C}_{N}\left(0\right)\\ {C}_{N}\left(1\right)& {C}_{N}\left(1\right)\\ {C}_{N}\left(1\right)& -{C}_{N}\left(1\right)\\ ...& ...\\ {C}_{N}\left(N-1\right)& {C}_{N}\left(N-1\right)\\ {C}_{N}\left(N-1\right)& -{C}_{N}\left(N-1\right)\end{array}\right]
Note that CN is only defined for N a power of 2. It follows by induction that the rows of CN are orthogonal.
The OVSF codes can also be defined recursively by a tree structure, as shown in the following figure.
If [C] is a code length 2r at depth r in the tree, where the root has depth 0, the two branches leading out of C are labeled by the sequences [C C] and [C -C], which have length 2r+1. The codes at depth r in the tree are the rows of the matrix CN, where N = 2r.
Note that two OVSF codes are orthogonal if and only if neither code lies on the path from the other code to the root. Since codes assigned to different users in the same cell must be orthogonal, this restricts the number of available codes for a given cell. For example, if the code C41 in the tree is assigned to a user, the codes C10, C20, C82, C83, and so on, cannot be assigned to any other user in the same cell.
You specify the code the OVSF Code Generator block outputs by two parameters in the block's dialog: the Spreading factor, which is the length of the code, and the Code index, which must be an integer in the range [0, 1, ... , N - 1], where N is the spreading factor. If the code appears at depth r in the preceding tree, the Spreading factor is 2r. The Code index specifies how far down the column of the tree at depth r the code appears, counting from 0 to N - 1. For CN, k in the preceding diagram, N is the Spreading factor and k is the Code index.
You can recover the code from the Spreading factor and the Code index as follows. Convert the Code index to the corresponding binary number, and then add 0s to the left, if necessary, so that the resulting binary sequence x1 x2 ... xr has length r, where r is the logarithm base 2 of the Spreading factor. This sequence describes the path from the root to the code. The path takes the upper branch from the code at depth i if xi = 0, and the lower branch if xi = 1.
To reconstruct the code, recursively define a sequence of codes Ci for as follows. Let C0 be the root [1]. Assuming that Ci has been defined, for i < r, define Ci+1 by
{C}_{i+1}=\left\{\begin{array}{ll}{C}_{i}{C}_{i}\hfill & \text{if }{x}_{i}=0\hfill \\ {C}_{i}\left(-{C}_{i}\right)\hfill & \text{if }{x}_{i}=1\hfill \end{array}
The code CN has the specified Spreading factor and Code index.
For example, to find the code with Spreading factor 16 and Code index 6, do the following:
Convert 6 to the binary number 110.
Add one 0 to the left to obtain 0110, which has length 4 = log2 16.
Construct the sequences Ci according to the following table.
0 C0 = [1]
1 0 C1 = C0 C0 = [1] [1]
2 1 C2 = C1 -C1 = [1 1] [-1 -1]
3 1 C3 = C2 -C2 = [1 1 -1 -1] [-1 -1 1 1]
4 0 C4 = C3 C3 = [1 1 -1 -1 -1 -1 1 1] [1 1 -1 -1 -1 -1 1 1]
The code C4 has Spreading factor 16 and Code index 6.
Positive integer that is a power of 2, specifying the length of the code.
Integer in the range [0, 1, ... , N - 1] specifying the code, where N is the Spreading factor.
Starting in R2020a, Simulink® no longer allows you to use the OVSF Code Generator block version available before R2015b.
Existing models automatically update to load the OVSF Code Generator block version announced in Source blocks output frames of contiguous time samples but do not use the frame attribute in the R2015b Release Notes. For more information on block forwarding, see Maintain Compatibility of Library Blocks Using Forwarding Tables (Simulink).
Hadamard Code Generator | Walsh Code Generator
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Gear Ratio RPM Calculator
What is the difference between gear rpm and gear speed?
How to use our gear ratio rpm calculator?
More gear related calculators
Are you wondering how to find the rpm of your gear system? Don't fret! Our gear ratio rpm calculator will help you do that in a flash!
After reading the article below, you'll have learnt more about:
The relation between the gear ratio and the gear speed in rpm; and
How to use our gear ratio rpm calculator.
Gears are ubiquitous mechanisms, and you can find them in watches, bicycles, cars, etc. The gear ratio is an important parameter that describes the characteristics of a gear system.
The gear ratio is the ratio of the number of teeth of the driver gear to the number of teeth of the driven gear.
Now that we know what does gear ratio mean, the relationship between gear ratio and gear rpm is given as:
\footnotesize \text{Output gear rpm} = \frac{ \text{Input gear rpm} }{ \text{Gear ratio} }
Using the above relation, we can calculate the gear rpm by using the gear ratio.
The gear rpm and gear speed are the quantities used to represent how fast an object rotates.
Gear speed indicates how much angle is spanned by a rotating object per second. To convert from gear rpm to gear speed in
\text{rad}/\text{s}
: multiply gear rpm by
\pi/30
Gear rpm indicates how many rotations a rotating object makes per minute. To convert from gear speed in
\text{rad}/\text{s}
to gear speed in rpm: multiply gear speed in
\text{rad}/\text{s}
30/\pi
To use our calculator, use the following guidelines:
First, you must know about the number of teeth present in your gear system. The gear system includes the input (driver) and output (driven) gear. Input gear is the gear that drives the output gear.
Insert the number of teeth for input and output gear in Input gear teeth and Output gear teeth fields.
Right after you insert the teeth information, you will obtain the Gear ratio.
Now insert the rpm of your input gear to obtain the output gear rpm.
Great! Now you know how to use our calculator to find output gear rpm quickly.
Do you have any other problems related to gears that you want solve? If yes, then check out our other gear-related calculators:
Gear ratio calculator;
Gear ratio speed calculator;
What is the gear speed of 30 rpm in rad/s?
The gear speed in rad/s is π. To obtain gear speed in rad/s from rpm: multiply the gear speed in rpm by 2π and divide by 60.
How to calculate the output torque from gear ratio?
To calculate the output torque from gear ratio, use the following guidelines:
Obtain the information about the number of teeth present in the input and the output gears;
Take the ratio of the number of teeth of the input gear to the number of teeth of output gear to find the gear ratio; and
Find output torque by multiplying the input torque by the gear ratio.
Awesome, now you know how to find the output torque given you know the gear ratio and input torque of the gear system.
How do I find the gear speed in rpm given its speed in rad/s ?
To calculate the gear speed in rpm, use the following steps:
Multiply the gear speed in rad/s by 60;
Now divide the obtained quantity from step 1 by 2π to obtain the gear speed in revolution per minute (rpm); and
Cool!, Now you can easily convert the gear speed in rpm to rad/s.
The Ideal Transformer Calculator is an easy tool that helps you to see how a transformer works.
Use the velocity addition calculator to discover that we cannot just add the velocities.
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Piece of Cake | Toph
By shajia, jackal_1586 · Limits 1s, 512 MB
Diya has got a nice birthday cake for Arshiya and Alayna, her twin daughters. The two sisters were amazed to see the strange triangular-shaped birthday cake. On the cake ABC, there is a line from one of the vertex A to the midpoint of the opposite line BC. On her daughter's birthday, Diya declared a quiz competition. She told her daughters that she will make two more square-shaped cake as the prize of that quiz competition. Each side length of Arshiya's cake will be equal to AB, one side of the triangle-shaped cake. Each side length of Alayna's cake will be equal to AC, one side of the triangle-shaped cake. For getting the prize, Arshiya and Alayna have to calculate the sum of the area of these two square-shaped cake. As they want to win the prize, they want your help.
You will be given some information about the triangle-shaped cake for solving the challenge. You will be given AB, the one side of the triangular-shaped cake, AD, the line from vertex A to the midpoint of BC and the angle DAB.
\pi = cos^{-1}(-1.0)
π=cos−1(−1.0) if needed.
First line of the input contains an integer T(1 < T < 100000) which denotes the number of test cases. Then for each test case, there will be three real number X which denotes the length of AB, Y which denotes the length of AD and A which denotes the angle DAB. All lengths are less than 104, also angles and lengths are greater than zero.
It is guaranteed that the input forms a valid triangle with positive area.
For each test case, output the total sum of the area of the two square-shaped cake. Absolute error less than 10-6 will be ignored.
100 86.60254037844386 30
ApolloneasTheorem, Geometry
JU_FIREW0RKSEarliest, Nov '19
subah_noshinFastest, 0.1s
Noshin_1703086Lightest, 1.3 MB
extrem_crackShortest, 138B
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Evaluate the integral without using tables. \int_{0}^{1}x\ln x dx
{\int }_{0}^{1}x\mathrm{ln}xdx
Evaluate the integrals as follows:
u=\mathrm{ln}x,du=\frac{1}{x}dx,{v}^{\prime }=x,v=\frac{{x}^{2}}{2}\text{ }\in \text{ }\int udv=uv-\int vdu
{\int }_{0}^{1}x\mathrm{ln}\left(x\right)dx={\left[\frac{1}{2}{x}^{2}\mathrm{ln}\left(x\right)-\int \frac{x}{2}dx\right]}_{0}^{1}
={\left[\frac{1}{2}{x}^{2}\mathrm{ln}\left(x\right)-\frac{{x}^{2}}{4}\right]}_{0}^{1}
=-\frac{1}{4}
\int \frac{5}{2}\mathrm{cos}\left(4x\right)\mathrm{ln}\left(\mathrm{sin}\left(4x\right)+1\right)dx
\int {x}^{\frac{5}{7}}dx
\frac{8}{3}\mathrm{sin}\left(t\right){\mathrm{sec}}^{3/2}\left(t\right)
\int {\mathrm{sin}}^{2}0{\mathrm{cos}}^{5}0d0
{\int }_{-1}^{1}{\left(x-3\right)}^{2}dx
\frac{4{e}^{{\mathrm{tan}}^{-1}\left(t\right)}}{7\left({t}^{2}+1\right)}
{\int }_{0}^{1}{\mathrm{sin}}^{-1}\left\{\frac{{x}^{2}}{1+{x}^{2}}\right\}dx
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The Bayesian Counterpart of Pearson's Correlation Test - Publishable Stuff
The Bayesian Counterpart of Pearson's Correlation Test
Except for maybe the t test, a contender for the title “most used and abused statistical test” is Pearson’s correlation test. Whenever someone wants to check if two variables relate somehow it is a safe bet (at least in psychology) that the first thing to be tested is the strength of a Pearson’s correlation. Only if that doesn’t work a more sophisticated analysis is attempted (“That p-value is still to big, maybe a mixed models logistic regression will make it smaller…”). One reason for this is that the Pearson’s correlation test is conceptually quite simple and have assumption that makes it applicable in many situations (but it is definitely also used in many situations where underlying assumption are violated).
Since I’ve converted to “Bayesianism” I’ve been trying to figure out what Bayesian analyses correspond to the classical analyses. For t tests, chi-square tests and Anovas I’ve found Bayesian versions that, at least conceptually, are testing the same thing. Here are links to Bayesian versions of the t test, a chi-square test and an Anova, if you’re interested. But for some reason I’ve never encountered a discussion of what a Pearson’s correlation test would correspond to in a Bayesian context. Maybe this is because regression modeling often can fill the same roll as correlation testing (quantifying relations between continuous variables) or perhaps I’ve been looking in the wrong places.
The aim of this post is to explain how one can run the Bayesian counterpart of Pearson’s correlation test using R and JAGS. The model that a classical Pearson’s correlation test assumes is that the data follows a bivariate normal distribution. That is, if we have a list $x$ of pairs of data points
[[x_{1,1},x_{1,2}],[x_{2,1},x_{2,2}],[x_{3,1},x_{3,2}],...]
x_{i,1} \text{s}
x_{i,2} \text{s}
are each assumed to be normally distributed with a possible linear dependency between them. This dependency is quantified by the correlation parameter $\rho$ which is what we want to estimate in a correlation analysis. A good visualization of a bivariate normal distribution with $\rho = 0.3$ can be found on the the wikipedia page on the multivariate normal distribution :
We will assume the same model and our Bayesian correlation analysis then reduces to estimating the parameters of a bivariate normal distribution given some data. One problem is that the bivariate normal distribution and, more general, the multivariate normal distribution isn’t parameterized using $\rho$, that is, we cannot estimate $\rho$ directly. The bivariate normal is parameterized using
\mu_1
\mu_2
the means of the two marginal distributions (the red and blue normal distribution in the graph above) and a covariance matrix $\Sigma$ which defines
\sigma_1^2
\sigma_2^2
, the variances of the two marginal distributions, and the covariance, how much the marginal distributions vary together. So the covariance is another measure of how much two variables vary together and the covariance corresponding to a correlation of $\rho$ can be calculated as
\rho \cdot \sigma_1 \cdot \sigma_2
. So here is the model we want to estimate:
Add some flat priors on this (which could, of course, be made more informative) and we’re ready to roll:
So, how to implement this model? I’m going to do it with R and the JAGS sampler interfaced with R using the rjags package. First I’m going to simulate some data with a correlation of 0.7 to test the model with.
library(mvtnorm) # to generate correlated data with rmvnorm.
library(car) # To plot the estimated bivariate normal distribution.
mu <- c(10, 30)
sigma <- c(20, 40)
cov_mat <- rbind(c( sigma[1]^2 , sigma[1]*sigma[2]*rho ),
c( sigma[1]*sigma[2]*rho, sigma[2]^2 ))
x <- rmvnorm(30, mu, cov_mat)
plot(x, xlim=c(-125, 125), ylim=c(-100, 150))
The simulated data looks quite correlated and a classical Pearson’s correlation test confirms this:
cor.test(x[, 1], x[, 2])
## data: x[, 1] and x[, 2]
## t = -6.6, df = 28, p-value = 3.704e-07
The JAGS model below implements the bivariate normal model described above. One difference is that JAGS parameterizes the normal and multivariate normal distributions with precision instead of standard deviation or variance. The precision is the inverse of the variance so in order to use a covariance matrix as a parameter to dmnorm we have to inverse it first using the inverse function.
x[i,1:2] ~ dmnorm(mu[], prec[ , ])
# Constructing the covariance matrix and the corresponding precision matrix.
# Uninformative priors on all parameters which could, of course, be made more informative.
mu[1] ~ dnorm(0, 0.001)
# Generate random draws from the estimated bivariate normal distribution
x_rand ~ dmnorm(mu[], prec[ , ])
Update: Note that while the priors defined in model_string are relatively uninformative for my data, they are not necessary uninformative in general. For example, the priors above considers a value of mu[1] as large as 10000 extremely improbable and a value of sigma[1] above 1000 impossible. An example of more vague priors would for example be:
An extra feature is that the model above generates random samples (x_rand) from the estimated bivariate normal distribution. These samples can be compared to the actual data in order to get a sense of how well the model fits the data. Now let’s use JAGS to sample from this model. I’m using the textConnection trick (described here) in order to run the model without having to first save the model string to a file.
data_list = list(x = x, n = nrow(x))
inits_list = list(mu = c(mean(x[, 1]), mean(x[, 2])),
rho = cor(x[, 1], x[, 2]),
sigma = c(sd(x[, 1]), sd(x[, 1])))
n.adapt = 500, n.chains = 3, quiet = T)
mcmc_samples <- coda.samples(jags_model, c("mu", "rho", "sigma", "x_rand"),
Now let’s plot trace plots and density plots of the MCMC parameter estimates:
plot(mcmc_samples, auto.layout = FALSE)
The trace plots look sufficiently furry and stationary and looking at the density plots it looks like the model have captured the “true” parameters well (those used when we created the data). Looking at point estimates and credible intervals confirms this:
## mu[1] 9.621 4.3713 0.035692 0.07107
## mu[2] 31.988 8.1653 0.066669 0.13381
## rho -0.755 0.0816 0.000666 0.00142
## sigma[1] 24.531 3.3368 0.027245 0.05816
## x_rand[1] 9.513 24.9313 0.203563 0.21112
## x_rand[2] 32.353 47.1290 0.384806 0.39614
## 2.5% 25% 50% 75% 97.5%
## mu[1] 0.970 6.779 9.640 12.485 18.178
## mu[2] 15.671 26.551 31.969 37.371 48.264
## rho -0.881 -0.814 -0.766 -0.708 -0.564
## sigma[1] 19.042 22.181 24.174 26.517 32.003
## x_rand[1] -39.002 -6.852 9.658 25.964 58.649
## x_rand[2] -60.836 1.619 32.223 63.050 125.009
The median of the rho parameter is -0.76 (95% CI: [-0.88, -0.55]) close to the true parameter value -0.7. Even if this is not that strange, we did use a bivariate normal distribution both to generate and model the data, it is nice when things work out :) Now let’s calculate the probability that there actually is a negative correlation.
mean(samples_mat[, "rho"] < 0)
Given the model and the data it seems like the probability that there is a negative correlation is 100%. Accounting for the MCMC error I think it is fair to downgrade this probability to at least 99.9%. We can also use the random samples from the bivariate normal distribution estimated by the model to compare how well the model fits the data. The following code plots the original data with two superimposed ellipses where the inner ellipse covers 50% of the density of the distribution and the outer ellipse covers 95%.
Quite a good fit!
Update: If you are more of a Python person, check out Philipp Singer’s Bayesian Correlation with PyMC, which implements the model above in Python.
Analysis of Some “Real” Data
So let’s use this model on some real data. The data.frame below contains the names, weights in kg and finishing times for all participants of the men’s 100 m semi-finals in the 2013 World Championships in Athletics. Well, those I could find the weights of anyway…
d <- data.frame(runner = c("Usain Bolt", "Justin Gatlin", "Nesta Carter", "Kemar Bailey-Cole",
"Nickel Ashmeade", "Mike Rodgers", "Christophe Lemaitre", "James Dasaolu",
"Zhang Peimeng", "Jimmy Vicaut", "Keston Bledman", "Churandy Martina", "Dwain Chambers",
"Jason Rogers", "Antoine Adams", "Anaso Jobodwana", "Richard Thompson",
"Gavin Smellie", "Ramon Gittens", "Harry Aikines-Aryeetey"), time = c(9.92,
9.94, 9.97, 9.93, 9.9, 9.93, 10, 9.97, 10, 10.01, 10.08, 10.09, 10.15, 10.15,
10.17, 10.17, 10.19, 10.3, 10.31, 10.34), weight = c(94, 79, 78, 83, 77,
So, I know nothing about running (and I’m not sure this is a very representative data set…) but my hypothesis is that there should be a positive correlation between weight and finishing time. That is, the more you weigh the slower you run. Sounds like logic, right? Let’s look at the data…
At a first glance it seems like my hypothesis is not supported by the data. I wonder what our model has to say about that?
mcmc_samples <- coda.samples(jags_model, c("rho"), n.iter = 5000)
## -0.09905 0.22054 0.00180 0.00243
Seems like there is no support for my hypothesis, the posterior distribution of rho is centered around zero and, if anything, there might be a tiny negative correlation. So your weight doesn’t seem to influence how fast you run (if you are a runner in the 100 m semi finals, at least).
« Modeling Match Results in La Liga Using a Hierarchical Bayesian Poisson Model: Part three. More Bayesian Football Models »
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L-attribute grammars.
S-attribute grammars.
Equivalence of L-attribute and S-attribute grammars.
Given an SDD, it is difficult to tell whether there are any parse trees whose dependency graphs have cycles.
Here we discuss two classes of definitions that can efficiently be implemented in connection to top-down or bottom-up parsing.
We also look at attribute grammars that result from serious restrictions.
The first is the L-attribute grammar whereby an inherited attribute of a child of a non-terminal N may depend on synthesized attributes of children to the left of it in the production rule for N and on the inherited attributes of N itself.
The second is the S-attribute grammar that cannot have inherited attributes at all.
An SDD is S-attributed if every attribute is synthesized.
If an SDD is S-attributed, we evaluate its attributes in any bottom-up ordering of the parse tree nodes.
It is simpler to perform a post-order tree traversal and evaluate the attributes at a node N when the traversal leaves N for the last time.
We apply the following postorder function to the root of the tree;
postorder(N){
for(each child C of N, from the left)
postorder(C):
evaluate attribute associated with node N;
These definitions are implemented during bottom-up parsing because a bottom-up parse corresponds to a post-order traversal, in other words, a postorder traversal corresponds to the order that an LR parser reduces the production body to its head.
The idea is that between attributes associated with a production body, the edges of a dependency graph can go from right to left but not the other way round(left to right), hence the name 'L-attributed'.
In other words, each attribute must either be;
Synthesized, or,
Inherited but with limited rules, i.e Suppose there is a production A →
{\mathrm{X}}_{1}
{\mathrm{X}}_{2}
{\mathrm{X}}_{n}
and an inherited attribute
{\mathrm{X}}_{i}
.a computed by a rule associated with this production, then the rule only uses;
** inherited attributes that are associated with head A.
** Either inherited attribute or synthesized attributes associated with the occurrences of symbols
{\mathrm{X}}_{1}
{\mathrm{X}}_{2}
{\mathrm{X}}_{i-1}
located to the left of
{\mathrm{X}}_{i}
** Inherited or synthesized attributes that are associated with such an occurrence of
{\mathrm{X}}_{i}
itself, only in such a way that no cycles exist in the dependency graph formed by
{\mathrm{X}}_{i}
An SDD with the following production and rules is not L-Attributed.
Looking at the first rule, A.s = B.b is legitimate in either an S-attributed or L-attributed SDD since it defines a synthesized attribute A.s in terms of an attribute at a child.
Looking at the second rule, it defines an inherited attribute B.i therefore, the entire SDD cannot be S-attributed since attribute C.c helps to define B.i and C is to the right of B in the body of the production.
While attributes at siblings in a parse tree may be used in L-attributed SDDs, keep in mind that they must be to the left of the symbol whose attribute is being defined.
During parsing, nodes are constructed from left to right, first the parent node, then the children in top-down parsing, and in bottom-up parsing, we start with the children then the parent nodes.
An L-attribute grammar allows the evaluation of attributes in one left to right traversal of the syntax tree. It is characterized by no dependency graph of any of its production rules having a data-flow edge pointing from a child to that child or to a child to its left.
Most programming language grammars are L-attributed, this is because the left to right flow of data assists programmers in reading and understanding the resulting programs.
The L-attributed property has an important consequence for processing a syntax tree in that once work starts, no part of the compiler needs to return to one of the node's siblings on the left to perform processing there.
The parser will have finished with them and their attributes already computed, only data that the nodes contain in the form of synthesized attributes remain important.
An example of data flow in part of a parse tree for an L-attributed grammar.
Assume an attribute evaluator is currently working on node C2 - the second child of node B3 - the third child of node A, whether A is the top or a child of another node is immaterial.
The upward arrow represents the data flow of synthesized attributes of children that all point to the right or to the synthesized attributes of the parent.
Inherited attributes are available when work on a node starts and therefore can be passed to any child that needs them. From the image, they are the ones with arrows pointing downwards.
We also see that when an evaluator is working on a node C2, two sets of attributes play a role,
The first are all attributes of the nodes lying on the path from the top to the node currently being processed: C2, B3, A.
The second are the synthesized attributes of the left siblings of those nodes: C1, B1, B2 and any left siblings of A(these are not shown).
In other words, no role is played by the children of the left siblings of C2, B3, A because all computations in them have already been performed and results summarized in their synthesized attributes, furthermore, the right siblings of C2, B3, A don't play a role since their synthesized attributes have no influence yet.
Attributes of C2, B3, A live in the corresponding nodes, work on such nodes is started but not completed. Conversely, work on their left siblings is completed.
Now, all that is left of them are their synthesized attributes
If we found a place to store data synthesized by the left siblings, we would discard each node in the left to right order after the parser has created it and the attribute evaluator has computed its attributes, meaning, we don't need to build the entire syntax tree but will always restrict ourselves to the nodes lying on the path from the top to the node being processed.
Everything remaining on the path has been processed except for synthesized attributes of the left siblings which have been discarded, everything to the right has not yet been touched.
We store the synthesized attributes in the parent node while the inherited ones remain in the nodes they belong to while their values are transported down along the path from the top to the node being processed.
If inherited attributes pose a problem we get rid of them and the result is an S-attribute grammar that is characterized as having no inherited attributes.
Anything that can be done with an L-attribute grammar is still possible with this grammar.
For the bottom-up parser, the complexity reduces since each node stacks its synthesized attributes, and the code at the end of an alternative of the parent scopes them all up, processes them, and replaces them with the resulting synthesized attributes of the parent.
It is easy to convert an L-attribute into an S-attribute grammar although this doesn't improve the looks.
The idea is to delay any computation that cannot be done now to a later occasion when it can be done.
In particular, any computation that needs inherited attributes is replaced by the creation of a data structure that specifies the computation and all its synthesized attributes up to the level where the missing inherited attributes are available as constants or as synthesized attributes of nodes at the current level, after which we perform the computation.
Transforming an L-attribute into an S-attribute grammar is attractive as it allows better bottom-up parsing methods to be implemented for more convenient attribute L-attribute grammars.
Unfortunately, this transformation is only feasible for small problems.
In an L-attributed SDD attributes may be inherited or synthesized, this is referred to as an L-attribute definition.
In an S-attributed SDD, attributes all attributes are synthesized - S-attribute definition.
An L-attributed grammar is a grammar that node dependency graph of any of its production rules has data-flow arrow pointing from an attribute to an attribute to its left. Such grammars allow attributes to be evaluated in a left-to-right traversal.
An S-attribute grammars don't have inherited attributes at all, here attributes need to be retained only for non-terminal nodes that haven't yet been reduced to other non-terminals.
Everything that is possible with the former is possible with the latter.
Compilers Principles, Techniques, & Tools. Chapter 5.
Modern Compiler Design Dick Grune, Kees van Reeuwijk, Henri E. Bal. Part II.
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What is an expansion wave?
What is the Prandtl Meyer expansion theory?
How to use the Prandtl Meyer expansion calculator?
If you are thinking of finding the supersonic flow Mach number after an expansion wave, this Prandtl Meyer expansion calculator is the perfect match for you. Our calculator will help you find the Mach number along with downstream flow properties.
Please read the following article to learn about:
What is an expansion wave, and how are they generated?
What is the Prandtl Meyer theory, and how to evaluate Prandtl Meyer function?
When a supersonic flow encounters a surface that folds outward, the flow will have more space to move, leading to an expansion of the flow. This expansion is assisted by a fan of waves called an expansion wave, which acts as a continuous boundary where the expansion happens.
Next time, take a closer look at the rocket launch!
Shock waves, in particular, are essential characteristics of rocket engines. The rocket engine’s aft part is called a nozzle. During the ascending phase of a rocket, the flow coming out of the nozzle will go through different conditions. They are:
Optimal expansion: The flow coming out from the nozzle will have the same area as the nozzle cross-sectional area. In this case, we will not see any shock waves.
Under expansion: The flow coming out from the nozzle will have a reduced cross-sectional area than the nozzle cross-sectional area. In this condition, we will be able to see shock waves.
Over expansion: The flow coming out from the nozzle will have an increased cross-sectional area compared to the nozzle cross-sectional area. In this condition, we will see an expansion wave helping the supersonic waves expand after coming out of the nozzle.
So, keep an eye on the launcher’s nozzle and look for shock and expansion waves!
In 1907, German fluid dynamicist Ludwig Prandtl studied the expansion waves, followed by his student Theodor Meyer in 1908. They developed a theory called Prandtl Meyer expansion to explain the behavior of expansion waves. We will briefly look at the theory and the equations behind our Prandtl Meyer expansion calculator in the following paragraphs.
Below, in the figure, you can observe a surface folding in the outward direction. A flow with Mach number
M_1 ( M_1>1)
flows along the parallel surface (a-b). Because of the outward folded surface (after surface a-b), we notice the generation of an expansion wave. This expansion wave lends itself as a fan of waves. The flow passing through this fan of waves changes the direction of the flow. After the expansion wave (region 2), the flow will have a Mach number
M_2
with its direction changed by an angle
\mu-\theta
. The properties in the region after expansion wave are denoted by
T_2
P_2
\rho_2
Schematic demonstration of an expansion wave generation via a surface turning outward.
According to Prandtl Meyer theory, you can find the downstream flow properties by using the following steps:
Find the Prandtl Meyer function
\nu(M1)
for the upstream Mach number
M_1
with known specific heat ratio
\gamma
of the fluid medium . We can use the following equation to do that:
\scriptsize \begin{align*} \nu(M_2)&=\sqrt \frac{\gamma+1}{\gamma-1} \arctan\sqrt\frac{{(\gamma-1)} (M_1^2-1)}{(\gamma+1)} \\ & -\arctan\sqrt{M_1^2-1} \end{align*}
Now use the deflection of the surface
\theta
\nu(M2)
\footnotesize \qquad \nu(M_2)=\theta+\nu(M_1)
\nu(M2)
to find the Mach number in the downstream region with known
\gamma
of the fluid medium. You can again make use of the Prandtl Meyer function:
\scriptsize \begin{align*} \nu(M_2)&=\sqrt \frac{\gamma+1}{\gamma-1} \arctan\sqrt \frac{{(\gamma-1)} (M_2^2-1)}{(\gamma+1)} \\ & -\arctan\sqrt{M_2^2-1} \end{align*}
The above equation is transcendental, and you can solve for
M_2
either by iterative method or by using the Prandtl Meyer function – Mach number tables.
As expansion wave is isentropic, we can use the isentropic flow equations to find the flow properties. Once you know the upstream properties, you can use the following equations to find pressure, temperature, and density, respectively:
\scriptsize \qquad \frac{T2}{T1}=\left(\frac{1+[(\gamma-1)/2]M_1^2}{1+[(\gamma-1)/2]M_2^2}\right) <p></p>
\scriptsize \qquad \frac{P2}{P1}=\left(\frac{1+[(\gamma-1)/2]M_1^2}{1+[(\gamma-1)/2]M_2^2}\right)^{\gamma/(\gamma-1)}
\scriptsize \qquad \frac{\rho_2}{\rho_1}=\left(\frac{1+[(\gamma-1)/2]M_1^2}{1+[(\gamma-1)/2]M_2^2}\right)^{1/(\gamma-1)}
Good job! You have successfully understood how to find the downstream properties of an expansion wave. Use our calculator to avoid all these tricky calculation steps and get answers at supersonic speed ;).
To use our Prandtl Meyer expansion calculator, use the following instructions:
First, based on your requirement, you can choose YES/NO for the "Do you want flow properties ?" option. If you want only the Mach number and Mach angles, you can set it to "NO": you need only the upstream Mach number
M_1
and deflection angle
\theta
. If you want downstream properties: pressure, temperature, and density, you should choose "YES": along with upstream Mach number
M_1
\theta
, you must provide the upstream temperature, pressure, and density.
Let us assume that we choose YES for the "Do you want flow properties ?" option as this is more general.
Now, you can insert upstream Mach number, deflection angle, and upstream flow properties: temperature
T_1
P_1
, and density
\rho_1
Our calculator will provide you with the following results:
Downstream Mach number or Mach number after the expansion wave
M_2
Downstream flow properties: temperature
T_2
P_2
\rho_2
Mach angles and Prandtl meyer function evaluations for
M_1
M_2
Hooray! You learned to use our calculator to find flow properties after the expansion waves. If you are curious about oblique shock waves, please check our oblique shock wave calculator. We are sure you will like it ;).
Now, it's time to look at an example.
Let us consider an upstream flow with the following properties:
M_1
P_1
= 1 atm;
T_1
\rho_1
= 1.22586 kg/m3; and
Deflection angle
\theta
= 15 degrees.
For our example, let us assume we need all the downstream properties. Therefore, first, choose "YES" for the "Do you want flow properties ?" option. Now you can insert the inputs, and our calculator will provide you with the results right away. Click the advanced mode button to see forward and rearward Mach angles and Prandtl Meyer function evaluations. Following are the results from the calculator:
M_2
\nu_2
= 26.91 degrees (advanced mode);
\nu_1
P_2
= 0.469 atm;
T_2
= 230.0336 K;
\rho_2
= 0.69896 kg}/m3;
Forward Mach angle
\mu_1
Rearward Mach angle
\mu_2
= 29.6874 degrees (advanced mode); and
Flow deflection with respect to horizontal plane =
\nu_2-\theta
= 14.68737 degrees.
Cool! We hope now you are confident to tackle any challenges that arise when you want to deal with Prandtl Meyer expansion waves!
How do I calculate the pressure downstream of an expansion fan?
To find the pressure downstream of the expansion wave, follow these steps:
Find the downstream Prandtl Meyer function using the upstream Mach number and the deflection angle.
Using the result from step 1, use the Prandtl Meyer function – Mach number table or solve the Prandtl Meyer equation for downstream Mach number to obtain the downstream Mach number.
Calculate the total pressure using the upstream pressure and isentropic pressure ratio – Mach number table.
Obtain the downstream pressure by using the total pressure and the isentropic pressure ratio, and Mach number table.
What is the Mach angle for Mach number of 1.5?
Mach angle is 41.81 degrees for Mach number 1.5. You can calculate the Mach angle by taking arcsin of one divided by the Mach number.
What happens to total pressure through an expansion wave?
The total pressure remains constant through an expansion wave. This constancy is due to the fact that there is no discontinuity in the expansion wave, unlike shock waves where discontinuity changes the total pressure.
Are expansion waves isentropic?
Yes, expansion waves are isentropic. Due to the continuous nature of the expansion wave, there is no loss due to dissipation; as a result, entropy and stagnation properties remain constant. This condition makes expansion waves isentropic.
Upstream: region 1
Do you want flow properties?
Pressure (P₁)
Temperature (T₁)
Density (ρ₁)
Downstream: region 2
Pressure (P₂)
Temperature (T₂)
Density (ρ₂)
Check out the dew point calculator to calculate the highest temperature at which water vapor condenses.
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Welcome to the second tutorial. Now I will step you through a deep learning framework that will allow you to build neural networks more easily
Welcome to the second tutorial. Until now, we have used NumPy to build neural networks. Now I will step you through a deep learning framework that will allow you to build neural networks more easily. Machine learning frameworks like TensorFlow can speed up your machine learning development significantly. These frameworks have a lot of documentation, which you should feel free to read. In this tutorial, I will teach to do the following in TensorFlow:
Initialize variables;
Start your own session.
Programing frameworks can not only shorten our coding time but sometimes also perform optimizations that speed up our coding.
1 - Exploring Tensorflow basics:
Let's begin with imports:
We will start with an example, where I'll compute for you the loss of one training example:
loss=L\left(\stackrel{^}{y}, y\right)={\left({\stackrel{^}{y}}^{\left(i\right)}-{y}^{\left(i\right)}\right)}^{2}
# Define y_hat constant. Set to 36.
y_hat = tf.constant(36, name='y_hat')
# Define y. Set to 39
y = tf.constant(39, name='y')
# Create a variable for the loss
loss = tf.Variable((y - y_hat)**2, name='loss')
# When init is run later (session.run(init)),
# Create a session and print the output
# Initializes the variables
# Prints the loss
print(session.run(loss))
1. Create Tensors (variables) that are not yet executed/evaluated;
2. Write operations between those Tensors;
3. Initialize your Tensors;
4. Create a Session;
5. Run the Session. This will run the operations you'd written above.
Therefore, when we created a variable for the loss, we defined the loss as a function of other quantities but did not evaluate its value. To evaluate it, we had to run init=tf.global_variables_initializer(). This line initialized the loss variable, and in the last line, we were finally able to evaluate the value of loss and print its value.
Now let's look at an easy example. Run the cell below:
Tensor("Mul_4:0", shape=(), dtype=int32)
As expected, you will not see 50! You got a tensor saying that the result is a tensor that does not have the shape attribute and type "int32". All you did was put in the 'computation graph', but you have not run this computation yet. To actually multiply the two numbers, you will have to create a session and run it:
Next, we also have to know about placeholders. A placeholder is an object whose value you can specify only later. To specify values for a placeholder, you can pass in values using a "feed dictionary" (feed_dict variable). Below, I created a placeholder for x. This allows us to pass in a number later when we run the session.
When we first defined x, we didn't have to specify a value for it. A placeholder is simply a variable that you will assign data to only later when running the session. We say that you feed data to these placeholders when running the session.
Here's what's happening: When you specify the operations needed for computation, you tell TensorFlow how to construct a computation graph. The computation graph can have some placeholders whose values you will specify only later. Finally, when you run the session, you are telling TensorFlow to execute the computation graph.
1.1 - Linear function:
Let's compute the following equation: Y=WX+b, where W and X are random matrices and b is a random vector.
W is of shape (4, 3), X is (3,1), and b is (4,1).
# Initializes W to be a random tensor of shape (4,3)
# Initializes X to be a random tensor of shape (3,1)
# Initializes b to be a random tensor of shape (4,1)
Y = tf.add(tf.matmul(W,X),b)
# Create the session using tf.Session() and run it with sess.run(...) on the variable we want to calculate
print( "result = ",linear_function())
result = [[ 3.30652566]
1.2 - Computing the sigmoid:
We just implemented a linear function. Tensorflow offers a variety of commonly used neural network functions like tf.sigmoid and tf.softmax. For this exercise, let's compute the sigmoid function of the input.
We will do this using a placeholder variable x. When running the session, we should use the feed dictionary to pass in the input z. So, we will have to:
1. create a placeholder x;
2. define the operations needed to compute the sigmoid using tf.sigmoid, and then;
3. run the session.
Now we'll use 2nd method:
x = tf.placeholder(tf.float32, name = 'x')
# Create a session, and run it.
# We should use a feed_dict to pass z's value to x.
result = sess.run(sigmoid, feed_dict = {x: z})
print("sigmoid(0) = " + str(sigmoid(0)))
print("sigmoid(12) = " + str(sigmoid(12)))
sigmoid(12) = 0.9999938
To summarize, now we know how to:
1. Create placeholders;
2. Specify the computation graph corresponding to operations you want to compute;
3. Create the session;
4. Run the session and using a feed dictionary to specify placeholder variable values.
1.3 - Computing the cost:
We can also use a built-in function to compute the cost of our neural network. So instead of needing to write code to compute this as a function of a and y for i=1...m:
J=-\frac{1}{m}\sum _{i=1}^{m}\left({y}^{\left(i\right)}\mathrm{log}\left({a}^{\left[2\right]\left(i\right)}\right)+\left(1-{y}^{\left(i\right)}\right)\mathrm{log}\left(1-{a}^{\left[2\right]\left(i\right)}\right)\right)
We can do it in one line of code in TensorFlow!
The function we will use is:
Our code should input z, compute the sigmoid (to get a), and then compute the cross-entropy cost J. All this can be done using one call to tf.nn.sigmoid_cross_entropy_with_logits, which computes the above-given cost formula of J:
# Create the placeholders for "logits" (z) and "labels" (y)
z = tf.placeholder(tf.float32, name = 'z')
# Use the loss function
cost = tf.nn.sigmoid_cross_entropy_with_logits(logits = z, labels = y)
# Create a session (method 1 above)
cost = sess.run(cost, feed_dict = {z: logits, y: labels})
# Close the session (method 1 above)
logits = sigmoid(np.array([0.2,0.4,0.7,0.9]))
print ("cost = ", cost)
cost = [1.0053872 1.0366408 0.41385433 0.39956617]
1.4 - Using One Hot encoding:
In deep learning, we will have a y vector with numbers ranging from 0 to C-1, where C is the number of classes. If C is, for example, 4, then you might have the following y vector, which you will need to convert as follows:
This is called a "one-hot" encoding because, in the converted representation, exactly one element of each column is "hot" (meaning set to 1). To do this conversion in NumPy, you might have to write a few lines of code. In TensorFlow, we can use one line of code:
We'll implement the function below to take one vector of labels and the total number of classes 𝐶 and return the one-hot encoding.
# Create a tf.constant equal to C (depth), with name 'C'.
C = tf.constant(C, name = "C")
# Use tf.one_hot, be careful with the axis
one_hot_matrix = tf.one_hot(labels, C, axis=0)
print ("one_hot = ")
print (one_hot)
1.5 - Initialize with zeros and ones:
Now we will learn how to initialize a vector of zeros and ones. The function we will be calling is tf.ones(). To initialize with zeros, we could use tf.zeros() instead. These functions take in shape and return an array of dimension shape full of zeros and ones, respectively.
# Create "ones" tensor using tf.ones(...).
# Run the session to compute 'ones'
print("ones = ")
print(ones((5,3)))
What you should remember after this tutorial:
1. Tensorflow is a programming framework used in deep learning;
2. The two main object classes in TensorFlow are Tensors and Operators;
3. When you code in TensorFlow, you have to take the following steps:
3.1. Create a graph containing Tensors (Variables, Placeholders ...) and Operations (tf.matmul, tf.add, ...);
3.2. Create a session;
3.4. Initialize the session;
3.5 Run the session to execute the graph.
Jupyter file you can download from GitHub.
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Angle of Banking Calculator
Turning a vehicle
Tilted turns, a.k.a. how not to lose control
Tilted turns with a side of friction
In the sky: the angle of banking of an aircraft
How to find the angle of banking?
How to use our angle of banking calculator
A turning road is not flat, it's tilted: with our angle of banking calculator, you will discover why!
Some simple physical considerations and some math will be enough to understand something that seems trivial but has consequences for our safety when driving: tilted turns. With a world going faster and faster by the day, a slightly oblique road can make the difference between staying on the road and not. Here you will learn:
How a vehicle behaves in a turn;
How to solve the problem of turns at high speed;
The mathematical derivation of the angle of banking equation;
What is the angle of banking in aircraft;
How to use our angle of banking calculator.
When turning a vehicle, like a car, we move it from a straight trajectory. Physics tells us that we need to apply a force in the desired direction: in the case of a turn, this is the centripetal force, which points to the turn's center.
🔎 Centripetal comes from the Latin words centrum (center) and petere (to look for).
Consider now a turn on a flat surface and the absence of forces besides gravity and friction. The only force that can counteract the centripetal one is friction. The frictional force
F_f
is a function of the mass
m
of the object and the friction coefficient
\mu
F_f=\mu\cdot m\cdot g
To keep the car on track, the force due to friction must be bigger (or equal, at least) to the centripetal force:
F_f = \mu\cdot m\cdot g > \frac{m\cdot v^2}{r}
v
is the vehicle's speed and
r
is the radius of the turn.
How can we facilitate things? Enter the tilted turn.
Suppose you tilt the road at a certain angle
\theta
, the angle of banking. In that case, the forces acting on the car will break into components, and by carefully tuning the angle, engineers can design the safest road for a given speed.
We can consider two situations: with and without friction. In the first one, the only forces acting on the vehicle are gravity and centripetal force: there is no friction to help to steer.
🔎 Remember that centrifugal force doesn't exist: it is an apparent force that comes from inertia. The magnitude of this apparent force equals the one of the centripetal force, which, pointing inward, causes the turning.
We will study the tilted turn using the formalism of the inclined plane but from a different perspective. Consider the normal force
F_N
, perpendicular to the plane, and consider its vertical and horizontal components. Usually, we consider the normal force a component itself.
The two components (obtained with the sine and cosine trigonometric functions) are:
The vertical component, which is nothing but the weight of the vehicle;
The horizontal component, which equals the centripetal force.
Let's consider the horizontal one first:
F_{N}\sin{(\theta)}=\frac{m\cdot v^2}{r}
And the vertical:
F_{N}\cos{(\theta)} = m\cdot g
Rearranging them to isolate
F_{N}
results in the equality:
\frac{m\cdot v^2}{r}\cdot\frac{1}{\sin{(\theta)}}=\frac{m\cdot g}{\cos{(\theta)}}
Now we can find the velocity at which we can engage a turn with a specified range and angle of banking.
v=\sqrt{r\cdot g\cdot \tan{(\theta)}}
Notice how the mass cancels out — this often happens in physics. The daily experience suggests that the mass is important, but this is often the result of real-life, more complex phenomena. If you ignore outside forces and assume spherical cows, the mass disappears from the equation, like in a free fall.
🔎 All of the results on this page depend somehow on Newton's second law: find out more with our Newton's second law calculator!
Let's introduce friction in our tilted turn! We will write the formulas first and then take a closer look at them.
We can now see a friction force parallel to the turn. We can decompose that one too in horizontal and vertical components, which will eventually contribute to the total force acting on the vehicle. When friction appears, we can identify a range of speeds at which the vehicle can approach the turn. When driving at the maximum speed, the friction points inward: going above that speed would make the vehicle skid out of the turn. The minimal speed sees the vehicle "falling" toward the center of the turn: the friction points outward.
Let's analyze the maximum speed case first.
The horizontal component, the one contributing to the centripetal force, is:
\footnotesize F_{N}\sin{(\theta)}+\mu\cdot F_{N}\cos{(\theta)}=\frac{m\cdot v^2}{r}
The centripetal force now receives two contributions: as a result, the angle can be smaller while keeping the speed unvaried.
To isolate the speed, we need to consider the vertical component too: in this case, the friction force adds a downward contribution:
\footnotesize F_{N}\cos{(\theta)} =\mu\cdot F_{N}\sin{(\theta)} + m\cdot g
To calculate the minimum speed equations, simply invert the signs of the friction terms. Now rearrange the equations to isolate the speed as we've seen in the frictionless case:
v=\sqrt{\frac{r\cdot g \cdot (\tan{(\theta)}\pm\mu)}{1\mp\mu\tan{(\theta)}}}
Where choosing
+
at the numerator gives the maximum speed, whereas
-
returns the minimum one. Remember to choose the correct sign at the denominator, too!
🙋 Notice how by setting
\mu=0
, we can simplify the equation for
v
to the frictionless case's one.
What if it's not the road to tilt, but the vehicle? That's what happens with planes. The set of equations to calculate the angle of banking for an aircraft is slightly different.
First thing: we need to think in terms of lift, the vertical force that counteracts the weight of a plane. The lift point upwards and equals the aircraft's weight in level flight. In a turn, the lift changes direction, pointing towards the center of the turn, thus creating the component of centripetal force we need.
The sequence of actions needed to steer an aircraft is:
Roll the aircraft of an angle
\theta
, lowering the wing in the direction of the center of the turn;
Pull the yoke.
Congrats, you now know how to steer a plane!
Pulling on the cloche increases the lift, which now outmatches the weight. The excess horizontal component contributes to the centripetal force.
The two components of the lift are:
\begin{split} L_{v}&=m\cdot g\cos{(\theta)}\\ L_{h}&=\frac{m\cdot v^2}{r} \sin{(\theta)} \end{split}
Since an aircraft can't really get thrown off the road — there are no roads in the sky — the airspeed is usually specified in the calculation instead of the radius, leaving the latter as the unknown of the equation. The angle of banking equation is:
r=\frac{v^2}{g\tan{(\theta)}}
To find the angle of banking, you need to reverse the equations used to find either the speed or turning radius.
For a vehicle on a fixed-design road, the equation for the angle of banking is:
\theta = \arctan{\left(\frac{v^2 \mp r \cdot g \cdot \mu}{r\cdot g \pm v^2\cdot \mu}\right)},
\arctan
is the inverse tangent function. For an aircraft, the angle of banking equation is:
\theta = \arctan{\left(\frac{r\cdot g}{v^2}\right)}
A usual value for the tilt of a curve on highways is 7% — if snow is unlikely. This angle corresponds to
\theta=4\degree
. The radius of the turn is
500\ \text{m}
. It didn't rain, so we get a nice friction coefficient
\mu = 0.7
Let's first assume a flat road: what is the maximum speed we can go before falling off the tracks? Set
\theta = 0\degree
\begin{split} v&=\sqrt{500\cdot g \cdot 0.7}\\ &= 58.6\ \text{m}/\text{s} \\ &= 211\ \text{km}/\text{h} \end{split}
Rather high, isn't it? Engineers are known to play safe!
What about the minimum speed? It is undefined since you can't fall from a flat road! Let's check what happens when we tilt the road.
\begin{gather*} \small v=\sqrt{\frac{500\cdot g \cdot (\tan{(4\degree)}+0.7)}{1-0.7\tan{(4\degree)}}} =\\ \small = 63.0\ \frac{\text{m}}{\text{s}} = 226.8\ \frac{\text{km}}{\text{h}} \end{gather*}
Well, that's a noticeable increase! Let's check the minimum speed now: again undefined. That's because of the relatively low bank angle. Take a look at the circuits of NASCAR races: the tilt is noticeable there, and the risk of sliding down is not negligible!
What about planes? Let's say you are flying in an SR71, at the ludicrous speed of
3,951\ \text{km}/\text{s}
— more than three times the speed of sound! If you want to turn remaining inside the territory of Ohio, you have to keep your turning radius at
150\ \text{km}
. Let's find the banking angle!
\small \theta = \arctan{\left(\frac{150,\!000\cdot g}{1097.5^2}\right)} = 50.7\degree
This is a lot! But we guess it's not that hard to go fast with those engines!
Everything you read in the article is in our angle of banking calculator!
First, choose if you are a car or an airplane. Erm... that's if you are even studying a problem involving a car or an airplane. This will give you the right set of equations to work on.
Next choice, the direction: choose if you want to find the speed (or radius) or the angle of banking.
Now you are all set! Insert the values you know in our calculator, and we will do the math for you. 😀
We hope our angle of banking calculator helped you, both in your physics problems and in satisfying your curiosity!
What is the angle of banking?
The angle of banking is the angle at which a road is tilted to guarantee safety for cars driving on it at high speed. This intelligent solution uses simple physics to trade some of the friction force for an increased centripetal force, thus allowing for higher speeds.
To find the angle of banking, you have to know the speed and the radius of the turn, plus the friction coefficient. Considering the maximum allowed speed on a turn, that's how you find the angle of banking:
θ = arctan((v² - r × g × μ) / (r × g + v² × μ)),
v is the speed;
r is the turning radius;
g the gravitational parameter; and
μ the friction coefficient.
What is the advantage of the angle of banking?
Tilting a road at a certain angle allows increasing the maximum speed allowed on that particular turn. The increase in maximum speed may be negligible in normal conditions, but when the weather is adverse, a tilted road can counterbalance the decrease in traction due to the reduced friction between tires and road.
Why do bikers lean during a turn?
Bikers and cyclists lean when making a turn — and they lean more as the turn gets "smaller" — to increase their centripetal force. It is a natural action that allows performing the turn more safely and at higher speeds!
Where are we turning?
Maximum or minimum speed?
Turn radius (r)
Banking angle (θ)
Tire, dry asphalt
AccelerationBelt lengthBrake Mean Effective Pressure (BMEP)… 75 more
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Different Results for Same Beta
Calculation of Beta Using Excel
Low Beta–High Beta
Peering through Yahoo (YHOO) Finance, Google (GOOG) Finance, or other financial data feeders, one may see a variable called beta amid other financial data, such as stock price or market value.
In finance, the beta of a firm refers to the sensitivity of its share price with respect to an index or benchmark. Generally, the index of 1.0 is selected for the market index (usually the S&P 500 index), and if the stock behaved with more volatility than the market, its beta value will be greater than one. If the opposite is the case, its beta will be a value less than one. A company with a beta of greater than one will tend to amplify market movements (for instance the case for the banking sector), and a business with a beta of less than one will tend to ease market movements.
Beta can be seen as a measure of risk: the higher the beta of a company, the higher the expected return should be to compensate for the excess risk caused by volatility.
Therefore, from a portfolio management or investment perspective, one wants to analyze any measures of risk associated with a company to gain a better estimation of its expected return.
Beta is a measure of how sensitive a firm's stock price is to an index or benchmark.
A beta greater than 1 indicates that the firm's stock price is more volatile than the market, and a beta less than 1 indicates that the firm's stock price is less volatile than the market.
A beta may produce different results because of the variations in estimating it, such as different time spans used to calculate data.
Microsoft Excel serves as a tool to quickly organize data and calculate beta.
Low beta stocks are less volatile than high beta stocks and offer more protection during turbulent times.
Different Results for the Same Beta
Incidentally, it is important to differentiate the reasons why the beta value that is provided on Google Finance may be different from the beta on Yahoo Finance or Reuters.
This is because there are several ways to estimate beta. Multiple factors, such as the duration of the period taken into account, are included in the computation of the beta, which creates various results that could portray a different picture. For example, some calculations base their data on a three-year span, while others may use a five-year time horizon. Those two extra years may be the cause of two vastly different results. Therefore, the idea is to select the same beta methodology when comparing different stocks.
It's simple to calculate the beta coefficient over a certain time period. The beta coefficient needs a historical series of share prices for the company that you are analyzing. In our historical example, we will use Apple (AAPL) stock prices from 2012 through 2015 as our object of analysis and the S&P 500 as our historical index. To get this data, go to:
Yahoo! Finance –> Historical prices, and download the time series "Adj Close" for the S&P 500 and the firm Apple.
We only provide a small snippet of the data over 750 rows as it is extensive:
Once we have the Excel table, we can reduce the table data to three columns: the first is the date, the second is the Apple stock, and the third is the price of the S&P 500.
There are then two ways to determine beta. The first is to use the formula for beta, which is calculated as the covariance between the return (ra) of the stock and the return (rb) of the index divided by the variance of the index (over a period of three years).
\begin{aligned} &\beta_a = \frac { \text{Cov} ( r_a, r_b ) }{ \text{Var} ( r_b ) } \\ \end{aligned}
βa=Var(rb)Cov(ra,rb)
To do so, we first add two columns to our spreadsheet; one with the index return r (daily in our case), (column D in Excel), and with the performance of Apple stock (column E in Excel).
At first, we only consider the values of the last three years (about 750 days of trading) and a formula in Excel, to calculate beta.
BETA FORMULA = COVAR (D1: D749; E1: E749) / VAR (E1: E749)
The second method is to perform a linear regression, with the dependent variable performance of Apple stock over the last three years as an explanatory variable and the performance of the index over the same period.
Now that we have the results of our regression, the coefficient of the explanatory variable is our beta (the covariance divided by variance).
With Excel, we can pick a cell and enter the formula: "SLOPE" which represents the linear regression applied between the two variables; the first for the series of daily returns of Apple (here: 750 periods), and the second for the daily performance series of the index, which follows the formula:
BETA FORMULA = SLOPE (E1: E749; D1:D749)
Here, we have just computed a beta value for Apple's stock (0.77 in our example, taking daily data and an estimated period of three years, from April 9, 2012, to April 9, 2015).
Many investors found themselves with heavy losing positions as part of the global financial crisis that began in 2007. As part of those collapses, low beta stocks dove down much less than higher beta stocks during periods of market turbulence. This is because their market correlation was much lower, and thus the swings orchestrated through the index were not felt as acutely for those low beta stocks.
However, there are always exceptions given the industry or sectors of low beta stocks, and so, they might have a low beta with the index but a high beta within their sector or industry.
Therefore, incorporating low beta stocks versus higher beta stocks could serve as a form of downside protection in times of adverse market conditions. Low beta stocks are much less volatile; however, another analysis must be done with intra-industry factors in mind.
On the other hand, higher beta stocks are selected by investors who are keen and focused on short-term market swings. They wish to turn this volatility into profit, albeit with higher risks. Such investors would select stocks with a higher beta, which offer more ups and downs and entry points for trades than stocks with lower beta and lower volatility.
What Does a Stock's Beta Tell You?
The Beta of a stock indicates its relative volatility compared to the broader equity market, as measured by the S&PO 500 (which has a beta of 1.0). A beta greater than one would indicate that the stock will go up more (in percentage terms) than the index when the index goes up, but also fall more than the index when it declines. A beta of less than one would suggest more muted movements relative to the index.
How Is Beta Computed?
Beta is essentially the regression coefficient of a stock's historical returns compared to those of the S&P 500 index. This coefficient represents the slope of a line of best fit correlating the stock's returns against the index's. Because regression coefficients are called "betas" (β) in statistics, the terminology was carried over to investing.
How Is Beta Used in Practice?
Beta is used to gauge the relative riskiness of a stock. As an example, consider the hypothetical firm US CORP (USCS). Financial websites provide a current beta for this company at 5.48, which means that with respect to the historical variations of the stock compared to the Standard & Poor's 500, US CORP increased on average by 5.48% if the S&P 500 rose by 1%. Conversely, when the S&P 500 is down 1%, US CORP Stock would tend to average a decline of 5.48%. If the index rose by 0,2%, USGC rose, on average, by 1.1%. As a result, one may conclude that USGC is a fairly risky investment.
It is important to follow strict trading strategies and rules and apply a long-term money management discipline in all beta cases. Employing beta strategies can be useful as part of a broader investment plan to limit downside risk or realize short-term gains, but it's important to remember that it is also subject to the same levels of market volatility as any other trading strategy.
Calculating Beta w/Excel: Portfolio Math For The Average Investor
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Recurrent Neural Network | Brilliant Math & Science Wiki
John McGonagle, Christopher Williams, and Jimin Khim contributed
A simple recurrent neural network
Recurrent neural networks are artificial neural networks where the computation graph contains directed cycles. Unlike feedforward neural networks, where information flows strictly in one direction from layer to layer, in recurrent neural networks (RNNs), information travels in loops from layer to layer so that the state of the model is influenced by its previous states. While feedforward neural networks can be thought of as stateless, RNNs have a memory which allows the model to store information about its past computations. This allows recurrent neural networks to exhibit dynamic temporal behavior and model sequences of input-output pairs.
Because they can model temporal sequences of input-output pairs, recurrent neural networks have found enormous success in natural language processing (NLP) applications. This includes machine translation, speech recognition, and language modeling. RNNs have also been used in reinforcement learning to solve very difficult problems at a level better than humans. A recent example is AlphaGo, which beat world champion Go player Lee Sedol in 2016. An interactive example of an RNN for generating handwriting samples can be found here.
Problems with Modeling Sequences
Unrolling RNNs
Vanishing/Exploding Gradients Problem
Consider an application that needs to predict an output sequence
y = \left(y_1, y_2, \dots, y_n\right)
for a given input sequence
x = \left(x_1, x_2, \dots, x_m\right)
. For example, in an application for translating English to Spanish, the input
x
might be the English sentence
\text{"i like pizza"}
and the associated output sequence
y
would be the Spanish sentence
\text{"me gusta comer pizza"}
. Thus, if the sequence was broken up by character, then
x_1=\text{"i"}
x_2=\text{" "}
x_3=\text{"l"}
x_4=\text{"i"}
x_5=\text{"k"}
, all the way up to
x_{12}=\text{"a"}
y_1=\text{"m"}
y_2=\text{"e"}
y_3=\text{" "}
y_4=\text{"g"}
y_{20}=\text{"a"}
. Obviously, other input-output pair sentences are possible, such as
(\text{"it is hot today"}, \text{"hoy hace calor"})
(\text{"my dog is hungry"}, \text{"mi perro tiene hambre"})
It might be tempting to try to solve this problem using feedforward neural networks, but two problems become apparent upon investigation. The first issue is that the sizes of an input
x
and an output
y
are different for different input-output pairs. In the example above, the input-output pair
(\text{"it is hot today"}, \text{"hoy hace calor"})
has an input of length
15
and an output of length
14
while the input-output pair
(\text{"my dog is hungry"}, \text{"mi perro tiene hambre"})
16
21
. Feedforward neural networks have fixed-size inputs and outputs, and thus cannot be automatically applied to temporal sequences of arbitrary length.
The second issue is a bit more subtle. One can imagine trying to circumvent the above issue by specifying a max input-output size, and then padding inputs and outputs that are shorter than this maximum size with some special null character. Then, a feedforward neural network could be trained that learns to produce
y_i
x_i
. Thus, in the example
(\text{"it is hot today"}, \text{"hoy hace calor"})
, the training pairs would be
\big\{(x_1=\text{"i"}, y_1=\text{"h"}), (x_2=\text{"t"}, y_2=\text{"o"}), \dots, (x_{14}=\text{"a"}, y_{14}=\text{"r"}), (x_{15}=\text{"y"}, x_{15}=\text{"*"})\big\},
where the maximum size is
15
and the padding character is
\text{"*"}
, used to pad the output, which at length
14
is one short of the maximum length
15
The problem with this is that there is no reason to believe that
x_1
has anything to do with
y_1
. In many Spanish sentences, the order of the words (and thus characters) in the English translation is different. Thus, if the first word in an English sentence is the last word in the Spanish translation, it stands to reason that any network that hopes to perform the translation will need to remember that first word (or some representation of it) until it outputs the end of the Spanish sentence. Any neural network that computes sequences needs a way to remember past inputs and computations, since they might be needed for computing later parts of the sequence output. One might say that the neural network needs a way to remember its context, i.e. the relation between its past and its present.
Both of the issues outlined in the above section can be solved by using recurrent neural networks. Recurrent neural networks, like feedforward layers, have hidden layers. However, unlike feedforward neural networks, hidden layers have connections back to themselves, allowing the states of the hidden layers at one time instant to be used as input to the hidden layers at the next time instant. This provides the aforementioned memory, which, if properly trained, allows hidden states to capture information about the temporal relation between input sequences and output sequences.
RNNs are called recurrent because they perform the same computation (determined by the weights, biases, and activation functions) for every element in the input sequence. The difference between the outputs for different elements of the input sequence comes from the different hidden states, which are dependent on the current element in the input sequence and the value of the hidden states at the last time step.
In simplest terms, the following equations define how an RNN evolves over time:
\begin{aligned} o^t &= f\big(h^t; \theta\big)\\ h^t &= g\big(h^{t-1}, x^t; \theta\big), \end{aligned}
o^t
is the output of the RNN at time
t,
x^t
is the input to the RNN at time
t,
h^t
is the state of the hidden layer(s) at time
t.
The image below outlines a simple graphical model to illustrate the relation between these three variables in an RNN's computation graph.
A graphical model for an RNN. The values
\theta_i
\theta_h
\theta_o
represent the parameters associated with the inputs, previous hidden layer states, and outputs, respectively.
The first equation says that, given parameters
\theta
(which encapsulates the weights and biases for the network), the output at time
t
depends only on the state of the hidden layer at time
t
, much like a feedforward neural network. The second equation says that, given the same parameters
\theta
, the hidden layer at time
t
depends on the hidden layer at time
t-1
and the input at time
t
. This second equation demonstrates that the RNN can remember its past by allowing past computations
h^{t-1}
to influence the present computations
h^{t}
Thus, the goal of training the RNN is to get the sequence
o^{t+\tau}
to match the sequence
y_t
\tau
represents the time lag
(
it's possible that
\tau=0)
between the first meaningful RNN output
o^{\tau + 1}
and the first target output
y_t
. A time lag is sometimes introduced to allow the RNN to reach an informative hidden state
h^{\tau + 1}
before it starts producing elements of the output sequence. This is analogous to how humans translate English to Spanish, which often starts by reading the first few words in order to provide context for translating the rest of the sentence. A simple case when this is actually required is when the last word in the input sequence corresponds to the first word in the output sequence. Then, it would be necessary to delay the output sequence until the entire input sequence is read.
RNNs can be difficult to understand because of the cyclic connections between layers. A common visualization method for RNNs is known as unrolling or unfolding. An RNN is unrolled by expanding its computation graph over time, effectively "removing" the cyclic connections. This is done by capturing the state of the entire RNN (called a slice) at each time instant
and treating it similar to how layers are treated in feedforward neural networks. This turns the computation graph into a directed acyclic graph, with information flowing in one direction only. The catch is that, unlike a feedforward neural network, which has a fixed number of layers, an unfolded RNN has a size that is dependent on the size of its input sequence and output sequence. This means that RNNs designed for very long sequences produce very long unrollings. The image below illustrates unrolling for the RNN model outlined in the image above at times
t-1
t
t+1
An unfolded RNN at time steps
t-1
t
t+1
One thing to keep in mind is that, unlike a feedforward neural network's layers, each of which has its own unique parameters (weights and biases), the slices in an unrolled RNN all have the same parameters
\theta_i
\theta_h
\theta_o
. This is because RNNs are recurrent, and thus the computation is the same for different elements of the input sequence. As mentioned earlier, the differences in the output sequence arise from the context preserved by the previous, hidden layer state
h^{t-1}
Furthermore, while each slice in the unrolling may appear to be similar to a layer in the computation graph of a feedforward graph, in practice the variable
h^t
in an RNN can have many internal hidden layers. This allows the RNN to learn more hierarchal features since a hidden layer's feature outputs can be another hidden layer's inputs. Thus, each variable
h^t
in the unrolling is more akin to the entirety of hidden layers in a feedforward neural network. This allows RNNs to learn complex "static" relationships between the input and output sequences in addition to the temporal relationship captured by cyclic connections.
Training recurrent neural networks is very similar to training feedforward neural networks. In fact, there is a variant of the backpropagation algorithm for feedforward neural networks that works for RNNs, called backpropagation through time (often denoted BPTT). As the name suggests, this is simply the backpropagation algorithm applied to the RNN backwards through time.
Backpropagation through time works by applying the backpropagation algorithm to the unrolled RNN. Since the unrolled RNN is akin to a feedforward neural network with all elements
o_t
as the output layer and all elements
x_t
from the input sequence
x
as the input layer, the entire input sequence
x
and output sequence
o
are needed at the time of training.
BPTT starts similarly to backpropagation, calculating the forward phase first to determine the values of
o_t
and then backpropagating (backwards in time) from
o_t
o_1
to determine the gradients of some error function with respect to the parameters
\theta
. Since the parameters are replicated across slices in the unrolling, gradients are calculated for each parameter at each time slice
t
. The final gradients output by BPTT are calculated by taking the average of the individual, slice-dependent gradients. This ensures that the effects of the gradient update on the outputs for each time slice are roughly balanced.
One issue with RNNs in general is known as the vanishing/exploding gradients problem. This problem states that, for long input-output sequences, RNNs have trouble modeling long-term dependencies, that is, the relationship between elements in the sequence that are separated by large periods of time.
For example, in the sentence
\text{"The quick brown fox jumped over the lazy dog"}
\text{"fox"}
\text{"dog"}
are separated by a large amount of space in the sequence. In the unrolling of an RNN for this sequence, this would be modeled by a large difference
\Delta t
x_a
for the start of the word
\text{"fox"}
x_a + \Delta t
for the end of the word
\text{"dog"}
. Thus, if an RNN was attempting to learn how to identify subjects and objects in sentences, it would need to remember the word
\text{"fox"}
(or some hidden state representing it), the subject, up until it reads the word
\text{"dog"}
, the object. Only then would the RNN be able to output the pair
(\text{"fox"}, \text{"dog"})
, having finally identified both a subject and an object.
This problem arises due to the use of the chain rule in the backpropagation algorithm. The actual proof is a bit messy, but the idea is that, because the unrolled RNN for long sequences is so deep and the chain rule for backpropagation involves the products of partial derivatives, the gradient at early time slices is the product of many partial derivatives. In fact, the number of factors in the product for early slices is proportional to the length of the input-output sequence. This is a problem because, unless the partial derivatives are all close in value to
1
, their product will either become very small, i.e. vanishing, when the partial derivatives are
\lt 1
, or very large, i.e. exploding, when the partial derivatives are
\gt 1
. This causes learning to become either very slow (in the vanishing case) or wildly unstable (in the exploding case).
Luckily, recent RNN variants such as LSTM (Long Short-Term Memory) have been able to overcome the vanishing/exploding gradient problem, so RNNs can safely be applied to extremely long sequences, even ones that contain millions of elements. In fact, LSTMs addressing the gradient problem have been largely responsible for the recent successes in very deep NLP applications such as speech recognition, language modeling, and machine translation.
LSTM RNNs work by allowing the input
x_t
to influence the storing or overwriting of "memories" stored in something called the cell. This decision is determined by two different functions, called the input gate for storing new memories, and the forget gate for forgetting old memories. A final output gate determines when to output the value stored in the memory cell to the hidden layer. These gates are all controlled by the current values of the input
x_t
and cell
c_t
t
, plus some gate-specific parameters. The image below illustrates the computation graph for the memory portion of an LSTM RNN (i.e. it does not include the hidden layer or output layer).
Computation graph for an LSTM RNN, with the cell denoted by
c_t
. Note that, in this illustration,
o_t
is not the output of the RNN, but the output of the cell to the hidden layer
h_t
While the general RNN formulation can theoretically learn the same functions as an LSTM RNN, by constraining the form that memories can take and how they are modified, LSTM RNNs can learn long-term dependencies quickly and stably, and thus are much more useful in practice.
, B. Long_Short_Term_Memory. Retrieved October 4, 2015, from https://commons.wikimedia.org/wiki/File:Long_Short_Term_Memory.png
Cite as: Recurrent Neural Network. Brilliant.org. Retrieved from https://brilliant.org/wiki/recurrent-neural-network/
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How the density of gas differ from other states of matter
How to find the density of gas with the ideal gas law
How our gas density calculator works
List of related calculators
Our gas density calculator helps you to calculate the density of gas at a definite pressure and temperature.
What density is;
How the density of gas differs from liquid and solid;
Whether natural gas is heavier than air; and
Whether air is a liquid or not.
Density refers to how tightly packed a substance is within a particular space.
The higher the density, the more tightly packed the substance is. Solid and liquid typically have a higher density than gas.
The density of gases is affected by temperature and pressure.
Density is represented by the Greek letter
\rho
Unlike the density of solids or liquids, the density of gases is changeable. This is because gas is not compact, and its molecules are affected by temperature and pressure.
Because of this characteristic, gas density is calculated differently from liquids and solids. Two ways that the density of gas differs drastically from solids are:
When the temperature of gas increases, the molecules move further apart. This causes a decrease in density.
When we apply pressure to gas, its molecules draw closer together. This increases the density.
You typically need to know the mass and volume to find the density. With gas, as the volume goes up, the density goes down and vice versa.
To calculate the density of gas, we use the ideal gas law:
\footnotesize PV = nRT
P
represents pressure.
V
represents volume.
n
represents the number of moles of gas.
R
is the universal gas constant. It is equal to
8.314462618\ \text{J}\cdot\text{mol}^{-1}\cdot\text{K}^{-1}
T
is the absolute temperature in Kelvin (
\text{K}
To calculate the volume of the gas, we need to make volume the subject of the formula. So the gas law rewritten to find the volume is:
\footnotesize V = nRT/P
To find the mass of the gas, you use the number of moles of the gas divided by the molecular mass (
M
). So, this means that:
\footnotesize n = m/M
Now that we have established that
n = m/M
, we can replace
n
with the mass value in the equation we use for volume. So we now have:
\footnotesize V = mRT/MP
Since density is mass per volume, we need to divide both sides by mass to get to the formula for density. So our new equation is now:
\footnotesize V/m = RT/MP
To get the mass per volume, we need to invert this equation. With this, we get:
\footnotesize \begin{split} m/V &= MP/RT \\ \therefore \rho &= MP/RT \end{split}
🙋 You can also calculate density without knowing the molar mass. In our ideal gas density calculator we calculate the density using pressure, temperature, and a specific gas constant.
Our gas density calculator employs this formula:
\rho = MP/RT
to find the density of gas. It takes in the pressure, temperature, and molar mass of gas and calculates the density. Because
R
is a constant, we do not require you to enter this value.
When you enter these values, the molecular mass is divided by
1,\!000
(to convert it to
\text{kg}/\text{mol}
, the actual unit for the molecular mass of gas) and then multiplied by pressure. This is further divided by the universal gas constant times the temperature.
So if the given molecular mass is
28\ \text{g}/\text{mol}
, the temperature is
50\ \text{K}
, and the pressure is
10\ \text{Pa}
. We will substitute these values into the density equation to calculate the density. So now we will have:
\footnotesize \begin{split} \rho &= \frac{(28\ \text{g}/\text{mol})/ 1,\!000 \times 10\ \text{Pa}}{8.3145 \times 50\ \text{K}} \end{split}
If you are interested in other similar calculators you should check out this list:
Density of a cylinder calculator;
Sphere density calculator; and
Liquid ethylene density calculator.
Is natural gas heavier than air?
No, natural gas is not heavier than air.
The molecular weight of natural gas ranges from 16 to 18 g/mol, while the air we breathe weighs about 29 g/mol.
No, air is not a liquid, it is a gas. In fact, the air we breathe is made up of several gasses. These are:
Check this centripetal force calculator if you want to estimate the centripetal force acting upon a body in circular motion.
Use Omni's inductors in series calculator to work out the equivalent inductance of a series circuit.
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What is the Collatz conjecture?
Definition of the Collatz conjecture
Behavior of Collatz sequences
How does our Collatz conjecture calculator work?
Simple things are sometimes surprising: explore a problem that makes mathematicians say, "I'm not going to do that" with our Collatz conjecture calculator!
What the Collatz conjecture is;
Why it is driving mathematicians crazy;
How to calculate the Collatz (or hailstone) sequences; and
How to use our Collatz conjecture calculator.
Ready to learn more about this mathematical mystery? Just be careful — trying to break the conjecture is dangerous. Remember to stop after a while!
A "conjecture" is mathematics lingo for something we're pretty sure is true, but we can't find a way to prove it. Quite frustrating, probably! The Collatz sequence is formed by starting at a given integer number and continually:
Dividing the previous number by 2 if it's even; or
Multiplying the previous number by 3 and adding 1 if it's odd.
The Collatz conjecture states that this sequence eventually reaches the value 1.
It is wonderfully simple, and yet every initial number ever tried returned 1 sooner or later — but no one has been able to prove it in almost a century!
The rules of the Collatz conjecture are, formally speaking:
x_{n+1} = \begin{cases} \frac{1}{2}x_{n} & \textrm{if}\ x_{n}\textrm{ mod }2=0 \\ 3x_{n}+1 & \textrm{if}\ x_{n}\textrm{ mod } 2=1 \end{cases}
We use the modulus operation to distinguish odd and even numbers. The second equation explains why the Collatz conjecture is sometimes also called the 3x+1 problem.
Starting with any number, how will the sequence behave? Will it go up to infinity, or down to zero? Will it loop forever?
Choose an initial number first. Let's say 11. The sequence goes up and down:
11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1
See the end? Once it reaches 4, it collapses into the loop
4, 2, 1, 4, 2, 1
which it will never escape!
Feeding 11 to the Collatz rules return this nice hailstone sequence!
Choose any other number, and eventually, the sequence will end up in the
4, 2, 1
loop. The oscillations may be big, as for
x_1 = 31
which reaches a peak of more than 9,000 before falling down to 1.
This wildly unpredictably oscillating behavior earned these sequences the name "hailstone sequences", because the path of the sequence resembles that of a hailstone in a cloud before reaching the ground: swinging up and down before falling (preferably not on our heads).
The sequences follow a random pattern, and by just looking at the initial term, it's impossible to say how its sequence will behave without computing the next steps: mathematicians say that it is an undecidable problem. In layman's terms, there's no computer program that can take a number and say if it will or will not reach one.
This is why the 3x+1 problem is such a problem for mathematicians: at the moment, the only thing we can do is to brute force our way through numbers, trying to find one of them that would escape the
4, 2, 1
loop. And things don't look promising: researchers have tried numbers up to (are you ready?)
295,147,905,179,352,825,856
— more seconds than have passed since the Big Bang! And not a single number didn't end up at 1.
And in an attempt to show the Collatz conjecture who's boss, researchers tried the number
2^{100,000}- 1
(the minus one was used to give the final flex, apparently). That number is 30,000 digits long, and guess what? After almost a million and a half steps, it ended up at 1.
XKCD's Cueball was caught by the Collatz conjecture, and his social life suffered because of it.
The search for a counterexample (one that doesn't end up in the
4, 2, 1
loop) is still undergoing, but many mathematicians think that this problem is out of the reach of our knowledge at this time.
The Collatz conjecture and its hailstone sequences are good examples of chaotic behaviour. If you need to balance it with some order, try out or Fibonacci sequence calculator or golden ratio calculator!
All you have to do, is choose a number! Our Collatz conjecture calculator will show you:
A chart of the sequence;
The stopping time (the number of steps before reaching 1 for the first time); and
A table with all terms of the sequence.
Did you try a negative number to see if you could break the tool? We covered that too, but mathematicians are even more worried about it: there are three loops (starting at
-1
-5
-17
) for negative integers, and no-one knows why they exist!
If you move to the advanced mode of the Collatz conjecture calculator, you can modify the parameters of the conjecture, but we don't guarantee that something interesting will come out of it. Try anyway — maybe the next mathematical nightmare will be named after you!
Now that you know what the Collatz conjecture is, you can also understand that it is unlikely you are going to use it in real life. There are more useful (but maybe less thought-provoking) sequences in math: we have some calculators for some of the more "conventional" of them:
Arithmetic sequence calculator;
Geometric sequence calculator.
What is the Collatz's conjecture?
The Collatz's conjecture is an open problem in mathematics which asks if there are numbers that, given a simple set of rules, don't fall to 1 at the end of the sequence that is obtained by applying these rules.
Even if tested for amazingly big numbers, the sequences always reach 1: mathematicians still lack the tools to explain this, if it even can be explained!
Why a result of the Collatz's conjecture is called "hailstone sequence"?
Feeding a number to the rules of the Collatz's conjecture may result in the sequence oscillating wildly before finally reaching 1. Imagining 1 to be the ground, that "motion" resembles the path of a hailstone in a cloud during the process of growing, before the final fall to Earth.
Is there a solution to the Collatz's conjecture?
No, the Collatz's conjecture doesn't have a solution — yet! The best that mathematicians can do, is either to investigate the behavior of a sequence for increasingly big numbers, or to find an upper boundary under which all of the numbers collapse to 1.
How do I calculate Collatz's sequences?
The rules of the Collatz's sequence depend on the parity of the number itself. If the number is even, then the rule returns half the original number. If the number is odd, then we multiply the number by three, and add one.
How do I compute Collatz's sequence for 6?
The Collatz's sequence starting with 6 proceeds with 3 (half the previous number, since 6 is even), then 10 (3×3 + 1). It continues with 5, then 16. 16 is a power of two, and so it collapses with 8, 4, 2 and finally 1.
Here is the sequence: 6, 3, 10, 5, 16, 8, 4, 2, 1.
Collatz sequence rules:
xn+1 = 0.5xn if xn is even
xn+1 = 3xn + 1 if xn is odd
To modify the rules, check out the advanced mode below.
Starting number (x₁)
Collatz conjecture: the sequence will eventually hit 1.
And it does hit 1 after 7 steps! This is its stopping time.
Show the table of sequence values?
Arithmetic sequenceFibonacciGeometric sequence… 3 more
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Bus has become an integral traffic mode for urban residents along with the development of public transport system; and as a kind of public service, urban public transport is the basic infrastructure closely related to people, and is also required to constantly improve its service levels to better serve people. Therefore, how to evaluate its service level has become one of the important projects that need to be studied. Based on establishing the mathematical model of fuzzy comprehensive evaluation and illustrated by the case of Xi’an, this paper verifies the model rationality, assesses the bus development level in Xi’an, and further puts forward countermeasures for its bus services according to the results.
Public Transport, Passenger Satisfaction, Fuzzy Comprehensive Evaluation
Yin, D. (2018) Research on Fuzzy Comprehensive Evaluation of Passenger Satisfaction in Urban Public Transport. Modern Economy, 9, 528-535. doi: 10.4236/me.2018.93034.
U=\left\{{u}_{1},{u}_{2},\cdots ,{u}_{m}\right\}
{u}_{i}\left(i=1,2,\cdots \right)
{\alpha }_{i}\left(i=1,2,\cdots m\right)
A=\left\{{\alpha }_{1},{\alpha }_{2},\cdots ,{\alpha }_{m}\right\}
\underset{i=1}{\overset{m}{\sum }}{\alpha }_{i}=1,{\alpha }_{i}\ge 0\left(i=1,2,\cdots ,m\right)
A=\frac{{\alpha }_{1}}{{u}_{1}}+\frac{{\alpha }_{2}}{{u}_{2}}+\cdots +\frac{{\alpha }_{m}}{{u}_{m}}
V=\left\{{V}_{1},{V}_{2},\cdots ,{V}_{n}\right\}
{R}_{i}=\frac{{\gamma }_{i1}}{{v}_{1}}+\frac{{\gamma }_{i2}}{{v}_{2}}+\cdots +\frac{{\gamma }_{in}}{{v}_{n}}\left(i=1,2,\cdots ,m\right)
R=\left[\begin{array}{cccc}{\gamma }_{11}& {\gamma }_{12}& \cdots & {\gamma }_{1n}\\ {\gamma }_{21}& {\gamma }_{22}& \cdots & {\gamma }_{2n}\\ \cdots & \cdots & \cdots & \cdots \\ {\gamma }_{m1}& {\gamma }_{m2}& \cdots & {\gamma }_{mn}\end{array}\right]
B=A*R=\left({\alpha }_{1},{\alpha }_{2},\cdots ,{\alpha }_{m}\right)\left[\begin{array}{cccc}{\gamma }_{11}& {\gamma }_{12}& \cdots & {\gamma }_{1n}\\ {\gamma }_{21}& {\gamma }_{22}& \cdots & {\gamma }_{2n}\\ \cdots & \cdots & \cdots & \cdots \\ {\gamma }_{m1}& {\gamma }_{m2}& \cdots & {\gamma }_{mn}\end{array}\right]
\begin{array}{l}{R}_{1}=\frac{0.4}{{V}_{1}}+\frac{0.3}{{V}_{2}}+\frac{0.21}{{V}_{3}}+\frac{0.05}{{V}_{4}}+\frac{0.04}{{V}_{5}}\\ {R}_{2}=\frac{0.24}{{V}_{1}}+\frac{0.17}{{V}_{2}}+\frac{0.27}{{V}_{3}}+\frac{0.21}{{V}_{4}}+\frac{0.11}{{V}_{5}}\\ {R}_{3}=\frac{0.17}{{V}_{1}}+\frac{0.3}{{V}_{2}}+\frac{0.31}{{V}_{3}}+\frac{0.14}{{V}_{4}}+\frac{0.08}{{V}_{5}}\\ {R}_{4}=\frac{0.04}{{V}_{1}}+\frac{0.12}{{V}_{2}}+\frac{0.32}{{V}_{3}}+\frac{0.42}{{V}_{4}}+\frac{0.10}{{V}_{5}}\\ {R}_{5}=\frac{0.05}{{V}_{1}}+\frac{0.25}{{V}_{2}}+\frac{0.48}{{V}_{3}}+\frac{0.17}{{V}_{4}}+\frac{0.05}{{V}_{5}}\\ {R}_{6}=\frac{0.09}{{V}_{1}}+\frac{0.31}{{V}_{2}}+\frac{0.39}{{V}_{3}}+\frac{0.17}{{V}_{4}}+\frac{0.04}{{V}_{5}}\end{array}
R=\left[\begin{array}{ccccc}0.4& 0.3& 0.21& 0.05& 0.04\\ 0.24& 0.17& 0.27& 0.21& 0.11\\ 0.17& 0.3& 0.31& 0.14& 0.08\\ 0.04& 0.12& 0.32& 0.42& 0.10\\ 0.05& 0.25& 0.48& 0.17& 0.05\\ 0.09& 0.31& 0.39& 0.17& 0.04\end{array}\right]
\begin{array}{c}B=A*R=\left(0.03,0.15,0.04,0.15,0.44,0.09\right)\left[\begin{array}{ccccc}0.4& 0.3& 0.21& 0.05& 0.04\\ 0.24& 0.17& 0.27& 0.21& 0.11\\ 0.17& 0.3& 0.31& 0.14& 0.08\\ 0.04& 0.12& 0.32& 0.42& 0.10\\ 0.05& 0.25& 0.48& 0.17& 0.05\\ 0.09& 0.31& 0.39& 0.17& 0.04\end{array}\right]\\ =\left(0.19,0.2,0.35,0.19,0.07\right)\end{array}
{B}_{1}*N=\left[\begin{array}{ccccc}0.4& 0.3& 0.21& 0.05& 0.04\end{array}\right]\left[\begin{array}{c}95\\ 85\\ 75\\ 65\\ 55\end{array}\right]=84.7
{B}_{2}*N=\left[\begin{array}{ccccc}0.24& 0.17& 0.27& 0.21& 0.11\end{array}\right]\left[\begin{array}{c}95\\ 85\\ 75\\ 65\\ 55\end{array}\right]=77.2
{B}_{3}*N=\left[\begin{array}{ccccc}0.17& 0.3& 0.31& 0.14& 0.08\end{array}\right]\left[\begin{array}{c}95\\ 85\\ 75\\ 65\\ 55\end{array}\right]=78.4
{B}_{4}*N=\left[\begin{array}{ccccc}0.04& 0.12& 0.32& 0.42& 0.10\end{array}\right]\left[\begin{array}{c}95\\ 85\\ 75\\ 65\\ 55\end{array}\right]=70.8
{B}_{5}*N=\left[\begin{array}{ccccc}0.05& 0.25& 0.48& 0.17& 0.05\end{array}\right]\left[\begin{array}{c}95\\ 85\\ 75\\ 65\\ 55\end{array}\right]=75.8
{B}_{6}*N=\left[\begin{array}{ccccc}0.09& 0.31& 0.39& 0.17& 0.04\end{array}\right]\left[\begin{array}{c}95\\ 85\\ 75\\ 65\\ 55\end{array}\right]=77.4
B*N=\left[\begin{array}{ccccc}0.19& 0.2& 0.35& 0.19& 0.07\end{array}\right]\left[\begin{array}{c}95\\ 85\\ 75\\ 65\\ 55\end{array}\right]=77.5
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Average Price Call Defined
An average price call is a call option whose profit is determined by comparing the strike price to the average price of the asset that occurred during the option's term. Therefore, for a three-month average price call, the holder of the option would receive a positive payout if the average closing price for the underlying asset traded above the strike price during the three-month term of the option.
By contrast, the profit for a traditional call option would be calculated by comparing the strike price to the price occurring on the specific day when the option is exercised, or at the contract's expiration if it remains unexercised.
Average price call options are also known as Asian options and are considered a type of exotic option.
Average price calls are a modification of a traditional call option where the payoff depends on the average price of the underlying asset over a certain period.
This is opposed to standard call options whose payoff depends on the price of the underlying asset at a specific point in time - at exercise or expiry.
Also known as Asian options, average price options are used when hedgers or speculators are interested in smoothing the effects of volatility and not rely on a single point of time for valuation.
Understanding Average Price Calls
Average price call options are part of a broader category of derivative instruments known as average price options (APOs), which are sometimes also referred to as average rate options (AROs). They are mostly traded OTC, but some exchanges, such as the Intercontinental Exchange (ICE), also trade them as listed contracts. These kinds of exchange-listed APOs are cash-settled and can only be exercised on the expiration date, which is the last trading day of the month.
Some investors prefer average price calls to traditional call options because they reduce the option's volatility. Because volatility increases the likelihood that an option holder will be able to exercise the option during its term, this means that average price call options are generally less expensive than their traditional counterparts.
The complement of an average price call is an average price put, in which the payoff is positive if the average price of the underlying asset is less than the strike price during the option's term.
Real World Example of an Average Price Call
To illustrate, suppose you believe that interest rates are set to decline and therefore wish to hedge your exposure to Treasury bills (T-bills). Specifically, you wish to hedge $1 million worth of interest rate exposure for a period of one month.
You begin considering your options and observe that T-bill futures are currently trading in the market at $145.09. To hedge your interest rate exposure, you purchase an average price call option whose underlying asset is T-bill futures, in which the notional value is $1 million, the strike price is $145.00, and the expiration date is one month in the future. You pay for the option with a $45,500 premium.
One month later, the option is about to expire and the average price of the T-bills futures over the previous month has been $146.00. Realizing that your option is in the money, you exercise your call option, buying for $145.00 and selling at the average price of $146.00. Because the average price call option had a notional value of $1 million, your profit is $954,500, calculated as follows:
\begin{aligned}&\text{Profit}\ = \ (\text{Average Price}\ - \ \text{Strike Price})\\&\qquad\qquad \times\ \text{Notional Value}\ - \ \text{Option Premium Paid}\\&\text{Profit}\ = \ (\$146.00\ - \ \$145.00)\\&\qquad\qquad \times\ \$1,000,000\ - \ \$45,500\\&\text{Profit}\ =\ \$954,500\end{aligned}
Profit = (Average Price − Strike Price)× Notional Value − Option Premium PaidProfit = ($146.00 − $145.00)× $1,000,000 − $45,500
Alternatively, if the average price of T-bills over this period had been $144.20 instead of $146.00, then the option would have expired worthless. In that scenario, you would have experienced a loss equal to the option premium, or $45,500.
An average strike option is an option where the payoff depends on the average price of the underlying asset instead of a single price at expiration.
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CORDIC-based four quadrant inverse tangent - MATLAB cordicatan2 - MathWorks Switzerland
CORDIC-based four quadrant inverse tangent
theta = cordicatan2(y,x)
theta = cordicatan2(y,x,niters)
theta = cordicatan2(y,x) computes the four quadrant arctangent of y and x using a CORDIC algorithm approximation.
theta = cordicatan2(y,x,niters) performs niters iterations of the algorithm.
y,x are Cartesian coordinates. y and x must be the same size. If they are not the same size, at least one value must be a scalar value. Both y and x must have the same data type.
niters is the number of iterations the CORDIC algorithm performs. This is an optional argument. When specified, niters must be a positive, integer-valued scalar. If you do not specify niters or if you specify a value that is too large, the algorithm uses a maximum value. For fixed-point operation, the maximum number of iterations is one less than the word length of y or x. For floating-point operation, the maximum value is 52 for double or 23 for single. Increasing the number of iterations can produce more accurate results but also increases the expense of the computation and adds latency.
theta is the arctangent value, which is in the range [-pi, pi] radians. If y and x are floating-point numbers, then theta has the same data type as y and x. Otherwise, theta is a fixed-point data type with the same word length as y and x and with a best-precision fraction length for the [-pi, pi] range.
Floating-point CORDIC arctangent calculation.
theta_cdat2_float = cordicatan2(0.5,-0.5)
theta_cdat2_float =
Fixed- point CORDIC arctangent calculation.
theta_cdat2_fixpt = cordicatan2(fi(0.5,1,16,15),fi(-0.5,1,16,15));
theta_cdat2_fixpt =
\begin{array}{l}{x}_{0}\text{ is initialized to the }x\text{ input value}\\ {y}_{0}\text{ is initialized to the }y\text{ input value}\\ {z}_{0}\text{ is initialized to }0\end{array}
atan2 | atan2 | cordicsin | cordiccos
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Anharmonicity - Wikipedia @ WordDisk
Period of oscillations
In classical mechanics, anharmonicity is the deviation of a system from being a harmonic oscillator. An oscillator that is not oscillating in harmonic motion is known as an anharmonic oscillator where the system can be approximated to a harmonic oscillator and the anharmonicity can be calculated using perturbation theory. If the anharmonicity is large, then other numerical techniques have to be used. In reality all oscillating systems are anharmonic, but most approximate the harmonic oscillator the smaller the amplitude of the oscillation is.
Deviation of a physical system from being a harmonic oscillator
This article is about anharmonic oscillators. For the anharmonic ratio, see Cross-ratio.
Not to be confused with Enharmonicity or Inharmonicity.
Potential energy of a diatomic molecule as a function of atomic spacing. When the molecules are too close or too far away, they experience a restoring force back towards u0. (Imagine a marble rolling back and forth in the depression.) The blue curve is close in shape to the molecule's actual potential well, while the red parabola is a good approximation for small oscillations. The red approximation treats the molecule as a harmonic oscillator, because the restoring force, -V'(u), is linear with respect to the displacement u.
As a result, oscillations with frequencies
{\displaystyle 2\omega }
{\displaystyle 3\omega }
{\displaystyle \omega }
is the fundamental frequency of the oscillator, appear. Furthermore, the frequency
{\displaystyle \omega }
deviates from the frequency
{\displaystyle \omega _{0}}
of the harmonic oscillations. See also intermodulation and combination tones. As a first approximation, the frequency shift
{\displaystyle \Delta \omega =\omega -\omega _{0}}
is proportional to the square of the oscillation amplitude
{\displaystyle A}
{\displaystyle \Delta \omega \propto A^{2}}
In a system of oscillators with natural frequencies
{\displaystyle \omega _{\alpha }}
{\displaystyle \omega _{\beta }}
, ... anharmonicity results in additional oscillations with frequencies
{\displaystyle \omega _{\alpha }\pm \omega _{\beta }}
Anharmonicity also modifies the energy profile of the resonance curve, leading to interesting phenomena such as the foldover effect and superharmonic resonance.
This article uses material from the Wikipedia article Anharmonicity, and is written by contributors. Text is available under a CC BY-SA 4.0 International License; additional terms may apply. Images, videos and audio are available under their respective licenses.
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Created by AbdulRafay Moeen
How to use the voltage regulation calculator
Calculate voltage regulation of linear and switching regulators
Linear and switching voltage regulator diagrams
What is power dissipation in a voltage regulator?
Using our voltage regulation calculator, you can find the voltage ratio, i.e., the voltage difference between no-load and full-load, and the percentage change of that voltage ratio on the secondary voltage terminals of your regulator.
In the following article, we share:
What voltage regulation is;
Diagrams of voltage regulators;
How to calculate power dissipation in voltage regulators; and
How to calculate voltage regulation in linear and switching regulators.
Voltage regulation maintains a constant output voltage to match the needs of the components of an electrical device, even when the input voltage or load conditions change.
We measure the voltage regulation in two parameters:
It is the ratio of change in micro voltage output per unit change in the input voltage, i.e.,
\Delta V_{\text{output}} / \Delta V_{\text{input}}
\Delta V_{\text{output}}
– Change in the output voltage; and
\Delta V_{\text{input}}
– Change in the input voltage.
It is the ratio of change in the output voltage when going from no-load to full-load:
\Delta V_{\text{output}} / \Delta I_{\text{load}}
V_{\text{no-load}} - V_{\text{full-load}} / \Delta I_{\text{load}}
\because
\Delta V_{output} = V_{\text{no-load}} - V_{\text{full-load}}
\Delta I_{\text{load}}
– Change in the load current;
V_{\text{no-load}}
– Voltage when there is no load; and
V_{\text{full-load}}
– Voltage when there is full load.
💡 In an ideal case scenario, the output voltage remains constant with and without the load. Thus, the line and load regulation are always zero.
There are two types of voltage regulators:
Linear type regulators
They are buck (step-down) regulators. These linear-type regulators are less efficient, with basic integrated circuits and voltage dividers to drop the voltage at the desired level while shedding the rest as heat.
Benefits of linear regulators:
Low resistor noise;
Simple to add to circuit;
Fast response time; and
Low output voltage ripple.
The output of a linear regulator is always lower than the input and drops out if the input voltage is too low.
V_{\text{input}} > V_{\text{output}} + V_{\text{drop-out}}
Switching type regulators
Switching type regulators can be buck (step-down), boost (step-up), or buck-boost (mixture of both). They are more advanced and challenging to design with the arrangement of capacitors, diodes, and inductors to determine whether the output voltage should increase or decrease.
They rapidly switch the input voltage on and off to produce desirable changes in voltage and current. And this switching of frequency creates a possibility to obtain a wide range of voltages from the same input source.
Benefits of switching type regulators:
Handling large voltage spikes;
Reverse polarity protection; and
Remove unwanted signal noise.
🙋 Every device in an electrical system can have a different voltage regulator based on its needs. We commonly use switching regulators for conversion from DC to DC electrical power of different voltages.
Our voltage regulation calculator helps you calculate the voltage regulation of linear and switching regulators as follows:
In the V ɴᴏ-ʟᴏᴀᴅ field, enter the measured voltage when there is no load on the regulator, e.g., 230 V.
Then, in the V ғᴜʟʟ-ʟᴏᴀᴅ field, enter the voltage when there is a full load on the regulator, e.g., 220 V.
Once you've entered your voltage in both fields, the calculator will present:
Your step-down voltage regulation is 0.0435.
With its percentage change value is 4.3%.
Your step-up voltage regulation is 0.0454.
Along with its percentage change value is 4.5%.
Linear regulators are only step-down, i.e., buck regulators. Switching regulators can be step-down, step-up, or both, i.e., buck, boost, or buck-boost.
We use the following formula to calculate step-down voltage regulation:
\small VR = \frac{V_{\text{no-load}} - V_{\text{full-load}}}{V_{\text{no-load}}}
And the following formula to calculate step-up voltage regulation:
\small VR = \frac{V_{\text{no-load}} - V_{\text{full-load}}}{V_{\text{full-load}}}
And when we multiply voltage regulation by 100, we get its percentage change:
\small PC = VR \times 100
VR
– Voltage regulation; and
PC
- Percentage change from no-load to full-load.
For example, let's find a buck regulator's voltage regulation and percentage change, with the voltage on no-load being 140 V, and on full-load, 120 V.
Placing the values in the formula, we get:
\small \begin{align*} VR &= \frac{140 - 120}{140} = 0.143\\\\ PC &= 0.143 \times 100 = 14.3% \end{align*}
Thus, our step-down voltage regulation is 0.143, and the percentage change is 14.3%.
The following diagrams help us better perceive step-up and step-down converters.
Buck converters: Reduce voltage.
V_{\text{input}} > V_{\text{output}}
Boost converters: Increase voltage.
V_{\text{input}} < V_{\text{output}}
Buck-boost converters: Increase or decrease voltage but reverse the polarity.
V_{\text{output}}< 0
The efficiency of a regulator depends on the difference between its input and output voltages and how much the circuit draws current. The greater the difference, or the more the current, the more the heat or power dissipation by the regulator.
We can obtain this value using the following formula:
\small PD = ( V_{\text{input}} - V_{\text{output}} ) \times I_{\text{output}}
PD
– Power dissipation from the regulator;
V_{\text{input}}
– Regulator voltage input;
V_{\text{output}}
– Regulator voltage output; and
I_{\text{output}}
– Regulator current output.
An adjustable voltage regulator is a variable or buck-boost switching regulator that can be adjusted to increase or reduce its output voltage to meet the electrical system requirements.
They are generally DC to DC power converters.
What is the purpose of a DC voltage regulator?
The purpose of a DC voltage regulator is to function as a power supply and provide a stable input voltage for the devices to operate. A DC voltage regulator can also stabilize the output voltage, avoiding input voltage and current fluctuations.
How many types of voltage regulators are there?
There are two main types of voltage regulators:
Linear regulators; and
Both regulate the voltage, but linear regulators have low efficiency, thus dissipating more power as heat. In comparison, switching regulators are highly efficient, as most of their input power transfers as output with minimum dissipation.
How do I calculate power dissipation in a voltage regulator?
To calculate the power dissipation in a voltage regulator:
Subtract the output voltage from the input voltage.
Multiply the result with the output current.
That is the power dissipation of your voltage regulator.
PD = ( Vi - Vo ) × Io
PD – Power dissipation from the regulator;
Vi – Regulator voltage input;
Vo – Regulator voltage output; and
Io – Regulator current output.
AbdulRafay Moeen
Enter regulator's "no-load" and "full-load" voltage
V ɴᴏ-ʟᴏᴀᴅ
V ғᴜʟʟ-ʟᴏᴀᴅ
Linear or step-down voltage regulation
Step-down voltage regulation
Step-up voltage regulation
Our Gauss's law calculator gives you the exact electrical flux through a closed surface around an electric charge.
Use the Snell's law calculator to analyze the refraction of a ray.
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To learn about Logistic Regression, at first, we need to learn Logistic Regression basic properties. Only then will we be able to build a machine learning model on a real-world application. So we will do everything step by step.
Classification techniques are an essential part of machine learning and data mining applications. Most problems in Data Science are classification problems. There are many classification problems available, but logistics regression is common and is a useful regression method for solving binary classification problems.
Logistic Regression can be used for various classification problems such as spam detection, prediction if a customer will purchase a particular product or choosing another competitor, whether the user will click on a given advertisement link or not, and many more examples.
Logistic Regression is one of the most simple and commonly used Machine Learning algorithms for two-class classification. It is easy to implement and can be used as the baseline for any binary classification problem. Its basic fundamental concepts are also constructive in deep learning. Logistic regression describes and estimates the relationship between one dependent binary variable and independent variables.
Numpy is the main and the most used package for scientific computing in Python. It is maintained by a large community (www.numpy.org). In this tutorial, we will learn several key NumPy functions such as np.exp and np.reshape. You will need to know how to use these functions for future deep learning tutorials.
At first, we must learn to implement a sigmoid function. It is a logistic function that gives an ‘S’ shaped curve that can take any real-valued number and map it into a value between 0 and 1. If the curve goes to positive infinity, y predicted will become 1, and if the curve goes to negative infinity, y predicted will become 0. If the output of the sigmoid function is more than 0.5, we can classify the outcome as 1 or YES, and if it is less than 0.5, we can classify it as 0 or NO. For example: If the output is 0.75, we can say in terms of the probability that there is a 75 percent chance that the patient will have cancer.
Before using np.exp(), you will use math.exp() to implement the sigmoid function. You will then see why np.exp() is preferable to math.exp(). So here is the sigmoid activation function:
f\left(x\right)=\frac{1}{1+{e}^{-x}}\phantom{\rule{0ex}{0ex}}
Sigmoid is also known as the logistic function. It is a non-linear function used in Machine Learning (Logistic Regression) and Deep Learning. The sigmoid function curve looks like an S-shape:
Let's write the code to see an example with math.exp().
s = 1/(1+math.exp(-x))
Let's try to run the above function: basic_sigmoid(1). As a result, we should receive "0.7310585786300049".
Actually, we rarely use the "math" library in deep learning because the functions' inputs are real numbers. In deep learning, we mostly use matrices and vectors. This is why NumPy is more powerful and useful. For example, if we'll try to run a list into the above function:
print(basic_sigmoid(x))
If we run the above code, we will receive an error about a bad operand. If we would like to run such lists, we would need to modify our basic_sigmoid() function, inserting for loop, but then losing computation power. So, in this case, we'll write a sigmoid function to use vectors instead of scalars:
x could now be either a real number, a vector, or a matrix. Data structures we use in NumPy to represent these shapes are vectors or matrices called NumPy arrays. Now we can call, for example, this function:
print(sigmoid(x))
As a result, we should see: "[0.95257413 0.88079708 0.73105858]".
When constructing Neural Network (NN) models, one of the primary considerations is choosing activation functions for hidden and differentiable output layers. This is because calculating the backpropagated error signal used to determine NN parameter updates requires the gradient of the activation function gradient. Three of the most commonly-used activation functions used in NNs are the relu function, the logistic sigmoid function, and the tangent function. Simply talking, we need to compute gradients to optimize loss functions using backpropagation. Let's code your first sigmoid gradient function:
sigmoid_derivative\left(x\right)=\sigma \text{'}\left(x\right)=\sigma \left(x\right)·\left(1-\sigma \left(x\right)\right)
We'll code the above function in two steps:
1. Set s to be the sigmoid of x. we'll use sigmoid(x) function.
2. Then we compute σ′(x)=s(1−s):
ds = s*(1-s)
Above, we compute the gradient (also called the slope or derivative) of the sigmoid function concerning its input x. We can store the output of the sigmoid function into variables and then use it to calculate the gradient.
print(sigmoid_derivative(x))
As a result, we receive "[0.04517666 0.10499359 0.19661193]"
To visualize our sigmoid and sigmoid_derivative functions, we can generate data from -10 to 10 and use matplotlib to plot these functions. Below is the full code used to print sigmoid and sigmoid_derivative functions:
plt.plot(values, sigmoid(values), 'r')
plt.plot(values, sigmoid_derivative(values), 'b')
As a result, we receive the following graph:
The above curve in red is a plot of our sigmoid function, and the curve in red color is our sigmoid_derivative function.
In this tutorial, we reviewed sigmoid activation functions used in neural network literature and sigmoid derivative calculation. Note that many other activation functions are not covered here: e.g., tanh, relu, softmax, etc. Before writing the Logistic Regression classification code, we still need to cover array reshaping, rows normalization, broadcasting, and vectorization. We will cover them in our second tutorial.
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In the previous tutorial, we wrote an optimization function that will output the learned w and b parameters. Now we can use w and b to predict the labels for our dataset X. So, in this tutorial, we will implement the predict() function. There will be two steps computing predictions:
\stackrel{^}{Y}=A=\sigma \left({W}^{T}X+b\right)
Convert the entries of a into 0 (if activation <= 0.5 we'll get a dog) or 1 (if activation > 0.5 we'll get a cat). We will store the predictions in a vector 'Y_prediction'.
X - data of size (ROWS * COLS * CHANNELS, number of examples).
Y_prediction - a NumPy array (vector) containing all predictions (0/1) for the examples in X.
if A[0,i] > 0.5:
If we'll run our new function on previous values "predict(w, b, X)" we should receive the following results:
From our results, we could say that we predicted two cats and one dog. But because input was not real images but just simple random test numbers, our predictions also don't mean anything.
print(predict(w, b, X))
Up to this point, now we know how to prepare our training data, how to optimize the loss iteratively to learn w and b parameters (computing the cost and its gradient, updating the parameters using gradient descent). And in this tutorial, we learned (w,b) to predict the labels for a given set of examples. So in the next tutorial, we'll merge all functions into a model, and we'll train it to predicts cats vs. dogs.
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In this part, we'll design a simple algorithm to distinguish cat images from dog images. We will build a Logistic Regression using a Neural Network mindset. Figure bellow explains why Logistic Regression is actually a very simple Neural Network (one neuron):
Parts of our algorithm:
The main steps we will use to build a "Neural Network" are:
• Define the model structure (data shape).
• Initialize model parameters.
- Calculate current loss (forward propagation).
- Calculate current gradient (backward propagation).
- Update parameters (gradient descent).
• Use the learned parameters to make predictions (on the test set).
We will build the above parts separately, and then we will integrate them into one function called model().
In our first tutorial part, we already wrote a sigmoid function so that I will copy it from there:
First, weight and bias values are propagated forward through the model to arrive at a predicted output. At each neuron/node, the linear combination of the inputs is then multiplied by an activation function — the sigmoid function in our example. In this process, weights and biases are propagated from inputs to output is called forward propagation. After arriving at the predicted output, the loss for the training example is calculated.
The mathematical expression of the forward propagation algorithm for one example:
{z}^{\left(i\right)}={W}^{T}·{X}^{\left(i\right)}+b
{y}^{\left(i\right)}={a}^{\left(i\right)}=sigmoid\left({z}^{\left(i\right)}\right)
L\left({a}^{\left(i\right)}, {y}^{\left(i\right)}\right)=-{y}^{\left(i\right)}\mathrm{log}\left({a}^{\left(i\right)}\right)-\left(1-{y}^{\left(i\right)}\right)\mathrm{log}\left(1-{a}^{\left(i\right)}\right)
Then the cost is computed by summing over all training examples:
J=\frac{1}{m}\sum _{i=1}^{m}L\left({a}^{\left(i\right)}, {y}^{\left(i\right)}\right)
And our final forward propagation cost function will look like this:
J=-\frac{1}{m}\sum _{i=1}^{m}{y}^{\left(i\right)}\mathrm{log}\left({a}^{\left(i\right)}\right)-\left(1-{y}^{\left(i\right)}\right)\mathrm{log}\left(1-{a}^{\left(i\right)}\right)
Backpropagation is the process of calculating the partial derivatives from the loss function back to the inputs. We are updating the values of w and b that lead us to the minimum. It’s helpful to write out the partial derivatives starting from dA to see how to arrive at dw and db.
The mathematical expression of backward propagation (calculating derivatives):
\frac{\partial J}{\partial b}=\frac{1}{m}X{\left(A-Y\right)}^{T}\phantom{\rule{0ex}{0ex}}\frac{\partial J}{\partial b}=\frac{1}{m}\sum _{i=1}^{m}{\left({a}^{\left(i\right)}-{y}^{\left(i\right)}\right)}^{T}
Coding forward and backward propagation:
So we will implement the function explained above, but first, let's see what the inputs and outputs are:
Y - true "label" vector (containing 0 if a dog, 1 if cat) of size (1, number of examples).
cost - cost for logistic regression;
dw - gradient of the loss with respect to w, the same shape as w;
db - gradient of the loss with respect to b, the same shape as b.
Here is the code we wrote in the video tutorial:
Let's test the above function with sample data:
db = 2.934645119504845e-11
z = np.dot(w.T, X)+b
cost = (-np.sum(Y*np.log(A)+(1-Y)*np.log(1-A)))/m
dw = (np.dot(X,(A-Y).T))/m
db = np.average(A-Y)
So in this tutorial, we defined general learning architecture and defined steps needed to implement the learning model. I explained what is forward and backward propagation and we learned how to implement them in code. In the next tutorial, we will continue with the optimization algorithm.
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How do I use the gear ratio speed calculator?
How to calculate the gear ratio for gear speed?
The relationship between the gear ratio and the speed ratio
Welcome to the gear ratio speed calculator! Here, we'll show you:
How to calculate the gear ratio and the speed at which your output gear rotates relative to your input gear; and
How to calculate speed from the gear ratio.
The gear ratio speed calculator is easy to use and understand. Follow these simple steps:
Enter the number of teeth in each of your two gears. Make sure you know which is the driving gear and which one is driven!
Alternatively, to obtain the number of teeth for either gear for a suitable speed ratio, enter one gear's number of teeth and a suitable gear ratio instead.
Under the field for the gear ratio, we explain how the gear ratio influences the speed ratio between the input and output gears.
You can also enter an input speed to find the output speed corresponding to the two-gear system you've created above.
And that's it! You now know how to use the gear ratio speed calculator!
When you have two connected gears, the gear ratio is the ratio between the input (driving) gear's number of teeth and the output (driven) gear's number of teeth. It allows the system designer to slow or speed up a rotational speed, which has other useful effects like influencing the torque. The gear ratio is also the factor by which the input speed is multiplied to deliver the output speed.
So, if you want to know how to calculate the gear ratio and the speed ratio mathematically, the equation would be:
\footnotesize \text{Gear ratio} = \frac{ \text{\# teeth on input gear} }{ \text{\# teeth on output gear} }
For example, a two-gear system featuring an input gear with
10
teeth and an output gear with
20
teeth would have a gear ratio of
10/20 = 0.5
0.5:1
. The output speed would then be
0.5
times faster (so
2
times slower) than the input speed.
The speed ratio is all about angles. The input and output gears interlock, and so they each rotate one tooth at a time, together.
If the output gear has more teeth than the input gear, the output gear has a smaller angle between teeth, resulting in a lower rotational speed on the output than the input. The gear speed ratio is lesser than 1.
If the output gear has fewer teeth than the input gear, the angle between the output's teeth is larger, and the output gear rotates faster than the input. The gear speed ratio is greater than 1.
See the animation below: Having more teeth on the output (right) gear means a smaller angle to turn per tooth compared to the input (left) gear. This results in a gear speed ratio of <1, and a slower rotational speed on the output compared to the input.
Haven't found quite what you're looking for with our gear ratio speed calculator? Why not try one of our other gear calculators:
Gear ratio calculator; or
Gear ratio rpm calculator.
How do I calculate a vehicle's speed from its gear ratio?
To calculate the speed of the vehicle, follow these steps:
Determine the engine's speed in rotations per minute (rpm). Consult the car's tachometer for this value.
Multiply it with 3.6 × π × r, where r is the wheel radius in meters.
Divide it by 30 × g, where g is the gear ratio at the gearbox.
If the vehicle has a differential gear ratio, divide your result by it.
The result is the vehicle's speed in km/h.
How do I calculate the speed ratio of gears?
The speed ratio is the inverse of the normal gear ratio. The gear ratio is the ratio of input teeth to output teeth (e.g., with 10 teeth on the input and 20 teeth on the output, the gear ratio is 10/20 or 0.5 : 1. The speed ratio is the ratio of output speed to input speed (e.g., with a gear ratio of 0.5, the output speed will be 0.5 that of the input speed).
What is the speed ratio for a gear ratio of 0.5?
The output gear's speed will be 0.5 times as fast (so 2 times as slow) as the input's speed in a gear system with a total gear ratio of 0.5.
Enter the number of teeth in your gears, and we'll work out the resulting gear speed ratio.
Input gear teeth
Output gear teeth
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Logical_biconditional Knowpia
In logic and mathematics, the logical biconditional, sometimes known as the material biconditional, is the logical connective (
{\displaystyle \leftrightarrow }
) used to conjoin two statements P and Q to form the statement "P if and only if Q", where P is known as the antecedent, and Q the consequent.[1][2] This is often abbreviated as "P iff Q".[3] Other ways of denoting this operator may be seen occasionally, as a double-headed arrow (↔[4] or ⇔[5] may be represented in Unicode in various ways), a prefixed E "Epq" (in Łukasiewicz notation or Bocheński notation), an equality sign (=), an equivalence sign (≡),[3] or EQV. It is logically equivalent to both
{\displaystyle (P\rightarrow Q)\land (Q\rightarrow P)}
{\displaystyle (P\land Q)\lor (\neg P\land \neg Q)}
, and the XNOR (exclusive nor) boolean operator, which means "both or neither".
{\displaystyle P\leftrightarrow Q}
(true part in red)
Semantically, the only case where a logical biconditional is different from a material conditional is the case where the hypothesis is false but the conclusion is true. In this case, the result is true for the conditional, but false for the biconditional.[1]
In the conceptual interpretation, P = Q means "All P's are Q's and all Q's are P's". In other words, the sets P and Q coincide: they are identical. However, this does not mean that P and Q need to have the same meaning (e.g., P could be "equiangular trilateral" and Q could be "equilateral triangle"). When phrased as a sentence, the antecedent is the subject and the consequent is the predicate of a universal affirmative proposition (e.g., in the phrase "all men are mortal", "men" is the subject and "mortal" is the predicate).
In the propositional interpretation,
{\displaystyle P\leftrightarrow Q}
means that P implies Q and Q implies P; in other words, the propositions are logically equivalent, in the sense that both are either jointly true or jointly false. Again, this does not mean that they need to have the same meaning, as P could be "the triangle ABC has two equal sides" and Q could be "the triangle ABC has two equal angles". In general, the antecedent is the premise, or the cause, and the consequent is the consequence. When an implication is translated by a hypothetical (or conditional) judgment, the antecedent is called the hypothesis (or the condition) and the consequent is called the thesis.
A common way of demonstrating a biconditional of the form
{\displaystyle P\leftrightarrow Q}
is to demonstrate that
{\displaystyle P\rightarrow Q}
{\displaystyle Q\rightarrow P}
separately (due to its equivalence to the conjunction of the two converse conditionals[1]). Yet another way of demonstrating the same biconditional is by demonstrating that
{\displaystyle P\rightarrow Q}
{\displaystyle \neg P\rightarrow \neg Q}
When both members of the biconditional are propositions, it can be separated into two conditionals, of which one is called a theorem and the other its reciprocal.[citation needed] Thus whenever a theorem and its reciprocal are true, we have a biconditional. A simple theorem gives rise to an implication, whose antecedent is the hypothesis and whose consequent is the thesis of the theorem.
Logical equality (also known as biconditional) is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both operands are false or both operands are true.[1]
The following is truth table for
{\displaystyle P\leftrightarrow Q}
{\displaystyle P\equiv Q}
, P = Q, or P EQ Q):
{\displaystyle P}
{\displaystyle Q}
{\displaystyle P\leftrightarrow Q}
When more than two statements are involved, combining them with
{\displaystyle \leftrightarrow }
{\displaystyle x_{1}\leftrightarrow x_{2}\leftrightarrow x_{3}\leftrightarrow \cdots \leftrightarrow x_{n}}
{\displaystyle (((x_{1}\leftrightarrow x_{2})\leftrightarrow x_{3})\leftrightarrow \cdots )\leftrightarrow x_{n}}
or may be interpreted as saying that all xi are jointly true or jointly false:
{\displaystyle (x_{1}\land \cdots \land x_{n})\lor (\neg x_{1}\land \cdots \land \neg x_{n})}
{\displaystyle ~x_{1}\leftrightarrow \cdots \leftrightarrow x_{n}}
meant as equivalent to
{\displaystyle \neg ~(\neg x_{1}\oplus \cdots \oplus \neg x_{n})}
The central Venn diagram below,
and line (ABC ) in this matrix
represent the same operation.
{\displaystyle ~x_{1}\leftrightarrow \cdots \leftrightarrow x_{n}}
meant as shorthand for
{\displaystyle (~x_{1}\land \cdots \land x_{n}~)}
{\displaystyle \lor ~(\neg x_{1}\land \cdots \land \neg x_{n})}
The Venn diagram directly below,
Red areas stand for true (as in
for and).
The biconditional of two statements
is the negation of the exclusive or:
{\displaystyle ~A\leftrightarrow B~~\Leftrightarrow ~~\neg (A\oplus B)}
{\displaystyle \Leftrightarrow \neg }
The biconditional and the
exclusive or of three statements
{\displaystyle ~A\leftrightarrow B\leftrightarrow C~~\Leftrightarrow }
{\displaystyle ~A\oplus B\oplus C}
{\displaystyle \leftrightarrow }
{\displaystyle ~~\Leftrightarrow ~~}
{\displaystyle \oplus }
{\displaystyle ~~\Leftrightarrow ~~}
{\displaystyle ~A\leftrightarrow B\leftrightarrow C}
may also be used as an abbreviation
{\displaystyle (A\leftrightarrow B)\land (B\leftrightarrow C)}
{\displaystyle \land }
{\displaystyle ~~\Leftrightarrow ~~}
{\displaystyle A\leftrightarrow B}
{\displaystyle \Leftrightarrow }
{\displaystyle B\leftrightarrow A}
{\displaystyle \Leftrightarrow }
{\displaystyle ~A}
{\displaystyle ~~~\leftrightarrow ~~~}
{\displaystyle (B\leftrightarrow C)}
{\displaystyle \Leftrightarrow }
{\displaystyle (A\leftrightarrow B)}
{\displaystyle ~~~\leftrightarrow ~~~}
{\displaystyle ~C}
{\displaystyle ~~~\leftrightarrow ~~~}
{\displaystyle \Leftrightarrow }
{\displaystyle \Leftrightarrow }
{\displaystyle ~~~\leftrightarrow ~~~}
Distributivity: Biconditional doesn't distribute over any binary function (not even itself), but logical disjunction distributes over biconditional.
{\displaystyle ~A~}
{\displaystyle ~\leftrightarrow ~}
{\displaystyle ~A~}
{\displaystyle \Leftrightarrow }
{\displaystyle ~1~}
{\displaystyle \nLeftrightarrow }
{\displaystyle ~A~}
{\displaystyle ~\leftrightarrow ~}
{\displaystyle \Leftrightarrow }
{\displaystyle \nLeftrightarrow }
{\displaystyle A\rightarrow B}
{\displaystyle \nRightarrow }
{\displaystyle (A\leftrightarrow C)}
{\displaystyle \rightarrow }
{\displaystyle (B\leftrightarrow C)}
{\displaystyle \nRightarrow }
{\displaystyle \Leftrightarrow }
{\displaystyle \rightarrow }
{\displaystyle A\land B}
{\displaystyle \Rightarrow }
{\displaystyle A\leftrightarrow B}
{\displaystyle \Rightarrow }
{\displaystyle A\leftrightarrow B}
{\displaystyle \nRightarrow }
{\displaystyle A\lor B}
{\displaystyle \nRightarrow }
Walsh spectrum: (2,0,0,2)
Like all connectives in first-order logic, the biconditional has rules of inference that govern its use in formal proofs.
Biconditional introductionEdit
Biconditional introduction allows one to infer that if B follows from A and A follows from B, then A if and only if B.
∴ B ↔ A
Biconditional eliminationEdit
∴ A → B
∴ B → A
^ a b c d Peil, Timothy. "Conditionals and Biconditionals". web.mnstate.edu. Retrieved 2019-11-25.
^ Brennan, Joseph G. (1961). Handbook of Logic (2nd ed.). Harper & Row. p. 81.
^ a b Weisstein, Eric W. "Iff". mathworld.wolfram.com. Retrieved 2019-11-25.
^ "Biconditional Statements | Math Goodies". www.mathgoodies.com. Retrieved 2019-11-25.
^ "2.4: Biconditional Statements". Mathematics LibreTexts. 2018-04-25. Retrieved 2019-11-25.
^ In fact, such is the style adopted by Wikipedia's manual of style in mathematics.
Media related to Logical biconditional at Wikimedia Commons
This article incorporates material from Biconditional on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
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In the previous tutorial, we learned how to code Sigmoid and Sigmoid gradient functions. In this tutorial, we'll learn how to reshape arrays, normalize rows, what is broadcasting, and softmax.
Two common NumPy functions used in deep learning are np.shape and np.reshape(). The shape function is used to get the shape (dimension) of a matrix or vector X. Reshape(...) is used to reshape the matrix or vector into another dimension.
For example, in computer science, a standard image is represented by a 3D array of shape (length, height, depth). However, when you read an image as the input of an algorithm, you convert it to a vector of shape (length*height*depth,1). In other words, you "unroll", or reshape, the 3D array into a 1D vector:
So we will implement a function that takes an input of shape (length, height, depth) and returns a vector of shape (length*height*depth,1). For example, if you would like to reshape an array A of shape (a, b, c) into a vector of shape (a*b,c), you would do:
A = A.reshape((A.shape[0]*A.shape[1], A.shape[2])) # A.shape[0] = a ; A.shape[1] = b ; A.shape[2] = c
To implement the above function, we write simple few lines of code:
A = image.reshape( image.shape[0]*image.shape[1]*image.shape[2],1)
To test our above function, we will create a 3 by 3 by 2 array. Typically images will be (num_px_x, num_px_y,3) where 3 represents the RGB values:
print(image2vector(image))
print(image2vector(image).shape)
As a result, we will receive:
As you can see, the image's shape is (3, 3, 2), and after we call our function, it is reshaped to a 1D array of shape (18, 1).
Normalizing rows:
Another common technique used in Machine Learning and Deep Learning is to normalize our data. Here, by normalization, we mean changing x to x/∥x∥ (dividing each row vector of x by its norm). It often leads to a better performance because gradient descent converges faster after normalization.
x=\left[\begin{array}{ccc}0& 3& 4\\ 2& 6& 4\end{array}\right]
||x||=np.linalg.norm\left(x, axis=1, keepdims=True\right) = \left[\begin{array}{c}5\\ \sqrt{56}\end{array}\right]
\mathrm{x_normalised} = \frac{\mathrm{x}}{||\mathrm{x}||}= \left[\begin{array}{ccc}0& \frac{3}{5}& \frac{4}{5}\\ \frac{2}{\sqrt{56}}& \frac{6}{\sqrt{56}}& \frac{4}{\sqrt{56}}\end{array}\right]
If you ask how we received division by 5 or division by sqrt(56), answer:
\sqrt{{0}^{2}+{3}^{2}+{4}^{2}}=5
\sqrt{{2}^{2}+{6}^{2}+{4}^{2}}=\sqrt{56}
Next, we will implement a function that normalizes each row of the matrix x (unit length). After applying 2nd function to an input matrix x, each row of x should be a vector of unit length:
x = x/x_norm
To test our function, we'll call it with a simple array:
print(normalizeRows(x))
We can try to print the shapes of x_norm and x. You'll find out that they have different shapes. This is normal given that x_norm takes the norm of each row of x. So x_norm has the same number of rows but only 1 column. So how did it worked when you divided x by x_norm? This is called broadcasting.
Now we will implement a softmax function using NumPy. You can think of softmax as a normalizing function when your algorithm needs to classify two or more classes. You will learn more about softmax in future tutorials.
Mathematical softmax functions:
for\mathit{ }x\mathit{\in }{R}^{n}\mathit{,}\mathit{ }softmax\mathit{\left(}x\mathit{\right)}\mathit{=}softmax\mathit{\left(}\left[\begin{array}{cccc}{x}_{1}& {x}_{2}& ...& {x}_{n}\end{array}\right]\mathit{\right)}\mathit{=}\mathit{\left[}\begin{array}{cccc}\frac{{\mathit{e}}^{{\mathit{x}}_{\mathit{1}}}}{\mathit{\sum }_{\mathit{j}}{\mathit{e}}^{{\mathit{x}}_{\mathit{j}}}}& \frac{{\mathit{e}}^{{\mathit{x}}_{\mathit{2}}}}{\mathit{\sum }_{\mathit{j}}{\mathit{e}}^{{\mathit{x}}_{\mathit{j}}}}& \mathit{.}\mathit{.}\mathit{.}& \frac{{\mathit{e}}^{{\mathit{x}}_{\mathit{n}}}}{\mathit{\sum }_{\mathit{j}}{\mathit{e}}^{{\mathit{x}}_{\mathit{j}}}}\end{array}\mathit{\right]}
\text{for a matrix} x\mathit{\in }{R}^{m×n}, {x}_{ij}\text{ maps to the elemnt in the} {i}^{th} \text{row and} {j}^{th} \text{column of x, thus we have}:\phantom{\rule{0ex}{0ex}}\text{softmax(x)=softmax}\left[\begin{array}{ccccc}{x}_{11}& {x}_{12}& {x}_{13}& ...& {x}_{1n}\\ {x}_{21}& {x}_{22}& {x}_{23}& ...& {x}_{2n}\\ ...& ...& ...& ...& ...\\ {x}_{m1}& {x}_{m2}& {x}_{m3}& ...& {x}_{mn}\end{array}\right]=
=\left[\begin{array}{ccccc}\frac{{e}^{{x}_{\mathit{11}}}}{\mathit{\sum }_{j}{e}^{{x}_{1j}}}& \frac{{e}^{{x}_{\mathit{12}}}}{\mathit{\sum }_{j}{e}^{{x}_{1j}}}& \frac{{e}^{{x}_{\mathit{13}}}}{\mathit{\sum }_{j}{e}^{{x}_{1j}}}& ...& \frac{{e}^{{x}_{\mathit{1}n}}}{\mathit{\sum }_{j}{e}^{{x}_{1j}}}\\ \frac{{e}^{{x}_{\mathit{21}}}}{\mathit{\sum }_{j}{e}^{{x}_{2j}}}& \frac{{e}^{{x}_{\mathit{22}}}}{\mathit{\sum }_{j}{e}^{{x}_{2j}}}& \frac{{e}^{{x}_{\mathit{23}}}}{\mathit{\sum }_{j}{e}^{{x}_{2j}}}& ...& \frac{{e}^{{x}_{2n}}}{\mathit{\sum }_{j}{e}^{{x}_{2j}}}\\ ...& ...& ...& ...& ...\\ \frac{{e}^{{x}_{m\mathit{1}}}}{\mathit{\sum }_{j}{e}^{{x}_{mj}}}& \frac{{e}^{{x}_{m\mathit{2}}}}{\mathit{\sum }_{j}{e}^{{x}_{mj}}}& \frac{{e}^{{x}_{m\mathit{3}}}}{\mathit{\sum }_{j}{e}^{{x}_{mj}}}& ...& \frac{{e}^{{x}_{mn}}}{\mathit{\sum }_{j}{e}^{{x}_{mj}}}\end{array}\right]=\left(\begin{array}{c}\mathrm{softmax}\left(\mathrm{first} \mathrm{row} \mathrm{of} \mathrm{x}\right)\\ \mathrm{softmax}\left(\mathrm{second} \mathrm{row} \mathrm{of} \mathrm{x}\right)\\ ...\\ \mathrm{softmax}\left(\mathrm{last} \mathrm{row} \mathrm{of} \mathrm{x}\right)\end{array}\right)
We will create a softmax function that calculates the softmax for each row of the input x:
# We exp() element-wise to x.
# We create a vector x_sum that sums each row of x_exp.
# We compute softmax(x) by dividing x_exp by x_sum. It should automatically use numpy broadcasting.
[4, 9, 1, 0 ,5]])
If we try to print the shapes of x_exp, x_sum, and s above, you will see that x_sum is of shape (2,1) while x_exp and s are of shape (2, 5). x_exp/x_sum works due to python broadcasting.
We now have a pretty good understanding of the python NumPy library and have implemented a few useful functions that we will be using in future deep learning tutorials.
From this tutorial, we must remember that np.exp(x) works for any np.array(x) and applies the exponential function to every coordinate. Sigmoid function and its gradient image2vector are two commonly used functions in deep learning. np.reshape is also widely used. In the future, you'll see that keeping our matrix and vector dimensions straight will go toward eliminating a lot of bugs. You will see that NumPy has efficient built-in functions - broadcasting that is extremely useful in machine learning.
Up to this point, we learned nice stuff about the NumPy library. We'll learn about vectorization in the next tutorial, and then we will start coding our first gradient descent algorithm.
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PedalTriangle - Maple Help
Home : Support : Online Help : Mathematics : Geometry : 2-D Euclidean : Triangle Geometry : PedalTriangle
find the pedal triangle of a point with respect to a triangle
PedalTriangle(pT, P, T, n)
the PedalTriangle triangle to be created
(optional) list of three names denoting the names of three vertices of the pedal triangle
The pedal triangle pT of point P with respect to triangle T is the triangle formed by the feet of the perpendiculars drawn from point P to the sides of T (or their extensions).
If the optional argument is given and is a list of three names, these three names will be assigned to the three vertices of the pedal triangle pT
For a detailed description of the pedal triangle pT, use the routine detail (i.e., detail(pT))
The command with(geometry,PedalTriangle) allows the use of the abbreviated form of this command.
\mathrm{with}\left(\mathrm{geometry}\right):
\mathrm{triangle}\left(T,[\mathrm{point}\left(A,0,0\right),\mathrm{point}\left(B,2,0\right),\mathrm{point}\left(C,1,3\right)]\right):
\mathrm{point}\left(P,4,4\right):
\mathrm{PedalTriangle}\left(\mathrm{pT},P,T,[\mathrm{A1},\mathrm{B1},\mathrm{C1}]\right)
\textcolor[rgb]{0,0,1}{\mathrm{pT}}
\mathrm{detail}\left(\mathrm{pT}\right)
\begin{array}{ll}\textcolor[rgb]{0,0,1}{\text{name of the object}}& \textcolor[rgb]{0,0,1}{\mathrm{pT}}\\ \textcolor[rgb]{0,0,1}{\text{form of the object}}& \textcolor[rgb]{0,0,1}{\mathrm{triangle2d}}\\ \textcolor[rgb]{0,0,1}{\text{method to define the triangle}}& \textcolor[rgb]{0,0,1}{\mathrm{points}}\\ \textcolor[rgb]{0,0,1}{\text{the three vertices}}& [[\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3}]\textcolor[rgb]{0,0,1}{,}[\frac{\textcolor[rgb]{0,0,1}{8}}{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{,}\frac{\textcolor[rgb]{0,0,1}{24}}{\textcolor[rgb]{0,0,1}{5}}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{0}]]\end{array}
\mathrm{draw}\left({P,T\left(\mathrm{color}=\mathrm{blue}\right),\mathrm{pT}\left(\mathrm{color}=\mathrm{green}\right)},\mathrm{printtext}=\mathrm{true}\right)
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Characterization of Rapidly Consolidated Titanium Diboride | J. Eng. Mater. Technol. | ASME Digital Collection
Kunal Kothari,
e-mail: kotharik@gmail.com
Ramachandran Radhakrishnan,
e-mail: radha38@hotmail.com
Kunal Kothari Graduate Research Assistant
Ramachandran Radhakrishnan Assistant Research Scientist
Norman M. Wereley Professor and Associate Chair
Kothari, K., Radhakrishnan, R., and Wereley, N. M. (March 23, 2011). "Characterization of Rapidly Consolidated Titanium Diboride." ASME. J. Eng. Mater. Technol. April 2011; 133(2): 024501. https://doi.org/10.1115/1.4003599
This study reports on the microstructure and mechanical properties of titanium diboride
(TiB2)
that was rapidly consolidated via plasma pressure compaction
(P2C®)
. Titanium diboride powder with
2.5 wt %
of silicon nitride
(Si3N4)
, as a sintering aid, was rapidly consolidated using
P2C®
to inhibit grain growth and ensure high strength and ductility of the consolidated material. The consolidated specimens were 93.5% of theoretical density. The microstructure of the consolidated material was characterized using optical microscopy, scanning electron microscopy, and wavelength dispersive spectroscopy. Oxygen concentration in the consolidated sample was 13.55% less than in the as-received powders. The flexure strength, fracture toughness, Young’s modulus, and Vickers hardness of the consolidated specimens were measured at room temperature and were found to be equivalent if not superior to those reported in the literature.
bending strength, compaction, crystal microstructure, density, ductility, fracture toughness, optical microscopy, plasma materials processing, powder technology, powders, scanning electron microscopy, sintering, titanium compounds, Vickers hardness, Young's modulus
Compacting, Density, Ductility, Fracture toughness, Mechanical properties, Optical microscopy, Plasmas (Ionized gases), Pressure, Scanning electron microscopy, Sintering, Temperature, Titanium, Vickers hardness testing, Young's modulus, Oxygen, Bending (Stress), Silicon nitride ceramics, Bending strength, Powder metallurgy, Spectroscopy, Wavelength
Sintering and Properties of Titanium Diboride Made From Powder Synthesized in a Plasma-Arc Heater
Effect of Microstructure on the Properties of TiB2 Ceramics
Densification and Mechanical Properties of Titanium Diboride With Silicon Nitride as a Sintering Aid
The Microhardness and Microstructural Characteristics of Bulk Molybdenum Samples Obtained by Consolidating Nanopowders by Plasma Pressure Compaction
An Investigation of the Influence of Powder Particle Size on Microstructure and Hardness of Bulk Samples of Tungsten Carbide
Effect of Pulsed Current on Reactively Synthesized TiB2 Consolidated by Plasma Pressure Compaction
Mechanical Properties of Titanium Diboride
Effect of Oxygen Contamination on Densification of TiB2
Elastic Properties of TiB2 and MgB2
Rapid Consolidation of TiB 2 via Plasma Pressure Compaction
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ExternalBisector - Maple Help
Home : Support : Online Help : Mathematics : Geometry : 2-D Euclidean : Triangle Geometry : ExternalBisector
find the external bisector of a given triangle
ExternalBisector(bA, A, ABC)
A-external-bisector of ABC
The external bisector bA of the angle at A of the triangle ABC is the line perpendicular to the internal bisector of the angle at A.
For a detailed description of the external bisector bA, use the routine detail (i.e., detail(bA))
The command with(geometry,ExternalBisector) allows the use of the abbreviated form of this command.
\mathrm{with}\left(\mathrm{geometry}\right):
\mathrm{triangle}\left(\mathrm{ABC},[\mathrm{point}\left(A,0,0\right),\mathrm{point}\left(B,2,0\right),\mathrm{point}\left(C,1,3\right)]\right):
define the external bisector bA
\mathrm{ExternalBisector}\left(\mathrm{bA},A,\mathrm{ABC}\right)
\textcolor[rgb]{0,0,1}{\mathrm{bA}}
\mathrm{detail}\left(\mathrm{bA}\right)
\begin{array}{ll}\textcolor[rgb]{0,0,1}{\text{name of the object}}& \textcolor[rgb]{0,0,1}{\mathrm{bA}}\\ \textcolor[rgb]{0,0,1}{\text{form of the object}}& \textcolor[rgb]{0,0,1}{\mathrm{line2d}}\\ \textcolor[rgb]{0,0,1}{\text{equation of the line}}& \textcolor[rgb]{0,0,1}{\mathrm{_x}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{10}}\textcolor[rgb]{0,0,1}{+}\sqrt{\textcolor[rgb]{0,0,1}{4}}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{_y}}\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}\end{array}
\mathrm{bisector}\left(\mathrm{ibA},A,\mathrm{ABC}\right):
\mathrm{ArePerpendicular}\left(\mathrm{bA},\mathrm{ibA}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
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How does a Helmholtz resonator work?
How to calculate the Helmholtz resonator frequency
Examples of Helmholtz resonators — and an experiment!
How to use our Helmholtz resonator calculator?
An example of a Helmholtz resonator
Take a box, poke a hole in it: that's how you can build a rudimental resonating chamber, then use our Helmholtz resonator calculator to find the frequency it operates!
Since antiquity, humanity has built tools that help amplify or absorb sounds: the Helmholtz resonator is one of these devices. If you want to discover more about them, keep reading! You will learn:
What a Helmholtz resonator is;
How a Helmholtz resonator works;
How to calculate the frequency of a Helmholtz resonator;
The applications of Helmholtz resonance: from exhausts to musical instruments; and
A neat experiment you can try at home.
Tune in to our Helmholtz frequency calculator!
Resonators are devices that use resonance, the property of objects to prefer a specific frequency of oscillations at which the energy transfer is particularly effective to enhance or dampen a wave.
🙋 Resonators can exist wherever there is a wave-like behavior: water, electromagnetism, acoustics, the list goes on. Our calculator focuses on acoustic resonators!
A Helmholtz resonator is a closed (or partially closed) cavity where air oscillates at a particular standing frequency controlled by a few parameters. The result is a resonance effect widely used in acoustics: from sound absorbers to musical instruments. Let's take a look at the design of a Helmholtz resonator!
The shape of the resonator itself doesn't really matter; theoretically, any shape of the cavity works. Helmholtz (a German physicist) invented the original resonator, which used an almost spherical cavity to pick up a specific frequency from a complex sound, isolating it from the rest. A small opening in the cavity allowed the experimenter to listen to that specific frequency.
When the Helmholtz resonator is used to pick up a sound, the cavity should have a larger opening from which the sound "enters". Inside the cavity, the pressure increases because of the soundwave entering it, and decreases thanks to the inertia of air when it bounces in the cavity, doing so at a specific frequency. The user can use a smaller opening to listen to the sound inside the cavity.
Helmholtz resonators can also work as sound absorbers: in this case, there is no "listening" opening. The resonator design makes it "focus" at a specific frequency, trapping it in the chamber where it eventually gets dissipated after multiple reflections inside the resonator.
If appropriately designed, a Helmholtz resonator can almost entirely erase a frequency from a room where a complex sound is reproduced.
To calculate the Helmholtz resonator frequency, you need to know a few parameters of the resonator's design. Let's take at the formula first:
f_{\text{res}} = \frac{c}{2\pi}\sqrt{\frac{A_0}{V \cdot L_0}}
c
is the speed of the sound in the medium (i.e., air). Its value is taken as
344\ \text{m}\!\cdot\!\text{s}^{-1}
A_0
is the cross-section of the opening that picks the sound;
V
is the volume of the resonator; and
L_0
is the depth of the opening.
🔎 The frequency is associated with the geometrical measurements of the resonator: remember that you can describe oscillating phenomena in terms of wavelengths!
The depth of the opening may take into account the end correction of a hole in acoustics: that's a factor added to the length to justify the different starting point of the soundwave in the opening itself. The following formula gives the end correction:
L_0=L+\Delta L
L
is the true length of the opening and
\Delta L
is the end correction.
🔎 The concept of resonance frequency is not limited to acoustics: try our resonant frequency calculator to learn how engineers (and not only) design LC circuits!
The name Helmholtz resonator may sound highly technical, but we are almost sure you met them already without knowing it!
The first example of a Helmholtz resonator is... a bottle! When you take a bottle, and blow over the open neck, you hear a sound, a single note, usually pretty low: that's the resonating frequency of the bottle. Blowing right above the neck creates the difference in pressure necessary to make the air inside the bottle vibrate: that's an interesting variation on the functioning of a Helmholtz resonator.
Fill the bottle (or empty it) with a liquid, and you will hear the frequency shift: the emptier the bottle, the lower the frequency. Check the formula for the resonating frequency: the volume of the resonator is at the denominator of the fraction!
Many musical instruments make use of Helmholtz resonators. The examples range from acoustic guitars, where the wooden body helps amplify the sounds, to ocarinas, which are almost ideal Helmholtz resonators.
In ocarinas, the different notes are obtained by closing or opening holes with the fingers, thus changing the value of
A_0
Helmholtz resonators excel in muffling sounds when designing concert rooms or studios. In those cases, once sound engineers identify a problematic frequency, they design the proper resonator that absorbs that particular tone. Another application of Helmholtz resonance is in the exhausts of vehicles: creating a suitably sized exhaust allows to filter the unwanted noises from the engine. Or to amplify them, even if no one likes that.
One last example! Have you ever tried listening to a seashell? The sound you hear, commonly called the "sound of the sea", is the effect of Helmholtz's resonance: the internal structure of the shell makes it an astonishingly good resonator, and you can hear only some specific frequencies. 🐚
Our Helmholtz resonator calculator allows you to calculate the value of the Helmholtz resonance frequency for various combinations of shapes and openings. Let's take a look at it.
Choose the type of chamber you are considering. We offer you four different possibilities:
Arbitrary shape;
Parallelepipedal shape;
Spherical shape; and
In the first one, you have to manually insert the volume, while for the others, we are the ones doing the math — you only have to insert the measurement of the chamber.
Then you have to choose the type of opening you are considering. Here you have three choices:
Arbitrarily shaped opening;
Circular opening; and
Choose the one that fits your problem better and insert the measurements if necessary.
🙋 We added the end correction for a circular opening. We set it by default at
0.00
, but if you want to modify, simply click on advanced mode and tune its value!
Finally, insert the value of the length of the opening. It is crucial for a satisfying outcome of the calculations.
We got really excited to see that there is an exceptionally simple experiment you can make at home to test our Helmholtz resonator calculator. You will only need an empty bottle and a tape meter.
The bottle should have a shape made of a composition of elementary shapes, like sphere and cylinder. Take the measurements you will need:
Circumference of the bottle;
Height of the cylindrical part;
Diameter of the opening (where the cork sits, to be clear);
Now calculate the volume of the bottle. You can use our calculators to do it faster: go to our sphere volume calculator and cylinder volume calculator!
We found that our bottle, with a circumference of
23.5\ \text{cm}
, and a height of the cylindrical section equal to
16.5\ \text{cm}
has a volume of:
\begin{align*} \footnotesize V_\text{bottle}& = \footnotesize V_\text{sph}+V_\text{cyl}\\ & \footnotesize = 108.58\ \text{cm}^{3}+725.12\ \text{cm}^{3} \\ & \footnotesize = 833.7\ \text{cm}^{3} \end{align*}
The neck has the radius of
0.95\ \text{cm}
7.5\ \text{cm}
Insert these measurements in our calculator after selecting arbitrary shape in the resonator shape section, and circular opening in the opening section. The result you will find is:
\begin{align*} \footnotesize f_\text{res} & \footnotesize =\frac{344}{2\pi} \sqrt{\frac{A_0}{V\cdot L_0}}\\ & \footnotesize = \frac{344}{2\pi}\sqrt{\frac{2.835\cdot 10^{-4}}{8.337\cdot 10^{-4}\cdot 0.075}} \\ & \footnotesize =116.58\ \text{Hz} \end{align*}
Install an application that allows you to detect the frequency of a sound. We used this application for Android, but you can easily use an online tool like this pitch detector.
Blow on the bottle and read the measurement: we got
117\ \text{Hz}
! Science works!
🔎 Fill the bottle with a bit of tap water and change the height of the cylindrical part according to the new measurement. Blowing on the bottle will produce a sound with a higher pitch: this is an adjustable Helmholtz resonator. For a new height of
12.2\ \text{cm}
, we calculated a resonant frequency of
132.57\ \text{Hz}
, and measured
134\ \text{Hz}
. Mind-blowing bottle-blowing!
How do I calculate the Helmholtz resonance frequency?
The Helmholtz resonator frequency equation is fᵣₑ= c/2π × √(A₀/(V × L₀)), where c is the speed of sound in air (344 m/s), and A₀, L₀, V are the geometrical parameters of the resonator: the area and length of the opening, and the volume of the chamber.
A Helmholtz resonator is a device able to pick up a single frequency (the Helmholtz resonance frequency), and amplify or suppress it, depending on the configuration of the resonance chamber.
The chamber's volume and one or more openings determine the frequency at which the air inside the resonator vibrates according to the standing waves contained there.
Find out more at omnicalculator.com!
What is a Helmholtz resonator exhaust?
The noise of an engine can be reduced noticeably using a carefully designed Helmholtz resonator exhaust able to absorb and dissipate the problematic frequencies. Helmholtz resonators in exhausts are also used to enhance a particular note from the engine: this is a common modification in car tuning.
Where do you find the Helmholtz resonance?
Helmholtz resonance is a widespread acoustic phenomenon. You can experience it in many situations: from blowing on a bottle to keeping your car window open while driving, the appearance of a fixed frequency sound is associated with the existence of a resonance frequency for every cavity, which can produce a sound in the right conditions.
Length of the opening
Use the Cyclotrone Frequency Calculator to explore the world of the first particle's accelerators.
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The Division Process - MATLAB & Simulink - MathWorks France
The C programming language provides access to integer division only for fixed-point data types. Depending on the size of the numerator, you can obtain some of the fractional bits by performing a shift prior to the integer division.
Suppose you want to divide two numbers. Each of these numbers is represented by an 8-bit word, and each has a binary-point-only scaling of 2-4. Additionally, the output is restricted to an 8-bit word with binary-point-only scaling of 2-4.
The division of 9.1875 by 1.5000 is shown in the following model.
\begin{array}{c}{Q}_{a}={2}^{-4-\left(-4\right)-\left(-4\right)}\left({Q}_{b}/{Q}_{c}\right)\\ ={2}^{4}\left({Q}_{b}/{Q}_{c}\right).\end{array}
Assuming a large data type was available, this could be implemented as
{Q}_{a}=\frac{\left({2}^{4}{Q}_{b}\right)}{{Q}_{c}},
where the numerator uses the larger data type. If a larger data type was not available, integer division combined with four repeated subtractions would be used. Both approaches produce the same result, with the former being more efficient.
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Nested Functions | Brilliant Math & Science Wiki
Patrick Corn, Geoff Pilling, A Former Brilliant Member, and
Nested functions are expressions such as nested radicals and continued fractions involving infinitely recursive expressions. Examples include
\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{\cdots}}}} \qquad \text{and} \qquad 1+\frac2{2+\frac3{3+\frac4{4+\ddots}}}.
A sufficiently general and natural definition of nested function, such as the one given below, includes more familiar expressions such as infinite series and infinite products as well.
Determining the value of nested functions when they converge leads to many striking identities. Some of the most beautiful of these are due to the famous Indian mathematician Srinivasa Ramanujan (1887-1920).
Other Nested Functions
General Definition and Convergence Properties
Nested radicals involve recursive expressions with repeated square roots. A common problem-solving strategy for evaluating nested radicals is to find a copy of the expression inside itself.
\sqrt{n+\sqrt{n+\sqrt{n+\cdots}}},
assuming it converges.
x
x = \sqrt{n+x}
x^2-x-n = 0
. By the quadratic formula,
x = \frac{1\pm\sqrt{4n+1}}2
, and clearly the
+
sign is indicated since
x
should be positive, so we get
x = \frac{1 + \sqrt{4n+1}}2
_\square
This approach can be directly applied to the following problem:
\sqrt{156 + \sqrt{156 + \sqrt{156 + \cdots}}} = \, ?
Some nested functions require more elaborate manipulations.
and
be positive real numbers. Show that
a+b = \sqrt{b^2+a\sqrt{b^2+(a+b)\sqrt{b^2+(a+2b)\sqrt{\cdots}}}}
if it converges. This identity is due to Ramanujan.
(b+x)^2 = b^2+ 2bx+x^2 = b^2 + x\big(b + (x+b)\big)
, taking square roots gives
b+x = \sqrt{b^2+x\big(b+(x+b)\big)}.
The idea is to apply this expression repeatedly, with
x=a,a+b,a+2b,\ldots
\begin{aligned} b+a &= \sqrt{b^2+a(\color{#D61F06}{b+a+b})} \qquad \text{(plugging in } x=a\text{)} \\ &= \sqrt{b^2+a\sqrt{b^2+(a+b)(\color{#D61F06}{b+a+2b})}} \qquad \text{(plugging in } x=a+b\text{)} \\ &= \sqrt{b^2+a\sqrt{b^2+(a+b)\sqrt{b^2+(a+2b)(\color{#D61F06}{b+a+3b})}}}. \qquad \text{(plugging in } x=a+2b\text{)} \\ \end{aligned}
_\square
Use the identity
(2^n+x)^2 = 4^n+x(2^{n+1}+x)
\sqrt{4+\sqrt{16+\sqrt{64+\sqrt{\cdots}}}}.
Take the square root of both sides of the identity to get
2^n+x = \sqrt{4^n+x(2^{n+1}+x)}.
This identity works for any integer
n
; replacing
n
n+1
2^{n+1}+x = \sqrt{4^{n+1}+x(2^{n+2}+x)}.
2^{n+1}+x
was inside the first square root, so applying the identity repeatedly (increasing the exponent by 1 each time) leads to an infinite nested radical:
\begin{aligned} 2^n+x &= \sqrt{4^n+x(\color{#D61F06}{2^{n+1}+x})} \\ &= \sqrt{4^n+x\sqrt{4^{n+1}+x(\color{#D61F06}{2^{n+2}+x})}} \\ &= \sqrt{4^n+x\sqrt{4^{n+1}+x\sqrt{4^{n+2}+x(\color{#D61F06}{2^{n+3}+x})}}}. \end{aligned}
The identity is applied to the quantities in red, replacing
n
n+1
n+2
, and so on. This gives an expression of the form
2^n+x = \sqrt{4^n+x\sqrt{4^{n+1}+x\sqrt{4^{n+2}+x\sqrt{\cdots}}}}.
n = x = 1,
3 = \sqrt{4+\sqrt{16+\sqrt{64+\sqrt{\cdots}}}}.\ _\square
The strategy of looking for copies of a nested function inside itself works in other contexts as well.
\sqrt{5}+\log 5
e^{\sqrt{2}}
1+\sqrt{2}
\large x=2+\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{2+\ddots}}}}
Given the above, what is the value of
x?
\large x = 6+\frac1{2+\frac1{2+\frac1{12+\frac1{2+\frac1{2+\frac1{12+\cdots}}}}}}
x.
\begin{aligned} x-6 &= \frac1{2+\frac1{2+\frac1{12+(x-6)}}} \\ &= \frac1{2+\frac1{2+\frac1{x+6}}} \\ &= \frac1{2+\frac{x+6}{2x+13}}\\ &= \frac{2x+13}{5x+32}\\ 5x^2+2x-192 &= 2x+13 \\ x^2 &= 41, \end{aligned}
x
is clearly positive. Therefore,
x = \sqrt{41}
_\square
\sqrt{14+\sqrt[4]{14+\sqrt[4]{14+\sqrt[4]{\cdots}}}}.
Call the expression
x
x = \sqrt{14+\sqrt{x}},
x^2=14+\sqrt{x}
. The graphs of
y = x^2
y = 14+\sqrt{x}
intersect in exactly one place, and by inspection it happens when
x = 4
_\square
Consider the infinitely nested exponential equation
\large x^{x^{x^{\cdot^{\cdot^{\cdot}}}}} = N.
One might naively say, "Easy, just substitute in," and
x^N = N, \ \text{ so }\ x = \sqrt[N]{N}.
However, this doesn't converge for all
N
. What is the highest
N
for which it does?
Nested radicals and continued fractions can be considered as two special cases of the general definition, which includes quite a few familiar expressions as other special cases as well.
A nested function is an expression of the form
x_0 + y_0\Big(x_1+y_1\big(x_2+y_2(\cdots)^p\big)^p\Big)^p
p, x_i, y_i
. We say it converges if the sequence of partial expressions
x_0+y_0\Big(x_1+y_1\big(x_2+y_2(\cdots+y_{k-1})^p\cdots\big)^p\Big)^p
p=1
y_i = 1
, the nested function is an infinite sum of the
x_i
p = 1
x_i = 0
, the nested function is an infinite product of the
y_i
p = -1
, the nested function is a continued fraction
x_0+\frac{y_0}{x_1+\frac{y_1}{x_2+\cdots}}.
p = \frac12
, the nested function is of the form
x_0+y_0\sqrt{x_1+y_1\sqrt{x_2+\cdots}}.
It is generally not hard to show that nested radicals of the kind considered above converge, as the sequence of partial expressions is often monotonically increasing and bounded above by the proposed limit. (The sequence of partial expressions is what is obtained by replacing the red terms in the above examples by 1.) Showing that the sequence converges to the expected limit is usually harder but straightforward.
A theorem of Herschfeld (1935) states that for a nested function with
0 < p < 1
y_i = 1
, the nested power
x_0+\Big(x_1+\big(x_2+(\cdots)^p\big)^p\Big)^p
converges if and only if
x_i^{p^i}
is a bounded infinite sequence of real numbers.
Cite as: Nested Functions. Brilliant.org. Retrieved from https://brilliant.org/wiki/nested-functions/
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A Bayesian Twist on Tukey's Flogs - Publishable Stuff
A Bayesian Twist on Tukey's Flogs
In the last post I described flogs, a useful transform on proportions data introduced by John Tukey in his Exploratory Data Analysis. Flogging a proportion (such as, two out of three computers were Macs) consisted of two steps: first we “started” the proportion by adding 1/6 to each of the counts and then we “folded” it using what was basically a rescaled log odds transform. So for the proportion two out of three computers were Macs we would first add 1/6 to both the Mac and non-Mac counts resulting in the “started” proportion (2 + 1/6) / (2 + 1/6 + 1 + 1/6) = 0.65. Then we would take this proportion and transform it to the log odds scale.
The last log odds transform is fine, I’ll buy that, but what are we really doing when we are “starting” the proportions? And why start them by the “magic” number 1/6? Maybe John Tukey has the answer? From page 496 in Tukey’s EDA:
The desirability of treating “none seen below” as something other than zero is less clear, but also important. Here practice has varied, and a number of different usages exist, some with excuses for being and others without. The one we recommend does have an excuse but since this excuse (i) is indirect and (ii) involves more sophisticated considerations, we shall say no more about it. What we recommend is adding 1/6 to all […] counts, thus “starting” them.
Not particularly helpful… I don’t know what John Tukey’s original reason was (If you know, please tell me in the comments bellow!) but yesterday I figured out a reason that I’m happy with. Turns out that starting proportions by adding 1/6 to the counts is an approximation to the median of the posterior probability of the $\theta$ parameter of a Bernouli distribution when using Jeffrey’s prior on $\theta$. I’ll show you in a minute why this is the case but first I just want to point out that intuitively this is roughly what we would want to get when “starting” proportions.
The reason for starting proportions in the first place was to gracefully handle cases such as 1/1 and 0/1 where the “raw” proportions would result in the edge proportions 100 % and 0 %. But surely we don’t believe that a person is a 100 % Apple fanboy just after getting to know that that person has one Mac computer out of in total one computer. That person is probably more likely to purchase a new Mac rather that a new PC but to estimate this probability as 100 % just after seeing one data point seems a bit rash. This kind of thinking is easily accommodated in a Bayesian framework by using a prior distribution on the proportion/probability parameter $\theta$ of the Bernoulli distribution that puts some prior probability on all possible proportions. One such prior is Jeffrey’s prior which is $\theta \sim Beta(0.5,0.5)$ and looks like this:
theta <- seq(0, 1, length.out = 1000)
plot(theta, dbeta(theta, 0.5, 0.5), type = "l", ylim = c(0, 3.5), col = "blue",
So instead of calculating the raw proportions we could estimate them using the following model:
Here $x_i$ is a vector of zeroes and ones coding for, e.g., non-Macs (0) and Macs (1). This is easy to estimate using the fact that the posterior distribution of $\theta$ is given by
\text{Beta}(0.5 + n_1, 0.5 + n_0)
n_0
n_1
are the number of zeroes and ones respectively. Running with our example of the guy with the one single Mac the estimated posterior distribution of the proportion $\theta$ would be:
plot(theta, dbeta(theta, 0.5 + 1, 0.5 + 0), type = "l", ylim = c(0, 3.5), col = "blue",
abline(v = median(rbeta(99999, 0.5 + 1, 0.5 + 0)), col = "red", lty = 2, lwd = 3)
Now in order to get a single, point estimate out of this posterior distribution we have to summarize it in some way. One way is by taking the median as our “best guess” of the underlying value of $\theta$. The median of the distribution above is marked by the red, dashed line. It turns out that there is no easy, closed form expression for the median of a Beta distribution, however, the median can be approximated (according to Kerman , 2011) by
where $\alpha$ and $\beta$ are the values of the first and second parameters to the Beta distribution. In our case, with Jeffrey’s prior adding 0.5 to both parameters, we have:
Substituting this into the expression above we get:
this is exactly the same expression as Tukey used to “start” proportions! Thus, the way Tukey calculate proportions is justifiable as the point estimate of a Bayesian estimation procedure.
Knowing this, it is possible to play around with the flog function I presented in the last post:
flog <- function(successes, failures, total_count = successes + failures) {
p <- (successes + 1/6)/(total_count + 1/3)
(1/2 * log(p) - 1/2 * log(1 - p))
Instead of using Jeffrey’s prior we might want to use a flat prior on $\theta$, as we believe that, initially, all values of $\theta$ are equally likely. A flat prior is given by the $\text{Beta}(1,1)$ distribution:
plot(theta, dbeta(theta, 1, 1), type = "l", ylim = c(0, 2), col = "blue", lwd = 3)
Instead of using the median of the resulting posterior to produce a point value we might prefer taking the mean instead. The mean of a Beta distribution is just $\alpha / (\alpha + \beta)$. The flog function can easily be changed to accommodate these two changes as follows:
p <- (successes + 1)/(total_count + 2)
So all of this is great, but I still wonder what Tukey’s original justification was for “starting” proportions in the way he did…
Kerman, J. (2011). A closed-form approximation for the median of the beta distribution. arXiv preprint arXiv:1111.0433. http://arxiv.org/abs/1111.0433
Posted by Rasmus Bååth Sep 30th, 2013 Bayesian, R, Statistics
« Going to Plot Some Proportions? Why not Flog 'em First? How Do You Write Your Model Definitions? »
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How do I use the high-pass filter calculator?
Different high-pass filters — passive vs. active high-pass filters
Inverting op-amp high-pass filter
Non-inverting op-amp high-pass-filter
Welcome to the high pass filter calculator. If you want to remove some low-frequency noise or you're trying to get more treble, you've come to the right place. Together, we'll learn:
What a high-pass filter is;
The different types of high-pass filter circuits;
Passive high-pass filters and active high-pass filters;
How to tell high-pass and low-pass filters apart; and
How capacitors can be used in high-pass filters.
Using the high-pass filter calculator is easy! Here's how:
Select the filter type you're designing. The high-pass filter calculator covers the following filter types:
RC high-pass filter;
RL high-pass filter;
Non-inverting op-amp high-pass filter; and
Inverting op-amp high-pass filter.
Input the values for which you are designing. The passive (RC and RL) filters let you fine-tune the component values and the desired cutoff frequency, while the active (op-amp) filters also let you adjust the gain on your output.
Keep reading if you want to learn how they work!
💡 The high-pass filter calculator lets you enter whatever values you know, and it will calculate the other values for you. It adjusts its equations according to your choice of filter type.
A high-pass filter is an electronic circuit that removes low-frequency components from a given AC signal. In other words, it blocks low frequencies and lets high frequencies pass through it. That's why we call it a "high-pass filter".
What does this mean in practice? Take a look at the Bode plot for high-pass filters below.
The Bode plot (frequency response) of a high-pass filter. The blue line represents an ideal filter, while the red line represents a real filter.
A Bode plot (like the one above) illustrates a circuit's frequency response, which is another word for how it amplifies signals of certain frequencies and damps others. A high-pass filter's frequency response suppresses low-frequency signals, which we can see all the way up to
f = f_c
. That frequency is called the cutoff frequency, and it's what defines any high-pass filter:
Frequencies below
f_c
are damped; and
Frequencies above
f_c
are left untouched.
💡 The Bode plot above has two lines, one representing ideal theoretical filter behavior and the other representing real-world filter behavior. Ideal filters would slope straight down at
f_c
. For real-world filters, the cutoff frequency is the frequency at which a signal is damped to
-3\ \text{dB}
The slope at which the frequency response drops for increasingly low frequencies is
-20\ \text{dB}/\text{decade}
. This means that every time the frequency is reduced by a factor of 10, the amplitude is reduced by 20 decibels. This slope is made even steeper if the order of the filter is increased: first-order filters have a 20-decibel slope, second-order filters slope at 40 decibels, third-order descends at 60, etc.
Real-world signals rarely consist of just one frequency, and that's partly why we need high-pass filters in signal-handling circuits. Here's how a high-pass filter affects signals comprised of more than one frequency:
A high-pass filter can be seen as a system that removes low-frequency components from input signals, leaving behind only the high-frequency components.
There are two main categories of high-pass filters:
Passive high-pass filters exclusively use passive components (which are resistors, capacitors and inductors). The two common passive high-pass filters are RC high-pass filters and RL high-pass filters.
Active high-pass filters use some active component, typically an operational amplifier (or "op-amp"). Thanks to their adjustable gains, they can be modified more than passive filters.
The RC high-pass filter consists of a resistor and a capacitor in the configuration shown below.
An RC high-pass filter, built with a resistor and a capacitor.
It's probably the most well-known yet basic high-pass filter, partly thanks to the easy formula for its cutoff frequency:
f_c = \frac{1}{2\pi RC}
The RC high-pass filter works thanks to its capacitor. A capacitor's impedance
Z_c
decreases with frequency, making it act like a short-circuit for high-frequency signals. Inversely, a capacitor acts as an open circuit for low frequencies.
Z_C = \frac{1}{j\cdot (2\pi f)\cdot C}
j
is the unit imaginary number.
Thanks to this frequency-dependent impedance, we can use capacitors for high-pass filters.
💡 Engineers frequently bridge their constant voltage rails to ground with capacitors, which act as high-pass filters and remove any shakiness in the voltage supply.
The RL high-pass filter uses an inductor and a resistor in the configuration below:
An RL high-pass filter, built with a resistor and an inductor.
Its cutoff frequency is also very simple to calculate:
f_c = \frac{R}{2\pi L}
An inductor
L
acts the opposite way a capacitor does: An inductor becomes a short-circuit for low-frequency signals and an open circuit for high frequencies. Because of this, any low-frequency components of
v_\text{in}
pass through the inductor, but high-frequency components are blocked. Instead, high-frequency components must travel through the load at
v_\text{out}
instead. Here is the equation for the impedance of an inductor:
Z_L = \ j\cdot(2\pi f)\cdot L
The inverting op-amp high-pass filter is an active filter that incorporates an operational amplifier (op-amp). The op-amp feeds back into itself with the feedback resistor
R_f
, with the high-pass filtering performed by the capacitor
C
. It's designed as follows:
An inverting op-amp high-pass filter, built with resistors and a capacitor.
Its cutoff frequency is calculated much like the RC filter's but with the feedback resistor
R_f
f_c = \frac{1}{2\pi R_f C}
And the gain
G
introduced by the op-amp is calculated with
\begin{split} G &= -\tfrac{R_f}{R_i} \\ v_\text{out} &= G\cdot v_\text{in} \\ \therefore v_\text{out} &= -\tfrac{R_f}{R_i}\cdot v_\text{in} \\ \end{split}
See that minus sign in the equation for
G
? It means that we're flipping the signal around the time-axis, or in more technical terms, we're introducing a 180° phase shift. If this behavior is undesirable, then head on over to the next section on NON-inverting op-amp high-pass filters.
The non-inverting op-amp high-pass filter is more complicated than its inverting sibling, but it has the useful property of not inverting the input. We connect an RC high-pass filter to the negative input of the op-amp and create a feedback loop between the op-amp's output
v_\text{out}
and the op-amp's positive input. See its circuit diagram below:
A non-inverting op-amp high-pass filter, built with resistors and a capacitor.
Because the only filtering is done by the passive RC circuit at
v_\text{in}
, we can calculate the filter's cutoff frequency by:
f_c = \frac{1}{2\pi R_i C}
G
(the gain from
v_\text{in}
v_\text{out}
) with:
\begin{split} G &= 1 + \tfrac{R_f}{R_g} \\ v_\text{out} &= G\cdot v_\text{in} \\ \therefore v_\text{out} &= (1+\tfrac{R_f}{R_g})\cdot v_\text{in} \\ \end{split}
G
is positive, but also that
G \ge 1
, as resistance can never be negative.
What components do I need for a 1 kHz high-pass filter?
To build an RC high-pass filter with a cutoff frequency of 1 kHz, use a 3.3kΩ resistor and a 47nF capacitor. Such a high-pass filter circuit will have a cutoff frequency of precisely.
fc = 1 / (2π × 3.3 kΩ × 47 nF) = 1.0261 kHz
Remember to keep components' tolerances in mind — consider measuring them with a multimeter!
How do I tell a high-pass filter from a low-pass filter?
You can differentiate high-pass vs. low-pass filters in a few ways.
Analyze the circuit. Using techniques like Laplacian transformation, calculate the frequency-dependent impedance of each component and determine the filter's frequency response.
Test it with signals. Feed waveforms of varying frequencies to the circuit and monitor the amplitudes at the outputs.
Compare the circuit to known filter layouts. If you see your circuit matches a well-known filter design, you can guess what filter type it is.
How do I build a high-pass filter?
To build a high-pass filter, follow these easy steps:
Select a suitable filter type (RC, RL, op-amp, etc.).
Determine your desired cutoff frequency, fc.
Calculate the components' values based on the above choices. The equations are dependent on the filter type you selected.
If you need more help, come to omnicalculator.com's high-pass filter calculator!
RC (passive)
The RC passive high-pass filter
Cutoff frequency (Fc)
The simple pendulum calculator finds the period and frequency of a pendulum.
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Logistic Regression cost optimization function
In this tutorial, we will learn how to update learning parameters (gradient descent). We'll use parameters from the forward and backward propagation
Like tutorial Must be logged in to Like Like 0
Let's begin with steps we defined in the previous tutorial and what's left to do:
• Define the model structure (data shape) (done);
• Initialize model parameters;
• Learn the parameters for the model by minimizing the cost:
- Calculate current loss (forward propagation) (done);
- Calculate current gradient (backward propagation) (done);
- Update parameters (gradient descent);
• Use the learned parameters to make predictions (on the test set);
• Analyse the results and conclude the tutorial.
In the previous tutorial, we defined our model structure, learned to compute a cost function and its gradient. In this tutorial, we will write an optimization function to update the parameters using gradient descent.
So we'll write the optimization function that will learn w and b by minimizing the cost function J. For a parameter θ, the update rule is (α is the learning rate):
\theta =\theta -\alpha d\theta
The cost function in logistic regression:
One of the reasons we use the cost function for logistic regression is that it’s a convex function with a single global optimum. You can imagine rolling a ball down the bowl-shaped function (image bellow) - it would settle at the bottom.
Similarly, to find the minimum cost function, we need to get to the lowest point. To do that, we can start from anywhere on the function and iteratively move down in the direction of the steepest slope, adjusting the values of w and b that lead us to the minimum. For this, we use the following two formulas:
w=w-\alpha \frac{\partial J\left(w, b\right)}{\partial w}\phantom{\rule{0ex}{0ex}}b=b-\alpha \frac{\partial J\left(w, b\right)}{\partial b}
In these two equations, the partial derivatives dw and db represent the effect that a change in w and b have on the cost function, respectively. By finding the slope and taking the negative of that slope, we ensure that we will always move in the minimum direction. To get a better understanding, let’s see this graphically for dw:
When the derivative term is positive, we move in the opposite direction towards a decreasing value of w. When the derivative is negative, we move toward increasing w, thereby ensuring that we’re always moving toward the minimum.
The alpha term in front of the partial derivative is called the learning rate and measures how big a step to take at each iteration. The choice of learning parameters is an important one - too small, and the model will take very long to find the minimum, too large, and the model might overshoot the minimum and fail to find the minimum.
Gradient descent is the essence of the learning process - through it, the machine learns what values of weights and biases minimize the cost function. It does this by iteratively comparing its predicted output for a set of data to the true output in the training process.
Coding optimization function:
So we will implement an optimization function, but first, let's see what are the inputs and outputs to it:
w - weights, a NumPy array of size (ROWS * COLS * CHANNELS, 1);
b - bias, a scalar;
X - data of size (ROWS * COLS * CHANNELS, number of examples);
Y - true "label" vector (containing 0 if a dog, 1 if cat) of size (1, number of examples);
num_iterations - number of iterations of the optimization loop;
learning_rate - learning rate of the gradient descent update rule;
print_cost - True to print the loss every 100 steps.
params - a dictionary containing the weights w and bias b;
grads - a dictionary containing the gradients of the weights and bias concerning the cost function;
costs - list of all the costs computed during the optimization.
# update w and b to dictionary
# update derivatives to dictionary
Let's test the above function with variables from our previous tutorial where we were writing propogate() function:
params, grads, costs = optimize(w, b, X, Y, num_iterations = 100, learning_rate = 0.009, print_cost = False)
print("w = " + str(params["w"]))
print("b = " + str(params["b"]))
print("dw = " + str(grads["dw"]))
print("db = " + str(grads["db"]))
If everything is fine as a result, you should get:
w = [[-0.49157334]
Full tutorial code:
#train_images = [TRAIN_DIR+i for i in os.listdir(TRAIN_DIR)]
#test_images = [TEST_DIR+i for i in os.listdir(TEST_DIR)]
img = cv2.imread(file_path, cv2.IMREAD_COLOR)
def prepare_data(images):
m = len(images)
X = np.zeros((m, ROWS, COLS, CHANNELS), dtype=np.uint8)
y = np.zeros((1, m))
X[i,:] = read_image(image_file)
if 'dog' in image_file.lower():
elif 'cat' in image_file.lower():
z = np.dot(w.T, X)+b # tag 1
A = sigmoid(z) # tag 2
cost = (-np.sum(Y*np.log(A)+(1-Y)*np.log(1-A)))/m # tag 5
dw = (np.dot(X,(A-Y).T))/m # tag 6
db = np.average(A-Y) # tag 7
w = np.array([[1.],[2.]])
X = np.array([[5., 6., -7.],[8., 9., -10.]])
print(grads["dw"])
print(grads["db"])
train_set_x, train_set_y = prepare_data(train_images)
test_set_x, test_set_y = prepare_data(test_images)
train_set_x_flatten = train_set_x.reshape(train_set_x.shape[0], ROWS*COLS*CHANNELS).T
test_set_x_flatten = test_set_x.reshape(test_set_x.shape[0], -1).T
print("train_set_x shape " + str(train_set_x.shape))
print("train_set_x_flatten shape: " + str(train_set_x_flatten.shape))
print("train_set_y shape: " + str(train_set_y.shape))
print("test_set_x shape " + str(test_set_x.shape))
print("test_set_x_flatten shape: " + str(test_set_x_flatten.shape))
print("test_set_y shape: " + str(test_set_y.shape))
So in this tutorial, we learned how to update learning parameters (gradient descent). You saw how we use parameters from forward and backward propagation to teach our model. In the next tutorial, we'll write a function to compute prediction.
Understanding Logistic Regression Sigmoid function
Reshaping arrays, normalizing rows and softmax function in machine learning
Vectorized and non vectorized mathematical computations
Prepare logistic regression data with Neural Networks mindset
Logistic Regressions architecture of the learning rate
Logistic Regression predict function
Final cats vs dogs logistic regression model
Best choice of learning rate in Logistic Regression
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Difference between revisions of "Dictionary:Kirchhoff’s equation" - SEG Wiki
Difference between revisions of "Dictionary:Kirchhoff’s equation"
(Fixed Equations.)
{{DISPLAYTITLE:Dictionary:Kirchhoff’s equation}}{{#category_index:K|Kirchhoff’s equation}}
<b>1</b>. An integral form of the wave equation expressing the wave function ψ<sub>''p''</sub> at the point ''P'' as the sum of wave contributions from the surroundings. Wave contributions have to allow for the traveltime from the sources to ''P'', that is, what the source does at an earlier time τ=(''t''–''r''/''V'') affects ''P'' at time ''t'', where ''r'' is the distance from the source to ''P'' and ''V'' is the velocity. The earlier time τ is called <b>retarded time</b>. ψ<sub>''p''</sub> is expressed as an integral over the volume surrounding ''P'' (to accommodate sources within the volume) plus an integral over the surface surrounding the volume (to accommodate sources from outside). In source-free space in terms of the values of ψ and its derivative on a surrounding surface ''S'' at the preceding time (''t''–''r''/''V''):
'''1'''. An integral form of the wave equation expressing the wave function <math>\psi_p </math> at the point ''P'' as the sum of wave contributions from the surroundings. Wave contributions have to allow for the traveltime from the sources to ''P'', that is, what the source does at an earlier time <math> \tau = (t-r/V) </math> affects ''P'' at time ''t'', where ''r'' is the distance from the source to ''P'' and ''V'' is the velocity. The earlier time <math>\tau </math> is called <b>retarded time</b>. <math>\psi _p </math> is expressed as an integral over the volume surrounding ''P'' (to accommodate sources within the volume) plus an integral over the surface surrounding the volume (to accommodate sources from outside). In source-free space in terms of the values of <math> \psi </math> and its derivative on a surrounding surface ''S'' at the preceding time <math>(t-r/V) </math>:
<center>ψ<sub>''p''</sub>=(1/4π)∫∫{[ψ]∂(1/''r'')/∂''n''–(1/''Vr'')∂''r''/∂''n''[∂ψ/∂''t'']–(1/''r'')[∂ψ/∂''n'']}''ds''.</center>
<center><math> \psi _p = \frac{1}{4 \pi} \iint\left\{ [\psi] \frac{\partial\frac{1}{r}}{\partial n} - \frac{1}{Vr}\frac{\partial r}{\partial n }\left[\frac{\partial \psi}{\partial t}\right] - \frac {1} {r} \left[\frac{\partial \psi}{\partial n}\right] \right\} ds </math>.</center>
The terms in brackets are evaluated at the retarded time <math>\tau = (t-r/V)</math>, ''r'' is the distance from ''P'' to points on the surface ''S'', and <b>n</b> is a unit vector normal to ''S''. The Kirchhoff integral equation used in migration can be written
<center><math>\psi(x,z,t)=\frac{z}{\pi}\int \left[1/r^3-(2/Vr^2)\times \left(\frac{\partial}{\partial t} \right)\psi(x',0,t+T)\right]dx'</math>,</center>
The terms in brackets are evaluated at the retarded time τ=(''t''–''r''/''V''); ''r'' is the distance from ''P'' to points on the surface ''S'', and <b>n</b> is a unit vector normal to ''S''. The Kirchhoff integral equation used in migration can be written
where ''x'' '; is position at ''z''=0, <math>\tau </math>,is the two-way time 2''r''/''V'', and ''r'' is the distance from ''x'' ' to ''x''. For ''r'' much longer than a wavelength this simplifies to the Rayleigh-Sommerfeld approximation,
<center>ψ(''x'',''z'',''t'')=(''z''/π) ∫ [1/''r''<sup>3</sup>–(2/''Vr''<sup>2</sup>)(∂/∂''t'')ψ(''x''′,0,''t''+τ)]''dx''′,</center>
<center><math>\psi_p(x,T,t)=-\frac{2T}{\pi V^2} \int \frac {1}{T^2} \frac {\partial}{\partial t} \psi (x',0,t+T) dx' </math>,</center>
where ''T''=2''z''/''V''=vertical traveltime. This expresses migration by integration along a diffraction curve.
where ''x''′ is position at ''z''=0, τ is the two-way time 2''r''/''V'', and ''r'' is the distance from ''x''′ to ''x''. For ''r'' much longer than a wavelength this simplifies to the Rayleigh-Sommerfeld approximation,
'''2'''. The radiation law that the ratio of emissivity to absorptance depends only on the wavelength and temperature, or that it is the same for all bodies as for an ideal blackbody for any wavelength at the given temperature.
<center>ψ<sub>''p''</sub>(''x'',''T'',''t'')=–(2''T''/π''V''<sup>2</sup>) ∫ (1/<b>T</b><sup>2</sup>)(∂/∂''t'')ψ(''x''′,0,''t''+''T'')''dx''′,</center>
where ''T''=2''z''/''V''=vertical traveltime. This expresses migration by integration along a diffraction curve. <b>2</b>. The radiation law that the ratio of emissivity to absorptance depends only on the wavelength and temperature, or that it is the same for all bodies as for an ideal blackbody for any wavelength at the given temperature.
1. An integral form of the wave equation expressing the wave function
{\displaystyle \psi _{p}}
at the point P as the sum of wave contributions from the surroundings. Wave contributions have to allow for the traveltime from the sources to P, that is, what the source does at an earlier time
{\displaystyle \tau =(t-r/V)}
affects P at time t, where r is the distance from the source to P and V is the velocity. The earlier time
{\displaystyle \tau }
is called retarded time.
{\displaystyle \psi _{p}}
is expressed as an integral over the volume surrounding P (to accommodate sources within the volume) plus an integral over the surface surrounding the volume (to accommodate sources from outside). In source-free space in terms of the values of
{\displaystyle \psi }
and its derivative on a surrounding surface S at the preceding time
{\displaystyle (t-r/V)}
{\displaystyle \psi _{p}={\frac {1}{4\pi }}\iint \left\{[\psi ]{\frac {\partial {\frac {1}{r}}}{\partial n}}-{\frac {1}{Vr}}{\frac {\partial r}{\partial n}}\left[{\frac {\partial \psi }{\partial t}}\right]-{\frac {1}{r}}\left[{\frac {\partial \psi }{\partial n}}\right]\right\}ds}
The terms in brackets are evaluated at the retarded time
{\displaystyle \tau =(t-r/V)}
, r is the distance from P to points on the surface S, and n is a unit vector normal to S. The Kirchhoff integral equation used in migration can be written
{\displaystyle \psi (x,z,t)={\frac {z}{\pi }}\int \left[1/r^{3}-(2/Vr^{2})\times \left({\frac {\partial }{\partial t}}\right)\psi (x',0,t+T)\right]dx'}
where x '; is position at z=0,
{\displaystyle \tau }
,is the two-way time 2r/V, and r is the distance from x ' to x. For r much longer than a wavelength this simplifies to the Rayleigh-Sommerfeld approximation,
{\displaystyle \psi _{p}(x,T,t)=-{\frac {2T}{\pi V^{2}}}\int {\frac {1}{T^{2}}}{\frac {\partial }{\partial t}}\psi (x',0,t+T)dx'}
where T=2z/V=vertical traveltime. This expresses migration by integration along a diffraction curve.
2. The radiation law that the ratio of emissivity to absorptance depends only on the wavelength and temperature, or that it is the same for all bodies as for an ideal blackbody for any wavelength at the given temperature.
Retrieved from "https://wiki.seg.org/index.php?title=Dictionary:Kirchhoff’s_equation&oldid=22426"
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What is the Archimedes' principle? – Archimedes' principle definition
Buoyancy and the law of floatation
Buoyancy force calculation example
How to use the Archimedes' principle calculator
Omni's Archimedes' principle calculator helps you understand the concept of buoyancy. You can also use it to determine objects' densities and changes in mass and weight when they are submerged in a fluid.
Have you ever wondered why certain objects float and others sink? If you have, you have come to the right place.
Continue reading this article to learn about Archimedes' principle and his formula. You will also find examples of buoyancy force calculations and some interesting applications of Archimedes' principle.
Let us first start with learning what Archimedes' principle is.
When we immerse an object in any fluid, there is a net upward force on the object. This upward force is called the buoyant force or buoyancy.
The origin of the buoyant force lies in the fact that the pressure increases with an increase in depth in a fluid (remember the hydrostatic pressure equation). Therefore, the upward force applied by the fluid to the bottom of the object is greater than the downward force applied to the top of the object. Check out our dedicated tool to learn more about buoyancy and perform buoyancy experiments at home.
According to Archimedes' principle, when an object is partially or wholly immersed in a fluid, it displaces some of the fluid. The buoyant force acting on the object is equal to the weight of the fluid displaced by it.
To understand what the principle means, let us try to derive a formula for the Archimedes principle.
Let us consider an object of height
h-x
M
immersed in a fluid of density
\rho
(see figure 1). If
a
is the cross-sectional area of its top and bottom face, we can write expressions for the vertically downward (
F_1
) and upward force (
F_2
) on the object as:
Fig 1: Pressure on an object immersed in a fluid increases with depth.
\small \begin{align*} F_1 &= p_1 a = x \rho g a\\ F_2 &= p_2 a = h \rho g a \end{align*}
p_1 = x \rho g
– Pressure on the top face of the object;
p_2 = h \rho g
– Pressure on the bottom face of the object;
x
– Depth of the top surface of the object in the fluid; and
h
– Depth of the bottom surface of the object in the fluid.
h>x
F_2 > F_1
. The net upward force or buoyant force acting on the object is:
\small \begin{align*} F_B & = F_2 - F_1\\ & = (h -x) \rho g a\\ & = [(h-x)\times a]\rho g\\ & = V\rho g \end{align*}
V
is the volume of the object, and it is equal to the volume of the displaced fluid
V_{fl}
m
is the mass of the displaced fluid, we can rewrite the above equation as:
\small \begin{align*} F_B & = V_{fl} \rho g \\ & = m g \end{align*}
In other words, the buoyant force acting on the object is equal to the weight of the fluid displaced by the object.
Since the true weight of the object is
W_{obj} = M g
, and the upward force on it is
F_B= m g
, the observed weight of the object in the fluid is (see figure 2):
Fig 2: Buoyant force experienced by an object immersed in a fluid.
\small \begin{align*} W_{obs} &= W_{obj} - F_B\\ &= Mg - m g \end{align*}
This means that when an object is immersed in a fluid, its observed weight becomes less than its true weight by an amount equal to the weight of the fluid displaced by the object.
Now, if the weight of the fluid displaced by the immersed body is greater than or equal to the body's weight, the body will float. Otherwise, it will sink. This is also known as the law of floatation.
Another fact that follows from the law mentioned above is that if the average density of an object is less than that of the fluid it is immersed in, it will float. In contrast, objects with an average density greater than the fluid will sink.
Now, as an example, let us calculate the buoyant force on a
1.00 \times 10^6 \ \rm{kg}
aluminum block wholly immersed in water.
First, we need to find the volume (
V_{\text{Al}}
) of the block. We can calculate this by substituting the density of aluminium (
\rho_{\text{Al}} = 2.7 \times 10^3\ \rm{kg/m^3}
) in the given formula:
\quad\scriptsize \quad \begin{align*} V_{\text{Al}} &= \frac{M_{\text{Al}}}{\rho_{\text{Al}}}\\\\ &= \frac{1.00 \times 10^6}{2.7 \times 10^3} = 370.4\ \rm{m^3} \end{align*}
Since the block is completely immersed in water, the volume of water displaced is the same as the volume of the block:
\quad\scriptsize \quad V_{\text{Al}} = V_{\rm{H_2O}} = 370.4 \ \rm{m^3}
Now, we will determine the mass of water displaced by the block by substituting the density of water (
\rho_{\rm{H_2O}} = 1.0 \times 10^3\ \rm{kg/m^3}
) in the formula for density:
\quad\scriptsize \quad \begin{align*} M_{\rm{H_2O}} &= V_{\rm{H_2O}} \times \rho_{\rm{H_2O}} \\ &= 370.4 \times 1.0 \times 10^3 \ \rm {kg}\\ & = 3.70 \times 10^5\ \rm{kg} \end{align*}
Since the buoyant force acting on the block is equal to the weight of water displaced, we can calculate the buoyancy force using Archimedes' principle as:
\quad\scriptsize \quad \begin{align*} F_B&= M_{\rm{H_2O}} \times g \\ &= 3.70 \times 10^5\ \rm{kg} \times 9.8\ \rm {m/s^2} \\ &= 3.63 \times 10^6\ \rm{N} \\ \end{align*}
You can also use our Archimedes' principle calculator to solve the same problem with just a few clicks.
In this section, we will try to determine the density of an unknown object using our Archimedes' principle calculator. Let the true mass (mass in the air) and apparent mass (when immersed in water) of a rock be 540 g and 340 g, respectively. We will calculate the average density of this rock as follows:
Enter the true mass and the apparent mass of the rock as 540 g and 340 g in the respective fields.
Using the drop-down menu, choose fluid type as water. The density field will auto-populate. You can also enter the density of the fluid manually.
The Archimedes' principle calculator will display the density of the rock as 2.70 g/cm3 and the volume of the rock as 200 cm3. It will also show the buoyant force acting on the rock as 1.96 N and a message whether the rock will float or sink.
Let us consider another example, where we drop a 5 kg rock into a cylindrical container with a base surface area of 700 cm2. If the height of the water inside the container changes by 2.86 cm, we can calculate the buoyant force as:
Click on the Advanced mode button beneath the calculator and type the change in the height of the fluid (2.9 cm) and surface area of the fluid (700 cm2).
Enter the true mass of the object (5 kg).
The calculator will display the volume of the fluid displaced (2030 cm3) and the buoyant force (19.89 N).
Note: You can enter whatever variables you know, and the calculator will compute the rest.
How do I calculate density using Archimedes' principle?
To calculate the density of an object using Archimedes' principle, follow the given instructions:
Measure the object's mass in the air (ma) and when it is completely submerged in water (mw).
Calculate the loss in mass (ma - mw), which is also the mass of displaced water.
Determine the volume of displaced water by dividing the mass of displaced water by the density of water, i.e., 1000 kg/m3. This value is also the volume of the object.
Find out the object's density by dividing its mass by volume.
What is the buoyant force on a coin immersed in water if it displaces 200 g of liquid?
1.96 N. The buoyant force acting on the coin is equal to the weight of the water displaced by it. The weight of 200 g (0.2 kg) of water is W = 0.2 kg × 9.8 m/s2 = 1.96 N. Hence the buoyant force is also 1.96 N.
What is the formula for calculating buoyant force?
The formula for calculating buoyant force FB on a body immersed in a fluid is:
FB = V × g × d
V – Volume of the fluid displaced by the immersed body;
g – Acceleration due to gravity; and
d – Density of the fluid.
Some applications of Archimedes' principle are:
Designing ships and submarines;
Hydrometers to determine the specific gravity or density of fluids;
Geology to determine the densities and hence purity of unknown objects; and
How is floating explained by the Archimedes principle?
According to the Archimedes principle, whether an object will float or sink in a fluid depends on the following criteria:
If the true weight of a body immersed in a fluid is greater than the buoyant force acting on it, the body will sink to the bottom of the liquid.
If the true weight is less than the buoyant force, the body will float.
🤔 Enter the value of two or more known variables and the calculator will compute the rest.
True mass of the object
True weight of the object
Apparent mass of the object
Apparent weight of the object
Volume of the fluid displaced
Mass of the fluid displaced
API gravityBallistic coefficientBernoulli equation… 36 more
The Newton's law of cooling calculator lets you find the time it takes for different objects to cool down.
Use this kVA to amperage calculator to find how strong an electric current is at a given apparent power in kVA, whether from a single-phase or 3-phase load distribution.
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TB67S128FTG Stepper Motor Driver Carrier Pololu 2998 – MakerSupplies Singapore
TB67S128FTG Stepper Motor Driver Carrier Pololu 2998
TB67S128FTG Stepper Motor Driver Carrier, bottom view with dimensions.
This product is a carrier board or breakout board for Toshiba’s TB67S128FTG stepper motor driver; we therefore recommend careful reading of the TB67S128FTG datasheet (2MB pdf) before using this product. This stepper motor driver offers microstep resolutions down to 1/128 of a step, and it lets you control one bipolar stepper motor at up to approximately 2.1 A per phase continuously (5 A peak) without a heat sink or forced air flow (see the Power Dissipation Considerations section below for more information.) The board breaks out every control pin and output of the TB67S128FTG, making all of the driver’s features available to the user.
Two interface modes to select from:
clock mode for simple step and direction control
serial mode for controlling the driver’s many features through a serial interface (this mode also allows for serial control of the current limit)
Eight different step modes: full-step, half-step, 1/4-step, 1/8-step, 1/16-step, 1/32-step, 1/64-step, and 1/128-step
Four different decay modes: mixed decay (two timing ratios), fast decay, or Advanced Dynamic Mixed Decay (ADMD), which dynamically switches between slow and fast decay modes by monitoring the state of current decay (not according to fixed timing)
Can deliver up to approximately 2.1 A per phase continuously (5 A peak) without additional cooling
Protection against over-current/short-circuit and over-temperature
Carrier board breaks out all of the TB67S128FTG pins in a compact size (1.2″ × 1.6″)
This product ships with all surface-mount components installed as shown in the product picture. However, soldering is required for assembly of the included through-hole parts. The following through-hole parts are included:
Two 1×16-pin breakaway 0.1″ male header strips
One 0.1″ shorting block (for optionally connecting IOREF to the neighboring VCC pin when using this driver in 5 V systems)
The 0.1″ male headers can be broken or cut into smaller pieces as desired and soldered into the smaller through-holes. These headers are compatible with solderless breadboards, 0.1″ female connectors, and our premium and pre-crimped jumper wires. The terminal blocks can be soldered into the larger holes to allow for convenient temporary connections of unterminated power and motor wires (see our short video on terminal block installation). You can also solder your motor leads and other connections directly to the board for the most compact installation.
Minimal wiring diagram for connecting a microcontroller to a TB67S128FTG stepper motor driver carrier.
The driver requires a motor supply voltage of 6.5 V to 44 V to be connected across VIN and GND. This supply should be capable of delivering the expected stepper motor current.
A 5 V output from the TB67S128FTG’s internal regulator is made available on the VCC pin. This output can supply up to 5 mA to external loads, and it can optionally be used to supply the neighboring IOREF pin.
Four, six, and eight-wire stepper motors can be driven by the TB67S128FTG if they are properly connected; a FAQ answer explains the proper wirings in detail.
VCC Regulated 5 V output: this pin gives access to the voltage from the internal regulator of the TB67S128FTG. The regulator can only provide a few milliamps, so the VCC output should only be used for logic inputs on the board (such as the neighboring IOREF pin), not for powering external devices.
IOREF All of the board signal outputs are open-drain outputs that are pulled up to IOREF, so this pin should be supplied with the logic voltage of the controlling system (e.g. 3.3 V for use in 3.3 V systems). For convenience, it can be connected to the neighboring V5 (OUT) pin when it is being used in a 5 V system.
VREF Voltage reference pin for setting the current limit. This pin is connected to the potentiometer. See the Current limiting section below for more information.
MODE2 LOW Step resolution selection pins.
STANDBY LOW Standby mode input. By default, the driver pulls this pin low, disabling the motor outputs and internal oscillating circuit; it must be driven high to enable the driver.
MO This open-drain output is low when the driver’s internal electrical angle is at its initial value (the value after reset); otherwise, the board pulls it up to IOREF.
LO1 HIGH Error outputs: these pins drive low to indicate that an error condition has occurred; otherwise, the board pulls them up to IOREF. The specific error can be determined by the state of both error pins.
IF_SEL LOW Interface select pin. By default, the driver pulls this pin low, setting the driver in CLK mode, where the CLK input steps the electrical angle of the stepper motor. When driven high, the driver is in serial input mode, where settings can be configured and the motor can be controlled through a serial interface.
RS_SEL LOW RS mode select pin. By default, the driver pulls this pin low, enabling internal current sensing. When driven high, current is sensed through external resistors added to the RS_x pins.
EDG_SEL LOW CLK edge setting pin. By default, the driver pulls this pin low, causing the driver to take a step (advance the motor’s electrical angle) on each rising edge of the CLK signal. When driven high, the driver takes a step on both the rising and falling edges of the CLK signal.
GAIN_SEL LOW VREF gain setting pin; see Current limiting below.
AGC HIGH This pin determines whether Active Gain Control (AGC) is enabled. See the datasheet and the Active Gain Control section below for details about the AGC feature.
MDT1 LOW Decay mode selection pins; see Decay modes below.
TORQE0,
TORQE2 LOW Digital current control pins; see Current limiting below.
RS_A,
RS_B Current sense resistor connection pins. Optional external current-sensing resistors can be added to these pins; see Current limiting below.
Stepper motors typically have a step size specification (e.g. 1.8° or 200 steps per revolution), which applies to full steps. A microstepping driver such as the TB67S128FTG allows higher resolutions by allowing intermediate step locations, which are achieved by energizing the coils with intermediate current levels. For instance, driving a motor in quarter-step mode will give the 200-step-per-revolution motor 800 microsteps per revolution by using four different current levels.
The resolution (step size) selector inputs (MODE0, MODE1, and MODE2) enable selection from the seven step resolutions according to the table below. These three pins have internal 100 kΩ pull-down resistors, so leaving these three microstep selection pins disconnected results in full-step mode. For the microstep modes to function correctly, the current limit must be set low enough (see below) so that current limiting gets engaged. Otherwise, the intermediate current levels will not be correctly maintained, and the motor will skip microsteps.
Low Low High Half step
High Low Low 1/16 step
The TB67S128FTG supports four different decay modes that can be selected using the MDT0 and MDT1 pins according to the table below. Both of these pins have internal 100 kΩ pull-down resistors, so the default decay mode is 37.5% mixed decay.
Low Low 37.5% mixed decay Starts as slow decay; switches to fast decay for the last 37.5% of each PWM cycle
Low High 50% mixed decay Starts as slow decay; switches to fast decay for the last 50% of each PWM cycle
High Low Fast decay
High High Advanced Dynamic Mixed Decay (ADMD) Dynamically switches between slow and fast decay modes by monitoring the state of current decay (not according to fixed timing)
See the datasheet for more details about these decay modes. We recommend tying both MDT pins high to enable Advanced Dynamic Mixed Decay for most applications.
The chip has two different inputs for controlling its power states: STANDBY and ENABLE. (The chip’s datasheet uses the name STANDBY, but we call the pin STANDBY on our board based on the logic of how it works.) For details about these power states, see the datasheet. Please note that the driver pulls both of these pins low through internal 100 kΩ pull-down resistors. The default states of these pins prevent the driver from operating; both must be high to enable the driver (they can be connected directly to a logic high voltage between 2 V and 5.5 V, such as the driver’s own VCC output, or they can be dynamically controlled via connections to digital outputs of an MCU).
When the RESET pin is driven high, the driver resets its internal electrical angle (the state in the translator table that it is outputting) to an initial value of 45°. This corresponds to +100% of the current limit on both coils in full step mode and +71% on both coils in other microstep modes. Note that, unlike the reset pin on many other stepper drivers, the RESET pin on the TB67S128FTG does not disable the motor outputs when it is asserted: when RESET is high, the driver will continue supplying current to the motor, but it will not respond to step inputs on the CLK pin.
The MO pin drives low to indicates when the driver’s electrical angle is equal to the initial value of 45° (immediately after reset and whenever the driver has stepped a full cycle through the translator table after that); it is pulled up to IOREF otherwise.
The TB67S128FTG can detect several fault (error) states that it reports by driving one or both of the LO pins low (the datasheet describes what each combination of LO0 and LO1 means). Otherwise, these pins are pulled up to IOREF by the board. Errors are latched, so the outputs will stay off and the error flag(s) will stay asserted until the error is cleared by toggling standby mode with the STANDBY pin or disconnecting power to the driver.
The TB67S128FTG supports such active current limiting, and the trimmer potentiometer on the board can be used to set the current limit:
Another way to set the current limit is to measure the VREF voltage and calculate the resulting current limit. The VREF voltage is accessible on the VREF pin. The driver’s RS_SEL and GAIN_SEL pins are pulled low by default, selecting internal current sensing and making the current limit relate to VREF as follows:
\text{current limit}=\text{VREF}×1.56 \frac{\text{A}}{\text{V}}
(GAIN_SEL = L)
So, for example, if you have a stepper motor rated for 1 A you can set the current limit to 1 A by setting the reference voltage to about 0.64 V.
If the GAIN_SEL pin is high, the VREF gain (multiplier) is reduced by half, and the relationship between the current limit and VREF instead becomes:
\text{current limit}=\text{VREF}×0.78 \frac{\text{A}}{\text{V}}
(GAIN_SEL = H)
Alternatively, the driver can measure motor current with external sense resistors instead of using internal current sensing. To use external sensing, cut the connections between the RS_A and RS_B pins and the adjacent GND pins, connect appropriate resistors between each RS pin and GND, and drive the RS_SEL pin high. See the TB67S128FTG datasheet for information about setting the current limit in this mode.
The TB67S128FTG also features three inputs (TORQE0, TORQE1, and TORQE2) that can be used for digital control of the current limit, applying a multiplier between 10% and 100% (the default) to the current limit set by the VREF voltage. See the driver’s datasheet for details about these pins and their available settings.
The TB67S128FTG has a feature called Active Gain Control, or AGC, that automatically optimizes the motor current by sensing the load torque applied to the motor and dynamically reducing the current below the full amount. This allows it to minimize power consumption and heat generation when the motor is lightly loaded, but if the driver senses an increased load, it will quickly ramp the current back up to the full amount to try to prevent a stall.
AGC is configured with six pins (AGC, CLIM0, CLIM1, FLIM, BOOST, and LTH) that are brought out along the bottom edge of the board, and all of the pins except LTH are also connected to surface-mount jumpers on the back side of the board that let you reconfigure them without external components or connections. See the driver’s datasheet for details about what each pin does and what input states it accepts.
By default, AGC and CLIM0 are pulled up to IOREF through 10 kΩ pull-up resistors. Cutting the trace that goes between the pads of each jumper allows the chip’s internal 100 kΩ pull-down to pull that pin low. Alternatively, you can simply drive or tie the pin low with the corresponding through-hole.
CLIM1, FLIM, and BOOST (BST) are four-state logic inputs that can be tied high to VCC, pulled high through a 100 kΩ resistor, pulled low through a 100 kΩ resistor, or tied low to GND. Our carrier board connects each of these pins to VCC through a 100 kΩ pull-up resistor by default. As shown in the picture below, the trace between the VCC pad and the pad labeled “R” (connected to the pin through the 100 kΩ resistor) should be cut before selecting a different state by shorting across the desired two pads (although it is also possible to override the 100 kΩ pull-up by tying the pin to VCC or GND without cutting the trace).
AGC HIGH AGC enabled
Finally, the board pulls the LTH pin low through a 100 kΩ resistor to set a normal AGC detection threshold.
The TB67S128FTG driver IC has a maximum current rating of 5 A per coil, but the actual current you can deliver depends on how well you can keep the IC cool. The carrier’s printed circuit board is designed to draw heat out of the IC, but to supply more than the specified continuous current per coil, a heat sink or other cooling method is required.
Schematic diagram of the TB67S128FTG Stepper Motor Driver Carrier.
This diagram is also available as a downloadable pdf: TB67S128FTG stepper motor driver carrier schematic (183k pdf)
potentiometer, digital, SPI-programmable
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A Novel Technique for Assessing Turbine Cooling System Performance | J. Turbomach. | ASME Digital Collection
S. Luque,
, Parks Road, Oxford OX1 3PJ,
Luque, S., and Povey, T. (November 15, 2010). "A Novel Technique for Assessing Turbine Cooling System Performance." ASME. J. Turbomach. July 2011; 133(3): 031013. https://doi.org/10.1115/1.4001232
A new experimental technique for the accurate measurement of steady-state metal temperature surface distributions of modern heavily film-cooled turbine vanes has been developed and is described in this paper. The technique is analogous to the thermal paint test but has been designed for fundamental research. The experimental facility consists of an annular sector cascade of high pressure (HP) turbine vanes from a current production engine. Flow conditioning is achieved by using an annular sector of deswirl vanes downstream of the test section, being both connected by a three-dimensionally contoured duct. As a result, a transonic and periodic flow, highly representative of the engine aerodynamic field, is established: Inlet turbulence levels, mainstream Mach and Reynolds numbers, and coolant-to-mainstream total pressure ratio are matched. Since the fully three-dimensional nozzle guide vane (NGV) geometry is used, the correct radial pressure gradient and secondary flow development are simulated and the cooling flow redistribution is engine-realistic. To allow heat transfer measurements to be performed, a mainstream-to-coolant temperature difference (up to
33.5°C
) is generated by using two steel-wire mesh heaters, operated in series. NGV surface metal temperatures are measured (between
20°C
40°C
) by wide-band thermochromic liquid crystals. These are calibrated in situ and on a per-pixel basis against vane surface thermocouples, in a heating process that spans the entire color play and during which the turbine vanes can be assumed to slowly follow a succession of isothermal states. Experimental surface distributions of overall cooling effectiveness are presented in this paper. By employing resin vanes of the same geometry and cooling configuration (to implement adiabatic wall thermal boundary conditions) and the transient liquid crystal technique, surface distributions of external heat transfer coefficient and film cooling effectiveness can be acquired. By combining these measurements with those from the metal vanes, the results can be scaled to engine conditions with a good level of accuracy.
blades, cooling, hydraulic turbines, nozzles, temperature measurement, turbines, turbulence
Coolants, Cooling, Cooling systems, Engines, Flow (Dynamics), Heat transfer, Liquid crystals, Nozzle guide vanes, Pressure, Temperature, Turbines, Pressure gradient, Turbulence, Film cooling, Thermocouples, Cascades (Fluid dynamics), Calibration, Metals, Reynolds number, Steady state
Progress Towards the Understanding and Predicting Heat Transfer in the Turbine Gas Path
An Experimental Study of Endwall Airfoil Surface Heat Transfer in a Large Scale Turbine Blade Cascade
Technical Memorandum Report No. ARL-TR-2029.
Influence of Surface Roughness on Heat Transfer and Effectiveness for a Fully Film Cooled Nozzle Guide Vane Measured With Wide Band Liquid Crystals and Direct Heat Flux Gages
Effect of Unsteady Wake on Detailed Heat Transfer Coefficient and Film Cooling Effectiveness Distributions for a Gas Turbine Blade
Experimental Study on the Cooling Performance of a Turbine Nozzle With an Innovative Internal Cooling Structure
Experimental Verification of Film-Cooling Concepts on a Turbine Vane
Rådeklint
A New Test Facility for Testing of Cooled Gas Turbine Components
A New Annular Sector Cascade Test Facility to Investigate Steady State Cooling Effects
Proceedings of the XIV Bi-Annual Symposium on Measuring Techniques in Transonic and Supersonic Flow in Cascades and Turbomachines
, University of Limerick, Ireland, Sept. 2–4.
Experimental Investigation of the Periodicity in a Sector of an Annular Turbine Cascade
Proceedings of the XV Bi-Annual Symposium on Measuring Techniques in Transonic and Supersonic Flow in Cascades and Turbomachines
, Florence, Italy, Sept. 21–22.
Proceedings of the XVI Bi-Annual Symposium on Measuring Techniques in Transonic and Supersonic Flow in Cascades and Turbomachines
, Cambridge, UK, Sept.
The Design and Performance of a Transonic Flow Deswirling System
Proceedings of the IMechE Advances of CFD in Fluid Machinery Design Seminar
, London, UK, Jun. 13.
Proceedings of the Eighth European Turbomachinery Conference
, Graz, Austria, Mar. 23–27.
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How to use the thermal resistance calculator?
Thermal resistance challenge!
Thermal resistance formula for common shapes
Thermal and electrical resistance analogy
How to use the thermal resistance calculator to tackle the challenge
If you're looking for a faster way to find the thermal resistance of a plate, hollow cylinder, or hollow sphere, then our thermal resistance calculator is the right place. Once you select the shape of your object and insert geometric and material properties into our tool, you will end up getting the thermal resistance value. In essence, you can use this tool as a thermal conductivity to thermal resistance calculator for a given shape.
As an extension to thermal resistance calculation, if you are dealing with an object with a hollow geometrical shape, our calculator estimates an extra parameter called the critical radius of insulation, which acts as a limiting outer radius beyond which the object will serve as an insulator.
Now that you know how to use our calculator, would you like to understand the exciting concepts and equations behind it? Please read on to learn more about thermal resistance and the associated thermal resistance formula.
The following points will guide you on how to use our calculator to find the thermal resistance of an object:
First, select the shape of your object from the drop-down menu. Our calculator has three shapes: a plate, a hollow cylinder, and a hollow sphere.
After selecting the shape of your object, you can choose an available material from the drop-down menu; it will automatically fill the "Conductance (k)" field. If you can't find the required material, please enter the thermal conductance value directly into the "Conductance (k)" field.
Based on the selected object's shape, our calculator will show you different geometrical options:
First, enter the thickness of your plate.
Now, enter the plate's cross-sectional area (A). If you have the explicit values for length and width, you can use the advanced mode at the bottom of the calculator to insert them.
A hollow cylinder
Enter the length of the hollow cylinder, following that enter inner and outer radii.
Enter inner and outer radii of the hollow sphere.
After entering the above values, you will get the thermal resistance of your object.
In addition, if your object is a hollow cylinder or a hollow sphere, the following points will guide you to find the critical radius of insulation:
Insert thermal conductance value in the "Conductance (k)" field; and
Now, insert the convective heat transfer coefficient value of your hollow object in the "Heat transfer coefficient" field to get the critical radius of insulation in the "Critical radius" field.
To understand the importance of thermal resistance, let's tackle a challenge. Let's assume somebody told you to choose the best insulating container out of three containers to preserve the coldness of your cold drink. And imagine that it's a hot day, so you need good insulation to keep the cold drink at cold temperature. As shown in the figure, you have three containers; glass, wood (oak), and aluminum.
Three containers for thermal resistance challenge made of glass, wood, and aluminium.
We assume all three containers have the same shape, dimensions and have the following thermal conductivity values:
\text{Glass = 0.78 W/(m}\cdot\text K)
\text{Wood (oak) = 0.17 W/(m}\cdot\text K)
\text{Aluminum = 237 W/(m}\cdot\text K)
At the end of this article, we will find the answer by using our thermal resistance calculator. Firstly, let's try to understand what thermal resistance is.
Formally, thermal resistance
R
is defined as the ratio of the temperature difference
T_2-T_1
and the heat flow
Q_{1-2}
between two points
1
2
\begin{equation} \quad R=\frac{T_2-T_1}{Q_{1-2}} \end{equation}
The SI unit for thermal resistance is K/W.
Informally, we define thermal resistance as the ability of the object of specific geometry and material to resist the flow of heat. Thermal is derived from the Greek word therme, which means heat. Combined with resistance becomes thermal resistance, which means heat obstruction.
Unlike thermal conductance, which depends only on the object's material, thermal resistance depends both on the material and shape of the object. Good insulators are objects which have a higher thermal resistance. The following section will look at various shapes and corresponding thermal resistance formulas.
Common shapes are the geometry of the objects present all around us. This article focuses on three common shapes: a plate, a hollow cylinder, and a hollow sphere. You can find each object's geometry and corresponding thermal resistance formula below:
A plate is a thin rectangular block whose thickness (
t
) is lesser than its length (
l
) and width (
w
l>>t
w>>t
as shown in the figure below. The figure shows a plate with thickness (
t
l
), and width (
w
); with temperature
T_1
T_2
across the thickness (
t
A rectangular plate with different temperatures on both sides and indicated dimensions.
Some of the examples of a plate are a windowpane and a wall. Below is the thermal resistance formula:
\begin{equation} \quad R_{\text{plate}}=\frac{t}{kA} \end{equation}
R_{\text{plate}}
– Thermal resistance in
\text{K/W}
k
– Thermal conductivity of the material
\text{W/m}\cdot\text K
t
– Length of the plate in
m
A
– Cross-sectional area, i.e.,
A=l \times w
\text m^2
A hollow cylinder and hollow sphere
A hollow cylinder is a tubular structure of finite length
L
r_1
r_2
as its internal and outer radii. When you cut out a sphere of radius
r_1
from the sphere of radius
r_2
, the obtained spherical geometry is called a hollow sphere.
In the below figure we can see a hollow cylinder of length
L
with inner radius
r_1
r_2
; with temperature
T_1
T_2
r_1
r_2
A hollow cylinder with different temperatures on both surfaces and indicated radii.
In the next figure, we have a hollow sphere with inner radius
r_1
r_2
T_1
T_2
r_1
r_2
A hollow sphere with different temperatures on both surfaces and indicated radii.
The thermal resistance equation for both geometries is given by:
\begin{equation} \quad R_{\text{cylinder}} = \frac{\ln(r_2/r_1)}{2\pi Lk} \end{equation}
\begin{equation} \quad\ \ R_{\text{sphere}} = \frac{r_2-r_1}{4\pi r_1r_2k} \end{equation}
R_{\text{cylinder}}
R_{\text{sphere}}
– Thermal resistances for hollow cylinder and hollow sphere respectively in
\text{K/W}
r_1
r_2
– Internal and outer radius respectively of both the geometries in
\text m
(refer to the figure);
L
– Length of the hollow cylinder in
m
k
\text{W/(m}\cdot\text K)
💡 The thermal resistance concept is an essential aspect of rocket launchers that primarily use cryogenic propellants. Cryogenic propellants are fuels and oxidizers kept at very low temperatures. For example, liquid oxygen is stored at 90 K, and liquid hydrogen is kept at 20 K. As a result, these tanks must be insulated to keep the temperature very low. One of the famous examples of a cryogenic launcher is SpaceX's Starship which uses liquid oxygen and liquid methane as its fuels.
Many domains in science provide us with equations that are surprisingly similar. For example, coulomb force in electromagnetism and gravitational force in Newtonian mechanics have similar governing equations. In the same line, we can analyze the thermal resistance equation in heat transfer by drawing an analogy with the electric current flow equation. This section will briefly look at how we can achieve that.
In the 17th century, a French mathematician called Jean-Baptiste Joseph Fourier developed an empirical relation for conduction heat transfer, known as Fourier's law of heat conduction. A simple form (the one-dimensional case) of Fourier's law is the equation (5), which describes the heat flow across a medium with temperature difference:
\begin{equation} \quad Q=kA\frac{\Delta T}{\Delta x} \end{equation}
Q
– Rate of heat flow given in
\text W
across the temperature difference
\Delta T
\text K
A
– Cross-section area perpendicular to the direction of heat flow in
\text m^2
\Delta x
– Distance over which the heat flows.
If our object is a plate, then in the above equation, we replace
\Delta x
by thickness
L
, with some rearrangement of terms, the equation becomes:
\begin{equation} \quad L/kA=\Delta T/Q \end{equation}
Now, let us look at Ohm's law, which describes the current flow across a voltage difference:
\begin{equation} \quad \Delta U= IR \end{equation}
\Delta U
– Electric potential difference between two terminals in
\text V
(volts);
I
– Current flow across the terminal in
\text A
(amperes); and
R
– Resistance offered by the object against the current flow given in
\Omega
(ohms).
\begin{equation} \quad R=\Delta U/I \end{equation}
By comparing equation (6) and equation (8), we can notice the following analogies:
Voltage difference across the terminal is similar to the temperature difference across the medium (or the object);
Current is similar to heat flow; and
Thus, the term
L/(kA)
is equivalent to having a thermal resistance for Fourier's law similar to electrical resistance for Ohm's law.
For the plate case, if we increase the thickness
L
, the cross-section area
A
remains the same. As a result, it continually reduces heat flow across the plate. Whereas for hollow geometries such as a hollow sphere or a hollow cylinder, as the thickness (or outer radius) increases, the convective heat transfer becomes dominant, increasing heat flow. Thus, a quantity called critical radius is defined to establish a condition on the outer radius.
The critical radius of insulation is a threshold parameter that determines the limit on the outer radius of a hollow cylinder or sphere to reduce the heat transfer.
Variation of heat flux vs. radius with a maximum and critical radius.
The figure shows the variation of heat flux from the hollow object vs. radius.
r_1
indicates the internal radius. As we increase the thickness, the outer radius will increase. This increase in radius, in turn, increases the heat flux; this occurs because the area available for convective heat transfer has increased. At a certain outer radius value called critical radius
r_{\text{cr}}
, we notice a maximum heat flux. After
r_{\text{cr}}
, heat flow is smaller. Thus, to reduce the heat flow, we need to satisfy the
r_2>r_{\text{cr}}
The following are the critical radius of insulation equations for a hollow cylinder and a hollow sphere:
\begin{equation} \quad r_{\text{cr-cylinder}}=\frac{k}{h} \end{equation}
\begin{equation} \quad\ \ r_{\text{cr-sphere}}= \frac{2k}{h} \end{equation}
h
– Heat transfer coefficient between the hollow structure and the medium surrounding it in
\text{W/(m}^2 \text K)
Let us go back to our thermal resistance challenge, which asked you to choose the best container to preserve the coldness of our cold drink. Now that you have learned what thermal resistance is and how to calculate it for different shapes, it shouldn't be a problem for you! The following steps summarize the solutions to the challenge:
Firstly, you can approximate the container shape as a hollow cylinder (no way as a plate!).
After that, choose the selected shape, insert the geometric property of the container.
In our challenge, let's assume the following geometric properties of the container: length
L=0.5\ \text m
r_1=0.1\ \text m
r_2=0.2\ \text m
Finally, insert the thermal conductivity of the material. In the following table, you will see the material property and corresponding thermal resistance value of each container. You obtained these values by using the thermal resistance formula embedded inside our thermal conductivity to the thermal resistance calculator.
Thermal conductivity and thermal resistance for three containers made of different materials.
\text{W/(m}\cdot\text K)
\text{K/W}
The table shows that a wood container has a higher thermal resistance than the other two. As a result, the wood container is the best insulation container that can preserve the coldness of the cold drink.
Hurray! We hope you have learned a lot about thermal resistance. If you want to extend the knowledge gained from this article, please check out our heat transfer coefficient calculator, which provides an opportunity to apply the knowledge that you gained here to multilayer plates!
What are convective and conductive thermal resistances?
Convective thermal resistance is the resistance offered by the medium to the heat flow through the convection phenomenon. Similarly, conductive thermal resistance is the resistance provided by the object/medium to the heat flow due to the conduction phenomenon.
What are thermal resistance units?
The units for thermal resistance are Kelvins per watts (K/W). By definition, thermal resistance R is the ratio of the temperature difference T2 - T1 and the heat flow Q1-2 between two points. The thermal resistance formula is the following:
R = (T2 - T1) / Q1-2
What is convective heat transfer coefficient?
Convective heat transfer coefficient is the rate of heat transfer between a solid surface and a fluid medium per unit surface area per unit temperature difference.
From the definition, we can easily infer that this heat transfer coefficient depends on the properties of both the solid object and the surrounding fluid medium.
What is critical radius of insulation?
The critical radius rc of insulation is a parameter that acts as a threshold on the outer radius of the hollow objects. If r2 is the outer radius, then the condition at which heat flow becomes maximum is r2 = rc.
If r2 > rc, the heat flow will be reduced, and this is the condition required to have good insulation.
Conductance (k)
°F/W
With this Bragg's law calculator you can compute the angle of an incident X-ray for which the reflected wave from a crystal has the maximum intensity.
Use our escape velocity calculator to find out the speed required to leave the surface of any planet.
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The energy to wavelength formula
How do I calculate wavelength from energy?
Our other photon energy calculators
Many, many problems in physics and chemistry require you to use an energy to wavelength calculator. Do you have an energy transition between two states and wonder what wavelength of light this corresponds to? Have you the energy of two waves that have undergone a constructive or destructive interaction and want to find the new wavelength? If so, you've found the right place in Omni's energy to wavelength calculator, which will help you learn how to calculate wavelength from energy!
The energy to wavelength formula relies upon two other formulae, namely, the wave speed equation and the Planck–Einstein relation. The wave speed equation looks like this:
v = f \times \lambda
v
- Wave speed, in m/s;
f
- Frequency of the wave, in s; and
\lambda
- Wavelength, in m.
In most cases, we are looking at how a wave travels through a vacuum. In this case
v
is actually equal to the speed of light, 299,792,458 meters per second, which we denote as
c
The next equation we need to consider is the Planck-Einstein relation:
E = h \times f
E
- Photon energy, in J;
h
- Planck's constant, equal to 6.62607015×10−34 J Hz−1; and
f
- Frequency of the wave, in s.
Manipulation of the formulas is then required. First, replace
v
c
. Then rearrange the wave speed equation so that frequency is in terms of wavelength. We then substitute this formula into the Planck-Einstein equation. Finally, we must rearrange this substituted formula in order to get wavelength in terms of energy:
\lambda = \frac{h \times c}{E}
To calculate wavelength from the energy of a photon:
Convert the photon's energy into joules.
Divide the speed of light, equal to 299,792,458 meters per second, by the photon's energy.
Multiply the resulting number by Planck's constant, which is 6.626×10−34 J/Hz.
Congratulations, you have just found your photon's wavelength in meters.
We hope that this tool provides you with everything you need to solve your problem. However, if for some reason it does not, please check out our other relevant tools:
Wavelength to energy.
To calculate photon energy from wavelength:
Make sure your wavelength is in meters.
Divide the speed of light, approximately 300,000,000 m/s, by the wavelength to get the wave's frequency.
Multiply the frequency by Planck's constant, 6.626×10−34 J/Hz.
The resulting number is the energy of a photon!
What happens to energy when the wavelength is shortened?
When the wavelength is shortened, the photon's energy increases. This is because photon energy is proportional to a constant over the wavelength. Therefore, as the wavelength gets smaller, the constant is divided up by less, and so photon energy increases.
When wavelength increases what happens to the energy?
As the wavelength increases, the energy of the photon decreases. This can be seen if you examine the relationship between the two values - energy is inversely proportional to wavelength. When you increase wavelength, then, the constant is divided by more, and so the energy decreases.
Use our Magnus force calculator to find the total force generated by the Magnus effect on a spinning cylinder.
Magnus Force Calculator
|
Measure (mathematics) — Wikipedia Republished // WIKI 2
For the coalgebraic concept, see Measuring coalgebra.
Not to be confused with Metric (mathematics).
Informally, a measure has the property of being monotone in the sense that if A is a subset of B, the measure of A is less than or equal to the measure of B. Furthermore, the measure of the empty set is required to be 0.
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general.
The intuition behind this concept dates back to Ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, Johann Radon, Constantin Carathéodory, and Maurice Fréchet, among others.
Electrician Math
Episode 39 - Using Ohm's Law In The Field - ELECTRICIAN MATH REAL WORLD EXAMPLES
Unit Conversion Maths | How To Convert Units | Convert mm, cm, m and km | Measurement Table
Introduction to Electrical Math Principles and Applications
3 Basic properties
3.1 Monotonicity
3.2 Measure of countable unions and intersections
3.2.1 Subadditivity
3.2.2 Continuity from below
3.2.3 Continuity from above
4.2 μ{x : f(x) ≥ t} = μ{x : f(x) > t} (a.e.)
4.4 Sigma-finite measures
4.5 s-finite measures
5 Non-measurable sets
Let X be a set and Σ a σ-algebra over X. A function μ from Σ to the extended real number line is called a measure if it satisfies the following properties:
{\displaystyle \mu (\varnothing )=0}
{\displaystyle \{E_{k}\}_{k=1}^{\infty }}
of pairwise disjoint sets in Σ,
{\displaystyle \mu \left(\bigcup _{k=1}^{\infty }E_{k}\right)=\sum _{k=1}^{\infty }\mu (E_{k}).}
{\displaystyle E}
{\displaystyle \mu (\varnothing )=0}
{\displaystyle \mu (E)=\mu (E\cup \varnothing )=\mu (E)+\mu (\varnothing ),}
{\displaystyle \mu (\varnothing )=0.}
If the condition of non-negativity is omitted but the second and third of these conditions are met, and μ takes on at most one of the values ±∞, then μ is called a signed measure.
The pair (X, Σ) is called a measurable space, the members of Σ are called measurable sets.
A triple (X, Σ, μ) is called a measure space. A probability measure is a measure with total measure one – i.e. μ(X) = 1. A probability space is a measure space with a probability measure.
For measure spaces that are also topological spaces various compatibility conditions can be placed for the measure and the topology. Most measures met in practice in analysis (and in many cases also in probability theory) are Radon measures. Radon measures have an alternative definition in terms of linear functionals on the locally convex space of continuous functions with compact support. This approach is taken by Bourbaki (2004) and a number of other sources. For more details, see the article on Radon measures.
Main category: Measures (measure theory)
The counting measure is defined by μ(S) = number of elements in S.
The Lebesgue measure on ℝ is a complete translation-invariant measure on a σ-algebra containing the intervals in ℝ such that μ([0, 1]) = 1; and every other measure with these properties extends Lebesgue measure.
Circular angle measure is invariant under rotation, and hyperbolic angle measure is invariant under squeeze mapping.
The Haar measure for a locally compact topological group is a generalization of the Lebesgue measure (and also of counting measure and circular angle measure) and has similar uniqueness properties.
The Hausdorff measure is a generalization of the Lebesgue measure to sets with non-integer dimension, in particular, fractal sets.
Every probability space gives rise to a measure which takes the value 1 on the whole space (and therefore takes all its values in the unit interval [0, 1]). Such a measure is called a probability measure. See probability axioms.
The Dirac measure δa (cf. Dirac delta function) is given by δa(S) = χS(a), where χS is the indicator function of S. The measure of a set is 1 if it contains the point a and 0 otherwise.
Other 'named' measures used in various theories include: Borel measure, Jordan measure, ergodic measure, Gaussian measure, Baire measure, Radon measure, Young measure, and Loeb measure.
In physics an example of a measure is spatial distribution of mass (see e.g., gravity potential), or another non-negative extensive property, conserved (see conservation law for a list of these) or not. Negative values lead to signed measures, see "generalizations" below.
Liouville measure, known also as the natural volume form on a symplectic manifold, is useful in classical statistical and Hamiltonian mechanics.
Gibbs measure is widely used in statistical mechanics, often under the name canonical ensemble.
{\displaystyle \mu (E_{1})\leq \mu (E_{2}).}
Measure of countable unions and intersections
{\displaystyle \mu \left(\bigcup _{i=1}^{\infty }E_{i}\right)\leq \sum _{i=1}^{\infty }\mu (E_{i}).}
Continuity from below
{\displaystyle E_{n}\subseteq E_{n+1},}
{\displaystyle \mu \left(\bigcup _{i=1}^{\infty }E_{i}\right)=\lim _{i\to \infty }\mu (E_{i})=\sup _{i\geq 1}\mu (E_{i}).}
Continuity from above
{\displaystyle E_{n+1}\subseteq E_{n},}
{\displaystyle \mu \left(\bigcap _{i=1}^{\infty }E_{i}\right)=\lim _{i\to \infty }\mu (E_{i})=\inf _{i\geq 1}\mu (E_{i}).}
Main article: Complete measure
A measurable set X is called a null set if μ(X) = 0. A subset of a null set is called a negligible set. A negligible set need not be measurable, but every measurable negligible set is automatically a null set. A measure is called complete if every negligible set is measurable.
A measure can be extended to a complete one by considering the σ-algebra of subsets Y which differ by a negligible set from a measurable set X, that is, such that the symmetric difference of X and Y is contained in a null set. One defines μ(Y) to equal μ(X).
μ{x : f(x) ≥ t} = μ{x : f(x) > t} (a.e.)
{\displaystyle \mu }
{\displaystyle f}
{\displaystyle [0,\infty ],}
{\displaystyle \mu \{x:f(x)\geq t\}=\mu \{x:f(x)>t\},}
for almost all
{\displaystyle t\in \mathbb {R} ,}
with respect to the Lebesgue measure.[1] This property is used in connection with Lebesgue integral.
{\displaystyle \mu \{x:f(x)>t\}}
{\displaystyle \mu \{x:f(x)\geq t\}}
{\displaystyle t,}
{\displaystyle \mu \{x:f(x)>t\}=\infty }
{\displaystyle t}
{\displaystyle \mu \{x:f(x)\geq t\}=\mu \{x:f(x)>t\}+\mu \{x:f(x)=t\}=\infty ,}
{\displaystyle \mu \{x:f(x)>t\}\neq \infty ,}
{\displaystyle t,}
{\displaystyle t_{0}\in \{-\infty \}\cup [0,\infty )}
{\displaystyle t}
{\displaystyle t_{0}\geq 0)}
{\displaystyle \mu \{x:f(x)\geq t\}=\infty }
{\displaystyle t<t_{0}.}
{\displaystyle t>t_{0},}
{\displaystyle t_{n}}
{\displaystyle t.}
{\displaystyle \{x:f(x)>t_{n}\}}
{\displaystyle \mu }
{\displaystyle \mu }
{\displaystyle \{x:f(x)\geq t\}=\bigcap _{n}\{x:f(x)>t_{n}\}.}
{\displaystyle \mu \{x:f(x)\geq t\}=\lim _{t_{n}\uparrow t}\mu \{x:f(x)>t_{n}\}.}
{\displaystyle \mu \{x:f(x)>t\}}
{\displaystyle t}
{\displaystyle I}
{\displaystyle r_{i},i\in I}
{\displaystyle \sum _{i\in I}r_{i}=\sup \left\lbrace \sum _{i\in J}r_{i}:|J|<\aleph _{0},J\subseteq I\right\rbrace .}
{\displaystyle r_{i}}
{\displaystyle \mu }
{\displaystyle \Sigma }
{\displaystyle \kappa }
{\displaystyle \lambda <\kappa }
{\displaystyle X_{\alpha },\alpha <\lambda }
{\displaystyle \bigcup _{\alpha \in \lambda }X_{\alpha }\in \Sigma }
{\displaystyle \mu \left(\bigcup _{\alpha \in \lambda }X_{\alpha }\right)=\sum _{\alpha \in \lambda }\mu \left(X_{\alpha }\right).}
{\displaystyle \kappa }
Main article: Sigma-finite measure
A measure space (X, Σ, μ) is called finite if μ(X) is a finite real number (rather than ∞). Nonzero finite measures are analogous to probability measures in the sense that any finite measure μ is proportional to the probability measure
{\displaystyle {\frac {1}{\mu (X)}}\mu }
For example, the real numbers with the standard Lebesgue measure are σ-finite but not finite. Consider the closed intervals [k, k+1] for all integers k; there are countably many such intervals, each has measure 1, and their union is the entire real line. Alternatively, consider the real numbers with the counting measure, which assigns to each finite set of reals the number of points in the set. This measure space is not σ-finite, because every set with finite measure contains only finitely many points, and it would take uncountably many such sets to cover the entire real line. The σ-finite measure spaces have some very convenient properties; σ-finiteness can be compared in this respect to the Lindelöf property of topological spaces.[original research?] They can be also thought of as a vague generalization of the idea that a measure space may have 'uncountable measure'.
s-finite measures
Main article: s-finite measure
A measure is said to be s-finite if it is a countable sum of bounded measures. S-finite measures are more general than sigma-finite ones and have applications in the theory of stochastic processes.
Main article: Non-measurable set
If the axiom of choice is assumed to be true, it can be proved that not all subsets of Euclidean space are Lebesgue measurable; examples of such sets include the Vitali set, and the non-measurable sets postulated by the Hausdorff paradox and the Banach–Tarski paradox.
For certain purposes, it is useful to have a "measure" whose values are not restricted to the non-negative reals or infinity. For instance, a countably additive set function with values in the (signed) real numbers is called a signed measure, while such a function with values in the complex numbers is called a complex measure. Observe, however, that complex measure is necessarily of finite variation, hence complex measures include finite signed measures but not, for example, the Lebesgue measure.
Measures that take values in Banach spaces have been studied extensively.[2] A measure that takes values in the set of self-adjoint projections on a Hilbert space is called a projection-valued measure; these are used in functional analysis for the spectral theorem. When it is necessary to distinguish the usual measures which take non-negative values from generalizations, the term positive measure is used. Positive measures are closed under conical combination but not general linear combination, while signed measures are the linear closure of positive measures.
Another generalization is the finitely additive measure, also known as a content. This is the same as a measure except that instead of requiring countable additivity we require only finite additivity. Historically, this definition was used first. It turns out that in general, finitely additive measures are connected with notions such as Banach limits, the dual of L∞ and the Stone–Čech compactification. All these are linked in one way or another to the axiom of choice. Contents remain useful in certain technical problems in geometric measure theory; this is the theory of Banach measures.
Abelian von Neumann algebra
Almost everywhere
Carathéodory's extension theorem
Content (measure theory)
Fubini's theorem
Fatou's lemma
Fuzzy measure theory
Geometric measure theory
Hausdorff measure
Inner measure
Lebesgue measure
Lorentz space
Lifting theory
Measurable cardinal
Measurable function
Minkowski content
Outer measure
Product measure
Pushforward measure
Regular measure
Vector measure
Valuation (measure theory)
^ Rao, M. M. (2012), Random and Vector Measures, Series on Multivariate Analysis, vol. 9, World Scientific, ISBN 978-981-4350-81-5, MR 2840012 .
^ Bhaskara Rao, K. P. S. (1983). Theory of charges : a study of finitely additive measures. M. Bhaskara Rao. London: Academic Press. p. 35. ISBN 0-12-095780-9. OCLC 21196971.
Robert G. Bartle (1995) The Elements of Integration and Lebesgue Measure, Wiley Interscience.
Bourbaki, Nicolas (2004), Integration I, Springer Verlag, ISBN 3-540-41129-1 Chapter III.
D. H. Fremlin, 2000. Measure Theory. Torres Fremlin.
Jech, Thomas (2003), Set Theory: The Third Millennium Edition, Revised and Expanded, Springer Verlag, ISBN 3-540-44085-2
R. Duncan Luce and Louis Narens (1987). "measurement, theory of," The New Palgrave: A Dictionary of Economics, v. 3, pp. 428–32.
Shilov, G. E., and Gurevich, B. L., 1978. Integral, Measure, and Derivative: A Unified Approach, Richard A. Silverman, trans. Dover Publications. ISBN 0-486-63519-8. Emphasizes the Daniell integral.
Teschl, Gerald, Topics in Real and Functional Analysis, (lecture notes)
Tao, Terence (2011). An Introduction to Measure Theory. Providence, R.I.: American Mathematical Society. ISBN 9780821869192.
Weaver, Nik (2013). Measure Theory and Functional Analysis. World Scientific. ISBN 9789814508568.
"Measure", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Tutorial: Measure Theory for Dummies
Lp spaces
Measure space
Measurable space/function
Borel set
Convergence in measure
Essential range
Locally measurable
Non-measurable set
Null set
Transverse measure
Locally finite
Metric outer
Borel regular
Inner regular
Outer regular
Set function
Strictly positive
Particular measures
Spherical measure
Main results
Hölder's inequality
Minkowski inequality
Riesz–Markov–Kakutani representation theorem
Disintegration theorem
Lebesgue's density theorem
Lebesgue differentiation theorem
Spectral theory
|
Inversion-Based Feedforward Approach to Broadband Acoustic Noise Reduction | Journal of Vibration and Acoustics | ASME Digital Collection
Tom C. Waite,
Tom C. Waite
, 2025 Black Engineering Building, Ames, IA 50011
e-mail: twaite@iastate.edu
e-mail: qzzou@iastate.edu
e-mail: akelkar@iastate.edu
Qingze Zou Assistant Professor
Atul Kelkar Professor
Waite, T. C., Zou, Q., and Kelkar, A. (August 14, 2008). "Inversion-Based Feedforward Approach to Broadband Acoustic Noise Reduction." ASME. J. Vib. Acoust. October 2008; 130(5): 051010. https://doi.org/10.1115/1.2948411
In this article, an inversion-based feedforward control approach to achieve broadband active-noise control is investigated. Broadband active-noise control is needed in many areas, from heating, ventilation and air conditioning (HVAC) ducts to aircraft cabins. Achieving broadband active-noise control, however, is very challenging due to issues such as the complexity of acoustic dynamics (which has no natural roll-off at high frequency, and is often nonminimum phase), the wide frequency spectrum of the acoustic noise, and the critical requirement to overcome the delay of the control input relative to the noise signal. These issues have limited the success of existing feedforward control techniques to the low-frequency range of
[0,1]kHz
. The modeling issues in capturing the complex acoustic dynamics coupled with its nonminimum-phase characteristic also prevent the use of high-gain feedback methods, making the design of an effective controller to combat broadband noises challenging. In this article, we explore, through experiments, the potential of inversion-based feedforward control approach for noise control over the
1kHz
low-frequency range limit. Then we account for the effect of modeling errors on the feedforward input by a recently developed inversion-based iterative control technique. Experimental results presented show that noise reduction of over
10–15dB
can be achieved in a broad frequency range of
5kHz
by using the inversion-based feedforward control technique.
active noise control, aeroacoustics, feedforward, iterative methods
Acoustics, Dynamics (Mechanics), Feedforward control, Noise (Sound), Noise control, Errors
Feedforward Active Noise Controller Design in Ducts Without Independent Noise Source Measurements
Active Control of Aircraft Cabin Noise Using Collocated Structural Actuators and Sensors
Lqg-Based Robust Broadband Control of Acoustic-Structure Interaction in 3-D Enclosure
A New Approach to Scan Trajectory Design and Track: AFM Force Measurement Example
Robust Broadband Control of Acoustic Duct
, Invited Paper, (
Modelling and Control of Acoustic Ducts
Modeling and Control of Acoustic-Structure Interaction in 3-D Enclosures
IEEE Conference on Decision and Controls
Adaptive Resonant Mode Control for High Frequency Tonal Noise
,” MS thesis, Iowa State University, Ames, IA.
H2∕hinf Active Control of Sound in a Headrest: Design And Implementation
Frequency-Domain Periodic Active Noise Control and Equalization
Process of Silencing Sound Oscillation
Feedforward Piezoelectric Structural Control: An Application to Aircraft Cabin Noise Reduction
Performance Comparison Between the Filtered-Error lms and the Filtered-x lms Algorithms [anc]
Hybrid Filtered Error lms Algorithm: Another Alternative to Filtered-x lms
Demonstration of Active Control of Broadband Sound
An Algorithm for Designing a Broadband Active Sound Control System
Self-Adaptive Broadband Active Sound Control System
Performance Limitations of Non-Minimum Phase Systems in the Servomechanism Problem
Preview-Based Optimal Inversion for Output Tracking: Application to Scanning Tunneling Microscopy
Preview-Based Stable-Inversion for Output Tracking of Nonlinear Nonminimum-Phase Systems: The VTOL Example
Control of Dynamics-Coupling Effects in Piezo-Actuator for High-Speed afm Operation
Vander Giessen
Precision Tracking of Driving Waveforms for Inertial Reaction Devices
Woon-Seng
A Model-Less Inversion-Based Iterative Control Technique for Output Tracking: Piezo Actuator Example
|
Quantization - Algorithm details | CatBoost
Before learning, the possible values of objects are divided into disjoint ranges (buckets) delimited by the threshold values (splits). The size of the quantization (the number of splits) is determined by the starting parameters (separately for numerical features and numbers obtained as a result of converting categorical features into numerical features).
Quantization is also used to split the label values when working with categorical features. А random subset of the dataset is used for this purpose on large datasets.
The table below shows the quantization modes provided in CatBoost.
How splits are chosen
Median Include an approximately equal number of objects in every bucket.
Uniform Generate splits by dividing the [min_feature_value, max_feature_value] segment into subsegments of equal length. Absolute values of the feature are used in this case.
UniformAndQuantiles Combine the splits obtained in the following modes, after first halving the quantization size provided by the starting parameters for each of them:
- Median.
MaxLogSum Maximize the value of the following expression inside each bucket:
\sum\limits_{i=1}^{n}\log(weight){ , where}
n
— The number of distinct objects in the bucket.
weight
— The number of times an object in the bucket is repeated.
MinEntropy Minimize the value of the following expression inside each bucket:
\sum \limits_{i=1}^{n} weight \cdot log (weight) { ,<br/> where}
n
weight
GreedyLogSum Maximize the greedy approximation of the following expression inside every bucket:
\sum\limits_{i=1}^{n}\log(weight){ , where}
n
weight
|
{\displaystyle {\frac {Q^{2}}{4\pi \epsilon _{0}}}+{\frac {c^{2}J^{2}}{GM^{2}}}\leq GM^{2}}
{\displaystyle J\leq {\frac {GM^{2}}{c}},}
{\displaystyle 0\leq {\frac {cJ}{GM^{2}}}\leq 1.}
{\displaystyle r_{\mathrm {s} }={\frac {2GM}{c^{2}}}\approx 2.95\,{\frac {M}{M_{\odot }}}~\mathrm {km,} }
{\displaystyle r_{\mathrm {+} }={\frac {GM}{c^{2}}}.}
{\displaystyle r_{\rm {ISCO}}=3\,r_{s}={\frac {6\,GM}{c^{2}}},}
{\displaystyle z\sim 7}
{\displaystyle M+{\sqrt {M^{2}-{(J/M)}^{2}-Q^{2}}}.}
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^ Abbott, Benjamin P.; et al. (LIGO Scientific Collaboration & Virgo Collaboration) (11 February 2016). "Properties of the binary black hole merger GW150914". Physical Review Letters. 116 (24): 241102. arXiv:1602.03840. Bibcode:2016PhRvL.116x1102A. doi:10.1103/PhysRevLett.116.241102. PMID 27367378. S2CID 217406416.
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^ Merali, Zeeya (3 April 2013). "Astrophysics: Fire in the hole!". Nature. 496 (7443): 20–23. Bibcode:2013Natur.496...20M. doi:10.1038/496020a. PMID 23552926.
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Carter, B. (1973). "Black hole equilibrium states". In DeWitt, B. S.; DeWitt, C. (eds.). Black Holes.
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Black holeat Wikipedia's sister projects
Retrieved from "https://en.wikipedia.org/w/index.php?title=Black_hole&oldid=1088791000"
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Rule of sum - Wikipedia
5+0=5 illustrated with collections of dots.
In combinatorics, the rule of sum[1][2][verification needed] or addition principle[3][4] is a basic counting principle. Stated simply, it is the intuitive idea that if we have A number of ways of doing something and B number of ways of doing another thing and we can not do both at the same time, then there are A + B ways to choose one of the actions.[1][3]
3+2=5 illustrated with shapes.[relevance questioned]
More formally, the rule of sum is a fact about set theory.[verification needed] It states that sum of the sizes of a finite collection of pairwise disjoint sets is the size of the union of these sets. That is, if
{\displaystyle S_{1},S_{2},...,S_{n}}
are pairwise disjoint sets, then we have:[3][4]
{\displaystyle |S_{1}|+|S_{2}|+\cdots +|S_{n}|=|S_{1}\cup S_{2}\cup \cdots \cup S_{n}|.}
2 Inclusion–exclusion principle
A person has decided to shop at one store today, either in the north part of town or the south part of town. If they visit the north part of town, they will shop at either a mall, a furniture store, or a jewelry store (3 ways). If they visit the south part of town then they will shop at either a clothing store or a shoe store (2 ways).
Thus there are 3+2=5 possible shops the person could end up shopping at today.[citation needed]
A series of Venn diagrams illustrating the principle of inclusion-exclusion.
The inclusion–exclusion principle (also known as the sieve principle[3]) can be thought of as a generalization of the rule of sum in that it too enumerates the number of elements in the union of some sets (but does not require the sets to be disjoint). It states that if A1, ..., An are finite sets, then[3]
{\displaystyle \left|\bigcup _{i=1}^{n}A_{i}\right|=\sum _{i=1}^{n}\left|A_{i}\right|-\sum _{i,j\,:\,1\leq i<j\leq n}\left|A_{i}\cap A_{j}\right|+\sum _{i,j,k\,:\,1\leq i<j<k\leq n}\left|A_{i}\cap A_{j}\cap A_{k}\right|-\ \cdots \ +\left(-1\right)^{n-1}\left|A_{1}\cap \cdots \cap A_{n}\right|.}
^ a b Leung, K. T.; Cheung, P. H. (1988-04-01). Fundamental Concepts of Mathematics. Hong Kong University Press. ISBN 978-962-209-181-8.
^ Penner, R. C. (1999). Discrete Mathematics: Proof Techniques and Mathematical Structures. World Scientific. ISBN 978-981-02-4088-2.
^ a b c d e Biggs, Norman L. (2002). Discrete Mathematics. India: Oxford University Press. pp. 91, 112. ISBN 978-0-19-871369-2.
^ a b "enumerative combinatorics". planetmath.org. 22 March 2013. Archived from the original on 23 July 2014. Retrieved 14 August 2021.
Combinatorial principle
Retrieved from "https://en.wikipedia.org/w/index.php?title=Rule_of_sum&oldid=1076969263"
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Enable TLS for DM | PingCAP Docs
This document describes how to enable TLS between components of the DM cluster in Kubernetes and how to use DM to migrate data between MySQL/TiDB databases that enable TLS for the MySQL client.
Enable TLS between DM components
Starting from v1.2, TiDB Operator supports enabling TLS between components of the DM cluster in Kubernetes.
To enable TLS between components of the DM cluster, perform the following steps:
Generate certificates for each component of the DM cluster to be created:
A set of server-side certificates for the DM-master/DM-worker component, saved as the Kubernetes Secret objects: ${cluster_name}-${component_name}-cluster-secret
A set of shared client-side certificates for the various clients of each component, saved as the Kubernetes Secret objects: ${cluster_name}-dm-client-secret.
The Secret objects you created must follow the above naming convention. Otherwise, the deployment of the DM cluster will fail.
Deploy the cluster, and set .spec.tlsCluster.enabled to true.
After the cluster is created, do not modify this field; otherwise, the cluster will fail to upgrade. If you need to modify this field, delete the cluster and create a new one.
Configure dmctl to connect to the cluster.
Certificates can be issued in multiple methods. This document describes two methods. You can choose either of them to issue certificates for the DM cluster:
Generate certificates for components of the DM cluster
This section describes how to issue certificates using two methods: cfssl and cert-manager.
Generate the ca-config.json configuration file:
Generate the ca-csr.json configuration file:
Generate the server-side certificates:
In this step, a set of server-side certificate is created for each component of the DM cluster.
First, generate the default dm-master-server.json file:
cfssl print-defaults csr > dm-master-server.json
Then, edit this file to change the CN and hosts attributes:
"${cluster_name}-dm-master",
"${cluster_name}-dm-master.${namespace}",
"${cluster_name}-dm-master.${namespace}.svc",
"${cluster_name}-dm-master-peer",
"${cluster_name}-dm-master-peer.${namespace}",
"${cluster_name}-dm-master-peer.${namespace}.svc",
"*.${cluster_name}-dm-master-peer",
"*.${cluster_name}-dm-master-peer.${namespace}",
"*.${cluster_name}-dm-master-peer.${namespace}.svc"
${cluster_name} is the name of the DM cluster. ${namespace} is the namespace in which the DM cluster is deployed. You can also add your customized hosts.
Finally, generate the DM-master server-side certificate:
cfssl gencert -ca=ca.pem -ca-key=ca-key.pem -config=ca-config.json -profile=internal dm-master-server.json | cfssljson -bare dm-master-server
First, generate the default dm-worker-server.json file:
cfssl print-defaults csr > dm-worker-server.json
"${cluster_name}-dm-worker",
"${cluster_name}-dm-worker.${namespace}",
"${cluster_name}-dm-worker.${namespace}.svc",
"${cluster_name}-dm-worker-peer",
"${cluster_name}-dm-worker-peer.${namespace}",
"${cluster_name}-dm-worker-peer.${namespace}.svc",
"*.${cluster_name}-dm-worker-peer",
"*.${cluster_name}-dm-worker-peer.${namespace}",
"*.${cluster_name}-dm-worker-peer.${namespace}.svc"
${cluster_name} is the name of the cluster. ${namespace} is the namespace in which the DM cluster is deployed. You can also add your customized hosts.
Finally, generate the DM-worker server-side certificate:
cfssl gencert -ca=ca.pem -ca-key=ca-key.pem -config=ca-config.json -profile=internal dm-worker-server.json | cfssljson -bare dm-worker-server
Generate the client-side certificates:
First, generate the default client.json file:
Create the Kubernetes Secret object:
If you have already generated a set of certificates for each component and a set of client-side certificate for each client as described in the above steps, create the Secret objects for the DM cluster by executing the following command:
The DM-master cluster certificate Secret:
kubectl create secret generic ${cluster_name}-dm-master-cluster-secret --namespace=${namespace} --from-file=tls.crt=dm-master-server.pem --from-file=tls.key=dm-master-server-key.pem --from-file=ca.crt=ca.pem
The DM-worker cluster certificate Secret：
kubectl create secret generic ${cluster_name}-dm-worker-cluster-secret --namespace=${namespace} --from-file=tls.crt=dm-worker-server.pem --from-file=tls.key=dm-worker-server-key.pem --from-file=ca.crt=ca.pem
Client certificate Secret：
kubectl create secret generic ${cluster_name}-dm-client-secret --namespace=${namespace} --from-file=tls.crt=client.pem --from-file=tls.key=client-key.pem --from-file=ca.crt=ca.pem
You have created two Secret objects:
One Secret object for each DM-master/DM-worker server-side certificate to load when the server is started;
One Secret object for their clients to connect.
Refer to cert-manager installation in Kubernetes for details.
Create an Issuer to issue certificates to the DM cluster.
Then, create a dm-cluster-issuer.yaml file with the following content:
name: ${cluster_name}-dm-issuer
${cluster_name} is the name of the cluster. The above YAML file creates three objects:
An Issuer object of the SelfSigned type, used to generate the CA certificate needed by Issuer of the CA type;
A Certificate object, whose isCa is set to true.
An Issuer, used to issue TLS certificates between components of the DM cluster.
kubectl apply -f dm-cluster-issuer.yaml
Each component needs a server-side certificate, and all components need a shared client-side certificate for their clients.
The DM-master server-side certificate
name: ${cluster_name}-dm-master-cluster-secret
secretName: ${cluster_name}-dm-master-cluster-secret
- "${cluster_name}-dm-master"
- "${cluster_name}-dm-master.${namespace}"
- "${cluster_name}-dm-master.${namespace}.svc"
- "${cluster_name}-dm-master-peer"
- "${cluster_name}-dm-master-peer.${namespace}"
- "${cluster_name}-dm-master-peer.${namespace}.svc"
- "*.${cluster_name}-dm-master-peer"
- "*.${cluster_name}-dm-master-peer.${namespace}"
- "*.${cluster_name}-dm-master-peer.${namespace}.svc"
Set spec.secretName to ${cluster_name}-dm-master-cluster-secret.
Add server auth and client auth in usages.
Add the following DNSs in dnsNames. You can also add other DNSs according to your needs:
"${cluster_name}-dm-master"
{cluster_name}-dm-master.
{namespace}"
{cluster_name}-dm-master.
{namespace}.svc"
"${cluster_name}-dm-master-peer"
{cluster_name}-dm-master-peer.
{cluster_name}-dm-master-peer.
"*.${cluster_name}-dm-master-peer"
"*.
{cluster_name}-dm-master-peer.
{cluster_name}-dm-master-peer.
Add the following two IPs in ipAddresses. You can also add other IPs according to your needs:
Add the Issuer created above in issuerRef.
After the object is created, cert-manager generates a ${cluster_name}-dm-master-cluster-secret Secret object to be used by the DM-master component of the DM cluster.
The DM-worker server-side certificate
name: ${cluster_name}-dm-worker-cluster-secret
secretName: ${cluster_name}-dm-worker-cluster-secret
- "${cluster_name}-dm-worker"
- "${cluster_name}-dm-worker.${namespace}"
- "${cluster_name}-dm-worker.${namespace}.svc"
- "${cluster_name}-dm-worker-peer"
- "${cluster_name}-dm-worker-peer.${namespace}"
- "${cluster_name}-dm-worker-peer.${namespace}.svc"
- "*.${cluster_name}-dm-worker-peer"
- "*.${cluster_name}-dm-worker-peer.${namespace}"
- "*.${cluster_name}-dm-worker-peer.${namespace}.svc"
Set spec.secretName to ${cluster_name}-dm-worker-cluster-secret.
"${cluster_name}-dm-worker"
{cluster_name}-dm-worker.
{cluster_name}-dm-worker.
"${cluster_name}-dm-worker-peer"
{cluster_name}-dm-worker-peer.
{cluster_name}-dm-worker-peer.
"*.${cluster_name}-dm-worker-peer"
{cluster_name}-dm-worker-peer.
{cluster_name}-dm-worker-peer.
After the object is created, cert-manager generates a ${cluster_name}-dm-cluster-secret Secret object to be used by the DM-worker component of the DM cluster.
A set of client-side certificates of DM cluster components.
name: ${cluster_name}-dm-client-secret
secretName: ${cluster_name}-dm-client-secret
After the object is created, cert-manager generates a ${cluster_name}-cluster-client-secret Secret object to be used by the clients of the DM components.
Deploy the DM cluster
When you deploy a DM cluster, you can enable TLS between DM components, and set the cert-allowed-cn configuration item to verify the CN (Common Name) of each component's certificate.
Currently, you can set only one value for the cert-allowed-cn configuration item of DM-master. Therefore, the commonName of all Certificate objects must be the same.
Create the dm-cluster.yaml file:
Use the kubectl apply -f dm-cluster.yaml file to create a DM cluster.
Configure dmctl and connect to the cluster
Get into the DM-master Pod:
kubectl exec -it ${cluster_name}-dm-master-0 -n ${namespace} sh
Use dmctl:
cd /var/lib/dm-master-tls
/dmctl --ssl-ca=ca.crt --ssl-cert=tls.crt --ssl-key=tls.key --master-addr 127.0.0.1:8261 list-member
Use DM to migrate data between MySQL/TiDB databases that enable TLS for the MySQL client
This section describes how to configure DM to migrate data between MySQL/TiDB databases that enable TLS for the MySQL client.
To learn how to enable TLS for the MySQL client of TiDB, refer to Enable TLS for the MySQL Client.
Step 1: Create the Kubernetes Secret object for each TLS-enabled MySQL
Suppose you have deployed a MySQL/TiDB database with TLS-enabled for the MySQL client. To create Secret objects for the TiDB cluster, execute the following command:
kubectl create secret generic ${mysql_secret_name1} --namespace=${namespace} --from-file=tls.crt=client.pem --from-file=tls.key=client-key.pem --from-file=ca.crt=ca.pem
kubectl create secret generic ${tidb_secret_name} --namespace=${namespace} --from-file=tls.crt=client.pem --from-file=tls.key=client-key.pem --from-file=ca.crt=ca.pem
Step 2: Mount the Secret objects to the DM cluster
After creating the Kubernetes Secret objects for the upstream and downstream databases, you need to set spec.tlsClientSecretNames so that you can mount the Secret objects to the Pod of DM-master/DM-worker.
- ${mysql_secret_name1}
- ${tidb_secret_name}
Step 3: Modify the data source and migration task configuration
After configuring spec.tlsClientSecretNames, TiDB Operator will mount the Secret objects ${secret_name} to the path /var/lib/source-tls/${secret_name}.
Configure from.security in the source1.yaml file as described in the data source configuration:
relay-dir: /var/lib/dm-worker/relay
host: ${mysql_host1}
user: dm
ssl-ca: /var/lib/source-tls/${mysql_secret_name1}/ca.crt
ssl-cert: /var/lib/source-tls/${mysql_secret_name1}/tls.crt
ssl-key: /var/lib/source-tls/${mysql_secret_name1}/tls.key
Configure target-database.security in the task.yaml file as described in the Configure Migration Tasks:
ssl-ca: /var/lib/source-tls/${tidb_secret_name}/ca.crt
ssl-cert: /var/lib/source-tls/${tidb_secret_name}/tls.crt
ssl-key: /var/lib/source-tls/${tidb_secret_name}/tls.key
- source-id: "replica-01"
dir: "/var/lib/dm-worker/dumped_data"
Step 4: Start the migration tasks
Refer to Start the migration tasks.
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Transient Goal - MATLAB & Simulink
Initial Signal Selection
Desired Transient Response
Shape how the closed-loop system responds to a specific input signal when using Control System Tuner. Use a reference model to specify the desired transient response.
Transient Goal constrains the transient response from specified input locations to specified output locations. This requirement specifies that the transient response closely match the response of a reference model. The constraint is satisfied when the relative difference between the tuned and target responses falls within the tolerance you specify.
You can constrain the response to an impulse, step, or ramp input signal. You can also constrain the response to an input signal that is given by the impulse response of an input filter you specify.
In the Tuning tab of Control System Tuner, select New Goal > Transient response matching to create a Transient Goal.
When tuning control systems at the command line, use TuningGoal.Transient to specify a step response goal.
Specify response inputs
Select one or more signal locations in your model at which to apply the input. To constrain a SISO response, select a single-valued input signal. For example, to constrain the transient response from a location named 'u' to a location named 'y', click Add signal to list and select 'u'. To constrain a MIMO response, select multiple signals or a vector-valued signal.
Specify response outputs
Select one or more signal locations in your model at which to measure the transient response. To constrain a SISO response, select a single-valued output signal. For example, to constrain the transient response from a location named 'u' to a location named 'y', click Add signal to list and select 'y'. To constrain a MIMO response, select multiple signals or a vector-valued signal. For MIMO systems, the number of outputs must equal the number of inputs.
Select the input signal shape for the transient response you want to constrain in Control System Tuner.
Impulse — Constrain the response to a unit impulse.
Step — Constrain the response to a unit step. Using Step is equivalent to using a Step Tracking Goal.
Ramp — Constrain the response to a unit ramp, u = t.
Other — Constrain the response to a custom input signal. Specify the custom input signal by entering a transfer function (tf or zpkmodel) in the Use impulse response of filter field. The custom input signal is the response of this transfer function to a unit impulse.
This transfer function represents the Laplace transform of the desired custom input signal. For example, to constrain the transient response to a unit-amplitude sine wave of frequency w, enter tf(w,[1,0,w^2]). This transfer function is the Laplace transform of sin(wt).
The transfer function you enter must be continuous, and can have no poles in the open right-half plane. The series connection of this transfer function with the reference system for the desired transient response must have no feedthrough term.
Specify the reference system for the desired transient response as a dynamic system model, such as a tf, zpk, or ss model. The Transient Goal constrains the system response to closely match the response of this system to the input signal you specify in Initial Signal Selection.
Enter the name of the reference model in the MATLAB® workspace in the Reference Model field. Alternatively, enter a command to create a suitable reference model, such as tf(1,[1 1.414 1]). The reference model must be stable, and the series connection of the reference model with the input shaping filter must have no feedthrough term.
Use this section of the dialog box to specify additional characteristics of the transient response goal.
Keep % mismatch below
Specify the relative matching error between the actual (tuned) transient response and the target response. Increase this value to loosen the matching tolerance. The relative matching error, erel, is defined as:
\text{gap}=\frac{{‖y\left(t\right)-{y}_{ref}\left(t\right)‖}_{2}}{{‖{y}_{ref\left(tr\right)}\left(t\right)‖}_{2}}.
{‖\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}‖}_{2}
denotes the signal energy (2-norm). The gap can be understood as the ratio of the root-mean-square (RMS) of the mismatch to the RMS of the reference transient.
Adjust for amplitude of input signals and Adjust for amplitude of output signals
For a MIMO tuning goal, when the choice of units results in a mix of small and large signals in different channels of the response, this option allows you to specify the relative amplitude of each entry in the vector-valued signals. This information is used to scale the off-diagonal terms in the transfer function from the tuning goal inputs to outputs. This scaling ensures that cross-couplings are measured relative to the amplitude of each reference signal.
When these options are set to No, the closed-loop transfer function being constrained is not scaled for relative signal amplitudes. When the choice of units results in a mix of small and large signals, using an unscaled transfer function can lead to poor tuning results. Set the option to Yes to provide the relative amplitudes of the input signals and output signals of your transfer function.
For example, suppose the tuning goal constrains a 2-input, 2-output transfer function. Suppose further that second input signal to the transfer function tends to be about 100 times greater than the first signal. In that case, select Yes and enter [1,100] in the Amplitudes of input signals text box.
Adjusting signal amplitude causes the tuning goal to be evaluated on the scaled transfer function Do–1T(s)Di, where T(s) is the unscaled transfer function. Do and Di are diagonal matrices with the Amplitudes of output signals and Amplitudes of input signals values on the diagonal, respectively.
The default value, No, means no scaling is applied.
When you use this requirement to tune a control system, Control System Tuner attempts to enforce zero feedthrough (D = 0) on the transfer that the requirement constrains. Zero feedthrough is imposed because the H2 norm, and therefore the value of the tuning goal (see Algorithms), is infinite for continuous-time systems with nonzero feedthrough.
Control System Tuner enforces zero feedthrough by fixing to zero all tunable parameters that contribute to the feedthrough term. Control System Tuner returns an error when fixing these tunable parameters is insufficient to enforce zero feedthrough. In such cases, you must modify the requirement or the control structure, or manually fix some tunable parameters of your system to values that eliminate the feedthrough term.
To fix parameters of tunable blocks to specified values, see View and Change Block Parameterization in Control System Tuner.
This tuning goal also imposes an implicit stability constraint on the closed-loop transfer function between the specified inputs to outputs, evaluated with loops opened at the specified loop-opening locations. The dynamics affected by this implicit constraint are the stabilized dynamics for this tuning goal. The Minimum decay rate and Maximum natural frequency tuning options control the lower and upper bounds on these implicitly constrained dynamics. If the optimization fails to meet the default bounds, or if the default bounds conflict with other requirements, on the Tuning tab, use Tuning Options to change the defaults.
When you tune a control system, the software converts each tuning goal into a normalized scalar value f(x). Here, x is the vector of free (tunable) parameters in the control system. The software then adjusts the parameter values to minimize f(x) or to drive f(x) below 1 if the tuning requirement is a hard constraint.
For Transient Goal, f(x) is based upon the relative gap between the tuned response and the target response:
\text{gap}=\frac{{‖y\left(t\right)-{y}_{ref}\left(t\right)‖}_{2}}{{‖{y}_{ref\left(tr\right)}\left(t\right)‖}_{2}}.
{‖\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}‖}_{2}
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Projectiles - Maple Help
Home : Support : Online Help : Math Apps : Natural Sciences : Physics : Projectiles
Two objects of the same mass are located on top of a tower. Both objects are released from the tower at the same time. One is pushed parallel to the ground, while the other drops vertically. Which ball reaches the ground first?
Use the following animation to check your hypothesis.
\mathrm{Velocity}=
If the effect of air resistance is neglected, both balls will reach the ground at the same time.
Each ball falls the same vertical distance and both accelerate due to gravity. The ball moving horizontally is traveling faster overall, but the vertical component of its motion is the same as that for the ball falling vertically. Notice that the initial speed in the vertical direction is 0 for both objects.
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Regularity of Functions on the Reduced Quaternion Field in Clifford Analysis
Ji Eun Kim, Su Jin Lim, Kwang Ho Shon, "Regularity of Functions on the Reduced Quaternion Field in Clifford Analysis", Abstract and Applied Analysis, vol. 2014, Article ID 654798, 8 pages, 2014. https://doi.org/10.1155/2014/654798
Ji Eun Kim ,1 Su Jin Lim ,1 and Kwang Ho Shon1
We define a new hypercomplex structure of and a regular function with values in that structure. From the properties of regular functions, we research the exponential function on the reduced quaternion field and represent the corresponding Cauchy-Riemann equations in hypercomplex structures of .
Meglihzon [1], Sudbery [2], and Fueter [3] demonstrated that there are three possible approaches (the Cauchy approach, Weierstrass approach, and Riemann approach) in the theories of functions that would generalize holomorphic functions with respect to several complex variables. Sudbery [2], Soucek [4], and Sommen [5] attempted to research the Cauchy approach using differential forms and differential operators in Clifford analysis. Fueter [3] and Naser [6] studied the properties of quaternionic differential equations as a generalization of the extended Cauchy-Riemann equations in the complex holomorphic function theory. Nôno [7–9] and Sudbery [2] gave a definition and the development of regular functions over the quaternion field. Ryan [10, 11] developed the theories of regular functions in a complex Clifford analysis using a generalization of the Cauchy-Riemann equation. Malonek [12] considered analogously the function theory of hypercomplex variables. He defined the hypercomplex differentiability for the existence of a function over the Clifford algebra and monogenicity based on a generalized Cauchy-Riemann system. Gotô and Nôno [13] and Koriyama et al. [14] dealt with differential operators with the derivative of regular functions in quaternion.
We shall denote by , , and , respectively, the field of complex numbers, the field of real numbers, and the set of all integers. We [15, 16] showed that any complex-valued harmonic function in a pseudoconvex domain of has a hyperconjugate harmonic function in such that the quaternion-valued function is hyperholomorphic in and gave a regeneration theorem in quaternion analysis in the view of complex and Clifford analysis. Further, we [17, 18] investigated the existence of the hyperconjugate harmonic functions of the octonion number system and some properties of dual quaternion functions.
In this paper, we introduce the Fueter variables on and investigate a hypercomplex structure of . We define regular functions and obtain the representation of the corresponding Cauchy-Riemann equations for regular functions in the reduced quaternion field.
A three-dimensional, noncommutative, and associative real field, called a ternary number system, is constructed by three base elements , , and which satisfy In addition, let be the identity of a ternary number system and identifies the imaginary unit in the complex field, and where and are real variables. They satisfy the equations where , , , and are real variables.
For any two elements and of , their product is given by where the corresponding commutative inner product satisfies and the corresponding noncommutative outer product satisfies The conjugation , the corresponding norm , and the inverse of in are given by
For any element in , we have the corresponding exponential function denoted by
Theorem 1. Let be an arbitrary number in . Then the corresponding exponential function of in is given as where .
Furthermore, as hyperbolic functions, one has where .
Proof. For any element of , Since a scalar part of is , a vector part of is , and , by [19], and, similarly, we have Then we have Also, we obtain Since (15) has to be equal to (14), , that is, or . Therefore, or , and then or , where . If , then Similarly, if , then Further, by the Euler formula and the addition rule of trigonometric functions, Since and , we have Since we obtain and, similarly, Since (22) has to be equal to (21), , that is, or . Therefore, or , and then or , where . If , then Similarly, if , then
Remark 2. By Theorem 1 and the properties of the Euler formula, if , then we can write also, if , then where and are the conjugate Fueter variables of (see [20]).
Let be an open subset of and let a function be defined by the following form on with values in : satisfying where , and are real-valued functions.
From the chain rule, we use the following differential operators: where in . We have the following equations: and then, the operator operates to as follows: Thus, we have a corresponding Laplacian in the reduced quaternion :
Remark 3. Let Ω be an open set of . From the definition of the differential operators in , we have and, therefore, Similarly, we have and, therefore,
Definition 4. Let be an open set in and for any element in . A function is said to be L(R)-regular on if the following conditions are satisfied:(i) are continuously differential functions on , and(ii) on .
In particular, the equation of Definition 4 is equivalent to Moreover, (38) is equivalent to the following system: The above system is a corresponding Cauchy-Riemann system in .
Remark 5. From the multiplications of , the equation of Definition 4 is equivalent to Also, the above equation (40) is equivalent to the following system: Further, the above system (41) is also a corresponding Cauchy-Riemann system in . Since the system (39) is equivalent to the system (41), we say that of Definition 4 is a regular function on . When the function is either an L-regular function or an R-regular function on , we simply say that is a regular function on .
3. Properties of Regular Functions with Values in
We define the derivative of by the following:
Proposition 6. Let be an open set in and let a function be a regular function defined on . Then
Proof. From the definition of a regular function , we have Therefore, Hence, we obtain the equation Similarly, by calculating the derivative of , Therefore, we have the equation Further, using the same procedure, we obtain the equations
Proposition 7. Let be an open set in . If is a regular function on , then we have where is a positive integer.
Proof. Since is a regular function on with values in , by Definition 4, Hence, is a regular function with values in . From Proposition 6, we have By repeating the above process, we can obtain the equation
We let on an open set in .
Theorem 8. Let be an open set in . If is a regular function on , then the following equation holds true:
Proof. Since is a regular function on , we have the following system: By the definition of , we have From Proposition 7, we have . Hence, by calculating and comparing the above polynomials, we obtain that is equal to .
Next, we consider a differential form
Theorem 9. Let be an open set in and let be any domain on with a smooth distinguished boundary such that . If is a regular function on , then one has where is the reduced quaternionic product of the form on the function .
Proof. Since , we have where in . From the corresponding Cauchy-Riemann system (39) for in , we have the system (56). Hence, and, therefore, by Stokes theorem, we obtain the following result:
The third author was supported by a 2-Year Research Grant of Pusan National University.
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Copyright © 2014 Ji Eun Kim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Kontsevich quantization formula - Wikipedia
In mathematics, the Kontsevich quantization formula describes how to construct a generalized ★-product operator algebra from a given arbitrary finite-dimensional Poisson manifold. This operator algebra amounts to the deformation quantization of the corresponding Poisson algebra. It is due to Maxim Kontsevich.[1][2]
1 Deformation quantization of a Poisson algebra
2 Kontsevich graphs
2.1 Associated bidifferential operator
2.2 Associated weight
3.1 Explicit formula up to second order
Deformation quantization of a Poisson algebra[edit]
Given a Poisson algebra (A, {⋅, ⋅}), a deformation quantization is an associative unital product ★ on the algebra of formal power series in ħ, A[[ħ]], subject to the following two axioms,
{\displaystyle {\begin{aligned}f*g&=fg+{\mathcal {O}}(\hbar )\\{}[f,g]&=f*g-g*f=i\hbar \{f,g\}+{\mathcal {O}}(\hbar ^{2})\end{aligned}}}
If one were given a Poisson manifold (M, {⋅, ⋅}), one could ask, in addition, that
{\displaystyle f*g=fg+\sum _{k=1}^{\infty }\hbar ^{k}B_{k}(f\otimes g),}
where the Bk are linear bidifferential operators of degree at most k.
Two deformations are said to be equivalent iff they are related by a gauge transformation of the type,
{\displaystyle {\begin{cases}D:A[[\hbar ]]\to A[[\hbar ]]\\\sum _{k=0}^{\infty }\hbar ^{k}f_{k}\mapsto \sum _{k=0}^{\infty }\hbar ^{k}f_{k}+\sum _{n\geq 1,k\geq 0}D_{n}(f_{k})\hbar ^{n+k}\end{cases}}}
where Dn are differential operators of order at most n. The corresponding induced ★-product, ★′, is then
{\displaystyle f\,{*}'\,g=D\left(\left(D^{-1}f\right)*\left(D^{-1}g\right)\right).}
For the archetypal example, one may well consider Groenewold's original "Moyal–Weyl" ★-product.
Kontsevich graphs[edit]
A Kontsevich graph is a simple directed graph without loops on 2 external vertices, labeled f and g; and n internal vertices, labeled Π. From each internal vertex originate two edges. All (equivalence classes of) graphs with n internal vertices are accumulated in the set Gn(2).
An example on two internal vertices is the following graph,
Associated bidifferential operator[edit]
Associated to each graph Γ, there is a bidifferential operator BΓ( f, g) defined as follows. For each edge there is a partial derivative on the symbol of the target vertex. It is contracted with the corresponding index from the source symbol. The term for the graph Γ is the product of all its symbols together with their partial derivatives. Here f and g stand for smooth functions on the manifold, and Π is the Poisson bivector of the Poisson manifold.
The term for the example graph is
{\displaystyle \Pi ^{i_{2}j_{2}}\partial _{i_{2}}\Pi ^{i_{1}j_{1}}\partial _{i_{1}}f\,\partial _{j_{1}}\partial _{j_{2}}g.}
Associated weight[edit]
For adding up these bidifferential operators there are the weights wΓ of the graph Γ. First of all, to each graph there is a multiplicity m(Γ) which counts how many equivalent configurations there are for one graph. The rule is that the sum of the multiplicities for all graphs with n internal vertices is (n(n + 1))n. The sample graph above has the multiplicity m(Γ) = 8. For this, it is helpful to enumerate the internal vertices from 1 to n.
In order to compute the weight we have to integrate products of the angle in the upper half-plane, H, as follows. The upper half-plane is H ⊂ ℂ, endowed with a metric
{\displaystyle ds^{2}={\frac {dx^{2}+dy^{2}}{y^{2}}};}
and, for two points z, w ∈ H with z ≠ w, we measure the angle φ between the geodesic from z to i∞ and from z to w counterclockwise. This is
{\displaystyle \phi (z,w)={\frac {1}{2i}}\log {\frac {(z-w)(z-{\bar {w}})}{({\bar {z}}-w)({\bar {z}}-{\bar {w}})}}.}
The integration domain is Cn(H) the space
{\displaystyle C_{n}(H):=\{(u_{1},\dots ,u_{n})\in H^{n}:u_{i}\neq u_{j}\forall i\neq j\}.}
The formula amounts
{\displaystyle w_{\Gamma }:={\frac {m(\Gamma )}{(2\pi )^{2n}n!}}\int _{C_{n}(H)}\bigwedge _{j=1}^{n}\mathrm {d} \phi (u_{j},u_{t1(j)})\wedge \mathrm {d} \phi (u_{j},u_{t2(j)})}
where t1(j) and t2(j) are the first and second target vertex of the internal vertex j. The vertices f and g are at the fixed positions 0 and 1 in H.
Given the above three definitions, the Kontsevich formula for a star product is now
{\displaystyle f*g=fg+\sum _{n=1}^{\infty }\left({\frac {i\hbar }{2}}\right)^{n}\sum _{\Gamma \in G_{n}(2)}w_{\Gamma }B_{\Gamma }(f\otimes g).}
Explicit formula up to second order[edit]
Enforcing associativity of the ★-product, it is straightforward to check directly that the Kontsevich formula must reduce, to second order in ħ, to just
{\displaystyle f*g=fg+{\tfrac {i\hbar }{2}}\Pi ^{ij}\partial _{i}f\,\partial _{j}g-{\tfrac {\hbar ^{2}}{8}}\Pi ^{i_{1}j_{1}}\Pi ^{i_{2}j_{2}}\partial _{i_{1}}\,\partial _{i_{2}}f\partial _{j_{1}}\,\partial _{j_{2}}g-{\tfrac {\hbar ^{2}}{12}}\Pi ^{i_{1}j_{1}}\partial _{j_{1}}\Pi ^{i_{2}j_{2}}(\partial _{i_{1}}\partial _{i_{2}}f\,\partial _{j_{2}}g-\partial _{i_{2}}f\,\partial _{i_{1}}\partial _{j_{2}}g)+{\mathcal {O}}(\hbar ^{3})}
^ M. Kontsevich (2003), Deformation Quantization of Poisson Manifolds, Letters of Mathematical Physics 66, pp. 157–216.
^ Cattaneo, Alberto and Felder, Giovanni (2000). "A Path Integral Approach to the Kontsevich Quantization Formula". Communications in Mathematical Physics. 212 (3): 591. arXiv:math/9902090. Bibcode:2000CMaPh.212..591C. doi:10.1007/s002200000229. {{cite journal}}: CS1 maint: uses authors parameter (link)
Retrieved from "https://en.wikipedia.org/w/index.php?title=Kontsevich_quantization_formula&oldid=1037895312"
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Temperature vs. altitude chart analysis. Why does temperature decrease with higher altitude?
Example using the temperature at altitude calculator: temperature at 35000 feet altitude
With this temperature at altitude calculator, you can easily find an approximation of the temperature 🌡 at any given altitude.
Have you ever wondered what the temperature at cruising altitude is? ✈ Or why does temperature decrease with higher altitude? Then, we have the perfect tool for you!
In a few paragraphs, we will answer all those questions and more. We will also give you some examples while showing and explaining the temperature vs. altitude chart.
Most people think that the higher you go within Earth's atmosphere, the colder it gets. Well, they're right... kind of. In reality, the atmosphere is a complex thermodynamic system that needs constant monitoring through satellite or radar information to produce accurate predictions.
One of the models describing Earth's atmosphere is the ISA, International Standard Atmosphere model, which utilizes geopotential altitude to obtain the temperature.
According to this model, temperature either increases, decreases, or remains constant as you climb up in the atmosphere, contrary to pressure, which only decreases with higher altitude (and so does air density). See the temperature vs altitude section for a full explanation.
Why use geopotential altitude?
Geopotential altitude is used instead of geometric altitude (distance above a surface) because gravity on Earth is not exactly the same everywhere. It varies with height (Newton's law of gravity), latitude (due to centrifugal forces), longitude (uneven distribution of Earth's mass), and some other parameters.
Because of that, geopotential altitude is a more accurate variable to quantify the properties of large masses of air. It's described by the following formula to account for gravity variations:
Z_{g}(h) = \frac{\Phi(h)}{g_{0}}
\Phi(h)
– Geopotential energy at
h
g_{0}
– Gravitational acceleration at mean sea level.
💡 Don't worry! You don't need a Ph.D. in meteorology to use the temperature at altitude calculator. You can just input the normal altitude, and there will be little difference in the results (more significant at higher altitudes).
As we said, this calculator is based on the ISA model, but what does the model actually say? Let's take a look at the temperature vs altitude chart:
Temperature (°C) vs altitude (km) based on the ISA model (1976).
At first look, it seems that asking 'Why does temperature decrease with higher altitude' wouldn't be precise. As the graph shows, temperature either decreases, remains constant, or increases with higher altitude. Let's break down each part of the graph to learn how altitude affects temperature:
Troposphere (0 to ∼12 km, or 0 to ∼7 mi): this is the lowest part of Earth's atmosphere, and it's heated by the Earth's surface. So, as you go up, you get further away from the surface and the temperature decreases by about 6.5 °C per km, or ∼39 °F per mi.
Tropopause (between the troposphere and stratosphere): in this layer, the temperature remains constant at -55 °C, or 131°F.
Stratosphere (∼12 to ∼51 km, or ∼7 to ∼32 mi): on the contrary, in this layer, temperature increases by 1 to 2.8 °C per km, or 33 to 35 °F per mi. UV absorption by the ozone layer produces this heating effect.
Stratopause (∼48 to ∼51 km, or ∼30 to ∼32 mi): here, the temperature is again constant at ∼-1 °C, or ∼30 °F.
Mesosphere (∼51 to ∼86 km, or ∼32 to ∼53 mi): this part of the atmosphere is challenging to study. Aircraft can't reach high enough, and atmospheric drag renders satellites unusable. Here, the temperature decreases by 2 to 2.8 °C per km, or 34 to 35 °F.
Mesopause (∼86 km, or ∼53 mi): the coldest place on Earth. Here, temperatures lie around -87 °C, or -124.6 °F and can be as low as -100 °C, or -148 °F.
💡 Now you know how altitude affects temperature! Feel free to use Omni's temperature at altitude calculator to find an approximation for input altitudes between 0-90 km based on the ISA model.
Let's see how we can obtain the temperature at cruising altitude/temperature at 35000 feet altitude:
First, we need to write down the temperature and altitude at your location.
Let's assume the temperature to be 59 °F and we are at 2640 ft high.
Now, we need to find where 35000 ft lay on the temperature vs. altitude chart. As we can see, a 35000 ft altitude is within the troposphere where temperature decreases with altitude.
We subtract the altitude at our current location from 35000 ft and multiply the result by 0.00356.
The result of the product will be the temperature difference: 32360 * 0.00356 = 115.2 °F.
Since the temperature decreases in this layer, we subtract 115.2 °F from the temperature at 2640 ft to get the temperature at our desired altitude.
59 - 115.2 = -56.2 °F.
Temperature at 35000 feet : -56.2 °F.
Remember: this is for 59 °F at 2640 ft, not 59 °F at sea level.
Or simply input the temperature at sea level in our tool, and it will automatically get the result for you. The temperature at altitude calculator even has its own chart!
So far, in the temperature at altitude calculator, we've covered:
How does altitude affect temperature?;
Why does temperature decrease with higher altitude?;
What is the temperature at cruising altitude?; and
Insight into the ISA model with the temperature vs. altitude chart.
If you still have questions, check the FAQ section where we added some more answers to understand the subject thoroughly.
How do you calculate temperature with altitude?
To calculate temperature with altitude:
Write down the current temperature at your location.
Convert the height (from your current altitude) at which you want to obtain the temperature to m or ft.
0.00650 if using the metric system; or
0.00356 if using the imperial or US customary system.
Subtract the result from the temperature in step 1. This number is the temperature at your chosen altitude.
Why does temperature increase with altitude through the stratosphere?
Temperature increases with altitude through the stratosphere because the ozone layer present there absorbs most of the UV radiation coming from the Sun. This layer acts as a shield from incoming ultraviolet radiation.
How does temperature change with altitude in the troposphere?
In the troposphere, temperature decreases by 6.5 °C per 1000 m, or ∼3.56 °F per 1000 ft. Since this layer is heated directly by Earth's surface, the higher it extends, the colder the air within the layer gets.
In the troposphere, temperature decreases with altitude because this layer of the atmosphere is heated through direct contact with Earth's surface. Because of that, the farthest you are from the ground, the colder it gets.
What is the farthest layer of the atmosphere?
The exosphere is the farthest layer of Earth's atmosphere. It extends from the thermosphere up to 10,000 km and gradually fades into outer space. It's primarily composed of hydrogen and helium.
How cold is it 1000 feet up?
55.44 °F. At 1000 feet, the temperature drops by 3.56 °F compared to the surface temperature. Assuming 59 °F at sea level, the temperature at 1000 feet would be 55.44 °F.
Temperature at sea level (T0)
Show chart?
Use this 3-phase motor amperage calculator to quickly estimate how much amperage your 3-phase motor draws, given its needed voltage, power rating, power factor, and efficiency.
This crossover calculator will help you design amazing sounding speaker units.
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Electron volt definition and associated terms
How to convert volts to electron volts
What is the formula for electron volt to joule conversion?
If you are looking for a volt to electron volt calculator or just hoping to learn about the process used to convert volts to electron volts, you have come to the right place. You can use this tool to change the electrical voltage, measured in volts (V), to energy measured in electron volts (eV). You can also see the answer in joules by changing the units of the last parameter of the volt to electron volt calculator. Please do not confuse this tool with the one we use for energy conversion.
Continue reading to learn the definition of the volt and electron volt, the different formulas used to convert volt to electron volts, and how to apply them. We also show the equation used in the conversion of electron volt to joule and vice versa. You may also check out our Ohm's law calculator to learn more about electrical power.
Below, you can find some definitions of terms we use in this article:
Definition of a volt (V): Volt is the SI unit of voltage, the potential difference or electromotive force; 1 volt = 1 joule per coulomb.
Definition of an electron volt (eV): An electron volt or electron-volt is an amount of kinetic energy gained or lost by an electron when it moves from rest through a potential of 1 volt. An electron volt is a unit of energy. One electron volt is equal to 1.602176634 × 10-19 J.
Definition of elementary charge (e): An elementary charge is the smallest unit of electricity. It is a positive constant and equivalent to 1.602176634 × 10-19 coulombs. We represent the elementary charge by the symbol e.
Definition of coulombs (C): Coulomb is the name for the electric charge in the SI unit system. It is the amount of electricity transported in a single second by one ampere.
Do you wish to learn more about coulombs? Check out our Coulomb's law calculator.
Volts can be converted to electron volts using either elementary charge or coulombs. To do this, you must choose whether you will be using coulombs or elementary charge, then use the relevant formula to do the volt conversion.
We have explained both methods of volt conversion below.
How to calculate electron volts from elementary charge
Our volt to electron volt calculator uses the following formula:
eV = V × e
To better understand the formula above, let us consider the following problem:
Find the energy in electron volts used in a circuit where the voltage was 36 volts, and the charge flow was 30 electron charges.
eV = 36 × 30 = 1080 eV
How to calculate electron volts from coulombs
The second option in this volt to electron volt calculator is to use coulombs instead of the elementary charge. The formula is:
\footnotesize\text{eV} = \text{V} \times \text{C} / (1.602176634 \times 10^{-19})
Let’s look at the example below to understand it better:
Calculate the energy consumed in an electric circuit where the charge flow was 5 coulombs with a voltage supply of 15 volts.
\footnotesize\text{eV} = 15 \times 5 / (1.602176634 \times 10^{-19})
\footnotesize\quad\ \ = 4.68 \times 10^{20} \text{eV}
If you are studying electricity or need to do other calculations involving electricity, this voltage drop calculator may interest you.
An electron volt is a unit of energy, but a joule is the energy base SI unit. Our calculator would not be complete without showing the answer in joules as well. Here is the formula we use in our calculator for the electron volt to joule conversion:
1 J = 6.24150907 × 1018 eV
Conversely, to see how much is 1 electron volt we use the formula:
1 eV = 1.602176634 × 10-19 J
To convert electron volt to joules, enter the value in electron volts and change the units to joules using the drop-down list on the right.
The momentum calculator evaluates the linear momentum of an object based on its mass and velocity.
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* 1 Feb: HW 3 is graded. Average 37/40. Solutions will be posted soon.
<tr><td> 1 Feb (W) <td> [[CDS 110b: Kalman Filtering|Kalman Filters]] <td> Friedland, Ch 11 <td rowspan=2> Midterm (due 7 Feb)
<td> 6 Feb (M) <td> [[CDS 110b:Sensor Fusion|Sensor Fusion]] <td> Friedland, Ch 11 + notes <td rowspan=2> [[CDS 110b: Homework 5|HW 5]]
{\displaystyle H_{\infty }}
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The inverse spin-Hall effect generates the electric charge current JC (electro motive force) toward the direction of vector product of the spin current (JS) and the magnetization (σ), as expressed in eq. (1).
Jc//{J}_{s}×\sigma
Therefore, polarity of the electro motive force also inverts as a magnetic direction against the sample inverts. It is an interesting property that the angular dependence of the inverse spin-Hall electro motive force shows the extremely different behavior from the ordinal FMR one.
FMR spectra were measured with the step size of 15 degree against the applied magnetic field using the angular rotation device (ES-UCR3X in Fig. 1).
The sample was the same metallic bilayer thin film reported on JEOL application note[1].
The angular dependence of the VISHE spectra
The dependence of VISHE signal on the magnetic field shows a drastic inverting phenomena, as shown in Fig. 2. This is clearly different from FMR behavior. Substituting the obtained parameters from the angular experiment of FMR to eq. (2), the behavior of VISHE against the angular can be simulated[2, 3].
Angle [deg.] 0-360
Temp. Room Temp.(26C)
NW Frequency [MHz] 9441.523
MW Power [mW] 160
Ho [mT] Sweep width 150mT(corrected by Mn2+)
Mod. Width [mT] 0.0002/0.1
Mod. Phase [deg.] 0
Amp. Gain 10(FMR)
ISHE 40 dB(CA-261F2)+0.6uF(LPF)→CN115
Fig. 2 an experimental result and analysis of the angular dependence of VISHE spectra.
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The term dividends per share (DPS) refers to the total dividend a company pays out over a 12-month period, divided by the total number of outstanding shares. A company uses this calculation to share profits with its shareholders. DPS can indicate how profitable a company is over a fiscal period and it can tell an investor about the company's past financial health and its current financial stability.
Dividends per share (DPS) is an important financial ratio in understanding the financial health and long-term growth prospects of a company.
A steady or growing dividend payment by a company can be a signal of stability and growth.
A declining DPS may be due to reinvestment in a firm's operations or debt reduction, but may also indicate poor earnings and be a red flag for financial hardship.
What Dividends Per Share Tells You
DPS is an important metric to investors because the amount a firm pays out in dividends directly translates to income for the shareholder, and the DPS is one of the most straightforward figures an investor can use to calculate his or her dividend payments from owning shares of a stock over time. Meanwhile, a growing DPS over time can also be a sign that a company's management believes that its earnings growth can be sustained.
For example, suppose company ABC had a DPS of 60 cents last year, but this year, it doesn't pay a dividend to its shareholders. This can signal to investors the company may be in poor financial health and cannot withstand the current market conditions. A decrease in DPS can thus cause investors to sell their stake in the company, driving the market value of ABC down further.
However, a decrease in dividend per share does not always signal a company is not financially stable. For example, suppose ABC did not pay out a dividend to its shareholders because it is using its profit to reinvest into the company to create a new product. This reinvestment into the business can potentially produce higher dividends in the long term.
How to Calculate Dividends Per Share
Dividend per share is the sum of declared dividends issued by a company for every ordinary share outstanding. The figure is calculated by dividing the total dividends paid out by a business, including interim dividends, over a period of time by the number of outstanding ordinary shares issued. A company's DPS is often derived using the dividend paid in the most recent quarter, which is also used to calculate the dividend yield.
DPS can be calculated by using the following formula, where the variables are defined as:
\begin{aligned} &\text{DPS} = \frac { \text{D} - \text{SD} }{ \text{S} } \\ &\textbf{where:} \\ &\text{D} = \text{sum of dividends over a period (usually} \\ &\text{a quarter or year)} \\ &\text{SD} = \text{special, one-time dividends in the period} \\ &\text{S} = \text{ordinary shares outstanding for the period} \\ \end{aligned}
DPS=SD−SDwhere:D=sum of dividends over a period (usuallya quarter or year)SD=special, one-time dividends in the periodS=ordinary shares outstanding for the period
Suppose company YXZ has been paying a steady dividend of 90 cents per share. The next year, company YXZ raises its dividend to $1.10 per share. This signals the company is financially stable and performing well in its current market condition. An increase in DPS also signals the management team is confident in the company's future profits.
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Easier Than You Thought! | Toph
By ridzz007 · Limits 1s, 512 MB
CSE Programming Club of Comilla University was created for the betterment of the students of Computer Science and Engineering department. Since then, it has helped programmers of the department in various ways possible. Arranging programming contests, training sessions, discussion sessions along with LAB facilities are to name a few. Following the tradition, the club is here with yet another programming contest for you to test your programming skills. In this problem, you’ll have to answer several test cases. In each case, you’ll be given a non-empty string containing English letters A-Z, a-z, spaces and special characters ( comma ( , ) and dot ( . ) ). You have to answer if you can obtain the string "Programming Club, CSE CoU." (without quotes), by taking characters from the input string.
Input starts with an integer T, the number of test cases. In each case, you’ll be given a non-empty string of characters.
1 ≤ T ≤
10^{5}
sum of characters over all test cases does not exceed
10^{6}
Print “YES” if you can obtain the string "Programming Club, CSE CoU."(without quotes) By taking characters from the input string, otherwise print “NO”.
Programming Club is created by CSE Society, CoU.
Programming Club helps Programmers a lot.
steinumEarliest, Oct '20
Sakib_UddinFastest, 0.0s
Peal_HassanLightest, 1.1 MB
Intra CoU Programming Contest 2020
Replay of Intra CoU Programming Contest 2020
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In this tutorial, I’ll cover the Yolo v3 loss function and model training. We’ll train a custom object detector on Mnist dataset
In a previous tutorial, I introduced you to the Yolo v3 algorithm background, network structure, feature extraction, and finally, we made a simple detection with original weights. In this part, I’ll cover the Yolo v3 loss function and model training. We’ll train a custom object detector on the Mnist dataset.
In YOLO v3, the author regards the target detection task as the regression problem of target area prediction and category prediction, so its loss function is somewhat different. For the loss function, Redmon J did not explain in detail in the Yolo v3 paper. But I found Yolo loss explanation in this link; the loss function looks following:
However, through the interpretation of the darknet source code, the loss function of YOLO v3 can be summarized as follows:
Confidence loss, determine whether there are objects in the prediction frame;
Box regression loss, calculated only when the prediction box contains objects;
Classification loss, decide which category the things in the prediction frame belong to.
YOLO v3 directly optimizes the confidence loss to let the model learn to distinguish the background and foreground areas of the picture, which is similar to the RPN function in Faster R-CNN:
The rule of determination is simple: if the IoU of a prediction box and all actual boxes is less than a certain threshold, it is determined to be the background. Otherwise, it is the foreground (including objects).
The classification loss used here is the cross-entropy of the two classifications. The classification problem of all categories is reduced to whether it belongs to this category so that multi-classification is regarded as a two-classification problem. The advantage of this is to exclude the mutual exclusion of the classes, especially to solve the problem of missed detection due to the overlapping of multiple categories of objects.
Loss=\frac{-1}{n}\sum _{x}\left(y*\mathrm{ln}\left(a\right)+\left(1-y\right)*\mathrm{ln}\left(1-a\right)\right)
respond_bbox = label[:,:,:,:, 4:5]
prob_loss = respond_bbox * tf.nn.sigmoid_cross_entropy_with_logits (labels = label_prob, logits = conv_raw_prob
Box regression loss
Box regression loss in code looks as follows:
respond_bbox = label[:, :, :, :, 4:5]
bbox_loss_scale = 2.0 - 1.0 * label_xywh[:, :, :, :, 2:3] * label_xywh[:, :, :, :, 3:4] / (input_size ** 2)
giou_loss = respond_bbox * bbox_loss_scale * (1 - giou)
giou_loss = tf.reduce_mean(tf.reduce_sum(giou_loss, axis=[1,2,3,4]))
The smaller the size of the bounding box, the larger the value of bbox_loss_scale. We know that the anchors in YOLO v1 have done root and width processing in the Loss, which is to weaken the impact of the size of the bounding box on the loss value;
Respond_bbox means that if the grid cell contains objects, then the bounding box loss will be calculated;
The larger the value of GIoU between the two bounding boxes, the smaller the loss value of GIoU, so the network will optimize towards the direction of higher overlap between the prediction box and the actual box.
In my implementation, the original IoU loss was replaced with GIoU loss. This improved the detection accuracy by about 1%. The advantage of GIoU is that it enhances the distance measurement method between the prediction box and the anchor box.
GIoU background introduction
This is quite a new proposed way to optimize the bounding box-GIoU (Generalized IoU). The bounding box is generally represented by the coordinates of the upper left corner and the lower right corner, namely (x1, y1, x2, y2). Well, you find that this is a vector. The L1 norm or L2 norm can generally measure the distance of the vector. However, the detection effect is very different when the L1 and L2 norms take the same value. The direct performance is that the IoU value of the prediction and the actual detection frame changes significantly, which shows that the L1 and L2 norms are not very good to Reflect the detection effect.
When the L1 or L2 norms are the same, it is found that the values of IoU and GIoU are very different, which indicates that it is not appropriate to use the L norm to measure the distance of the bounding box. In this case, the academic community generally uses the IoU to calculate the similarity between two bounding boxes. The author found that using IoU has two disadvantages, making it less suitable for loss function:
When there is no coincidence between the prediction box and the actual box, the IoU value is 0, which results in a gradient of 0 when optimizing the loss function, which means that it cannot be optimized. For example, the IoU value of scene A and scene B are both 0, but the prediction effect of scene B is better than A because the distance between the two bounding boxes is closer (the L norm is smaller):
Even when the prediction box and the actual box coincide and have the same IoU value, the detection effect has a significant difference, as shown in the following figure:
The three images above have IoU = 0.33, but the GIoU values are 0.33, 0.24, and -0.1, respectively. This indicates that the better the two bounding boxes overlap and align, the higher the GIOU value will be.
GIoU calculation formula:
And here how it looks in code (part from original code):
def bbox_giou (boxes1, boxes2):
# Calculate the iou value between the two bounding boxes
# Calculate the coordinates of the upper left corner and the lower right corner of the smallest closed convex surface
enclose_left_up = tf.minimum (boxes1 [..., :2], boxes2 [..., :2])
enclose_right_down = tf.maximum (boxes1 [..., 2:], boxes2 [..., 2:])
enclose = tf.maximum(enclose_right_down-enclose_left_up, 0.0 )
# Calculate the area of the smallest closed convex surface C
enclose_area = enclose [..., 0 ] * enclose [..., 1 ]
# Calculate the GIoU value according to the GioU formula
giou = iou- 1.0 * (enclose_area-union_area) / enclose_area
One problem training neural networks faces, especially Deep Neural Networks, is that the gradient disappears or the gradient explodes. This means that when we train a Deep Network, the derivative or slope sometimes becomes very large, very small, or even decreases exponentially. At this time, the Loss we see will become NaN. Suppose we are training such an extremely Deep Neural Network. The activation function g(z)=z is used for simplicity, and the bias parameter is ignored.
First, we assume that g(z)=z, b[l]=0, so for the target output:
\text{If} {W}^{\left[l\right]}=\left[\begin{array}{cccc}1.5& 0& 0& 1.5\end{array}\right], \text{then} \stackrel{^}{y}={W}^{\left[L\right]}{\left[\begin{array}{cc}1.5& 0\\ 0& 1.5\end{array}\right]}^{\left[L-1\right]}X, \text{the value of the activation function will increase exponentially;}
\text{If} {W}^{\left[l\right]}=\left[\begin{array}{cccc}0.5& 0& 0& 0.5\end{array}\right], \text{then} \stackrel{^}{y}={W}^{\left[L\right]}{\left[\begin{array}{cc}0.5& 0\\ 0& 0.5\end{array}\right]}^{\left[L-1\right]}X, \text{the value of the activation function will decrease exponentially;}
The intuitive understanding here is: if the weight W is only slightly larger than one or the unit matrix, the output of the deep neural network will explode. And if W is somewhat smaller than 1, it may be 0.9; the output value of each layer of the network will decrease exponentially. This means that the proper weight value initialization is particularly important! The following is a simple code to show this:
x = np.random.randn(2000, 800) * 0.01 # Create input data
stds = [0.25, 0.2, 0.15, 0.1, 0.05, 0.04, 0.03, 0.02] # Try to use different standard deviations so that the initial weights are different
for i, std in enumerate(stds):
# First layer - fully connected layer
dense_1 = tf.keras.layers.Dense(750, kernel_initializer=tf.random_normal_initializer(stddev=std), activation='tanh')
output_1 = dense_1(x)
# Second layer - fully connected layer
output_2 = dense_2(output_1)
# Third layer - fully connected layer
output_3 = dense_3(output_2).numpy().flatten()
plt.subplot(1, len(stds), i+1)
plt.hist(output_3, bins=600, range=[-1, 1])
plt.xlabel('std = %.3f' %std)
After running this above code, you should see the following chart:
We can see that when the standard deviation is large (std => 0.25), almost all output values are concentrated near -1 or 1, which indicates that the Neural Network has a gradient explosion at this time. When the standard deviation is small (std = 0.03 and 0.02), we see that the output value is quickly approaching 0, which indicates that the gradient of the neural network has disappeared at this time. If the standard deviation of the initialization weight is too large or too small, NaN may appear while training the network.
The learning rate is one of the hyperparameters that the most affect performance. If we can only adjust one hyperparameter, then the best choice is the learning rate. In most cases, the improper selection of learning rate causes Loss to become NaN. The following image shows that gradient descent can be slow if the learning rate is too low. If the learning rate is too large, gradient descent can overshoot the minimum. It may fail to converge or even diverge.
Since the Neural Network is very unstable at the beginning of training, the learning rate should be set very low to ensure that the network can have good convergence. But the lower learning rate will make the training process very slow, so here we will use a way to gradually increase the lower learning rate to a higher learning rate to achieve the “warmup” stage of network training, called the warmup stage. About warmup, you can read on this paper.
But suppose we minimize the Loss of Network training. In that case, it is not appropriate to always use a higher learning rate because it will make the weight gradient oscillate back and forth, and it is difficult to make the Loss of training reach the global minimum. Therefore, the cosine attenuation method in the same paper is adopted in the end. This stage can be called a consensus decay stage. This is how our learning rate chart will look like:
Pre-trained Yolo v3 model weights
The current mainstream approach to target detection is to extract features based on the pre-trained model of the Imagenet dataset and then perform fine-tuning training on target detection (such as the YOLO algorithm) on the COCO dataset referred to as transfer learning. Transfer learning is based on a similar distribution of the data set. For example, the Mnist is entirely different from the COCO dataset distribution. There is no need to load the COCO pre-training model.
Quick training for custom Mnist dataset
To test if custom Yolo v3 object detection training works for you, you must first complete the tutorial steps to ensure that simple detection with original weights works for you.
When you have cloned the GitHub repository, you should see the “mnist” folder containing mnist images. From these images, we create mnist training data with the following command:
This make_data.py script creates training and testing images in the correct format. Also, this makes an annotation file. One line for one image, in the form like the following:
image_absolute_path xmin,ymin,xmax,ymax,label_index xmin2,ymin2,xmax2,ymax2,label_index2 ...
The origin of coordinates is at the left top corner, left top => (xmin, ymin), right bottom => (xmax, ymax), label_index is in range [0, class_num — 1]. We’ll talk more about this in the next tutorial, where I will show you how to train the YOLO model with your custom data.
./yolov3/configs.py file is already configured for Mnist training.
Now, you can train it and then evaluate your model running these commands from a terminal: python train.py tensorboard --logdir ./log
Track your training progress in Tensorboard by going to http://localhost:6006/; after a while, you should see similar results to this:
When the training process is finished, you can test detection with detect_mnist.py script: python detect_mnist.py
Then you should see similar results to the following:
That’s it for this tutorial. We learned how the Loss works in the Yolo v3 algorithm, and we trained our first custom object detector with the Mnist dataset. This was relatively easy because I prepared all files to test training with only a few commands.
In the next part, I will show you how to configure everything for custom objects training, transfer weights from the original weights file, and finally fine-tuning with your classes. See you in the next part!
|
Price Elasticity of Supply | Boundless Economics | Course Hero
Definition of Price Elasticity of Supply
The price elasticity of supply is the measure of the responsiveness in quantity supplied to a change in price for a specific good.
Differentiate between the price elasticity of demand for elastic and inelastic goods
Elasticity is defined as a proportionate change in one variable over the proportionate change in another variable:
Elasticity \;= \; \frac{\%\; Change\; in\; quantity}{\%\; Change\; in\; price}
The impact that a price change has on the elasticity of supply also directly impacts the elasticity of demand.
Inelastic goods are often described as necessities, while elastic goods are considered luxury items.
The elasticity of a good will be labelled as perfectly elastic, relatively elastic, unit elastic, relatively inelastic, or perfectly inelastic.
luxury: Something very pleasant but not really needed in life.
supply: The amount of some product that producers are willing and able to sell at a given price, all other factors being held constant.
In economics, elasticity is a summary measure of how the supply or demand of a particular good is influenced by changes in price. Elasticity is defined as a proportionate change in one variable over the proportionate change in another variable:
Elasticity \;= \; \frac{\%\; Change\; in\; quantity}{\%\; Change\; in\; price}
The price elasticity of supply (PES) is the measure of the responsiveness in quantity supplied (QS) to a change in price for a specific good (% Change QS / % Change in Price). There are numerous factors that directly impact the elasticity of supply for a good including stock, time period, availability of substitutes, and spare capacity. The state of these factors for a particular good will determine if the price elasticity of supply is elastic or inelastic in regards to a change in price.
The price elasticity of supply has a range of values:
PES > 1: Supply is elastic.
PES < 1: Supply is inelastic.
PES = 0: The supply curve is vertical; there is no response of demand to prices. Supply is "perfectly inelastic."
\infty
(i.e., infinity): The supply curve is horizontal; there is extreme change in demand in response to very small change in prices. Supply is "perfectly elastic."
Inelastic goods are often described as necessities. A shift in price does not drastically impact consumer demand or the overall supply of the good because it is not something people are able or willing to go without. Examples of inelastic goods would be water, gasoline, housing, and food.
Elastic goods are usually viewed as luxury items. An increase in price for an elastic good has a noticeable impact on consumption. The good is viewed as something that individuals are willing to sacrifice in order to save money. An example of an elastic good is movie tickets, which are viewed as entertainment and not a necessity.
The price elasticity of supply is determined by:
Number of producers: ease of entry into the market.
Spare capacity: it is easy to increase production if there is a shift in demand.
Ease of switching: if production of goods can be varied, supply is more elastic.
Ease of storage: when goods can be stored easily, the elastic response increases demand.
Length of production period: quick production responds to a price increase easier.
Time period of training: when a firm invests in capital the supply is more elastic in its response to price increases.
Factor mobility: when moving resources into the industry is easier, the supply curve in more elastic.
Reaction of costs: if costs rise slowly it will stimulate an increase in quantity supplied. If cost rise rapidly the stimulus to production will be choked off quickly.
The result of calculating the elasticity of the supply and demand of a product according to price changes illustrates consumer preferences and needs. The elasticity of a good will be labelled as perfectly elastic, relatively elastic, unit elastic, relatively inelastic, or perfectly inelastic.
Price elasticity over time: This graph illustrates how the supply and demand of a product are measured over time to show the price elasticity.
Perfectly Inelastic Supply: A graphical representation of perfectly inelastic supply.
The price elasticity of supply is the measure of the responsiveness of the quantity supplied of a particular good to a change in price.
Calculate elasticities and describe their meaning
When calculating the price elasticity of supply, economists determine whether the quantity supplied of a good is elastic or inelastic.
PES > 1: Supply is elastic. PES < 1: Supply is inelastic. PES = 0: if the supply curve is vertical, and there is no response to prices. PES = infinity: if the supply curve is horizontal.
mobility: The ability for economic factors to move between actors or conditions.
capacity: The maximum that can be produced on a machine or in a facility or group.
The price elasticity of supply (PES) is the measure of the responsiveness of the quantity supplied of a particular good to a change in price (PES = % Change in QS / % Change in Price). The intent of determining the price elasticity of supply is to show how a change in price impacts the amount of a good that is supplied to consumers. The price elasticity of supply is directly related to consumer demand.
The elasticity of a good provides a measure of how sensitive one variable is to changes in another variable. In this case, the price elasticity of supply determines how sensitive the quantity supplied is to the price of the good.
Calculating the PES
When calculating the price elasticity of supply, economists determine whether the quantity supplied of a good is elastic or inelastic. The percentage of change in supply is divided by the percentage of change in price. The results are analyzed using the following range of values:
PES = 0: Supply is perfectly inelastic. There is no change in quantity if prices change.
PES = infinity: Supply is perfectly elastic. An decrease in prices will lead to zero units produced.
The price elasticity of supply is calculated and can be graphed on a demand curve to illustrate the relationship between the supply and price of the good.
Supply and Demand Curves: A demand curve is used to graph the impact that a change in price has on the supply and demand of a good.
Applications of Elasticities
In economics, elasticity refers to how the supply and demand of a product changes in relation to a change in the price.
Give examples of inelastic and elastic supply in the real world
To determine the elasticity of a product, the proportionate change of one variable is placed over the proportionate change of another variable (Elasticity = % change of supply or demand / % change in price ).
For elastic demand, a change in price significantly impacts the supply and demand of the product.
For inelastic demand, a change in the price does not substantially impact the supply and demand of the product.
Economists use demand curves in order to document and study elasticity.
elastic: Sensitive to changes in price.
inelastic: Not sensitive to changes in price.
In economics, elasticity refers to the responsiveness of the demand or supply of a product when the price changes.
The technical definition of elasticity is the proportionate change in one variable over the proportionate change in another variable. For example, to determine how a change in the supply or demand of a product is impacted by a change in the price, the following equation is used: Elasticity = % change in supply or demand / % change in price.
The price is a variable that can directly impact the supply and demand of a product. If a change in the price of a product significantly influences the supply and demand, it is considered "elastic." Likewise, if a change in product price does not significantly change the supply and demand, it is considered "inelastic."
For elastic demand, when the price of a product increases the demand goes down. When the price decreases the demand goes up. Elastic products are usually luxury items that individuals feel they can do without. An example would be forms of entertainment such as going to the movies or attending a sports event. A change in prices can have a significant impact on consumer trends as well as economic profits. For companies and businesses, an increase in demand will increase profit and revenue, while a decrease in demand will result in lower profit and revenue.
For inelastic demand, the overall supply and demand of a product is not substantially impacted by an increase in price. Products that are usually inelastic consist of necessities like food, water, housing, and gasoline. Whether or not a product is elastic or inelastic is directly related to consumer needs and preferences. If demand is perfectly inelastic, then the same amount of the product will be purchased regardless of the price.
Economists study elasticity and use demand curves in order to diagram and study consumer trends and preferences. An elastic demand curve shows that an increase in the supply or demand of a product is significantly impacted by a change in the price. An inelastic demand curve shows that an increase in the price of a product does not substantially change the supply or demand of the product.
Inelastic Demand: For inelastic demand, when there is an outward shift in supply and prices fall, there is no substantial change in the quantity demanded.
Elastic Demand: For elastic demand, when there is an outward shift in supply, prices fall which causes a large increase in quantity demanded.
A-level Economics/AQA/Markets and Market failure. Provided by: Wikibooks. Located at: http://en.wikibooks.org/wiki/A-level_Economics/AQA/Markets_and_Market_failure%23Price_elasticity_of_supply. License: CC BY-SA: Attribution-ShareAlike
IB Economics/Microeconomics/Elasticities. Provided by: Wikibooks. Located at: http://en.wikibooks.org/wiki/IB_Economics/Microeconomics/Elasticities. License: CC BY-SA: Attribution-ShareAlike
Elasticity, Consumers, Producers, and Market Efficiency. Provided by: Wikiversity. Located at: http://en.wikiversity.org/wiki/Elasticity,_Consumers,_Producers,_and_Market_Efficiency. License: CC BY-SA: Attribution-ShareAlike
Transportation Economics/Demand. Provided by: Wikibooks. Located at: http://en.wikibooks.org/wiki/Transportation_Economics/Demand%23Elasticity. License: CC BY-SA: Attribution-ShareAlike
luxury. Provided by: Wiktionary. Located at: http://en.wiktionary.org/wiki/luxury. License: CC BY-SA: Attribution-ShareAlike
supply. Provided by: Wikipedia. Located at: http://en.wikipedia.org/wiki/supply. License: CC BY-SA: Attribution-ShareAlike
Price elasticity over time. Provided by: Wikimedia. Located at: http://commons.wikimedia.org/wiki/File:Price_elasticity_over_time.png. License: CC BY-SA: Attribution-ShareAlike
Boundless. Provided by: Boundless Learning. Located at: http://www.boundless.com//economics/definition/mobility. License: CC BY-SA: Attribution-ShareAlike
capacity. Provided by: Wiktionary. Located at: http://en.wiktionary.org/wiki/capacity. License: CC BY-SA: Attribution-ShareAlike
Fig5 Supply and demand curves. Provided by: Wikimedia. Located at: http://commons.wikimedia.org/wiki/File:Fig5_Supply_and_demand_curves.jpg. License: Public Domain: No Known Copyright
inelastic. Provided by: Wikipedia. Located at: http://en.wikipedia.org/wiki/inelastic. License: CC BY-SA: Attribution-ShareAlike
elastic. Provided by: Wiktionary. Located at: http://en.wiktionary.org/wiki/elastic. License: CC BY-SA: Attribution-ShareAlike
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|
Delta V Calculator | Spaceflight Dynamics
What is the difference between exhaust velocity and impulse?
How to calculate delta-v?
What is a delta-v budget?
How to use our delta-v calculator
Some practical examples of dubious use 😉
Moving in space is quite different from doing so on the surface of a planet — discover a little bit of spacecraft flight dynamics with our delta-v calculator. How much can you change your speed up there?
Whether studying orbital maneuvers or just playing Kerbal Space Program, you may need to know what delta-v is. Keep on reading to find it out:
How to find delta-v?
Some examples and how to use our delta-v calculator!
Space is dark and silent, but also fantastic to navigate! Don your suit, climb in your spaceship, and let us take care of guiding you there.
Ignition sequence start. 3, 2, 1, liftoff! 🚀
Delta-v, or
\Delta v
, is a technical way to say "difference in velocity". You can commonly find this quantity in physics, and it has a particular importance when moving in space, between orbits, bodies, or just arbitrary points.
In space, there's no air resistance nor any other source of friction. Once your starship reaches a certain speed, there is nothing to slow it down, and you can just coast until you reach your destination — and when you arrive, you have to brake, or risk overshooting! The concept of distance, then, is not that important: it only matters to humanity because it means we have to wait for our probe to reach the icy moons of Neptune, or because we are getting bored waiting to land on Mars!
When planning your space trip, it's helpful to think in terms of the propellant required to reach your destination more than the distance covered. This quantity translates easily, given the characteristics of the engine in use, to the difference of speed that can be attained while traveling.
When calculating delta-v, you can use either the effective exhaust velocity
v_e
or the specific impulse
I_{sp}
of the engine. Those are different quantities, but they are still closely related. Let's check them out.
The specific impulse
I_{sp}
of a reaction engine is a quantity defining the efficiency of the generation of thrust. It equals the change in momentum (in physics lingo, impulse) per unit mass of fuel — or using different words, the time for which an engine can generate thrust equal to its mass at
1\ g_0
of acceleration. We have a whole calculator dedicated to it — go check it out!
v_e
is a way to tackle the same problem, but omitting the reference to Earth's gravity (
g_0
). This is a measure of the speed of the gases produced by the engine when they exit the exhaust cone. The term effective is added because, in practice, the value of exhaust velocity varies across the cone's surface. The value used is a good approximation of a uniform velocity.
If we substitute the weight of the propellant with its flow — hence with the exhaust velocity (the two things are similar: more fuel in means more speed out), it is possible to cancel out the factor
g_0
. We use the relation:
v_e=I_{sp} \cdot g_0
Now we have all the tools to calculate delta-v. We need to introduce the backbone of rocket science, the Tsiolkovsky rocket equation.
In a propulsive system like a rocket, the engine consumes propellant, reducing the mass of the vehicle as it moves. This decrease in mass translates into the desired change of speed following the equation:
\begin{split} \Delta v & = v_e \cdot \ln\left({\frac{m_0}{m_t}}\right) \\ & =I_{sp}\cdot g_0 \cdot \ln\left({\frac{m_0}{m_t}}\right) \end{split}
m_0
is the initial mass, while
m_t
is the mass at the end of the engine's operations. We used the equality between specific impulse and exhaust velocity, see above.
We can derive this equation ourselves — it's not hard (we wanted to say that it's not rocket science, but...). First, define a system where a rocket with initial mass
m+\Delta m
t_0
, traveling with velocity
V
, expels a portion
\Delta m
of its mass (the propellant) with velocity
v_e
(the exhaust velocity). We define the momentum at the time
t
p_{t_0} = V \cdot (m+\Delta m)
t = t_0 + \Delta t
the value of the momentum changed:
p_t= (V+\Delta V) \cdot m +V_e \cdot \Delta m
V_e
is the velocity of the expelled mass from the outside of the rocket's reference system:
V_e=V-v_e
. From the rocket's perspective, the expelled mass travels at
-v_e
Given the absence of external forces, the momentum obeys the law of conservation, and using Newton's second law, it's possible to write:
F_\text{net} = \lim_{\Delta t\rightarrow 0} \frac{p_t - p_{t_0}}{\Delta t} = 0
We can substitute and then exchange
\Delta m
-dm
since we are considering a decrease in mass. Erasing two pairs of equal terms and taking the limit, we have:
\begin{split} \frac{m\cdot dV+v_e\cdot dm}{dt}&=0\\ -m\frac{dV}{dt}&=v_e\frac{dm}{dt}\\ -\frac{dV}{dt}&=v_e\frac{dm}{m \cdot dt} \end{split}
Finally, let's integrate. We change the limits of integration for the mass to
m_0
m_t
; thus we can write:
-\int^{V}_{V+\Delta V} dV=v_e\int^{m_t}_{m_0}\frac{1}{m}dm
which finally yields the delta-v:
\Delta V = v_e \cdot \ln{\frac{m_0}{m_t}}
Here you can obviously substitute the value of
v_e
with the specific impulse. That's all you need to know on how to calculate the delta-v!
Well, you know how to calculate the delta-v, now what? Let's see how to use it.
When you launch a space vehicle, you load a certain amount of fuel with you: this way, you set in advance the amount of thrust you can generate overall, which sets the amount of velocity you can acquire (or lose) during the flight. We call this quantity delta-v budget.
Maneuvers in space are elegant, to say the least. There are no roads and routes — only geometry, gravity, and a lot of math. Freeing yourself from the Earth's gravity well requires a huge delta-v of about
9\ \text{km}/\text{s}
just to reach low Earth orbit, or LEO), but once you're there, it takes a relatively small amount of fuel to move yourself to higher orbits, or in trajectories to other bodies.
However, refuelling in space is hard — if it's even feasible at all. Engineers need to carefully consider every eventuality before launching a spacecraft on its way. Think of the James Webb Space Telescope. It sits in an orbit around a point called Lagrange point L2, which requires constant corrections to the trajectory to be stable. The telescope has enough fuel for about ten years of operations, equalling a delta-v budget of less than
30\ \text{m}/\text{s}
Using our delta-v calculator is extremely easy. First, choose if you want to input the specific impulse or the exhaust speed, then fill the other required fields: the initial and final mass. Be careful to write them in the correct order!
🙋 You can use our delta-v calculator in reverse, too — input the delta-v and find out how much fuel you need, or if you need to change your engine!
You are in orbit around Earth, at the altitude of the International Space Station: you want to reach the Moon. How much fuel do you need to get there?
We plan on using a Hohmann transfer, a set of two burns described by this sequence:
The departure from a circular orbit;
The first burn, that accelerates the vehicle from the circular orbit to an elliptic one, at its perigee;
The coasting phase;
The second burn to transfer the vehicle on the target circular orbit from the apogee of the elliptic one by changing its speed to match the circular orbit's one;
The destination: a circular orbit with a larger radius.
🔎 The delta-v of a maneuver depends on the trajectory the vehicle will follow. There are many types of maneuvers, with different values of delta-v. The Hohmann transfer is generally a good choice in terms of efficiency, but it's not necessarily the best one!
The first circular orbit, at 400 km of height over the surface of Earth, requires a speed of about
v_{c_1} =7.67\ \text{km}/\text{s}
— you can calculate the orbital speeds by using our orbital velocity calculator.
The transfer orbit, with elliptic shape, has two characteristic velocities:
Velocity at the perigee
(v_\text{elliptic})_p=10.76\ \text{km}/\text{s}
, higher than the one in the circular orbit;
Velocity at the apogee
(v_\text{elliptic})_a =1.82\ \text{km}/\text{s}
, lower than the one of the target orbit.
The final orbit is circular, at a distance of about
40,\!000\ \text{km}
. The corresponding orbital velocity is about
1\ \text{km}/\text{s}
It is straightforward to calculate the required delta-v in both burn:
\Delta v_1 = 3.09\ \text{km}/\text{s}
\Delta v_2 = 0.82\ \text{km}/\text{s}
Which sum to a total delta-v of
\Delta v= \Delta v_1 + \Delta v_2 = 3.91\ \text{km}/\text{s}
Let's now take the parameters of a classic rocket engine, the third stadium of the Saturn V, which moved the Apollo spacecrafts on the lunar transfer orbit in real life. We are going to use it for both burns.
The engine has a specific impulse of
421\ \text{s}
in the vacuum (where it will be used). Then we consider the dry mass of the Apollo craft, assuming we are going to reach the Moon tapping into the fuel reserve. The value is
11,\!900\ \text{kg}
Input all of these values in the calculator, with the final mass of the spacecraft as
m_t
. The resulting initial mass
m_0
31,\!000\ \text{kg}
, almost three times the mass of the dry capsule. This is a lot of fuel!
🔎 The vast majority of our delta-v budget is usually spent on escaping the gravity attraction of Earth. As we explained, the
\Delta v
required to reach a low Earth orbit is about
9\ \text{km}/\text{s}
— compare it with the
4\ \text{km}/\text{s}
needed to reach the Moon! Once we are far enough from this possessive planet, the amount of fuel required is never excessive!
Let's go the other way. Let's assume you are aboard a rocket equipped with a NERVA engine, a colossal nuclear-powered engine studied in the '70s. The specific impulse of that beast was
I_{sp}=841\ \text{s}
. Your spacecraft weights
40\ \text{t}
, and you loaded a mere
20\ \text{t}
of fuel — and you are going to use it all. We suppose that it will be enough for quite a journey with that engine!
We calculate the delta-v with the rocket equation:
\begin{align*} \Delta v & = 841\ \text{s} \cdot 9.81\ \frac{\text{m}}{\text{s}^2} \cdot \ln\left({\frac{60,\!000}{40,\!000}}\right) \\ & = 3.345\ \text{km}/\text{s} \end{align*}
That is more than enough to reach Jupiter. It's such a pity that we didn't pursue this kind of engine.
We are sure that this calculator will be useful for your travels. Maybe you can try our UFO calculator too, or other spaceflight dynamics tools like the orbital period calculator or the escape velocity calculator!
Now, off to explore the final frontier! 👨🚀
What is the delta-v?
The delta-v is the difference of velocity that a rocket engine can impose on a spacecraft as a function of the specific impulse and the variation in the mass of the vehicle itself. It is a fundamental value in planning a journey in space, where distance (even if astronomical 😉) is less of a problem than mass is.
How do I calculate the delta-v?
The delta-v can be calculated using the rocket equation: you can use either the specific impulse or the effective exhaust speed in the calculations. The formula is:
Δv = Iₛₚ * g₀* ln(m₀/mₜ) = vₑ * ln(m₀/mₜ)
What is the delta-v to reach the Moon?
Departing from a low Earth orbit to reach the surface of the Moon, the required delta-v is about 6 km/s. This value consider the capture from the Moon's gravity and the landing. A simple change of orbit would take slightly less: 4 km/s.
What is the meaning of delta-v?
Delta-v means a change in velocity. The naming comes from the Greek letter Δ and from the early use in calculus of the letter d to express a differential.
How much delta-v is needed to enter Earth's orbit?
The value of delta-v required to achieve Earth's orbit varies with the type of orbit, the location of the launch, and many other conditions, but it generally starts at about 9 km/s. We need to subtract a fraction of velocity that is lost (due to air resistance during the atmospheric portion of the journey) from the achieved orbital speed.
Specific impulse (Iₛₚ)
Initial mass (m₀)
Final mass (mₜ)
With this thin lens equation calculator you can find focal length of your lens.
Estimate the thrust to weight ratio for real aircraft or your designs using the thrust to weight ratio calculator.
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Gisbert Hasenjaeger - Wikipedia
Gisbert F. R. Hasenjaeger
Picture of Gisbert Hasenjaeger in his identity papers during his time at OKW/Chi
Testing the Enigma encryption machine for cryptographic weaknesses.
Developing a proof of the completeness theorem in 1949.
Dieter Rödding
Peter Schroeder-Heister[1]
Gisbert F. R. Hasenjaeger (June 1, 1919 – September 2, 2006) was a German mathematical logician. Independently and simultaneously with Leon Henkin in 1949, he developed a new proof of the completeness theorem of Kurt Gödel for predicate logic.[2][3] He worked as an assistant to Heinrich Scholz at Section IVa of Oberkommando der Wehrmacht Chiffrierabteilung, and was responsible for the security of the Enigma machine.[4]
2.1 Safety Testing the Enigma Machine
2.2 Proof of Gödel's completeness theorem
3 Construction of Turing Machines
3.1 Comments on the Enigma Machine weakness
Gisbert Hasenjaeger went to high school in Mülheim, where his father Edwin Renatus Hasenjaeger [de] was a lawyer and local politician. After completing school in 1936, Gisbert volunteered for labor service. He was drafted for military service in World War II, and fought as an artillerist in the Russian campaign, where he was badly wounded in January 1942. After his recovery, in October 1942, Heinrich Scholz[5] got him employment in the Cipher Department of the High Command of the Wehrmacht (OKW/Chi), where he was the youngest member at 24. He attended a cryptography training course by Erich Hüttenhain, and was put into the recently founded Section IVa "Security check of own Encoding Procedures" under Karl Stein, who assigned him the security check of the Enigma machine.[6][7] At the end of the war as OKW/Chi disintegrated, Hasenjaeger managed to escape TICOM, the United States effort to roundup and seize captured German intelligence people and material.[6]
From the end of 1945, he studied mathematics and especially mathematical logic with Heinrich Scholz at the Westfälische Wilhelms-Universität University in Münster. In 1950 received his doctorate Topological studies on the semantics and syntax of an extended predicate calculus and completed his habilitation in 1953.[3]
In Münster, Hasenjaeger worked as an assistant to Scholz and later co-author, to write the textbook Fundamentals of Mathematical Logic in Springer's Grundlehren series (Yellow series of Springer-Verlag), which he published in 1961 fully 6 years after Scholz's death. In 1962, he became a professor at the University of Bonn, where he was Director of the newly created Department of Logic.[3]
In 1962, Dr Hasenjaeger left Münster University to take a full professorship at Bonn University, where he became Director of the newly established Department of Logic and Basic Research. In 1964/65, he spent a year at Princeton University at the Institute for Advanced Study[8] His doctoral students at Bonn included Ronald B. Jensen, his most famous pupil.[3]
Hasenjaeger became professor emeritus in 1984.[9]
Safety Testing the Enigma Machine[edit]
In October 1942, after starting work at OKW/Chi, Hasenjaeger was trained in cryptology, given by the mathematician, Erich Hüttenhain, who was widely considered the most important German cryptologist of his time. Hasenjaeger was put into a newly formed department, whose principal responsibility was the defensive testing and security control of their own methods and devices.[6][10] Hasenjaeger was ordered, by the mathematician Karl Stein who was also conscripted at OKW/Chi, to examine the Enigma machine for cryptologic weaknesses, while Stein was to examine the Siemens and Halske T52 and the Lorenz SZ-42.[10] The Enigma machine that Hasenjaeger examined was a variation that worked with 3 rotors and had no plugboard. Germany sold this version to neutral countries to accrue foreign exchange. Hasenjaeger was presented with a 100 character encrypted message for analysis and found a weakness which enabled the identification of the correct wiring rotors and also the appropriate rotor positions, to decrypt the messages. Further success eluded him, however. He crucially failed to identify the most important weakness of the Enigma machine: the lack of fixed points (letters encrypting to themselves) due to the reflector. Hasenjaeger could take some comfort from the fact that even Alan Turing missed this weakness. Instead, the honour was attributed to Gordon Welchman, who used the knowledge to decrypt several hundred thousand Enigma messages during the war.[6][10] In fact fixed points were earlier used by Polish codebreaker, Henryk Zygalski, as the basis for his method of attack on Enigma cipher, referred to by the Poles as "Zygalski sheets" (Zygalski sheets) (płachty Zygalskiego) and by the British as the "Netz method".
Proof of Gödel's completeness theorem[edit]
It was while Hasenjaeger was working at Westfälische Wilhelms-Universität University in Münster in the period between 1946 and 1953 that Hasenjaeger made a most amazing discovery - a proof of Kurt Gödel's Gödel's completeness theorem for full predicate logic with identity and function symbols.[3] Gödel's proof of 1930 for predicate logic did not automatically establish a procedure for the general case. When he had solved the problem in late 1949, he was frustrated to find that a young American mathematician Leon Henkin, had also created a proof.[3] Both construct from extension of a term model, which is then the model for the initial theory. Although the Henkin proof was considered by Hasenjaeger and his peers to be more flexible, Hasenjaeger' is considered simpler and more transparent.[3]
Hasenjaeger continued to refine his proof through to 1953 when he made a breakthrough. According to the mathematicians Alfred Tarski, Stephen Cole Kleene and Andrzej Mostowski, the Arithmetical hierarchy of formulas is the set of arithmetical propositions that are true in the standard model, but not arithmetically definable. So, what does the concept of truth for the term model mean, the results for the recursively axiomatized Peano arithmetic from the Hasenjaeger method? The result was the truth predicate is well arithmetically, it is even
{\displaystyle \Delta _{2}^{0}}
.[3] So far down in the arithmetic hierarchy, and that goes for any recursively axiomatized (countable, consistent) theories. Even if you are true in all the natural numbers
{\displaystyle \Pi _{1}^{0}}
formulas to the axioms.
This classic proof is a very early, original application of the arithmetic hierarchy theory to a general-logical problem. It appeared in 1953 in the Journal of Symbolic Logic.[11]
Construction of Turing Machines[edit]
In 1963, Hasenjaeger built a Universal Turing machine out of old telephone relays. Although Hasenjaeger's work on UTMs was largely unknown and he never published any details of the machinery during his lifetime, his family decided to donate the machine to the Heinz Nixdorf Museum in Paderborn, Germany, after his death.[12][13] In an academic paper presented at the International Conference of History and Philosophy of Computing in 2012.[12] Rainer Glaschick, Turlough Neary, Damien Woods, Niall Murphy had examined Hasenjaeger's UTM machine at the request of Hasenjaeger family and found that the UTM was remarkably small and efficiently universal. Hasenjaeger UTM contained 3-tapes, 4 states, 2 symbols and was an evolution of ideas from Edward F. Moore's first universal machine and Hao Wang's B-machine. Hasenjaeger went on to build a small efficient Wang B-machine simulator. This was again proven by the team assembled by Rainer Glaschick to be efficiently universal.
Comments on the Enigma Machine weakness[edit]
It was only in the 1970s that Hasenjaeger learned that the Enigma Machine had been so comprehensively broken.[6] It impressed him that Alan Turing himself, considered one of the greatest mathematicians of the 20th century, had worked on breaking the device. The fact that the Germans had so comprehensively underestimated the weaknesses of the device, in contrast to Turing and Welchman's work, was seen by Hasenjaeger today as entirely positive. Hasenjaeger stated:
Would it not been so, then the war would have lasted probably longer and the first atomic bomb had not fallen on Japan, but on Germany.[6]
Schmeh, Klaus (18 September 2009). "Enigma's Contemporary Witness: Gisbert Hasenjaeger". Cryptologia. 33 (4): 343–346. doi:10.1080/01611190903186003.
^ Gisbert Hasenjaeger at the Mathematics Genealogy Project
^ "Past Professors at Münster University" (PDF). wwmath.uni-muenster.de. Retrieved 6 January 2014.
^ a b c d e f g h "Laudatio anläßlich der Erneuerung der Doktorurkunde". WWU Münster Mathematik: Logik. Archived from the original on 31 August 2012. Retrieved 17 February 2014.
^ Schmeh, Klaus (15 September 2009). "Enigma's Contemporary Witness: Gisbert Hasenjaeger". Cryptologia. 33 (4): 343–346. doi:10.1080/01611190903186003. ISSN 0161-1194. S2CID 205487783.
^ Hasenjaeger knew Scholz since his school days and corresponded with him during his time as a conscript.
^ a b c d e f "Enigma Contemporary Witness - Enigma Vulnerability Part 3". Heise Online. Klaus Schmeh. 29 August 2005. Retrieved 2 March 2014.
^ Friedrich L. Bauer (2000). Entzifferte Geheimnisse — Methoden und Maximen der Kryptologie (3 ed.). Heidelberg: Springer. ISBN 978-3-540-67931-8. Cited from German Wikipedia
^ "IAS - Gisbert Hasenjeager". www.ias.edu. IAS. Retrieved 20 July 2016.
^ Wirth, Claus-Peter (4 March 2018). A Most Interesting Draft for Hilbert and Bernays’ “Grundlagen der Mathematik” that never found its way into any publication, and two CV of Gisbert Hasenjaeger. Saarland University.
^ a b c Cooper, S. Barry; Leeuwen, J. van (3 Jun 2013). Alan Turing: His Work and Impact: His Work and Impact. Elsevier Science. p. 936. ISBN 978-0-12-386980-7.
^ Hasenjaeger, G. (1953). "Eine Bemerkung zu Henkin's Beweis für die Vollständigkeit des Prädikatenkalküls der ersten Stufe". Journal of Symbolic Logic. 18 (1): 42–48. doi:10.2307/2266326. JSTOR 2266326. Gödel proof.
^ a b Neary, Turlough; Woods, Damien; Murphy, Niall; Glaschick, Rainer (October 2014). "Wang's B machines are efficiently universal, as is Hasenjaeger's small universal electromechanical toy". Journal of Complexity. 30 (5): 634–646. arXiv:1304.0053. Bibcode:2013arXiv1304.0053N. doi:10.1016/j.jco.2014.02.003. S2CID 18828226.
^ "Hasenjaeger's electromechanical small universal Turing machine is time efficient" (PDF). Department of History an Philosophy Universiteit Gent. Retrieved 18 March 2014.
Rebecca Ratcliffe: Searching for Security. The German Investigations into Enigma's security. In: Intelligence and National Security 14 (1999) Issue 1 (Special Issue) S.146–167.
Rebecca Ratcliffe: How Statistics led the Germans to believe Enigma Secure and Why They Were Wrong: neglecting the practical Mathematics of Cipher machines Add:. Brian J. angle (eds.) The German Enigma Cipher Machine. Artech House: Boston, London of 2005.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Gisbert_Hasenjaeger&oldid=1076119065"
German cryptographers
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Catenary Curve Calculator | Suspended Rope
The catenary curve equation
Applications of the catenary curve in architecture — and not only
A mathematical curiosity!
How to use our catenary curve calculator?
Suspended ropes have a simple yet fascinating mathematical definition: discover it with our catenary curve calculator!
The catenary curve is the graph generated by the catenary function. It describes the ideal behavior of a rope hanging in a gravitational field under its own weight. Catenary curves find applications in many fields, so it's worth learning about them. Keep reading to learn:
What a catenary curve is;
The catenary curve equation;
Applications of the catenary curve in architecture;
Where to find the catenary curve in nature; and
How to use our catenary curve calculator.
Take a rope and hang it between two supports so that it sags a little. That's the definition of catenary curve — our job is done!
We are joking, but not entirely: that's exactly how you describe a catenary curve. However, there is more to say! 😄
The word catenary comes from the Latin "catēna", chain ⛓. The curve describes not only ropes, but also chains. In the past, many mathematicians tried to describe the behavior of a hanging rope. They first compared the catenary curve to a parabola, but soon they shifted to more complex functions once they realized the parabola fails to capture the catenary curve perfectly.
🔎 Two famous scientists of the past, Galileo Galilei and Robert Hooke (yes, the guy who came up with the spring constant) tried their luck with catenary curves. Galileo noted that the curve designed by a hanging rope was not a parabola; Hooke is considered the first to have found the mathematical expression for building arches following the catenary.
A catenary curve follows a simple mathematical formula:
y=a\cdot \cosh{\frac{x}{a}}
\cosh
is the hyperbolic cosine, a function part of the family of hyperbolic functions. It is defined as:
\cosh{x}=\frac{e^x+e^{-x}}{2}
This makes possible to write the catenary curve formula as:
y=a\cdot \cosh{\frac{x}{a}}=\frac{a}{2}(e^{\frac{x}{a}}+e^{-\frac{x}{a}})
If you want to learn more about hyperbolic functions, check our hyperbolic functions calculator!
It is possible to define a generalization of the catenary curve equation, the weighted catenary. It requires two parameters, instead of one:
y=b\cdot{}\cosh{\frac{x}{a}}=\frac{b}{2}(e^{\frac{x}{a}}+e^{-\frac{x}{a}})
🔎 Weighted catenaries are not that common! However, you can find one in St. Louis, Missouri: the Gateway Arch, which is often mistaken for a parabola!
The catenary curve is so simple that it has conquered the heart of humanity from the beginning of civilization. Its interesting properties made it a perfect choice for a particular field: here, we introduce the love story between the catenary curve and architecture.
The shape of a catenary curve facilitates unloading the weight of the suspended structure on the lateral supports. Catenary curves appear in suspended bridges, either by coincidence (lay down a rope bridge and it will assume that shape) or by design.
🔎 The ropes of bridges where they act as a support for the road below don't follow a catenary shape, but rather a parabolic one. But don't worry, they are almost identical: for centuries catenaries hid behind parables, their more famous "cousins"!
Since antiquity, architects applied the catenary curve when designing arches and domes. Many examples can be found — from the Tāq Kasrām (the millenary imperial complex still standing in Iraq) to the Clocháin (the tiny stone houses in Skelling Michael) to the Brunelleschi Dome to the mud huts of the Musgun people in Cameroon.
The Tāq Kasrām in Iraq is a long standing example of the strength of catenary arches.
But obviously, it doesn't stop here: catenary curves find many other applications. If you take a train (an electric one is the best way to commute), the electric lines hanging overhead are catenary curves. And speaking of electric lines, almost every landscape is populated by power lines that are — as you now know — catenary curves.
A string of helium-filled balloons (a balloon arch) hangs as an upside-down catenary curve too!
Nature is the world's best engineer, and the catenary curve appears in many more or less unexpected places. Take a closer look at a spider web: the ropes anchored to other strings are catenaries. Many arches found in nature assume catenary shape due to erosion, too.
But opening the fridge would give you the easiest example of catenary curves in nature: eggs! An egg's particular strength is associated to their ability to dissipate forces applied on them, thanks to their double catenary shape.
Catenary curves are the only shapes on which regular polygons can roll smoothly. Choose a point on the polygon and trace its movement: you will obtain a curve called a roulette: we talked about it in our involute function calculator.
In the particular case of polygons rolled on catenaries, we call those curves unduloids.
The generation of the unduloid of the square (in red). The repeated blue curve at the bottom is the catenary.
Our catenary curve calculator is easy to use and still offers you many functionalities.
First, you have to choose the type of catenary you want to use in the tool: weighted or not We show the respective catenary curve formula there, don't worry!
We offer you four modes of operation:
Choose the one you need. In the last three, you can change the function domain and the sampling frequency by changing the parameters hidden in the advanced mode.
🙋 If you are using the calculator in value mode, remember that you can input the value of the catenary function to find the associated value of
x
The definition of catenary curve is: "a mathematical curve representing the shape of a rope hanging between two supports under the sole influence of its own weight".
Find more with the catenary curve calculator on omnicalculator.com
How do I calculate the catenary curve?
The equation of the catenary curve makes use of hyperbolic functions. Given the parameter a, the sag of the rope, we find the catenary with the formula y = a * cosh(x/a), where cosh is the hyperbolic cosine.
Which are the applications of the catenary curve?
Architecture and civilian engineering are the most common applications of catenary curves. The fact that a catenary unloads the weight it carries very efficiently on its supports makes these curves a perfect choice to suspend bridges or create the most resistant shape of arches!
What is the difference between catenary curve and parabola?
In a catenary curve, the force is uniform on the length of the rope, while in a parabola, the force is uniform when you consider the horizontal length. This is why the catenary is more common in self-suspending structures, while you can find the parabolic shape when the cable supports something else (think of the Golden Gate Bridge).
Sag parameter (a)
Change the domain/step by clicking on advanced mode!
Average rate of changeBilinear interpolationCross product… 35 more
Law of sines calculator finds the side lengths and angles of a triangle using the law of sines.
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\mathrm{with}\left(\mathrm{MultivariatePowerSeries}\right):
We define a univariate polynomial over power series and evaluate it at the origin (in this case, at
x=0
f≔\mathrm{UnivariatePolynomialOverPowerSeries}\left(\left(z-1\right)\left(z-2\right)\left(z-3\right)+x\left({z}^{2}+z\right),z\right):
\mathrm{EvaluateAtOrigin}\left(f\right)
{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{11}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{6}
We define another univariate polynomial over power series and evaluate it at the origin (in this case, at
x=0
y=0
g≔\mathrm{UnivariatePolynomialOverPowerSeries}\left({y}^{2}+{x}^{2}+\left(y+1\right){z}^{2}+{z}^{3},z\right):
\mathrm{EvaluateAtOrigin}\left(g\right)
{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{2}}
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Surface area of a rectangle equation
How to find the surface area of a rectangle?
How do I use this surface area of a rectangle calculator?
This surface area of a rectangle calculator is the perfect tool for you to calculate the surface area of a rectangle from its length and width. In this article, we shall go over the surface area of a rectangle, its equation, how to find the surface area of a rectangle, and some frequently asked questions.
A rectangle is a closed figure with four sides, whose opposite sides are equal. Hence, two sides or dimensions are sufficient to describe a rectangle — its length (sometimes called height) and its width (sometimes called breadth).
The equation for the surface area of a rectangle is given by:
\text{A} = \ell \times w
\text{A}
is the surface area of the rectangle;
\ell
is the length of the rectangle. Usually, it refers to the longest side of the rectangle.
w
is the width of the rectangle. Usually, it refers to the shortest side of the rectangle.
The SI units for area are
\text{m}^2
\text{sq m}
(square meters), and its imperial units are
\text{ft}^2
\text{sq ft}
(square feet). Always ensure that the length and width are expressed in the same units before using the surface area formula of a rectangle.
For example, a rectangle with
5 \text{ m}
length and
2 \text{ m}
breadth (or width) would have a surface area of
5 \times 2 = 10 \text{ m}^2
🔎 Since a rectangle is a 2D geometry, it is sufficient to refer to its surface area as area.
To find the surface area of a rectangle, you require its length and width:
Convert the length and width into the same unit. We must measure them both in the same unit. For example, if the length is 5 m, and width is 2 ft, convert both to either m or ft.
Multiply this length and width to obtain the area in corresponding units.
Double-check your answer using the surface area of a rectangle calculator.
This surface area of a rectangle calculator is simple to use:
Enter the length of the rectangle in its appropriate field. Ensure you've picked the suitable units.
Enter the width (or breadth) of the rectangle in its appropriate field. Again, ensure that you've picked the correct units.
This surface area of a rectangle will automatically find the surface area of the rectangle and display the result in the appropriate field.
So now you know how to find the area of a rectangle!
How do I calculate the area of my monitor screen?
Most monitor screens are rectangular in shape. To calculate the area of your monitor screen, follow these simple steps:
Measure the length and width of your monitor with a monitor or measuring scale. Alternatively, you can get these dimensions from the product description.
Multiply the length and width together to obtain the surface area of your monitor screen.
Rejoice in your newfound knowledge of your screen's area.
How many acres to one hectare?
There are 2.471 acres to one hectare. An acre is a unit of land and is roughly equal to 4047 sq m. A hectare is a unit of land equal to 10,000 sq m.
Your rectangle has...
Length (ℓ)
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Term representation and simplification
Performance of symbolic simplification depends on the datastructures used to represent terms. Efficient datastructures often have the advantage of automatic simplification, and of efficient storage.
The most basic term representation simply holds a function call and stores the function and the arguments it is called with. This is done by the Term type in SymbolicUtils. Functions that aren't commutative or associative, such as sin or hypot are stored as Terms. Commutatative and associative operations like +, *, and their supporting operations like -, / and ^, when used on terms of type <:Number, stand to gain from the use of more efficient datastrucutres.
All term representations must support operation and arguments functions. And they must define istree to return true when called with an instance of the type. Generic term-manipulation programs such as the rule-based rewriter make use of this interface to inspect expressions. In this way, the interface wins back the generality lost by having a zoo of term representations instead of one. (see interface section for more on this.)
Preliminary representation of arithmetic
Linear combinations such as
\alpha_1 x_1 + \alpha_2 x_2 +...+ \alpha_n x_n
are represented by Add(Dict(x₁ => α₁, x₂ => α₂, ..., xₙ => αₙ)). Here, any
x_i
may itself be other types mentioned here, except for Add. When an Add is added to an Add, we merge their dictionaries and add up matching coefficients to create a single "flattened" Add.
x_1^{m_1}x_2^{m_2}...x_{m_n}
is represented by Mul(Dict(x₁ => m₁, x₂ => m₂, ..., xₙ => mₙ)).
x_i
may not themselves be Mul, multiplying a Mul with another Mul returns a "flattened" Mul.
Note that Add and Mul types perform a preliminary simplification which suffices to simplify numeric expressions to a large extent during construction.
p / q
is represented by Div(p, q). The result of * on Div is maintainted as a Div. For example, Div(p_1, q_1) * Div(p_2, q_2) results in Div(p_1 * p_2, q_1 * q_2) and so on. The effect is, in Div(p, q), p or q or, if they are Mul, any of their multiplicands is not a Div. So Muls must always be nested inside a Div and can never show up immediately wrapping it. This rule sets up an expression so that it helps the simplify_fractions procedure described two sections below.
Packages like DynamicPolynomials.jl provide representations that are even more efficient than the Add and Mul types mentioned above. They are designed specifically for multi-variate polynomials. They provide common algorithms such as multi-variate polynomial GCD. The restrictions that make it fast also mean some things are not possible: Firstly, DynamicPolynomials can only represent flat polynomials. For example, (x-3)*(x+5) can only be represented as (x^2) + 15 - 8x. Secondly, DynamicPolynomials does not have ways to represent generic Terms such as sin(x-y) in the tree.
To reconcile these differences while being able to use the efficient algorithms of DynamicPolynomials we have the PolyForm type. This type holds a polynomial and the mappings necessary to present the polynomial as a SymbolicUtils expression (i.e. by defining operation and arguments). The mappings constructed for the conversion are 1) a bijection from DynamicPolynomials PolyVar type to a Symbolics Sym, and 2) a mapping from Syms to non-polynomial terms that the Syms stand-in for. These terms may themselves contain PolyForm if there are polynomials inside them. The mappings are transiently global, that is, when all references to the mappings go out of scope, they are released and re-created.
julia> @syms x y
julia> PolyForm((x-3)*(y-5))
x*y + 15 - 5x - 3y
Terms for which the operation is not +, *, or ^ are replaced with a generated symbol when representing the polynomial, and a mapping from this new symbol to the original expression it stands-in for is maintained as stated above.
julia> p = PolyForm((sin(x) + cos(x))^2)
(cos(x)^2) + 2cos(x)*sin(x) + (sin(x)^2)
julia> p.p # this is the actual DynamicPolynomial stored
cos_3658410937268741549² + 2cos_3658410937268741549sin_10720964503106793468 + sin_10720964503106793468²
By default, polynomials inside non-polynomial terms are not also converted to PolyForm. For example,
julia> PolyForm(sin((x-3)*(y-5)))
sin((x - 3)*(y - 5))
But you can pass in the recurse=true keyword argument to do this.
julia> PolyForm(sin((x-3)*(y-5)), recurse=true)
sin(x*y + 15 - 5x - 3y)
Polynomials are constructed by first turning symbols and non-polynomial terms into DynamicPolynomials-style variables and then applying the +, *, ^ operations on these variables. You can control the list of the polynomial operations with the Fs keyword argument. It is a Union type of the functions to apply. For example, let's say you want to turn terms into polynomials by only applying the + and ^ operations, and want to preserve * operations as-is, you could pass in Fs=Union{typeof(+), typeof(^)}
julia> PolyForm((x+y)^2*(x-y), Fs=Union{typeof(+), typeof(^)}, recurse=true)
((x - (y))*((x^2) + 2x*y + (y^2)))
Note that in this case recurse=true was necessary as otherwise the polynomialization would not descend into the * operation as it is now considered a generic operation.
simplify_fractions(expr) recurses through expr finding Divs and simplifying them using polynomial divison.
First the factors of the numerators and the denominators are converted into PolyForm objects, then numerators and denominators are divided by their respective pairwise GCDs. The conversion of the numerator and denominator into PolyForm is set up so that simplify_fractions does not result in increase in the expression size due to polynomial expansion. Specifically, the factors are individually converted into PolyForm objects, and any powers of polynomial is not expanded, but the divison process repeatedly divides them as many times as the power.
julia> simplify_fractions((x*y+5x+3y+15)/((x+3)*(x-4)))
(5.0 + y) / (x - 4)
julia> simplify_fractions((x*y+5x+3y+15)^2/((x+3)*(x-4)))
((5.0 + y)*(15 + 5x + x*y + 3y)) / (x - 4)
julia> simplify_fractions(3/(x+3) + x/(x+3))
julia> simplify_fractions(sin(x)/cos(x) + cos(x)/sin(x))
(cos(x)^2 + sin(x)^2) / (cos(x)*sin(x))
julia> simplify(ans)
1 / (cos(x)*sin(x))
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Central_angle Knowpia
A central angle is an angle whose apex (vertex) is the center O of a circle and whose legs (sides) are radii intersecting the circle in two distinct points A and B. Central angles are subtended by an arc between those two points, and the arc length is the central angle of a circle of radius one (measured in radians).[1] The central angle is also known as the arc's angular distance.
Angle AOB is a central angle
The size of a central angle Θ is 0° < Θ < 360° or 0 < Θ < 2π (radians). When defining or drawing a central angle, in addition to specifying the points A and B, one must specify whether the angle being defined is the convex angle (<180°) or the reflex angle (>180°). Equivalently, one must specify whether the movement from point A to point B is clockwise or counterclockwise.
If the intersection points A and B of the legs of the angle with the circle form a diameter, then Θ = 180° is a straight angle. (In radians, Θ = π.)
Let L be the minor arc of the circle between points A and B, and let R be the radius of the circle.[2]
Central angle. Convex. Is subtended by minor arc L
If the central angle Θ is subtended by L, then
{\displaystyle 0^{\circ }<\Theta <180^{\circ }\,,\,\,\Theta =\left({\frac {180L}{\pi R}}\right)^{\circ }={\frac {L}{R}}.}
Proof (for degrees)
The circumference of a circle with radius R is 2πR, and the minor arc L is the (Θ/360°) proportional part of the whole circumference (see arc). So:
{\displaystyle L={\frac {\Theta }{360^{\circ }}}\cdot 2\pi R\,\Rightarrow \,\Theta =\left({\frac {180L}{\pi R}}\right)^{\circ }.}
Central angle. Reflex. Is not subtended by L
Proof (for radians)
The circumference of a circle with radius R is 2πR, and the minor arc L is the (Θ/2π) proportional part of the whole circumference (see arc). So
{\displaystyle L={\frac {\Theta }{2\pi }}\cdot 2\pi R\,\Rightarrow \,\Theta ={\frac {L}{R}}.}
If the central angle Θ is not subtended by the minor arc L, then Θ is a reflex angle and
{\displaystyle 180^{\circ }<\Theta <360^{\circ }\,,\,\,\Theta =\left(360-{\frac {180L}{\pi R}}\right)^{\circ }=2\pi -{\frac {L}{R}}.}
If a tangent at A and a tangent at B intersect at the exterior point P, then denoting the center as O, the angles ∠BOA (convex) and ∠BPA are supplementary (sum to 180°).
Central angle of a regular polygonEdit
A regular polygon with n sides has a circumscribed circle upon which all its vertices lie, and the center of the circle is also the center of the polygon. The central angle of the regular polygon is formed at the center by the radii to two adjacent vertices. The measure of this angle is
{\displaystyle 2\pi /n.}
^ Clapham, C.; Nicholson, J. (2009). "Oxford Concise Dictionary of Mathematics, Central Angle" (PDF). Addison-Wesley. p. 122. Retrieved December 30, 2013.
^ "Central angle (of a circle)". Math Open Reference. 2009. Retrieved December 30, 2013. interactive
"Central angle (of a circle)". Math Open Reference. 2009. Retrieved December 30, 2013. interactive
"Central Angle Theorem". Math Open Reference. 2009. Retrieved December 30, 2013. interactive
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Home : Support : Online Help : Science and Engineering : Units : Environments : Simple : Root Functions
the square root function in the Simple Units environment
the nth root function in the Simple Units environment
the non-principal root function in the Simple Units environment
\sqrt{x}
{\mathrm{root}}_{n}\left(x\right)
The sqrt(x) function takes the square root of the unit-free portion of x and multiplies it by the unit raised to the power 1/2.
The root(x,n) and root[n](x) functions take the nth root of the unit-free portion of x and multiply it by the unit raised to the power 1/n.
The surd(x,n) function takes the nth root of the unit-free portion of x, whose (complex) argument is closest to the unit-free portion of x, and multiplies it by the unit raised to the power 1/n.
\mathrm{unit}
\mathrm{with}\left(\mathrm{Units}[\mathrm{Simple}]\right):
\mathrm{sqrt}\left(3.532\mathrm{Unit}\left({'m'}^{2}\right)\right)
\textcolor[rgb]{0,0,1}{1.879361594}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{m}⟧
\mathrm{root}\left(3.532\mathrm{Unit}\left({'m'}^{3}\right),3\right)
\textcolor[rgb]{0,0,1}{1.522907638}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{m}⟧
\mathrm{root}[3]\left(-3.532\mathrm{Unit}\left({'m'}^{3}\right)\right)
\left(\textcolor[rgb]{0,0,1}{0.7614538193}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1.318876702}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{I}\right)\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{m}⟧
\mathrm{surd}\left(-3.532\mathrm{Unit}\left({'m'}^{3}\right),3\right)
\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1.522907638}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{m}⟧
\mathrm{surd}\left(16.532\mathrm{Unit}\left({'m'}^{4}\right),4\right)
\textcolor[rgb]{0,0,1}{2.016421638}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{m}⟧
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