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https://stacks.math.columbia.edu/tag/06ZR
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Lemma 46.7.2. Let $U = \mathop{\mathrm{Spec}}(A)$ be an affine scheme. The bounded below derived category $D^+(\textit{Adeq}(\mathcal{O}))$ is the localization of $K^+(\mathit{QCoh}(\mathcal{O}_ U))$ at the multiplicative subset of universal quasi-isomorphisms.
Proof. If $\varphi : \mathcal{F}^\bullet \to \mathcal{G}^\bullet$ is a morphism of complexes of quasi-coherent $\mathcal{O}_ U$-modules, then $u\varphi : u\mathcal{F}^\bullet \to u\mathcal{G}^\bullet$ is a quasi-isomorphism if and only if $\varphi$ is a universal quasi-isomorphism. Hence the collection $S$ of universal quasi-isomorphisms is a saturated multiplicative system compatible with the triangulated structure by Derived Categories, Lemma 13.5.3. Hence $S^{-1}K^+(\mathit{QCoh}(\mathcal{O}_ U))$ exists and is a triangulated category, see Derived Categories, Proposition 13.5.5. We obtain a canonical functor $can : S^{-1}K^+(\mathit{QCoh}(\mathcal{O}_ U)) \to D^{+}(\textit{Adeq}(\mathcal{O}))$ by Derived Categories, Lemma 13.5.6.
Note that, almost by definition, every adequate module on $U$ has an embedding into a quasi-coherent sheaf, see Lemma 46.5.5. Hence by Derived Categories, Lemma 13.15.5 given $\mathcal{F}^\bullet \in \mathop{\mathrm{Ob}}\nolimits (K^+(\textit{Adeq}(\mathcal{O})))$ there exists a quasi-isomorphism $\mathcal{F}^\bullet \to u\mathcal{G}^\bullet$ where $\mathcal{G}^\bullet \in \mathop{\mathrm{Ob}}\nolimits (K^+(\mathit{QCoh}(\mathcal{O}_ U)))$. This proves that $can$ is essentially surjective.
Similarly, suppose that $\mathcal{F}^\bullet$ and $\mathcal{G}^\bullet$ are bounded below complexes of quasi-coherent $\mathcal{O}_ U$-modules. A morphism in $D^+(\textit{Adeq}(\mathcal{O}))$ between these consists of a pair $f : u\mathcal{F}^\bullet \to \mathcal{H}^\bullet$ and $s : u\mathcal{G}^\bullet \to \mathcal{H}^\bullet$ where $s$ is a quasi-isomorphism. Pick a quasi-isomorphism $s' : \mathcal{H}^\bullet \to u\mathcal{E}^\bullet$. Then we see that $s' \circ f : \mathcal{F} \to \mathcal{E}^\bullet$ and the universal quasi-isomorphism $s' \circ s : \mathcal{G}^\bullet \to \mathcal{E}^\bullet$ give a morphism in $S^{-1}K^{+}(\mathit{QCoh}(\mathcal{O}_ U))$ mapping to the given morphism. This proves the "fully" part of full faithfulness. Faithfulness is proved similarly. $\square$
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
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2022-06-30 13:46:11
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https://courses.lumenlearning.com/cuny-hunter-collegealgebra/chapter/applications_of_linear_equations/
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## 1.5 – Applications of Linear Equations
### Learning Objectives
• (1.5.1) – Set up a linear equation to solve an application
• Translate words into algebraic expressions and equations
• Solve an application using a formula
• (1.5.2) – Solve distance, rate, and time problems
• (1.5.3) – Solve area and perimeter problems
• (1.5.4) – Rearrange formulas to isolate specific variables
• (1.5.5) – Solve temperature conversion problems
Many real-world applications can be modeled by linear equations. For example, a cell phone package may include a monthly service fee plus an additional charge if you exceed your data plan; a car rental company charges a daily fee plus an amount per mile driven; Chipotle offers a base price for your burrito plus additional charges for extra toppings, like guacamole and sour cream. These are examples of applications we come across every day that are modeled by linear equations. In this section, we will set up and use linear equations to solve such problems.
# (1.5.1) – Set Up a Linear Equation to Solve an Application
To set up or model a linear equation to fit a real-world application, we must first determine the known quantities and define the unknown quantity as a variable. Then, we begin to interpret the words as mathematical expressions using mathematical symbols. Let use an example of a car rental company. The company charges $0.10/mi in addition to a flat rate. In this case, a known cost, such as$0.10/mi, is multiplied by an unknown quantity, the number of miles driven. Therefore, we can write $0.10x$. This expression represents a variable cost because it changes according to the number of miles driven.
If a quantity is independent of a variable, we usually just add or subtract it, according to the problem. As these amounts do not change, we call them fixed costs. Consider a car rental agency that charges $0.10/mi plus a daily fee of$50. We can use these quantities to model an equation that can be used to find the daily car rental cost $C$.
$C=0.10x+50$
### Translate words into algebraic expressions and equations
When dealing with real-world applications, there are certain expressions that we can translate directly into math. The table lists some common verbal expressions and their equivalent mathematical expressions.
Verbal Translation to Math Operations
One number exceeds another by a $x,\text{ }x+a$
Twice a number $2x$
One number is a more than another number $x,\text{ }x+a$
One number is a less than twice another number $x,2x-a$
The product of a number and a, decreased by b $ax-b$
The quotient of a number and the number plus a is three times the number $\frac{x}{x+a}=3x$
The product of three times a number and the number decreased by b is c $3x\left(x-b\right)=c$
### How To: Given a real-world problem, model a linear equation to fit it.
1. Identify known quantities.
2. Assign a variable to represent the unknown quantity.
3. If there is more than one unknown quantity, find a way to write the second unknown in terms of the first.
4. Write an equation interpreting the words as mathematical operations.
5. Solve the equation. Be sure the solution can be explained in words, including the units of measure.
### Example
Find a linear equation to solve for the following unknown quantities: One number exceeds another number by $17$ and their sum is $31$. Find the two numbers.
In the following video, we show another example of how to translate an expression in english into a mathematical equation that can then be solved.
In the next example we will write equations that will help us compare cell phone plans.
### Example
There are two cell phone companies that offer different packages. Company A charges a monthly service fee of $34 plus$.05/min talk-time. Company B charges a monthly service fee of $40 plus$.04/min talk-time.
1. Write a linear equation that models the packages offered by both companies.
2. If the average number of minutes used each month is 1,160, which company offers the better plan?
3. If the average number of minutes used each month is 420, which company offers the better plan?
4. How many minutes of talk-time would yield equal monthly statements from both companies?
The following video shows another example of writing two equations that will allow you to compare two different cell phone plans.
### Solve an Application Using a Formula
Many applications are solved using known formulas. The problem is stated, a formula is identified, the known quantities are substituted into the formula, the equation is solved for the unknown, and the problem’s question is answered. Typically, these problems involve two equations representing two trips, two investments, two areas, and so on. Examples of formulas include the area of a rectangular region, $A=LW$; the perimeter of a rectangle, $P=2L+2W$; and the volume of a rectangular solid, $V=LWH$. When there are two unknowns, we find a way to write one in terms of the other because we can solve for only one variable at a time.
# (1.5.2) – Solve distance, rate, and time problems
### Example
It takes Andrew 30 min to drive to work in the morning. He drives home using the same route, but it takes 10 min longer, and he averages 10 mi/h less than in the morning. How far does Andrew drive to work?
# (1.5.3) – Solve area and perimeter problems
In the next example, we will find the lengths of a rectangular field given it’s perimeter and a relationship between it’s side lengths.
### Example
The perimeter of a rectangular outdoor patio is $54$ ft. The length is $3$ ft greater than the width. What are the dimensions of the patio?
The following video shows an example of finding the dimensions of a rectangular field given it’s perimeter.
### Example
The perimeter of a tablet of graph paper is 48 in. The length is $6$ in. more than the width. Find the area of the graph paper.
The following video shows another example of finding the area of a rectangle given it’s perimeter and the relationship between it’s side lengths.
# (1.5.4) – Rearrange formulas to isolate specific variables
Sometimes, it is easier to isolate the variable you you are solving for when you are using a formula. This is especially helpful if you have to perform the same calculation repeatedly, or you are having a computer perform the calculation repeatedly. In the next examples, we will use algebraic properties to isolate a variable in a formula.
### Example
Isolate the term containing the variable, w, from the formula for the perimeter of a rectangle:
${P}=2\left({L}\right)+2\left({W}\right)$.
### Example
Use the multiplication and division properties of equality to isolate the variable b given $A=\frac{1}{2}bh$
Use the multiplication and division properties of equality to isolate the variable given $A=\frac{1}{2}bh$
# (1.5.5) – Solve temperature conversion problems
Let’s look at another formula that includes parentheses and fractions, the formula for converting from the Fahrenheit temperature scale to the Celsius scale.
$C=\left(F-32\right)\cdot \frac{5}{9}$
### Example
Given a temperature of $12^{\circ}{C}$, find the equivalent in ${}^{\circ}{F}$.
As with the other formulas we have worked with, we could have isolated the variable F first, then substituted in the given temperature in Celsius.
### Example
Solve the formula shown below for converting from the Fahrenheit scale to the Celsius scale for F.
$C=\left(F-32\right)\cdot \frac{5}{9}$
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2021-11-28 23:36:34
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http://www.physicsforums.com/showthread.php?t=61054
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# Specific Heat Ratio - Rocketry
by mrjeffy321
Tags: heat, ratio, rocketry, specific
Sci Advisor P: 882 Hello, I know that the Specific Heat Ratio when refering to the gass(es) comming out the back of a rocket engine is the ratio of the specific heats at a constant pressure/volume and then what I am trying to do is use that to help me solve for what the theoretical force I can expect from the rocket using a rather long equation. Here is my question, I dont know which specific heat ratios to use and for which gasses? I have found many tables that list common gasses specific heat rations, the most meaningful being Oxygen gas, Carbon Dioxide, "Air", and Water Vapor/Steam, all of which are involved with a combustion reaction that is taking place inside my solid fueled rocket (propelent is Potassium Chlorate + Sucrose), the reaction is a combination os two reactions actually, first the chlorate decomposes: 2KClO3 --> KCl + 3O2 second, the oxygen made during the first reaction is used to combust the sucrose: 12O2 + C12H22O11 + 11H2O which is then combined to this reaction: 8KClO3 + C12H22O11 --> 12CO2(g) + 11H20(g) so I know the ratio of gasses to eachother, their molar wieghts, each individuals specific heat ratios, but how do I combine them all to find the overall specific heat ratio of the whole reaction? is it just the average so to speak of carbon dioxide and water vapor? would I just find out the ratio of carbon dioxide's weight to water vapor's weight and then use that to determine the average specific heat ratio?
Sci Advisor PF Gold P: 1,479 If I were you I'd do: $$\gamma_{average}=\frac{\sum X_i C_{pi}}{\sum X_i C{vi}}$$ where Xi is the molar fraction of the i-th chemical product.
Sci Advisor P: 882 So I was correct in thinking that I needed to take the average of the gasses in the product, but it is the molar heat ratio, and not the weight ratio? are you sure? even though my specific heats are given in kJ/kg*degree kelvin, but then the units would cancel I suppose when you divided them. so for the above reaction, I would take the average ( i use italics because it really isnt a true average), specific heat ratios of carbon dioxide and water vapor, using the ratios 12/23 for carbon dioxide and 11/23 for oxygen, and we just disreguard the oxygen in the middle that is comming off the chlorate and going right back into another reaction, rather than doing the ratio of thier weights
Sci Advisor PF Gold P: 1,479 Specific Heat Ratio - Rocketry Sorry, I forgot to say that Cp_i and Cv_i are the molar heat capacities. I think you will obtain the same number (or at least a very close number to that) if you calculate it by means of the mass fraction and the mass heat capacities. The fact is such coefficients are calculated assuming an equal energetic state of both the complete mixture and the sum of the components: $$U=N_{total}C_{Vaverage}T)_{average of the mixture}=\sum U_i$$ If you start from that point, using a mass weighted average or molar weighted average you will obtain an equivalent number for Cv (molar or mass based).
Sci Advisor P: 882 OK, thanks, so I will go good to continue with my mass ratioed specific heat ratios, that makes my life slightly easier since I had already worked out the math and I wont have to look for a new table with the substances' molar spcific heat ratios (even though it should be the same in theory, since it is the ratio I want, not the actual specific heats).
Related Discussions Introductory Physics Homework 3 Introductory Physics Homework 5 Biology, Chemistry & Other Homework 1 Introductory Physics Homework 4 Classical Physics 2
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2014-09-02 19:26:45
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https://miniature-unicycle-firmware.readthedocs.io/en/latest/hardware.html
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# Hardware¶
The controller for the robot is a chipKIT Max32, which is an arduino-compatible board with a PIC32 microcontroller on board. The specific microcontroller on the board is a PIC32MX795F512, which is relevant when looking for specific hardware features such as timers and PWM.
Power comes from a 7.4V Li-ion battery.
## Flashing a program¶
The trace under JP5 has been deliberately cut on the board. This jumper connects the DTR line of the RS232 interface (from the FTDI232RQ chip that tunnels RS232 over USB), to the reset pin of the microcontroller. The flash programmer normally uses this line to send the chip into the bootloader, so its code can be changed.
Unfortunately, this line is also unconditionally asserted when plugged into a computer, making it impossible to attach to a long-running program. The fact that this is a problem at all is a design flaw in the Max32 - were the CTS line used instead, there would be no problem.
To reprogram the board then, JP5 must be temporarily closed. This could be done with a jumper, but this requires that the top layer of circuit board be removed. In practive, this is best done using a screwdriver to short the two pins.
Danger
Do not reprogram the board while the battery is attached! Shorting JP5 is clumsy, and has a risk of shorting other parts of the board. Your USB port is probably protected against this, but the battery may do unspeakably bad things.
## Motors¶
There are a pair of motors on the unicycle, driver with some custom and unfortunately poorly-documented hardware. The interface to the microcontroller is a pair of pwm lines, one for the forward direction, and one for the reverse direction.
Each motor is a Maxon 110134 with a Maxon 134158 gearbox attached. These motors also have attached Encoders.
Functions
void setMotorTurntable(float cmd)
Set the speed of the turntable.
Parameters
• cmd: The fraction of maximum speed, in [-1 1]. Positive is counter-clockwise around the positive Z axis
void setMotorWheel(float cmd)
Set the speed of the wheel
Parameters
• cmd: The fraction of maximum speed, in [-1 1].
void setupMotors()
Initialize the timers and PWM needed for the motors.
void beep(float freq, int duration)
Play a tone, using the turntable motor.
Not all frequencies resonate well. The built in <ToneNotes.h> is handy for turning note names to frequencies. Octaves 6 and 7 are loud and audible.
Parameters
• freq: The frequency in Hz
• duration: The duration in milliseconds
## Sensors¶
There is a limit switch attached to the top of the robot, for use as basic human input. This switch must be connected to a CN pin, as only those pins support pull-up resistors.
### Inertial¶
The robot has a combined accelerometer and gyro board that is sold by Sparkfun. The accelerometer is an ADXL345, and the gyroscope is an ITG-3200.
Both of these sensors use the I2C protocol - in fact, they share a bus. Unfortunately, the builtin arduino Wire interface that implements this protocol does not appear to work on our microcontroller board.
Functions
void gyroAccelSetup()
Initialize the connection to the accelerometer and gyro.
template <typename T>
Vector3<T> chipToRobotFrame(Vector3<T> v_chip)
Convert from the chip coordinate frame to the robot frame.
The robot frame has x pointing forwards (the side with the microcontroller) z pointing left y pointing up
template <typename T>
Vector3<float> gyroRawToSI(Vector3<T> raw)
Convert from the raw chip reading to radians per second, in the robot frame.
template <typename T>
Vector3<float> accRawToSI(Vector3<T> raw)
Convert from the raw chip reading to meters per second squared, in the robot frame.
Vector3<int16_t> accelReadRaw()
Read the raw values of the accelerometer, in internal frame and units.
Vector3<float> accelRead()
Get the acceleration in m s^-2, in the robot frame.
Vector3<int16_t> gyroReadRaw()
Read the raw values of the gyroscope, without subtracting initial values.
Vector3<float> gyroCalibrate(int N)
calibrate the offset for the gyro. Returns the stdev of each component
Vector3<float> gyroRead()
Get the angular velocity in the robot frame.
quat accelOrient(Vector3<float> acc)
Get the robot orientation based on the accelerometer reading. Only accurate when static
quat accelOrient()
### Encoders¶
The robot has an encoder on each motor. These go via some circuitry on the board that convert them into two lines - a tick, and a direction.
We count the ticks using the builtin hardware timers, but in order to deal with the direction reversing, we have to monitor the direction pin. We use the “change notifier” hardware to fire an interrupt whenever these pins change, and correct the sign accordingly.
Each encoder is a Maxon 201937, “Encoder MR, Type M, 512 CPT, 2 Channels, with Line Driver”.
Functions
void setupEncoders()
Initialize the hardware required by the encoders.
void resetEncoders()
Reset the counts of the encoders.
wrapping<uint16_t> getTTangle()
Get the angle of the turntable, in encoder ticks. Increases with counter-clockwise in the vertical axis
wrapping<uint16_t> getWangle()
Get the angle of the wheel, in encoder ticks. Increases with forwards motion
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2021-04-12 16:31:48
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https://www.semanticscholar.org/paper/J%2F%CF%88-production-as-a-function-of-charged-particle-in-Collaboration/1a7411aa5086cfa2fb4dea396de8c28104a3c500
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# J/ψ production as a function of charged-particle multiplicity in p-Pb collisions at $$\sqrt{s_{\mathrm{NN}}}$$ = 8.16 TeV
@inproceedings{Collaboration2020JPA,
title={J/$\psi$ production as a function of charged-particle multiplicity in p-Pb collisions at \$\$ \sqrt\{s\_\{\mathrm\{NN\}\}\} \$\$ = 8.16 TeV},
author={Alice Collaboration},
year={2020}
}
Inclusive J/$\psi$ yields and average transverse momenta in p-Pb collisions at a center-of-mass energy per nucleon pair $\sqrt{s_{\rm NN}}$ = 8.16 TeV are measured as a function of the charged-particle pseudorapidity density with ALICE. The J/$\psi$ mesons are reconstructed at forward $(2.03<y_{\rm cms}<3.53)$ and backward ($-4.46<y_{\rm cms}<-2.96$) center-of-mass rapidity in their dimuon decay channel while the charged-particle pseudorapidity density is measured around midrapidity. The J… Expand
#### References
SHOWING 1-7 OF 7 REFERENCES
Double, triple, and $n$-parton scatterings in high-energy proton and nuclear collisions
• Physics
• 2017
The framework to compute the cross sections for the production of particles with high mass and/or large transverse momentum in double- (DPS), triple- (TPS), and in general $n$-parton scatterings,Expand
The quest for the quark–gluon plasma
• Physics, Medicine
• Nature
• 2007
Experiments at the Large Hadron Collider should consolidate the evidence for this exotic medium's existence, and allow its properties to be characterized. Expand
The EvtGen particle decay simulation package
With several new B-physics experiments now taking data, the physics of B-meson decays will be studied in greater detail than previously possible. It is important to have a simulation of theExpand
The Monte Carlo Event Generator DPMJET-III
• Physics, Mathematics
• 2000
A new version of the Monte Carlo event generator DPMJET is presented. It is a code system based on the Dual Parton Model and unifies all features of the DTUNUC-2, DPMJET-II and PHOJET1.12 eventExpand
PHOTOS - a universal Monte Carlo for QED radiative corrections: version 2.0
• Physics
• 1994
This paper documents an update of the PHOTOS algorithm, by E. Barberio, B. van Eijk and Z. Was, for the Monte Carlo simulation of QED photon radiative corrections in decays. The algorithm isExpand
$J/\psi$ Suppression by Quark-Gluon Plasma Formation
• Physics
• 1986
If high energy heavy ion collisions lead to the formation of a hot quark-gluon plasma, then colour screening prevents ce binding in the deconfined interior of the interaction region. To study thisExpand
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2021-10-21 00:17:00
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https://math.stackexchange.com/questions/702165/continuous-probability-angles
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Continuous probability. Angles
3 points are randomly chosen on circumference of the circle. Points are connected to form a triangle. What's the probability that at least on angle will be less than 42 degrees? What's the probability that the sum of any 2 angles is greater than 147 degrees? I would appreciate a solution, but some hints would be fine too, thank you.
Since the circle is symmetric, we can decide to measure positions around the circle clockwise from the first random points. Then the position of the two other points are still independent and uniformly distributed between $0$ and $360$.
This is useful because our sample space is then a 2-dimensional square and easy to visualize. We can draw a diagram of it with $x$-axis giving the position of the second point and the $y$-axis giving the position of the third point. Then for each of the questions, it is simple enough (though perhaps slightly tedious) to sketch out those areas of the sample space that the event covers and compute their area (they will be polygons with axis-parallel or 45° lines).
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2019-12-14 10:40:03
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https://abagoffruit.wordpress.com/2003/12/16/compulsory-update/
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## Compulsory Update
Almost nothing happens around here. All I ever get done is a bit of reading on the third book on the Riemann Hypothesis, The Music of the Primes by Marcus du Sautoy. I should start doing some music theory since I still have three quarters to learn by January. I want to do math though. I want to understand something at the conference, but I probably won’t.
My algebra notes are now on the web here in pdf and here in dvi. If you find any errors, please tell me. I want these notes to be perfect.
Happy 233rd birthday to Beethoven. It’s prime!
Hi. I'm Simon Rubinstein-Salzedo. I'm a mathematics postdoc at Dartmouth College. I'm also a musician; I play piano and cello, and I also sometimes compose music and study musicology. I also like to play chess and write calligraphy. This blog is a catalogue of some of my thoughts. I write them down so that I understand them better. But sometimes other people find them interesting as well, so I happily share them with my small corner of the world.
This entry was posted in Uncategorized. Bookmark the permalink.
### 4 Responses to Compulsory Update
1. Anonymous says:
erkerkerk
erk, Simon is afraid he won’t understand anything. What is the world coming to?
btw, what level of students was the conference designed for?
-LV
• Simon says:
Re: erkerkerk
It wasn’t designed for students at all.
2. yuethomas says:
A couple of questions. TeX notations are used to facilitate clearer understanding.
1. Definition 1.5. $f(j) = j$ if $j \neq n_k$ for $1 \le k \le r$ means $\nexists k \in \mathbb{N}$ such that $j = n_k$ right?
2. Example of Proposition 1.6. Is multiplication of non-disjoint cycles from right to left? (Sorta like $(f \cdot g)(a) = f(g(a))$) Because if I write (1,2)(2,3) as one cycle from left to right, I get (1,3,2) which implies that 3 -> 1 not 3 -> 2. Same for (2,3)(1,2): I get the identity permutation in which case 3 -> 1.
3. Proof of Theorem 1.7. In the 6th line (not counting the whitespace) of the proof, you say “for if $i_m$ is an $i$…” Do you mean “for if $j_m$ is an $i$…”, since $i_m$ is by definition an $i$?
That’ll do for now – thanks for the file! I needed a pick-me-up in group theory. 😉
• Simon says:
I agree with all. Thanks.
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2017-06-23 22:32:45
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https://ncertmcq.com/mcq-questions-for-class-8-maths-chapter-1/
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Check the below Online Education NCERT MCQ Questions for Class 8 Maths Chapter 1 Rational Numbers with Answers Pdf free download. MCQ Questions for Class 8 Maths with Answers were prepared based on the latest exam pattern. We have provided Rational Numbers Class 8 Maths MCQs Questions with Answers to help students understand the concept very well. https://ncertmcq.com/mcq-questions-for-class-8-maths-with-answers/
Students can also refer to NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers for better exam preparation and score more marks.
## Online Education for Rational Numbers Class 8 MCQs Questions with Answers
Class 8 Maths Chapter 1 MCQ Question 1.
Which of the following statements is false ?
(a) Natural numbers are closed under addition
(b) Whole numbers are closed under addition
(c) Integers are closed under addition
(d) Rational numbers are not closed under addition.
Rational Numbers Class 8 MCQ Question 2.
Which of the following statements is false ?
(a) Natural numbers are closed under subtraction
(b) Whole numbers are not closed under subtraction
(c) Integers are closed under subtraction
(d) Rational numbers are closed under subtraction.
Class 8 Maths MCQ Chapter 1 Question 3.
Which of the following statements is true ?
(a) Natural numbers are closed under multiplication
(b) Whole numbers are not closed under multiplication
(c) Integers are not closed under multiplication
(d) Rational numbers are not closed under multiplication.
Rational Numbers MCQ Class 8 Question 4.
Which of the following statements is true ?
(a) Natural numbers are closed under division
(b) Whole numbers are not closed under division
(c) Integers are closed under division
(d) Rational numbers are closed under division.
Class 8 Math Chapter 1 MCQ Question 5.
Which of the following statements is false ?
(a) Natural numbers are commutative for addition
(b) Whole numbers are commutative for addition
(c) Integers are not commutative for addition
(d) Rational numbers are commutative for addition.
Class 8 Maths Ch 1 MCQ Question 6.
Which of the following statements is true ?
(a) Natural numbers are commutative for subtraction
(b) Whole numbers are commutative for subtraction
(c) Integers are commutative for subtraction
(d) Rational numbers are not commutative for subtraction.
MCQ On Rational Numbers For Class 8 Question 7.
Which of the following statements is false ?
(a) Natural numbers are commutative for multiplication
(b) Whole numbers are commutative for multiplication
(c) Integers are not commutative for multiplication
(d) Rational numbers are commutative for multiplication.
MCQ Class 8 Maths Chapter 1 Question 8.
Which of the following statements is true ?
(a) Natural numbers are commutative for division
(b) Whole numbers are not commutative for division
(c) Integers are commutative for division
(d) Rational numbers are commutative for division.
MCQ Questions For Class 8 Maths Chapter 1 With Answers Question 9.
Which of the following statements is true ?
(a) Natural numbers are associative for addition
(b) Whole numbers are not associative for addition
(c) Integers are not associative for addition
(d ) Rational numbers are not associative for addition.
MCQ Questions For Class 8 Maths Chapter 1 Question 10.
Which of the following statements is true ?
(a) Natural numbers are associative for subtraction
(b) Whole numbers are not associative for subtraction
(c) Integers are associative for subtraction
(d) Rational numbers are associative for subtraction.
Maths Class 8 Chapter 1 MCQ Question 11.
Which of the following statements is true ?
(a) Natural numbers are not associative for multiplication
(b) Whole numbers are not associative for multiplication
(c) Integers are associative for multiplication
(d) Rational numbers are not associative for multiplication.
Ch 1 Maths Class 8 MCQ Question 12.
Which of the following statements is true ?
(a) Natural numbers are associative for division
(b) Whole numbers are associative for division
(c) Integers are associative for division
(d) Rational numbers are not associative for division.
Class 8 Chapter 1 Maths MCQ Question 13.
0 is not
(a) a natural number
(b) a whole number
(c) an integer
(d) a rational number.
Class 8 Rational Numbers MCQ Question 14.
$$\frac{1}{2}$$ is 2
(a) a natural number
(b) a whole number
(c) an integer
(d) a rational number.
MCQ Of Rational Numbers Class 8 Question 15.
a + b = b + a is called
(a) commutative law of addition
(b) associative law of addition
(c) distributive law of addition
(d) none of these.
Question 16.
a × b = b × a is called
(a) commutative law for addition
(b) commutative law for multiplication
(c) associative law for addition
id) associative law for multiplication.
Question 17.
(a + b) + c = a + (b + c) is called
(a) commutative law for multiplication
(b) commutative law for addition
(c) associative law for addition
id) associative law for multiplication.
Question 18.
a × (b × c) = (a × b) × c is called
(a) associative law for addition
(b) associative law for multiplication
(c) commutative law for addition
(d) commutative law for multiplication.
Question 19.
a(b + c) = ab + ac is called
(a) commutative law
(b) associative law
(c) distributive law
(d) none of these.
Question 20.
The additive identity for rational numbers is
(a) 1
(b) -1
(c) 0
(d) none of these.
Question 21.
The multiplicative identity for rational numbers is
(a) -1
(b) 1
(c) 0
(d) none of these.
Question 22.
The additive inverse of $$\frac{2}{3}$$ is
(a) –$$\frac{2}{3}$$
(b) $$\frac{3}{2}$$
(c) –$$\frac{3}{2}$$
(d) 1
(d) 1
Question 23.
The additive inverse of –$$\frac{3}{4}$$ is
(a) –$$\frac{3}{4}$$
(b) 1
(c) 0
(d) $$\frac{3}{4}$$
Question 24.
The multiplicative inverse of $$\frac{1}{2}$$ is
(a) 1
(b) -1
(c) 2
(d) 0
Question 25.
The multiplicative inverse of –$$\frac{2}{5}$$ is
(a) –$$\frac{2}{5}$$
(b) –$$\frac{5}{2}$$
(c) $$\frac{5}{2}$$
(d) 1
Question 26.
The multiplicative inverse of 1 is
(a) 0
(b) -1
(c) 1
(d) none of these.
Question 27.
The multiplicative inverse of -1 is
(a) 0
(b) -1
(c) 1
(d) none of these.
Question 28.
How many rational numbers are there between any two given rational numbers?
(a) Only one
(b) Only two
(c) Countless
(d) Nothing can be said.
Question 29.
The negative of 2 is
(a) 2
(b) $$\frac{1}{2}$$
(c) -2
(d) –$$\frac{1}{2}$$
Question 30.
The negative of -2 is
(a) -2
(b) 2
(c) –$$\frac{1}{2}$$
(d) $$\frac{1}{2}$$
Question 31.
If a and b are two rational numbers, then
(a) $$\frac{a+b}{2}$$ < a
(b) $$\frac{a+b}{2}$$ < b
(c) $$\frac{a+b}{2}$$ = a
(d) $$\frac{a+b}{2}$$ > b
Question 32.
The rational number that does not have a reciprocal is
(a) 0
(b) 1
(c) -1
(d) $$\frac{1}{2}$$
Question 33.
The rational number which is equal to its negative is
(a) 0
(b) -1
(c) 1
(d) $$\frac{1}{2}$$
Question 34.
The reciprocal of $$\frac{1}{x}$$ (x ≠ 0) is
(a) x
(b) $$\frac{1}{x}$$
(c) 1
(d) 0
Question 35.
The reciprocal of a positive rational number is
(a) a positive rational number
(b) a negative rational number
(c) 0
(d) 1.
Question 36.
The reciprocal of a negative rational number is
(a) a positive rational number
(b) a negative rational number
(c) 0
(d) -1
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2023-03-27 09:51:46
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https://learn.sarthaks.com/direct-and-inverse-proportion/
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# Direct and inverse proportion:
Let us consider some examples to understand direct and inverse proportion:
1. More you ride your bicycle the more distance you will cover.
2. More the packets of biscuits more the number of biscuits.
3. If there are many shops of the same kind then less will be the crowd.
4. If there is more advertisement in newspaper then less will be the news.
From the above examples, it is clear that change in one quantity leads to change in the other quantity.
## Types of proportions:
There are two proportions i.e. direct and inverse proportion.
1. ## Direct proportion:
• We can say two quantities x and y in direct proportion if they increase (decrease) together in the same manner.
• Also, two quantities x and y in direct proportion if they increase (decrease) together in such a manner that the ratio of their corresponding values remains constant. That is if $$\frac{x}{y}$$ = k [k is a positive number], then x and y are said to vary directly.
• In such a case if y1, y2 are the values of y corresponding to the values x1 , x2 of x respectively then $$\frac{x_{1}}{y_{1}}$$ = $$\frac{x_{2}}{y_{2}}$$
• If x is directly proportional to y, we can write it as x ∝ y.
### Variables increasing (or decreasing) together need not always be in direct proportion. For example:
• In human beings, physical changes occur with time but not necessarily in a predetermined ratio.
• Among individuals when we talk about changes in weight and height are not in any known proportion.
• In natural phenomena such as there is no direct relationship or ratio between the height of a tree and the number of leaves growing on its branches.
1. ## Inverse proportion:
• We can say two quantities x and y in direct proportion if they increase (decrease) together in the same manner.
• Also, two quantities x and y are said to be in inverse proportion if an increase in x causes a proportional decrease in y (and vice-versa) in such a manner that the product of their corresponding values remains constant. That is, if xy = k, then x and y are said to vary inversely.
• In such a case if y1, y2 are the values of y corresponding to the values x1 , x2 of x respectively then $$\frac{x_{1}}{x_{2}}$$ = $$\frac{y_{2}}{y_{1}}$$
• If x is inversly proportional to y, we can write it as x ∝ $$\frac{1}{y}$$.
## Direct and inverse proportion examples:
1. An electric pole, 14 metres high, casts a shadow of 10 metres. Find the height of a tree that casts a shadow of 15 metres under similar conditions.
Solution:
More the height of an object, the more would be the length of its shadow.
Thus, x1 : x2 = y1 : y2
14 : x = 10 : 15
10 × x = 15 × 14
thus, x = 21
1. 6 pipes are required to fill a tank in 1 hour 20 minutes. How long will it take if we use only 5 pipes of the same type?
Solution:
Number of pipes 6 5 Time (in minutes) 80 x
This is a case of inverse proportion as less number of pipes, it will require more time to fill the tank.
Hence, 80 × 6 = x × 5 as [x1y1 = x2 y2]
or, $$\frac{80×6}{5}$$
x = 96
Thus, time taken to fill the tank by 5 pipes is 96 minutes or 1 hour 36 minutes.
Ratio and Proportion: Learn, Take Notes, Revise, Examples
## Be First to Comment
error: Content is protected !!
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2019-02-17 18:10:50
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https://docs.sctl.xyz/tokenomics/tokenomics
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# Tokenomics
The Societal network will have an inflation rate on the $SCTL token. The purpose of the inflation rate is to pay for the block production and uptime of the network. The two items the inflation will pay for are: incentivizing collators to provide block production services and to reward$SCTL token holders for staking their tokens with collators.
The inflation rate (I) of the Societal network can be calculated by the following formula:
$I = I npos - I slashing - I burn$
Where:
• I = yearly inflation rate
• I NPoS = inflation caused by token creation to reward collators and delegators
• I slashing = deflation from the burning of tokens due to misconduct by a collator
• I burn = The purchase and burning of $SCTL tokens in the secondary market In the above formula, I NPoS has a maximum value of 5% and will vary based on the staking rate of the network. Therefore the maximum annual inflation of the$SCTL token is 5%, however this does not mean that the network will always be in inflation. In order to reduce the overall inflation of the $SCTL token, Societal will use part of its revenue source for purchasing the$SCTL token in the secondary market and burning the token. When the Societal network is prosperous enough, the purchasing of the tokens in the secondary market and burning them will act as a deflationary force and will accrue value to the $SCTL holders. Once this point is reached, there will be enough utilization on the network to potentially allow for$SCTL to become deflationary.
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2023-03-28 12:50:14
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https://www.authorea.com/users/4485/articles/4705-photometric-science-alerts-from-gaia/_show_article
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# Photometric Science Alerts From Gaia
10/18/2012
Abstract
Gaia is a European Space Agency (ESA) astrometry space mission, and a successor to the ESA Hipparcos mission. The main goal of the Gaia mission is to collect high-precision astrometric data (i.e. positions, parallaxes, and proper motions) for the brightest one billion objects in the sky. This data, complemented with G band, multi-epoch photometric and low resolution (lowers) spectroscopic data collected from the same observing platform, will allow astronomers to reconstruct the formation history, structure, and evolution of the Galaxy.
In addition, the Gaia satellite is an excellent transient discovery instrument, covering the whole sky (including the Galactic plane) for the next 5 years, at high spatial resolution (50 to 100 mas, similar to the Hubble space telescope (HST)) with precise photometry (1$$\%$$ at G=19) and milliarcsecond astrometry (down to $$\sim$$20mag). Thus, Gaia provides a unique opportunity for the discovery of large numbers of transient and anomalous events, e.g. supernovae, black hole binaries and tidal disruption events. We discuss the validation of the alerts stream for the first six months of the Gaia observations, in particular noting how a significant ground based campaign involving photometric and spectroscopic followup of early Gaia alerts is now in place. We discuss the validation approach, and highlight in more detail the specific case of Type Ia supernova (SNe Ia) to be discovered by Gaia. The intense initial ground based validation campaign will ensure that the Gaia alerts stream for the remainder of the Gaia mission, are well classified.
# What is a Photometric Science Alert?
A photometric science alert is the appearance of a new source, or a change in flux, which suggests we could learn something from prompt ground-based follow-up. This does not include: periodic variable stars (these sources may be better left to the end of the mission) and moving objects (however, astrometric microlensing would be an exception). The science alerts will be made public, within one to two days of Gaia detection, most of this time is due to downloading the data from the satellite.
# Potential Triggers
Potential triggers for the the Gaia science alerts are objects of scientific interest which would benefit from fast ground based follow-up, as just discussed. Some examples of sources which maybe potential triggers include supernovae, super-luminous supernovae, tidal disruption events, cataclysmic variables, outbursts and eclipses from young stellar objects, X-ray binaries, microlensing events and other theoretical or unexpected phenomena. Figure \ref{fig:triggers} shows some of these potential triggers and the area of pars space they occupy for their brightness as a function of duration.
This shows the amplitude and duration of a range of potential triggers for the Gaia science alerts. \label{fig:triggers}
# Gaia as a Transient Search Machine
Gaia is comparable to other transient search machines, such as the Catalina Sky Survey and the Palomar Transient Factory, as shown in Table \ref{tab:transient_machine}, which covers similar areas each day and similar limiting magnitude. The disadvantage of the Gaia survey is that the average cadence is only $$\sim$$30days whereas transient surveys usually have a cadence of approximately 3 to 5 days. However, there is also a shorter cadence of 106.5 mins from the two mirrors in the satellite, also sometimes a 253.5 mins cadence, and sometimes 3 or more observations are thus obtained (when close to the 45 degrees ecliptic latitude zones for example). This 106.5 mins cadence is a huge advantage and means that changes in brightness should be detected quickly. Also, Gaia will cover the whole sky (including the Galactic plane), which is a significant survey area increase over other transient searches. The Gaia transient alerts will also have high spatial resolution with precise photometry (1$$\%$$ at G=19) and milliarcsecond astrometry (down to $$\sim$$20mag), lowres spectra for all objects brighter than $$\sim$$19mag and colours for fainter objects (see \citet{Jordi:2010} for details of the photometry and lowres spectra).
Patient Gaia Catalina Sky Survey Palomar Transient Factory
deg2 day-1 $$\sim$$1230 1500 1000
Avg Cadence $$\sim$$30 days 14 days 5 days
Limiting mag 20 (21?) 19.5 21
fsky all sky 0.6 0.2
# Time line
The Gaia satellite was launched on the 19th December 2013, and has now successfully been placed into orbit around the second Lagrange point. Over the next few months the telescope will undergo system shake-down and ESA commissioning (Figure \ref{fig:timeline}). It is planned that in June the Gaia satellite will spend a month scanning the Ecliptic Poles internally verifying the data, and learning how to identify large amplitude variable stars (potential contaminants of the Gaia Science Alerts stream).
\label{fig:timeline} Current timeline for Gaia operations and data accumulation.
Then in July Gaia will switch to nominal scanning and history of the whole sky will begin to be accumulated. In Figure \ref{fig:scanning} we show the expected coverage of Gaia by the end of July and then the end of September 2014. This will give some history of each patch of sky in the Gaia passbands and allow detection of transient objects. We propose to begin Gaia Alerts Spectroscopic Follow-up in the last weeks of August and the first week of September.
# Scanning law
The Gaia satellite consist of two telescopes, which are projected onto one focal plane. The time between the two fields of view being observed is 106.5 mins and then the time between subsequent scans is 6 hours. After these initial observations the field will be revisited every $$\sim$$10-30 days. Over the full mission each patch of sky will be measured, on average, approximately 70 times. The densest coverage is at 45 degrees to the ecliptic plane and this region is covered with approximately 200 epochs.
\label{fig:scanning} By 30 days 11.6$$\%$$ of the sky has been observed at least 3 times by Gaia. By 90 days, 52.03$$\%$$ of the sky has been observed at least 3 times by Gaia.
# SNa discovery rates
Simulations, Belokurov (2003) and updated by Altavilla et al. (2012), predict Gaia will see $$\sim$$6000 SNe down to G=19 (3/day), and twice this to G=20. One SN per day will be brighter than 18th magnitude (see Fig \ref{fig:SN_rate}). For cataclysmic variables (CVs) the rate will be approximately similar, and Breedt, (priv. comm) predict Gaia should find 1000 new CVs. Blagorodnova (in prep) predict that Gaia will find of order 20 Tidal disruption event’s (TDE’s) per year. Young stellar objects outbursts will be less common and Gaia will probably only find a few per year.
\label{fig:SN_rate} Predicted SN detections with Gaia as a function of G-band magnitude.
Alerts are expected to be discovered and published to the world within $$\sim$$24-48 hours of observation by the satellite. The Alert Stream will go live once Gaia has mapped at least 10$$\%$$ of the sky, a minimum of 3 times, which takes approximately one month (see Fig \ref{fig:scanning}). Once the Gaia alert stream is fully operational all alerts will be made publicly available, and thus accessible for use by the community in their dedicated followup campaigns (see Section \ref{followup}). During the commissioning, initialisation and early operations phase of Gaia (January - August 2014) - there will be systematic validation of the Gaia alerts, whereby the operational system will be assessed before going ‘Live’. The science alerts will be available to the community in web-based and email-based formats and will be produced in Virtual Observatory Event (VOEvent) - machine-readable format.
Each Alert package will consist of: coordinates, magnitudes, light curves, spectra, colours, proper motions, parallaxes (when available), astrophysical parameters (pars) (when available), features (random forest classifier see Section \ref{class}), classifier probabilities, cross match results.
# Classification
\label{class} Gaia is predicted to detect 44 million transits per day,which is $$\sim$$150 - 800 GByte/day of data. Within this huge volume of data we expect 100s -1000s of potential interesting astrophysical triggers per day (real variables/moving objects). This precludes visual classification of a rich data stream and thus automated methods which are fast, repeatable and tuneable are essential. The Gaia alerts classification pipeline uses random forest classification. The random forest will use all the information available, and its features will include: light curve photometry (gradient, amplitude, historic rms, magnitude, signal-to-noise ratio, transit rms), lowres spectra (flux v lambda, colours, spectral shape coefficients, spectral type), auxiliary information (neighbour star, shape pars, motion pars, coords, crowding, calibration offset, correlations, QC pars) and crossmatch environment (near known star mags, near known variable class, near galaxy, near galaxy redshift and circumnuclear).
To build up a sufficient sample of classification labels in order to train the random forest classifier (e.g. \citet{Ofek_Cenko_Butler_et_al__2012}) we aim to observe $$\sim$$500s homogenous high-quality spectra in the first year of the mission, spread across each broad class of transient phenomena (active galactic nuclei, core collapse SN, TDE, SN, Novae, CV and variable stars).
The light curve classification utilises the flux gradient of the transient object. The Gaia observations with 106.5 mins cadence are used to indicate the type of object. The lowers (BP/RP) spectra provide far more information to aid classification \citet{Blagorodnova_in_prep_2014} and provide robust class for most objects, at $$>$$19mag, when the classifier is fully trained on representative data. In addition, the transient object will be cross matched with archival catalogues, for example, Sloan Digital Sky Survey (SDSS), Two Micron All Sky Survey (2MASS), HST and Visible and Infrared Survey Telescope for Astronomy (VISTA). This will help remove known variable star contaminates and provide environmental information for the transient events, e.g. is there a host galaxy associated with the source and if so what is the type and magnitude.
\label{followup} We are also co-ordinating a large program of photometric follow-up to improve the light curve sampling of Gaia transients. 47 x (7cm-2m) telescopes are listed as currently active (http://bit.ly/1aHNXzy) and 13 observatories are already doing tests (http://bit.ly/17ViW7s). All make use of our photometric calibration server (a tool developed to maximise the usefulness of the photometric followup data) to place the disparate data onto the same system (Wyrzykowski et al. 2013 $$ATEL\#5245$$). Additionally, Las Cumbres Observatory Global Telescope Network (LCOGT) are expected to play a key role in the follow-up especially of $$\mu$$lensing and young star transients. We point out the strong synergies with external facilities operating at different wavelengths. We will be able to confirm and characterise e.g. Low Frequency Radio Array (LOFAR) transients, and we may also trigger prompt SWIFT follow-up for particularly interesting events.
There is also a large educational (mostly utilising the Faulkes telescopes) and amateur involvement planned in the followup of these transient events, to assist in compiling light curves and increase the public evolvement and interest.
We need a large sample of well-exposed (S/N$$\sim$$20$$-$$50), medium-dispersion (R$$\sim$$500$$-$$1000) spectra, over a wide range of classes and magnitudes, to build classification training sets, in order for our (Random Forest) machine learning algorithms (discussed in Section \ref{class}) to perform well for the Gaia spectra for the remainder of the mission. Therefore we aim to obtain 1.5-4m telescope time to build this training set. It is important to invest time at the beginning of the Gaia mission to understand and characterise the transients that will be discovered with Gaia, so that we can optimise the process, and ensure that the rest of the mission is as productive as possible. We also intend to archive and release our spectroscopic classifications promptly after processing each night’s observing.
# Summary
The alert stream is non-proprietary and will be (some of) the first data from Gaia Summer 2014. We have planned an extensive follow-up program for classifying large numbers of transients: e.g. 10,000 SNe Ia over the whole sky. The alerts will be published one to two days after the event was initially detected (most of this time is due to the time taken for the data to be down linked from the satellite and processed). The alerts will be preliminarily classified using random forest classifiers based on the Gaia photometry and lowres spectra with additional cross match information from existing surveys. These classifications should improve after the first few months of ground based followup and retraining of the Bayesian classifiers. The alerts will be published in the VO format. For more information visit: http://www.ast.cam.ac.uk/ioa/wikis/gsawgwiki.
# Acknowledgements
Material used in this work has been provided by the Coordination Unit 5 (CU5) of the Gaia Data Processing and Analysis Consortium (DPAC). They are gratefully acknowledged for their contribution.
HCC, STH, GG, NAW, LW are members of the Gaia Data Processing and Analysis Consortium (DPAC) and this work has been supported by the UK Space Agency. NB has been supported by the The Gaia Research for European Astronomy Training (GREAT-ITN) network, funded through the European Union Seventh Framework Programme ([FP7/2007-2013] under grant agreement n$$\,^{\circ}$$ 264895.
### References
1. C. Jordi, M. Gebran, J. M. Carrasco, J. de Bruijne, H. Voss, C. Fabricius, J. Knude, A. Vallenari, R. Kohley, A. Mora. Gaia broad band photometry. Astronomy & Astrophysics 523, A48 EDP Sciences, 2010. Link
2. V. Belokurov. Supernovae with super-Hipparcos. 341, 569-576 Springer-Verlag, 2003. Link
3. Giuseppe Altavilla, Maria Teresa Botticella, Enrico Cappellaro, Massimo Turatto. Supernovae and Gaia. Astrophysics and Space Science 341, 163-178 Springer-Verlag, 2012. Link
4. N. Blagorodnova. BANGST! Bayesian Analysis for Nearby Gaia Supernovae Transients. (in prep). Link
5. J. S. Bloom, J. W. Richards, P. E. Nugent, R. M. Quimby, M. M. Kasliwal, D. L. Starr, D. Poznanski, E. O. Ofek, S. B. Cenko, N. R. Butler, et al.. Automating Discovery and Classification of Transients and Variable Stars in the Synoptic Survey Era. Publications of the Astronomical Society of the Pacific 124, 1175-1196 The University of Chicago Press, 2012. Link
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2019-05-25 00:52:27
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https://math.stackexchange.com/questions/2543203/prove-that-f-mathbbz-to-mathbbr-x-mapsto-x%C2%B2-is-uniformly-continuous
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# Prove that $f: \mathbb{Z} \to \mathbb{R}: x \mapsto x²$ is uniformly continuous (or not)
I have to prove that $f: \mathbb{Z} \to \mathbb{R}: x \mapsto x²$ is uniformly continuous (or not)
My attempt:
Suppose $f$ is not uniformly continuous.
Then $\exists \epsilon >0: \forall \delta >0: \exists x,y \in \mathbb{Z}: |x-y| < \delta \land |x² - y²| \geq \epsilon$
Choose such an $\epsilon > 0$ and $\delta = 1/2$
Then, there exists $x,y \in \mathbb{Z}$ such that both $|x-y| < 1/2$ and $|x²-y²| \geq \epsilon$
But from $|x-y| < 1/2$, we find that $x=y$. Hence $|x²-y²| = 0$, contradicting the fact that $|x²-y²| \geq \epsilon$
Hence, $f$ must be uniformly continuous
Is this correct?
To prove it directly, given $\varepsilon > 0$ we need to find $\delta > 0$ such that $|x^2 - y^2| < \varepsilon$ for all $x,y \in \mathbb{Z}$ satisfying $|x - y| < \delta$. Now, for any $\varepsilon > 0$, $\delta = 1/2$ works for the same reason as in your proof: $|x-y| < 1/2$ and $x,y \in \mathbb{Z}$ imply that $x=y$, and so $|x^2 - y^2| = 0 < \varepsilon$ for any $\varepsilon > 0$. In particular, $\delta$ is independent of $x$ and $y$, and so it is uniformly continuous.
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2021-07-27 16:12:43
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http://bradleyallen.org/2724-fx-swap-implied-yield.php
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# Eikon.
How to obtain indicative data on government bond-yield derived Zero coupon rates. 101. How to get Indicative Data on Currency Basis Swap. Implied volatility is the trader's estimate of how volatile spot FX will be.In fact, it is a combination of an FX spot and an FX swap. Cross currency swaps, or basis, where one bets on the difference between the FX swap implied 3 months rate. spread of the 2 currencies and the spread of the respective IBOR 3 month fixings, every 3 months, over the. length of the swap.Current Treasuries and Swap Rates. U. S. Treasury yields and swap rates, including the benchmark 10 year U. S. Treasury Bond, different tenors of the USD London Interbank Offered Rate LIBOR, the Secured Overnight Financing Rate SOFR, the Fed Funds Effective Rate, Prime and SIFMA.Because the day count of your inquired date is 366 days Hkd daycount is act/365 therefore 366/365; Usd daycount is act/360 therefore 366/360. Assets. Thus, important information can be obtained from the development of outstanding amounts and implied FX swap yields in relation to strategies adopted.In this article we cover how to calculate forex swap and rollover points computed using the Interest Rate Parity.Rates from a government bond yield curve describe the risk-free rates of return available in the market today, however they also imply risk-free rates of return for future time periods. These implied future rates, known as implied forward rates, or simply forward rates, can be derived from a given spot yield curve using boot-strapping.
### Yr Swap Rates, Treasuries, LIBOR, SOFR - Chatham Financial
If we assume the currency convention is AB, it will be spot(AB) 4%.- In operations, when the need for Won currency is temporarily made at the end of month or quarter, etc., the company holding US Dollars can easily raise the Won currency without FX risk needless of selling of the US Dollars they hold.- Swap price in FX Swap deal means the difference between the Spot rate and the Forward rate that are applied on Swap deal. Hedging mechanism than swaps when used to hedge the foreign exchange risk. effect, the higher yielding currency will be discounted going forward and vice.Implied interest rate from FX swap. This is not homework. I am trying to calculate the implied interest rate of one currency C2 using an FX swap and the interest rate of another currency C1 - base. I have the following I have a borrowing in C1 for 0.9650% for the year. I solve for $r_{C2} = 0.8349\%$.Implied yields surged to 2-year high even after RBI cut rates. leading to a mismatch in the foreign-exchange market, which then pushes up. commonly known as a sell-buy swap in the forwards market to offset the liquidity.
### Implied interest rate from FX swap - Quantitative Finance Stack.
In an FX swap, one party borrows one currency from, and simultaneously lends another currency to, a second party (see also Baba et al (2008)).The borrowed amounts are exchanged at the spot rate, ).If the party lending a currency via FX swaps makes a higher or lower return than implied by the interest rate differential in the two currencies, then CIP fails to hold. Binare optionen handel ab 5 euro münze. Typically, the US dollar has tended to command a premium in FX swaps.In this case, rearranging the CIP equation yields the following relationship between (), above, indicates that a party lending US dollars sells the foreign currency forward at a higher dollar price than warranted by the interest differential.Equivalently, a party borrowing US dollars via an FX swap - say, to hedge its US dollar asset - is effectively paying a higher interest rate on the swapped dollars than is paid in the cash market.A cross-currency swap is a longer-term instrument, typically above one year, in which the two parties also simultaneously borrow and lend an equivalent amount of funds in two different currencies.
### The role of currency swaps in the domestic banking. - MNB
The turbulence in money markets has spilled over to FX swap markets amid a. Deviations of 12-month FX-swaps-implied US dollar rate from dollar LIBOR and. rate less the Overnight Index Swap rate and the interbank rate less the yield of.Major types of swaps. Foreign exchange swap simultaneous spot purchase and future sale of. the spot or forward yield curve. Floating-rate. In emerging markets with less-liquid money markets, forward-implied interest rate.Interest and principal payments. The latter is more often covered with a cross currency swap. In practice, however, forwards are sometimes favored as a more affordable, albeit less effective, hedging mechanism than swaps when used to hedge the foreign exchange risk of the principal of a loan, while leaving interest payments uncovered. S& p 500 futures trading strategies horses. The collection of all R fi for i = 1. n constitutes a yield curve of fictitious foreign interest rates – referred as the fx-implied foreign yield curve.I thought NIBOR implied from FX was a great idea because. The Consultation considers a pure “Foreign Exchange Swap Rate”. From our perspective, it is the final answers that yield the most interesting conclusions.Cross currency swaps, or basis, where one bets on the difference between the FX swap implied 3 months rate spread of the 2 currencies and.
More generally, suppose Even if risk premia in the underlying transaction are low, CIP deviations can arise if the demand to hedge one of the currencies is large.Then, even small risk premia can have big effects when scaled by the large size of the balance sheet exposures needed to meet the hedgers' demand.For example, Sushko et al (2016) show that CIP deviation can be proportional to the hedging demand multiplied by the per-dollar balance sheet costs of FX derivatives exposures: in this example. Brokers de forex en peru. Since markets have to clear, the aggregate position of CIP arbitrageurs when the US dollar is at a premium in FX swaps will be equal to the aggregate net position of currency hedgers.The latter will be paying the forward points, will reflect any costs that banks or other participants assign to deploying their balance sheet in CIP arbitrage, which in turn will reflect their risk management practices.For individual players, these practices may even include absolute credit limits that would set a maximum for the underlying exposures to the underlying instruments and counterparties.
Even without strict limits, the funding cost of the capital allocated to the arbitrage activity, notably to the (current and potential future) derivatives exposures involved, will prevent the basis from closing when it opens up owing to changes in hedging demand.The specific constraints, and hence the instruments involved, will also depend on the players acting as arbitrageurs.For instance, for highly rated supranational and quasi-government agencies, which can arbitrage the long-term basis thanks to their top credit rating by issuing bonds in US dollars at attractive rates and then swapping them out, is more closely related to the costs of placing bonds in different currencies. Software bisnis forex. For hedge funds, which rely on collateralised markets to fund CIP arbitrage, the price and availability of repo market funding will play a significant role.The implied rate is the difference between the spot interest rate and the interest rate for the forward or futures delivery date. Or, if the spot price for a currency is 1.050 and the futures contract price is 1.110, the difference of 5.71% is the implied interest rate. dollar deposit rate is 1% for spot and 1.5% in one year's time, the implied rate is the difference of 0.5%.
The implied interest rate gives investors a way to compare returns across investments and evaluate the risk and return characteristics of that particular security.An implied interest rate can be calculated for any type of security that also has an option or futures contract.To calculate the implied rate, take the ratio of the forward price over the spot price. Raise that ratio to the power of 1 divided by the length of time until the expiration of the forward contract. Divide the futures price of $71 by the spot price of$68.Since this is a one-year contract, the ratio is simply raised to the power of 1 (1 / time).Subtract 1 from the ratio and find the implied interest rate of 4.41 percent.
### Computing Swap Points and Forward Prices Forex Trader.
[KEYPART-[URAND-102-201]]
While forward pricing typically represents the interest-rate differential between two countries, they are also influenced by demand and supply for the currencies. There is speculation that the RBI has been intervening heavily in the spot market to buy dollars to curb rupee appreciation, and then doing what is commonly known as a sell-buy swap in the forwards market to offset the liquidity impact, according to Anindya Banerjee, a currency strategist at Kotak.As companies pay taxes, it drains cash from India’s financial system, leading to a mismatch in the foreign-exchange market, which then pushes up the forward premiums, according to Madhavi Arora, an economist at Edelweiss Securities Ltd. Under the arrangement, the RBI buys dollars in the spot market to inject rupee.To offset the inflows of the Indian currency, it goes to the forwards market and sells these greenbacks, for say a three-month period, to banks. Calculate the ratio of the forward price over the spot price by dividing 1.2655 by 1.2291.Since this is a one-year forward contract, the ratio is simply raised to the power of 1.Subtracting 1 from the ratio of the forward price over the spot price results in an implied interest rate of 2.96 percent.
### ILLUSTRATING SPOT AND FORWARD INTEREST RATES Learning.
Implied AUD Interest Rate from USDAUD FX Swap and USD Interest Rate.
Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. I am trying to calculate the implied interest rate of one currency (C2) using an FX swap and the interest rate of another currency (C1 - base).I have the following: Spot: 7.7587 (C2 per unit C1) Buy Notional (spot) C1: 12,888,757.14 Sell Notional (spot) C2: 100,000,000.00 Start date: 6-May-13 End date: 7-May-14 Buy Notional (forward) C2: 100,000,000.00 Sell Notional (forward) C1: 12,905,390,58 Forward FX rate: 7.7487 I have a borrowing in C1 for 0.9650% for the year.Using interest rate parity: $$F_0 = S_0 \frac$$I solve for $r_ = 0.8349\%$. C1 is USD C2 is HKD (I believe these are the correct day-count convention based on a paper by UBS). Broker unternehmen gründen. However, I am told that the right answer is $0.8486\%$. EDIT If I use ACT/360 for C1 and ACT/365 for C2 with $ACT=365$ I get actually pretty close $(0.8483\%)$. Which should be the implied interest rate in currency C1. By Subhadip Sircar An anomaly in India’s currency forwards market is piquing the curiosity of traders.
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2020-07-10 05:11:08
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https://www.semanticscholar.org/paper/The-QCD-renormalization-group-equation-and-the-of-Wu-Shen/c9d317307ddd52c05fa51e420338c21eced85312
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# The QCD renormalization group equation and the elimination of fixed-order scheme-and-scale ambiguities using the principle of maximum conformality
@article{Wu2019TheQR,
title={The QCD renormalization group equation and the elimination of fixed-order scheme-and-scale ambiguities using the principle of maximum conformality},
author={Xing-Gang Wu and Jian-Ming Shen and Bo-Lun Du and Xu-Dong Huang and Sheng-quan Wang and Stanley J. Brodsky},
journal={Progress in Particle and Nuclear Physics},
year={2019}
}
• X. Wu, +3 authors S. Brodsky
• Published 2019
• Physics
• Progress in Particle and Nuclear Physics
Abstract The conventional scale setting approach to fixed-order perturbative QCD (pQCD) predictions is based on a guessed renormalization scale, usually taking as the one to eliminate the large log-terms of the pQCD series, together with an arbitrary range to estimate its uncertainty. This ad hoc assignment of the renormalization scale causes the coefficients of the QCD running coupling at each perturbative order to be strongly dependent on the choices of both the renormalization scale and the… Expand
A novel determination of non-perturbative contributions to Bjorken sum rule
• Physics
• 2021
In the present paper, we first give a detailed study on the pQCD corrections to the leading-twist part of BSR. Previous pQCD corrections to the leading-twist part derived under conventionalExpand
The scheme-independent generalized Crewther relation in QCD using the single-scale approach of the principle of maximum conformality.
• Physics
• 2020
In the paper, we study the generalized Crewther Relation (GCR) between the Adler function ($D$) and Gross-Llewellyn Smith sum rule for the polarized deep-inelastic electron scattering ($C^{\rm GLS}$)Expand
Z-boson hadronic decay width up to $${{\mathcal {O}}}(\alpha _s^4)$$-order QCD corrections using the single-scale approach of the principle of maximum conformality
In the paper, we study the properties of the Z-boson hadronic decay width by using the $$\mathcal {O}(\alpha _s^4)$$ -order quantum chromodynamics (QCD) corrections with the help of the principle ofExpand
Precise determination of the top-quark pole mass from the $t\bar{t}$ production cross-section at the LHC.
• Physics
• 2020
The top-quark is the heaviest known particle of the Standard Model (SM); its heavy mass plays a crucial role in testing the electroweak symmetry breaking mechanism and for searching for new physicsExpand
Computation of effective front form Hamiltonians for massive Abelian gauge theory
Renormalization group procedure for effective particles (RGPEP) is applied in terms of a second-order perturbative computation to an Abelian gauge theory, as an example of application worth studyingExpand
QCD effective charges and the structure function F2 at small-x: Higher twist effects
• Physics
• 2020
Abstract We consider the effect of higher twist operators of the Wilson operator product expansion in the structure function F 2 ( x , Q 2 ) at small-x, taking into account QCD effective chargesExpand
Scale-Fixed Predictions for $\gamma + \eta_c$ production in electron-positron collisions at NNLO in perturbative QCD
• Physics
• 2020
In the paper, we present QCD predictions for $\eta_{c} + \gamma$ production at an electron-position collider up to next-to-next-to-leading order (NNLO) accuracy without renormalization scaleExpand
The $$\Upsilon (1S)$$ leptonic decay using the principle of maximum conformality
• Physics
• The European Physical Journal C
• 2019
In the paper, we study the $$\Upsilon (1S)$$ leptonic decay width $$\Gamma (\Upsilon (1S)\rightarrow \ell ^+\ell ^-)$$ by using the principle of maximum conformality (PMC) scale-setting approach.Expand
The $Z$-boson hadronic decay width up to $\mathcal{O}(\alpha_s^4)$-order QCD corrections using the principle of maximum conformality
• Physics
• 2020
In the paper, we study the properties of the $Z$-boson hadronic decay width by using the $\mathcal{O}(\alpha_s^4)$-order QCD corrections with the help of the principle of maximum conformality (PMC).Expand
Color Confinement and Supersymmetric Properties of Hadron Physics from Light-Front Holography
I review applications of superconformal algebra. light-front holography, and an extended form of conformal symmetry to hadron spectroscopy and dynamics. QCD is not supersymmetrical in the traditionalExpand
#### References
SHOWING 1-10 OF 389 REFERENCES
Scale Setting Using the Extended Renormalization Group and the Principle of Maximum Conformality: the QCD Coupling Constant at Four Loops
• Physics
• 2012
A key problem in making precise perturbative QCD predictions is to set the proper renormalization scale of the running coupling. The extended renormalization group equations, which express theExpand
The renormalization scale-setting problem in QCD
• Physics
• 2013
A key problem in making precise perturbative QCD predictions is to set the proper renormalization scale of the running coupling. The conventional scale-setting procedure assigns an arbitrary rangeExpand
Renormalization group invariance and optimal QCD renormalization scale-setting: a key issues review.
• X. Wu, +4 authors M. Mojaza
• Physics, Medicine
• Reports on progress in physics. Physical Society
• 2015
A detailed comparison of the PMC and PMS procedures is presented by analyzing two physical observables R(e+e-) and [Formula: see text] up to four-loop order in pQCD. Expand
Extending the predictive power of perturbative QCD
• Physics
• The European Physical Journal C
• 2019
The predictive power of perturbative QCD (pQCD) depends on two important issues: (1) how to eliminate the renormalization scheme-and-scale ambiguities at fixed order, and (2) how to reliably estimateExpand
Self-consistency requirements of the renormalization group for setting the renormalization scale
• Physics
• 2012
In conventional treatments, predictions from fixed-order perturbative QCD calculations cannot be fixed with certainty due to ambiguities in the choice of the renormalization scale as well as theExpand
Eliminating the renormalization scale ambiguity for top-pair production using the principle of maximum conformality.
• Physics, Medicine
• Physical review letters
• 2012
The PMC procedure is applied to obtain next-to-next- to-leading-order (NNLO) predictions for the tt-pair production at the Tevatron and LHC colliders and it is verified that the initial scale independence of the PMC prediction is satisfied to high accuracy at the NNLO level. Expand
Novel All-Orders Single-Scale Approach to QCD Renormalization Scale-Setting
• Physics
• 2017
The Principle of Maximal Conformality (PMC) provides a rigorous method for eliminating renormalization scheme-and-scale ambiguities for perturbative QCD predictions. The PMC uses the renormalizationExpand
Systematic Scale-Setting to All Orders: The Principle of Maximum Conformality and Commensurate Scale Relations
• Physics
• 2014
We present in detail a new systematic method which can be used to automatically eliminate the renormalization scheme and scale ambiguities in perturbative QCD predictions at all orders. We show thatExpand
Renormalization scheme dependence of high-order perturbative QCD predictions
• Physics
• 2018
Conventionally, one adopts typical momentum flow of a physical observable as the renormalization scale for its perturbative QCD (pQCD) approximant. This simple treatment leads to renormalizationExpand
Renormalization-scheme-invariant QCD and QED: The method of effective charges
We review, extend, and give some further applications of a method recently suggested to solve the renormalization-scheme-dependence problem in perturbative field theories. The use of a couplingExpand
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2021-10-22 09:56:47
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https://math.stackexchange.com/questions/3839729/how-to-solve-this-question-of-parabola-by-only-using-euclidean-geometry
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# how to solve this question of parabola by only using euclidean geometry?
The original question was to prove that $$A_1A_2$$ and $$B_1B_2$$ intersect at directrix given $$A_1B_1$$ and $$A_2B_2$$ are focal chords of the parabola.
I solved it using the parametric form of parabola and got the answer. But then when I tried using geometry, I'm stuck, in the figure A1B1 is a focal chord, and A2 is a point on parabola, A2A1 intersects directrix at O. Then B2 is the intersection of A1F and B1O,B2' is foot of perp of B2 on directrix. The main target was to prove B2F=B2B'.how am I supposed to prove that the marked angles $$x$$ in the figure are equal? I'm stuck here!
Let $$O$$ be the intersection between line $$A_1A_2$$ and the directrix. From the similitude of triangles $$OA_1A_1'$$ and $$OA_2A_2'$$ one gets: $$OA_1:OA_2=A_1A_1':A_2A_2'=FA_1:FA_2.$$ By the converse of exterior angle bisector theorem we then obtain that line $$FO$$ is the bisector of external angle $$\angle A_1FB_2$$ of triangle $$FA_1A_2$$.
The same reasoning can be repeated for triangle $$FB_1B_2$$: the line joining focus $$F$$ with the intersection of line $$B_1B_2$$ with the directrix is also the bisector of exterior angle $$\angle B_2FA_1$$. But those exterior angles coincide and have thus the same bisector: it follows that line $$B_1B_2$$ also intersects the directrix at point $$O$$. That completes the proof.
EDIT.
The argument given above can be expanded a bit, to prove some interesting properties of the tangents of a parabola.
If we let $$A_2\to A_1$$ in figure above, then lines $$A_1A_2$$ and $$B_1B_2$$ become the lines tangent to the parabola at $$A_1$$ and $$B_1$$ respectively, endpoints of a focal chord (see figure below). The results proven above still hold, hence those tangents meet at a point $$O$$ on the directrix, and line $$FO$$ is the bisector of $$\angle A_1FB_1$$, i.e. $$FO\perp A_1B_1$$.
It follows that triangles $$A_1FO$$ and $$A_1A_1'O$$ are congruent, implying that the line tangent at $$A_1$$ is the bisector of $$\angle A_1'A_1F$$, a first well known result:
The line tangent to a parabola at a point $$P$$ is the bisector of the angle formed by the line passing through $$P$$ and the focus, with the line through $$P$$ parallel to the axis.
Finally, from $$\angle A_1'A_1F+\angle B_1'B_1F=180°$$ one gets $$\angle OA_1F+\angle OB_1F=90°$$ and consequently $$\angle A_1OB_1=90°$$, which is another famous property:
The lines tangent to a parabola at the endpoints of a focal chord are perpendicular and meet on the directrix.
• Great proof! I wonder if one can also infer somehow that $\angle A_1 O B_1=90^{\circ}$ – highgardener Sep 25 '20 at 9:48
• indeed a great proof! Didn't think of it that way....@highgardner won't that be true only when A1O and B1O are tangents?? – Karthik Nambiar Sep 25 '20 at 10:10
• @highgardener That angle becomes $90°$ only in the limit $A_1\to A_2$, when lines $A_1A_2$ and $B_1B_2$ become tangent to the parabola. And this proof is indeed the first step for proving that. – Intelligenti pauca Sep 25 '20 at 10:54
• @KarthikNambiar Yes, my bad. Was misled by the figure. Considering the standard parabola $y^2-4ax=0$ and two points $A_1=(at_1^2, 2at_1)$ and $A_2=(at_2^2, 2at_2)$ on it, where $t_1>t_2>0$ (same side of axis). The slope of this chord is $\frac{2}{t_1+t_2}$. Consider the points $B_1, B_2$, their focal opposites. Their parameters are $\frac{-1}{t_1}$ and $\frac{-1}{t_2}$ resp. Thus the slope as before is: $\frac{-2}{\frac{1}{t_1}+\frac{1}{t_2}}$. Their product will be $-1$ only when $t_1=t_2$ – highgardener Sep 25 '20 at 11:04
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2021-05-12 06:14:32
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https://www.allaboutcircuits.com/news/californias-piezoelectric-energy-harvesting-aims-turn-traffic-green-energy/
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Is high-volume traffic a good source for energy harvesting? The California Energy Commission intends to find out. Here's how piezoelectric devices could turn nightmare traffic into the dream of renewable energy.
Energy harvesting methods have been growing in popularity lately, especially with regards to renewable sources of energy. Renewable sources are not just limited to solar; there are others such as wind, geothermal, and tidal. The only problem with all these forms of energy is that they are either location-dependent or weather-dependent and are not always available.
But what about human activity as a source of energy? Could energy be harvested from our day to day activities to help power our modern society?
There's a particular form of human activity that California has in spades: traffic. California has decided to run an experiment to see if energy can be harvested from vehicle vibrations on the road.
### The Piezoelectric Idea
California has funded an experiment whereby roads which experience heavy traffic will be fitted with piezoelectric transducers to convert the vibration generated by vehicles (both stationary and in motion), into electricity. This electricity can then be fed into the grid to provide power for everything including buildings, homes, and street lights.
Piezoelectric energy harvesting is not “new” technology. It's actually been around for some time. In fact, energy harvesting from foot traffic has not only been proven but is in use in some places. For example, several train stations in Tokyo use piezoelectric energy to generate the power needed to run the ticket machines and electronic displays. Another example is a Dutch nightclub which uses piezoelectric tiles on the floor to power lights.
##### Tokyo uses piezoelectric tiles in some of its stations to power billboards. Image courtesy of the Japan Railways Group
However, will such a system work in traffic conditions and will enough electricity be generated to compensate for the cost?
### Cost For Piezoelectric Devices and Energy Output
Piezoelectric devices are more commonly used for generating sounds and measuring vibrations as opposed to generating electricity for energy consumption. This makes finding energy data for piezoelectric energy harvesting rather difficult. Luckily, there is a transducer for sale that is specifically designed for generating electricity in energy harvesters. Here are some of the specs given:
• Open circuit voltage at rated deflection = 20.9V
• Closed circuit current at rated deflection = 57 microamps
• Power output at rated deflection = 7.1mW
• Operating temperature = -20°C to 90°C
• Dimension = 70mm x 31.8mm (height 1.5mm)
• 1 piece = $301 : 100 pieces =$132
Given this data, we can roughly project that the amount of energy generated by piezo devices (50Hz vibration) is 3.189 W / square meter.
The cost for a single transducer (if purchased in bulk) would be $132 per device which gives a total cost of$60,000 per square meter (450 devices in a square meter). While this may seem like an unusually hefty sum of money, this is for a square meter of material that can stretch a long way if made narrow. In the case of California, the idea is to create a 60 meter stretch of road and use 2 cm wide piezoelectric generators in stacks. However, this suggests that the piezo devices used in the project will be of a more common type found in everyday electronics as opposed to devices specifically designed for energy harvesting (such as the one produced by Piezo Systems Inc).
Ler's look at an example where the pavement that is to be fitted with energy harvesters will be a complete strip of piezo material. Taking a pavement width of 1.5 meters, the total area of such a strip would be 90 meters square which results in a piezo cost of $5,400,000 and a total energy generation (assuming that all paving is occupied by people jumping at 50Hz) of 287.01W. It's more likely, however, that piezo devices would be scattered so, if we change the example to require a 10% coverage of piezo devices, the cost would be$540,000 with an overall energy output of 28.7W.
Considering that a single 1KW solar panel can cost as little as \$1000, the price for harvesting electricity from walking seems extreme.
### A Place for Cars?
There are issues beyond the costs of such a system; there is the practical side to consider, too.
Piezo generators work well in night clubs and train stations because people are generally unlikely to destroy their pathways. Since piezo devices rely on deflection to generate electricity, the footpath needs to allow for movement and vibration to maximize efficiently of the energy harvesting devices. This is not a problem for pedestrian areas as people are not very destructive, but roads that handle vehicle traffic are a different matter.
##### Piezo discs: Great for buzzing, not so much for cars. Image courtesy of Adam Wysocki [CC-BY-SA-3.0]
Roads must be built to resist damage from tires and debris if they are to be safe and reliable. As a result, roads have a tendency to not move around and deform under impact (instead they break into pieces). This makes placing transducers under roads problematic on two fronts. Either the road is built to allow vibrational energy to reach the transducers which potentially makes the road weaker or the transducers do not reach anywhere near their potential making them a costly investment.
Also, the world is shifting from internal combustion engines to electric batteries. Combustion engines tend to vibrate a lot, even when stationary. This is what the Californian system would rely on during heavy traffic. However, electric cars do not vibrate nearly as much when stationary, thus rendering the energy harvesters less effective.
This raises the question of whether this piezo road idea will age well as more and more consumers turn to electric cars. As a matter of fact, California has arguably been the global leader in the electric vehicle movement, giving China a run for its money in electric car sales. This isn't to say that electric vehicles would render traffic energy harvesting moot, but it's a factor that should be considered when investing millions of state dollars into infrastructure.
### Summary
Perhaps the researchers working on this proposal will use a new type of device that can generate more electricity or maybe the stacks will be much larger. It's difficult to project how effective this system will be without knowing more details. It's difficult to imagine, however, how such a system could be successful enough to warrant expanding.
• AEKron 2017-06-19
I thought solar feakin’ roadways were bad enough, but piezo freakin’ roadways? why not stack the two together, and get double government grants?
• I do not Consent 2017-06-22
Be aware though, that these piezo devices are also used as microphones.
Imagine the possibilities of microphones everywhere, matched with voice recognition systems…
Arrays of microphones, which can determine where a person is, where they are, and how fast they are going - imagine the applications…
• ozjon69 2017-06-22
Well, it doesn’t read like a practical concept - certainly not with Piezo generators at the prices suggested.
However, MUCH cheaper piezo generators can be made with pieces of solid-dielectric coaxial cable, made so that the dielectric has high residual stress.
Maybe, that could improve the economics enough to make the concept viable?
This is proven technology (at low power levels) - stressed cable has been used for many years on intruder detecting security fences.
• pmd34 2017-06-23
So energy from nowhere huh?! Oh but wait, the deflection causes added road resistance, so the car fuel consumption has to go up to compensate. Still im sure they will get lots of research funding from the usual committees! Maybe you could make them out of graphine! Thats a sure fire application winner!
• viki2000 2017-06-23
I find this idea not very smart and without a real future.
The piezo ceramic membranes or buzzers used to produce a sound when a voltage is applied, the known application, suffers of low lifetime due to one reason: vibration. Most of the time the contact between membrane and the wire is cracked, the wire are broken right in that point. We speak here about thousands of hour’s lifetime under continuous vibration.
Here is the opposite effect, with vibration due to mechanical tension in the membrane to get the voltage, but the connection is the same, the continuous vibration will reduce the lifetime of the transducer. It is not going to work for long time under continuous vibrations.
• viki2000 2017-06-23
Sorry for the wrong place to place the comment.
• viki2000 2017-06-23
I find this idea not very smart and without a real future.
The piezo ceramic membranes or buzzers used to produce a sound when a voltage is applied, the known application, suffers of low lifetime due to one reason: vibration. Most of the time the contact between membrane and the wire is cracked, the wire are broken right in that point. We speak here about thousands of hour’s lifetime under continuous vibration.
Here is the opposite effect, with vibration due to mechanical tension in the membrane to get the voltage, but the connection is the same, the continuous vibration will reduce the lifetime of the transducer. It is not going to work for long time under continuous vibrations.
• theace 2017-06-23
This idea is so stupid that it cannot be described as it should using words (and yes, it’s on the same level as solar freakin’ roadways), but I’ll just make two remarks:
1) you want your roads to be as stable and as long lasting as possible. Promoting something as vibration is extraordinary counter productive. You don’t want your roads to vibrate at all. Because vibration goes to potholes and in order to fix them you’ll spend more money than what you get on the energy you produce.
2) where would that energy come from? you guessed it: from the cars. So, you’ll get the energy from burning fuel to the car to the road to the power grid. How stupid is that? So my car has to bounce up and down to give energy to the road. So I have to spend money on fuel to power the road with losses instead of producing more energy from the same fuel in a power plant.
Yes, yes, the vibration could be collected from static cars (engine vibration), but wouldn’t be efficient to put this vibration generators on the cars themselves and power the battery instead of using fuel to spin the alternator?
An the discussion can continue, but is pointless (see solar roadways).
Now I’m asking the reporters on this site to stop posting this kind of bullshit on an otherwise interesting site. Just because it contains electronics, doesn’t mean it actually has a place here. You can do all sort of things with electronics, but this doesn’t mean that they belong here.
• pmd34 2017-06-23
I wouldn’t be too hard on the site, its important to know what people are up to, but more a brief “look at the crazy idea” angle, or debunking it would be better!
• pmd34 2017-06-23
How do they get 7.1mW from Open circuit voltage 20.9V & Closed circuit current 57 microamps? Surely that’s an absolute maximum 1.1mW?!
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2017-06-23 10:36:49
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https://mathoverflow.net/questions/247266/cohen-macaulay-ring-of-sections
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# Cohen-Macaulay ring of sections
Given a smooth algebraic variety $X$ and a Q-Cartier Q-divisor $D$ which is semiample and big. Assume that $Y=Spec ( \oplus_{m\geq 0}H^0 (X, mD))$ is Cohen-Macaulay. Do it follows that $H^i (X,mD)=0$ for $0 <i<\dim (X)$?
I know the answer is yes if $D$ is Q-ample and this characterize Cohen-Macaulayness for ring of sections of Q-ample divisors.
If this is true, what is a reference in the literature?
This is not true in this form.
Think about it this way: For simplicity let us assume that $D$ is already basepoint-free (and to give a counter-example this is certainly enough), so there exists a proper surjective birational morphism $f:X\to Z$ and a very ample line bundle $\mathscr L$ on $Z$ such that $\mathscr N:=\mathscr O_X(D)=f^*\mathscr L$. We may assume that $Z$ is normal and $f_*\mathscr O_X\simeq \mathscr O_Z$. Then for any $m\in\mathbb Z$, $$\tag{\star} H^0(X, \mathscr N^{\otimes m})\simeq H^0(Z, \mathscr L^{\otimes m}),$$ so in particular $Y\simeq \rm{Spec} (\oplus_m H^0(Z, \mathscr L^{\otimes m}))$ and since $\mathscr L$ is very ample, this means that $Y$ is a cone over $Z$.
Then you can connect the vanishing of the cohomology groups $H^i(Z, \mathscr L^{\otimes m}))$ to depth conditions on $Z$. In other words, it follows almost as you desire that $$H^i(Z, \mathscr L^{\otimes m})=0$$ for all $m\in \mathbb Z$ and $0<i<\dim Z=\dim X$. However, this is exactly where the problem comes from.
While we do have $(\star)$ for $i=0$, in general we do not have that for arbitrary $i$'s. Instead what we have is a Leray spectral sequence, which essentially says that
$$%\tag{\star} H^i(X, \mathscr N^{\otimes m})\simeq H^i(Z, \mathscr L^{\otimes m}\otimes Rf_*\mathscr O_X).$$
So, as long as $\mathscr O_X$ have higher direct images, you're out of luck.
However, this tells you how to fix the statement. You need to assume that $Z$ has rational singularities. Based on the above you should be able to put together a proof of that.
Finally, here is an explicit simple example when your desired statement fails: Let $Z\subseteq \mathbb P^3$ be a normal projective surface with a non-rational singularity. For example a cone over an elliptic curve. Let $f:X\to Z$ be a resolution of singularities. Choose a very very ample line bundle $\mathscr L$ on Z such that it has no higher cohomology and let $D$ be the divisor corresponding to $f^*\mathscr L$. Clearly this is semi-ample and big. Since $Z$ is a hypersurface, the cone over $Z$ (corresponding to the very ample line bundle $\mathscr L$) is Cohen-Macaulay, so your assumption on $Y$ is satisfied. Now let $\mathscr N:=\mathscr O_X(D)=f^*\mathscr L$ and consider the exact sequence provided by the obvious Leray spectral sequence: $$0 \to H^1(Z, \mathscr L) \to H^1(X, \mathscr N) \to H^0(Z, \mathscr L \otimes R^1f_*\mathscr O_X) \to H^2(Z, \mathscr L) \to \dots$$ By assumption $H^1(Z, \mathscr L) =H^2(Z, \mathscr L) =0$, so $H^1(X, \mathscr N) \simeq H^0(Z, \mathscr L \otimes R^1f_*\mathscr O_X)$, but the latter is strictly non-zero by the assumption that $Z$ has a non-rational singularity (which is necessarily isolated).
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2021-10-17 09:48:47
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https://joningram.org/questions/Algebra/200259
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# Graph f(x)=x^3
Graph f(x)=x^3
Find the point at .
Replace the variable with in the expression.
Simplify the result.
Raise to the power of .
The final answer is .
Convert to decimal.
Find the point at .
Replace the variable with in the expression.
Simplify the result.
Raise to the power of .
The final answer is .
Convert to decimal.
Find the point at .
Replace the variable with in the expression.
Simplify the result.
Raising to any positive power yields .
The final answer is .
Convert to decimal.
Find the point at .
Replace the variable with in the expression.
Simplify the result.
One to any power is one.
The final answer is .
Convert to decimal.
Find the point at .
Replace the variable with in the expression.
Simplify the result.
Raise to the power of .
The final answer is .
Convert to decimal.
The cubic function can be graphed using the function behavior and the points.
The cubic function can be graphed using the function behavior and the selected points.
Falls to the left and rises to the right
Do you know how to Graph f(x)=x^3? If not, you can write to our math experts in our application. The best solution for your task you can find above on this page.
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2023-03-24 13:11:48
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https://calculator.academy/coupon-rate-calculator/
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Enter the total annual coupon payment, and the par value of the bond into the calculator to determine the coupon rate. This calculator can also evaluate the annual payment or par value if the other variables are known.
## Coupon Rate Formula
The following is the coupon rate formula:
CR = AP / PV * 100
• Where CR is the coupon rate (%)
• AP is the annual coupon payment ($) • PV is the par value of the bond ($)
To calculate a coupon rate, divide the annual coupon payment by the par value of the bond, then multiply by 100.
## Coupon Rate Defintion
A coupon rate is defined as the rate of interest paid to the bondholders by the bond issuers of any given bond.
## Can a coupon rate change?
A coupon rate can change if the bond or security is variable rate security. The most common type of bond is a fixed rate bond, which would not have its coupon rate changed.
## What affects coupon rate?
The formula above shows that only the annual coupon payment and the par value of the bond affect the coupon rate. Those two variables, however, can be influenced by other factors at the time of purchase.
## Is coupon rate and yield to maturity the same?
A coupon rate and yield to maturity can be the same if the bond is purchased at face value, but not if the bond is purchased at more or less than the face value.
## Why is the coupon rate higher than the yield?
A coupon rate can be higher than a yield when an investor purchases a bond at a premium over its face value. A coupon rate can be lower than a yield when the bond is purchased at a lower price than face value.
## How to calculate coupon rate?
1. First, determine the face value of the bond. Calculate or determine the market price of the face value of the bond.
2. Next, determine the annual coupon payment received.
3. Finally, calculate the annual coupon rate using the formula shown above.
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2022-12-02 11:54:53
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https://msp.org/gt/2018/22-3/b06.xhtml
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#### Volume 22, issue 3 (2018)
Download this article For screen For printing
Recent Issues
The Journal About the Journal Editorial Board Subscriptions Editorial Interests Editorial Procedure Submission Guidelines Submission Page Ethics Statement ISSN (electronic): 1364-0380 ISSN (print): 1465-3060 Author Index To Appear Other MSP Journals
Group trisections and smooth $4$–manifolds
### Aaron Abrams, David T Gay and Robion Kirby
Geometry & Topology 22 (2018) 1537–1545
##### Bibliography
1 D Gay, R Kirby, Trisecting 4–manifolds, Geom. Topol. 20 (2016) 3097 MR3590351 2 J Hempel, 3–Manifolds, 86, Princeton Univ. Press (1976) MR0415619 3 F Laudenbach, V Poénaru, A note on 4–dimensional handlebodies, Bull. Soc. Math. France 100 (1972) 337 MR0317343 4 C J Leininger, A W Reid, The co-rank conjecture for 3–manifold groups, Algebr. Geom. Topol. 2 (2002) 37 MR1885215 5 J Meier, T Schirmer, A Zupan, Classification of trisections and the generalized property R conjecture, Proc. Amer. Math. Soc. 144 (2016) 4983 MR3544545 6 J Meier, A Zupan, Genus-two trisections are standard, Geom. Topol. 21 (2017) 1583 MR3650079 7 J Morgan, G Tian, The geometrization conjecture, 5, Amer. Math. Soc. (2014) MR3186136 8 C D Papakyriakopoulos, On Dehn’s lemma and the asphericity of knots, Ann. of Math. 66 (1957) 1 MR0090053 9 J R Stallings, Some topological proofs and extensions of Grusko’s theorem, PhD thesis, Princeton University (1959) MR2612898 10 J Stallings, How not to prove the Poincaré conjecture, from: "Topology seminar" (editors R H Bing, R J Bean), Ann. of Math. Stud. 60, Princeton Univ. Press (1966) 83 MR2906378 11 F Waldhausen, Heegaard–Zerlegungen der 3–Sphäre, Topology 7 (1968) 195 MR0227992
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2020-08-13 00:26:18
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http://math.stackexchange.com/questions/159493/can-there-be-a-function-thats-even-and-odd-at-the-same-time/159500
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# Can there be a function that's even and odd at the same time?
I woke up this morning and had this question in mind. Just curious if such function can exist.
-
In case anyone has forgotten what “even” and “odd” functions are, $f$ is even if $f(x) = f(-x)$ and odd if $-f(x) = f(-x)$. See also Wikipedia on even and odd functions. – Rory O'Kane Jun 17 '12 at 18:54
You might find it interesting that I often used to ask this as an extra credit question on precalculus tests when even/odd function properties were covered, typically worth an extra 3 points on a 100 point scale (so a score of 103/100 was possible). I'd usually get about 2 to 5 students getting the extra points (out of a total of maybe 25-35 students) in a U.S. college precalculus class, and about half the class getting the extra points in U.S. honors level high school classes I used to teach. – Dave L. Renfro Jun 18 '12 at 15:56
Others have mentioned that $f(x)=0$ is an example. In fact, we can prove that it is the only example of a function from $\mathbb{R}\to \mathbb{R}$ (i.e a function which takes in real values and outputs real values) that is both odd and even. Suppose $f(x)$ is any function which is both odd and even. Then $f(-x) = -f(x)$ by odd-ness, and $f(-x)=f(x)$ by even-ness. Thus $-f(x) = f(x)$, so $f(x)=0.$
-
Of course, one could argue that restrictions of the constant $0$ function to different domains symmetric about the origin are different functions, set-theoretically speaking. – Cameron Buie Jun 17 '12 at 15:30
@CameronBuie That is true, I will make my answer more precise to indicate this. Thank you. – Ragib Zaman Jun 17 '12 at 15:31
Funny, I never thought of f(x) = 0 as a possibility. Thanks for the answers everyone! – bodacydo Jun 17 '12 at 21:06
If $K$ is a field of characteristic 2, every function $K\to K$ is both even and odd.
-
i'm sorry, wouldn't that be "unequal to 2"? – akkkk Jun 17 '12 at 15:31
@Auke: No. I won't spoil the joke by spelling it out, sorry. – Harald Hanche-Olsen Jun 17 '12 at 15:40
Actually, you don't even need a field, any ring of characteristic 2 will do. – Ilmari Karonen Jun 17 '12 at 15:42
@HaraldHanche-Olsen, oh, I am sorry, I misread your answer, you are completely right :) nice one – akkkk Jun 17 '12 at 15:43
This is a wonderful answer! – Edward Hughes Jun 17 '12 at 23:52
Yes. The constant function $f(x) = 0$ satisfies both conditions.
Even: $$f(-x) = 0 = f(x)$$
Odd: $$f(-x) = 0 = -f(x)$$
Furthermore, it's the only real function that satisfies both conditions:
$$f(-x) = f(x) = -f(x) \Rightarrow 2f(x) = 0 \Rightarrow f(x) = 0$$
-
Hint $\rm\ f\:$ is even and odd $\rm\iff f(x) = f(-x) = -f(x)\:\Rightarrow\: 2\,f(x) = 0.\:$ This is true if $\rm\:f = 0,\:$ but may also have other solutions, e.g. $\rm\:f = n\:$ in $\rm\:\mathbb Z/2n =\:$ integers mod $\rm 2n,$ where $\rm\: -n \equiv n.$
-
+1, but note that your last $\iff$ applies (in the backwards, i.e. 'if' direction) only to $f(x) = -f(x)$, and not to the part where $f(-x)$ equals both of them. – ShreevatsaR Jun 17 '12 at 18:04
Yes, I meant to write $\:\Rightarrow\:$ but it was lost in editing. Now fixed. Thanks. – Bill Dubuque Jun 17 '12 at 18:19
Suppose $f$ odd an even. Let $x \in D$ ( D is set definition of $f$) then you have : $f(x)=f(-x)=-f(x)$. What can you conclude about $f$ ?
As other people have mentioned already, the real function $f(x)$ which maps every real number to zero (i.e.$f(x) = 0 \space \forall x \in \mathbb{R}$) is both even and odd because $$f(x) - f(-x) = 0 \space \space , f(x)+f(-x) = 0\space \forall x \in \mathbb{R} .$$ Also it is the only function defined over $\mathbb{R}$ to possess this property.
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2014-04-20 22:22:57
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https://plainmath.net/32410/critical-thinking-suppose-you-take-random-sample-from-normal-population
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# Critical Thinking Suppose you take a random sample from a normal population and
Critical Thinking Suppose you take a random sample from a normal population and you want to determine whether there is sufficient statistical evidence to claim that the population variance differs from a corresponding variance specified in a goverment contract. Which type of test is appropriate, a test of one variance or a test of two variances?
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Gennenzip
The claim is to test whether the population variance is different from government contract variance or not. In the study, the population variance is compared with the fixed variance value. This shows that, using the test of one variance is appropriate.
The test of two variances is used for comparing variances of two populations. This shows that, using test of two variances is not appropriate.
Hence, the test that would be appropriate to determine whether the population variance differs from variance specified in a goverment contract is test of one variance.
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2022-08-09 11:14:12
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https://labs.tib.eu/arxiv/?author=S.%20Niemi
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• ### VIS: the visible imager for Euclid(1608.08603)
Aug. 30, 2016 astro-ph.IM
Euclid-VIS is the large format visible imager for the ESA Euclid space mission in their Cosmic Vision program, scheduled for launch in 2020. Together with the near infrared imaging within the NISP instrument, it forms the basis of the weak lensing measurements of Euclid. VIS will image in a single r+i+z band from 550-900 nm over a field of view of ~0.5 deg2. By combining 4 exposures with a total of 2260 sec, VIS will reach to deeper than mAB=24.5 (10sigma) for sources with extent ~0.3 arcsec. The image sampling is 0.1 arcsec. VIS will provide deep imaging with a tightly controlled and stable point spread function (PSF) over a wide survey area of 15000 deg2 to measure the cosmic shear from nearly 1.5 billion galaxies to high levels of accuracy, from which the cosmological parameters will be measured. In addition, VIS will also provide a legacy dataset with an unprecedented combination of spatial resolution, depth and area covering most of the extra-Galactic sky. Here we will present the results of the study carried out by the Euclid Consortium during the period up to the Critical Design Review.
• ### On Scale-Dependent Cosmic Shear Systematic Effects(1507.05334)
July 19, 2015 astro-ph.CO
In this paper we investigate the impact that realistic scale-dependence systematic effects may have on cosmic shear tomography. We model spatially varying residual ellipticity and size variations in weak lensing measurements and propagate these through to predicted changes in the uncertainty and bias of cosmological parameters. We show that the survey strategy - whether it is regular or randomised - is an important factor in determining the impact of a systematic effect: a purely randomised survey strategy produces the smallest biases, at the expense of larger parameter uncertainties, and a very regularised survey strategy produces large biases, but unaffected uncertainties. However, by removing, or modelling, the affected scales (l-modes) in the regular cases the biases are reduced to negligible levels. We find that the integral of the systematic power spectrum is not a good metric for dark energy performance, and we advocate that systematic effects should be modelled accurately in real space, where they enter the measurement process, and their effect subsequently propagated into power spectrum contributions.
• ### Euclid space mission: a cosmological challenge for the next 15 years(1501.04908)
Jan. 20, 2015 astro-ph.CO
Euclid is the next ESA mission devoted to cosmology. It aims at observing most of the extragalactic sky, studying both gravitational lensing and clustering over $\sim$15,000 square degrees. The mission is expected to be launched in year 2020 and to last six years. The sheer amount of data of different kinds, the variety of (un)known systematic effects and the complexity of measures require efforts both in sophisticated simulations and techniques of data analysis. We review the mission main characteristics, some aspects of the the survey and highlight some of the areas of interest to this meeting
• ### The Cosmic Origins Spectrograph: On-Orbit Instrument Performance(1012.5827)
Dec. 28, 2010 astro-ph.IM
The Cosmic Origins Spectrograph (COS) was installed in the Hubble Space Telescope in May, 2009 as part of Servicing Mission 4 to provide high sensitivity, medium and low resolution spectroscopy at far- and near-ultraviolet wavelengths (FUV, NUV). COS is the most sensitive FUV/NUV spectrograph flown to date, spanning the wavelength range from 900{\AA} to 3200{\AA} with peak effective area approaching 3000 cm^2. This paper describes instrument design, the results of the Servicing Mission Orbital Verification (SMOV), and the ongoing performance monitoring program.
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2021-04-10 11:31:43
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https://www.jobilize.com/physics3/course/2-3-images-formed-by-refraction-by-openstax?qcr=www.quizover.com
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# 2.3 Images formed by refraction
Page 1 / 5
By the end of this section, you will be able to:
• Describe image formation by a single refracting surface
• Determine the location of an image and calculate its properties by using a ray diagram
• Determine the location of an image and calculate its properties by using the equation for a single refracting surface
When rays of light propagate from one medium to another, these rays undergo refraction, which is when light waves are bent at the interface between two media. The refracting surface can form an image in a similar fashion to a reflecting surface, except that the law of refraction (Snell’s law) is at the heart of the process instead of the law of reflection.
## Refraction at a plane interface—apparent depth
If you look at a straight rod partially submerged in water, it appears to bend at the surface ( [link] ). The reason behind this curious effect is that the image of the rod inside the water forms a little closer to the surface than the actual position of the rod, so it does not line up with the part of the rod that is above the water. The same phenomenon explains why a fish in water appears to be closer to the surface than it actually is.
To study image formation as a result of refraction, consider the following questions:
1. What happens to the rays of light when they enter or pass through a different medium?
2. Do the refracted rays originating from a single point meet at some point or diverge away from each other?
To be concrete, we consider a simple system consisting of two media separated by a plane interface ( [link] ). The object is in one medium and the observer is in the other. For instance, when you look at a fish from above the water surface, the fish is in medium 1 (the water) with refractive index 1.33, and your eye is in medium 2 (the air) with refractive index 1.00, and the surface of the water is the interface. The depth that you “see” is the image height ${h}_{\text{i}}$ and is called the apparent depth . The actual depth of the fish is the object height ${h}_{\text{o}}$ .
The apparent depth ${h}_{\text{i}}$ depends on the angle at which you view the image. For a view from above (the so-called “normal” view), we can approximate the refraction angle $\theta$ to be small, and replace sin $\theta$ in Snell’s law by tan $\theta$ . With this approximation, you can use the triangles $\text{Δ}OPR$ and $\text{Δ}OQR$ to show that the apparent depth is given by
${h}_{\text{i}}=\left(\frac{{n}_{2}}{{n}_{1}}\right){h}_{\text{o}}.$
The derivation of this result is left as an exercise. Thus, a fish appears at 3/4 of the real depth when viewed from above.
## Refraction at a spherical interface
Spherical shapes play an important role in optics primarily because high-quality spherical shapes are far easier to manufacture than other curved surfaces. To study refraction at a single spherical surface, we assume that the medium with the spherical surface at one end continues indefinitely (a “semi-infinite” medium).
What is photoelectric
If you lie on a beach looking at the water with your head tipped slightly sideways, your polarized sunglasses do not work very well.Why not?
it has everything to do with the angle the UV sunlight strikes your sunglasses.
Jallal
this is known as optical physics. it describes how visible light, ultraviolet light and infrared light interact when they come into contact with physical matter. usually the photons or light upon interaction result in either reflection refraction diffraction or interference of the light.
Jallal
I hope I'm clear if I'm not please tell me to clarify further or rephrase
Jallal
what is bohrs model for hydrogen atom
hi
Tr
Hello
Youte
Hi
Nwangwu-ike
hi
Siddiquee
hi
Omar
helo
Mcjoi
what is the value of speed of light
1.79×10_¹⁹ km per hour
Swagatika
what r dwarf planet
what is energy
কাজের একক কী
Jasim
কাজের একক কী
Jasim
friction ka direction Kaise pata karte hai
friction is always in the opposite of the direction of moving object
Punia
A twin paradox in the special theory of relativity arises due to.....? a) asymmetric of time only b) symmetric of time only c) only time
b) symmetric of time only
Swagatika
fundamental note of a vibrating string
every matter made up of particles and particles are also subdivided which are themselves subdivided and so on ,and the basic and smallest smallest smallest division is energy which vibrates to become particles and thats why particles have wave nature
Alvin
what are matter waves? Give some examples
according to de Broglie any matter particles by attaining the higher velocity as compared to light'ill show the wave nature and equation of wave will applicable on it but in practical life people see it is impossible however it is practicaly true and possible while looking at the earth matter at far
Manikant
a centeral part of theory of quantum mechanics example:just like a beam of light or a water wave
Swagatika
Mathematical expression of principle of relativity
given that the velocity v of wave depends on the tension f in the spring, it's length 'I' and it's mass 'm'. derive using dimension the equation of the wave
What is the importance of de-broglie's wavelength?
he related wave to matter
Zahid
at subatomic level wave and matter are associated. this refering to mass energy equivalence
Zahid
it is key of quantum
Manikant
how those weight effect a stable motion at equilibrium
how do I differentiate this equation- A sinwt with respect to t
just use the chain rule : let u =wt , the dy/dt = dy/du × du/dt : wA × cos(wt)
Jerry
I see my message got garbled , anyway use the chain rule with u= wt , etc...
Jerry
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2020-11-29 15:40:20
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https://questioncove.com/updates/4df825680b8b370c28bda1ba
|
Mathematics
OpenStudy (anonymous):
Solve by completing the square: $3x^2 - 6x - 5 = 0$
OpenStudy (anonymous):
OK
OpenStudy (anonymous):
$3x^2-6x=5$ $x^2-2x=5/3$ $(x-1)^2=5/3 +1$ $x-1=\pm \sqrt{(8/3)}$ $x=1\pm \sqrt{8/3}$
OpenStudy (anonymous):
you failed to complete the square...
OpenStudy (anonymous):
why?
OpenStudy (anonymous):
completing the square means that you take b/2 and square it to get c...
OpenStudy (anonymous):
I am pretty sure that I did what the question asked
OpenStudy (anonymous):
completing the square is not shown in your work. you moved the 5 to the other side and divided by three. this is not completing the square
OpenStudy (anonymous):
completing the square would start like: ${-6 \over 2}, -3^2 = 9$
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2021-09-21 19:55:50
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https://codeforces.com/blog/entry/111769?locale=ru
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Автор awoo, история, 2 недели назад,
Привет, Codeforces!
Продолжается серия образовательных раундов в рамках инициативы Harbour.Space University! Подробности о сотрудничестве Harbour.Space University и Codeforces можно прочитать в посте.
Этот раунд будет рейтинговым для участников с рейтингом менее 2100. Соревнование будет проводиться по немного расширенным правилам ICPC. Штраф за каждую неверную посылку до посылки, являющейся полным решением, равен 10 минутам. После окончания раунда будет период времени длительностью в 12 часов, в течение которого вы можете попробовать взломать абсолютно любое решение (в том числе свое). Причем исходный код будет предоставлен не только для чтения, но и для копирования.
Вам будет предложено 6 или 7 задач на 2 часа. Мы надеемся, что вам они покажутся интересными.
Задачи вместе со мной придумывали и готовили Адилбек adedalic Далабаев, Владимир vovuh Петров, Иван BledDest Андросов и Максим Neon Мещеряков. Также большое спасибо Михаилу MikeMirzayanov Мирзаянову за системы Polygon и Codeforces.
Удачи в раунде! Успешных решений!
Также от наших друзей и партнёров из Harbour.Space есть сообщение для вас:
Привет, Codeforces!
Полным ходом идет подготовка ко второму буткемпу по программированию «Hello Muscat 2023», продолжении серии буткемпов «Hello», организованном Harbour.Space University в сотрудничестве с PhazeRo, Gutech, UK Oman Digital Club, Leagues of Code, Gutech CS Club и Codeforces!
Довольно впечатляюще, не так ли? Пришло время глубже погрузиться в мир соревновательного программирования с 8-дневным интенсивом Hello Muscat 2023. Он будет проходить в Маскате, Оман, и онлайн с 8 по 16 марта 2023 года, доступны оба формата участия.
Наш тренерский состав сочетает в себе талант и опыт, в нем участвуют чемпионы мира ICPC, победители и финалисты, а также легендарные имена из области соревновательного программирования: Михаил Мирзаянов MikeMirzayanov, Егор Дубовик 244mhq, Артем Плоткин Rox, Максим Обозный MaksymOboznyi и Николай Будин budalnik.
Буткемп будет разделен на три дивизиона:
• Дивизион A. станет зеркалом Петрозаводского лагеря программирования. Подходит для команд, которые уже прошли квалификацию на Финал ICPC или стремятся к этому.
• Дивизион B. Разработан, чтобы помочь командам подготовиться к следующему сезону региональных соревнований ICPC. Подходит в качестве введения для команд и студентов, которые только начинают свой путь в мир ICPC и соревнований по программированию в целом.
• Дивизион C. Предназначен для новичков в мире соревновательного программирования.
Типы участия: очное и онлайн
_Мы считаем, что участие в нашем буткемпе должно быть доступно для всех команд, где бы они ни находились, и именно поэтому мы сделали очную и онлайн-формы участия. Скидка 20% на раннее бронирование предоставляется университетам и участникам, которые зарегистрируются и оплатят до 31 января 2023 года.
• Очное:
Цена: 1500 € — 1200 €
Что включено: обучение, контесты, доступ к записям лекций, проживание в течение 9 ночей в 4-звездочном отеле Mysk, завтрак и обед, трансфер из гостиницы к месту проведения буткемпа каждый день, развлечения и приветственный подарок.
• Онлайн
Цена: 100 € — 80 €
Что включено: обучение, контесты и доступ к записям лекций.
Узнать больше о Hello Muscat 2023→
Удачи в раунде,
Harbour.Space University Team
• +222
» 2 недели назад, # | +72 Welcome to slow-rating-update round #142!
• » » 2 недели назад, # ^ | 0 True. These rounds and div3 rounds take a lot of time. maybe because of hacking phase.
• » » 2 недели назад, # ^ | 0 Welcome to slow-rating-update round #142!
• » » 2 недели назад, # ^ | +29 Me after solving ABCD:
• » » 2 недели назад, # ^ | ← Rev. 2 → -29 HI!:)
• » » 2 недели назад, # ^ | +2 I'm not aware of this: I don't see this contest taken into account on rating because it's not yet processed?
» 2 недели назад, # | 0 Anybody noticed that quite hell unusual time of 26th Feb contest??!!
• » » 2 недели назад, # ^ | 0 Yes, it's unbelievable.
• » » 2 недели назад, # ^ | ← Rev. 2 → 0 Yeah,12 at night
» 2 недели назад, # | 0 OMG red round!!!
» 2 недели назад, # | 0 Hope for the best and prepare for the rest.(★ᴗ★)
» 2 недели назад, # | ← Rev. 2 → +8 Hope to get BLUE today!
• » » 2 недели назад, # ^ | +11 Uchiha Madara, you have to be LGM
• » » 2 недели назад, # ^ | 0 me too but the rating update is a bit slow
» 2 недели назад, # | 0 I feel like Educational Rounds are more Mathematical than usual rounds, is this the case?
» 2 недели назад, # | +11
• » » 2 недели назад, # ^ | +2 sure
» 2 недели назад, # | +3 Good luck! Wish every participant have higher ratings!
» 2 недели назад, # | ← Rev. 2 → -22 no
» 2 недели назад, # | -33
• » » 2 недели назад, # ^ | -12 hahah
» 2 недели назад, # | -62 Classic Mathforces.None of A,B and C had anything to do with DSA
• » » 2 недели назад, # ^ | +20 A was greedy, B was a math problem, C use binary search. 1 out of the first 3 problems of a div 2 round being math problems doesn't imply that a contest is "Mathforces" dumbass. Also, atleast don't be green and talk shit about a contest not having enough DSA problems.
• » » » 2 недели назад, # ^ | -37 Mind your language, this is not reddit/twitter. As for rating, I don't take these contest seriously nowadays. I at one point was Blue unlike someone who has never crossed 1500 xD
• » » » » 2 недели назад, # ^ | ← Rev. 2 → +4 "Mind your language"- What you say matters as much as how you say it, so that's pretty rich coming from a guy who spouts shit like there's no tomorrow. Also, I don't want to get into petty comparisons, but I have given like 7 contests till now, so I'm pretty sure my peak will be way above yours :)Edit: The part about "someone who has never crossed 1500" didn't age well
• » » » » » 2 недели назад, # ^ | -10 Bro called me a dumbass for sharing my take on the contest and then preaching me.And then Bro says he doesnt do pity comparisons but advices me to get out of green and then give my analysis on the contest.Hypocrisy at its finest!
• » » 2 недели назад, # ^ | +4 Considering your past rating you shouldn't have had any trouble doing A, B, or C... idk man skill issue
• » » » 2 недели назад, # ^ | 0 It's not about whether I am able to solve or not, its about the quality of questions. I was able to solve A and B but still complaining because there was no satisfaction solving those problems
• » » » » 2 недели назад, # ^ | ← Rev. 2 → +5 of course you didn't have any satisfaction solving them, they aren't supposed to be hard for anyone above 1k rating
• » » » » » 2 недели назад, # ^ | 0 idk B was kinda annoying for me, C was good tho
» 2 недели назад, # | 0 what is "educational" about the problems in this round?
» 2 недели назад, # | 0 Does anyone know which test case was giving a lot of wrong answers in problem C?
• » » 2 недели назад, # ^ | 0 mine were 51 2 3 5 4and 33 2 1
» 2 недели назад, # | ← Rev. 2 → +10 Короче, Меченый, я тебе высрал div2 и в благородство играть не буду: посчитаешь n-е число A000311 — и мы в расчете. Заодно посмотрим, как быстро у тебя пукан после раунда загорится. А по твоей теме постараюсь узнать. Хрен его знает, на кой ляд тебе эти числа Эйлера 2 рода сдались, но я в чужие дела не лезу, хочешь отглиномесить, значит, есть за что....
» 2 недели назад, # | 0 how to solve c?
» 2 недели назад, # | -28 Problem C is the literal definition of million corner cases and i love it lol
• » » 2 недели назад, # ^ | +4 Think about binary search approach
• » » 2 недели назад, # ^ | ← Rev. 2 → +32 Huh? I have 0 corner cases.(now someone will definitely hack me xd)
• » » » 2 недели назад, # ^ | +1 well, my first approach is to find middle positions not numbers, and when i realised that, i also realised the sample test cases has outsmarted me ._.
• » » 2 недели назад, # ^ | 0 190436527 No corner cases whatsoever.
» 2 недели назад, # | +27 Nice problem D, it took a while before I noticed the name of the problem :) great hint
» 2 недели назад, # | 0 Hints for question D Hint 1It's a Tree problem Hint 2Try to make a tree such that we can do a dfs for each premutation can get maximum length My solution[submission:190390614]
• » » 2 недели назад, # ^ | +3 It can be easily solved via hashes. Great example 190367922
» 2 недели назад, # | 0 I quite liked problem C. I am pretty proud of an elegant solution I came up with inspired by some monotonic stack problems I was solving the other day.
• » » 2 недели назад, # ^ | 0 Can you share the link to the problem which you are talking about?
• » » 2 недели назад, # ^ | 0 can you share the approach to solve it, i completely didn't get the idea to appraoch this
» 2 недели назад, # | 0 how to solve 1st ?My approach was to count the number of frequency and take their sum, and ans should be max to max n what is wrong in this?
• » » 2 недели назад, # ^ | ← Rev. 2 → 0 My solution is to count number of 1-s in array and print n — (count_1 / 2).
• » » 2 недели назад, # ^ | +5 Only double 1-s pair in the array can decrease operation times, so all you need to do is just count the pair's amount.
• » » 2 недели назад, # ^ | +5 You can kill monsters of health 1 in pairs and remaining you can kill individually.
» 2 недели назад, # | +2 After finishing it agrees with me that the problem D is easier than the problem C, lol
» 2 недели назад, # | +1 I was stuck with problem C. My code outputs correctly for samples and inputs I give. Perhaps I can't find the edge cases. Here is my code https://codeforces.com/contest/1792/submission/190399259 .
• » » 2 недели назад, # ^ | 0 Counterexample: 8 2 3 1 7 8 4 5 6 Output should be 2. Pick (2, 7), then pick (1, 8). Your code seems to output 3.
• » » » 2 недели назад, # ^ | 0 Thanks, man. I didn't account if the numbers were not continuous.
» 2 недели назад, # | 0 Can anyone give hint how to solve third problem?
» 2 недели назад, # | ← Rev. 2 → -19 leave it there is an easy way to solve D then my way
» 2 недели назад, # | -10 A nice perfectly balanced contest. Kudos to the authors.
» 2 недели назад, # | ← Rev. 4 → +55 OMGGG solved E on 01.59.54 les goo
» 2 недели назад, # | 0 I tried to solve problem C with a monotonic stack but no luck.
» 2 недели назад, # | 0 how to solve 2nd problem my approach was if we can have a > 0 then one joke from b and one joke from c and alternating it till either b or c runs out . so it's min of (b , c) + min jokes in total . now these jokes will balance it and bring it back to a . now if d > 0 i can take the min of d , a extra jokes . if( b or c ) are not 0 . add one more joke .
• » » 2 недели назад, # ^ | ← Rev. 3 → +6 If $a = 0$, output $min(1, a + b + c + d)$.Else the answer is $a + min(b, c) * 2 + min(a + 1, b + c + d - min(b, c) * 2)$
• » » » 2 недели назад, # ^ | 0 Can u explain how min(a+1,b+c+d-min(b,c)*2) come up .Beacause I am not able to figure it out
• » » » » 2 недели назад, # ^ | ← Rev. 2 → 0 Take all $a$ Taking $1$ $b$ then $1$ $c$ until $b = 0$ or $c = 0$. Now both of them will have $a$ points and at least one of them cannot get anymore points. You can take at most $a + 1$ from $b + c + d$.
• » » » » 2 недели назад, # ^ | 0 both alice and bob will have a mood of "a" initially . now since we alternate between the other 2 jokes until one runs out, the mood for both will increase and decrease and end up with "a" again .now finally if a is smaller than the remaining jokes you can have a+1 extra jokes else if a is much bigger than the remaining jokes it's just that remaining .
• » » 2 недели назад, # ^ | 0 Dude if ur approach was right then why the heck did u NOT solve it during the contest.
• » » » 2 недели назад, # ^ | 0 screwed up with the implementation
» 2 недели назад, # | ← Rev. 2 → 0 D was nice combination of DP and trie.. loved the question
• » » 2 недели назад, # ^ | 0 How is DP being involved? Do you store the answers so that you do not insert a permutation again?
• » » » 2 недели назад, # ^ | 0 no, You create dp[i][j] = Indices of all the permutations, which has '1' on i'th index and 2 on j'th index . Then, for each permutation, u can loop. this will give you ( 5 * 10^4 * 8! ) roughly... Also, u need to put break statement if you already found the ans of length 'm' . You can follow my solution here. ALAS, I was just 2 minutes late from finishing my final solution :( ... If you need understanding my solution, feel free to ping me.
• » » » » 2 недели назад, # ^ | 0 Unfortunately your solution got TLE at 31
• » » » » » 2 недели назад, # ^ | 0 yeah, it got TLE, I am resolving the errors. Sorry. I will update once done. Thanks :)
» 2 недели назад, # | 0 i freaking could not implement trie, i chocked and even forgot dynamic allocation !!!
• » » 2 недели назад, # ^ | +20 If you are talking about D, straightforward polynomial hashing (without the mod) would do.
• » » » 2 недели назад, # ^ | 0 you mean rolling hash right? I could not think of this in contest, wasted an hour googling dynamic allocation and still could not do because of the stress of the contest.
• » » » 2 недели назад, # ^ | 0 Can you explain what polynomial hashing is? or post a link please. I'm having a hard time implementing D as well
• » » » » 2 недели назад, # ^ | 0 polynomial hashing is basically encoding an array/string into a number. So for example the permutation $5, 3, 1, 4, 2$, you just encode it into the number $53142$. This way, you can easily compare array in $O(1)$, in order to binary search/use map
• » » 2 недели назад, # ^ | 0 If you use map with vector it also works
• » » » 2 недели назад, # ^ | 0 Can you explain how that would work, I saw many submissions with maps but I can't seem to comprehend them? Im not able to understand how the map solutions are working.
• » » » » 2 недели назад, # ^ | 0 Map remembers a value by their key, a key can be almost anything, even vector, so when you are itterating over an array and you are looking at what prefix should the other array have to get beauty of A with the array we are looking at, you can remember that prefix as a vector and use that vector as a key in map, and remember value 1 in map at that key. after doing that for all arrays, you are now looking for an answer for an array lets say C, you itterate over its prefixes, push them in vector, and for every prefix vector you check if you have written 1 in the map at the place of this vector( so you use this prefix vector as a key) if you have then you can get the beauty of the lenght of the prefix vector
• » » 2 недели назад, # ^ | 0 If you use bitset it also works(of course you need to do some prework!)
» 2 недели назад, # | ← Rev. 3 → 0 I think I had an interesting approach for C, curious if anyone else had the same idea: Push all elements in the permutation into a queue. Initialize variable 'min' as 1, 'ans' as 0. For i = 1 : n: While queue.front() == min or queue.front() == selected, queue.pop(), min++ if queue.front() == min Selected.insert(i, n-i-1), ans++; Done! Return answer.
» 2 недели назад, # | +20 Damn. beethoven97 hacked LGM turmax's solution
» 2 недели назад, # | +3 For me C > E > D. I Spent more than 40 min on C and am still not able to solve it. What's the solution?
• » » 2 недели назад, # ^ | 0 Think in terms of binary search. Can we solve the problem with $i$ operations? Now think what should the last operation be? What should the second to last operation be, and so on so forth ...
• » » 2 недели назад, # ^ | 0 My reasoning was that if we select some elements, we will always select 1, n, 2, n-1, etc... The order we select them should not matter because some optimal ordering should exist anyways. So I created a queue and popped off elements while it was sorted. Then, whenever it was unsorted, I removed elements i and n-1-i from the array and continued to pop while it was sorted. The number of times you remove i and n-1-i is the answer.Submission: 190357765
• » » 2 недели назад, # ^ | 0 You can see that you can not use operation on numbers that are close to each other and ordered for example 4,5,6. 4,5,7 and 6,5,4 will not go. So if array is like this 4,8,7,5,3,2,1,6 you can do operation on every other number except these 3 numbers = 4,5,6, and answer will be max(min(numbers)-1, n-max(numbers)) = max(4-1, 8-6) = 3.
• » » 2 недели назад, # ^ | 0 Just do n/2,n/2+1;n/2-1,n/2+2;... if n%2==0 or do n/2,n/2+2;... if n%2==1 and skip the first operations if they are already in place.
• » » 2 недели назад, # ^ | +17 You have to find the longest MIDDLE subarray. 5 1 2 5 3 4 The longest you can find is 2 3 4.
• » » » 2 недели назад, # ^ | +3 yeah that's what i did too, O(n) is great
» 2 недели назад, # | 0 Amazing system testing.So fast.
» 2 недели назад, # | 0 Would C Problem be solved by 2 Pointers ? if not, then what?
• » » 2 недели назад, # ^ | +3 Yes, 2 pointers is fine my submission.
• » » 2 недели назад, # ^ | ← Rev. 10 → 0 Yes
» 2 недели назад, # | 0 Can someone give stress test for 2nd testcase on E? 190401625
» 2 недели назад, # | ← Rev. 3 → +1 A: Use first spell for 1-health monsters and second spell for others.B: First tell all type-1 jokes, then tell type-2 and type-3 jokes alternately, until one type of jokes run out. Then tell all remaining jokes until someone leaves.C: Take n=6 as example. First let ans=3=n/2, and check if the order of {3,4} is correct (which means, 3 appear earlier than 4 in the initial permutation), here 3,4 are "central" elements of 1-6. If it's not, we need ans=3 operations. Otherwise, we need to do ans--, and check the order of {2,3,4,5}. Do this repeatly until we fail at any check or we've checked the whole permutation. If n id odd, let k=(n-1)/2, start at ans=k and checking {k,k+1,k+2}.D: We need to check the longest common prefix of ai and aj^(-1) (where aj^(-1) is the inverse of aj), we could store all aj^(-1) in a trie and find for all ai.E: Didn't solved. Maybe we can let set s={d: m1*m2%d==0 && 1<=d && d<=n} and do loop for(d1:s) for(d2:s && d2>=d1) but I don't know if this approach will get TLE (for cases like n=1e9, m1=m2=735134400).
• » » 2 недели назад, # ^ | 0 Stored all aj in a trie and searched for aj^-1, didn't realize the solution during contest. :(
• » » 2 недели назад, # ^ | 0 D: Sort the permutations and For every prefix of every permutation, binary search permutations that match this prefix and use segment tree to update range (l <= i <= r) a[i] = max(a[i], x). 190359788
• » » » 2 недели назад, # ^ | 0 there is much more simple solution with trie :)190383705
» 2 недели назад, # | +3 Any hints for E?
• » » 2 недели назад, # ^ | -34 two pointers
• » » » 2 недели назад, # ^ | 0 In defense of this hint, my submission passed system testing with a two pointer-like approach: https://codeforces.com/contest/1792/submission/190397103. Although upon further reflection, the analysis I had of its time complexity was not correct, so if anyone can come up with a proof of correctness (or a hack), that would be cool!The main idea was to process the divisors in ascending order. Let the current divisor be $a$. We will maintain a pointer to the minimum divisor, $b$ such that $a / b <= n$. Then we just search from $b$ until the last divisor <= $n$. It feels like there might be an argument that the average number of elements we check is not too high, but I can't find it. The dp solution seems much more straightforward to understand, so apologies for the misdirection.
• » » 2 недели назад, # ^ | 0 Don't know for sure that this is the intended solution.First, find all the divisors by brute force as the maximum number of divisors for (large) n cannot exceed cube root n. And for every divisor x below 1e9, remove the divisors of x until x*1e9 iteratively and update the answer.
• » » 2 недели назад, # ^ | +63 Generate all divisors of $m$ (there's about $10^5$ of them in the worst case). For every divisor, instead of the minimum row where it appears, let's search for the maximum column (it's easy to see that these two are equivalent). So, for every divisor, we need to find its maximum divisor which is not greater than $n$.This can be done with the following dynamic: $dp[d]$ is maximum divisor of $d$ not exceeding $n$. If $d \le n$, then $dp[d] = d$, otherwise iterate on the prime divisor $p$ in the factorization of $d$ and find the maximum of $dp[d/p]$.
• » » » 2 недели назад, # ^ | 0 Could you mention the time complexity of this approach? It's not immediately clear this solution can fit into time limit?
• » » » » 2 недели назад, # ^ | ← Rev. 3 → +15 Something like $O(D \log D \log m)$, where $D$ is the number of divisors of $m$. There are $D$ states in the dynamic programming, each state has up to $\log m$ transitions (each transition corresponds to dividing by a prime from the factorization of $m$), and an extra logarithm because everything is stored in a map.upd: Plus $\sqrt{m_1} + \sqrt{m_2}$ to factorize $m$.
• » » » » » 2 недели назад, # ^ | +20 More accurately, $D \le 105\,000$, "$\log{m}$ transitions" is actually $\le 15$.And don't use maps for $dp$, use vectors.
• » » » » » » 2 недели назад, # ^ | 0 How to not use maps? The factors of m could be large right?
• » » » 2 недели назад, # ^ | 0 I generated all divisors of $m$ . For each divisor $a$ , I brutely searched minimal row number within the range $[\lceil \frac a n \rceil,min(n, \sqrt a)]$ among divisors of $m$ .This naive solution seems to run very fast.Is it reasonable to let this solution pass? I mean, this solution is unbelievably too simple, meanwhile hard to know exact time complexity.
• » » » » 2 недели назад, # ^ | 0 My best guess is that for each divisor $a$, there's a big chance that you have to go through like $50$ divisors on average (maybe much less I don't really know) before finding an answer. You see, for highly composite numbers, aka numbers that has a lot of divisors, its divisors are also expected to have a lot of divisors, thus it is likely for the algorithm to encounter a divisor of $a$ in very few loops. For number that has less divisors, I think that there simply isn't enough divisors to make a simple $O(n^2)$ getting TLE
• » » » 2 недели назад, # ^ | ← Rev. 2 → 0 Why dp[d] = std::max(dp[d], dp[d / p]), I don't understand this, please explain it to me.
• » » » » 2 недели назад, # ^ | 0 I think dp[d] must equal std::max(dp[d], dp[all_of_divisors(d)])
• » » » » » 2 недели назад, # ^ | +8 If you do dp[d] in order then dp[d/p1/p2] would've already been considered during the transition of dp[d/p1]
• » » » » » » 2 недели назад, # ^ | +3 Oh, thank you so much
» 2 недели назад, # | 0 D could have been a really nice binary search + trie task if bounds were N<=1e5, NxM <= 1e5
• » » 2 недели назад, # ^ | 0 Hmm bs, i think only Trie + dfs
• » » 2 недели назад, # ^ | 0 Why binary search tho? You can just DFS down the trie, and the answer would simply be where the DFS end.
» 2 недели назад, # | 0 I feel bad when I heard that $O(n^2)$ solution can pass F2.I feel worse when I really pass it after the contest. HereI guess the author think D&C + fft is too slow, but it is not that slow, and is it reasonable to let $n^2$ solutions pass?
• » » 2 недели назад, # ^ | 0 The problem F of last contest also be passed by some O(n^2) solutions.
• » » 2 недели назад, # ^ | ← Rev. 2 → +22 Unfortunately, looks like really fast templates for modular arithmetics do the trick. I haven't come up with the D&C+FFT solution, the model one has slower asymptotics than D&C+FFT. So, basically, I could try one of the following two things: let only solutions with very optimized FFT and modular arithmetics pass; let solutions with more "normal" implementations of FFT and modular arithmetics pass, but risk that someone with a very strong modular template will squeeze $O(n^2)$ in I still think that 2nd is better choice. Maybe my mistake was even trying to distinguish $O(n^2)$ and $O(n \sqrt{n \log n})$. I am sorry for that, but I hope not a lot of participants were affected by the issue.
• » » » 2 недели назад, # ^ | ← Rev. 2 → 0 The formula is like $f_{i} = \sum{f_{j}\times f_{i-j}}$, in my memory it can be solve by D&C and fft(since $f_{i}=\sum{f_{j}\times g_{i-j}}$ for fixed $g$ can be solved) , but maybe I remember it wrong(I feel sorry), and I didn't figure it out during the contest.However, OEIS A000311 shows that $ans = exp(f(x)) = 2f(x) - x + 1$, thus we can solve it by Polynomial Newtonian iteration($O(nlogn)$ maybe?).
• » » » 2 недели назад, # ^ | +10 I squeezed in $O(n^2)$ by precomputing for biggest n and putting it into the source code and running the $n^2$ normally for small $n$. I didn't use any very optimized modular arithmetic. 190408679 (there seems to be no test with a number in range of [3.5e4,4e4), so the runtime is misleading but testing the worstcase on custom invocation gives 4500 ms.
• » » 2 недели назад, # ^ | 0 In fact we can directly get a formula using lagrange inversion. the final result (plus 2) is the $x^{n-1}$ coefficient of $2(n-1)!(\frac{x}{2x+1-e^x})^n$
» 2 недели назад, # | +1 Did any1 use strings for D?
• » » 2 недели назад, # ^ | 0 I concatenated values in array and stored them in hashmap, but I did it because golang doesn't support custom key types in hashmap.
• » » 2 недели назад, # ^ | 0 Yup! I did.
» 2 недели назад, # | 0 https://codeforces.com/contest/1792/submission/190405242Could anyone help me understand why my code gives incorrect output in this question?
» 2 недели назад, # | ← Rev. 2 → 0 Could anyone help me understand why my code for D gives incorrect output here: https://codeforces.com/contest/1792/submission/190405242
• » » 2 недели назад, # ^ | +12 your answer would have been fine if rj = pqj
• » » » 2 недели назад, # ^ | 0 Thanks for helping. What makes this hurt more is that I would have got last 5 second AC instead of WA, had I not done that silly mistake xD.
• » » » » 2 недели назад, # ^ | +9 It happens 😹 😹
• » » » 2 недели назад, # ^ | ← Rev. 3 → +1 Wait a minute, but wont p(q(j)) and q(p(j)) be like the inverse of each other, ie. they will produce each other?For example, wouldn't the product of p = 3142 and q = 2413 be 1234 whether you take r = p.q or r = q.p?
• » » » » 2 недели назад, # ^ | +2 No take this for example 2 4 1 3 2 1 4 3
• » » » » » 2 недели назад, # ^ | ← Rev. 2 → +1 In the first sample test case, the optimal p for i = 1 such that k for ai * p is maximised will be [3 1 4 2] right? But none of the given arrays have a prefix 3, So how is the answer 1 and not 0?I'm extremely sorry if I'm asking dumb questions rn, I'm a bit sleep deprived.
• » » » » » » 2 недели назад, # ^ | 0 You are getting confused 💀, take a vector of pair store the values of "p" in that vector along with index ({value,index}), sort it and then take the prefix of indexes , what you are doing is you are still finding pqj 💀 💀
• » » » » » » » 2 недели назад, # ^ | +1 I must sleep. Urgently.
• » » » » » » » » 2 недели назад, # ^ | +3 Good night 🛌
» 2 недели назад, # | 0 The One Piece is real!!!
» 2 недели назад, # | +1 Is E solved by observing numbers of divisors is relatively little(milion or so cause max 20 diferent primes in numbers) and then searching through primes?
» 2 недели назад, # | 0 Trie method for problem D https://codeforces.com/contest/1792/submission/190408398
» 2 недели назад, # | 0 What's wrong with my solution for problem C? 190408656
• » » 2 недели назад, # ^ | +8 Test Case 1 5 4 1 5 3 2 your output3 expected output2, as you can first select 2 and 4, and in the next select 1 and 5
• » » » 2 недели назад, # ^ | 0 Thank You
» 2 недели назад, # | 0 In C what I did for every index I find the fine permutation till that then I will find the left portions towards left and right of the this range 3 1 2 4 5 here fine permutation of index 4 will be 3 4 and then in final permutation 1 2 should come on left of it and 5 in right so greedily we pick 2 and 5 first then 1 and 5 My submission https://codeforces.com/contest/1792/submission/190407907
• » » 2 недели назад, # ^ | +8 Google punctuation mate, it'll change your life.
» 2 недели назад, # | +6 Why so many hacks of B? Is there any edge case?
• » » 2 недели назад, # ^ | +20 I think people are simulating it, which is too slow. Most of the hacks are TLEs.
• » » 2 недели назад, # ^ | 0 I think it's becaues of the $1e18$ upper limit of $a1,a2,a3,a4$, which causes the TLE
• » » » 2 недели назад, # ^ | +5 You scared the heck out of me when I read 1e18 because I used ints in my program, but thankfully I rechecked the constraints and saw that they were 1e8.
• » » » » 2 недели назад, # ^ | +5 oops, I guess I misread it xd, anyway it will still TLE regardless
» 2 недели назад, # | 0 Am I the only one who solve D with trie!
» 2 недели назад, # | ← Rev. 2 → +4 Am I the only one who solve problem D with trie and later see that it has an easy solution?
• » » 2 недели назад, # ^ | -8 I used a trie too, I found it to be somewhat intuitive. What's the easier way?
• » » » 2 недели назад, # ^ | 0 maybe bitset?
• » » 2 недели назад, # ^ | +1 You are not alone bro.
• » » 2 недели назад, # ^ | +2 yeah I firstly thought of trie but then realised that a map would just suffice because of the low constraints.
» 2 недели назад, # | ← Rev. 2 → 0 Can someone give a failing TC for this submission of Problem B?190391460Thanks.Upd: Found the TC.
• » » 2 недели назад, # ^ | 0 if a==0 then answer at max can be 1 only.you can check this https://youtu.be/TOotS4TDzTI
» 2 недели назад, # | ← Rev. 2 → +53 F1 can be cheesed since $n$ is small and the answer is required modulo a fixed number, You can pre-compute the answers in $O(n ^ 3)$ and copy the array into your code.My solution 190420429
» 2 недели назад, # | ← Rev. 2 → +6 Yesterday I was practicing hashing, but I tried to don't think biased and made a solution with custom sorting and binary search in D :) My solutionI sorted all permutations, first by the order of key 1, then by the order of key 2 and so on. with a binary search I found lower and upper bound of each number in the permutation order, and updating the range of the search to it, until range equals 0
» 2 недели назад, # | +8 D was really standard...even using simple map for counting will pass for me C>D
» 2 недели назад, # | +4 I passed E in 15 minutes after the match，it didn't seem like a difficult problem，what a pity
» 2 недели назад, # | 0 I understood that problem D can be solved by trie, and I was having some difficulty so I looked at some submissions. I see that many people have implemented trie (or something similar) using map. Can anyone explain the logic behind the implementation?
• » » 2 недели назад, # ^ | +1 I haven't implemented using trie but the intuition is somewhat similar to using a map. you can check the solution here — https://youtu.be/TOotS4TDzTI
» 2 недели назад, # | 0 any hints for D
» 2 недели назад, # | 0 190389330 Who could help me about the question C?I use two pointers but TLE
» 2 недели назад, # | +3 C can also be solved using DP: 190339025
• » » 2 недели назад, # ^ | 0 what's the meaning of vector d and statues shifting funcion
• » » » 2 недели назад, # ^ | 0 $d_i$ means the length of Longest Continious Subsequence ($...,i-2,i-1,i$) (End with number $i$)
• » » 2 недели назад, # ^ | 0 Oh this $O(n)$ is better than std's $O(n\log n)$ :)
» 2 недели назад, # | ← Rev. 2 → +1 Hi guys if you are still stuck on the problems or want a editorial on it you might wanna check this out: https://youtu.be/TOotS4TDzTIhappy coding!
» 2 недели назад, # | ← Rev. 2 → 0 B seemed quiet simple but at the end I did it out of intuition, eager to see how to properly solve it. I found problem C really interesting too!! dying to see how to solve it, thanks for the round, it was fun!!Edit: Any hints for C?
» 2 недели назад, # | 0 Bad F due to oeis.org/A006351.We can search exsamples+2 to get this
• » » 2 недели назад, # ^ | ← Rev. 2 → -8 how is that related to the problem, ur solution to the problem (F1) doesn't use the formula mentioned in the link, how could someone possibly use it ?also i dont think someone possibly would search first two samples + 2 in oeis and if so, it displays 316 results found, i dont see any reason calling it "Bad" because of such a reasoning.
• » » » 2 недели назад, # ^ | 0 i dont think someone possibly would search first two samples + 2 in oeis Several participants did search the answers for $n$ between $1$ and $4$, found the formula, and had their solutions accepted. Moreover, one could find these values with simple combinatorial considerations.
• » » » 2 недели назад, # ^ | ← Rev. 2 → 0 look at this: a(1) = 1; a(n) = a(n-1) + Sum_{k=1..n-1} binomial(n-1,k) * a(k) * a(n-k). — Ilya Gutkovskiy, Aug 28 2020use this recursive, we can solve F1 and then the recursive is a convolution, while module is 998244353, which means it is easy to brainstorm NTT. And let me tell you why I searched answer+2. the problem said that at least one edge should be painted as red/blue, that means when n>=2, there are two illegal answer(all red or all blue) being removed, while n=1, one illegal answer(no edge) will be removed, so let's search 1,2,8,52(see samples) in oeis.
» 2 недели назад, # | 0 Did anyone try to solve D using polynomial hashing and got AC? My solution keeps getting TLE in test case 6. I'm trying to maintain HashSet over all possible subsequences keeping their own position intact. After that, for each i I'm calculating q putting numbers one by one, and calculating the hash. 190454887
• » » 2 недели назад, # ^ | 0 Please look at my sending, the system tests have passed completely
» 2 недели назад, # | 0 hope to get green
• » » 2 недели назад, # ^ | 0 Good luck!
» 2 недели назад, # | ← Rev. 2 → 0 About E.Divisors Does this problem have too strict time limit?This is my solution https://codeforces.com/contest/1792/submission/190443658 and it has been hacked for 3 times, I think this problem may need to have more loose time limit like 4 seconods.
• » » 2 недели назад, # ^ | +3 There are some better solutions that don't iterate over divisors of every single divisor of m. See this. Also your code is T^2 (T is the number of divisors of m), which is even worse...
• » » » 2 недели назад, # ^ | 0 thanks，But I think number of divisors of m will not exceeded 10^4.
• » » » » 2 недели назад, # ^ | 0 Number of divisors can go upto 10^6.
• » » » » » 2 недели назад, # ^ | 0 thanks so much，i even donot know this before
• » » » » » 2 недели назад, # ^ | 0 Nah, there's at most $1e5$ divisor. Upper Bound for number of divisors
• » » » » 2 недели назад, # ^ | 0 Number 614889782588491410 has 32768 divisors. But anyway ((10^4) ^ 2) * 10 = 1e9, which isn't good (there are 10 testcases)
• » » » 2 недели назад, # ^ | +8 I will study better solution soon，thanks
• » » 2 недели назад, # ^ | ← Rev. 2 → 0 The question is, you only need to make a few changes to your code to pass all testdata.I raised my question here , but still no one answered. vector s3; for (const auto &x1 : s1) for (const auto &x2 : s2) s3.push_back(x1 * x2); sort(s3.begin(), s3.end()); s3.erase(std::unique(s3.begin(), s3.end()), s3.end()); int ans1 = 0, ans2 = 0; for (const auto &x : s3) { for (auto pos = std::lower_bound(s3.begin(), s3.end(), (x + n - 1) / n); pos != s3.end(); ++pos) if (*pos > n or *pos * *pos > x) break; elif (x % (*pos) == 0) { ans1++, ans2 ^= *pos; break; } }
• » » » 2 недели назад, # ^ | 0 We knew about such solution and it passes. We suspect that either the number of divisors is small (so $divs(m)^2$ is fine) or divisors are "packed" close, so it's near enough. Add here that divisors, you actually need to find their own divisors, are in the segment $(n, n^2]$. So if $n$ is big, many divisors $\le n$. If $n$ is small, many divisors $> n^2$.Anyway, we decided that it's okay if some participants gamble and get AC.P.S.: Even so, you need to write it accurately enough.
• » » » » 2 недели назад, # ^ | ← Rev. 2 → 0 I'm not saying that a naive solution needs no skill. I'm saying that, such solution needs obviously less skill than people always expect for E.If you attribute my not_pass to my cowardice or bad strategy, that's right.
• » » » » 2 недели назад, # ^ | 0 In addition, the naive solution is not only "near enough", but far more.I found all factors during the contest. Now I added two for-loops. Guess what, it costs 156 ms only.156 msIf you say that this solution requires some advanced mathematical knowledge, which you had already used to prove the time complexity, so this solution can be an alternative. I would blame me for my lack of knowledge.But you said otherwise. Emm... whatever, it was history.
• » » » » 2 недели назад, # ^ | 0 thanks，i will write it more detailed soon
• » » » 2 недели назад, # ^ | +5 OK，i will try it soon，thanks so much
» 2 недели назад, # | +3 There will be only 6 hrs before next round begin. Maybe the rating update of the next round will be earlier than this round.
» 2 недели назад, # | 0 I am sensing Problem B is a broken copied problem. I don't know source but its just my hunch
» 2 недели назад, # | ← Rev. 2 → 0 .
• » » 2 недели назад, # ^ | 0 wait.System Testing is running.After system testing everything will be fine.
» 2 недели назад, # | 0 Too slow system testing.
» 2 недели назад, # | +9 When will the Editorial be published ?
» 2 недели назад, # | 0 when will ratings update?
» 2 недели назад, # | 0 Why am I always
» 2 недели назад, # | +4 This is to address the concerned authorities about my submission 190353992, which got skipped because the system considered it similar to another submission by BoosterImpulse , submission id 190386013.I think the major reason why these 2 codes appear similar, is because of the use of a template of MOD INT (MINT), which I have used many times in my code and is clearly written before the contest and you can clearly see my code is completely different from the other person's code, my code is a normal implementation of two pointers. You can also check the timings of the submission and other persons' submissions. Please look into this matter as soon as possible cause I will become an expert if my solutions are been judged fairly. awoo BledDest Neon MikeMirzayanov adedalic
• » » 2 недели назад, # ^ | 0 Yes I too am pretty confused as to why the solutions are plagiarised.
» 2 недели назад, # | ← Rev. 2 → 0 .
• » » 2 недели назад, # ^ | 0 Have you ever read the rules?
» 2 недели назад, # | +8 Rating updated :D
» 2 недели назад, # | 0 When will the editorial be published?
» 2 недели назад, # | +79 System said that I cheated but I did not.wsyear/190368291, LXH-cat/190370101 we used a similar template of modint and checked oeis to find a formula:a(0)=0, a(1)=a(2)=1; for n >= 2, a(n+1) = (n+2)*a(n) + 2*Sum_{k=2..n-1} binomial(n, k)*a(k)*a(n-k+1).With the same formula, our codes are similar.The code that I used modint before this contest.
• » » 2 недели назад, # ^ | +58
» 2 недели назад, # | 0 When the editorial will be updated?
» 2 недели назад, # | 0 Auto comment: topic has been updated by awoo (previous revision, new revision, compare).
» 2 недели назад, # | 0 Hi my submission 190325251 is said to coincide with submission 190323813. I believe the logic to this problem is very straight forward and the solve() template we both used is very standard. Any of my previous submission uses a similar template + style and I believe this is just a mistake from the automated plagiarism checking system. Please update I dont want to be flagged for cheating :(
» 2 недели назад, # | 0 IMO the problems were great. I enjoyed thinking about and solving them. Also I got caught on the special case for B (a1 == 0) haha.
» 2 недели назад, # | 0 Through this comment, I want to address the cheating charges levied on me and Numinous , for having a coinciding solution to problem C in Educational Round 142. The submissions are 190386013 and 190353992. The reason for getting plagiarised was having a similar template, going through solve function for both speaks out the difference clearly. I hereby in my humble opinion ask MikeMirzayanov Neon BledDest adedalic vovuh awoo to provide us justice and the ratings we deserve . Thank You for your time and effort .
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2023-02-09 12:40:02
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https://support.bioconductor.org/p/133581/
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Issue with DECIPHER::IdTaxa
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0
Entering edit mode
@sebastianmynott-7437
Last seen 15 months ago
United Kingdom
I've encountered an issue which hasn't occurred previously. I’m running DECIPHER v2.16.1 in R v4.0.2 and RStudio v1.3.959. and have managed to downgrade to R v3.6.3 to see if it made any difference but it hasn't.
I get the following error, even when running the Examples from the IdTaxa function:
data("TrainingSet_16S")
fas <- system.file("extdata", "Bacteria_175seqs.fas", package="DECIPHER")
dna <- RemoveGaps(dna)
ids <- IdTaxa(dna, TrainingSet_16S, strand="top")
Error in getMethod(f, c("XRawList", "XRawList")) :
no method found for function 'match' and signature XRawList, XRawList
The issue seems to present itself as a result of certain packages loading and getting in the way of the IdTaxa function but I can't seem to figure out a workaround. I've noticed that I can sometimes get the IdTaxa function to run if I restart the R session, load the DECIPHER library and run it before loading other libraries. However, even this fails if the last bits of the restart (see below) manage to run before I can run the function.
Registered S3 method overwritten by 'spdep':
method from
plot.mst ape
Registered S3 method overwritten by 'pegas':
method from
I've also tried explicitly loading the XVector package as suggested here but to no avail.
Any assistance would be very much appreciated!
DECIPHER IdTaxa • 651 views
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Entering edit mode
UPDATE: I've done a clean reinstall of the latest version of R and reinstalled the latest versions only the packages that are used in my script (and their dependencies).
At first I had the same problem but then I removed the packages spdep, ape, pegas and ade4, which got IdTaxa running again.
However, now it won't multithread! I've checked parallel is correctly loaded.
Again, any assistance would be appreciated.
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Hi Sebastian. If you are using a Mac, the compiler was changed so that multi-threading is no longer supported by default. See the recent thread here for how to enable it.
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Erik,
Thanks for posting the link to these workarounds. I've followed the steps you used via Homebrew. My install path (Step 2) was the same as yours but Step 3 didn't work for me. I restarted RStudio and R. I also tried using these lines from http://mac.r-project.org/openmp:
CPPFLAGS += -Xclang -fopenmp
LDFLAGS += -lomp
Strangely enough, multithreading in other packages seems to be working. Do you have any suggestions for what I might try next?
Many thanks! Seb
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Entering edit mode
Using Homebrew worked for me, but this is not the recommended way of enabling OpenMP. I haven't tried the recommended way, so I can't help you there.
Note that not all multithreaded packages use OpenMP, which might be why you observed multithreading on the Mac working in some of your other packages.
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Entering edit mode
Ok. I'll keep trying options on my end to get it multithreading again.
The recommended way you mention is clearly intended as a workaround which could stop working at any point. Are you planning on updating DECIPHER to use another multithreading package that doesn't require OpenMP support?
Cheers, Seb
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Entering edit mode
So it seems this issue has been affecting other packages, namely multithreading in data.table. So finally this solution has worked for me: https://github.com/Rdatatable/data.table/wiki/Installation#openmp-enabled-compiler-for-mac.
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2022-01-19 17:33:20
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https://www.edaboard.com/threads/logic-design-interview-questions.338339/
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Logic Design -> Interview Questions
Status
Not open for further replies.
Hi All,
Thank you!
Warpspeed
If you really want to make some nervous young applicant really sweat, ask about metastability.
ivlsi
I know about metastability and many other things...
Like this one: "There is a stream of bits. Suppose LSB comes first. Implement FSM, which detect when the incoming binary number is modulo-5. The stream is infinite."
Super Moderator
Staff member
I know about metastability and many other things...
Like this one: "There is a stream of bits. Suppose LSB comes first. Implement FSM, which detect when the incoming binary number is modulo-5. The stream is infinite."
How is that a "tricky" question...it's just a design question.
I think tricky questions are more like open ended questions where the interviewee has to ask questions to clarify the question into something that can be answered. e.g. What is the net effect of a metastable event on the functionality of a system.
Now the interviewee is sweating wondering what is meant by the net effect on system functionality. Either the interviewee rambles on not answering anything or if they're savvy they ask some questions about what is meant by a functional failure and what is meant by net effects (i.e. they ask if we mean transition delay, missed transition, etc)
This kind of question is is typically used by some to check how someone thinks under pressure when they have a lack of information. Do they investigate or just try to act like they know something and spew nonsense. :wink:
ivlsi
Ok, I meant "not trivial complicated Logic Design questions, which might be asked on the interviews"
Sarathkumarkj
Newbie level 2
1. Verilog HDL originated at
A. AT&T Bell Laboratories
B. Defence Advanced Research Projects Agency (DARPA)
C. Gateway Design Automation
D. Institute of Electrical and Electronics Engineers (IEEE)
2. Verilog is an IEEE standard
A. IEEE 1346
B. IEEE 1364
C. IEEE 1394
D. IEEE 1349
3. Which level of abstraction level is available in Verilog but not in VHDL?
A. Behavioral level
B. Dataflow level
C. Gate level
D. Switch level
4. In verilog h1234 is a
D. It is invalid notation
5. Which logic level is not supported by verilog?
A. U
B. X
C. Z
D. None of the above
6. If a net has no driver, it gets the value
A. 0
B. X
C. Z
D. U
7. Default value of reg is
A. 0
B. X
C. Z
D. U
8. The task $stop is provided to A. End simulation B. Suspend simulation C. Exit simulator D. None of the above 9. Externally, a output port must always connected to a A. net only B. a reg only C. either net or reg D. None of the above 10. If A= 4b011 and B= 4b0011, then the result of A**B will be A. 6 B. 9 C. 27 D. Invalid expression 11. If A= 4b001x and B= 4b1011, then result of A+B will be A. 110x B. 1100 C. xxxx D. None of the above 12. If A= 41xxz and B= 4b1xxx, then A= = =B will return A. 1 B. X C. Z D. 0 13. Result of 9% -2 will be A. 4 B. 4.5 C. -1 D. +1 14. Initial value of a=1 and b=2, then what will be final value if always @ (posedge clock) a=b; always @ (posedge clock) b=a; A. a= 2, b=1 B. a= 1, b=2 C. Both a and b will have same value either 0 or 1 D. None of the above 15. Initial value of a=1 and b=2, then what will be final value if always @ (posedge clock) a<=b; always @ (posedge clock) b<=a; A. a= 2, b=1 B. a= 1, b=2 C. Both a and b will have same value either 0 or 1 D. None of the above 16. Given the following Verilog code, what value of "a" is displayed? always @ (clock) begin a = 0; a <= 1;$display(a);
end
A. 0
B. 1
C. either 0 or 1 depending on depending on simulator implementation
D. None of the above
17. In a pure combinational circuit is it necessary to mention all the inputs in sensitivity list?
A. No
B. Yes
C. It depends on the coding style
D. None of these
18. How many flops will be synthesized by the given code?
always @ (posedge clock) begin
Q1<=d;
Q2<=q1;
Q3<=q2;
end
A. 1
B. 2
C. 3
D. None of the above
19. Which is not a correct method of specifying time scale in verilog?
A. 1ns/1ps
B. 10ns/1ps
C. 100ns/100ps
D. 100ns/110ps
20. If time scale is defined as timescale 10ns/1ns and #1.55 a = b; then 'a' gets 'b' after
A. 10ns
B. 11 ns
C. 15.5ns
D. 16ns
21. A task can have arguments of type
A. Input only
B. Output only
C. Both input and output
D. All input, output and inout
22. If a recursive function is called concurrently from two locations, then
A. Recursive function can have multiple calls concurrently
B. It will result give ambiguous results
C. It will result in an error
D. Simulation will hang up
23. Which operators has highest precedence in verilog
A. Unary
B. Multiplication
D. Conditional
24. In the given code snippet, statement 2 will executed at
initial
begin
#5 x= 1’b0; // statement 1
# 15 y= 1b’1; //statement 2
End
A. 15
B. 20
C. 5
D. Current simulation time
25. Variable and signal which will be updated first?
A. Variable
B. Signal
C. Can’t say
D. None of the above
dave_59
Sarathkumarkj, you need to update your quiz to the 21st century. There is no longer an IEEE Verilog standard. it has been replaced by IEEE 1800 in 2009, the current revision is 1800-2012
Super Moderator
Staff member
Sarathkumarkj, your post also does not address the OP's question, which isn't about generic questions that you give to someone fresh out of school.
Ok, I meant "not trivial complicated Logic Design questions, which might be asked on the interviews"
Basically this says I want a good design question that has to be solved to prove the interviewee can actually design something and not just talk a good show.
Questions like:
"show me how you would architect a 128-tap 256 channel fir filter that will run at 300 MHz in a Kintex7"
"design a LDPC encoder"
"design an MPEG4 encoder and decoder"
"design our next product for us, make sure you finish by the end of your interview, if you want the job"
;-)
FYI, I'm exaggerating a little bit, but I'm sure you get the picture.
Status
Not open for further replies.
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2021-06-19 15:23:58
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http://log.moonshile.com/post/2014-11/2014-11-19
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# Log - Moonshile
## 2014-11-19
Published on 2014-11-19
## Godson-T murphi-modeling (2)
### Model Rules in Godson-T
As we mentioned, all rules that need replace cached data to memory should be separated into two parts in our murphi model, which is inspired by interrupts in systems programming or function calls in programming languages. The two stages are listed as follows:
• Firstly, save current state to the assistant variables such as curNode, curCache, etc. Then REQUIRE the replacement.
• Secondly, after the replacement has been DONE, retrieve the stored states and resume the interrupted rules, and set the replacement state to NON.
There are also some tips we should keep careful.
• All normal rules except replacement stages and resumed stages, must be triggered under the condition that the replacement state is NON.
• All resumed stages must be triggered under the condition that the replacement state is DONE.
• The replacement state must be asserted in every guard part of each rule including invariants.
• If need to assert cache state and cache content, cache state must be asserted first.
• If need to assert attibutes of locks, the beUsed attribute must be asserted first.
The last three tips can be ignored if we initialize all variables in the start state.
### Deal with Counterexamples
#### Properties
There are two properties that the cache coherence protocol must hold.
• Deadlock-free Property: One node can only hold one lock at the same time.
• Coherence Property: The cached data must be the same with data in their associated memories. In Godson-T, this means Read operation in critical region must cache correct data.
#### Counterexamples and Solutions
With those strong properties above, our model will generate a counterexample, which implies that two different process could cache the same location of memory in critical region protected by different locks, and once one of them modified the memory (with write through strategy), the other would cache wrong data.
There are two solutions to this problem.
• Do not allow multiple locks, e.i., there is only one lock there
• Weaken the Coherence Property, so as to permit the counterexamples, and only require that the cached data must be correct in which critical region protected by the same lock.
## Linear Regression
Simple Linear Regression is the least squares estimator of a linear regression model with a single explanatory variable. In other words, simple linear regression fits a straight line through the set of n points in such a way that makes the sum of squared residuals of the model (that is, vertical distance between the points of the data set and the fitted line) as small as possible.
Suppose there are n data points {(xi, yi), i = 1, ..., n}. The function that describes x and y is:
yi = α + βxi + εi.
The goal is to find the equation of the straight line which would provide a "best" fit for the data points. Here the "best" will be understood as in the least-squares approach: a line that minimizes the sum of squared residuals of the linear regression model. In other words, this is a minimization problem:
Find minimum Q(α, β) in α and β, for Q(α, β) = ∑ni=1ε^ 2i = ∑ni=1(yi - α - βxi)2.
By using either calculus, the geometry of inner product spaces, or simply expanding to get a quadratic expression in α and β, it can be shown that the values of α and β that minimize the objective function Q are
where rxy is the sample correlation coefficient between x and y; sx is the standard deviation of x; and sy is correspondingly the standard deviation of y.
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2018-12-17 12:55:41
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https://www.intmath.com/blog/letters/intmath-newsletter-curvilinear-anchoring-pancakes-11810
|
# IntMath Newsletter: Curvilinear, anchoring, pancakes
By Murray Bourne, 16 Jan 2019
16 Jan 2019
1. New on IntMath: Curvilinear motion animated graphs
2. Estimation and the anchoring principle
3. Bill Gates and his pancakes
4. Math movies
5. Math puzzle: Who dies?
6. Final thought: Truth
## 1. New on IntMath: Curvilinear motion animated graphs
If the concepts are new to you, it's often hard to determine what's really happening in mathematics problems. I remember struggling with the concepts of x-position in space at time t, y-position at time t and then a particle's position in x-y space at time t. I added some animations to the Curvilinear Motion page to make things a bit clearer.
You'll see the animations towards the bottom of the page, after the explanation and examples. Go to:
Of course, in this work, like most mathematics, it's essential to label your axes correctly!
## 2. Estimation and the anchoring principle
Most of you know I'm a big fan of employing technological approaches to "doing math". It doesn't make a lot of sense to me that we still make students churn through endless algebra exercises, when such things are much more quickly and accurately done by computers.
However, such approaches come with a large danger - a small misunderstanding when setting up the computer to solve a problem can have drastic implications for the solution, which can turn out to be very wrong.
So even more these days, good estimation skills have become essential. Before we ever get a computer (or calculator) to solve a problem for us, we should step back and come up with a "ball park figure" for the final answer. If the computer gives a very different answer to what we expected, we need to investigate further.
With these thoughts in mind, I was interested to come across this article by the Norman Nielsen Group (user-experience experts) on the Anchoring Principle, an important concept for learning, but it can also be used by advertisers and political parties to sway opinions.
The first part of the article discusses some very interesting experiments in estimation. How did you go? Go to: The Anchoring Principle
(The remainder of the article talks about how to apply the results to better user experience - which is worth reading for anyone who ever uses - or will design - a computer interface!)
## 3. Bill Gates and his pancakes
Photo credit: St Marks Beaumont
In the photo, we see a stack of pancakes where some have a larger radius than others. What if we needed to flip the stack so the largest pancake is on the bottom, and each subsequent one is smaller as you go up? What is the minimum number of flips needed to achieve this?
This was a problem given to a Harvard undergraduate class in the 1970s, and Bill Gates was the first to solve it, with an elegant solution.
See the full story from National Public Radio:
Before Microsoft, Gates Solved A Pancake Problem
Here is the actual paper "William Gates" published in 1979.
Bounds for sorting by prefix reversal [PDF, 11 pages]
Of course, efficient sorting is essential in computer operations, and I suspect that's why he was interested to solve this.
## 4. Math Movies
### (a) How to spot a misleading graph
There are still some of us who regard finding the truth as an important human endeavor.
Many times, we can be distracted or confused by graphs that deliberately try to mislead.
### (b) How religion spread around the world
This animated map follows a timeline, showing how and when the major religions spread around the world.
## 5. Math puzzles
The puzzle in the last IntMath Newsletter asked about divisibility of a number. Correct answers with sufficient explanation were submitted by Ibrahim, Ajayshukla, Georgios, Alex, Elizabeth, Russell, Saikrishna, Aidan, Bandu, Vijay, Charles, Tugume, Kevin, Tomas, Gerard, and Alfredo.
Special mention to Ibrahim and Bandu who made use of the math entry facility in the blog comments.
### New math puzzle: Who dies?
You can leave your response here.
## 6. Final thought - Truth
Yolande Cornelia "Nikki" Giovanni, Jr. is an American poet, writer, commentator, activist, and educator. Her words are particularly relevant today.
"If now isn't a good time for the truth I don't see when we'll get to it." [Nikki Giovanni]
Until next time, enjoy whatever you learn.
1. Amusa says:
Nobody dies
2. Shivakumar Konnur says:
No one dies.
As the ball have less weight and one side oppen.
And ball will stop over.
Hence no one dies
3. Glenn says:
D and e
4. James says:
No one.
5. Richard says:
D will die
6. zeina says:
C&B&A will die if E pushes the stone
7. Eddie says:
No one dies. The cut-out on the boulder saves D, the radius of the teeter totter strikes the wall, plus the lower weight of the initial boulder cannot lift the second boulder, thereby saving C, so therefore the system comes to a halt, allowing all to live a long and prosperous life.
8. Elke Kelly says:
I would say, that if E pushes the stone C will die.
9. karam says:
The stone may kill any of them
10. zia qalandry says:
c and d.
11. Murray says:
There's quite a divergence of answers! Here's my take.
To get a better idea of what's happening, I overlaid a grid with the same dimensions given in the original image. This is the starting scenario:
The stone rolls 0.5 units before it heads down the slope:
This is the position ater travelling π = 3.14 units:
The slope has length 5 (it's a 3-4-5 triangle) and with another 0.8 of a unit, the total distance rolled by the time the stone reaches D is 2π = 6.28 units, making a mess of person D:
When the stone falls onto the seesaw, C has an unfortunate demise:
A key question is whether the rolling stone has enough momentum to project the other stone (which is heavier) high enough and far enough to land on B. (Notice there's also a small ledge at the left end of the seesaw, which will reduce the distance the stone will go to the left.) It's possible the heavier stone brings the motion to a stop, thus saving C.
I shall leave further speculation up to the reader.
12. karim azmy says:
C and E will be hurt
### Comment Preview
HTML: You can use simple tags like <b>, <a href="...">, etc.
To enter math, you can can either:
1. Use simple calculator-like input in the following format (surround your math in backticks, or qq on tablet or phone):
a^2 = sqrt(b^2 + c^2)
(See more on ASCIIMath syntax); or
2. Use simple LaTeX in the following format. Surround your math with $$ and $$.
$$\int g dx = \sqrt{\frac{a}{b}}$$
(This is standard simple LaTeX.)
NOTE: You can mix both types of math entry in your comment.
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2019-12-12 13:11:19
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https://im.kendallhunt.com/MS/teachers/1/7/16/preparation.html
|
# Lesson 16
Common Factors
### Lesson Narrative
In this lesson, students use contextual situations to learn about common factors and the greatest common factor of two whole numbers. They develop strategies for finding common multiples and least common multiples. They develop a definition of the terms common factor and greatest common factor for two whole numbers (MP6).
### Learning Goals
Teacher Facing
• Comprehend (orally and in writing) the terms “factor,” “common factor,” and “greatest common factor.”
• Explain (orally and in writing) how to determine the greatest common factor of two whole numbers less than 100.
• List the factors of a number and identify common factors for two numbers in a real-world situation.
### Student Facing
Let’s use factors to solve problems.
### Required Preparation
For the first classroom activity, "Diego's Bake Sale," provide access to two different colors of snap cubes (48 of one color and 64 of the other) for students who would benefit from manipulatives. For students with visual impairment, provide access to manipulatives that are distinguished by their shape rather than color.
In the second classroom activity, "Greatest Common Factor," it may be helpful for some students to have access to graph paper to make rectangles that will help them find all possible factors of a whole number.
### Student Facing
• I can explain what a common factor is.
• I can explain what the greatest common factor is.
• I can find the greatest common factor of two whole numbers.
### Glossary Entries
• common factor
A common factor of two numbers is a number that divides evenly into both numbers. For example, 5 is a common factor of 15 and 20, because $$15 \div 5 = 3$$ and $$20 \div 5 = 4$$. Both of the quotients, 3 and 4, are whole numbers.
• The factors of 15 are 1, 3, 5, and 15.
• The factors of 20 are 1, 2, 4, 5, 10, and 20.
• greatest common factor
The greatest common factor of two numbers is the largest number that divides evenly into both numbers. Sometimes we call this the GCF. For example, 15 is the greatest common factor of 45 and 60.
• The factors of 45 are 1, 3, 5, 9, 15, and 45.
• The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.
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2023-03-20 12:36:21
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https://www.onlinemathlearning.com/herons-formula.html
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# Heron's Formula
Related Topics:
More Lessons for High School Regents Exam
Math Worksheets
High School Math based on the topics required for the Regents Exam conducted by NYSED.
What is the Heron's Formula?
The Heron's Formula is used to determine the area of a triangle while given the lengths of the sides.
Given that the length of the sides of the triangle is a, b and c.
$$s = \frac{{a + b + c}}{2}$$
$$Area = \sqrt {s(s - a)(s - b)(s - c)}$$
Heron's Formula
Using Heron's Formula to determine the area of a triangle while only knowing the lengths of the sides
Using Heron formula
Find the area of a triangle using Heron formula
Heron's Formula This video explains how to determine the area of a triangle given the length of the three sides.
Derivation or Proof of Heron's Formula This video will help you to derive Heron's Formula to find the Area of Oblique Triangles using three sides
Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.
You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.
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2018-08-22 05:55:56
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https://math.stackexchange.com/questions/2906106/confusion-if-the-series-converges-or-not-alternating-series-test
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# Confusion if the series converges or not (alternating series test)
I have to test for convergence and absolute convergence for the following series:
$$\sum_{k=1}^{\infty} (-1)^k \frac{k}{1+2k^2}$$
Because of the alternating series test, I have to verify if the series decreases monotonically and then show that the limit goes to zero. I don't have any problems showing that it decreases monotonically, but I have trouble showing if the limit is zero.
$$\lim_{x\to\infty} \frac{k}{1+2k^2} = \frac{1}{2k} \rightarrow 0$$ Therefore it converges.
But with the direct comparison test it is: $$\mid (-1)^k \frac{k}{1+2k^2}\mid = \frac{k}{1+2k^2} =\frac{1}{\frac{1}{k}+2k}\geq \frac{1}{k+2k} = \frac{1}{3k}$$
Which is similar to the harmonic series$\sum_{k=1}^{\infty}\frac{1}{k}$ hence the series should diverge. So my question is, does it diverge or converge? And how do I know if it converges absolutely? Thank you for your time and help!
• Yes, the series diverges absolutely. The alternating series test tells you only if it converges. You must decide if it converges conditionally or absolutely... – PhysicsMathsLove Sep 5 '18 at 10:25
• You showed that the series converges, but it does not converge absolutely. – mechanodroid Sep 5 '18 at 10:25
• Indeed you have already proved that the series is convergent but not absolutely convergent. – Rigel Sep 5 '18 at 10:26
Since $|a_k|$ decreases monotonically, and $a_k \to 0$, you can conclude that $\sum a_k$ converges, by the alternating series test.
Now, we say that a series $\sum a_k$ coverges absolutely if $\sum |a_k|$ converges. But as you show, $\sum |a_k|$ doesn't converge, so we have a conditionally convergent series, one that converges, but not absolutely.
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2019-09-17 01:12:37
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https://solvedlib.com/n/obtnined-by-shllling-iho-grph-ol-w-sin-to-iho-nigm-urita,15248341
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# Obtnined by shllling iho grph ol W sin( = )to Iho nigm urita nnd downwnrd unita (1 poini) Find Ihe
###### Question:
obtnined by shllling iho grph ol W sin( = )to Iho nigm urita nnd downwnrd unita (1 poini) Find Ihe aquation ol sle Wavu Inat verlcally strutchod by Urclar ol "hn compntod to sin(-)
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http://www.danielhellerman.com/content/114/44/E9206.full?sid=73335c42-6016-4c26-a698-a76952e4153e
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# Methyl-compound use and slow growth characterize microbial life in 2-km-deep subseafloor coal and shale beds
1. aDivision of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125;
2. bGeomicrobiology Group, Kochi Institute for Core Sample Research, Japan Agency for Marine-Earth Science and Technology (JAMSTEC), Monobe B200, Nankoku, Kochi 783-8502, Japan;
3. cGeobiotechnology Group, Research and Development Center for Submarine Resources, JAMSTEC, Monobe B200, Nankoku, Kochi 783-8502, Japan;
4. dResearch and Development Center for Ocean Drilling Science, JAMSTEC, Yokohama, Kanagawa 236-0001, Japan
1. Edited by David M. Karl, University of Hawaii, Honolulu, HI, and approved September 6, 2017 (received for review May 5, 2017)
## Significance
Microbial cells are widespread in diverse deep subseafloor environments; however, the viability, growth, and ecophysiology of these low-abundance organisms are poorly understood. Using single-cell–targeted stable isotope probing incubations combined with nanometer-scale secondary ion mass spectrometry, we measured the metabolic activity and generation times of thermally adapted microorganisms within Miocene-aged coal and shale bed samples collected from 2 km below the seafloor during Integrated Ocean Drilling Program Expedition 337. Microorganisms from the shale and coal were capable of metabolizing methylated substrates, including methylamine and methanol, when incubated at their in situ temperature of 45 °C, but had exceedingly slow growth, with biomass generation times ranging from less than a year to hundreds of years as measured by the passive tracer deuterated water.
## Abstract
The past decade of scientific ocean drilling has revealed seemingly ubiquitous, slow-growing microbial life within a range of deep biosphere habitats. Integrated Ocean Drilling Program Expedition 337 expanded these studies by successfully coring Miocene-aged coal beds 2 km below the seafloor hypothesized to be “hot spots” for microbial life. To characterize the activity of coal-associated microorganisms from this site, a series of stable isotope probing (SIP) experiments were conducted using intact pieces of coal and overlying shale incubated at in situ temperatures (45 °C). The 30-month SIP incubations were amended with deuterated water as a passive tracer for growth and different combinations of 13C- or 15N-labeled methanol, methylamine, and ammonium added at low (micromolar) concentrations to investigate methylotrophy in the deep subseafloor biosphere. Although the cell densities were low (50–2,000 cells per cubic centimeter), bulk geochemical measurements and single-cell–targeted nanometer-scale secondary ion mass spectrometry demonstrated active metabolism of methylated substrates by the thermally adapted microbial assemblage, with differing substrate utilization profiles between coal and shale incubations. The conversion of labeled methylamine and methanol was predominantly through heterotrophic processes, with only minor stimulation of methanogenesis. These findings were consistent with in situ and incubation 16S rRNA gene surveys. Microbial growth estimates in the incubations ranged from several months to over 100 y, representing some of the slowest direct measurements of environmental microbial biosynthesis rates. Collectively, these data highlight a small, but viable, deep coal bed biosphere characterized by extremely slow-growing heterotrophs that can utilize a diverse range of carbon and nitrogen substrates.
Advances in deep-sea drilling technology and analytical methods for cored samples have led to the discovery of microbial life in a range of deep subseafloor habitats, from Earth’s most oligotrophic sediments (1) to its largest aquifer (2). To probe the limits for microbial life, Integrated Ocean Drilling Program (IODP) Expedition 337 sampled a deeply buried [2 km below seafloor (kmbsf)] coal bed system of terrestrial origin and low thermal alteration at Site C0020. In addition to increased depth, temperature, and organic carbon input from coal beds, Site C0020 lacked the dominant electron acceptors (oxygen and sulfate) that have been the focus of other deep-biosphere–focused IODP expeditions (3, 4). Initial results from Expedition 337 showed cell abundances no longer tracked the global depth trend based on previous cruises (5), suggesting that life-limiting conditions may have been reached at these extreme depths (6). The coal bed, however, contained comparatively high concentrations of microbial cells, 10- to 1,000-fold higher than in the adjacent shale and sandstone lithologies, which is a feature common to lignite-containing terrestrial sediments, where higher organic substrate concentrations sourced from the lignite can sustain higher cell numbers (7, 8). Recovered 16S rRNA gene diversity from the coal beds revealed an assemblage that phylogenetically resembled modern terrestrial environments (e.g., peat or forest soil), which was interpreted as representing indigenous microorganisms within Miocene-aged coal, rather than contaminants or overprinting of commonly observed marine sediment microbes (6).
While the concept of coal beds hosting microbial life for millions of years has existed for almost a century (9), microbial activity and metabolic potential through time remain largely unknown in these systems. Measuring activity in the deep biosphere requires a balance between stimulating life enough to observe a metabolic signal and overstimulation of the system such that experimental conditions no longer approximate in situ conditions (10). Subseafloor microbial metabolic activity, such as bacterial sulfate reduction, is often interpreted through geochemical profiles or microcosm incubations to determine rates of substrate utilization (11??14). Other approaches include traditional (13) and subseafloor-adapted (15) cultivation-based methods, analysis of actively transcribed genes (16), and biomolecule degradation modeling (17) from recovered core materials. These bulk methods have been valuable in developing a picture of the deep subseafloor biosphere characterized by viable, but extremely slow-growing, microorganisms with a range of metabolisms. There are limitations to these methods in some subseafloor habitats like the Shimokita coal bed, off Japan, where biomass is extremely low (1–1,000 cells per cubic centimeter), and bulk analyses may lack the sensitively to detect microbial activity and growth. Stable isotope probing (SIP) coupled to single-cell–targeted nanometer-scale secondary ion mass spectrometry (NanoSIMS) analysis is well suited for direct activity measurements in low biomass environments and has been applied in select deep subseafloor biosphere studies (18). This technique additionally offers a unique perspective on the cell-to-cell variation in growth and metabolic potential of co-occurring microorganisms in nature.
Here we used SIP-NanoSIMS to determine the viability and metabolic capability of microorganisms associated with the Shimokita Peninsula coal bed and surrounding lithologies since burial over 20 Mya. We focused specifically on methyl substrate metabolism, which could be hypothetically sourced from coal breakdown products, and rates of microbial growth in long-term, high-temperature microcosm experiments. Taking advantage of the ability to measure three different stable isotope tracers in parallel with NanoSIMS, we applied combinations of 13C- and 15N-labeled substrates in tandem with deuterated water (2H2O) as a passive tracer for growth. The use of 2H2O with NanoSIMS is an effective method for measuring single-cell microbial biosynthesis with high sensitivity, capable of detecting extraordinarily slow rates of growth (19) predicted to be characteristic of subseafloor biosphere ecosystems (20). This tracer has been successfully combined with lipid biomarker and NanoSIMS analyses for assessing rates of growth in pure culture and mammalian microbiome studies (19, 21, 22), soils (23), and seabed methane seep environments (24, 25), but it has not yet been applied to study microbial growth in low-biomass, deep subsurface ecosystems. This multiple-isotope SIP-NanoSIMS study of microbial life in deep subseafloor environments recovered evidence for methyl substrate metabolism in deep coal and shale beds and exceedingly slow growth within this low-abundance, high-temperature microbial community.
## Materials and Methods
### SIP Incubation Preparation.
IODP Expedition 337 operations commenced July 26 and continued through September 30, 2012, on the drilling research vessel Chikyu. Utilizing riser drilling, a sedimentary sequence was recovered down to 2,466 mbsf at Site C0020 Hole A (41°10′36′′N, 142°12′02′′E) in a water depth of 1,180 m off the Shimokita Peninsula, Japan. The drilled sequence transitioned from open marine (youngest; late Pliocene, ～5 Ma) to terrestrial (oldest; late Oligocene, ～30 Ma) with depth. Models for maximum temperature reached by Expedition 337 coring report 63.7 °C (26). Shipboard sedimentological, geochemical, and microbiological data and methods are available through IODP publications (27). Additional coal petrography is available in a report by Gross et al. (28).
A total of 52 incubation amendment conditions were prepared onboard to interrogate a range of potential deep biosphere metabolic strategies, and then incubated back in the laboratory at temperatures approximating in situ measurements (27). In this study, incubations from shale (core 8L4, 1,606 mbsf, 37 °C incubation temperature), coal (core 15R3, 1,921 mbsf, 45 °C incubation temperature), and mixed (homogenized mixture from multiple cores 19R1, 19R5, 19R7, 20R3, 23R6, 23R8, 24R3, 25R1, 25R2, and 25R3; 1,950–1,999 mbsf; 45 °C incubation temperature) with methanol, methylamine, and ammonium substrate additions were analyzed. Age estimates of these samples are early to middle Miocene (6). In situ temperatures ranged from 38–48 °C at these sample depths, with pressures of ～30 MPa (27). Two coal beds were included in these incubations: a shallower coal-only sample deposited under more marine-influenced conditions (～1,921 mbsf, core 15R3) and a deeper coal bed deposited under more limnic conditions included in the mixed lithology sample (～2,000 mbsf, cores 24R3 and 25R1).
Cores used for incubations were prepared by removal of outer drill-fluid–contaminated layers by means of a sterile ceramic knife as soon as possible after core recovery and stored at 4 °C until incubation preparation, while maintaining an anaerobic atmosphere during the entire process. For preparation of the SIP incubations, the interior portion of the core was manually crushed into centimeter-sized pieces under sterile, anaerobic conditions and distributed evenly into sterile 50-mL glass vials with butyl rubber stoppers and screw caps (Nichidenrika-Glass Co. Ltd.). Vials were flushed with argon and pressurized to 1 atm argon headspace. Sterile C-, N-, and S-free media (1% PBS, 30 g/L NaCl, 12 g/L MgCl2, and 3 g/L KCl) was prepared anaerobically with 2H2O [20 atom % (at. %) 2H2O]. This value (20 at. % 2H2O) was selected as the highest level of enrichment with little to no effect on the activity of microorganisms in pure culture (21, 22). Time point 1, time point 2, and autoclaved treatments were prepared for each substrate condition. Time point 1 incubations lasted for 6 mo, while time point 2 and autoclaved treatments were maintained at the in situ incubation temperature for 2.5 y. Due to low levels of activity ascertained from geochemical measurements, all NanoSIMS analyses were conducted on time point 2 and autoclaved samples. Amendments and incubation conditions for the methyl-substrate subset analyzed in this study are listed in Table 1. Equimolar amounts of substrate (30 μmol C, 1.5 mM final; 3 μmol N, 0.15 mM final) were added across incubation conditions at 50 at. % enrichment (Cambridge Isotopes). Hydrogen was added as 5 mL of 100% H2 overpressure to incubations (～0.15 atm H2). A full list of the additional incubation conditions prepared onboard is provided in cruise methods (29). Alkalinity (34.39–9.68 mM) and ammonium (2.80–1.83 mM) concentrations from formation fluid samples collected onboard exceed concentrations of C and N amendments (30). Concentrations of methylamine (0.05 mM) and methanol (1 mM) measured from another study of lignite coal (31) also suggested our substrate additions were environmentally relevant.
View this table:
Table 1.
Geochemical and cell enumeration for SIP-NanoSIMS incubations
After 30 months of incubation (March 2014), all treatments were sampled for geochemical analyses before preparation for NanoSIMS. Three milliliters of headspace gas was removed to a vial filled with 0.1 M NaOH for methane analysis. About 1 mL of liquid was filtered through a 0.1-μm 13-mm Whatman Polycarbonate Nuclepore Track-Etched Membrane (110405) for dissolved inorganic carbon (DIC) analysis. A detailed description of methane and DIC analyses is provided in Supporting Information.
### Sample Preparation for NanoSIMS Analysis.
To overcome technical challenges for NanoSIMS analysis of low-biomass samples, cell separation and fluorescence-activated cell sorting (FACS) were used to directly concentrate cells in a small analysis area, ～1–0.5 mm2 (32). NanoSIMS samples were prepared from paraformaldehyde (PFA)-fixed cell separates after 30 months of incubation. Cell preservation, separation, enumeration, and FACS were all conducted in the clean booth and clean room facilities at the Kochi Institute for Core Sample Research, Japan Agency for Marine-Earth Science and Technology. Half of the solid portion and half of the liquid portion of each sample were fixed overnight in a solution of 2% PFA, 3× PBS. Samples were then subjected to two washes, incubating in 3× PBS for 6 h and then for 2 h after each wash. Samples were centrifuged (3,500 × g), and supernatant was decanted after each wash. PFA-fixed samples were stored in 50% ethanol, 3× PBS. The other half of the sample was preserved in glyTE (70% glycerol, 100 mM Tris, and 10 mM EDTA; Bigelow Single Cell Genomics Center preservation protocol), frozen by a cell alive system (CAS; ABI Corporation, Ltd.), and stored at ?80 °C (33).
One milliliter of liquid and ～1 g of sediment chips were subsampled by pipette and sterile cell culture loop, respectively, from the PFA-fixed sample. Cell separation, microscopy, and sorting procedures followed the method of Morono et al. (32), with the following modifications to improve cell recovery: (i) Samples were sonicated (Bioruptor UCD-250; COSMO BIO) in an ice bath for 20 cycles of 30 s at 200 W on and 30 s off, and (ii) samples were incubated in hydrofluoric acid after initial sonication, rather than after first-density gradient separation. The cell detection limit was determined by no-sample–added controls run in parallel with samples. The same cell enumeration method was used for incubations and initial shipboard samples.
Cells were stained with SYBR Green I [1:40 dilution of SYBR Green in TE (10 mM Tris and 1 mM EDTA)] and sorted following the flow cytometry protocol of Morono et al. (32). Sorted cells were concentrated directly from the sorter onto NanoSIMS-compatible 0.2-μm polycarbonate filters coated with indium tin oxide (ITO) as described by Morono et al. (18) and Inagaki et al. (6). ITO coating on polycarbonate membranes (Isopore GTBP02500; Millipore) was prepared by a sputtering deposition technique at Astellatech Co. Ltd. Scanning electron microscopy of the filters was done on a Zeiss 1550 VP Field Emission Scanning Electron Microscope at the Geological and Planetary Sciences Division Analytical Facility at the California Institute of Technology (Caltech), and SYBR-stained cells were imaged with a BX51 epifluorescence microscope (Olympus) using 20× (UPlanFL N) dry, 60× (PlanApo N), and 100× (UPlanFL N) oil immersion objectives (Fig. S1).
View larger version:
Fig. S1.
SYBR-stained cells after separation and FACS concentration on ITO-coated 0.2-μm polycarbonate NanoSIMS membranes from the two highest cell abundance conditions in 15R3 coal and mixed lithology. SYBR and SEM images highlight some distinctions in cell morphology between the two methanol amended samples. White arrows indicate region of SYBR image on larger membrane target area. The 15R3 coal incubation amended with methylamine (A), SEM image of A (B), 15R3 coal amended with methanol (C), C magnified with false color (D), mixed lithology amended with methanol + H2 + ammonium (E), SEM image of E (F), mixed lithology amended with methanol (G), and mixed lithology amended with methanol magnified with false color (H; Inset, the same sample filtered before cell sorting to demonstrate cell density without concentration). (Scale bar: H, Inset, 20 μm.)
### NanoSIMS Instrument Tuning and Analysis.
Cell targets were identified (by SYBR stain) and marked on NanoSIMS membranes with a laser dissection microscope (LMD6000; Leica Microsystems) for ease of rediscovery on the NanoSIMS. Samples were analyzed by raster ion imaging with a CAMECA NanoSIMS 50L at the Caltech Microanalysis Center in the Division of Geological and Planetary Sciences. A focused primary Cs+ beam of ～1 pA was used for sample collection, with rasters of 256 × 256 or 512 × 512 pixels. Ions 1H (EM 1), 2H (EM 2), 12C2 (EM 3), 13C12C (EM 4), 12C14N (EM 5), and 12C15N (EM 6) were measured simultaneously. Technical development has been described by Kopf et al. (22). Collection began after a presputtering of equal intensity to one collection frame (～45 min). Recorded images and data were processed using [email protected] software (34). Images were deadtime-corrected, and individual ion image frames were merged and aligned using the 12C14N ion image to correct for drift during acquisition. Cell-based regions of interest (ROIs) were determined by “interactive thresholding” with the 12C14N ion image. Final ion images and counts per ROI were calculated by summation of ion counts for each pixel over all scans. Calculation of cell size from [email protected] software was based on the diameter of a circle (in micrometers) derived from the equivalent number of pixels from the ROI drawn on the NanoSIMS 14N12C ion image. We also confirmed that cell ROIs had a total C-to-total N ratio that was distinct from the background correction ROIs or coal to ensure drawn ROIs only included biomass targets.
### Sample Preparation for DNA and iTag Processing.
Cells for gDNA extraction were recovered from unfixed samples stored in glycerol using a modified version of the NanoSIMS protocol by omitting detergent and methanol treatments and only shaking (60 min at 500 rpm using a Shake Master; Bio Medical Science) before centrifugation. Nycodenz was also omitted from the gradient separation solution to prevent PCR inhibition (linear gradients of 10%, 30%, and 67% sodium polytungstate solutions were utilized instead). After separation, the staining and flow cytometry sorting protocol was identical to NanoSIMS processing, with the exception that the cells were collected into a UV sterilized 0.5-mL PCR tube rather than on the ITO filter. Sorted cells were stored at ?20 °C until DNA extraction. A proteinase K digestion mixture consisting of 20 mM Tris?HCl (pH 8.0), 5 mM MgCl2, 2 mM DTT, 4 U/mL heat-labile double-strand specific DNase (HL-dsDNase; ArcticZymes), and 0.4 mg/mL proteinase K was utilized for cell lysis. Before proteinase K application, potential endogenous DNA contamination in the proteinase K mixture was removed by digestion using HL-dsDNase. The HL-dsDNase extracellular DNA digestion was carried out at 25 °C for 2 h, followed by an inactivation step at 60 °C for 1 h. The sorted cells and negative controls (sheath fluid) were then lysed using the pretreated proteinase K. One-third volume of the HL-dsDnase–treated proteinase K solution was added to one volume of sheath fluid, which contained the sorted cells (total volume varied by cell sort time). Proteinase K digestion was performed at 60 °C for 8 h, followed by proteinase K inactivation at 95 °C for 10 min. The resultant DNA extract was then used directly for first-round PCR amplification with universal primers targeting the V4 region of the 16S rRNA gene (U515F: 5′-TGY CAG CMG CCG CGG TAA-3′, U806R: 5′-GGA CTA CHV GGG TWT CTA AT-3′). First-round PCR amplicons were quantified and purified via gel electrophoresis and then indexed and barcoded in the second PCR assay. The iTag sequencing was performed on a MiSeq Illumina platform following the methods outlined by Hoshino and Inagaki (35). Resultant paired-end reads were processed and assigned taxonomy using Qiime (36) and a custom version of the URef_NR99_119_SILVA database (37, 38) optimized for marine sedimentary microorganisms. The total sequences and relative abundance of most abundant taxa are provided in Dataset S1, and all taxa are provided in Dataset S2.
## Results
### Evaluation of Incubation Cellular Abundances and Sample Purity.
Stringent contamination monitoring during Expedition 337 suggested sandstone lithologies were the most susceptible to drilling mud contamination (27). Therefore, only shale and coal lithologies were used in this study. While cells were detected in the drilling fluids [2.66 × 108 cells per milliliter (30)] and have the potential to contaminate downstream analyses (6), two lines of evidence suggest exogenous cell contamination was minimal in our long-term stable isotope incubation study. First, molecular screening of the 16S rRNA sequences recovered from our 37 °C and 45 °C incubations was distinct from that of sequences associated with common drilling mud contaminants from Expedition 337 (6). Second, cultivation attempts to enrich for bacteria (39) or fungi (40) from drilling fluids were unsuccessful, suggesting that contaminant cells had low viability and were unlikely to overgrow the indigenous microorganisms in our moderately thermophilic, low-nutrient coal and shale incubations. The low microbial viability of exogenous microorganisms may be due to the chemical composition of the drilling mud, containing bentonite and detergent chemicals, as well as the extreme physical conditions experienced during circulation of the high pH (10.5–11) drilling fluids between the 2-kmbsf drilling formation and the ship, with repeated cycling between oxic and anoxic conditions and temperatures ranging from 2–60 °C.
Cell abundances were calculated from concentrates after cell separation from the rock matrix (Table 1). Cell numbers measured at the end of the incubations, including no amendment controls, were all elevated relative to cell concentrations calculated from shipboard-processed samples, reported as 7 cells per gram of shale, 200 cells per gram of coal, and 942 cells per gram of mixed lithology (6). The elevated cell concentration in the incubated no-amendment controls suggested that the addition of water alone could have been enough to stimulate growth in this system using substrates native to the coal and shale. Microbial cells stained with SYBR Green I appeared as small cocci (single or doublets) or rods, with the exception of the mixed incubation, which also contained filamentous microorganisms (Fig. S1). The range of microbial cell sizes determined from NanoSIMS ion images yielded median cell lengths ranging from 0.7–1.2 μm in shale, 0.3–0.9 μm in coal, and 0.4–1.0 μm in the mixed incubations, consistent with previous reports from deep subseafloor sediments (41).
### Assessing Methylated Substrate Metabolism by Bulk 13C and 2H Geochemical Analysis of Methane and DIC Measurements.
Total methane production at the end of the incubation period was higher in coal incubations, relative to shale and mixed samples, and highest overall in nonautoclaved coal samples (Table 1). Lower, but detectable, levels of methane production were also measured in the mixed incubations. The δ13C-CH4 was measured for coal incubations, which showed minimal 13C enrichment from the added 13C-methanol or methylamine (?42.3 to ?58.8‰) relative in situ methane values [approximately ?62‰ in both formation water and core gas sampling (6)]. Kinetic isotope effects during methane desorption could account for a portion of the methane enrichment [<5‰ (42, 43)], but the higher δ13C-CH4 enrichments (～15–20‰) are unlikely explained by kinetic fractionation alone, and most likely contain some 13C-substrate incorporation. If all isotopic enrichment in these heaviest samples were from 13C substrate, it would correspond to ～0.04% of total methane being sourced from the added 13C-methylamine or 13C-methanol. Since H2 in the incubations is predicted to exchange with 2H2O (44), any H2-derived methane produced over the course of the incubations should also be enriched in δ2H-CH4. As observed with 13C measurements, the values of δ2H-CH4 were also minimally enriched in deuterium (?96.5 to ?198‰) relative to the in situ δ2H-CH4 [approximately ?200‰ in both formation water and core gas sampling (6)]. The most enriched δ2H-CH4 was observed in incubations amended with 13C-methanol + 15N-ammonium + H2 (?96.5‰). Isotope mass balance indicates this level of enrichment could at most account for ～0.007% of the total methane recovered in the incubation. Therefore, both isotopic methods demonstrate consistently low, but detectable, levels of biogenic methane production in the coal incubations, with carbon isotope data supporting the presence of some methylotrophic methanogenesis.
Methylotrophic methanogenesis conversion of the 13C-labeled methanol or methylamine should also lead to the production of 13C-DIC in our incubations. Although shale incubations had overall higher DIC concentrations than incubations with only coal, DIC concentrations varied widely within each lithology. Methanol-amended shale incubations produced enriched δ13C-DIC values, with the highest values observed in the 13C-methanol + 15N-ammonium + H2 amendment (+430.03‰). In contrast, methylamine amendments to the shale incubations did not show any appreciable enrichment (?2.86 to +8.32‰). The opposite trend was observed in the coal incubations, where 13C-DIC production from 13C-methanol was minimal (?15.14 to +26.67‰), but some of the 13C-methylamine incubations (autoclaved treatment) produced enriched δ13C-DIC (+108.06‰). The 13C-DIC trends with the mixed incubations were similar to what was observed in the coal incubations, with the highest enrichment in 13C-methylamine amendments (+327.56‰) and lower activity observed with the addition of 13C-methanol (+6.32 to +44.64‰). By comparison, the in situ gaseous 13C-CO2 ranged from ?19 to +1‰, with the most 13C-enriched CO2 recovered from the coal beds (6). Assuming steady-state production during the incubation, the highest δ13C-DIC enrichment in coal would correspond to 2–0.2 pmol DIC per cm3?d?1 or 0.2–0.1% of total moles of DIC generated from 13C-methylamine and 13C-methanol, respectively. In summary, the observed minor contributions of labeled methane and DIC production in our incubations indicated the potential for both methane- and DIC-generating metabolisms (methylotrophic methanogenesis) in addition to DIC-only–generating metabolisms (methyl fermentation).
### Single-Cell Substrate Uptake and Generation Time Estimates of Deep Subseafloor Microorganisms by 15N and 2H2O.
Of the 12 different sample conditions prepared for each lithology, eight shale, 10 coal, and six mixed lithology incubations had cells detectible by NanoSIMS after the cell separation protocol. The number of cellular ROIs analyzed per incubation ranged from nine to 306 (Fig. 1). Due to low cellular 13C enrichment relative to carbon background on NanoSIMS filters, single-cell 13C-carbon enrichment was within instrument error of natural abundance, and was therefore not included. Although it cannot be determined if active cells were from the regeneration of existing biomass or the generation of new cells through division, at least one instance of potential cell division was observed via NanoSIMS from the coal incubation amended with 15N-methylamine (Fig. 2).
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Fig. 2.
1H ion (A), 2H ion (B), 14N12C ion (C), and 15N12C ion (D) counts of ROIs from 15R3 coal with 15N-methylamine and 2H2O. The pink arrow indicates biomass that is only present in 1H and 14N12C ion maps and connected to biomass present in all four ion maps, indicative of new biomass growth. Large circles are used for nitrogen background standard correction. There are also six other ROIs drawn on the 14N12C map that do not show 15N12C or 2H enrichment for comparison.
The cellular isotopic enrichment in 2H and 15N was highest in the 45 °C 15N-methylamine coal incubations, with both isotopes showing a similar bimodal distribution of ROIs (Fig. 1). This sample also had the highest cell number of all incubations. When compared with the paired 15N-methylamine sample set with H2 added, cellular activities (2H and 15N enrichment) were lower, suggesting the methylamine-utilizing microorganisms in the coal could be inhibited by H2. Although not as high as 15N-methylamine, 2H and 15N enrichment was observed in treatments with 13C-methylamine + 15N-ammonium and 2H enrichment was seen with 13C-methanol in coal samples. In shale incubations, the highest enrichment in bulk δ13C-DIC and cellular 2H and 15N was seen in treatments amended with 13C-methanol + 15N-ammonium + H2. Bulk δ13C-DIC was not enriched in 13C-methylamine–amended shale incubations, yet 2H and 15N (from 15N-ammonium) enrichment was still observed in cells from these incubations. The 15N-methylamine–amended shale incubations without hydrogen also showed high 2H and 15N incorporation. Finally, mixed lithology incubations had the highest δ13C-DIC (+327‰) in the 13C-methylamine + 15N-ammonium treatment, although this incubation did not yield enough cells to analyze by NanoSIMS. From the mixed incubations that could be analyzed by NanoSIMS, the 13C-methanol amendment had the highest cellular 2H enrichment (no 15N was added to this incubation).
In summary, both methanol and methylamine treatments produced 13C-DIC in coal and mixed incubations, but only methanol produced 13C-DIC in the shale incubations. Cells enriched with 15N were recovered from both 15N-methlyamine– and 15N-ammonium–amended incubations in shale and coal incubations, but only 15N-ammonium incubations showed cellular 15N uptake for mixed lithology incubations. The 2H-enriched cells were recovered across all substrate combinations and all lithologies. While bulk δ13C-DIC suggested methanol and shale, and methylamine and coal amendments and lithologies were correlated, single-cell incorporation suggests that viability and activity in these incubations were not uniformly linked to substrate addition or lithology.
Biomass generation times were calculated from the most robust NanoSIMS cellular ROI enrichments in 2H and 15N (Materials and Methods). As such, our estimates of microbial generation times represent values for the fastest growing cells within our incubations (Fig. 3), yet still result in generation time estimates from a few months to more than 100 y. The greatest overall 2H and 15N incorporation was from coal amended with 15N-methylamine, with much longer generation times for the rest of the incubations, which all had hydrogen added. This concurs with 13C-DIC production results, where samples containing coal had depressed activity when H2 was added. This sample also had the greatest offset between 2H and 15N enrichment, where generation time estimates calculated from cellular 2H2O enrichment were shorter than 15N-nitrogen enrichment from methylamine. The mixed lithology samples amended with 15N-ammonium or 15N-methylamine as the nitrogen source had more comparable uptake of 15N and 2H, with slightly higher enrichment observed in the 15N-ammonium treatment. This suggests that while ammonium and methylamine can both be assimilated into biomass, ammonium was more readily used for anabolism compared with methylamine.
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Fig. 3.
2H biomass generation time versus 15N biomass generation time, in years, calculated from ROIs where the 15N12N ion counts were above the Poisson error (Materials and Methods). The dashed line is where 2H and 15N generation time is at parity.
### Comparison of 16S rRNA Gene Profiles to Geochemically Predicted Metabolisms.
Microbial diversity surveys using 16S rRNA gene iTag sequencing were generated from FAC sorting of the seven most active shale (n = 2), coal (n = 3), and mixed lithology (n = 2) SIP incubations after 2.5 y of incubation at 37 °C and 45 °C (Datasets S1 and S2). The recovered microbial diversity was broadly similar to the in situ microbial assemblage reported from the same coal and shale beds described by Inagaki et al. (6), with an abundance of anaerobic heterotrophic groups that more closely resembled terrestrial and wetland microbial lineages than marine sediments, many of which are generally associated with heterotrophs described from coal bed methane systems (45?47). The fact that the microbial taxa recovered were derived from intact FACS cells rather than environmental sample DNA extracts suggests that the diversity within the deep coal and shale SIP incubations is likely associated with viable microorganisms that were capable of persisting or growing in the long-term incubations at elevated temperatures and anoxic or microoxic conditions.
The 16S rRNA gene sequences within the bacterial domain were dominant in our diversity surveys of the seven most active incubations, with rare archaeal sequences affiliated with marine benthic group B detected in only two of the shale incubations incubated at 37 °C (Dataset S2). Some recovered taxa in the 45 °C incubation were related to moderately thermophilic microorganisms [e.g., Thermomonas within the Gammaproteobacteria (48), Acidothermus within the Acinetobacteria (49)], although the majority of the lineages recovered are not known for being thermotolerant. Thermophilic and spore-forming microorganisms often show enhanced cultivation and growth after high-pressure temperature treatment (50), consistent with the dominance of Firmicutes from the autoclaved coal incubation amended with methylamine and ammonium, reaching 76% of the total diversity, relative to 2–3% Firmicutes within the parallel nonautoclaved incubation (Dataset S1).
Dominant bacterial groups in the incubations included Actinobacteria, Firmicutes, Bacteroidetes, Alphaproteobacteria, Betaproteobacteria, Gammaproteobacteria, and Deltaproteobacteria, with additional representatives belonging to the Chloroflexi (Dehalococcoidia), Deinococcus, Verucomicrobia (Opitatue), and Epsilonproteobacteria (Campylobacterales). At the genus level, microbial assemblages differed between samples, with no genus shared across all seven samples, and 37% of genera were found only in one of the seven samples. These differences were not statistically correlated with substrate amendment or lithology, despite bulk geochemistry analysis suggesting a preference for methanol utilization in shale and methylamine utilization in coal. There were also no statistically supported differences in the recovered diversity associated with lithology between the shale, coal, and mixed samples. When observed at lower taxonomic resolution and grouped by assigned taxonomic phylum, the microbial assemblages share the same dominant phyla across all incubations (Dataset S1). Therefore, while microdiversity exists at the genus level, the dominant phyla maintained over the 2.5-y incubation period were predominantly heterotrophic microorganisms (e.g., Gammaproteobacteria were at ≥10% relative abundance in all incubations).
Within these heterotrophic bacteria, there was some correlation between relative abundance of specific microbial taxa and hydrogen addition. For example, both shale and mixed lithology samples with 13C-methanol had a higher relative abundance of Firmicutes and Alphaproteobacteria in the absence of hydrogen addition to the headspace. In comparison, parallel hydrogen-amended incubations contained a higher percentage of Actinobacteria, suggesting potential utilization. While commonly described from deep subseafloor biosphere ecosystems, sequences affiliated with the Chloroflexi were only abundant (28% relative abundance) in the 13C-methanol mixed lithology incubation. Many members of the Chloroflexi have a characteristic filamentous morphology, making them likely candidates for the abundant filamentous cells recovered from this incubation (Fig. S1H).
While we did not recover sequences for methylotrophic methanogens, some of the bacterial lineages have cultured representatives reported to use methanol and/or methylamine as the sole carbon or energy source, including members of the Nocardia within the Actinobacteria (51), Rhizobiales [e.g., Rhodopseudomonas acidophilus (52)], Desulfovibrionales (Desulfovibrio fructosovorans) (53), and Firmicutes (54), especially many of the thermophilic spore-forming bacilli (55, 56). Other detected bacteria may contribute to methanol and methylamine production through breakdown of organic compounds, a process that has also been documented in marine sediments (57). For example, Verrucomicrobia (Opitutus sp.) recovered in some of the coal and shale incubations has cultured representatives that are capable of producing methanol during pectin degradation (58). Select members of the Clostridia also produce methanol from pectin and can degrade N-methyl compounds, such as betaine, to methylamines (54), while others within this class consume methanol and methoxylated compounds as sole carbon and energy sources.
## Discussion
IODP Expedition 337 off Shimokita Peninsula explored a pristine, deeply buried, Miocene-aged coal bed system in an attempt to characterize the deep subseafloor biosphere within this unique environment. The initial site characterization from this expedition, described by Inagaki et al. (6), demonstrated higher cell concentrations in the coal bed compared with surrounding lithologies but reported in situ cell counts for this site that were far below the predicted depth-based trends recorded in previous ocean drilling studies (5). Successful cultivation of a hydrogenotrophic methanogen (Methanobacterium sp.), combined with geochemical evidence for sulfate reducers and methanogens in isotopically enriched pyrite (14) and 13C-depleted methane, as well as detection of coenzyme F420 (6), indicated the potential for biogenic methane formation in the organic carbon- and H2-replete coal bed environment (6). The calculated Gibbs free energies of methylotrophic methanogenesis and methyl fermentation under in situ conditions indicate that both methylotrophic methanogenesis (approximately ?300 kJ/mol) and fermentation (approximately ?10 kJ/mol) were possible, but methylotrophic methanogenesis is more energetically favorable. Given these energetic predictions, it was unexpected that the coal and shale SIP experiments with 13C-labeled methylated substrates did not show significant biological methanogenic activity during the 2.5-y incubation period. Instead, our isotopic data suggested the majority of methane detected during these long-term experiments was associated with the coal before incubation and subsequently released from the sample matrix over time, with minimal contributions from methylotrophic methanogenesis via the 13C-labeled substrates (at most, 0.04% of total methane from labeled substrate). This could also explain why the autoclaved coal samples had lower methane production, as the heat and pressure treatment during autoclaving, followed by flushing with argon, may have accelerated degassing of methane compared to treatments without autoclaving. Our finding that the majority of methane produced was not enriched in 13C may be attributed to other carbon substrates native to the organic-rich coal bed. However, the methane produced in the incubations also did not show enrichment in 2H, which is predicted to occur during active biogenic methane production in the presence of 2H2O, as biogenic methane production requires water-derived hydrogen (59). From incubation δ2H-CH4 values, we calculated a maximum of 0.007% of the total methane could have been biologically produced during the coal incubations from utilization of any carbon substrate available in the coal, or added methylated substrate. The combined 13C and 2H data point to the majority of biogenic methane recovered over the 2.5-y experiment being produced before the start of the incubation, highlighting an apparent temporal offset between in situ generation and subsequent release from the coal matrix (60).
While we have evidence for low levels of methanogenesis occurring in our incubations that could be attributed to the amended substrates, the in situ geochemistry at this site indicates a potential temporal succession in the microbial community over geological time from a methanogenic period to a now quiescent period that has retained the geochemical record [～30 mM CH4 (30)] of a once more active in situ methanogenic assemblage. Since this Miocene-aged coal bed has experienced a wide range of temperatures, pressures, and geochemical conditions during burial over the past 20 Mya, it is possible that one or more of these variables changed at a rate, or to a degree, where the less metabolically versatile methanogens were impacted more than the comparatively metabolically versatile bacterial assemblage.
Overall, the microbial diversity recovered from our incubations was consistent with the bulk microbial diversity recovered in situ (6), including a dominance of Gammaproteobacteria, Betaproteobacteria, Alphaproteobacteria, Actinobacteria, Firmicutes, Chloroflexi, and Bacteroidetes. The combined multidisciplinary results from our stable isotope incubation experiments suggest that these deep subseafloor microorganisms are metabolic generalists that appear to be resistant to the unique stressors of the in situ environment beyond what has been described of their better characterized surface relatives in culture. While total cell numbers and sample sizes were low, as expected given this environment’s inherent constraints, there were general trends in relative abundance of specific taxonomic groups aligning with hydrogen amendment inhibition (Firmicutes, Alphaproteobacteria) or stimulation (Actinobacteria), as well as observations of increased numbers of putative spore-forming Firmicutes in the autoclave-treated sample. Gibbs free energies calculated for our incubation conditions with hydrogen addition (～10-fold in situ hydrogen, +10 kJ/mol) and no hydrogen addition (near zero hydrogen approaches favorability of methanogenesis) concur with the coupling of fermentation and hydrogen addition, where incubations without hydrogen amendment should be much more favorable for fermentation. It is also notable that methanol anabolism (modeled via conversion to formaldehyde, approximately ?30 kJ/mol at standard temperature and pressure) is more energetically favorable than catabolism, which may have elevated importance in subsurface environments, where most, if not all, cellular energy is likely devoted to cell maintenance (61, 62).
The highest rate of 13C-DIC production from methyl substrate amendments in our coal incubations (2 pmol of DIC per cm3?d?1) was 25-fold faster than potential sulfate reduction rates from the same coal bed with radiolabeled sulfate (14). Using a conversion factor to translate radiolabeled sulfate rates to DIC production rates [2 mol of C from 1 mol of acetate, per 1 mol of sulfate (61)], other subseafloor systems yield rates ranging from ～10?4 fmol of C per cell per day in Peru Margin sediments (13) to shallow sediment rates of ～10?2 fmol of C per cell per day (12). Converting 13C-DIC production in our coal incubations to cellular rates using final cell concentrations, DIC production was ～0.02 fmol of C per cell per day for both methylamine and methanol. Assuming a minimal extraction efficiency of only 10%, this would decrease rates to ～0.002 fmol of C per cell per day, which is still above Peru Margin sediment rates. Our NanoSIMS data also indicate that it is unlikely all cells are active in our incubations (Table 1), and when assuming only the active portion of cells in these coal incubations produced the labeled 13C-DIC, rates are estimated to be 30–60% higher. Accepting that both ex situ stable-isotope and radioisotope incubation methods likely caused some perturbation to the natural system, either through the addition of micromolar concentrations of carbon and nitrogen (this study) or through homogenization of the sample and shaking during 35S-sulfate incubation (14), these rates are unlikely exact representations of in situ subseafloor activity, but remain consistent with the multiple independent indications of slow rates of metabolism and growth of deep biosphere microorganisms relative to pure culture studies.
Both 15N-ammonium and 2H2O have been used to assess anabolic activity and generation times of diverse microorganisms in culture and the environment (18, 19, 21, 63, 64). In the context of this single-cell–targeted multiple-isotope labeling study, the passive tracer 2H2O was observed to be a more sensitive measure of overall anabolic activity compared with 15N assimilation from 15N-ammonium or methylamine within the deep subseafloor biosphere. However, when these assimilation rates were converted to biomass generation times, the paired 2H-based and 15N-based generation time estimates were consistent (Fig. 3). This suggests that the most active cells were not selectively producing different types of biomass, such as repairing lipids over generating new proteins, but were generally producing all forms of biomass.
Comparison of cellular enrichment from 2H2O incorporation across different incubations revealed a notable difference in cellular anabolic activity (generation time) in incubations supplied with H2 relative to those without. H2 concentrations have been reported as low as 10?2 μM in subsurface coal beds with active hydrogenotrophic methanogenesis (65), but in situ levels were much higher in the Shimokita coal bed (1–500 μM) (6), suggesting that methanogenesis was not the dominant control over environmental hydrogen concentrations (66). In contrast to many deep biosphere systems that are hypothesized to be fueled by hydrogen (67), the addition of hydrogen (0.15 atm H2) depressed microbial anabolic activity relative to those incubations without hydrogen for the coal incubations (Fig. 3), and suggests that fermentative microorganisms suggested by 16S rRNA gene sequencing are active in coal bed incubations. Microorganisms in the shale incubations from an overlying horizon produced the opposite trend. In this set of incubations, a larger fraction of active cells (ROIs) was detected with H2 amendment, relative to the paired incubation without H2, in both 13C-methylamine and 13C-methanol treatments (Fig. 1). These findings suggest the influence of H2 is complex in organic-rich systems, and as observed in the deep coal horizon, H2 may not always be a beneficial source of energy and, in fact, may depress microbial activity and growth.
Patterns of nitrogen source utilization (ammonia or methylamine) by the active microbial assemblage also differed between the coal and shale incubations. The 15N-ammonium incorporation was observed in the majority of incubations, independent of lithology. Methylotrophy via the microbial assimilation of 15N-methylamine, however, was predominately observed by the coal-associated microbial assemblage. It is expected that most anaerobes can use ammonium as a nitrogen source, but assimilation of methylamine is less common, and the observed difference in methylamine utilization between the coal and shale incubations is notable. Supporting these observations, the diversity recovered in the coal incubations included microbial taxa affiliated with bacterial lineages shown to be capable of methylamine metabolism. These findings may point to the potential use of methylated amines in lignite coal as a source of nitrogen in this deep biosphere ecosystem.
We compared the 15N-estimated generation times from previously published SIP-NanoSIMS data of 15N-ammonium uptake in cells from deep marine sediments (219 mbsf) that overlay the coal and shale beds studied here (18). In the study by Morono et al. (18), 15N-ammonium–based estimates of doubling times ranged from 63 to 252 d depending on added carbon source (e.g., glucose, amino acids), whereas the fastest 15N generation times calculated from our deep coal bed incubations were ～30-fold slower (～20 y; Fig. 3). The study by Morono et al. (18) did not include 2H2O, precluding direct comparison; however, estimates of 2H-based generation times from coal bed microorganisms did show some overlap with the slowest rates measured within the overlying sediment microbial assemblage (1–2 y) and other marine sediment studies using 2H2O [～20 y, 2H-lipid incorporation rates (25)]. Our inability to detect 13C-labeled biomass despite indication of carbon utilization from the bulk geochemical data also supports the findings of the study by Morono et al. (18), where nitrogen was shown to be assimilated to a greater extent than carbon by microorganisms in their sediment incubations.
Overall, the single-cell generation times from SIP-NanoSIMS data for carbon, nitrogen, and 2H2O in our study, as well as those reported by Morono et al. (18), yield estimates that are one to two orders of magnitude faster than deep biosphere turnover times determined from bulk methods, such as amino acid racemization and lipid degradation (17, 65). The simultaneous use of 15N-labeled substrates in combination with the passive 2H2O tracer offered two independent means to track single-cell anabolic activity within our incubations. Although our study focused on the most active cells in our generation time estimates, this targeted single-cell analysis illustrates the high level of heterogeneity in anabolic activity between cells within the subseafloor microbial community, with the active fraction of the community ranging between no cells above the enrichment threshold (10-fold natural abundance) to 76% of the total cells with values above the threshold. This suggests that bulk methods estimating average cellular activity through the incorporation of environmental cell counts are likely to underestimate microbial activity and generation times. The complementarity in these independent multiscale approaches offers an important frame of reference for setting minimum and maximum bounds for growth and metabolic activity estimates within deep subseafloor ecosystems and demonstrates that even at the fringes of microbial activity, survival is not a simple story of dead or alive.
## Conclusion
Despite cell abundances that were substantially lower than expected from global depth trends and sediment surface concentrations, this single-cell–targeted study successfully detected active microorganisms in terrestrial-sourced shale and coal buried at 2 kmbsf for millions of years. Incubations of intact shale and coal pieces maintained at in situ temperatures and amended with methyl compounds, ammonium, and 2H2O enabled the detection of some of the slowest measurements of growth using SIP-NanoSIMS. With biomass generation times of years to hundreds of years, these rates were much faster than deep biosphere activity estimates via other bulk methods (approximately thousands of years). Although methanogenic activity could be detected via geochemical analysis, methanogenic archaea were not recovered by 16S rRNA gene iTag sequencing. Instead, heterotrophy appears to be the predominant microbial metabolism for this system, which has been found in many deep subseafloor sedimentary environments (8, 68?70). Although variation in activity among incubations associated with methyl substrate amendment, presence/absence of hydrogen, high temperature/pressure (autoclave), and 16S rRNA diversity depict a heterogeneous and diverse heterotrophic community. Through this unique snapshot into a pristine coal bed, results from Expedition 337 have also opened new avenues for conceptualizing the residence time of carbon, nitrogen, and sulfur in coals that have never reached sterilization conditions. With global lignite reserves estimated at 839 gigatons (71), understanding what portion of this potential microbial energy source, assumed biologically inaccessible in the lithosphere, may be mobilized and returned to the surface biosphere is important for understanding both deep life and global biogeochemical cycles.
## Geochemical Analyses
DIC samples were injected into preweighed, He-flushed Exetainer vials (Labco Limited) containing 100 μL of 40% phosphoric acid. Concentrations and stable carbon isotopes of DIC were measured on a Gasbench II (Thermo Scientific) coupled to a Delta V Plus IRMS instrument (Thermo Scientific) in the Caltech Stable Isotope Facility, following the methods of Torres et al. (72). Briefly, Exetainers were sampled using a Gas Chromatograph Prep and Load autosampler (CTC Analytics) equipped with a doubled-holed needle that transferred headspace using a 0.5 mL?min?1 continuous flow of He to a 50-μm sample loop before separation by a PoraPlotQ fused silica column (25 m, i.d. = 0.32 mm) at 72 °C. A 20 mM NaHCO3 solution was used to establish a DIC concentration standard curve by comparing the average of the total peak area (masses 44, 45, and 46) of replicate sample injections with the volume of NaHCO3 added. Using the estimated error in the volume measurement and the SD of the peak area for the total carbon of bicarbonate standards, the estimated error of DIC concentrations was ±0.009 μmol. The δ13C values were corrected for sample size dependency and then normalized to the Vienna Pee Dee Belemnite scale with a three-point correction (73) using National Bureau of Standards standard 19, a previously calibrated internal laboratory standard, and the average of all of the NaHCO3 standards. Accuracy (0.11‰, n = 79) was determined by considering independent standards as samples, and precision (0.5‰) was determined from replicate measurements of all standard solutions.
Methane headspace concentrations were measured relative to argon (added to ca. equal partial pressure in all incubations) using a Hewlett Packard 5972 Series Mass Selective Detector and Hewlett Packard 5890 Series II Plus Gas Chromatograph at the Caltech Environmental Analysis Center. The δ13C-CH4 was analyzed as per the methods of Tsunogai et al. (74), and δ2H-CH4 was analyzed as per the methods of Kikuchi et al. (75) at the Kochi Institute for Core Sample Research, Japan Agency for Marine-Earth Science and Technology (JAMSTEC).
## Isotope Calculations
Isotope mass balance was calculated using Eqs. S1 and S2, where <mml:math><mml:mi>n</mml:mi></mml:math>n is the number of moles, <mml:math><mml:mi>F</mml:mi></mml:math>F is the fractional abundance of the rare isotope (Eq. S3), and <mml:math><mml:mi>R</mml:mi></mml:math>R is the ratio of ion counts of the rare isotope over ion counts of the more abundant isotope (76). <mml:math><mml:mrow><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mtext>total</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math>Ftotal, <mml:math><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mrow><mml:mtext>total</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math>ntotal, <mml:math><mml:mrow><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mtext>substrate</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math>Fsubstrate, and <mml:math><mml:mrow><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mtext>background</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math>Fbackground were known and used to solve for <mml:math><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mrow><mml:mtext>substrate</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math>nsubstrate. Fractional abundances are displayed as atom percent, which is 100-fold the fractional abundance (Eq. S4). DIC production rates were calculated by dividing the moles of DIC from substrate (<mml:math><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mrow><mml:mtext>substrate</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math>nsubstrate) by sample volume (4 cm3 for coal samples) and days of incubation (864 d). The percent molar methane from substrate was calculated from Eq. S5, a rearrangement of Eqs. S1 and S2:<mml:math display="block"><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mrow><mml:mtext>total</mml:mtext></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>n</mml:mi><mml:mrow><mml:mtext>substrate</mml:mtext></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>n</mml:mi><mml:mrow><mml:mtext>background</mml:mtext></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:mrow></mml:math>ntotal=nsubstrate+nbackground,[S1]<mml:math display="block"><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mrow><mml:mtext>total</mml:mtext></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mtext>total</mml:mtext></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>n</mml:mi><mml:mrow><mml:mtext>substrate</mml:mtext></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mtext>substrate</mml:mtext></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>n</mml:mi><mml:mrow><mml:mtext>background</mml:mtext></mml:mrow></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mtext>background</mml:mtext></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:mrow></mml:math>ntotal×Ftotal=nsubstrate×Fsubstrate+nbackground×Fbackground,[S2]<mml:math display="block"><mml:mrow><mml:mi>F</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mi>R</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>R</mml:mi></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:mrow></mml:math>F=R1+R,[S3]<mml:math display="block"><mml:mrow><mml:mtext>At</mml:mtext><mml:mo>.</mml:mo><mml:mo>%</mml:mo><mml:mo>=</mml:mo><mml:mi>F</mml:mi><mml:mo>×</mml:mo><mml:mn>100</mml:mn><mml:mo>,</mml:mo></mml:mrow></mml:math>At.%=F×100,[S4]<mml:math display="block"><mml:mrow><mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mtext>total</mml:mtext></mml:mrow></mml:msub><mml:mo>?</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mtext>background</mml:mtext></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mtext>substrate</mml:mtext></mml:mrow></mml:msub><mml:mo>?</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mtext>background</mml:mtext></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mrow><mml:mtext>substrate</mml:mtext></mml:mrow></mml:msub></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mrow><mml:mtext>total</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>×</mml:mo><mml:mn>100</mml:mn><mml:mo>.</mml:mo></mml:mrow></mml:math>(Ftotal?Fbackground)/(Fsubstrate?Fbackground)=(nsubstrate/ntotal)×100.[S5]Data analysis and display as violin plots of the kernel density function were done using R (77) with the “ggplot2” (78), “dplyr” (79), “gridExtra” (80), and “RColorBrewer” (81) packages. Cell ROIs with deuterium enrichment at least 10-fold above natural abundance were deemed “active.” This threshold was chosen as a number that was sufficiently higher than natural abundance to account for remnant isotope from hydrogen exchange and error in the NanoSIMS measurement. Our chosen threshold (0.115 at. %) is about half of that used by Kopf et al. (22) and a third of that reported by Berry et al. (21) for deuterium background in fixed pure culture studies.
For generation time calculations, a conservative approach was used that only included ROIs where 15N12C counts were above the Poisson error provided by [email protected] Poisson error is based on the theoretical precision of the mean for the minor ion. Eqs. S6 and S7 were used to calculate the rate of biomass generation via fractional abundance of isotope label in cellular biomass over the incubation period, where <mml:math><mml:mi>μ</mml:mi></mml:math>μ is the generation rate (encompassing both cell maintenance and generation of new cells), <mml:math><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mrow><mml:mtext>final</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math>Tfinal is the length of the incubation, <mml:math><mml:mrow><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mtext>label</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math>Flabel is the labeling strength, <mml:math><mml:mrow><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mtext>final</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math>Ffinal is the single-cell NanoSIMS measurement, and <mml:math><mml:mrow><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mtext>nat</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:math>Fnat is the natural abundance. We used Eq. S8 to estimate generation time, as the reciprocal of the rate, per Zilversmit et al. (82). NanoSIMS <mml:math><mml:mrow><mml:mmultiscripts><mml:mi>F</mml:mi><mml:mprescripts></mml:mprescripts><mml:none></mml:none><mml:mn>2</mml:mn></mml:mmultiscripts></mml:mrow></mml:math>F2 and <mml:math><mml:mrow><mml:mmultiscripts><mml:mi>F</mml:mi><mml:mprescripts></mml:mprescripts><mml:none></mml:none><mml:mrow><mml:mn>15</mml:mn></mml:mrow></mml:mmultiscripts></mml:mrow></mml:math>F15 values were multiplied by a conversion factor determined for single-cell to bulk isotope measurements by Kopf et al. (22) of 0.67 (2H) and 0.94 (15N). While lipid water assimilation values are not known for this system, the maximal value (aw of 0.83) from Zhang et al. (83) was used as an upper estimate of assimilation based on findings that lipid water assimilation increases as growth rate decreases (84). Nitrogen-based rate calculations assumed all nitrogen was derived from the substrate (methylamine or ammonium); therefore, the assimilation constant was excluded from Eq. S7:<mml:math display="block"><mml:mrow><mml:mmultiscripts><mml:mi>μ</mml:mi><mml:mprescripts></mml:mprescripts><mml:none></mml:none><mml:mn>2</mml:mn></mml:mmultiscripts><mml:mo>=</mml:mo><mml:mfrac bevelled="true"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mo>?</mml:mo><mml:mi>ln</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>?</mml:mo><mml:mfrac><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mmultiscripts><mml:mi>F</mml:mi><mml:mtext>final</mml:mtext><mml:none></mml:none><mml:mprescripts></mml:mprescripts><mml:none></mml:none><mml:mn>2</mml:mn></mml:mmultiscripts><mml:mo>?</mml:mo><mml:mmultiscripts><mml:mi>F</mml:mi><mml:mtext>nat</mml:mtext><mml:none></mml:none><mml:mprescripts></mml:mprescripts><mml:none></mml:none><mml:mn>2</mml:mn></mml:mmultiscripts></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:msub><mml:mi>a</mml:mi><mml:mi>w</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mmultiscripts><mml:mi>F</mml:mi><mml:mtext>label</mml:mtext><mml:none></mml:none><mml:mprescripts></mml:mprescripts><mml:none></mml:none><mml:mn>2</mml:mn></mml:mmultiscripts><mml:mo>?</mml:mo><mml:mmultiscripts><mml:mi>F</mml:mi><mml:mtext>nat</mml:mtext><mml:none></mml:none><mml:mprescripts></mml:mprescripts><mml:none></mml:none><mml:mn>2</mml:mn></mml:mmultiscripts></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mrow><mml:mtext>final</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:mrow></mml:math>μ2=(?ln(1?(Ffinal2?Fnat2)aw(Flabel2?Fnat2)))Tfinal,[S6]<mml:math display="block"><mml:mrow><mml:mmultiscripts><mml:mi>μ</mml:mi><mml:mprescripts></mml:mprescripts><mml:none></mml:none><mml:mn>15</mml:mn></mml:mmultiscripts><mml:mo>=</mml:mo><mml:mfrac bevelled="true"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mo>?</mml:mo><mml:mi>ln</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>?</mml:mo><mml:mfrac><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mmultiscripts><mml:mi>F</mml:mi><mml:mtext>final</mml:mtext><mml:none></mml:none><mml:mprescripts></mml:mprescripts><mml:none></mml:none><mml:mn>15</mml:mn></mml:mmultiscripts><mml:mo>?</mml:mo><mml:mmultiscripts><mml:mi>F</mml:mi><mml:mtext>nat</mml:mtext><mml:none></mml:none><mml:mprescripts></mml:mprescripts><mml:none></mml:none><mml:mn>15</mml:mn></mml:mmultiscripts></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mmultiscripts><mml:mi>F</mml:mi><mml:mtext>label</mml:mtext><mml:none></mml:none><mml:mprescripts></mml:mprescripts><mml:none></mml:none><mml:mn>15</mml:mn></mml:mmultiscripts><mml:mo>?</mml:mo><mml:mmultiscripts><mml:mi>F</mml:mi><mml:mtext>nat</mml:mtext><mml:none></mml:none><mml:mprescripts></mml:mprescripts><mml:none></mml:none><mml:mn>15</mml:mn></mml:mmultiscripts></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mrow><mml:mtext>final</mml:mtext></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:mrow></mml:math>μ15=(?ln(1?(Ffinal15?Fnat15)(Flabel15?Fnat15)))Tfinal,[S7]<mml:math display="block"><mml:mrow><mml:mi>τ</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>μ</mml:mi><mml:mrow><mml:mo>?</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>.</mml:mo></mml:mrow></mml:math>τ=μ?1.[S8]
## Acknowledgments
We thank the IODP for providing access and samples from the deep coalbed biosphere off Shimokita during Expedition 337. We thank the crew, drilling team members, laboratory technicians, and scientists on the drilling vessel Chikyu for supporting core sampling and onboard measurements. We also thank S. Fukunaga, S. Hashimoto, and A. Imajo [Japan Agency for Marine-Earth Science and Technology (JAMSTEC)] and T. Terada (Marine Works Japan, Ltd) for assistance in microbiological analyses; Y. Guan, F. Wu, C. Ma, and N. Dalleska (Caltech) for assistance with geochemical analyses; and A. L. Sessions, G. L. Chadwick, K. S. Metcalfe, M. K. Lloyd, and S. Kopf (Caltech) and H. Imachi (JAMSTEC) for feedback and valuable discussions. We appreciate the comments of two reviewers that also improved this manuscript. Funding for this work was provided by the Center for Dark Energy Biosphere (C-DEBI), NASA Astrobiology-Life Underground (NAI-LU; Award NNA13AA92A), the Gordon and Betty Moore Foundation Grant GBMF3780 (to V.J.O.), and Post Expedition Award (to E.T.-R. and V.J.O.), the Japan Society for the Promotion of Science (JSPS) Strategic Fund for Strengthening Leading-Edge Research and Development (F.I. and JAMSTEC), the JSPS Funding Program for Next Generation World-Leading Researchers (NEXT Program, Grant GR102 to F.I.), and JSPS Grants-in-Aid for Science Research (Grant 26251041 to F.I.; Grant 15K14907 to T.H.; and Grants 24687004, 15H05608, 24651018, 2665169, and 16K14817 to Y.M.). E.T.-R. was additionally supported, in part, by a Schlanger Ocean Drilling Fellowship, a C-DEBI travel grant for sample processing at the JAMSTEC Kochi Institute for Core Sample Research, and the Deep Life Cultivation Internship Program from the Deep Carbon Observatory (DCO). This is C-DEBI Grant contribution no. 389 and NAI-LU no. 314.
## Footnotes
• ?1To whom correspondence may be addressed. Email: eliztr{at}gmail.com or vorphan{at}gps.caltech.edu.
## References
1. ?
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5. ?
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6. ?
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7. ?
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8. ?
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9. ?
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10. ?
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11. ?
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12. ?
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13. ?
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14. ?
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15. ?
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16. ?
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17. ?
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18. ?
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19. ?
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20. ?
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21. ?
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22. ?
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23. ?
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24. ?
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25. ?
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26. ?
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27. ?
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28. ?
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29. ?
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30. ?
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31. ?
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32. ?
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33. ?
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34. ?
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35. ?
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36. ?
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37. ?
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38. ?
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48. ?
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51. ?
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72. ?
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73. ?
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74. ?
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75. ?
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76. ?
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77. ?
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78. ?
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79. ?
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80. ?
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81. ?
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82. ?
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83. ?
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84. ?
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2018-02-21 11:23:11
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https://www.physique.usherbrooke.ca/pages/node/7241
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Correlation-functions of the Hubbard-model At Low-density In A Crossing-symmetrical Approximation - Comparisons With Monte-carlo Simulations
Titre Correlation-functions of the Hubbard-model At Low-density In A Crossing-symmetrical Approximation - Comparisons With Monte-carlo Simulations Type de publication Journal Article Nouvelles publications 1994 Auteurs Daré, AM, Chen, L, Tremblay, A-MS Journal Physica B Volume 194 Pagination 1413–1414 Résumé The accuracy of a simple crossing-symmetric approximation for the fully reducible vertex is tested by comparisons of the spin, charge, and pairing correlations with those obtained by Monte Carlo simulations of the two-dimensional Hubbard model. The approximation under study consists in assuming that for parallel spins the fully reducible vertex vanishes, while for anti-parallel spins it is equal to the T-matrix. Up to quarter-filling, accuracies better than 10% are obtained. Texte complet
fichier:
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2022-01-26 14:45:20
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https://www.physicsforums.com/threads/calculate-the-change-of-volume-from-volume-expansion-coefficient.662035/
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Calculate the change of volume from volume expansion coefficient
1. Outrageous
375
β= (1/v)(∂v/∂T)constant pressure.
What is the v represent? molar volume?
If I am given the β and the change of temperature, how to calculate the change of volume? or it is not enough information to calculate it?
Thank you.
Staff: Mentor
it seems like you have enough if the change is relatively small so that you could use deltas:
beta * delta T * V =delta V
You'd have to decide on what relatively small means and you have to know what V is.
3. Outrageous
375
Thank you
Question : a container is filled with mercury at 0 degree Celcius. At temperature 50 degree Celcius , what is the volume of mercury that will spill out ?
β Of mercury is 18*10^(-5) /Celcius
Is this possible to do ?
Staff: Mentor
What do you think? A delta of 50 degrees is pretty significant.
What is the initial volume?
5. Outrageous
375
The initial volume is not given , so that question can't be solved?
Staff: Mentor
$$\frac{d\ln{v}}{dt}=\beta$$
$$v=v_0\exp(\beta(T - T_0))$$
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2015-10-07 10:05:42
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https://wtmaths.com/volume_cylinder.html
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Volume of a Cylinder
# Volume of a Cylinder
GCSE(F), GCSE(H),
A cylinder is a circular prism. As with other prisms, the volume of a cylinder can be calculated using the cross-sectional area x length.
Th cross-section of a cylinder is a circle. As the area of a circle =πr^2, the volume of a cylinder is πr^2l, where r is the radius of the circular end, and l is the length of the cylinder.
## Examples
1. What is the volume of a cylinder with a diameter of 4cm and a length of 42cm? Give your answer correct to 1 decimal place.
The radius of a circle is frac(1)(2) the diameter = frac(1)(2) xx 4 = 2cm
The cross-sectional area is pir^2 = pi xx 2^2 = 4pi
The volume of the cylinder = cross-sectional area x volume
= 4pi xx 42
= 527.79 cm3
2. What is the diameter of a cylinder with a volume of 1000cm3 and a length of 100cm? Give the answer correct to 1 decimal place.
Volume of a cylinder = pir^2l
1000 = pir^2 xx 100
10 = pir^2
r^2 = 3.183
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2019-09-23 05:23:24
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http://tex.stackexchange.com/tags/math-operators/new
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# Tag Info
6
Just scale it. \documentclass{article} % large ops, copied from shuffle font package \DeclareFontFamily{U}{bigshuffle}{} \DeclareFontShape{U}{bigshuffle}{m}{n}{ <5-8> s*[1.7] shuffle7 <8-> s*[1.7] shuffle10 }{} \DeclareSymbolFont{BigShuffle}{U}{bigshuffle}{m}{n} \DeclareMathSymbol\bigshuffle{\mathop}{BigShuffle}{"001} ...
7
For such a simple geometric symbol, scaling it seems the easiest road. You might want a larger factor than 1.2 for text style, adjust at will. However, 2 is the right scaling factor for display style. \documentclass{article} \usepackage{amssymb,graphicx} \newcommand{\bDiamond}{\mathbin{\Diamond}} \makeatletter ...
0
A bit more about "c or not to c"... The inverse hyperbolic sine sinh^(-1)z (Beyer 1987, p. 181; Zwillinger 1995, p. 481), sometimes called the area hyperbolic sine (Harris and Stocker 1998, p. 264) and sometimes denoted arcsinhz (Jeffrey 2000, p. 124), is the multivalued function that is the inverse function of the hyperbolic sine. The variants Arcsinhz or ...
8
The align environment expects a relation symbol after &, as the point of alignment, so there is an implicit {} at the beginning of the second column (and all other even numbered columns). This has the unfortunate consequence that, if a math operator follows &, a thin space is added, because of TeX's spacing rules: when a math operator follows an ...
8
Since the question already answers the cause of the space. The macro, defined by \DeclareMathOperator is defined as \mathop with additional spacing in some situations. Here align adds an empty math ordinary atom to get correct spacing for binary or relation symbols. Also a space is added between \mathord and \mathop. In this case the space can be avoided ...
Top 50 recent answers are included
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2014-07-13 13:31:39
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https://proofwiki.org/wiki/Union_with_Complement
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# Union with Complement
## Theorem
The union of a set and its complement is the universe:
$S \cup \map \complement S = \mathbb U$
## Proof
Substitute $\mathbb U$ for $S$ and $S$ for $T$ in $T \cup \relcomp S T = S$ from Union with Relative Complement.
$\blacksquare$
## Law of the Excluded Middle
This theorem depends on the Law of the Excluded Middle, by way of Union with Relative Complement.
This is one of the axioms of logic that was determined by Aristotle, and forms part of the backbone of classical (Aristotelian) logic.
However, the intuitionist school rejects the Law of the Excluded Middle as a valid logical axiom. This in turn invalidates this theorem from an intuitionistic perspective.
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2021-08-03 18:11:00
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http://www.reddit.com/r/learnmath/comments/1pgh8w/first_year_university_ms_unit_explanation/?sort=random
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[–] 7 points8 points ago
If you take a third of two, you get two-thirds.
1/3 × 2 = 2/3
2 × 1/3 = 2/3
More generally:
A × 1/B = A/B
It's the same with units:
m × 1/s = m/s
[–] 1 point2 points ago
This explains it well. The meters are the numerator. The seconds are the denominator.
When you take the meters you traveled and divide it by the time in seconds you get a numerical value for your velocity, or meters over seconds.
Additionally, when looking at acceleration, it is velocity divided by time in seconds. We know velocity is meters over seconds. We just have to divide that by seconds.
It looks like
m/s * 1/s = m/(s2)
s*s=s2
s2 is still the denominator. Its still on the bottom. You don't randomly flip fractions, and m/s does not imply that 1/s becomes s/1.
The math is that you times numerator by numerator and denominator by denominator. Since the time measured is constant (lets say you measured for 5 seconds). The amount of time is 5s regardless of whether you're measuring displacement, velocity, or acceleration.
A fraction is another way of expressing division without having to work the division out since sometimes you need to have the original numbers. A decimal is just the result of dividing any numerator by any denominator.
[–] 1 point2 points ago
Also, and just as importantly, it should be noted that
m/(s2)
means and implies:
m / s / s
read "meters per second per second," and it is the rate at which the original rate (m/s) is changing, or acceleration in this case.
This works the same going backward in steps, then, that
s * m/(s2) = m/s .
It should also be noted that when multiplying rates together with "coefficients" (the actual unit) other than one, the amounts are multiplied while at the same time the units cancel like in the aforementioned example.
An amount of time * the rate of acceleration =
20 s * 20 m/s/s =
400 m/s
This means that over 20 seconds, a velocity increasing at a rate of 20 meters per second every second would result in an increase of 400 m/s
[–] 0 points1 point ago
Lets suppose you want to calculate your average speed over some distance. So I'd go some distance, measure the distance and measure the time it took. Lets say I went 60m in 15s. Then to calculate my speed I'd divide the distance by the time:
v = d/t = 60m/15s = 4m/s
Basically, the m/s unit tells us that the value can be arrived at by dividing a distance by a time (not multiplying).
s/m and m x s are also a perfectly reasonable units which you might use in some other situation, but they don't represent velocities since they aren't the result of dividing a distance by a time.
[–] 1 point2 points ago
thats not really the question at hand though....
[–] 0 points1 point ago
I think OP just read the title of the post
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2014-03-10 05:59:42
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https://vbulletin.org/forum/showthread.php?s=40d5f1a2cbd4ecb9da5763fe71a96200&p=2579466
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Register Help Members List Search Today's Posts Mark Forums Read
Abbreviations/Acronyms (MathJax Compatible) Details »
Abbreviations/Acronyms (MathJax Compatible)
Mod Version: 1.4.2, by MarkFL (Coder)
Developer Last Online: Nov 2019
vB Version: 4.2.x Rating: (1 vote - 5.00 average) Installs: 15 Released: 27 Oct 2015 Last Update: 19 Feb 2017 Downloads: 89 Not Supported DB Changes Uses Plugins Additional Files Translations
Overview
This product allows you to define a set of abbreviations/acronyms that will have a tooltip explanation defined for your users who may not be familiar with these abbreviations and will find an explanation useful. You can define which areas of your site this product is active and your users can also decide if they wish for the product to be active for them in their settings. If you use MathJax on your site for mathematical expressions, you will find this product is completely compatible with it, and will not break such expressions.
Upon installation, a custom user profile field is auto-created to allow your users control over use of this product, and this field is deleted if you uninstall the product.
The product settings are divided into two primary groups: the first pertaining to the abbreviations/acronyms and the second to your implementation of MathJax. If you are not using MathJax, then you can simply ignore that section.
Abbreviations/Acronyms Details
Here you will find a textarea for defining your list of abbreviations and acronyms. Each one should be place on a separate line, with the abbreviation/acronym first, then a pipe symbol ("|"), followed by the explanation you want to appear in the tooltip. If you wish for your tooltip to contain line breaks, then use "\n" (without the quotes) for these breaks. When you save your settings, the list is alphabetically sorted to make locating these definitions easier in the future.
Next, you can set the areas of your site in which this product will be active, including Posts, Signatures, Private Messages, Visitor Messages and Group Messages.
You can then set whether you wish for this product to be active for guests.
MathJax Details
If you do not use MathJax, then you need not make any changes here.
If you do use MathJax, then first you need to define whether you have their script(s) manually defined in your templates or if you are using my product for its dynamic implementation. I recommend using my product for the most reliable and efficient compatibility.
If you use MathJax's default delimiters, then you need not change them, as the defaults are already defined by this product. However, if you have defined your own delimiters, then all delimiters you have defined need to be given in the delimiter settings.
Version 1.1:
• I discovered a MySQL function I was using was deprecated as of 5.5, so I replaced it with a query.
• Fixed bug whereby the preview of a post in a new thread now shows the acronyms.
Version 1.2:
• Fixed bug where "/" character in an abbreviation caused a PHP error.
• Fixed bug where acronym at the beginning of a text string will now work.
Version 1.3:
• Profile filed number of auto-created field cached so a db query is no longer required each time text is parsed.
Version 1.3.1:
• HTML special characters may now be used in abbreviations/acronyms.
Version 1.4:
• Admin options processing more efficient.
• Added setting for "Active Forums."
• Fixed issue where hex colors within COLOR BBCode containing abbreviations broke posts.
Version 1.4.1:
• Dotted underline is now the same color as the text.
Version 1.4.2:
• Fixed issue with non-Latin characters.
Compatibility:
Tested and working on VB 4.2.x and should work on all 4.x versions of vBulletin.
Backup/Warning:
This product alters your database, however it is always good practice to make regular backups and you should make a backup before installing ANY new mod.
As always, products are USE AT YOUR OWN RISK. I will provide support and do my best to help but no absolute guarantee is offered.
To Install:
4.) Click on "Choose File" and browse to the product .xml file that was packaged in the .zip file.
5.) Click "Import."
6.) You MUST enable the product in the settings before it will function.
7.) Configure the remaining settings to your liking. Each setting has a detailed explanation of its use.
Support for this product can be found here:
Screenshots
• This modification may not be copied, reproduced or published elsewhere without author's permission.
Similar Mod Mod Developer Type Replies Last Post Miscellaneous Hacks LaTeX By MathJax MarkFL vBulletin 4.x Add-ons 4 07 Jun 2017 03:55 Show Thread Enhancements Acronyms for vB RolandvonGilead vBulletin 3.6 Add-ons 55 02 Mar 2015 12:14
#16
17 Aug 2016, 07:56
bryanb Member Join Date: Oct 2003
Great mod! Are there any plans to get this to function properly with the mobile template?
__________________
"Master of my domain"
#17
17 Aug 2016, 09:18
bryanb Member Join Date: Oct 2003
By the way - it doesn't seem to work with what might be considered special characters: for example - T&Cs| Terms and Conditions.
Any way to fix this?
__________________
"Master of my domain"
#18
17 Aug 2016, 13:04
MarkFL Coder Join Date: Feb 2014 Real name: Mark
Originally Posted by bryanb
By the way - it doesn't seem to work with what might be considered special characters: for example - T&Cs| Terms and Conditions.
Any way to fix this?
Update - Version 1.3.1:
• HTML special characters may now be used in abbreviations/acronyms.
__________________
Former vBulletin.org Staff Member
Support for my products (as well as updates/new product publishing) has been moved to MHB - vBulletin Products and TAZ - Add-ons
#19
18 Aug 2016, 08:27
bryanb Member Join Date: Oct 2003
Originally Posted by MarkFL
Update - Version 1.3.1:
• HTML special characters may now be used in abbreviations/acronyms.
Many thanks! Works fine!
Any ideas to have it work in a mobile template? The underscore is there - but how do you mouse over?
__________________
"Master of my domain"
#20
18 Aug 2016, 14:15
MarkFL Coder Join Date: Feb 2014 Real name: Mark
Originally Posted by bryanb
Many thanks! Works fine!
Any ideas to have it work in a mobile template? The underscore is there - but how do you mouse over?
This product is designed to give mouseover tooltips...mobile gadgets that can't emulate a mouse I suppose just don't support tooltips.
__________________
Former vBulletin.org Staff Member
Support for my products (as well as updates/new product publishing) has been moved to MHB - vBulletin Products and TAZ - Add-ons
#21
15 Dec 2016, 04:19
KevinL Designer Join Date: Apr 2005
Great work once again!
Any future plans (or could it be added) that this is only allowed in certain forums? Some of the forums the abbreviations aren't really needed or mean different things.
Thanks again!
#22
15 Dec 2016, 04:26
MarkFL Coder Join Date: Feb 2014 Real name: Mark
Originally Posted by KevinL
Great work once again!
Any future plans (or could it be added) that this is only allowed in certain forums? Some of the forums the abbreviations aren't really needed or mean different things.
Thanks again!
I hadn't planned on it, but I can add a setting for active forums. However, having forum-specific lists of acronyms I don't plan on doing as that would be overly convoluted and simply not worth the effort to code.
__________________
Former vBulletin.org Staff Member
Support for my products (as well as updates/new product publishing) has been moved to MHB - vBulletin Products and TAZ - Add-ons
#23
15 Dec 2016, 04:30
KevinL Designer Join Date: Apr 2005
Originally Posted by MarkFL
I hadn't planned on it, but I can add a setting for active forums. However, having forum-specific lists of acronyms I don't plan on doing as that would be overly convoluted and simply not worth the effort to code.
Excellent! No. One set is perfect as long as it can be shut off - or not included in chosen forums.
Also I just noticed it trying to rewrite a color code. In the post the color was changed to [ COLOR=#0000CD ] And there was an abbreviation for CD so it broke the code and distorted the whole post. Not sure if there is a way to combat that. But I thought I would bring it to your attention.
Thanks Mark!
#24
15 Dec 2016, 04:41
MarkFL Coder Join Date: Feb 2014 Real name: Mark
Originally Posted by KevinL
Excellent! No. One set is perfect as long as it can be shut off - or not included in chosen forums.
Also I just noticed it trying to rewrite a color code. In the post the color was changed to [ COLOR=#0000CD ] And there was an abbreviation for CD so it broke the code and distorted the whole post. Not sure if there is a way to combat that. But I thought I would bring it to your attention.
Thanks Mark!
Thanks for letting me know...I will include a fix for this along with the active forums setting.
__________________
Former vBulletin.org Staff Member
Support for my products (as well as updates/new product publishing) has been moved to MHB - vBulletin Products and TAZ - Add-ons
#25
15 Dec 2016, 05:30
MarkFL Coder Join Date: Feb 2014 Real name: Mark
Update - Version 1.4:
• Admin options processing more efficient.
• Added setting for "Active Forums."
• Fixed issue where hex colors within COLOR BBCode containing abbreviations broke posts.
__________________
Former vBulletin.org Staff Member
Support for my products (as well as updates/new product publishing) has been moved to MHB - vBulletin Products and TAZ - Add-ons
#26
15 Dec 2016, 12:13
KevinL Designer Join Date: Apr 2005
Thanks for the update.
Another thing that was noticed if browsing the site on mobile or tablet the underlined words are now tiny compared to the normal words. It can get a little distracting if there are enough abbreviations.
Thanks again Mark
#27
15 Dec 2016, 12:26
MarkFL Coder Join Date: Feb 2014 Real name: Mark
I don't use any mobile devices, so I wouldn't be able to test that. That likely has to do with a HTML shortcoming in the browsers those devices use.
__________________
Former vBulletin.org Staff Member
Support for my products (as well as updates/new product publishing) has been moved to MHB - vBulletin Products and TAZ - Add-ons
#28
15 Dec 2016, 15:21
KevinL Designer Join Date: Apr 2005
The only way I could fix it if I added the following to the headinclude template. If I didn't include the If statement it distorted the rest of the forum. This issue only seems to come up on Chrome for mobile. Firefox mobile seems to work without issue
Block Disabled: (Update License Status) Suspended or Unlicensed Members Cannot View Code.
I was looking to use something like this https://osvaldas.info/elegant-css-an...obile-friendly within your mod but I didn't know where to start right now haha It seems to be working OK except for mobile devices but you already stated it wouldn't work so we're good.
Thanks for the great work Mark!
#29
07 Feb 2017, 13:43
hugoroger Member Join Date: Sep 2012
Nice @MarkFL! I installed this one to use it combined with the blurbs plugin. Question. Do you know if we can configure the abbreviation to be a hyperlink? As in, instead of just appearing the help text, can we add a link to it ? I tried using bbcode to add it but it didn't work. Is this something possible?
#30
07 Feb 2017, 14:01
hugoroger Member Join Date: Sep 2012
Hey again Mark! Just wanted to add a quick folloe up.
Okay.. so I went ahead and installed this plug in (FUZZY SEO Auto Linker) to try to create links in the blurbs but apparently it didn't work. :/
I have an example here. Where I used the blurb plug in to add the word : "joyXwriter" to the bottom of all posts. Now, you can see I configured the hyperlink correctly because in the OP, I added the same sample word by text and the auto linked picked up. Does the plugin(Abbreviations/Acronyms) support this option with HTML or BBcode?
Last edited by hugoroger; 07 Feb 2017 at 14:08.
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2020-02-23 18:15:25
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https://math.stackexchange.com/questions/1763888/residue-of-cot-z-at-z-0
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# Residue of $\cot z$ at $z= 0$
I need to calculate the Residue of $\cot z$ at the pole $0$ . Now, I think that the order of the pole is $1$. But I do not know how to show it. So, assuming the order of pole to be $1$, I calculated the residue as follows - $$\lim_{z \rightarrow 0} (z - 0) \cot z = \lim_{z \rightarrow 0} \frac{\cos z}{\frac{\sin z}{z}} = \frac{\lim_{z \rightarrow 0} \cos z}{\lim_{z \rightarrow 0} \frac{\sin z}{z}} = 1$$ Is my answer right? Again back to the main question, how do I determine or show the order of a pole(in general) ?
In general, assume to have two holomorphic functions $f$, $g$ with a zero at $z_0$. Assume $z_0$ is a zero of multiplicity $p$ of $f$ and multiplicity $q$ of $g$, then $z_0$ is a pole of order $q-p$ of $f/g$. If $q-p \leq 0$, the singularity is removable.
Here $0$ is a zero of multiplicity $1$ of $\sin(z)$ (since it doesn't cancel the derivative) and is a zero of multiplicity $0$ of $\cos(z)$ (since it doesn't cancel the cosine itself). Thus, it is a pole of order $1-0$ of $\frac{\cos(z)}{\sin(z)}.$
• By multiplicity you mean order? – Dark_Knight Apr 29 '16 at 9:36
• Yes, I mean $z_0$ is a zero of multiplicity/order $p$ of $f$ if it cancels $f$, $f'$, ... $f^{(p-1)}$ but not $f^{(p)}$. – C. Dubussy Apr 29 '16 at 9:38
Your answer is correct, and here is another way to try, in particular when taking derivatives or limits can be a little troublesome, and using the fact that we're only interested in low powers of $\;z\;$ in power or Laurent series since we want to find out something at $\;z=0\;$:
$$\frac{\cos z}{\sin z}=\frac{1-\frac{z^2}2+\ldots}{z-\frac{z^3}6+\ldots}=\frac{1-\frac{z^2}2+\ldots}{z\left(1-\frac{z^2}6+\ldots\right)}=\frac1z\left(1-\frac{z^2}2+\ldots\right)\left(1+\frac{z^2}6+\frac{z^4}{36}+\ldots\right)=$$
You can see the above two parentheses are power series and thus analytic, so in order to find out what the coefficient of $\;z^{-1}\;$ is we just do
$$=\frac1z\left(1-\frac{z^3}3+\ldots\right)=\frac1z+\ldots$$
and thus the residue is certainly $\;1\;$ .
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2019-08-17 16:55:22
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http://en.wikipedia.org/wiki/Row_and_column_spaces
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# Row and column spaces
The row space and column space of an m-by-n matrix are the linear subspaces generated by row vectors and column vectors, respectively, of the matrix. Its dimension is equal to the rank of the matrix and is at most min(m, n).[1]
The rest of article will consider matrices of real numbers: row and column spaces are subspace of Rn and Rm real spaces respectively. But row and column spaces can be constructed from matrices with components in any field and even a ring.
## Overview
Let A be a m-by-n matrix. Then
1. rank(A) = dim(rowsp(A)) = dim(colsp(A)),
2. rank(A) = number of pivots in any echelon form of A,
3. rank(A) = the maximum number of linearly independent rows or columns of A.
If one considers the matrix as a linear transformation from Rn to Rm, then the column space of the matrix equals the image of this linear transformation.
The column space of a matrix A is the set of all linear combinations of the columns in A. If A = [a1, ...., an], then colsp(A) = span {a1, ...., an}.
The concept of row space generalises to matrices to C, the field of complex numbers, or to any field.
Intuitively, given a matrix A, the action of the matrix A on a vector x will return a linear combination of the columns of A weighted by the coordinates of x as coefficients. Another way to look at this is that it will (1) first project x into the row space of A, (2) perform an invertible transformation, and (3) place the resulting vector y in the column space of A. Thus the result y =A x must reside in the column space of A. See the singular value decomposition for more details on this second interpretation.
## Example
Given a matrix J:
$J = \begin{bmatrix} 2 & 4 & 1 & 3 & 2\\ -1 & -2 & 1 & 0 & 5\\ 1 & 6 & 2 & 2 & 2\\ 3 & 6 & 2 & 5 & 1 \end{bmatrix}$
the rows are r1 = (2,4,1,3,2), r2 = (−1,−2,1,0,5), r3 = (1,6,2,2,2), r4 = (3,6,2,5,1). Consequently the row space of J is the subspace of R5 spanned by { r1, r2, r3, r4 }. Since these four row vectors are linearly independent, the row space is 4-dimensional. Moreover in this case it can be seen that they are all orthogonal to the vector n = (6,−1,4,−4,0), so it can be deduced that the row space consists of all vectors in R5 that are orthogonal to n.
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2014-03-17 14:51:07
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https://www.physicsforums.com/threads/tension-in-a-string-in-circular-motion.917354/
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# Homework Help: Tension in a string in circular motion
Tags:
1. Jun 12, 2017
1. The problem statement, all variables and given/known data
A string prq which is fixed at p and where q is vertically below p. r is a smooth ring threaded on the string which is made to rotate at an angular velocity ω rad/s in a horizontal circle centre q, the string being taut.
If |pq| = 0.12 m, |pr| + |rq| = 0.18 m,
show that ω = 294^(1/2) rad/s.
2. Relevant equations
m(ω^2)r
3. The attempt at a solution
I was able to solve this and get the correct answer, by setting the sum of the horizontal tension equal to the cent. force, but I had to assume that the tension in the qr part of the string is the same as the tension in the pr part. What I don't understand is how we know that the tension in both parts of the string are the same considering they are at different angles.
I was thinking that because the string is massless, it must be the same everywhere. But then I encountered another question:
For this question a ball is connected by one string (it's massless) and I had to find the tension in the lower and upper part of the string. I was able to solve this, but what's confusing me is when to know when the tension is the same everywhere in the string, and when it's different.
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2. Jun 12, 2017
### zwierz
is there gravity?
3. Jun 12, 2017
Yes for both.
4. Jun 12, 2017
### zwierz
As I understand the string is massless.
Then you have
Now break into the frame OXY the second Newton law: $m\boldsymbol a=\boldsymbol T_1+\boldsymbol T_2+m\boldsymbol g$
5. Jun 12, 2017
I was able to solve both questions and get the correct answer. What I'm wondering is why the tension was the same for the entire string in the first question and why for the second question the tension is different in the upper and lower part.
6. Jun 12, 2017
### zwierz
$|\boldsymbol T_1|$ can not be equal to $|\boldsymbol T_2|$ it is incompatible with Newton second law or with the configuration presented at the picture
7. Jun 12, 2017
### scottdave
How is the ball connected? It appears that the ball physically fixed to the rope at a position on the rope. Consider an extreme case: Ask yourself, if it stops spinning so that the ball is hanging straight down, what would be the tension in the upper string? and what would be the tension in the lower string? With the ball attached at a certain point on the string, you should be able to see that the bottom string would be hanging slack, while the upper string has m*g as it's tension.
Now look at the bead which can slide around on the string. Can you see how the tension in each string would be the same when it is hanging straight down? The same works when you consider strings over pulleys.
8. Jun 12, 2017
### zwierz
I guess that in case of entire string and when the bead can slide along the string freely then the triangle will not have the both sides equal
9. Jun 12, 2017
I was able to follow you up to the last two lines. In the original question, the ring is travelling in a horizontal circle. qr is parallel with the ground and pr is at an angle. I'm still unsure as to how the tension is the same and how the ring is "hanging straight down".
10. Jun 12, 2017
I posted two separate questions in my post. That picture applies only to the second question. The first question is a right angled triangle with a ring travelling in a horizontal circle. The only way I could get the answer 294^(1/2) was assuming that the tensions were the same.
11. Jun 12, 2017
### scottdave
Actually, I was referring to "if the pole stops spinning, what will happen to the bead?" Will it stay in the same place on the line, or will it slide down? The problem statement indicates that it can slide. With the pole stopped, it will hang straight down. What is the tension in the "two" strings (actually the same string). Now if the pole is spinning, you can expect the bead to slide up, and the tension on either side of the bead will be equal magnitude.
12. Jun 12, 2017
### scottdave
With the sliding bead, you should expect the tensions to be equal magnitude. If the bead were tied off, unable to slide, then the tensions will under most circumstances not be the same. I was demonstrating an extreme case of the pole stopping, to show how in the sliding bead, the tension remains the same on either side, while in the fixed ball, they are not.
13. Jun 12, 2017
How are the tensions equal in magnitude when the bead is sliding, shouldn't the upper string have more tension because of the gravity on the bead?
14. Jun 12, 2017
### scottdave
Let's say the pole is not moving. So the bead hangs straight down, and each side of the string has a tension of (m*g)/2. As it starts spinning, you will have a horizontal component (m*a). This will increase as the pole spins faster. Each side of the string has the same magnitude of tension, but in different directions: each one carries a different portion of the horizontal and vertical components.
15. Jun 12, 2017
Yes I kind of understand it better now. Thanks!
16. Jun 12, 2017
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2018-06-19 14:47:48
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http://hal.in2p3.fr/in2p3-01323815
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# High accuracy position response calibration method for a micro-channel plate ion detector
Abstract : We have developed a position response calibration method for a micro-channel plate (MCP) detector with a delay-line anode position readout scheme. Using an {\em in situ} calibration mask, an accuracy of 8~$\mu$m and a resolution of 85~$\mu$m (FWHM) have been achieved for MeV-scale $\alpha$ particles and ions with energies of $\sim$10~keV. For high statistics experiments, this method can be directly employed with the experimental data without any dedicated calibration runs. The improved performance of the MCP detector can find applications in many fields of AMO and nuclear physics. In our case, it helps reducing systematic uncertainties in a high-precision nuclear $\beta$-decay experiment.
Document type :
Journal articles
Complete list of metadatas
http://hal.in2p3.fr/in2p3-01323815
Contributor : Sandrine Guesnon <>
Submitted on : Tuesday, May 31, 2016 - 11:16:07 AM
Last modification on : Monday, January 27, 2020 - 11:42:13 AM
### Citation
R. Hong, A. Leredde, Y. Bagdasarova, X. Fléchard, A. Garcia, et al.. High accuracy position response calibration method for a micro-channel plate ion detector. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, Elsevier, 2016, 835, pp.42-51. ⟨10.1016/j.nima.2016.08.024⟩. ⟨in2p3-01323815⟩
Record views
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2020-09-19 19:20:45
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https://www.physicsforums.com/threads/calculate-pressure-drop-in-dense-phase-pneumatic-conveying.869131/
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# Calculate pressure drop in dense phase pneumatic conveying
I've came across many references for pressure drop calculations for dilute phase pneumatic conveying, but i've yet to encounter any for dense phase. Anyone has any idea how do i calculate that? Thank you
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2021-02-26 10:12:18
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https://lw2.issarice.com/posts/2LP4XkDvvrrjmk2jJ/on-charities-and-linear-utility
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# On Charities and Linear Utility
post by Anatoly_Vorobey · 2011-02-04T14:13:29.123Z · LW · GW · Legacy · 59 comments
Steven Landsburg argued, in an oft-quoted article, that the rational way to donate to charity is to give everything to the charity you consider most effective, rather than diversify; and that this is always true when your contribution is much smaller than the charities' endowments. Besides an informal argument, he provided a mathematical addendum for people who aren't intimidated by partial derivatives. This post will bank on your familiarity with both.
I submit that the math is sloppy and the words don't match the math. This isn't to say that the entire thing must be rejected; on the contrary, an improved set of assumptions will fix the math and make the argument whole. Yet it is useful to understand the assumptions better, whether you want to adopt or reject them.
And so, consider the math. We assume that our desire is to maximize some utility function U(X, Y, Z), where X, Y and Z are total endowments of three different charities. It's reasonable to assume U is smooth enough so we can take derivatives and apply basic calculus with impunity. We consider our own contributions Δx, Δy and Δz, and form a linear approximation to the updated value U(X+Δx, Y+Δy, Z+Δz). If this approximation is close enough to the true value, the rest of the argument goes through: given that the sum Δx+Δy+Δz is fixed, it's best to put everything into the charity with the largest partial derivative at (X,Y,Z).
The approximation, Landsburg says, is good "assuming that your contributions are small relative to the initial endowments". Here's the thing: why? Suppose Δx/X, Δy/Y and Δz/Z are indeed very small - what then? Why does it follow that the linear approximation works? There's no explanation, and if you think this is because it's immediately obvious - well, it isn't. It may sound plausible, but the math isn't there. We need to go deeper.
We don't need to go all that deep, actually. The tool which allows us to estimate our error is Taylor's theorem for several variables. If you stare into that for a bit, taking n=1, you'll see that the leftovers from U(X+Δx, Y+Δy, Z+Δz), after we take out U(X,Y,Z) and the linear terms with the partial derivatives, are a bunch of terms that are basically second derivatives and mixed derivatives of U times quadratic contributions. In other, scarier words, things like ∂U2/∂x2*(Δx)2 and ∂U2/∂x∂y*(ΔxΔy). The values of the second/mixed derivatives to be used are their maximal values over all the region from (X,Y,Z) to (X+Δx, Y+Δy, Z+Δz). And I've also ignored some small constant factors there, to keep things simple.
These leftovers are to be compared with the linear terms like ∂U(X, Y, Z)/∂x*(Δx). If the leftovers are much smaller than the linear terms, then the approximation is pretty good. When will that happen?
Let's look at the mixed derivatives like ∂U2/∂x∂y first. A partial derivative like ∂U/∂x measures the effectiveness of charity X - how much utility it brings per dollar, currently. A mixed derivative measures how much the effectiveness of X changes as we donate money to Y. If charities X and Y operate in different domains, this is likely to be very close to zero, and the mixed derivative terms can be ignored. If X and Y work on something related, this isn't likely to be zero at all, and the overall argument fails. So, we need a new assumption: X and Y operate in different domains, or more generally do not affect each other's effectiveness.
(Here's an artificial example: say you have X working on buying food for hungry Eurafsicans, and Y working on delivering food aid to Eurafsica. If things are perfectly balanced between them, then your $100 contribution to either is not of much use, but a$50 contribution to both is helpful. This is because X's effectiveness rises if Y gets a little more money, and vice versa. Note that it may also happen that the interference between X and Y hinders rather than helps their common goal, and in that case it may still be better to give money to only one of them, but not because of Landsburg's argument).
Now that the mixed derivate terms are gone, let's look at the second derivative terms like ∂U2/∂x2*(Δx)2 and compare them to the linear terms ∂U(X, Y, Z)/∂x*(Δx). Cancelling out the common factor Δx, we see that for the linear approximation to be good, the effectiveness of the charity (first derivative) must be much greater than maximum rate of effectiveness change (over the given interval) times the contribution. This gives us the practical criterion to follow. If our contribution is very large, or if it is liable to influence the charity's effectiveness considerably - and "large", "considerably" here are words that can in principle be measured and made precise - then the linear approximation isn't so good, and we can't infer automatically that banking everything on one charity is the right thing to do. It may still be, but that requires a deeper analysis, one that may or may not be feasible with our amount of uncertainty.
(Here's an artificial example: say X is a non-profit that has a program to match your contribution until they reach a set goal of donations, and they're $50 short of that goal. Say Y is another non-profit or charity that you consider to be 1.5 more effective than X, normally. Then, if your budget is$100, it's optimal to give $50 to X and$50 to Y, rather than everything to one of them. This is because X will undergo a radical change in effectiveness after the first $50, and the second derivative will be large). Where is the original criterion "keep Δx/X small", then? It failed to materialize; instead, the math tells us that we need to keep Δx small and ∂U2/∂x2 small. The endowment size, as expected, is irrelevant; but let's try to make a connection with it anyhow. We can do it with a heuristic argument that goes something like this. The second derivative measures the way our donations influence effectiveness of the charity, rather than the utility. If the charity is large and well-established, probably the way your money splits into administrative costs and actual good has stabilized and is unlikely to change; whereas if the charity is small and especially just starting out, your money may help it set up or change its infrastructure which will change its effectiveness. So the size of the endowment probably correlates well with the second derivative being small. And then the recipe becomes "keep Δx small and X large", which can be rephrased as the original "keep Δx/X small". Does this re-establish the correctness of the original argument, then? To some extent, yet to my mind not a large one. For one thing, the correlation is not ideal, it's easy to think of exceptions, and in fact if you're dealing with real charities that tell you something about how they operate, it may be easier for you to estimate that the rate of effectiveness change is or isn't zero, than it is to look at endowment size. But more importantly, the heuristic jump through the correlative hoop strips the argument of any numeric force that the correct version does have. Since we don't know how exactly X and ∂U2/∂x2 are related, e.g. what is the correlation factor, we can't give any specific meaning to "keep Δx/X small". We can't say how small, even approximately: 1/10? 1/1000? 1/106? This makes me suspect that the heuristic argument is not a good way to approach the truth, but may be a good way to convince someone to put everything into one charity, because normally Δx/X will appear to be rather small. Once we've worked out the math, some deficiencies of the original informal article become clear. For instance, the talk about "making a dent" in the problem is a little off the mark: So why is charity different? Here's the reason: An investment in Microsoft can make a serious dent in the problem of adding some high-tech stocks to your portfolio; now it's time to move on to other investment goals. Two hours on the golf course makes a serious dent in the problem of getting some exercise; maybe it's time to see what else in life is worthy of attention. But no matter how much you give to CARE, you will never make a serious dent in the problem of starving children. The problem is just too big; behind every starving child is another equally deserving child. But it isn't making the dent in the problem that's the issue; it's making the dent in effectiveness. It's conceivable that my donation doesn't make a noticeable dent in the problem, but changes the rate of dent-making enough so the argument falls through. The words don't match the math. Similarly, consider the analogy with investment. Why doesn't it in fact work - why doesn't the math argument work on investment portfolios? If you want to maximize profit, your small stock purchase is unlikely on its own to influence the stock (change its effectiveness) much. Landsburg's answer - putting it in terms of "making a dent in the problem of adding high-tech stocks" - is flawed: it presupposes that diversification is good by cordoning off different investment areas from each other - "high-tech stocks". The real reason is, of course, risk aversion - our utility function for investment is likely to be risk-averse. You may want to apply risk aversion to your charity donation as well, but in that case, Eliezer's advice to purchase fuzzies and utilons separately is persuasive. To sum up, these assumptions will let you conclude that the rational thing to do is donate all your charity budget to the one charity you consider most effective currently: 1. Assume that a unified single-currency utility function describes your idea of these charities' utilities. The crucial thing you're assuming here is that X, Y and Z can be compared in terms of the good they're doing; that their amounts of good can be computed in the same "utilons". This is a strong assumption that may work for some and not others. Certainly most people reject it when all goals are at stake, and not just charity-giving; Complexity of Value discusses this. Why are things different when we restrict to helping others, and are they necessarily different? Yvain has presented an excellent argument for the latter position in Efficient Charity, while Phil Goetz provided an excellent argument against it in the comments. 2. Assume that the charities you're choosing between operate in different domains, or more generally speaking do not affect each other's effectiveness. If they do, putting all into one may still be the right thing to do, but finer analysis is needed. 3. Assume that the effectiveness of the charities is influenced very little or not at all by your donation, and the donation itself is not too large. In case of doubt, apply more precise math: rate of effectiveness change times donation must be much smaller - say, a few percent of - effectiveness itself. ## 59 comments Comments sorted by top scores. comment by cousin_it · 2011-02-04T15:34:49.432Z · LW(p) · GW(p) Complexity of value doesn't necessarily imply that people have more than one kind of utilon, it just says that people can value 1 unit of food + 1 unit of sex higher than 2 units of either. In other words, it says the second derivative terms are significant :-) Replies from: rhollerith_dot_com comment by rhollerith_dot_com · 2011-02-04T18:07:16.796Z · LW(p) · GW(p) Thanks! Until I saw your comment, the third-from-last paragraph (combined with the impression of credibility created by the rest of the post) was making me very sad because of what it implies about the effectiveness of decision theory and the usefulness of formal goal systems. My probability of the success of the FAI project dropped sharply in a few minutes, then recovered when I read your comment. comment by [deleted] · 2011-02-05T16:59:08.724Z · LW(p) · GW(p) Do "finite charitable causes" require special consideration here? For example, Bill Gates is pushing hard for polio eradication. Polio is a special kind of problem: once solved, it will stay solved forever. I see two special features of finite problems: first, solving them earlier is better than solving them later. The eradications of smallpox and rinderpest are delivering benefits forever while consuming zero additional resources. Second, the marginal utility of donation is pretty unusual - for disease eradication, donating more helps to suppress it more (which is good in its own right), donating lots delivers the huge benefit you're looking for, and donating even more than that would do nothing. Other finite problems, like research, will feature different curves (notably without the property of "slipping away" if insufficient resources are devoted to them for a time) but still seem unusual. Note: By "finite", I mean that the solution is within sight. World hunger is a finite problem in the sense that bringing the world up to a post-scarcity economy, or even to the level of the industrialized economies (without changing the climate from original recipe to extra crispy), would solve it - but nobody has achieved the former yet, and even the latter is very far away. Commercial nuclear fusion is on the threshold of what I consider finite: we know it's possible, we know solving it is a matter of engineering and not discovering new physics, but at the same time it's very difficult and very expensive and will take a long time. Replies from: gwern comment by gwern · 2011-02-05T17:49:06.239Z · LW(p) · GW(p) Let's look at some relevant quotes I noticed when I read that a few days ago and was posting excerpts into #lesswrong: By contrast, the 14-year drive to wipe out smallpox, according to Dr. Donald A. Henderson, the former World Health Organization officer who began it, cost only$500 million in today’s dollars.
...
Right now, there are fewer than 2,000. The skeptics acknowledge that they are arguing for accepting more paralysis and death as the price of shifting that $1 billion to vaccines and other measures that prevent millions of deaths from pneumonia, diarrhea, measles, meningitis and malaria. “And think of all the money that would be saved,” Mr. Gates went on, turning sarcastic. “It’d be like 5 percent of the dog food market in the United States.” (I believe there is a fine Eliezer post on exactly the fallacious argument Mr. Gates is using there.) One injection stops smallpox, but in countries with open sewers, children need polio drops up to 10 times. Only one victim in every 200 shows symptoms, so when there are 500 paralysis cases, as in the recent Congo Republic outbreak, there are 100,000 more silent carriers. Other causes of paralysis, from food poisoning to Epstein-Barr virus, complicate surveillance. Also, in roughly one of every two million vaccinations, the live vaccine strain can mutate and paralyze the child getting it. And many poor families whose children are dying of other diseases are fed up with polio drives. Smallpox has no natural reservoirs; its vaccine is made from cowpox, not weakened or dead smallpox, while polio vaccines are made from weakened or dead polio viruses and may themselves undo eradication. No one speaks of eradicating anthrax because it's impossible to reach all the natural sources of anthrax spores in deserts in Mongolia or where-ever. Nor do we speak of eradicating ebola because we don't want to extinguish various primate species. Does squeezing polio jell-o offer the best marginal returns? Are we appealing to sunk-costs here? Eradicating polio may offer permanent benefits (though this is dubious for previously mentioned reasons), but still be a bad investment - similar to how one rarely invests in perpetual bonds. I see no reason why the usual apparatus of highest marginal return and discount rates do not cover your finite charity distinction. Replies from: None comment by [deleted] · 2011-02-05T19:15:41.371Z · LW(p) · GW(p) Correcting factual errors: Smallpox has no natural reservoirs polio vaccines are made from weakened or dead polio viruses and may themselves undo eradication. First, the inactivated/killed polio vaccine cannot come back to life. Second, that risk for the attenuated vaccine is well known. Eradicating polio may offer permanent benefits Will. You can disagree as to whether the benefits are worth the costs, of course. And perhaps finite problems can be analyzed within the usual framework - but I wanted to bring them up. Replies from: gwern comment by gwern · 2011-02-05T20:24:55.302Z · LW(p) · GW(p) No, may. Even if you have zero known cases, you have not eradicated polio for sure because of the carriers and natural reservoirs and obscure little hidden villages. So you still need vaccines. And your own links point out that the attenuated vaccine can still be infectious! This is believed to be a rare event, but outbreaks of vaccine-associated paralytic poliomyelitis (VAPP) have been reported, and tend to occur in areas of low coverage by OPV, presumably because the OPV is itself protective against the related outbreak strain. To quote from one of the references: The rate and pattern of VP1 divergence among the circulating vaccine-derived poliovirus (cVDPV) isolates suggested that all lineages were derived from a single OPV infection that occurred around 1983 and that progeny from the initiating infection circulated for approximately a decade within Egypt along several independent chains of transmission. Replies from: None comment by [deleted] · 2011-02-06T07:31:59.578Z · LW(p) · GW(p) What is a "contaminated natural source"? I am genuinely curious. Replies from: gwern comment by gwern · 2011-02-06T16:53:36.836Z · LW(p) · GW(p) A variation on natural reservoir. comment by Psychohistorian · 2011-02-04T21:09:32.403Z · LW(p) · GW(p) Your entire point seems to be that it's better to give to multiple charities when the joint utility of giving to those charities exceeds the benefit of giving all the money to one charity. This circumstance exists in the real world for most individuals so infrequently as to be properly ignored. It is extremely unlikely that there is some combination of charities such that giving$5,000 to each of them will generate substantially better returns than giving $10,000 to the best available charity. Unless I'm ignoring important evidence, charities just don't work together that comprehensively, and non-huge sums of money do not have dramatic enough effects that it would be efficient to split them up. Also, you chose an incredibly dense and inefficient way to make what seems like a very simple point. Replies from: komponisto, luminosity, Benquo comment by komponisto · 2011-02-04T23:08:12.576Z · LW(p) · GW(p) Also, you chose an incredibly dense and inefficient way to make what seems like a very simple point In general, I would caution against criticisms of this form for several reasons: • different thinking styles: what seems unnecessarily convoluted to one person may seem utterly natural to another; • hindsight bias: something may appear simple after you've worked it out, but that doesn't mean that working it out was easy while you were doing it; • incentives: one should think very carefully before writing any comment that sounds like "this post really ought not to have been written". Replies from: Psychohistorian comment by Psychohistorian · 2011-02-06T00:01:30.535Z · LW(p) · GW(p) incentives: one should think very carefully before writing any comment that sounds like "this post really ought not to have been written". While I don't necessarily hold that opinion of this particular post, it's a defensible position. I think that posts that use relatively complicated math where simple English would suffice substantially and negatively affect the quality of discourse. If someone has a OK point to make, it is arguably better that they not make it at all than that they make it in a convoluted manner, because that suggests to other people that it's OK to make such posts. It's certainly better that they start it out in the discussion section so that it turns into a more comprehensible post on the main page. Of course, the more original or interesting their actual idea, the more the benefits outweigh the costs. Your different thinking styles criticism is absolutely on point though, I admit, assuming it actually applies. comment by luminosity · 2011-02-05T13:06:40.459Z · LW(p) · GW(p) Off the top of my head, one charity might be worse overall but might need a small amount of funding to attempt an experimental strategy aimed at improving it. If the likelihood of finding a better way than the most efficient charity pursues is high enough, then small funding to the experimental charity and the rest of your donation to the other could be optimal. In general I am uneasy with suggestions that one should focus all their charitable energy in one direction, because people are far too prone to finding a local maximum then ceasing exploration for higher maximums. Replies from: Psychohistorian comment by Psychohistorian · 2011-02-06T00:04:29.842Z · LW(p) · GW(p) one charity might be worse overall but might need a small amount of funding to attempt an experimental strategy aimed at improving it. If this were true, that would mean that that charity had extremely high but rapidly diminishing marginal returns, in which case you should give it money until those diminishing MRs brought it below your next best option. I'm pretty confident that diverse investment is only proper where charities exhibit interactive returns (which is probably extremely rare for most people's value of charitable contributions) or whether you are trying to maximize something other than effective charity. comment by Benquo · 2011-02-05T02:07:19.837Z · LW(p) · GW(p) I agree; I'd be quite surprised if it were at all common for separate charities working in the same field to be so well-balanced in scale that proportional contributions to both outweigh contributions to just one. Since there's no clear feedback mechanism to help people maximize expected utility in giving, there's no reason to expect the MUs to be anywhere close. Therefore we should strongly expect that a "bullet" strategy will outperform diversification. I dub this the Inefficient Charity Markets Hypothesis. On the other hand, I wonder what percentage of charity contributions are given by the top 1% of donors, people who really can make a dent in these problems? Their impact probably dwarfs anything the vast majority of small donors in the audience would do. But I'd bet they're smart enough to realize this stuff doesn't apply to them. Replies from: TobyBartels comment by TobyBartels · 2011-02-05T05:47:24.300Z · LW(p) · GW(p) But I'd bet they're smart enough to realize this stuff doesn't apply to them. Or they've never heard Landsburg's argument anyway! comment by MatthewW · 2011-02-05T11:24:00.797Z · LW(p) · GW(p) I don't think there's much need for heuristics like "rate of effectiveness change times donation must be much smaller - say, a few percent of - effectiveness itself." If you're really using a Landsburg-style calculation to decide where to donate to, you've already estimated the effectiveness of the second-most effective charity, so you can just say that effectiveness drop must be no greater than the corresponding difference. Replies from: Anatoly_Vorobey comment by Anatoly_Vorobey · 2011-02-05T14:43:51.828Z · LW(p) · GW(p) That's an excellent point that I managed to completely miss. Thank you. I'll try to add an endnote to that effect. comment by whpearson · 2011-02-04T17:02:14.749Z · LW(p) · GW(p) I wondered for a while how the math would change if you assumed that a number of other agents had the same decision function as you. Even if you individual contribution is small, n rational agents see that charity X is optimal and give money to it might change the utility per dollar significantly. I haven't worked through the math though. Replies from: Snowyowl, jsalvatier, Benquo comment by Snowyowl · 2011-02-04T19:26:08.809Z · LW(p) · GW(p) Yes, but that only poses a problem if a large number of agents make large contributions at the same time. If they make individually large contributions at different times or if they spread their contributions out over a period of time, they will see the utility per dollar change and be able to adjust accordingly. Presumably some sort of equilibrium will eventually emerge. Anyway, this is probably pretty irrelevant to the real world, though I agree that the math is interesting. Replies from: David_Gerard, whpearson comment by David_Gerard · 2011-02-04T19:30:30.383Z · LW(p) · GW(p) Yes, but that only poses a problem if a large number of agents make large contributions at the same time. You mean, like donating to a funding drive with a specific aim? Replies from: Snowyowl comment by Snowyowl · 2011-02-04T22:11:49.900Z · LW(p) · GW(p) Point taken. comment by whpearson · 2011-02-04T20:54:30.899Z · LW(p) · GW(p) With perfect information. and infinity flexible charities (that could borrow off future giving if they weren't optimal that time period), then yep. I'd agree it is irrelevant to the real world because most people aren't following the "giving everything to one charity" strategy. If everyone followed givewell then things might get hairy for charities as they became and lost being flavour of the time period. comment by jsalvatier · 2011-02-04T20:14:26.911Z · LW(p) · GW(p) I'm not sure it's settled how to even do that math. Replies from: whpearson comment by whpearson · 2011-02-04T20:45:45.646Z · LW(p) · GW(p) There is a variety of math that could be done. It is relatively easy to show that certain strategies may not be optimal, which is what I was thinking about. I wasn't touching how to make optimal decisions, which would very much be in the TDT realm I think. comment by Benquo · 2011-02-05T01:49:05.024Z · LW(p) · GW(p) This should only matter to the extent that the agents have to act simultaneously or near-simultaneously. Otherwise, whoever goes second maximized utility conditioned on the choices of the first, and so on, so it's no worse than if a single person sought the local maximum for their giving. Of course, the difference between local and global maxima is important, but that has nothing to do with the OP, and everything to do with TDT. comment by Jonathan_Graehl · 2011-02-16T09:20:25.025Z · LW(p) · GW(p) Once everyone who gives is acting on the principle of "give each marginal charitable dollar so it does the most good", then you can worry about diversification, and only then if you're sure that the total contributions by everyone are actually over-concentrated. comment by timtyler · 2011-02-06T14:37:15.645Z · LW(p) · GW(p) You may want to apply risk aversion to your charity donation as well, but in that case, Eliezer's advice to purchase fuzzies and utilons separately is persuasive. That link presents a curious argument. I figure the reason most people give to charity is to affect their image - for signalling reasons. So, for example, we have Bill Gates - ex-boss one of the most unpopular IT companies ever (widely known as the evil empire) - trying to use his money to clean up his image by donating some of that money to charity. That such things result in good being done in the world is due to the entanglement of "fuzzies" and "utilons" - in the terminology of that post. If these become disentangled, surely most people would just buy the "fuzzies" - and fewer good deeds would be performed overall. Replies from: syllogism comment by syllogism · 2011-02-09T22:11:29.483Z · LW(p) · GW(p) I disagree. I think that individuals such as Gates have adopted making the world a better place as a terminal or near-terminal value. I see no evidence that he is acting in anything but the best of faith. I think he is sincerely trying to direct his money wherever it will gain the most utilons for the world, not the most utilons for him. Status-seeking charitable works look considerably different to me. They exhibit all the normal biases of people's emotional moral compass: they're not forward looking enough, they're too local, they focus on things the endower and their friends enjoy or make use of, such as the arts. You might say that the adoption of the value of doing good in the world is a status seeking behaviour. Maybe, but this is irrelevant as long as the value is to do good, rather than seem to do good. So long as the effort is in good faith, the advice to seek utilons and fuzzies separately applies. comment by AstroCJ · 2011-02-05T11:46:32.003Z · LW(p) · GW(p) Speaking from a physical perspective, assuming that "$\Delta x$is small" is a meaningless statement. Whenever we state that something is large or small, unless it's a nondimensionalised number, there is something against which we are comparing it. Simple example, which isn't the best example but is fast to construct. Comparing$1 to $(mean GDP from country to country) *$1 is a small amount of money in the USA. Even homeless people can scrape together a dollar, and it's not even enough to buy a cup of coffee from Starbucks. It's almost worthless.
*$1 is a large amount of money in Nigeria. The GNI is around$930 per capita per year[1], so if you're lucky enough to make the mean income, you'd better not be frittering away that $1; it's vital if you want to pay your rent and buy food. So we can't say$\Delta x$is a "small amount of money" without qualification; it seems like when you conclude that, we are actually concluding$\Delta x / X$is small, the original proposal. A better measure might be$\Delta x / \sqrt{XYZ,3}$, so that the scale in each direction doesn't change (but that's just choosing a different coordinate system, so not that relevant). Your argument seeks to confirm the original proposal, not refute it, and you've pointed out that sometimes higher derivatives can be important. (Incidentally, your second example - about nonmixed second derivatives - became clear to me only after some thought. You might want to include a clause like "Because after the first$50, the second derivatives represent a sudden jump down in net utility as we get less bang for our individual buck".)
Replies from: Anatoly_Vorobey
comment by Anatoly_Vorobey · 2011-02-05T14:30:16.163Z · LW(p) · GW(p)
I take your point about the meaninglessness of sizing up dimensional quantities without a referent. But sometimes the referent is inherently specified in different units. If you want to travel, with constant speed, no more than 10 miles - less is OK - then the time of your travel must be small - how small? - well, its product with your speed shouldn't exceed 10 miles. You could say, just divide 10 miles by the speed and use that as the upper bound, but that only works if the speed is fixed. If you're choosing between traveling on foot, on a bicycle, and in a car, you really are choosing on two different axes that are jointly constrained. So it is in my post: the second derivative times the donation is constrained, and the units work out. You can say "this works when the donation is small enough and the 2nd derivative is small enough" without comparing them to something in their own units, because the meaning of "small enough" is in that dimensional equation.
Besides, consider the following: why is it X that you're comparing Δx to? Sure, it's in the same units, but how is it relevant? In your analogy, GNI per capita is relevant to $1 because it represents the mean income I could expect to generate over the year. But note that you're not comparing$1 to the total GNI of the country, even though it's in the same unit, dollars, because the total population size, which drives that number, is not very relevant to the effect of $1 on one single person. With charities, how is the current endowment relevant to the contribution I hope to make with my own donation? It is not, after all, as if my goal was to maximize my donation's utility relative to other donors' in the same charity - because we stipulated that I'm only caring about the total absolute good I contribute... Thanks for the suggestion about my wording - I'll try to make that example a bit clearer along the lines you propose. comment by topynate · 2011-02-06T13:56:50.007Z · LW(p) · GW(p) Consider those charities that expect their mission to take years rather than months. These charities will rationally want to spread their spending out over time. Particularly for charities with large endowments, they will attempt to use the interest on their money rather than depleting the principal, although if they expect to receive more donations over time they can be more liberal. This means that a single donation slightly increases the rate at which such a charity does good, rather than enabling it to do things which it could not otherwise do. So the scaling factor of the endowment is restored: donating$1000 to a charity with a $10m endowment increases the rate at which it can sustainably spend by 1000/10^7 = 0.1%. This does not mean that a charity will say, look, if our sustainable spending rate was 0.1% higher we'd have enough available this year to fund the 'save a million kids from starvation' project, oh well. They'll save the million kids and spend a bit less next year, all other things being equal. In other words, the charity, by maximising the good it does with the money it has, smooths out the change in its utility for small differences in spending relative to the size of its endowment, i.e. the higher order derivatives are low. So long as the utility you get from a charity comes from it fulfilling its stated mission, your utility will also vary smoothly with small spending differences. Likewise, with rational collaborating charities, they will each adjust their spending to increase any mutually beneficial effects. So mixed derivatives are low, too. The upshot is that unless your donation is of a size that it can permanently and significantly raise the spending power of such a charity, you won't be leaving the approximately linear neighbourhood in utility-space. So if you're looking for counterexamples, you'll need to find one of: • charities with both low endowments and low donation rates, which nevertheless can produce massive positive effects with a smallish amount of money • charities which must fulfil their mission in a short time and are just short of having the money to do so. comment by PhilGoetz · 2011-02-06T05:06:48.775Z · LW(p) · GW(p) The approximation, Landsburg says, is good "assuming that your contributions are small relative to the initial endowments". Here's the thing: why? Suppose Δx/X, Δy/Y and Δz/Z are indeed very small - what then? Why does it follow that the linear approximation works? There's no explanation, and if you think this is because it's immediately obvious - well, it isn't. It may sound plausible, but the math isn't there. We need to go deeper. What shapes for U(X,Y,Z) could make the linear approximation not work? It would have to be a curve that had sudden local changes. It would be kinked or fractal. That would be surprising. If U(X,Y,Z) is continuous, smooth, monotonic, and its first and second derivatives are monotonic, I can't imagine how the linear approximation could fail. Replies from: Anatoly_Vorobey comment by Anatoly_Vorobey · 2011-02-06T14:50:18.209Z · LW(p) · GW(p) If U(X,Y,Z) is continuous, smooth, monotonic, and its first and second derivatives are monotonic, I can't imagine how the linear approximation could fail. There's an example later in the post, with mixed derivatives. Everything could be smooth and monotonic including all derivatives. Basically think of U(X,Y,Z) as containing a 100XY component. comment by Will_Sawin · 2011-02-04T15:30:56.714Z · LW(p) · GW(p) If U(X) follows a power-law distribution, that du/dx / d^2u/dx^2 is proportional to X. comment by XiXiDu · 2011-02-05T11:31:32.215Z · LW(p) · GW(p) If everyone was to take Landsburg's argument seriously, which would imply that all humans were rational, then everyone would solely donate to the SIAI. If everyone only donated to the SIAI, would something like Wikipedia even exist? I suppose the SIAI would have created Wikipedia if it was necessary. I'm just wondering how much important stuff out there was spawned by irrational contributions and how the world would look like if such contributions would have never been made. I'm also not sure how venture capitalist growth funding differs from the idea to diversify one's contributions to charity. Note that I do not doubt the correctness of Landsburg's math. I'm just not sure if it would have worked out given human shortcomings (even if everyone was maximally rational). If nobody was to diversify, contributing to what seems to be the most rational option given the current data, then being wrong would be a catastrophe. Even maximally rational humans can fail after all. This wouldn't likely be a problem if everyone contributed to a goal that could be verified rather quickly, but something like the SIAI could eat up the resources of the planet and still turn out to be not even wrong in the end. Since everyone would have concentrated on that one goal (no doubt being the most rational choice at the moment), might such a counterfactual world have been better off diversifying its contributions or would the SIAI have turned into some kind of financial management allocating those contributions and subsequently become itself a venture capitalist? Replies from: Larks, Anatoly_Vorobey, PhilGoetz, AstroCJ, nshepperd comment by Larks · 2011-02-05T14:27:54.468Z · LW(p) · GW(p) People don't make their decisions simultaneously and instantaneously; once SIAI suffers diminishing returns to the extent that it's no longer the best option, people can observe this and donate elsewhere. Replies from: XiXiDu comment by XiXiDu · 2011-02-05T16:39:42.234Z · LW(p) · GW(p) ...once SIAI suffers diminishing returns to the extent that it's no longer the best option, people can observe this and donate elsewhere. How would you observe that, what are the expected indications? comment by Anatoly_Vorobey · 2011-02-05T14:41:10.698Z · LW(p) · GW(p) It's consistent with Landsburg's analysis that everyone has their own utility function that emphasizes what that particular person considers important. So if everyone were a Landsburgian and donated only to a single charity, they would still donate all over the map to different charities - because, even if they all knew about SIAI, they either wouldn't care as much about SIAI's goals as other goals, or they would estimate SIAI's effectiveness in reaching those goals as very low. There probably would still be adverse impact to many charities which are second-choice for most their donors - and I'm sure there are many such - but not as catastrophic as you're outlining, I think. Personally, I believe that if everyone was presented with Landsburg's argument, most people would fail to be Landsburgians not because they couldn't stomach the math, or because they'd be wary of the more technical assumptions I wrote about in my post, but simply because they wouldn't agree to characterize their charitable utility in unified single-currency utilons. comment by PhilGoetz · 2011-02-06T05:09:36.713Z · LW(p) · GW(p) You shouldn't take it as an axiom that the SIAI is the most-beneficial charity in the world. You imply that anyone who thinks otherwise is irrational. Replies from: XiXiDu comment by XiXiDu · 2011-02-06T12:55:37.750Z · LW(p) · GW(p) I know, the Karma system made me overcompensate. I noticed that questions are often voted down so I tried to counter that by making it sound more agreeable. It was something that bothered me so I thought LW and this post would be the best place to get some feedback. I was unable to read the OP or Landsburg's proof but was still seeking answers before learning enough to come up with my own answers. I'm often trying to get feedback from experts without first studying that field myself. If I have an astronomy question I ask on an astronomy forum. Luckily most of the time people are kind enough to not demand that you first become an expert before you can ask questions. It would be pretty daunting if you would have to become a heart surgeon before you could ask about your heart surgery. But that's how it is on LW, I have to learn that the price you pay for any uninformed questions are downvotes. I acknowledge that the Karma system and general attitude here makes me dishonest in what I write and apologize for that, I know that it is wrong. Replies from: timtyler comment by timtyler · 2011-02-06T13:42:43.185Z · LW(p) · GW(p) You do have over 2000 karma. At this point, I figure you have earned the right to say more-or-less whatever you like, for quite a while, without bothering too much about keeping score. Replies from: Wei_Dai, PhilGoetz comment by Wei_Dai · 2011-02-06T23:41:01.225Z · LW(p) · GW(p) When I'm reading comments, I often skip over the ones that have low or negative score. I imagine other people do the same thing. So if you think your point is important enough to be read by more than a few people, you do want to try to have it voted up (but of course you shouldn't significantly compromise your other values/interests to do so). comment by PhilGoetz · 2011-02-06T16:53:18.354Z · LW(p) · GW(p) I'm curious why Tim's comment got downvoted 3 times. Replies from: gwern comment by gwern · 2011-02-06T16:56:04.627Z · LW(p) · GW(p) Karma isn't a license to act like a dick, make bad arguments, be sloppy, or commit sins of laziness. /checks karma; ~3469, good. Which should be obvious, you purblind bescumbered fen-sucked measle. Replies from: timtyler comment by timtyler · 2011-02-06T17:18:02.667Z · LW(p) · GW(p) Right - but the context was "the Karma system and general attitude here makes me dishonest". If you are not short of Karma, sugar-coating for the audience at the expense of the truth seems to be largely unnecessary. Replies from: gwern comment by gwern · 2011-02-06T18:14:15.726Z · LW(p) · GW(p) I looked at the context, but it seemed to me that Xi was just being sloppy. (Of course Landsburg's argument implies rational agents should donate solely to SIAI, if SIAI offers the greatest marginal return. A~>B, A, Q.E.D., B.) If Xi is being sloppy or stupid, then he should pay attention to what his karma is saying. That's what it's for! If you want to burn karma, it ought to be for something difficult that you're very sure about, where the community is wrong and you're right. Replies from: timtyler comment by timtyler · 2011-02-06T19:00:48.502Z · LW(p) · GW(p) Phil's: You shouldn't take it as an axiom that the SIAI is the most-beneficial charity in the world. You imply that anyone who thinks otherwise is irrational. ...was questioning XiXiDu's: If everyone was to take Landsburg's argument seriously, which would imply that all humans were rational, then everyone would solely donate to the SIAI. ...but it isn't clear that the SIAI is the best charity in the world!!! They are in an interesting space - but maybe they are attacking the problem all wrong, lacking in the required skills, occupying the niche of better players - or failing in other ways. XiXiDu justified making this highly-dubious claim by saying he was trying to avoid getting down-voted - and so wrote something which made his post "sound more agreeable". Replies from: FAWS, gwern comment by FAWS · 2011-02-06T20:08:37.827Z · LW(p) · GW(p) SIAI would probably be at least in competition for best charity in the world even if their chance for direct success was zero and their only actual success raising awareness of the problem. I did a wildly guessing back of the envelope type calculation on that a while ago and even with very conservative estimations of the chance of a negative singularity and completely discounting any effect on the far future as well as any possibility of a positive singularity SIAI scored about 1 saved life per$1000.
comment by gwern · 2011-02-06T19:11:16.373Z · LW(p) · GW(p)
Accepting the logical validity of an argument, and flatly denying its soundness, is not an interesting or worthwhile or even good contribution.
Replies from: timtyler
comment by timtyler · 2011-02-06T19:14:34.740Z · LW(p) · GW(p)
What? Where are you suggesting that someone is doing that?
If you are talking about me and your logical argument, that is just not what was being discussed.
The correctness of the axiom concerning charity quality was what was in dispute from the beginning - not any associated logical reasoning.
comment by AstroCJ · 2011-02-05T13:30:24.668Z · LW(p) · GW(p)
Downvoted.
For games where there are multiple agents interacting, the optimal strategy will usually involve some degree of weighted randomness. If there are noncommunicating rational agents A, B, C each with (an unsplittable) $1, and charities 1 and 2 - both of which fulfil a vital function but 1 requires$2 to function and 2 requires \$1 to function, I would expect the agents to donate to 1 with p = 2/3.
A rational agent is aware that other rational agents exist, and will take account of their actions.
comment by nshepperd · 2011-02-05T11:51:55.813Z · LW(p) · GW(p)
The entire resources of the world are somewhat large compared to a single person's donation. I expect the argument wouldn't apply in that situation (but you need TDT-like reasoning to realize that's relevant, or for the donations to be spread in time so each person can condition on what donations all previous people made.)
comment by Jordan · 2011-02-05T07:41:14.946Z · LW(p) · GW(p)
Using linear approximations is obviously a quick hack, and a better analysis could be done if we could make better approximations. That much is clear. You give examples where a better approximation might show that giving to a single charity is less optimal, but it is also possible that a better approximation would enhance the optimality of giving to a single charity. How can we be sure which way a better approximation will take us?
I have no useful information here, so a uniform prior seems reasonable, in which case the analysis from a linear model holds. This seems especially true when the differences in first derivatives between different charities is large, such that a second order correction would have to also be very large in order to sway the analysis.
Replies from: Anatoly_Vorobey
comment by Anatoly_Vorobey · 2011-02-05T14:52:45.049Z · LW(p) · GW(p)
I have no useful information here, so a uniform prior seems reasonable, in which case the analysis from a linear model holds.
I'm not sure what the uniform prior means in this case and how the conclusion follows - can you expand?
But anyway, granting this for the moment, in an actual real-life situation when you contemplate actual charities, you do have all sorts of useful information about them, for example the information that allows you to estimate their effectiveness. This information will probably also throw some light on how the effectiveness changes over time, and so let you determine whether the linear approximation is good.
This seems especially true when the differences in first derivatives between different charities is large, such that a second order correction would have to also be very large in order to sway the analysis.
I agree that when first derivatives are wildly different according to your utility function, it's a no-brainer (barring situations with huge second order effects that'll show up as very weird features of the landscape) to put all your budget into one of them. What I object to is slam-dunk arguing along the lines of "Landsburg has a solid math proof that the rational thing to do is to take first derivatives, compare them, and act on the result. If you don't agree, you're an obscurantist or you fail to grok the math".
Replies from: Jordan, Jordan
comment by Jordan · 2011-02-05T22:55:04.236Z · LW(p) · GW(p)
But anyway, granting this for the moment, in an actual real-life situation when you contemplate actual charities, you do have all sorts of useful information about them, for example the information that allows you to estimate their effectiveness. This information will probably also throw some light on how the effectiveness changes over time, and so let you determine whether the linear approximation is good.
If you have additional information beyond the first derivatives then by all means use it. Use all the information you have. However, in general you need more information to get an equally good approximation to higher order derivatives. Cross terms especially seem like they would be very difficulty to gauge empirically. In light of that I would be very skeptical of high confidence estimates for higher order terms, especially if they conveniently twist the math to allow for a desirable outcome.
comment by Jordan · 2011-02-05T22:45:47.758Z · LW(p) · GW(p)
Consider the simpler case with only two charities and total utility U(X,Y). For simplicity assume the second order derivatives are constant, and that the probability that
$\\frac\{\\partial^2 U\}\{\\partial x^2\} = z\_0,\\frac\{\\partial^2 U\}\{\\partial x\\partial y\} = z\_1,\\frac\{\\partial^2 U\}\{\\partial y^2\} = z\_2$
is given by
$\\phi\(z\_0,z\_1,z\_2\$=\phi(\mathbf{z}).)
Then the second order contribution to
$U\(X\+\\Delta X, Y \+ \\Delta Y\$)
is given by the integral over all possible second derivatives
$\\int\_\{R^3\} \\left\(\\frac\{1\}\{2\}\(\\Delta X\$%5E2z_0%20+%20\Delta%20X\Delta%20Yz_1%20+%20\frac{1}{2}(\Delta%20Y)%5E2z_2\right)\phi(\mathbf{z})%20d\mathbf{z},)
which equals
$\\frac\{1\}\{2\}\(\\Delta X\$%5E2\int_{R%5E3}z_0%20\phi(\mathbf{z})%20d\mathbf{z}%20+%0A\Delta%20X\Delta%20Y%20\int_{R%5E3}z_1%20\phi(\mathbf{z})%20d\mathbf{z}%20+%0A\frac{1}{2}(\Delta%20Y)%5E2\int_{R%5E3}z_2%20\phi(\mathbf{z})%20d\mathbf{z})
where R is some finite interval symmetric about 0. We can actually take R to be the whole real line, but the math becomes hairier. Now, each of these integrals is 0, because the uniform distribution is symmetric about each axis. The symmetry is all that is needed actually, not uniformity, so you could weaken the assumptions.
comment by Peter_de_Blanc · 2011-02-05T02:11:03.866Z · LW(p) · GW(p)
I voted this post down. You claim to have done math, and you tell a narrative of doing math, but for the most part your math is not shown. This makes it difficult for someone to form an opinion of your work without redoing the work from scratch.
[Edit: I was unnecessarily rude here, and I've removed the downvote.]
Replies from: Anatoly_Vorobey
comment by Anatoly_Vorobey · 2011-02-05T14:55:08.965Z · LW(p) · GW(p)
I'm unsure of what more I could have done, to be honest. The math involved is just Taylor's formula, and I pointed at its exact form in Wikipedia. Would it be better if I wrote out the exact result of substituting n=1 into the equation? I figured anyone who knows what a partial derivative is can do that on their own, and I wouldn't be helping much to those who don't know that, so it'd just be a token effort.
Replies from: Peter_de_Blanc
comment by Peter_de_Blanc · 2011-02-06T01:37:11.859Z · LW(p) · GW(p)
OK, I guess my biggest complaint is this:
"If this approximation is close enough to the true value, the rest of the argument goes through: given that the sum Δx+Δy+Δz is fixed, it's best to put everything into the charity with the largest partial derivative at (X,Y,Z)."
What does "close enough" mean? I don't see this established anywhere in your post.
I guess one sufficient condition would be that a single charity has the largest partial derivative everywhere in the space of reachable outcomes.
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2021-08-03 11:55:15
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https://www.gradesaver.com/textbooks/math/precalculus/precalculus-6th-edition-blitzer/chapter-10-section-10-5-the-binomial-theorum-exercise-set-page-1092/24
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Precalculus (6th Edition) Blitzer
The required solution is ${{\left( c+3 \right)}^{5}}={{c}^{5}}+15{{c}^{4}}+90{{c}^{3}}+270{{c}^{2}}+405c+243$
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2020-06-05 16:23:05
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https://zbmath.org/serials/?q=2231-346X
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## Ultra Scientist of Physical Sciences. Section A
### International Journal of Physical Sciences. Inter-Disciplinary Journal
Short Title: Ultra Sci. Phys. Sci., Sect. A Publisher: Dr. A. H. Ansari, Bhopal, Madhya Pradesh, India on behalf of Ansari Education and Research Society Bhopal ISSN: 2231-346X; 2319-8044/e Online: https://ultrascientist.org/previous-issue/ Predecessor: Ultra Scientist of Physical Sciences Comments: No longer indexed; This journal is available open access.
Documents Indexed: 168 Publications (2012–2019)
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### Latest Issues
31, No. 1 (2019) 30, No. 4 (2018) 30, No. 3 (2018) 30, No. 2 (2018) 30, No. 1 (2018) 29, No. 12 (2017) 29, No. 11 (2017) 29, No. 10 (2017) 29, No. 9 (2017) 29, No. 4 (2017) 29, No. 3 (2017) 29, No. 2 (2017) 29, No. 1 (2017) 28, No. 7 (2016) 28, No. 6 (2016) 28, No. 5 (2016) 28, No. 4 (2016) 28, No. 3 (2016) 28, No. 2 (2016) 28, No. 1 (2016) 27, No. 3 (2015) 27, No. 2 (2015) 26, No. 1 (2014) 24, No. 3 (2012) 24, No. 2 (2012) 24, No. 1 (2012)
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### Fields
48 General topology (54-XX) 22 Combinatorics (05-XX) 15 Operations research, mathematical programming (90-XX) 14 Special functions (33-XX) 14 Differential geometry (53-XX) 12 Linear and multilinear algebra; matrix theory (15-XX) 7 Real functions (26-XX) 7 Operator theory (47-XX) 6 Mathematical logic and foundations (03-XX) 5 Associative rings and algebras (16-XX) 5 Group theory and generalizations (20-XX) 4 Order, lattices, ordered algebraic structures (06-XX) 4 Number theory (11-XX) 4 Functions of a complex variable (30-XX) 4 Harmonic analysis on Euclidean spaces (42-XX) 4 Probability theory and stochastic processes (60-XX) 3 Ordinary differential equations (34-XX) 3 Sequences, series, summability (40-XX) 3 Integral equations (45-XX) 3 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 2 Approximations and expansions (41-XX) 2 Functional analysis (46-XX) 2 Statistics (62-XX) 2 Numerical analysis (65-XX) 2 Fluid mechanics (76-XX) 2 Biology and other natural sciences (92-XX) 1 General algebraic systems (08-XX) 1 Nonassociative rings and algebras (17-XX) 1 Category theory; homological algebra (18-XX) 1 Potential theory (31-XX) 1 Partial differential equations (35-XX) 1 Dynamical systems and ergodic theory (37-XX) 1 Integral transforms, operational calculus (44-XX) 1 Global analysis, analysis on manifolds (58-XX) 1 Computer science (68-XX) 1 Classical thermodynamics, heat transfer (80-XX) 1 Quantum theory (81-XX) 1 Statistical mechanics, structure of matter (82-XX) 1 Relativity and gravitational theory (83-XX) 1 Systems theory; control (93-XX) 1 Information and communication theory, circuits (94-XX)
### Citations contained in zbMATH Open
15 Publications have been cited 47 times in 47 Documents Cited by Year
On simultaneous approximation for Baskakov-Durrmeyer-Stancu type operators. Zbl 1339.41021
Mishra, Vishnu Narayan; Khatri, Kejal; Mishra, L. N.
2012
New sort of operators in nano ideal topology. Zbl 1347.54115
Thivagar, M. Lellis; Devi, V. Sutha
2016
Solving multi objective fuzzy fractional programming problem. Zbl 1339.90317
Dash, P. Durga Prasad; Dash, Rajani B.
2012
Dual Pell quaternions. Zbl 1390.11126
Aydin, Fügen Torunbalci; Yüce, Salim
2016
Simple properties of PUL-Stieltjes integral in Banach space. Zbl 1390.26014
Flores, Greig Bates C.; Benitez, Julius V.
2017
A heuristic approach to establish an algebraic structure on multi star granular nano topology. Zbl 1347.54116
Thivagar, M. Lellis; Priyalatha, S. P. R.
2016
$$(2,2)$$-total domination in graphs. Zbl 1347.05145
Kulli, V. R.
2014
Generalized binary closed sets in binary topological spaces. Zbl 1347.54007
Jothi, S. Nithyanantha; Thangavelu, P.
2014
Banach contraction principle on cone hexagonal metric space. Zbl 1347.54076
Garg, Manoj
2014
Prime cordial labeling of generalized prism graph $$Y_{m,n}$$. Zbl 1347.05214
Prajapati, U. M.; Gajjar, S. J.
2015
On weakly $$\varphi$$-Ricci symmetric Lorentzian Para (LP) $$\alpha$$-Sasakian manifolds. Zbl 1333.53059
Pujar, S. S.; Khairnar, V. J.
2012
Some special seminear-ring structures. II. Zbl 1333.16042
Perumal, R.; Balakrishnan, R.; Uma, S.
2012
On weak concircular symmetries of $$\in$$-Trans Sasakian manifolds. Zbl 1333.53060
Pujar, S. S.; Khairnar, V. J.
2012
Ricci solitons on three dimensional $$\beta$$-Kenmotsu manifolds with respect to Shouten-van Kampen connection. Zbl 1416.53026
Chakraborty, Debabrata; Mishra, Vishnu Narayan; Hui, Shyamal Kumar
2018
On pairwise minimal continuous maps in bitopological spaces. Zbl 1416.54014
Ittanagi, Basavaraj M.
2018
Ricci solitons on three dimensional $$\beta$$-Kenmotsu manifolds with respect to Shouten-van Kampen connection. Zbl 1416.53026
Chakraborty, Debabrata; Mishra, Vishnu Narayan; Hui, Shyamal Kumar
2018
On pairwise minimal continuous maps in bitopological spaces. Zbl 1416.54014
Ittanagi, Basavaraj M.
2018
Simple properties of PUL-Stieltjes integral in Banach space. Zbl 1390.26014
Flores, Greig Bates C.; Benitez, Julius V.
2017
New sort of operators in nano ideal topology. Zbl 1347.54115
Thivagar, M. Lellis; Devi, V. Sutha
2016
Dual Pell quaternions. Zbl 1390.11126
Aydin, Fügen Torunbalci; Yüce, Salim
2016
A heuristic approach to establish an algebraic structure on multi star granular nano topology. Zbl 1347.54116
Thivagar, M. Lellis; Priyalatha, S. P. R.
2016
Prime cordial labeling of generalized prism graph $$Y_{m,n}$$. Zbl 1347.05214
Prajapati, U. M.; Gajjar, S. J.
2015
$$(2,2)$$-total domination in graphs. Zbl 1347.05145
Kulli, V. R.
2014
Generalized binary closed sets in binary topological spaces. Zbl 1347.54007
Jothi, S. Nithyanantha; Thangavelu, P.
2014
Banach contraction principle on cone hexagonal metric space. Zbl 1347.54076
Garg, Manoj
2014
On simultaneous approximation for Baskakov-Durrmeyer-Stancu type operators. Zbl 1339.41021
Mishra, Vishnu Narayan; Khatri, Kejal; Mishra, L. N.
2012
Solving multi objective fuzzy fractional programming problem. Zbl 1339.90317
Dash, P. Durga Prasad; Dash, Rajani B.
2012
On weakly $$\varphi$$-Ricci symmetric Lorentzian Para (LP) $$\alpha$$-Sasakian manifolds. Zbl 1333.53059
Pujar, S. S.; Khairnar, V. J.
2012
Some special seminear-ring structures. II. Zbl 1333.16042
Perumal, R.; Balakrishnan, R.; Uma, S.
2012
On weak concircular symmetries of $$\in$$-Trans Sasakian manifolds. Zbl 1333.53060
Pujar, S. S.; Khairnar, V. J.
2012
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### Cited by 64 Authors
16 Mishra, Vishnu Narayan 5 Kumar, Alok 5 Mishra, Lakshmi Narayan 5 Patel, Prashantkumar Gordhanbhai 5 Sharma, Preeti 3 Rao, Nadeem 3 Wafi, Abdul 2 Deepmala, Vandana 2 Devdhara, Ankita R. 2 Gairola, Asha Ram 2 Priyalatha, S. P. R. 1 Acar, Tuncer 1 Acu, Ana Maria 1 Agrawal, Purshottam Narain 1 Ahangar, Hossein Abdollahzadeh 1 Amjadi, Jafar 1 Aydın, Fügen Torunbalcı 1 Benitez, Julius V. 1 Chacko, Tresa Mary 1 Chen, Xiaoyan 1 Chinnaraj, P. 1 El-Sheikh, Sobhy Ahmed 1 Flores, Greig Bates C. 1 Gandhi, Rajiv B. 1 Garg, Tarul 1 Han, Lingxiong 1 Hosny, Mona 1 Hui, Shyamal Kumar 1 Kalpana, G. 1 Kanaujia, Prerna 1 Kandil, Ali 1 Khurana, Dakshita 1 Kishor, Shyam 1 Kuziak, Dorota 1 Li, Wenhui 1 Liu, Zhi 1 Mandal, Yadab Chandra 1 Mishra, Umakanta 1 Mohapatra, Ram Narayan 1 Noor, Khalida Inayat 1 Pandey, Shikha 1 Perumal, R. 1 Qi, Feng 1 Raafat, Mahmoud 1 Schroeder, Justin Z. 1 Şenel, Güzide 1 Shahid, Humayoun 1 Sheikholeslami, Seyed Mahmoud 1 Siddiqi, Mohd Danish 1 Singh, Satya Prakash 1 Singh, Shobh Nath 1 Singh, Sujeet Kumar 1 Soylemez, Dilek 1 Susha, D. 1 Tan, Jieqing 1 Tapiawala, Dipti 1 Tasneem, Z. Sumaiya 1 Thivagar, Mariam Lellis 1 Umap, Hemant Pandurang 1 Ünver, Mehmet 1 Vijay, Yadav 1 Waliv, Rahul Hanmant 1 Xie, Jin 1 Yadav, Shiv Prasad
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### Cited in 36 Journals
3 Applied Mathematics and Computation 3 Cogent Mathematics 2 Iranian Journal of Science and Technology. Transaction A: Science 2 Thai Journal of Mathematics 2 Boletim da Sociedade Paranaense de Matemática. Terceira Série 2 European Journal of Pure and Applied Mathematics 2 Tbilisi Mathematical Journal 2 Journal of Applied Mathematics & Informatics 1 Journal of Mathematical Analysis and Applications 1 Periodica Mathematica Hungarica 1 Graphs and Combinatorics 1 Numerical Algorithms 1 The Journal of Analysis 1 Computational and Applied Mathematics 1 Advances in Applied Clifford Algebras 1 Journal of Discrete Mathematical Sciences & Cryptography 1 Communications of the Korean Mathematical Society 1 Proceedings of the National Academy of Sciences, India. Section A. Physical Sciences 1 South East Asian Journal of Mathematics and Mathematical Sciences 1 AKCE International Journal of Graphs and Combinatorics 1 Complex Analysis and Operator Theory 1 Bulletin of Mathematical Analysis and Applications 1 International Journal of Nonlinear Analysis and Applications 1 Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM 1 Journal of Advanced Studies in Topology 1 Afrika Matematika 1 Journal of Mathematical Extension 1 Iranian Journal of Mathematical Sciences and Informatics 1 Palestine Journal of Mathematics 1 Mathematical Sciences 1 International Journal of Analysis 1 Journal of Function Spaces 1 Journal of Calculus of Variations 1 International Journal of Applied and Computational Mathematics 1 Korean Journal of Mathematics 1 Electronic Research Archive
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### Cited in 16 Fields
28 Approximations and expansions (41-XX) 7 Real functions (26-XX) 6 General topology (54-XX) 4 Sequences, series, summability (40-XX) 3 Combinatorics (05-XX) 3 Special functions (33-XX) 3 Operator theory (47-XX) 3 Differential geometry (53-XX) 2 Computer science (68-XX) 2 Operations research, mathematical programming (90-XX) 1 Mathematical logic and foundations (03-XX) 1 Number theory (11-XX) 1 Associative rings and algebras (16-XX) 1 Measure and integration (28-XX) 1 Functions of a complex variable (30-XX) 1 Statistical mechanics, structure of matter (82-XX)
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2022-11-29 10:10:36
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https://codeforces.mn/suggestion/5695192468291584
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#### Жишээ тэстүүд
##### Оролт
3 1
2 1 4
11 3 16
##### Гаралт
4
##### Оролт
4 3
4 3 5 6
11 12 14 20
##### Гаралт
3
## Тэмдэглэл
In the first sample it is profitably for Apollinaria to make the existing 1 gram of her magic powder to ingredient with the index 2, then Apollinaria will be able to bake 4 cookies.
In the second sample Apollinaria should turn 1 gram of magic powder to ingredient with the index 1 and 1 gram of magic powder to ingredient with the index 3. Then Apollinaria will be able to bake 3 cookies. The remaining 1 gram of the magic powder can be left, because it can't be used to increase the answer.
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2019-09-17 19:45:53
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http://rybu.org/aggregator?page=15
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# Recent MathOverflow Questions
### Is there an easy way to prove the function is non-negative in a compact domain?
Math Overflow Recent Questions - Sat, 02/10/2018 - 18:48
I encounter a function $$f(x,y)=-4xy \pi( cos(y) sin^2(x\pi))+sin(y)(-2y^2 x \pi +2x^3 \pi^3+(y^2+x^2 \pi^2)sin(2x\pi))$$ defined for $x\times y\in[0,2]\times[0,\pi]$. This function seems to be nonnegative in this compact domain. However, I just can prove this result after tedious analysis on the local property of $f$ around the zero points. Since this function seems to be simple, I wonder whether there is a straightforward technique to do so .Any hints or references in this respect would be greatly appreciated!
### Is the Giry Monad also a Comonad and if not, is there a probability measures (Co)monad?
Math Overflow Recent Questions - Sat, 02/10/2018 - 18:41
The Giry monad captures probability measures. Is it also a comonad? If so, what are the natural transformations?
If not, is there a monad that is also a comonad that captures probability measures?
### Cutting Lemma in Discrete Geometry
Math Overflow Recent Questions - Sat, 02/10/2018 - 18:26
I'm looking for a survey or a source for Cutting Lemma. I looked at Matusek's Discrete Geometry textbook, but it only proved Cutting Lemma for lines in $\mathbb{R}^2.$ I need to know the proof in $\mathbb{R}^d$ and other variations of Cutting Lemma.
Thanks in advance for any help!
### Descent of isomorphisms between irreducible closed subschemes
Math Overflow Recent Questions - Sat, 02/10/2018 - 17:27
Let $S$ be an affine scheme, $X$ be a projective $S$-scheme, $W,Z\to X$ two reduced, irreducible closed $S$-subschemes, flat over $S$. Let $S'\to S$ be a faithfully flat map, with $S'$ affine.
Assume there exists an $S'$-isomorphism $g: W_{S'}\xrightarrow{\simeq} Z_{S'}$, commuting with the closed immersions into $X_{S'}$.
When does there exist an $S$-morphism $f : W\to Z$ such that $f_{S'} = g$, and (hence) $f$ is an $S$-isomorphism?
### A first order ODE problem
Math Overflow Recent Questions - Sat, 02/10/2018 - 17:05
Fix $a>0$ and $b>0$. Does the following ODE $$G(x)^2+2axG(x)G'(x)+2aG'(x)(x-b)=0 \tag{*}$$ have a solution, say, $F(x)$, that satisfies $F(x)>0$ and $F'(x)<0$ on $(b,\infty)$?
I tried to solve it by Mathematica, and it gives $G$ as solution to $$x=e^{-2a\left(\text{log}\,G(x)-\frac{1}{G(x)}\right)}\left(C_0+b(2a)^{2a}\text{Gamma}\left(1-2a,\frac{2a}{G(x)}\right)\right) \tag{**}$$
for some free parameter $C_0$, where Gamma$(\cdot,\cdot)$ is the upper incomplete gamma function. However I'm not sure if this is a proper inverse function or if $G$ given by this equation has domain $(b,\infty)$.
Thanks!!
### Region of attraction of simple ODE with perturbation
Math Overflow Recent Questions - Sat, 02/10/2018 - 16:18
Consider the following simplest example:
$$\dot{x} = x(x-1)(x+1)$$ $[-1,1]$ is the ROA.
Now consider the two dimensional case:
\begin{aligned} &\dot{x} = x(x-1)(x+1)\\ &\dot{y} = y(y-1)(y+1) \end{aligned}Obviously, ROA is a square. However, if I consider the following coupled ODE:
\begin{aligned} &\dot{x} = x(x-1)(x+1) + \epsilon (y-x)\\ &\dot{y} = y(y-1)(y+1) + \epsilon (x-y) \end{aligned} where $\epsilon$ is a very small number. Or
\begin{aligned} &\dot{x} = x(x-1)(x+1) + \epsilon (-y+x)\\ &\dot{y} = y(y-1)(y+1) + \epsilon (-x+y) \end{aligned} Then I have the following ROAs: (blue line-case three, black line-case two, red line-case one)
My questions are:
1. There are two different tilt directions for case two and three. I know this is because of the slope of the coupling term (for case two, the slope of $x$ and $y$ in the coupling terms are $-1$). But how could I analyze this formally?
2. Is it a good way to analyze 1. by perturbation method? (observe the sign of the leading order term of the solution obtained from perturbation method?) and how could I proceed it for the coupling term?
3. Are there any reference about my questions?
Note: It is simple to check that if you just use the linearization method to find the Jacobian matrix (w.r.t the point $(0,0)$), the ROA will be the whole $\mathbb{R}^2$ , which is not correct.
### Stratification of space of labelled circles in the plane
Math Overflow Recent Questions - Sat, 02/10/2018 - 15:50
Consider the space of $n$ round circles in the plane to be the open subset of $\mathbb R^{3n}$:
$$C_n = \{ (v_1, v_2, \cdots, v_n, r_1, r_2, \cdots, r_n ) : v_i \in \mathbb R^2, r_i \in (0, \infty) \ \ \forall i \}$$
The $i$-th circle corresponding to a point in the above space would be the solutions to the equation $|x-v_i| = r_i$. With this convention, you can think of $C_n$ as the space of $n$ labelled circles in the plane, where the circles are allowed to intersect or even coincide.
There is a natural stratification of this space induced by the co-dimension one subvarieties consisting of subspaces where two circles are tangent, or where three (or more) circles intersect in a non-empty set. This is a fairly natural stratification since the complement of this stratification in $C_n$ is what you might call the "isotopy classes" of "regular" circles, as the combinatorics of their intersections do not change in the path-components and it is a dense open subspace of $C_n$.
Is there much known about the basic combinatorics of this stratification? For example, labelling the path-connected co-dimension $k$ strata for $k=0,1,2, \cdots$ and enumerating them? It looks like a souped-up version of the partition problem from a fairly naive perspective.
It seems like a natural problem and there is much closely related to it (spaces of arrangements in homotopy theory). But I've never come across quite this problem before.
Any insights would be appreciated -- especially if there's papers out there on this topic that I'm unaware of.
### Shift of zeta functions
Math Overflow Recent Questions - Sat, 02/10/2018 - 14:46
Let $k$ be a global field, and $X$ a smooth projective variety over $K$.
Let $Z(X,s)$ be the completed Zeta function of $X$, defined via geometric $\ell$-adic cohomology if the characteristic of $K$ is positive, and by product of local factors, including Serre's Archimedean factors, if the characteristic of $K$ is zero.
Suppose we want to relate, for an integer $n$, the shifted Zeta function $Z(X,s-n)$ to $Z(X,s)$ this way: there is a smooth projective variety $P_n$ over $K$, such that
$$Z(X,s-n) = Z(X\times_KP_n, s)$$ for every $n$.
Does $P_n$ exist and what is it? More generally, how to "geometrically" modify $X$ to get $Z(X,s-n)$ from $Z(X,s)$?
### On minimality of semitopological and quasitopological groups
Math Overflow Recent Questions - Sat, 02/10/2018 - 10:47
The phenomemnon of minimality is well-studied in the realm of topological groups.
Let us recall that a topological group $X$ is minimal if each bijecive continuous homomorphism $h:X\to Y$ to a topological group $Y$ is a topological isomorphism.
I am interested if the notion of minimality was studied in the realm of semi-topological or quasi-topological groups.
A semi-topological group is a group $G$ endowed with a Hausdorff topology making the group operation $G\times G\to G$, $(x,y)\mapsto xy$, separately continuous.
A semitopological group is called a quasi-topological group if the operation of inversion $G\to G$, $x\mapsto x^{-1}$, is continuous.
Let us define a semi-topological group $X$ to be semi-minimal if each continuous bijective homomorphism $h:X\to Y$ to a semitopological group $Y$ is a topological isomorphism.
By analogy we can define a quasi-topological group $X$ to be quasi-minimal if each continuous bijective homomorphism $h:X\to Y$ to a quasi-topological group $Y$ is a topological isomorphism.
It is clear that each semi-minimal quasi-topological group is quasi-minimal and each quasi-minimal topological group is minimal.
It can be shown that each compact Hausdorff semitopological group is semi-minimal (topological group).
Problem 1. Is each semi-minimal semi-topological group compact?
Problem 2. Is each quasi-minimal quasi-topological group compact?
I even cannot prove or disprove the following
Conjecture. No countable Boolean semi-topological group is semi-minimal.
Remark. By the answer to this MO problem, for some submonoid $M$ of the monoid $\omega^\omega$ of self-maps of a countable set $X$ there is no minimal Hausdorff topology on $\omega$ in which all self-maps $f\in M$ are continuous. At the moment this is the unique (known to me) example of an algebraic systems with unary operations, admitting no minimal Hausdorff topology.
### Continuity of a real function defined on an orbit space
Math Overflow Recent Questions - Sat, 02/10/2018 - 10:12
I am reading some paper, where the authors claim the following:
Let $G$ be a compact (Hausdorff) group and let $X$ be a locally, compact, Hausdorff space. Assume that $G$ acts on $X$ continuously. Denote by $C_0(X)$ the continuous functions on $X$ with complex values. Denote by $X/G$ the orbit space (it is Hausdorff as well), and let $\pi: X\to X/G$ be the canonical quotient map.
claim: Let $f\in C_0(X)$. The map $\pi(x)\mapsto sup_{g\in G}|f(g\cdot x)|$ is a continuous map $X/G\to \mathbb{R}$.
The proof starts as follows: Let $f\in C_0(X)$, let $\epsilon>0$ and let $x\in X$. Use continuity of $f$ to find, for every $g\in G$, an open neighborhood $W_g$ of $g\cdot x$ such that $|f(g\cdot x)-f(y)|<\epsilon$ for all $y\in W_g$. By compactness of $G$, there exists an open neighborhood $W$ of $x$ such that $g\cdot W\subseteq W_g$ for all $g\in G$.
I can not see why the last claim is true or why it follows from compactness... it seems like they actually claim that $\bigcap\limits_{g\in G}g^{-1}\cdot W_g$ contains an open neighborhood of $x$ (of course, it contains $x$, but I don't see why more than that).
An alternative approach to the proof or counter example for the claim would be appreciated as well.
Thank for any help!
### Smooth functions which come from densely defined holomorphic functions
Math Overflow Recent Questions - Sat, 02/10/2018 - 08:23
The actual question might seem convoluted without context so let me first give a motivating example.
Consider the holomorphic function $f:\mathbb{C} \setminus \{0\} \to \mathbb{C}$ given by $f(z)=e^{-z^{-2}}$. Consider now $f$ restricted to the line $\mathbb{R} \setminus \{0\}$. It has a (unique) extension to a continuous function on $\mathbb{R}$ which is moreover smooth.
The aim of the question is to understand which functions arise in the above manner.
I'll try to give two versions of the question, one general and one specific.
Specific version:
Let $f: X \to \mathbb{C}$ where $X$ is either the punctured plane or the punctured disk. Let $\gamma:\mathbb{R} \setminus \{0\} \to X$ be a line s.t. $f \circ \gamma : \mathbb{R} \setminus \{0\} \to \mathbb{C}$ has a continuous (smooth) extension to $\mathbb{R}$.
Question: What kind of smooth functions $g: \mathbb{R} \to \mathbb{C}$ can be written as $g= f \circ \gamma$ for some $f$ and $\gamma$ as above? Is there some direct description of this class?
General version:
Let $M$ be an real analytic (resp. algebraic) manifold. Let $N$ be a complex analytic (resp. algebraic) manifold and $F:N \dashrightarrow \mathbb{C}$ be a holomorphic function on $N$ with singularities (meaning it is defined and holomorphic on some dense open subset of $N$). Let $j: M \to N$ be an analytic (resp. algebraic) embedding (considering $N$ as a real manifold) satisfying that $f: F \circ j: M \dashrightarrow \mathbb{C}$ is defined on a dense open subset of $M$ and has a continuous extension to all of $M$ (Must $f$ be smooth? if not we will assume from here on that it is).
Definition: For a given $M$ as above the functions which are obtained from the above procedure (lets stick with the analytic version of the above) for some choices of $(N,F,j)$ will be called "Really Smooth" functions.
Questions:
1. Is there a different (more direct perhaps) description of this class of "Really Smooth" functions? (maybe via hyperfunctions?)
2. Are "Really Smooth" functions closed under differentiation?
3. Are "Really Smooth" functions closed under pre-composition? Meaning: Let $\pi: M \to N$ be an analytic map between real analytic manifolds and let $f$ be a "Really Smooth" function on $N$, is $f \circ \pi$ "Really Smooth"?
4. Are "Really Smooth" real-valued functions on $\mathbb{R}$ closed under composition?
Aside: I have no idea how to tag this question so any help with this will be welcome, thanks.
### Computational Geometric Aspects of Greedy Tour Expansion
Math Overflow Recent Questions - Sat, 02/10/2018 - 05:38
Has the following problem already been investigated from the Computational Geometry point of view and what are the results regarding worst case complexity?
Given
• a finite set $\mathcal{P}$ of $n$ distinct points in the Euclidean plane,
• its convex hull $\mathcal{T_{\text{card(CH(}\mathcal{P}\text{))}}}\ :=\ \text{CH(}\mathcal{P}\text{)}$ as the initial tour.
while $m<n$
• chose $\ \left( p\in\mathcal{P}\setminus\mathcal{T_m},\ t_i\in\mathcal{T_m}\right):$
$\quad\quad\quad\|p-t_i\|+\|t_{\text{succ(}i\text{)}}-p\|-\|t_{\text{succ(}i\text{)}}-t_i\|$ $\quad\le\quad\|q-t_k\|+\|t_{\text{succ(}k\text{)}}-q\|-\|t_{\text{succ(}k\text{)}}-t_k\|$
$\forall q\in\mathcal{P}\setminus\mathcal{T_m},\quad\left(t_k,t_{\text{succ(}k\text{)}}\right)\in\mathcal{T_m}$
• $\mathcal{T_{m+1}} := \lbrace\mathcal{T_m}\setminus\left(t_{\text{i}},t_{\text{succ(}i\text{)}}\right)\rbrace\cup\lbrace\left(t_i,p\right),\left(p,t_{\text{succ(}i\text{)}}\right)\rbrace$
in this context, $t_{\text{succ(}i\text{)}}$ shall denote the tour-vertex that is encountered immediately after vertex $t_\text{i}$ when traversing the tour w.l.o.g in counter clockwise order.
Please note, that the objective of the task is not to create a simple polygon through all points or even an optimal tour; it is rather to determine the point, whose integration into the tour incurs the least length-increase, as fast as possible.
I am looking for techniques from Computational Geometry, i.e. algorithms and datastructures, that bring about provable improvements beyond updating least detour information w.r.t. the newly generated edges after the insertion of a further point.
Here is an example of a selfintersecting tour with 50 points, that was generated with the greedy insertion algorithm, answering a question of Joseph O'Rourke:
to indicate, that geometric concepts may indeed be promising, consider the problem of deciding, whether a point $p\notin \mathcal{T_i}$ is closer (in the sense of incurred elongation) to tour edge $(t_i,t_j)$ or to $(t_j,t_k)$.
in that special case, where the tour edges are adjacent, the separator equals the radical axis of the circles with center and radius $(t_i,\|t_j-t_i\|)$ and $(t_k,\|t_k-t_j\|)$; whether that is an actual improvement, depends of course on the computational model and the limitations on available resources - for very large instances it may not be viable to store all distances between pairs of points, whence deciding on which side of the radical line a point lies, is cheaper than calculating and comparing two detours.
for non-adjacent tour edges the situation is not so simple and yields non-linear algebraic curves as separators in general.
### Can the "Bisector" be represented by a holomorphic function?
Math Overflow Recent Questions - Sat, 02/10/2018 - 01:54
Note:
In this question, a complex number is counted as a vector initiated from the origin.
______________________________________________________________-
Is there a holomorphic function $B:\mathbb{C}^2 \to \mathbb{C}$ such that for every two non zero complex numbers $z,w$ with $z/w \notin \mathbb{R},$ the vector $B(z,w)$ is a non zero vector indicating to the direction of the bisector of the angle $\angle (z,w)$?
Motivation:
The initial formula for the "Bisector" of $\angle (z,w)$ is $B'(z,w)=|z|w+|w|z$. But it is not a holomorphic function.(It is not even smooth at $z=0$ or $w=0$). So we search for a holomorphic remedy, a holomorphic function $B$ defined on whole $\mathbb{C}^2$ such that $B(z,w)$ is real proportional to $(|z|w+|w|z)$ via a non constant real function $\lambda$.
What about if we require that such $\lambda$ be positive(Non negative)?
### Schur's Lemma for Quantized Universal Enveloping Algebra
Math Overflow Recent Questions - Fri, 02/09/2018 - 20:01
Let $U_q(\mathfrak{g})$ (defined over $\mathbb{C}(q)$) be the quantized universal enveloping algebra of a simple Lie algebra $\mathfrak{g}$. Let $M$ a finite-dimensional simple left $U_q(\mathfrak{g})$-module. Is it true that $\dim_{\mathbb{C}(q)}\mathrm{End}_{U_q(\mathfrak{g})}(M)=1$? How to sketch a proof? The problem is that the field $\mathbb{C}(q)$ is not algebraically closed.
Math Overflow Recent Questions - Fri, 02/09/2018 - 12:27
Let $E$ be an infinite-dimensional complex Hilbert space.
For $A = (A_1,\cdots,A_d)\in\mathcal{L}(E)^d$, the algebraic spectral radius of $A$ was given by $$r_a(A)=\lim_{n\to+\infty}\left\|\sum_{f\in F(n,d)} A_f^* A_f\right\|^{\frac{1}{2n}} ,$$ where $F(n,d):=\{f:\,\{1,\cdots,n\}\longrightarrow \{1,\cdots,d\}\}$ and $A_f:=A_{f(1)}\cdots A_{f(n)}$, for $f\in F(n,d)$.
Further the geometric spectral radius of $A$ was given by $$r_g(A)=\max\{\|\lambda\|_2,\;\lambda=(\lambda_1,\cdots,\lambda_d) \in \sigma(A)\},$$ with $\sigma(A)$ denotes the Taylor spectrum of $A$. Note that it was shown that $r_g(A)$ doesn't depend of the choice of the spectra.
Moreover, it was shown in (1) that if $A_iA_j=A_jA_i$ for all $i,j$, we have
$$r_g(A)=r_a(A)=\displaystyle\lim_{n\to \infty}\left\|\displaystyle\sum_{|\alpha|=n}\frac{n!}{\alpha!}{A^*}^{\alpha}A^{\alpha}\right\|^{\frac{1}{2n}}.$$
If $A_iA_j=A_jA_i$ for all $i,j$, it is true that $$r_a(A)\leq w_e(A):=\sup\left\{\bigg(\displaystyle\sum_{i=1}^d|\langle A_ix,x\rangle|^2\bigg)^{\frac{1}{2}},\;x\in E,\;\|x\|=1\;\right\}\;?$$ If $d=1$, the claim is true. For its proof one can see Theorem 1.1 in Moshe Goldberg, Eitan Tadmor: On the numerical radius and its applications, doi: 10.1016/0024-3795(82)90155-0.
Notice that Gelu Popescu has a big paper about multivariable operators (Memoirs of the AMS, arXiv) and he proves many things about $w_e(A)$ in section 2 of his paper.
My attempt: We can see that $$w_e(A):= \displaystyle\sup_{(\lambda_1,\cdots,\lambda_d)\in B_d}w(\lambda_1A_1+\cdots+\lambda_dA_d),$$ with $B_d$ is the open unit ball of $\mathbb{C}^d$. I try to show that, if $A_iA_j=A_jA_i$ for all $i,j$, we have $r_a(A)=r_e(A)$ where $$r_e(A):= \displaystyle\sup_{(\lambda_1,\cdots,\lambda_d)\in B_d}r(\lambda_1A_1+\cdots+\lambda_dA_d).$$ Finally we apply the well known result: for $T\in\mathcal{L}(E)$ we have $$r(T)\le w(T).$$
### What is the probability that a plane curve crosses a segment?
Math Overflow Recent Questions - Fri, 02/09/2018 - 12:01
Given
• a straight segment of length $l$ in the plane, and
• a random curve that starts at one end of this segment, that is smooth/differentiable everywhere, that has everywhere a minimum curvature radius $r<l$, and that reaches spatial infinity,
what is the probability $p$ that the random curve crosses the segment? Given that the random curves all start at one end of the segment, some curves will cross the segment later on, whereas others will not. The crossing probability $p$ will depend on the segment length $l$ and on the radius $r$. What is the dependence?
Possibly, the problem is not specific enough. What would be necessary to make it well defined?
Problem 2: What would the probability be if the segment is not a segment, but a full straight line across the plane (say the x-axis), and if all the curves start at the origin (0,0)? Is this problem well-defined?
Problem 3: What would the probability be if the "segment" is taken to be the positive x-axis, and if again, all curves start at the origin?
### Exercise 1.1.(c) in Hartshorne's Deformation Theory
Math Overflow Recent Questions - Fri, 02/09/2018 - 00:36
Exercise 1.1.(c) in Hartshorne's Deformation Theory:
Over an algebraically closed field $k$, we define a curve in $\mathbb P^2_k$ to be the closed subscheme, defined by a homogeneous polynomial $f(x,y,z)$ of degree $d$ in the coordinate ring $S=k[x,y,z]$.
(c) For any finitely generated $k$-algebra $A$, we define a family of curves of degree $d$ in $\mathbb P^2$ over $A$ to be a closed subscheme $X\subseteq\mathbb P^2_A$, flat over $A$, whose fibers above closed points of $\mathrm{Spec}\,A$ are curves in $\mathbb P^2$. Show that the ideal $I_X\subseteq A[x,y,z]$ is generated by a single homogeneous polynomial $f$ of degree $d$ in $A[x,y,z]$.
My attempts: the condition on fibers above closed points is equivalent to the condition that for any $\mathfrak m\in\mathrm{Specm}\,A$ the ideal $(I_X,\mathfrak m)/\mathfrak m$ is principal. Equivalently, for any $\mathfrak m$ there is $f_{\mathfrak m}\in I_X$ such that $I_X\subset (f_{\mathfrak m},\mathfrak m)$. Also, it is clearly sufficient to prove that $I_X$ contains a homogeneous element of degree $d$.
### What are the transitive extensions of finite representations of cyclic groups?
Math Overflow Recent Questions - Thu, 02/08/2018 - 20:41
This question is a generalisation of this one. Let $H$ be a finite, transitive permutation group of degree $n$. If the point stabiliser subgroup $H_n$ of degree $n-1$ is some faithful permutation representation of a cyclic group $C_k$ (with $k$ not necessarily equal to $n-1$), what can $H_n$ and $H$ be? For example, must $k$ divide $n-1$?
### Sheaf-theoretic Grothendieck groups
Math Overflow Recent Questions - Thu, 02/08/2018 - 19:55
Let $S$ be a scheme, $M\to S$ a commutative monoid object in algebraic $S$-spaces, ie. an algebraic $S$-space such that, functorially on $S$-schemes $T$, $M(T)$ is a commutative monoid with neutral element.
Each commutative monoid $M(T)$ has its own group completion $M(T)^{\rm gp}$, given by the classical Grothendieck group construction.
Formation of $M(T)^{\rm gp}$ is functorial in $T$, and we may define the fppf sheaf $M^{\rm gp}$ to be the fppf sheafification of $T\mapsto M(T)^{\rm gp}$.
Is $M^{\rm gp}$ an algebraic $S$-space, hence a commutative group object in algebraic $S$-spaces? Is it at least when $M$ is cancellative?
-----------------------------------------------------
Example
Suppose $M$ is, in addition, cancellative (ie. $M(T)$ is a cancellative commutative monoid with zero for all $S$-schemes $T$).
In this case, $M^{\rm gp}$ is the fppf quotient sheaf of $M\times_SM\times_SM\times_SM$ by the following equivalence relation:
$$R := (M\times_SM\times_SM\times_SM)\times_{\mu, M\times_SM,\Delta_{M/S}}M$$
with:
• $\mu : M\times_SM\times_SM\times_SM\to M\times_SM$ the map defined functorially on $T$-sections by sending $(a_T, b_T, c_T, d_T)$ to $(a_T+d_T, b_T+c_T)$.
• $\Delta_{M/S} : M\to M\times_SM$ is the diagonal.
• $s,t : R\to M\times_SM$ are defined to be the pullback along $\mu$ of $\text{pr}_1\circ\Delta_{M/S}$ and $\text{pr}_2\circ\Delta_{M/S}$.
On $T$-sections, $R(T)$ is the equivalence relation that identifies the pairs $((a,b),(c,d))$ with $((a,d),(b,c))$ treating $(a,b)$ and $(c,d)$ as $"a-b"$ and $"c-d"$, and ensuring that $"a-b = c-d"$ if and only if $"a+d = b+c"$. This way, the monoid operation $+$ on $M(T)\times M(T)$ descends to a group operation, where the inverse of $(a,0)$ is $(0,a)$.
$R$ is not an étale equivalence relation, unless $M$ is étale over $S$, as the fiber of $M\times_SM\to M^{\rm gp}$ over $0$ is the diagonal copy of $M$ in $M\times_SM$.
This is always going to be the case, so either the question has negative answer, or one should come up with a better presentation for $M^{\rm gp}$, or verify Artin's axioms.
The case of interest is when $M\to S$ is locally of finite presentation and separated, $S$ is affine.
### Weak-convergence of probability measures implies the convergence of the measure of a continuity set
Math Overflow Recent Questions - Thu, 02/08/2018 - 00:54
Let $\Omega$ be a Polish space and $\mathcal{B}(\Omega)$ be its Borel $\sigma$-algebra. Let $\{\mu_n\}$ be a sequence of probability measures on $\mathcal{B}(\Omega)$ such that $\mu_n$ weak-converges to $\mu$. Then in general, $\mu_n(E) \nrightarrow \mu(E)$ for $E \in \mathcal{B}(\Omega)$.
But if for some $F \in \mathcal{B}(\Omega)$ we have $\partial F = F$ and $\mu_n(F) = 0$ for all $n$, can we say that $\mu_n(F) \rightarrow \mu(F)$? If not, are there any conditions on $\Omega$ that would make this true?
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2018-03-23 03:26:45
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https://testbook.com/question-answer/in-each-of-the-following-question-below-are-given--5a633317478bfe37103fd1b5
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In each of the following question below are given some statements followed by some conclusions. Taking the given statements to be true even if they seem to be at variance from commonly known facts, read all the conclusions and then decide which of the given conclusion logically follows the given statements.Statements: I. Some bricks are stonesII. All cakes are bricksConclusions: I. Some stones are bricksII. Some stones are cakesIII. All bricks are stones
1. Only conclusion I follows.
2. Only conclusion I and II follow.
3. All conclusions follow.
4. No conclusion follows.
Option 1 : Only conclusion I follows.
Free
Cell
339239
10 Questions 10 Marks 7 Mins
Detailed Solution
Shortcut Trick
• If there are 2 conclusions for the same set of objects where one starts with some and the other starts with all, then only one of them will be true.
• However, one must remember that both can be false at a time.
Hint
• Using the trick, we can eliminate option 3 directly.
The possible Venn diagram is:
Explanation:
I. Some stones are bricks → It is a definite case, hence true.
II. Some stones are cakes → It is not a definite case, hence false.
III. All bricks are stones → It is not a definite case, hence false.
Thus, only conclusion (I) follows.
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2021-11-28 12:36:14
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http://www.iam.fmph.uniba.sk/amuc/_vol-71/_no_2/_mollin/mollin.html
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ACTA MATHEMATICA UNIVERSITATIS COMENIANAE
Vol. LXXI, 2(2002)
p. 121
The Diophantine Equation AX2-BY2=C Solved Via Continued Fractions
R. A. Mollin, K. Cheng and B. Goddard
Abstract. The purpose of this article is to provide criteria for the solvability of the Diophantine equation $a^2X^2-bY^2=c$ in terms of the simple continued fraction expansion of $\sqrt{a^2b}$, and to explore criteria for the solvability of $AX^2-BY^2=C$ for given $A,B,C\in\N$ in the general case. This continues work in [9]–[11].
AMS subject classification: 11A55 11R11 11D09
Keywords: Continued fractions, Diophantine equations, fundamental units
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2017-10-20 18:04:19
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https://www.biostars.org/p/9552968/#9553066
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2
0
Entering edit mode
6 weeks ago
I’m simulating shotgun reads for RNA data (e.g. I have a bunch of bacterial genomes and I want to generate simulated reads). Should I use as my input only the “coding sequences” of these reference genomes (from NCBI Nucelotide DB)? Or should I use the complete genomes? If I were simulating DNA reads, I’d use the complete fasta; for RNA I think I should use only the coding sequences. If I should use only coding sequences, should I concatenate the proteins and just keep one header?
Thanks!
2
Entering edit mode
6 weeks ago
Mensur Dlakic ★ 23k
If you are interested in strain differences only in coding DNA, then your simulation can be from predicted genes. That would be with an explicit understanding that only SNPs within coding regions would be measured, and that may be all you care about. I would not use complete genomes.
You will need to simulate more than 1000 reads. A general rule of thumb is that we need at least 50-100x coverage for short sequencing technologies to be able to assemble de novo. If the length of your total coding sequence is 1.2 million bases, for 1x coverage (on average) you need [1,200,000 / 150] reads (comes to 8000 reads of length 150 bp). Realistically, for these genomic parameters you would need at least 400,000 simulated reads, and 800,000 would be even better.
Both 85-88% and 92% coding densities are common for bacteria. Plasmids must have at least one origin of replication, and sometimes have two of them, which given the ratio of replication origin to plasmid size lowers the coding potential compared to genomic DNA.
1
Entering edit mode
Thanks very much for your time and explanation. I really appreciate it. If I could trouble you with one more question: how do people learn these rules of thumb? From reading and synthesizing many papers? Are there other sources for learning these things? I know I can learn them by designing and conducting small experiments with publicly-available data, but I find that I often don't have enough time to do experiments before my experiments...I do them later, in my spare time, but they don't help with the problem in front of me. I feel like every physics textbook has the information to solve my physics problems if I look and think hard enough, but the same is not true of bioinformatics, I think? Thank you again for any hints/recommendations. :)
1
Entering edit mode
I have learned too many things to be able to point out how exactly I learned this or that. Like most things in life, it is a combination of formal learning in the classroom and experiential learning by trial and error. Lucky for the younger generations, Google can find almost anything very quickly. It sure cuts down on time required to go to the library, browse through the shelves, and then still go through an actual book or journal.
A general solution is to research thing before doing them. Googling synthetic shotgun DNA reads or simulating shotgun DNA sequencing will give you plenty of information for all aspect of the process. There will be papers describing theoretical aspects of the process, guidelines for error percentage, depth of reads, etc. There will also be links to programs that can do this, and even to scripts that you can copy and modify as needed. I don't mean to sound like your parent, but trust me on this one: it is much easier to find information these days and learn something in do-it-yourself fashion than it was ever before.
1
Entering edit mode
6 weeks ago
Mensur Dlakic ★ 23k
I don't think you should use complete genomes. Still, not sure that using coding sequences will suffice either. I guess it depends on the purpose of your simulation: whether you want this to be very realistic, or just need a collection of sequences to test existing or new software.
First, transcripts are longer than just the coding sequence because of 5' and 3' UTRs. Second, many transcripts in bacteria are polycistronic, so there is a continuum between several coding sequences, including intergenic regions. Not sure how one would simulate that without knowing, or having predictions for, promoter and terminator sequences.
0
Entering edit mode
Thanks, Mensur. The purpose of this simulation is to see whether I can do strain typing for a specific strain of a bacteria species within a metatranscriptomic sample, and at what level of presence I can still detect it. I was thinking of doing several simulations (e.g., 1. generate 1,000 shotgun reads, 2. add specific numbers to my metatranscriptomic sample - say, 50, 100, 500, 1000 reads, 3. shuffle, 4. try to do strain typing by blasting against VFDB), and assess the results. I'm generating 150bp sequences. From what you say, it seems like if I use coding sequences I run the risk of underestimating because some of the simulated reads will be nonsensical (i.e. the end of one coding sequences running straight into the beginning of the next coding sequence, and not respecting the polycistronic nature of the transcripts). But if I use complete genomes, I have a lot of reads that wouldn't be present in an RNA-extracted sample. Right? Are there other limitations I've missed?
So, I calculated the percent of coding sequences for my bacterial genome as well as its plasmids (so I took number of lines in the complete genome - number of lines in the coding sequences, minus the headers of course). It's 92% coding sequence, and the plasmids are 85-88% coding sequence. Perhaps a good solution is use the complete genome, and generate 2,000 reads instead of 1,000?
A side note: is it common for plasmids to be 85-88% coding? I expected more than 92% coding sequence.
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2023-03-20 12:05:17
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https://stats.stackexchange.com/questions/349285/time-series-seasonality-cycle-not-constant
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# Time Series Seasonality (cycle) not constant
I'm having problems creating a reliable model to forecast this time series (quarterly data):
3235 3050 3045 2507 2346 2360 2396 2577 2518 2518 2683 2714 2711 2711 2731 2551 2474 2516 2526 3264 2569 2480 2885 3028 2913 3092 3091 3202 3143 3256 3106 2917 3184 3381 3343 3608 4152 3948 3839 3474
I'm using the library(forecast) from R. I specified several models like this:
model1 <- Arima(mydata, order=c(0,1,0), seasonal=list(order=c(0,1,1),period=4))
model2 <- Arima(mydata, order=c(0,0,0), seasonal=list(order=c(0,1,1),period=4),include.constant = TRUE)
Variations of the last one with mean and drift and many more.
I get poor results when I test the models with 80% of the data over the remaining 20%.
The issue is the seasonality (cycle) because my data begins with one peak, then in the 20th period has another peak and then in periods 37th, 38th and 39th peaks again. So the seasonality is not constant every 20 periods and based on latest data points I can have more than peak next to the other. I changed the parameter "period=4" to "period=17 or 18 or 19", but results still far from optimal.
I tried with external variables without success.
Should I be using something different from ARIMA? Maybe via intervention variables? (I don't know how to do it in R). Any smoothing technique?
Can anybody provide some code? Or point out to methods or libraries for this problem?
• is too extreme to take 2 seasonal differences and change the parameter "period=9"? – ceteris_paribus Jun 1 '18 at 4:28
• I got better results, but seems too much for the number of data points I have – ceteris_paribus Jun 1 '18 at 4:28
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2019-11-22 07:20:55
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http://math.stackexchange.com/questions/154497/derivative-of-a-function-with-variable-range
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# Derivative of a function with variable range
Suppose for example that I have a function $g_y(x)$ such that
$g_y(x) = \begin{cases} y+x &\mbox{if } -x<y<1-x \\ 1 & \mbox{if } y>1-x \\0 &\mbox{if } y<-x \end{cases}$
How, if possible, would I find $\frac{dg}{dx}$?
Edit: Also, I need to have that $x\in (0,1)$
-
$$g_y(x) = \begin{cases}0 & \text{if } x < -y\\ x+y & \text{if }x \in (-y,1-y)\\ 1 & \text{if }x > 1-y\end{cases}$$ You also need $x \in (0,1)$. We will split it into cases depending on where $y$ lies.
If $y \leq -1$, then we get that $g_y(x) = 0$ for all $x \in (0,1)$.
If $-1 \leq y \leq 0$, then we get that $$g_y(x) = \begin{cases}0 & \text{if } x \in (0,-y)\\ x+y & \text{if }x \in (-y,1)\end{cases}$$ for all $x \in (0,1)$.
If $0 \leq y \leq 1$, then we get that $$g_y(x) = \begin{cases} x+y & \text{if }x \in (0,1-y)\\ 1 & \text{if }x \in(1-y,1)\end{cases}$$ for all $x \in (0,1)$.
If $y \geq 1$, then we get that $g_y(x) = 1$ for all $x \in (0,1)$.
Now you should be able to work out the derivative with relative ease. Note that if $y \in [-1,0]$, then the function is not differentiable at $-y$ and for $y \in [0,1]$, the function is not differentiable at $1-y$.
-
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2015-11-28 15:24:19
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https://manasataramgini.wordpress.com/2018/07/27/reflections-on-our-journey-through-the-aliquot-sums-and-sequences/
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## Reflections on our journey through the aliquot sums and sequences
The numerology of aliquot sums and perfect numbers
The numerology of the Pythagorean sages among the old yavana-s is one of the foundations of science and mathematics as we know it. One remarkable class of numbers which they discovered were the perfect numbers — teleios as they termed it. What are these numbers? Let $n$ be a number and $d_j$ be all its proper divisors, i.e. those divisors of $n$, which are less than $n$. Then we can define an arithmetic function known as the aliquot sum $s(n)$ thus,
$\displaystyle s(n)=\sum_{j=1}^k d_j; d_k
For example, let us consider the number 10. Its divisors are 1, 2,5,10. Its proper divisors are 1, 2, 5. Hence, $s(10)=1+2+5=8$. Now, if $s(n)=n$ then $n$ is called a perfect number (termed pūrṇāṅka in Jagannātha’s Sanskrit edition of Euclid). From Figure 1, we can see that the numbers 6 and 28 are perfect numbers. The Pythagorean interest in them also becomes apparent from the fact that these numbers are associated with certain natural periodicities that have an old Indo-European significance. This becomes clear from their occurrence even in old Hindu tradition. The number 6 is associated with the 6 seasons in brāhmaṇa-s like the Śatapatha-brāhmaṇa and encoded into the śrauta altar. Similarly 28 is also encoded into the altar according to the Śatapatha-brāhmaṇa and corresponds to the count of the nakṣatra-s or days of the lunar month. The number 28 is also used in Vaidika tradition as one of the prescribed counts for the japa of the Savitṛ gāyatrī
Figure 1
Returning to the arithmetic, with our above example of $s(10)$, we can see that $s(n)n$. Such numbers are called abundant numbers. Now, a subset of the proper divisors of 20 add up to 20 (Figure 1). Hence, it is also a semi-perfect or a pseudo-perfect number. There is a rare set of abundant numbers for which no subset of their proper divisors add up to them; they are known as weird numbers. For example, $s(70)=74$; hence, it is an abundant number. However, no subset of its proper divisors, 1, 2, 5, 7, 10, 14, 35 add up to 70. Hence, 70 is a weird number. The next weird number is 836. Thus, due their rarity majority of abundant numbers are semiperfect numbers.
One of the high points of yavana brilliance was the discovery of a general formula for perfect numbers. In Book 9, proposition 36 Euclid states:
“ean apo monados hoposoioun arithmoi hexēs ektethōsin en tē diplasioni analogia, heōs hou ho sumpas suntetheis prōtos genētai, kai ho sumpas epi ton eskhaton pollaplasiastheis poiē tina, ho genomenos teleios estai.”
If as many numbers as we please beginning from an unit be set out continuously in double proportion, until the sum of all becomes prime, and if the sum multiplied into the last make some number, the product will be perfect.
In modern terms we can lay it out thus: If the sum of the series of the powers of 2, where the power are integers $\ge 0$, is a prime number then the product of that prime with the last power of 2 in the series is a perfect number:
$q=\displaystyle \sum_{j=0}^n 2^j$
If $q$ is a prime then $q\cdot 2^n$ is a perfect number. Let us take the first few examples.
$2^0+2^1=3\Rightarrow 2^1\times 3=6$
$2^0+2^1+2^2=7\Rightarrow 2^2\times 7=28$
$2^0+2^1+2^2+2^3=15 \Rightarrow$ Not a prime
$2^0+2^1+2^2+2^3+2^4=31\Rightarrow 2^4\times 31=496$
$2^0+2^1+2^2+2^3+2^4+2^5=63\Rightarrow$ Not a prime
$2^0+2^1+2^2+2^3+2^4+2^5+2^6=127\Rightarrow 2^6\times 127=8128$
After this point the perfect numbers become much less frequent and large.
$2^0...2^{12}=8191 \Rightarrow 33550336$
$2^0...2^{16}=131071 \Rightarrow 8589869056$
$2^0...2^{18}=524287 \Rightarrow 137438691328$
Another way of expressing this is thus: Given a prime number $p$ if $M_p=2^p-1$ is also a prime then $P=2^{p-1}\cdot(2^p-1)$ is a perfect number. The corresponding prime numbers $M_p$ are today famous as the Mersenne primes. For all $p <100$, the following $p$ yield $M_p$: 2, 3, 5, 7, 13, 17, 19, 31, 61, 89
The corresponding $M_p$ are:
3
7
31
127
8191
131071
524287
2147483647
2305843009213693951
618970019642690137449562111
The corresponding perfect numbers are:
6
28
496
8128
33550336
8589869056
137438691328
2305843008139952128
2658455991569831744654692615953842176
191561942608236107294793378084303638130997321548169216
These primes $M_p$ start getting huge and rare rapidly. We computed the above in a few seconds with a modern programming language and computer. However, some of them mark historic feats of arithmetic computation by the unaided human brain in the pre-computer era. For instance, Leonhard Euler computed the 8th $M_p$ (10 digits) and the corresponding perfect number. The Russian village mathematician I.M. Pervushin went further by computing the 9th of these numbers ( $M_p= 19$ digits). With the Great Internet Mersenne Prime Search we now have 50 of them and the corresponding perfect numbers. This also helps us to see why the $M_p$ always end in 1 or 7 and the corresponding perfect numbers end in 6 or 28. That the above are the only even perfect numbers was established by Euler. Unlikely as they seem, there has been no formal proof to show that no odd perfect numbers exist. The modern efforts also show that the perfect numbers are rarer than what the yavana sages thought them to be. The old Pythagorean numerologist Nicomachus states:
“It comes about that even as fair and excellent things are few and easily numerated, while ugly and vile ones are widespread, so also the abundant and deficient numbers are found in great multitude and irregularly placed – for the method of their discovery is irregular – but the perfect numbers are easily enumerated and arranged with suitable order; for only one is found among the units, 6, only one among the tens, 28, and a third in the rank of the hundreds, 496 alone, and a fourth within the limits of the thousands, that is, below ten thousand, 8128.”
The above statement suggests that Nicomachus might have thought that for each decade there is one perfect number. Alternatively, he might have simply stopped counting with the greatest perfect number he knew. That it was the former is strengthened by Platonic siddha (to use Gregory Shaw’s term), Iamblichus, even more explicitly claiming that there is one perfect number per decade. That, however, is clearly wrong as the next perfect number 33550336 is in the crores. In any case the modern confirmation of the real rarity of perfect numbers vindicates the philosophical analogy drawn from these numbers by the yavana siddha — the scarcity of truly perfect things as opposed to the profusion of the supernumerary and the deficient. With respect to the easy enumeration of the perfect numbers Nicomachus and Iamblichus likely meant Euclid’s proposition. This rarity of perfect numbers indicates that the sum of the reciprocals should converge to a constant:
$\displaystyle C_P=\sum_{j=1}^n \dfrac{1}{P_j}= \dfrac{1}{6}+\dfrac{1}{28}+\dfrac{1}{496}+\dfrac{1}{8128}... \approx 0.20452014283893...$
It would be immensely remarkable if this constant turns up somewhere in nature.
Abundant numbers
In any case, this $C_P$ provides a bound for the distribution of abundant numbers and thus, brings us to the point of whether Nicomachus’ statement regarding the abundant numbers is really so? This was what we set out to investigate in our youth armed with a rather meager arithmetic knowledge and a computer. Computing the first few abundant numbers yields a sequence like below:
12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 176, 180, 186, 192, 196, 198, 200, 204, 208, 210, 216…
A few rules become immediately apparent:
1) if $P_j$ is a perfect number then $k\cdot P_j$ is an abundant number for all $k \ge 2$. Thus, $P_1=6$ initiates a line of abundant numbers 12, 18, 24, 30…. $P_2=28$ initiates 56, 84, 112…
2) We notice that beyond these a few abundant numbers emerge in the interstices e.g. 20, 88 etc. A notable class of these emergent abundant numbers are of the form $2^k\cdot p_j$, where $p_j$ is the $j^{th}$ odd prime and $k \ge 2$. The first few of these are tabulated below.
These are all shown as red points in panel 1 of Figure 2. One notices that for each jump of $k$ we get a jump in these abundant numbers. Of these, some like 12, 56 and 992 are already accounted for as doubles of the perfect numbers 6, 28 and 496 but the rest are distinct. Their multiples, as with the perfect numbers, are also further abundant numbers.
3) A few further abundant numbers emerge in the interstices, which have a more complex description. They found new lineages of abundant numbers as their multiples continue to be abundant numbers. The first of these is $70=2\times 5 \times 7$. 70 is the product of successive odd primes 5, 7 with 2 leaving out 3, which would yield already accounted-for abundant numbers in a product with 2 via the perfect number 6. Another such is $2002=2\times 7 \times 11 \times 13$. Here 5 is left out because that will again result in already accounted-for abundants in a product with 7. Likewise, leaving out 7, we have $1430=2\times 5\times 11\times 13$. For further numbers in this category we need the next power of 2, e.g. $9724=2^2 \times 11\times 13\times 17$.
4) Till $n=1000$ odd abundant numbers are very rare. The $A[232]=945$ is the first odd abundant number and till $n=200000$ there are only 391 odd abundant numbers as opposed to 49090 even abundant numbers. Of these first 391 odd ones, 387 are divisible by 15. The remaining 4 which are not namely 81081, 153153, 171171, 189189 are divisible by 9009. All odd abundant numbers necessarily need to have the product of at least 3 distinct odd primes as a divisors. As with other abundant numbers, the multiples of odd abundant numbers are also abundant numbers. Based on the separation between the odd abundant numbers, we observed that the first of them 945 is the first of a lineage of odd abundant numbers, which are generated by the formula: $A_o=3 \times (315+210k)$, where $k=0,1,2...51$. Similarly, we observed that a related formula $A_o=11 \times (315+210k)$, where $k=0,1,2...192$ produces a continuous run of 193 odd abundant numbers starting from 3465. After $k=51$ and $k=192$ respectively these formulae do not necessarily produce abundant numbers; nevertheless numbers emerging from these rules continue to remain enriched in odd abundant numbers. Of course, the other emergent odd abundant numbers might have further less-apparent rules. In any case, it appears the difference between two successive odd abundant numbers is most of the times divisible by $2 \times 3 \times 5 =30$ and always divisible by the perfect number 6.
Thus, unlike what the yavana siddha-s thought, the abundant numbers are not entirely unruly. They have complex patterns, but can be described by some rules. The biggest bulk of them are children of perfect numbers — like mortal descendants of the immortal gods. This leads us to the growth of the sequence of abundant numbers, which exhibits striking regularity (Figure 2).
Figure 2.
Given that the sum of the reciprocals of the perfect numbers converges to a constant $C_P$ and given what we have seen above regarding the abundant numbers, they should grow at the upper bound by the equation $y=\tfrac{x}{C_P} \approx 4.89x$ (the dark red line panel 1 of Figure 2). However, since the progeny of perfect numbers are not the only abundant numbers and several emerge by the other rules mentioned above $A[n]$ should grow at a lower rate. During our initial foray into abundant numbers, by empirical examination we thought this growth rate might be exactly 4 (Figure 2, panel 1, light blue line). However, with better computers and computation of 49481 abundant numbers, we later realized that the rate was actually slightly more than 4 (Figure 2, panel 1, dark green line). Another way of visualizing the same is by defining the density of these numbers as the ratio of the number of abundant numbers $\le$ to a certain number $n$ to $n$,
$D= \dfrac{\#A\le n}{n}$
This is shown in Figure 2, panel 2, where we see $D$ converging to a value between 0.248 and 0.247 (red and blue lines). This has been a topic of deep investigation in modern mathematics, starting with the likes of Sarvadaman Chowla and Paul Erdős, and indeed the above has proven to be the approximate bound of the density of abundant numbers. We were quite pleased to have semi-empirically arrived at it for ourselves. Thus, even as the odd and even numbers are distributed in a 1:1 ratio, after initial jitters the ratio of deficient to abundant numbers converges to a constant of $\approx 4.04$ with new lineages of abundant numbers constantly emerging in the interstices of the established ones to maintain a linear growth at this rate.
The basic features of the aliquot sequences
The process of obtaining aliquot sums of a number $n$ can be applied iteratively (the aliquot map): $s(n)=n_1 \rightarrow s(n_1)=n_2 \rightarrow s(n_2)=n_3...$. The sequence $ali(n)=n, n_1, n_2, n_3...$ is called the aliquot sequence of $n$. Thus, if we start with $n= 20$ we get the sequence: 20, 22, 14, 10, 8, 7, 1, 0. More generally, by computing aliquot sequences for all numbers from 1:1000, we observe the following:
1) The number 1 converges in two steps to 0.
2) All primes converge in 3 steps, $p, 1, 0$.
3) Perfect numbers converge in 1 step to themselves.
4) Certain numbers converge to a perfect number. E.g. 25, 95, 119, 143, 417, 445, 565, 675, 685, 783, 909, 913… converge to 6; 608, 650, 652, 790 converge 496. We observe that these numbers have given the appellation “aspiring numbers”.
5) Notably, 220 converges to 284 and 284 converges to 220. $ali(562)=562, 284, 220$. Thus, if from whatever starting point if one reaches 284 or 220 one cycles between them. Such a pair is termed a pair of amicable numbers and was probably known to Platonists as indicated by the heathen Sabian from Harran, Thabit ibn Kurra’s knowledge of these numbers. Thabit discovered a rule to produce a lineage of these numbers, likely drawing on the rule to generate even perfect numbers: Let $a=3\cdot 2^{n-1}-1, b=3\cdot 2^n-1, c=9\cdot 2^{2n-1}-1$ and $n\ge 2$. If $a,b,c$ are primes then we can compose the amicable pair as $a_1=2^n \cdot ab, a_2= 2^n \cdot c$. For $n=2$ we get the pair 220, 284; $n=4$ produces 17296, 18416; $n=7$ produces the pair 9363584, 9437056. However, there are many more amicable numbers between these pairs which cannot be captured by this rule. For example, one can computationally show that 1184, 1210 are an amicable pair. Long after the days of Thabit, starting from Euler down to our times further rules have been found to capture more amicable pairs.
6) If perfect numbers are auto-cycles under the aliquot map, the amicable numbers can be considered 2-cycles. While we do not know if all higher cycles exist under the aliquot map, we can computationally find some of them. For example the pentad 12496, 14288, 15472, 14536, 14264 constitute a 5-cycle. Any number in this pentad will cycle through these values. Even more remarkable is this sequence 14316, 19116, 31704, 47616, 83328, 177792, 295488, 629072, 589786, 294896, 358336, 418904, 366556, 274924, 275444, 243760, 376736, 381028, 285778, 152990, 122410, 97946, 48976, 45946, 22976, 22744, 19916, 17716. These constitute a 28-cycle and any number in this 28-ad will cycle between these values. The lowest member of a 2- and greater cycle is always an abundant number.
7) Still other numbers will eventually converge to an odd prime and via that prime reach 0. There is a remarkable pattern in terms of the preference of the prime via which convergence occurs (Figure 3). 43 is the most preferred prime for convergence for $n=1:1000$ (any links to Heegner numbers or twin primes?). We are not clear as to whether this trend survives with increasing $n$.
Figure 3
8) If we leave out the starting $n$ will every other integer be reached by rest of the aliquot trajectory of some $n$? We can easily provide the answer to this question as no. For example, $2=1+1$. You cannot have two 1s as proper divisors, hence 2 can never be reached from any other number under the aliquot map. Similarly, $5=4+1=2+3$. But each of these sums leading to 5 cannot be a $s(n)$ because we would leave out divisors 2 and 1. Thus, 5 can never be reached from any other number under the aliquot map. On the other hand, a number $n=p+1$ (where $p$ is a prime number) is never untouchable because $s(p^2)=p+1$. Similarly, a number $n=p_o+3$ (where $p_o$ is an odd prime) is never untouchable because $s(2p_o)=p_o+3$. Nevertheless, some numbers are not reached easily until one of the above configurations is encountered for the first time. For example, $ali(1369=37^2)= 1369, 38, 22, 14, 10, 8, 7, 1, 0$ contains 38=37+1 for the first time. Similarly, $ali(2209=47^2)=2209, 48, 76, 64, 63, 41, 1, 0$ contains 48=47+1 for the first time. But the number 52 lies a in sweet spot as as neither 51=52-1 nor 49=52-3 are primes and is untouchable from any other number under the aliquot map. These are some obvious examples of untouchability and touchability; Erdős has shown that ultimately there are an $\infty$ of cases.
Patterns in the convergence length of aliquot sequences
Are these the only convergence patterns seen in the aliquot sequences? Is there any further pattern to the number of iterations needed to converge? To address these questions we can define a further sequence $f[n]$ where each element is the number of iterations taken by the integer $n$ to converge. The plot of $f[n]$ for the first 1000 elements is shown in Figure 4.
Figure 4
The picture is quite remarkable — the majority of values are rather pedestrian but from time to time there are huge eruptive values. The perfect numbers ( $f[n]=1$), 1 and the primitive aspiring numbers, e.g. 25 ( $f[n]=2$), and primes ( $f[n]=3$) form the lowest values of $f$. $f[n]=3$ might also be reached by certain secondary aspiring numbers like 95 or 119. Of these the primes are the most common. It is quite obvious that even numbers have significantly longer convergence paths than odd numbers ( $p=10^{-10}$; Figure 5). We also observed that on an average the abundant numbers have significantly longer convergence paths than the remaining numbers ( $p=5.3 \times 10^{-9}$ for $n=1:1000$). This holds even after the primes are removed from the non-abundant numbers ( $p= 1.9 \times 10^{-8}$ for $n=1:1000$). Similarly, if we compare even abundant numbers and even non-abundant numbers, which occur in the roughly similar counts, the abundant even numbers still have significantly longer trajectories the non-abundant even numbers under the aliquot map ( $p=3.2 \times 10^{-7}$). Due to the relative rarity of odd abundant numbers, these effects are likely to persist beyond $n=1000$. The basic statistics for the convergence trajectory lengths for different types of numbers from $n=1:1000$ are tabulated below and summed up in Figure 5.
$\begin{tabular}{|l|r|r|r|} \hline Number & Median & Mean & Max \\ \hline Even & 12 & 21.7 & 749 \\ \hline Odd & 4 & 4.9 & 17 \\ \hline Abundant & 15 & 34.04 & 749 \\ \hline Non-abundant & 5 & 6.76 & 25 \\ hline Non-abundant Non-p & 7 & 7.84 & 25 \\ \hline Even-abundant & 15 & 34.16 & 749 \\ \hline Even-non-abundant & 10 & 10.36 & 25 \\ \hline \end{tabular}$
Figure 5
During our initial foray into the first 200 terms of this sequence it appeared that certain integers, like 138, never converged under the aliquot map. Going on till $n=1000$ more such terms appeared; however, better computation showed us that the aliquot sequences for some of these $n$ indeed converge after a large number of steps after growing to monstrous values. It was then that we learned that the mathematicians have been studying this problem quite a bit computationally starting with Derrick Lehmer-II (the son of the father-son pair of arithmeticians), one of the doyens of computational mathematics. He was the first who, back in his days after what was a tough fight, showed that $ali(138)$ indeed converges. For us, the hardest nut to crack for $n=1:1000$ was $f[840]$. It simply would not complete in our laptop at home even after running it for over a day with an efficient divisor finding function. Nor did it complete on our primary work station used for most of routine computations. Then we had to bring out our Indrāstra, a 120 core machine that we use for big things. For that we had to first write a multi-threaded version, which was run on 100 cores of this machine and it completed $f[840]$ overnight. We term these large $f[n]$ as the monster numbers. For $n=1:1000$ there are 18 of them, which come in the below-tabulated families, all of which are abundant numbers descending from 6. After the first few terms the members of each family follow the same trajectory to convergence. The final prime via which they converge is termed $p_c$
Figure 6. The evolution of the first members of each family. $f_{840}$: dark red $f_{138}$: cyan; $f_{702}$: dark blue; $f_{720}$: dark green; $f_{858}$: hot pink; $f_{936}$: gray.
One notices that families $f_{138}$ and $f_{858}$ have a higher-order relationship because they converge via 59. However, we keep them as different families for, barring the last 8 elements of $ali(n)$, they followed very different trajectories (Figure 6). Their $p_c$ is the 6th most common $p_c$ for $n=1:1000$, with 45 numbers converging via it (Figure 3). However, 12 of them being monster values it is one $p_c$ over-represented in the aliquot sequences attaining monster values. In contrast, the most common $p_c$ in this range 43 (Figure 3) has only a single monster value (936) in this range. $ali(840)$ shows the most monstrous behavior in this range (Figure 6). It reaches a maximum value of the behemoth number: 3463982260143725017429794136098072146586526240388. I felt please on reaching this number and experiencing the magnificence of $ali(840)$ for myself. After that it remains in the high range for while with two prominent peaks before converging after 749 steps. Such a behavior, with a multiplicity of peaks before convergence, is also seen in the other monstrous families (Figure 6).
We wondered if there was any features of these sequences, which allowed their eruptive growth before finally converging. We observed the following:
1) They start as even abundant numbers, which means that they are statistically in the general group of numbers which are going to have longer trajectories to convergence.
2) Their trajectories tend to remain even, which again increases their probability of having a long convergence path. By specifically studying the first major eruptive case $ali(138)$, we see that it has a convergence trajectory of length 179. It remains even from $ali(138)[1]$ to $ali(138)[175]$. Thus, remaining even is a key factor for a long convergence path.
3) The next key feature observation regarding their growth came from examining the behavior of the first two cases which show above average behavior. These are $f(30)=16$ and $f(102)=19$. First, both are even abundant numbers satisfying the above condition. Then we note their behavior under the aliquot map:
$ali(30)=30, 42, 54, 66, 78, 90, 144, 259, 45, 33, 15, 9, 4, 3, 1, 0$
$ali(102)=102, 114, 126, 186, 198, 270, 450, 759, 393, 135, 105, 87, 33, 15, 9, 4, 3, 1, 0$
We see that both of them have a run of 7 continuous abundant numbers, which are multiples of the perfect number 6. This allows a monotonic increase. When they lose this divisor 6 in the 8th iteration they become odd and start falling rapidly and converge. We then looked if this pattern might play out in a truly eruptive example by taking the case of $ali(138)$. We see that from $ali(138)[1]=138$ to $ali(138)[30]=1467588$, then $ali(138)[65]=19406148$ to $ali(138)[72]=348957876$ and again $ali(138)[95]=1013592$ to $ali(138)[117]=100039354704$ each number is a multiple of the perfect number 6. The repeated generation of a multiple of a perfect number under the aliquot map means there can be stretches of unhindered growth with one abundant number succeeding the previous one until that perfect number is lost among the divisors. It is the final loss of 6 and then 2 among the divisors that allows $ali(138)$ to converge. In making these observations we had recapitulated Lehmer’s key findings in the context of the growth of the most mysterious class of aliquot sequences we shall talk about next.
There are 12 $f[n]$, for $n \le 1000$, which could not be determined because the corresponding $ali(n)$ never converged in our computation (the blue dots in Figure 4). A search on the Internet reveals that deep computational efforts by various investigators have not seen an end to these sequences. They belong to 5 families which are known after Lehmer-II who first studied them. Within each family (listed below), after the first few terms the evolution converges to the same trajectory (Figure 7). These families are:
$f_{276}=276, 306, 396, 696$
$f_{552}=552 ,888$
$f_{564}=564,780$
$f_{660}=660,828,996$
$f_{966}=966$
Figure 7. $f_{276}$: dark red; $f_{552}$: dark blue; $f_{564}$: dark green; $f_{660}$: hot pink; $f_{966}$: cyan
All these numbers are abundant numbers descending from 6, which is a tendency they share with the above monster numbers that eventually converge. The frequencies of their largest prime factors are tabulated below:
$\begin{tabular}{|r| r|r|r|r|r|r|r|r|r|} \hline p & 7 & 11 & 13 & 17 & 23 & 29 & 37 & 47 & 83 \\ \hline freq & 1 & 2 & 1 & 1 & 4 & 1 & 1 & 1 & 1 \\ \hline \end{tabular}$
Notably, in this set except $f_{564}$, 4 of the five families have a member with 23 as the highest prime factor. It is now conjectured by mathematicians that these define the founding members of a class of aliquot sequences that would never converge but eventually shoot off to infinity. Examining the eruptive growth of these sequences, we found that the same features that cause the eruptive growth of the converging but monster $n$ hold true here too. Thus, if we consider the first case $ali(276)$, which starts with a double of the first monster $n=178$, we notice that it starts off well with the perfect number 6 as the divisor for the first 5 terms successive terms — $ali(276)[1]=276$ to $ali(276)[5]=1872$. But then the sequence loses 6 in the 6th term a stutters for a few terms even going down from the high point it reached. But then in $ali(276)[9]=2716$ it acquires a new perfect number divisor 28, which drives the abundance of successive terms until the it reaches the giant 28 digit $ali(276)[170]=7165455981799192761913565252$. Lehmer’s study was the first which noted the role of such persistent divisors in driving up the aliquot sequences. Accordingly, the grand old man of arithmeticians, Richard Guy, called these ‘drivers’ and in addition to 2 and the perfect numbers defined a few more of such. What seems to happen is that in these apparently non-converging aliquot sequences the drivers drive the growth to some large number after which they are lost. But before the sequence can crash by attaining an odd number, a new driver emerges and pushes up its growth again. Thus, these sequences might not converge. On the other hand the case of the monstrous $ali(840)$ with $f=749$ tells us that even after many such runs there is some chance of hitting a number, which brings you crashing down though its even almost all the way through. Whatever the case, the aliquot sequence of these non-converging numbers tell us that there are several more numbers in their trajectories, which would similarly not converge. Hence, non-convergence will be encountered frequently as the starting numbers get large.
In conclusion, following along the lines of the ancients, we find that the aliquot sequences offer us some philosophical lessons and mysteries:
1) There is the predictable: The primes and perfects display very predictable and uniform behavior.
2) There is the somewhat predictable: like the amicable numbers, for some of which a rule was found as early as Thabit ibn Kurra. But not all of them can be captured by an easy rule.
3) There is the statistically tendency: We know that an even or abundant number is more likely to have a longer path to convergence though it is not clear at all if any rule can say which of them might be monstrous and which rather common place.
4) There is the unpredictable and mysterious: Why are some numbers “aspiring numbers” which converge to a perfect number? Is there any rule to describe them? Why is 43 the preferred prime for convergence at least for $n=1:1000$. Why do some numbers not converge? Can we predict which numbers will not converge? Do they really not converge at all? As far as I know these remain unanswered. Thus, a very simple arithmetic process gives rise to a whole range of unexpected behavior.
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2019-07-15 18:56:15
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AbstractA thesis template adapted for Algeria university's requirements (specially USTHB). The thesis divided into three main parts are: Front matter: This part contains: Title Page, Dedication, Acknowledgement, Abstract ( French, Arabic, and English), Table of Contents. (Without page numbering) List of Figures, Tables, Algorithms and Nomenclature.(Page numbering is Roman) Main matter: All the chapters of the thesis and the bibliography are included here, the page numbering Arabic was used in this part Back matter: The appendices are included here, the page numbering alphabetic was used in this part. This template has also the following properties: Automatic generation of Nomenclature and its divided into Roman, Greek, Mathematical symbols and Abbreviations. The main Language is French, but it's easy to switch to English Files management is easy and clear The most used packages are included in the Preamble
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# Is |x| < 1? (1) |x + 1| = 2|x - 1| (2) |x - 3| 0
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Is |x| < 1? (1) |x + 1| = 2|x - 1| (2) |x - 3| 0 [#permalink]
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15 Aug 2009, 14:58
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Is |x| < 1?
(1) |x + 1| = 2|x - 1|
(2) |x - 3| ≠ 0
[Reveal] Spoiler: OA
Last edited by Bunuel on 02 Mar 2012, 21:00, edited 1 time in total.
Edited the question and added the OA
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Re: Is |x| < 1? [#permalink]
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15 Aug 2009, 22:37
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C
1)
if x>1, solving the equality we get x=3.
if -1<x<1, we get x=1/3.
if x<-1, we get x=3.
Not suff
2)
we get x is not equal to 3.
Combining, x can only be 1/3
OA?
Last edited by Economist on 15 Aug 2009, 23:01, edited 1 time in total.
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Re: Is |x| < 1? [#permalink]
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15 Aug 2009, 22:51
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is -1<x<1 ?
1) |x + 1| = 2|x - 1|
two possibilities:
a) x+1=2(x-1) or x=3 (consider x+1 and x-1 to be of same sign)
b) x+1=2(1-x) or x=1/3 (consider x+1 and x-1 to be of different signs)
not sufficient, x may or may not be between -1 and 1
2) |x - 3| ≠ 0
=> x≠3
not sufficient, x may or may not be between -1 and 1
together, x≠3 so x is 1/3 which is between -1 and 1
sufficient
hence, C
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Re: Is |x| < 1? [#permalink]
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tejal777 wrote:
I am having trouble understanding how |x| < 1 translates to -1<x<1.Am missing something:
We have two case,
a. x>0 eg: x=5, |x|=5 so here |x|>0.Therefore,x < 1
b. when x<0,x=-5,|x| is still 5 how does this become x>-1??
Is $$|x| < 1$$?
Is $$|x| < 1$$, means is $$x$$ in the range (-1,1) or is $$-1<x<1$$ true?
(1) $$|x + 1| = 2|x - 1|$$
Two key points: $$x=-1$$ and $$x=1$$ (key points are the values of x when absolute values equal to zero), thus three ranges to check:
---------{-1}--------{1}---------
A. $$x<-1$$ (blue range) --> $$|x + 1| = 2|x - 1|$$ becomes: $$-x-1=2(-x+1)$$ --> $$x=3$$, not OK, as this value is not in the range we are checking ($$x<-1$$);
B. $$-1\leq{x}\leq{1}$$ (green range) --> $$|x + 1| = 2|x - 1|$$ becomes: $$x+1=2(-x+1)$$ --> $$x=\frac{1}{3}$$. OK, as this value is in the range we are checking ($$-1\leq{x}\leq{1}$$);
C. $$x>1$$ (red range) --> $$|x + 1| = 2|x - 1|$$ becomes: $$x+1=2(x-1)$$ --> $$x=3$$. OK, as this value is in the range we are checking ($$x>1$$).
So we got TWO values of $$x$$ (two solutions): $$\frac{1}{3}$$ and $$3$$, first is in the range (-1,1) but second is out of the range. Not sufficient.
(2) $$|x - 3|\neq{0}$$
Just says that $$x\neq{3}$$. But we don't know whether $$x$$ is in the range (-1,1) or not.
(1)+(2) $$x=\frac{1}{3}$$ or $$x=3$$ AND $$x\neq{3}$$ --> means $$x$$ can have only value $$\frac{1}{3}$$, which is in the range (-1,1). Sufficient.
Hope it helps.
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Re: Is |x| < 1? [#permalink]
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23 Oct 2009, 20:23
I got C too substituting values and evaluating each of those.
thanks Bunuel and Economist for elaborations
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09 Nov 2009, 22:26
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|x|<1 means x e (-1,1)
1)|x+1|=2|x-1|
There are 2 key points (x=-1 and x=1) where one of the expressions under modules passes 0 and change its sign.
x<-1: -(x+1)=-2(x-1) --> -x-1=-2x+2 ---> x = 3. But x=3 does not satisfy x<-1 condition.
-1<=x<1: x+1 = -2(x-1) ---> x+1= -2x+2 --> x=1/3. It satisfies the condition.
x>=1: x+1=2x-2 ---> x=3. It satisfies the condition.
Insufficient
2) |x - 3| ≠ 0
x≠3
Insufficient
1&2) Only x=1/3 satisfies both statements. Sufficient.
If you find it useful/useless, let me know
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09 Nov 2009, 22:32
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Answer is C - you beat me to it walker
For statement 1, you need to actually break it down to 2 separate statements to account for both scenarios:
Option #1 ->
x + 1 = 2(x-1)
x + 1 = 2x -2
x = 2x -3
3 + x = 2x
3 = x {or also Option #2}
Option #2 -> x + 1 = -2(x-1)
x + 1 = -2x +2
x = -2x + 1
3x = 1
x = 1/3
Statement (1) is insufficient because |x| could be 3 or 1/3. So the answer is sometimes yes, sometimes no which means insufficient.
Statement (2) is insufficient because we are told that |x -3| ≠ 0. This means that as long as x ≠ 3, then we're ok. There are far too many options so we certainly have a sometimes yes, and sometimes no.
But together, from Statement (1) we have either 3 or 1/3 and from statement (2) we have everything BUT 3. So the only overlap we have for possible values to make both statements true is 1/3. So the answer is Yes, |x| IS less than 1 and we have enough info to answer the question.
Is |x|< 1?
(1) |x + 1| = 2|x - 1|
(2) |x - 3| ≠ 0
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x| < 1? 1) |x+1| = 2|x-1| 2) |x-3| does not equal 0 [#permalink]
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01 May 2011, 20:59
dc123 wrote:
|x| < 1?
1) |x+1| = 2|x-1|
2) |x-3| does not equal 0
Therefore we are looking to see if -1<x<1????
Can someone plz help. I dont know why its C
In absolute equations, one should validate the equation for all possible cases w.r.t the signs of expression embedded in the modulus.
Q:
|X|<1
-1<X<1
(1) |x+1| = 2|x-1|
Case 1:
(x+1)>=0 AND (x-1)>=0
x+1=2(x-1)
-x=-3
x=3
Case 2:
(x+1)>=0 AND (x-1)<0
x+1=-2(x-1))
x+1=-2x+2
3x=1
x=1/3
Case 3:
(x+1)<0 AND (x-1)>=0
-(x+1)=2(x-1)
-x-1=2x-2
-3x=-1
x=1/3
Case 4:
(x+1)<0 AND (x-1)<0
-(x+1)=-2(x-1)
-x-1=-2x+2
x=3
Thus, x can be 3 or 1/3.
Not Sufficient.
2. x != 3
Not Sufficient.
Combining both;
x=1/3, which is between -1 and 1.
Sufficient.
Ans: "C"
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02 May 2011, 04:43
1. 3 regions to be checked x<-1, -1<x< 1 and x>1
for x>1 , (x+1) = 2(x-1)
x= 3
for x<-1, (x+1) = -2(x-1)
x= 1/3 which is not possible.Hence no solution in this region.
for -1<xx<1, (x+1) = -2(x-1)
x= 1/3
No sufficient.
2. x !=3 not sufficient
1+2 gives x = 1/3 Hence C
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02 May 2011, 10:22
i did it a bit different. plz tell me if u think im wrong doing so:
|x| < 1?
1) |x+1| = 2|x-1|
2) |x-3| does not equal 0
the question is - x^2<1
so
x^2+2x+1=4(x^2-2x+1)
====>
3x^2-10x+3=0
so x = 3 or 1/3
st. 2:
x<>3
not helping me.
combination
x=1/3
is that approach wrong?
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02 May 2011, 18:08
(1)
Square both sides :
x^+1+2x = 4(x^2 -2x + 1)
3x^2 -10x +3 = 0
3x^2 - 9x -x + 3 = 0
3x(x-3) -1(x-3) = 0
(x-3)(3x-1) = 0
x = 3, x = 1/3
Not sufficient
(2) is not sufficient as well
(1) + (2)
x = 1/3
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02 May 2011, 18:57
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dc123 wrote:
|x| < 1?
1) |x+1| = 2|x-1|
2) |x-3| does not equal 0
Therefore we are looking to see if -1<x<1????
Can someone plz help. I dont know why its C
You can do these questions in less than a minute if you understand the approach I discuss here:
http://www.veritasprep.com/blog/2011/01 ... edore-did/
Is |x| < 1?
Yes, essentially this just asks you whether x lies between -1 and 1. |x| represents the distance from 0 which will be less than 1 if x lies within -1 < x < 1.
Statement 1: |x+1| = 2|x-1|
Draw the number line. This equation says that distance from -1 is twice the distance from 1. At which point is the distance from -1, twice the distance from 1? If you split the distance (2 units) between -1 and 1 into 3 parts, 1 part away from 1 and 2 parts away from -1 will be the point 1/3. The diagram below will show you this situation. Similarly the distance between -1 and 1 is 2 so if you go 2 units further to the right of 1, you get the point 3 which will be 4 units away from -1 i.e. twice the distance from 1.
Attachment:
Ques6.jpg [ 4.74 KiB | Viewed 2105 times ]
x could be 1/3 or 3. Hence x may or may not lie between -1 and 1. Not sufficient.
Statement 2: |x-3| does not equal 0
This statement just says that x is not 3. It is not sufficient alone.
Both together, we get that x must be either 3 or 1/3 and it is not 3. Then x must be 1/3 and must lie between -1 and 1. Sufficient.
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Re: Is |x| < 1? (1) |x + 1| = 2|x - 1| (2) |x - 3| 0 [#permalink]
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16 Jan 2012, 08:48
+1 C
However, I used using other methods: a) Squaring both sides of the equation because |x| = $$(x^2)$$^(1/2), b) Combining different scenarios between these two possibilities: (x+1)>0, (x+1)<0, (x-1)>0, (x-1)<0. Therefore, I get four possible scenarios.
Bunuel, could you explain the logic behing your method to solve this problem?
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Re: Is |x| < 1? (1) |x + 1| = 2|x - 1| (2) |x - 3| 0 [#permalink]
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16 Jan 2012, 14:12
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metallicafan wrote:
+1 C
However, I used using other methods: a) Squaring both sides of the equation because |x| = $$(x^2)$$^(1/2), b) Combining different scenarios between these two possibilities: (x+1)>0, (x+1)<0, (x-1)>0, (x-1)<0. Therefore, I get four possible scenarios.
Bunuel, could you explain the logic behing your method to solve this problem?
It's about expanding the absolute value in different ranges. Discussed in Walker's post on absolute values: math-absolute-value-modulus-86462.html
If you're more comfortable with "squaring" method (note that it's not always applicable) then you can apply it to the first statement too (and not only to the stem).
Is $$|x| < 1$$?
Is $$|x| < 1$$ --> is $$-1<x<1$$?
(1) $$|x + 1| = 2|x - 1|$$ --> square both sides: $$(x+1)^2=4(x-1)^2$$ --> $$x^2+2x+1=4x^2-8x+4$$ --> $$3x^2-10x+3=0$$ --> $$x=\frac{1}{3}$$ or $$x=3$$ --> first is in the range (-1,1) but second is out of the range. Not sufficient.
(2) $$|x - 3|\neq{0}$$ --> just says that $$x\neq{3}$$. But we don't know whether $$x$$ is in the range (-1,1) or not.
(1)+(2) $$x=\frac{1}{3}$$ or $$x=3$$ AND $$x\neq{3}$$ --> means $$x$$ can have only value $$\frac{1}{3}$$, which is in the range (-1,1). Sufficient.
Hope it's clear.
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Re: Is |x| < 1? (1) |x + 1| = 2|x - 1| (2) |x - 3| 0 [#permalink]
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24 Apr 2012, 04:04
Hi Bunuel ,
I also used the squaring method....
In what all situations is the squaring method not applicable ???
Can you please explain why ?
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Is |x| < 1? (1) |x + 1| = 2|x - 1| (2) |x - 3| 0 [#permalink]
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24 Apr 2012, 05:43
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shikhar wrote:
Hi Bunuel ,
I also used the squaring method....
In what all situations is the squaring method not applicable ???
Can you please explain why ?
Sure, for example it's not always applicable for inequalities.
A. We can raise both parts of an inequality to an even power if we know that both parts of an inequality are non-negative (the same for taking an even root of both sides of an inequality).
For example:
$$2<4$$ --> we can square both sides and write: $$2^2<4^2$$;
$$0\leq{x}<{y}$$ --> we can square both sides and write: $$x^2<y^2$$;
But if either of side is negative then raising to even power doesn't always work.
For example: $$1>-2$$ if we square we'll get $$1>4$$ which is not right. So if given that $$x>y$$ then we can not square both sides and write $$x^2>y^2$$ if we are not certain that both $$x$$ and $$y$$ are non-negative.
B. We can always raise both parts of an inequality to an odd power (the same for taking an odd root of both sides of an inequality).
For example:
$$-2<-1$$ --> we can raise both sides to third power and write: $$-2^3=-8<-1=-1^3$$ or $$-5<1$$ --> $$(-5)^3=-125<1=1^3$$;
$$x<y$$ --> we can raise both sides to third power and write: $$x^3<y^3$$.
So for our question we can not square x/|x|< x as we don't know the sign of either of side.
Hope it helps.
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Re: Is |x| < 1? (1) |x + 1| = 2|x - 1| (2) |x - 3| 0 [#permalink]
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28 Apr 2012, 10:42
Nice explaination Bunuel, Thanks! 1+
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Re: Is |x| < 1? [#permalink]
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21 May 2012, 22:35
Bunuel wrote:
tejal777 wrote:
I am having trouble understanding how |x| < 1 translates to -1<x<1.Am missing something:
We have two case,
a. x>0 eg: x=5, |x|=5 so here |x|>0.Therefore,x < 1
b. when x<0,x=-5,|x| is still 5 how does this become x>-1??
Is $$|x| < 1$$?
Is $$|x| < 1$$, means is $$x$$ in the range (-1,1) or is $$-1<x<1$$ true?
(1) $$|x + 1| = 2|x - 1|$$
Two key points: $$x=-1$$ and $$x=1$$ (key points are the values of x when absolute values equal to zero), thus three ranges to check:
---------{-1}--------{1}---------
A. $$x<-1$$ (blue range) --> $$|x + 1| = 2|x - 1|$$ becomes: $$-x-1=2(-x+1)$$ --> $$x=3$$, not OK, as this value is not in the range we are checking ($$x<-1$$);
B. $$-1\leq{x}\leq{1}$$ (green range) --> $$|x + 1| = 2|x - 1|$$ becomes: $$x+1=2(-x+1)$$ --> $$x=\frac{1}{3}$$. OK, as this value is in the range we are checking ($$-1\leq{x}\leq{1}$$);
C. $$x>1$$ (red range) --> $$|x + 1| = 2|x - 1|$$ becomes: $$x+1=2(x-1)$$ --> $$x=3$$. OK, as this value is in the range we are checking ($$x>1$$).
So we got TWO values of $$x$$ (two solutions): $$\frac{1}{3}$$ and $$3$$, first is in the range (-1,1) but second is out of the range. Not sufficient.
(2) $$|x - 3|\neq{0}$$
Just says that $$x\neq{3}$$. But we don't know whether $$x$$ is in the range (-1,1) or not.
(1)+(2) $$x=\frac{1}{3}$$ or $$x=3$$ AND $$x\neq{3}$$ --> means $$x$$ can have only value $$\frac{1}{3}$$, which is in the range (-1,1). Sufficient.
Hope it helps.
How is the sign applied for different range?can someone help please
X<-1
-x-1=2(-x+1)
-1<=x<=1
x+1=2(-x+1)
X>1
x+1=2(x-1)
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Re: Is |x| < 1? [#permalink]
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21 May 2012, 23:18
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sdpkind wrote:
Bunuel wrote:
tejal777 wrote:
I am having trouble understanding how |x| < 1 translates to -1<x<1.Am missing something:
We have two case,
a. x>0 eg: x=5, |x|=5 so here |x|>0.Therefore,x < 1
b. when x<0,x=-5,|x| is still 5 how does this become x>-1??
Is $$|x| < 1$$?
Is $$|x| < 1$$, means is $$x$$ in the range (-1,1) or is $$-1<x<1$$ true?
(1) $$|x + 1| = 2|x - 1|$$
Two key points: $$x=-1$$ and $$x=1$$ (key points are the values of x when absolute values equal to zero), thus three ranges to check:
---------{-1}--------{1}---------
A. $$x<-1$$ (blue range) --> $$|x + 1| = 2|x - 1|$$ becomes: $$-x-1=2(-x+1)$$ --> $$x=3$$, not OK, as this value is not in the range we are checking ($$x<-1$$);
B. $$-1\leq{x}\leq{1}$$ (green range) --> $$|x + 1| = 2|x - 1|$$ becomes: $$x+1=2(-x+1)$$ --> $$x=\frac{1}{3}$$. OK, as this value is in the range we are checking ($$-1\leq{x}\leq{1}$$);
C. $$x>1$$ (red range) --> $$|x + 1| = 2|x - 1|$$ becomes: $$x+1=2(x-1)$$ --> $$x=3$$. OK, as this value is in the range we are checking ($$x>1$$).
So we got TWO values of $$x$$ (two solutions): $$\frac{1}{3}$$ and $$3$$, first is in the range (-1,1) but second is out of the range. Not sufficient.
(2) $$|x - 3|\neq{0}$$
Just says that $$x\neq{3}$$. But we don't know whether $$x$$ is in the range (-1,1) or not.
(1)+(2) $$x=\frac{1}{3}$$ or $$x=3$$ AND $$x\neq{3}$$ --> means $$x$$ can have only value $$\frac{1}{3}$$, which is in the range (-1,1). Sufficient.
Hope it helps.
How is the sign applied for different range?can someone help please
X<-1
-x-1=2(-x+1)
-1<=x<=1
x+1=2(-x+1)
X>1
x+1=2(x-1)
Absolute value properties:
When $$x\leq{0}$$ then $$|x|=-x$$, or more generally when $$some \ expression\leq{0}$$ then $$|some \ expression|={-(some \ expression)}$$. For example: $$|-5|=5=-(-5)$$;
When $$x\geq{0}$$ then $$|x|=x$$, or more generally when $$some \ expression\geq{0}$$ then $$|some \ expression|={some \ expression}$$. For example: $$|5|=5$$;
Applying this to $$|x + 1| = 2|x - 1|$$.
If $$x<-1$$ then $$x+1<0$$ and $$x-1<0$$ so $$|x + 1| =-(x+1)$$ and $$2|x - 1|=-2(x-1)$$ which means that $$|x + 1| = 2|x - 1|$$ becomes: $$-x-1=2(-x+1)$$ --> $$x=3$$.
Similarly for other ranges.
For more check Absolute Value chapter for Math Book: math-absolute-value-modulus-86462.html
Hope it helps.
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Re: Is |x| < 1? (1) |x + 1| = 2|x - 1| (2) |x - 3| 0 [#permalink]
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22 May 2012, 20:28
Hi Bunuel,
A. x<-1 (blue range) --> |x + 1| = 2|x - 1| becomes: -x-1=2(-x+1) --> x=3, not OK, as this value is not in the range we are checking (x<-1);
B. -1\leq{x}\leq{1} (green range) --> |x + 1| = 2|x - 1| becomes: x+1=2(-x+1) --> x=\frac{1}{3}. OK, as this value is in the range we are checking (-1\leq{x}\leq{1});
C. x>1 (red range) --> |x + 1| = 2|x - 1| becomes: x+1=2(x-1) --> x=3. OK, as this value is in the range we are checking (x>1).
If understand correctly for (A) you are putting '-' in front of both Left/Right Hand Side Equation. Hence coming to "-x-1=2(-x+1)"
Similarly in (C) I guess you are keeping the Left/Right Hand Side Equation both as '+'.
What about (B)?? value of x is between -1 and 1. How did you decide signs' here?
How are you deciding the sign and coming here: x+1=2(-x+1)
I couldn't understand this
I have seen this post: http://gmatclub.com/forum/math-absolute-value-modulus-86462.html as you referred above but ctill couldn't understand it.
I would really appreciate if you can throw a little light on this.
Thanks,
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Re: Is |x| < 1? (1) |x + 1| = 2|x - 1| (2) |x - 3| 0 [#permalink] 22 May 2012, 20:28
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# Demo system ID
Modified 2018-06-22 by Andrea Censi
TODO for Jacopo Tani: fix broken refs
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The following was marked as "todo".
TODO for Jacopo Tani: fix broken refs
File docs/docs-opmanual_duckiebot/book/opmanual_duckiebot/atoms_17_setup_duckiebot_DB17-jwd/30_demos/08_system_identification.md.
File book/opmanual_duckiebot/atoms_17_setup_duckiebot_DB17-jwd/30_demos/08_system_identification.md
in repo duckietown/docs-opmanual_duckiebot branch master commit ddcc1011
Created by function create_notes_from_elements in module mcdp_docs.task_markers.
This is the description of the wheels calibration procedure. In order to complete the procedure, you need your Duckiebot in configuration DB17-lc with its camera calibrated and the same chessboard as for the camera calibration. In the first step, you will put your Duckiebot in front of the chessboard and send specific commands to the wheels. By recording the chessboard, the Duckiebot will know its position at any time. On your computer, you will then use this informations to calculate the parameters of the kinematics of you Duckiebot. These parameters will be stored on a yaml file.
Duckiebot in configuration DB17-lc
Camera calibration completed
## Video of expected results
Modified 2018-06-22 by Andrea Censi
## Duckietown setup notes
Modified 2018-06-22 by Andrea Censi
The Duckietown is not needed for the wheels calibration.
## Duckiebot setup notes
Modified 2018-06-24 by Andrea Censi
Mount the USB drive.
The procedure is documented in Unit A-16 - Mounting USB drives.
## Pre-flight checklist
Modified 2018-06-22 by Andrea Censi
Check: the Duckiebot has sufficient battery
Check: the USB drive is mounted
Check: the camera is calibrated
Check: the chessboard has the correct dimensions (31 mm squares)
## Demo instructions
Modified 2018-06-24 by Andrea Censi
Everything should be run from branch: devel-sysid. When your are on the devel-sysid branch, rebuild the Workspace using:
duckiebot $catkin_make -C catkin_ws/ Step 1: Run the following commands: Make sure you are in the Duckietown folder: duckiebot$ cd ~/duckietown
Activate ROS:
duckiebot $source environment.sh Step 2: Mount the USB Storage: To do this,y ou can use procedure is documented in Unit A-16 - Mounting USB drives in the duckiebook or run the following commands on your duckiebot. duckiebot$ roscd calibration
duckiebot $bash mount_usb Step 3: Place the Duckiebot in front of the chessboard at a distance of slightly more than 1 meter in front of the checkerboard (~2 duckie tiles), as shown in the image (Figure 2.4).. The heading has to be set iteratively to maximize the time the duckiebot sees the checkerboard. Step 4: Run the calibration procedure duckiebot$ roslaunch calibration commands.launch veh:=robot name
The Duckietown should go forward and then stop.
Step 5 When the Duckiebot has stopped, you have 10 seconds to replace it again at a distance of approximately 1 meters of the chessboard. Wait for the Duckiebot to move forward again.
When the Duckiebot stops, and the node shuts down, you have 2 different alternatives to copy the rosbag to the computer. (6a or 6b)
Step 6a: duckiebot $sudo umount /media/logs And put the USB drive in your computer. Step 6b: cd to folder where you want to put the rosbag laptop$ sftp robot_name
laptop $cd /media/logs laptop$ get robot_name_calibration.bag
Step 7: On your computer, go in the Duckietown folder:
laptop $cd ~/duckietown Activate ROS: laptop$ source environment.sh
Step 8: Run the calibration process with
laptop $roslaunch calibration calibration.launch veh:=robot name path:=/absolute/path/to/the/rosbag/folder/ (path example: path:=/home/user_name/sysid/) Do not forget the backslash at the end of the path.(Common mistake: path not starting from /home, forgetting the last / in the path) Step 9: Once the command has finished, the parameters of your Duckiebot are stored in the folder DUCKIEFLEET_ROOT/calibrations/kinematics/robot name.yaml Step 10: Push the duckiefleet changes to git and pull from the duckiebot Step 11: Run the Validation: Duckiebot should first drive straight for 1m (in 5s) then turn a full circle to the left (in 8s) and then a full circle to the right (in 8s) duckiebot$ roslaunch calibration test.launch
known issue: the baseline is rather overestimated at the moment, thus the duckiebot will probably turn more than a circle
## Troubleshooting
Modified 2018-06-22 by Andrea Censi
No log have been recorded.
Try to mount the USB drive.
The Duckiebot deviates from the trajectory, so that the chessboard goes out of the camera’s field of view.
You can adjust the parameters of the voltage commands by passing arguments when launching the commands. You can change the parameter vFin and Nstep for the straight line, and the parameter k1, k2, omega and duration for the sinewave.
Sympton: Issues/bugs with copying from USB to computer. USB cannot be unmounted from the duckiebot (mine at least) Remounting USB is not possible without rebooting the duck After the first sftp get the USB drive becomes „read only“ and no further bags can be recorded
## Demo failure demonstration
Modified 2018-06-22 by Andrea Censi
No questions found. You can ask a question on the website.
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2018-07-16 02:30:58
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https://www.numerade.com/questions/graph-the-surfaces-z-x2-y2-and-z-1-y2-on-a-common-screen-using-the-domain-mid-x-mid-le-12-mid-y-mid-/
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💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!
TA
# Graph the surfaces $z = x^2 + y^2$ and $z = 1 - y^2$ on a common screen using the domain $\mid x \mid \le 1.2 , \mid y \mid \le 1.2$ and observe the curve of intersection of these surfaces. Show that the projection of this curve onto the $xy$-plane is an ellipse.
## ellipse.
Vectors
### Discussion
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##### Lily A.
Johns Hopkins University
##### Catherine R.
Missouri State University
##### Heather Z.
Oregon State University
Lectures
Join Bootcamp
### Video Transcript
Mhm. Yes, I think uh So what we want to do is it's good stuff. These are process it's Pendergraph G equal to that's the tablets like square and She got 1 -9 sq on a common screen Using and we want to come and the moment for us is absolute value picks is less than equal to 1.2. International number of varieties, these other domains. So what it means is that uh for us, x is really going to lie From 1.2 to -1.2. Similarly why is not a -1.2 for one point. Okay. So first let us draw the grabs. Okay? Yeah. So let's say this is my G access and this is my excitement, right? And this is my so no you can see that as the values of G. Right? And G cannot be negative because he is always positive. So I'll start with equal to zero, which means that point X and Y zero. Then I go to G equal to one. Let's say then what it means is that if G equals to all, then we get access for for us by historical the one who is a something This minister at equaled one. I get a circle. The circle is like this. The radius is one here Similarly equal to two. We get a circle of radius squared of two. So the radius length increases. What it means is that this is really something like this. Yeah. So really it is this cylinder that you join this? It becomes a cylinder, right decision that this opens to the trusted G. Thing. This is what we call texas purpose widespread looks like And then I have equal to one -Y school. Now if you recall this is safe equal to one minus y squared that we right here. So what you get is that why square equal to minus of g minus mint. So this is really a parabola on the plan and it is opening downwards. So how does it look like on the Y Z plane? I have a parabola. So this means that is going to right so it is going to look something like this and the white deeper imagining that this is in the white plains. Of course it's not going to pass to the origin. So what does it pass through G equal to one? We get Y 20 So it passes to the G. Access on the .1. You said this is 0.01. And then when G called zero we get Y two K plus or minus one. So this means that to do something like this, this is how it looks, was part of 0 1.0 At this point is 0 -1. This is how the graph today. Yeah. Oh okay. And then the next question is too so that if we present it onto the white to the X. Y plane, we should get an ellipse. So let us project onto the xy plane. So what does projection that protects yplan mean? Yeah. So projection onto the xy plane really means is that I plug in G two B zero. So look at the intersection. Look at the intersection. Mhm. Of these graphs and then you plug in. So let us look at the intersection to intersection. Really means that you're intersex actually means that I have G. These two guys must be equal to each other. So I showed them equal to each other. Success square plus Y squared must be equal to one minus y squared. Remember certain to be zero and looking at the intersection. Mhm. True. Yeah. So this means that extra square plus two Y squared called one. So this means that excess square upon one Plus y square upon to over one. This must be one. This is wonderful. And hello? So this is an ellipse what you can and yeah.
#### Topics
Vectors
##### Lily A.
Johns Hopkins University
##### Catherine R.
Missouri State University
##### Heather Z.
Oregon State University
Lectures
Join Bootcamp
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2021-10-28 04:03:24
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https://nbviewer.jupyter.org/github/pyGSTio/pyGSTi/blob/bugfix-tensor-model-load/jupyter_notebooks/Tutorials/algorithms/advanced/GST-FiducialPairReduction.ipynb
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# Circuit Reduction Tutorial¶
The circuits used in standard Long Sequence GST are more than what are needed to amplify every possible gate error. (Technically, this is due to the fact that the informationaly complete fiducial sub-sequences allow extraction of each germ's entire process matrix, when all that is needed is the part describing the amplified directions in model space.) Because of this over-completeness, fewer sequences, i.e. experiments, may be used whilst retaining the desired Heisenberg-like scaling ($\sim 1/L$, where $L$ is the maximum length sequence). The over-completeness can still be desirable, however, as it makes the GST optimization more robust to model violation and so can serve to stabilize the GST parameter optimization in the presence of significant non-Markovian noise. Recall that the form of a GST gate sequence is
$$S = F_i (g_k)^n F_j$$
where $F_i$ is a "preparation fiducial" sequence, $F_j$ is a "measurement fiducial" sequence, and "g_k" is a "germ" sequence. The repeated germ sequence $(g_k)^n$ we refer to as a "germ-power". There are currently three different ways to reduce a standard set of GST operation sequences within pyGSTi, each of which removes certain $(F_i,F_j)$ fiducial pairs for certain germ-powers.
• Global fiducial pair reduction (GFPR) removes the same intelligently-selected set of fiducial pairs for all germ-powers. This is a conceptually simple method of reducing the operation sequences, but it is the most computationally intensive since it repeatedly evaluates the number of amplified parameters for en entire germ set. In practice, while it can give very large sequence reductions, its long run can make it prohibitive, and the "per-germ" reduction discussed next is used instead.
• Per-germ fiducial pair reduction (PFPR) removes the same intelligently-selected set of fiducial pairs for all powers of a given germ, but different sets are removed for different germs. Since different germs amplify different directions in model space, it makes intuitive sense to specify different fiducial pair sets for different germs. Because this method only considers one germ at a time, it is less computationally intensive than GFPR, and thus more practical. Note, however, that PFPR usually results in less of a reduction of the operation sequences, since it does not (currently) take advantage overlaps in the amplified directions of different germs (i.e. if $g_1$ and $g_3$ both amplify two of the same directions, then GST doesn't need to know about these from both germs).
• Random fiducial pair reduction (RFPR) randomly chooses a different set of fiducial pairs to remove for each germ-power. It is extremly fast to perform, as pairs are just randomly selected for removal, and in practice works well (i.e. does not impair Heisenberg-scaling) up until some critical fraction of the pairs are removed. This reflects the fact that the direction detected by a fiducial pairs usually has some non-negligible overlap with each of the directions amplified by a germ, and it is the exceptional case that an amplified direction escapes undetected. As such, the "critical fraction" which can usually be safely removed equals the ratio of amplified-parameters to germ-process-matrix-elements (typically $\approx 1/d^2$ where $d$ is the Hilbert space dimension, so $1/4 = 25\%$ for 1 qubit and $1/16 = 6.25\%$ for 2 qubits). RFPR can be combined with GFPR or PFPR so that some number of randomly chosen pairs can be added on top of the "intelligently-chosen" pairs of GFPR or PFPR. In this way, one can vary the amount of sequence reduction (in order to trade off speed vs. robustness to non-Markovian noise) without inadvertently selecting too few or an especially bad set of random fiducial pairs.
## Preliminaries¶
We now demonstrate how to invoke each of these methods within pyGSTi for the case of a single qubit, using our standard $X(\pi/2)$, $Y(\pi/2)$, $I$ model. First, we retrieve a target Model as usual, along with corresponding sets of fiducial and germ sequences. We set the maximum length to be 32, roughly consistent with our data-generating model having gates depolarized by 10%.
In [ ]:
#Import pyGSTi and the "stardard 1-qubit quantities for a model with X(pi/2), Y(pi/2), and idle gates"
import pygsti
import pygsti.construction as pc
from pygsti.modelpacks import smq1Q_XYI
#Collect a target model, germ and fiducial strings, and set
# a list of maximum lengths.
target_model = smq1Q_XYI.target_model()
prep_fiducials = smq1Q_XYI.prep_fiducials()
meas_fiducials = smq1Q_XYI.meas_fiducials()
germs = smq1Q_XYI.germs()
maxLengths = [1,2,4,8,16,32]
opLabels = list(target_model.operations.keys())
print("Gate operation labels = ", opLabels)
## Sequence Reduction¶
Now let's generate a list of all the operation sequences for each maximum length - so a list of lists. We'll generate the full lists (without any reduction) and the lists for each of the three reduction types listed above. In the random reduction case, we'll keep 30% of the fiducial pairs, removing 70% of them.
### No Reduction ("standard" GST)¶
In [ ]:
#Make list-of-lists of GST operation sequences
fullStructs = pc.make_lsgst_structs(
opLabels, prep_fiducials, meas_fiducials, germs, maxLengths)
#Print the number of operation sequences for each maximum length
print("** Without any reduction ** ")
for L,strct in zip(maxLengths,fullStructs):
print("L=%d: %d operation sequences" % (L,len(strct)))
#Make a (single) list of all the GST sequences ever needed,
# that is, the list of all the experiments needed to perform GST.
fullExperiments = pc.create_lsgst_circuits(
opLabels, prep_fiducials, meas_fiducials, germs, maxLengths)
print("\n%d experiments to run GST." % len(fullExperiments))
### Global Fiducial Pair Reduction (GFPR)¶
In [ ]:
fid_pairs = pygsti.alg.find_sufficient_fiducial_pairs(
target_model, prep_fiducials, meas_fiducials, germs,
search_mode="random", n_random=100, seed=1234,
verbosity=1, mem_limit=int(2*(1024)**3), minimum_pairs=2)
# fid_pairs is a list of (prepIndex,measIndex) 2-tuples, where
# prepIndex indexes prep_fiducials and measIndex indexes meas_fiducials
print("Global FPR says we only need to keep the %d pairs:\n %s\n"
% (len(fid_pairs),fid_pairs))
gfprStructs = pc.make_lsgst_structs(
opLabels, prep_fiducials, meas_fiducials, germs, maxLengths,
fid_pairs=fid_pairs)
print("Global FPR reduction")
for L,strct in zip(maxLengths,gfprStructs):
print("L=%d: %d operation sequences" % (L,len(strct)))
gfprExperiments = pc.create_lsgst_circuits(
opLabels, prep_fiducials, meas_fiducials, germs, maxLengths,
fid_pairs=fid_pairs)
print("\n%d experiments to run GST." % len(gfprExperiments))
### Per-germ Fiducial Pair Reduction (PFPR)¶
In [ ]:
fid_pairsDict = pygsti.alg.find_sufficient_fiducial_pairs_per_germ(
target_model, prep_fiducials, meas_fiducials, germs,
search_mode="random", constrain_to_tp=True,
n_random=100, seed=1234, verbosity=1,
mem_limit=int(2*(1024)**3))
print("\nPer-germ FPR to keep the pairs:")
for germ,pairsToKeep in fid_pairsDict.items():
print("%s: %s" % (str(germ),pairsToKeep))
pfprStructs = pc.make_lsgst_structs(
opLabels, prep_fiducials, meas_fiducials, germs, maxLengths,
fid_pairs=fid_pairsDict) #note: fid_pairs arg can be a dict too!
print("\nPer-germ FPR reduction")
for L,strct in zip(maxLengths,pfprStructs):
print("L=%d: %d operation sequences" % (L,len(strct)))
pfprExperiments = pc.create_lsgst_circuits(
opLabels, prep_fiducials, meas_fiducials, germs, maxLengths,
fid_pairs=fid_pairsDict)
print("\n%d experiments to run GST." % len(pfprExperiments))
### Random Fiducial Pair Reduction (RFPR)¶
In [ ]:
#keep only 30% of the pairs
rfprStructs = pc.make_lsgst_structs(
opLabels, prep_fiducials, meas_fiducials, germs, maxLengths,
keep_fraction=0.30, keep_seed=1234)
print("Random FPR reduction")
for L,strct in zip(maxLengths,rfprStructs):
print("L=%d: %d operation sequences" % (L,len(strct)))
rfprExperiments = pc.create_lsgst_circuits(
opLabels, prep_fiducials, meas_fiducials, germs, maxLengths,
keep_fraction=0.30, keep_seed=1234)
print("\n%d experiments to run GST." % len(rfprExperiments))
## Running GST¶
In each case above, we constructed (1) a list-of-lists giving the GST operation sequences for each maximum-length stage, and (2) a list of the experiments. In what follows, we'll use the experiment list to generate some simulated ("fake") data for each case, and then run GST on it. Since this is done in exactly the same way for all three cases, we'll put all of the logic in a function. Note that the use of fiducial pair redution requires the use of run_long_sequence_gst_base, since run_long_sequence_gst internally builds a complete list of operation sequences.
In [ ]:
#use a depolarized version of the target gates to generate the data
mdl_datagen = target_model.depolarize(op_noise=0.1, spam_noise=0.001)
def runGST(gstStructs, exptList):
#Use list of experiments, expList, to generate some data
ds = pc.simulate_data(mdl_datagen, exptList,
num_samples=1000,sample_error="binomial", seed=1234)
#Use "base" driver to directly pass list of circuit structures
return pygsti.run_long_sequence_gst_base(
ds, target_model, gstStructs, verbosity=1)
print("\n------ GST with standard (full) sequences ------")
full_results = runGST(fullStructs, fullExperiments)
print("\n------ GST with GFPR sequences ------")
gfpr_results = runGST(gfprStructs, gfprExperiments)
print("\n------ GST with PFPR sequences ------")
pfpr_results = runGST(pfprStructs, pfprExperiments)
print("\n------ GST with RFPR sequences ------")
rfpr_results = runGST(rfprStructs, rfprExperiments)
Finally, one can generate reports using GST with reduced-sequences:
In [ ]:
pygsti.report.construct_standard_report(full_results, title="Standard GST Strings Example"
).write_html("tutorial_files/example_stdstrs_report")
pygsti.report.construct_standard_report(gfpr_results, title="Global FPR Report Example"
).write_html("tutorial_files/example_gfpr_report")
pygsti.report.construct_standard_report(pfpr_results, title="Per-germ FPR Report Example"
).write_html("tutorial_files/example_pfpr_report")
pygsti.report.construct_standard_report(rfpr_results, title="Random FPR Report Example"
).write_html("tutorial_files/example_rfpr_report")
If all has gone well, the Standard GST, GFPR, PFPR, and RFPR, reports may now be viewed. The only notable difference in the output are "gaps" in the color box plots which plot quantities such as the log-likelihood across all operation sequences, organized by germ and fiducials.
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2021-07-26 17:22:48
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https://www.physicsforums.com/threads/sequence-problem.220042/
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# Sequence Problem
1. Mar 5, 2008
### sylar
In a sequence with 101 elements show that one can find an increasing sequence of 11 terms.
Here is my approach: Pick the greatest element of the sequence (or one of the largest elements if there is more than one large element) and put it onto the end of the sequence we are forming. Then make ten groups consisting of the terms of the original sequence so that each group has ten elements. Now find the least elements of each group(if there is more than one least element in a group, then choose one of them). Finally, arranging these 10 least elements in an increasing order we get an increasing sequence of 11 terms.
Is this argument sound? Or can we solve this problem in a more elegant way?
2. Mar 6, 2008
### Gib Z
The statement isn't even true, though it may be if you apply stricter conditions. Take $$a_1 = a_2 = a_3 ... a_{101}$$ ie a constant sequence. No increasing sequence can be made out of those.
3. Mar 6, 2008
### sutupidmath
yeah, like Gib Z pointed out, that statement is not true, unless we put some further conditions, like for at least 11 terms not to be equal to each other or sth like that!
4. Mar 6, 2008
### sylar
Hey guys, i think you are missing one point. The question doesn't ask us to prove that there exists a strictly increasing sequence; it asks us to show that there exists an increasing sequence. That is, if all terms are equal to each other, then we have an increasing sequence.
5. Mar 6, 2008
### CRGreathouse
No, I don't think so. You're not supposed to re-order the terms, are you? If you were you would need only 11 terms.
6. Mar 6, 2008
### matticus
what is your definition of an increasing sequence...because a constant sequence is in no way increasing...it's constant.
7. Mar 6, 2008
### CRGreathouse
I imagine the intended definition is a sequence $a_1,a_2,\ldots$ with $a_1\le a_2\le\cdots$.
8. Mar 7, 2008
### Dragonfall
The actual theorem goes like this: if you have 101 distinct integers in a sequence, then you will have a 11-term subsequence that is INCREASING or DECREASING (like a lot of combinatorial theorems, it guarantees A or not A).
Or rather, if you want an increasing or decreasing sequence of n+1, then you need n^2+1 numbers.
The OP's statement can't possibly be right:
101, 100, 99, 98, ..., 1.
Last edited: Mar 7, 2008
9. Mar 7, 2008
### HallsofIvy
Staff Emeritus
Many texts use the word "increasing" to mean what others would call "non-decreasing" and "strictly increasing" to mean what others would call "increasing". I don't particularly like that terminology myself but I wouldn't take anyone to task for it!
But sylar has clearly misunderstood the question- it's not a matter of picking out 10 numbers from the sequence and putting them into increasing order. If that were what they were asking, you can obviously take any 101 number sequence and change the order so that the entire 101 were "increasing" (non-decreasing).
DragonFall has it right. His example, 101, 100, 99, ... does not have any increasing or non-decreasing subsequence in it. The problem must be to show that you can always find a sequence of 10 numbers that is either non-decreasing or non-increasing.
10. Mar 7, 2008
### Dragonfall
I was half thinking about the problem last night, and I almost worked out a sort of recursion on the general case (n+1, n^2+1); though I wasn't successful, I'm pretty sure the proof can be done by some sort of induction. I'd love to see a full proof if someone knows it.
11. Mar 7, 2008
### matticus
what about the sequence 1, 3, 2, 4, 6, 5, 7, 9, 8...there is no non-increasing/decreasing sequence greater than 3 in this, no matter how far out you go.
certainly the conjecture is not true.
12. Mar 7, 2008
### matticus
x_3n = 3n + 1, x_(3n + 1) = 3(n+1), x_(3n + 2) = 3n + 2. (n = 0, 1, 2...)
suppose the subsequence starts at x_3n. that means that x_3n through x_3n + 10 must either be non-decreasing or non-increasing. however, x_(3n+1) = 3n + 3 > 3n+2 = x_(3n + 2). so the subsequence is not non-decreasing. and x_(3n + 3) = x_3(n+1) = 3(n+1) + 1 = 3n + 4 > x_(3n+2). so the subsequence is not non-increasing.
starting at x_(3n + 1) and x_(3n + 2) leads to similar reasoning. But that exhausts the possibilities.
13. Mar 7, 2008
### Dragonfall
It's not a conjecture, it's a theorem, and the proof must go something like this:
Given n^2+1 numbers, there is an increasing or decreasing sequence of size n+1.
True for n=1.
Now suppose it's true for all n<k.
Now given (n+1)^2+1 numbers, partition the list into n^2+1 (A) and 2n+1(B). By induction hypothesis, WLOG, there is an increasing sequence within the first part of size n+1. Now if any of B is greater than the last element of one such sequence then we are done. Otherwise,
SOMETHING HAPPENS HERE.
QED.
14. Mar 7, 2008
### Dragonfall
A subsequence does not need to consist of consecutive elements of the sequence.
1 3 5 7 is a subsequence of 1 2 3 4 5 6 7
15. Mar 8, 2008
### HallsofIvy
Staff Emeritus
No, the conjecture, as DragonFall restated it, was to find either a non-decreasing or non-increasing sub-sequence (not required to be consecutive): 1, 2, 6, etc. gives an increasing sequence.
The original statement was simply for an increasing sub-sequence. That certainly applies here.
16. Mar 8, 2008
### sylar
Sorry for the delayed response. First, i want to thank all the posters who commented on these problem. Second, i admit that i misunderstood the problem and what i wrote here was not true. (Though still i'm right with the definition of an increasing sequence) The statement to prove is: In a sequence of (n^2)+1 distinct elements, there exists an increasing or decreasing subsequence of length n+1.
Our instructor asks questions at the end of the lectures just for fun and this was one of these problems. Yesterday, he told me that this is a very nice problem and he didn't know about it before he saw the proof in a mathematics journal. He added that the proof method was strong induction.
So if you prove the statement or see the proof somewhere, can you post here please? Thanks.
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2017-02-22 14:14:04
|
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https://codegolf.stackexchange.com/questions/108296/tips-for-golfing-in-brachylog?noredirect=1
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# Tips for golfing in Brachylog
Brachylog is a language that's beginning to rise in prominence in code-golfing recently (and just received a major update with a terser syntax). Like Prolog, it has the advantage that it can often solve a problem (typically via brute force) merely from a sufficiently accurate description of what a problem looks like, a feature that means that on the right sort of challenge, it's often comparable to the top golfing languages (and has been known to beat Jelly from time to time).
What tips do you have for golfing (i.e. writing the shortest possible programs in) Brachylog? This is mostly looking for advice that's specific to Brachylog in particular, rather than advice that's applicable to a wide range of languages. (Tips about golfing in declarative languages in general might potentially be appropriate here, depending on how much application they'll have to languages other than Brachylog, although see also Tips for golfing in Prolog.)
# Exploit nested predicates to create new variables
Brachylog has lots of special syntax cases to make its two special variables, ? (input / left parameter) and . (output / right parameter), terser to use. This means that if you don't need to access your predicate's ? and ., but do need to use variables, you can often save bytes via creating a nested predicate to use its ? and ..
As a simple example, consider a program that looks like this:
… A … ∧A … B … B …
This is a pretty common shape for a longer program; after all, there are lots of gaps that could contain anything. Suppose we have no need for ? or . inside the centre three gaps. Then we could rewrite it like this:
… { … & … . … } …
Here, the nested predicate's ? is serving the role of A, and its . is serving the role of B. We can observe that this is a byte shorter than the original code; changing AABB to {?.} has no change in terms of bytes, but this allowed us to simplify ∧? to the abbreviation &.
A related trick is to change
∧. … ?∧
to
~{ … }
(which is one byte shorter), although note that it's nearly always cheaper to get the caller to exchange the arguments instead (unless the predicate is called from at least three different places in the program, which is rare in Brachylog).
# Split up length-2 predicates inside metapredicates
This is best explained by example. To remove the first and last elements of a list, we behead and knife it:
bk
If we wanted to perform this operation on every element of a list, we can use a map operation:
{bk}ᵐ
However, it's a byte shorter to split the predicate into two, and map each part separately:
bᵐkᵐ
The same trick can be used with quite a few metapredicates:
{bk}ᵐ → bᵐkᵐ
{bk}ˢ → bˢkˢ
{bk}ᶠ → bᶠkˢ
~{bk} → ~k~b
Note that for some metapredicates, like ᵘ, there's no general-purpose way to split it into two parts, but it may nonetheless be possible to find a decomposition that works for the specific task you're working on.
# We do have an increment builtin... sometimes!
+₁ is the only reliable way to add 1 to something. However, under many circumstances, < does the job for a byte less--its output must be strictly greater than its input, so in the absence of external complications, its output will be the integer strictly greater than its input with the least absolute value.
Of course, this substitution can and will fall apart should further logic be employed, as well as in the case of non-integers (which are ceiled) and negative integers (which produce 0).
# s isn't longest first, but ⊇ is
I wanted to title this "(Ab)use all choice orders at your disposal", but I couldn't think of any other cases of note.
Sometimes choice order is pretty important. Sometimes the choice order your program already has is precisely what you need, just as often it's not; in the case of s generating substrings in the order of largest-first prefixes of largest-first suffixes, it's not infrequent that you might wish it was simply largest first instead. Fortunately, ⊇ generates its sublists longest first, where every possible substring is included--there's just some other stuff you also don't want. However, s's choice order is no problem at all if you're using it to check substring existence, in a program structured something like ⊇. [...] &s.
# Casting the empty list to the empty string
Sometimes, when working with strings, the algorithm we use might unify what we want with the empty list [], when we would rather want the empty string "".
We can cast the empty list to the empty string using ,Ẹ, which appends the empty string to its left variable (this is an exploit of the way , is implemented).
This also has the benefit that it does not do anything if the left variable is a string. So, if your program is
{
some predicate that should always output a string,
but actually outputs [] instead of "" in specific cases
}
Then
{
some predicate that should always output a string,
but actually outputs [] instead of "" in specific cases
},Ẹ
will work the way you want.
# Single-element runs in a list
Consider this snippet:
ḅ∋≠
If the input is a list or string, the output is unified with a sublist/substring of length 1 that's not part of a longer run of equal elements. It splits the list into blocks of equal elements and finds a block whose elements are all different. To get the elements themselves instead of singleton lists, tack h to the end. I used this construct here with o to find a character that occurs only once in the input string.
# Failing on empty lists
Empty lists can come up often, and sometimes you need to fail when you see one. The obvious means to do so may be ¬Ė, or even l>0, but there exist quite a few one-byte alternatives:
• b
• k
• h
• t
• ∋
• z
just off the top of my head. What inspired me to write this out is actually the discovery of a less obvious failure on an empty list: the metapredicate ⁿ, which one might expect to be vacuously successful on an empty input, fails instead. Metapredicates which produce failure on an empty list:
• ⁿ
• ᵛ
• ᵉ
• ᵈ (any list of length other than 2)
• ʰ
• ᵗ
The general idea to take away from this is that a lot of Brachylog's builtins that seem like they're just to produce an output from an input don't necessarily have an output for every conceivable input, which is more to our advantage than it is detrimental, and is (...usually) easy to think about once you're aware of it.
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2022-05-25 11:07:26
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https://leanprover-community.github.io/archive/stream/113488-general/topic/is_X_hom.html
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## Stream: general
### Topic: is_X_hom
#### Scott Morrison (Apr 09 2019 at 06:54):
Thanks, @Johan Commelin, for starting to clean these up in #911.
#### Scott Morrison (Apr 09 2019 at 06:55):
I've long been sad that is_ring_hom.map_one cannot be a simp lemma, because there's no fixed symbol for it to catch on.
#### Scott Morrison (Apr 09 2019 at 06:57):
For other lemmas, each map_mul and map_add, it's less clear that they should be simp lemmas. It depends whether you're trying to push all the ring homomorphisms down (in order to use ring, presumably), or pull them all up (hoping to get to a single homomorphism application).
#### Scott Morrison (Apr 09 2019 at 06:58):
We probably should have a tactic, however, for doing these rewrites.
#### Scott Morrison (Apr 09 2019 at 06:58):
I'm imagining hom by default would just push all homomorphisms down through algebraic operations.
#### Scott Morrison (Apr 09 2019 at 06:59):
It should probably take an argument, so for example hom ring would use the laws for ring homomorphisms.
#### Scott Morrison (Apr 09 2019 at 07:00):
An arrow might suffice for the mode where we want to pull the homomorphism out: either hom ← or hom ←monoid.
#### Johan Commelin (Apr 09 2019 at 07:00):
@Scott Morrison I think such a tactic is a very good idea.
#### Johan Commelin (Apr 09 2019 at 07:04):
Would it be hard to write such a tactic?
Let's do it. :-)
#### Scott Morrison (Apr 09 2019 at 07:04):
First of all, however, note the following:
#### Scott Morrison (Apr 09 2019 at 07:05):
the category theory library gives you an easy way to make these simp lemmas work with the simplifier directly! Observe the following:
#### Scott Morrison (Apr 09 2019 at 07:05):
@[simp] lemma map_mul {R S : Mon} (f : R ⟶ S) (x y : R) : f (x * y) = f x * f y :=
by rw is_monoid_hom.map_mul f
example {R S : Mon} (f : R ⟶ S) (x y : R) : f (x * y) = f x * f y :=
by simp --- Lo!
#### Scott Morrison (Apr 09 2019 at 07:06):
What's going on here? In (x y : R), R is being coerced automatically to its underlying type.
#### Scott Morrison (Apr 09 2019 at 07:06):
In f (x * y), the bundled ring morphism f is being coerced to its underlying function.
#### Scott Morrison (Apr 09 2019 at 07:06):
However, because the coercion is still sitting there, simp has something to hang on to now!
#### Johan Commelin (Apr 09 2019 at 07:07):
Yep. Bundled homs make simp work
#### Johan Commelin (Apr 09 2019 at 07:07):
With linear_map and alg_hom this has been quite pleasant.
#### Johan Commelin (Apr 09 2019 at 07:07):
The problem is that it seems to be non-trivial to make a bundled ring_hom also act as an add_group_hom and as a monoid_hom etc...
#### Scott Morrison (Apr 09 2019 at 07:08):
But really what I hope might happen is that everyone will stop talking about explicit functions, decorated is is_X_hom typeclasses, and just write R ⟶ S (with the \hom arrow)... :-)
#### Johan Commelin (Apr 09 2019 at 07:08):
Chris tried unification hints. That didn't work. I don't know if plain coercions would work...
#### Johan Commelin (Apr 09 2019 at 07:09):
Bundling and mixing are fighting with each other...
#### Scott Morrison (Apr 09 2019 at 07:09):
Can we define coercions along forget.obj and forget.map, for all the different forgetful functors?
#### Scott Morrison (Apr 09 2019 at 07:11):
(if they work, they could all be generated by @[derive] at the moment that each forget is defined)
#### Scott Morrison (Apr 09 2019 at 07:11):
Anyway, that's something to think about later.
#### Scott Morrison (Apr 09 2019 at 07:11):
How would we write the hom tactic?
#### Johan Commelin (Apr 09 2019 at 07:11):
Can we define coercions along forget.obj and forget.map, for all the different forgetful functors?
I've been meaning to experiment with that. But haven't had the time to do so yet.
#### Patrick Massot (Apr 09 2019 at 07:12):
I think you can take inspiration from https://github.com/leanprover-community/mathlib/commit/5fe470bb216e7b3fb6639d3b964c4eee36b88713
#### Patrick Massot (Apr 09 2019 at 07:12):
Which is a stuff pushing tactic
#### Johan Commelin (Apr 09 2019 at 07:15):
@Scott, how hard is it to gather all symbols from the context that have an is_X_hom instance?
#### Scott Morrison (Apr 09 2019 at 07:15):
do you mean local hypotheses that have an is_X_hom instance?
#### Scott Morrison (Apr 09 2019 at 07:16):
or just subexpressions of the goal?
#### Johan Commelin (Apr 09 2019 at 07:16):
I guess we only care about values of X in (add_) {semigroup, monoid, group}.
either is doable
#### Johan Commelin (Apr 09 2019 at 07:16):
The latter. The instance might not be a local hyp.
#### Johan Commelin (Apr 09 2019 at 07:16):
Maybe we need to infer it.
#### Scott Morrison (Apr 09 2019 at 07:17):
No -- the instance of course we'll find by inference. I was asking about the actual function.
#### Scott Morrison (Apr 09 2019 at 07:17):
By the way, the following works:
example (R S : CommRing) (f : R ⟶ S) : is_monoid_hom (f : R → S) := by apply_instance
#### Johan Commelin (Apr 09 2019 at 07:18):
After we have that list, we can start trying to erw (<-) along is_X_hom.map_Y the_symbol for Y in {zero, one, add, mul, pow, smul, gpow, sub, gsmul, gpow, sum, prod}.
Yes.
#### Scott Morrison (Apr 09 2019 at 07:19):
That example is an interesting combination of bundling and mixing getting along!
#### Scott Morrison (Apr 09 2019 at 07:19):
@Johan Commelin, do you want to try out the new VS Code live share mode and write this tactic? I have ~40 minutes now.
#### Johan Commelin (Apr 09 2019 at 07:20):
Cool. Let me grab a bigger screen.
#### Kevin Buzzard (Apr 09 2019 at 08:02):
How is this done in Coq?
#### Kevin Buzzard (Apr 09 2019 at 08:02):
I am spending the next few days with Gonthier but I do not understand the issues in this thread well enough to be able to ask him :-(
#### Kevin Buzzard (Apr 09 2019 at 08:03):
If people want to teach me, that would work. We could start with "there's no fixed symbol for it to catch on". I am already lost. I don't know how simp works and I don't know what a symbol is.
#### Scott Morrison (Apr 09 2019 at 08:59):
Johan, the code we wrote is on the hom-tactic branch. That was fun. :-) We need to write lots more porcelain, but the minimal plumbing is running!
#### Keeley Hoek (Apr 09 2019 at 10:54):
example (R S : CommRing) (f : R ⟶ S) : is_monoid_hom (f : R → S) := by apply_instance
(For my information) why does everyone do this instead of ... := infer_instance?
#### Johan Commelin (Apr 09 2019 at 10:56):
I've had cases where by apply_instance worked and infer_instance didn't. Don't remember the details though...
#### Mario Carneiro (Apr 09 2019 at 10:56):
I think by apply_instance predates the other one
#### Mario Carneiro (Apr 09 2019 at 10:57):
also infer_instance appears literally in the result term while by apply_instance only does instance search, just like inference for [] variables
#### Mario Carneiro (Apr 09 2019 at 10:58):
I think infer_instance is best used for #check and example things
#### Keeley Hoek (Apr 09 2019 at 10:59):
Cool, good to know
#### Sebastian Ullrich (Apr 09 2019 at 15:01):
I introduced infer_instance just so that I could use it before the tactic framework is defined
#### Kevin Buzzard (Apr 12 2019 at 19:28):
Here's a concrete example of what Mario is saying. I wanted to figure out where the category structure on opens X was coming from.
import category_theory.instances.Top.opens
open category_theory
open topological_space
variables (X : Type*) [topological_space X]
def XYZ : category (opens X) := by apply_instance
def ABC : category (opens X) := infer_instance
#print ABC -- useless
#print XYZ -- what I wanted to know i.e. category_theory.preorder_category
#check category_theory.preorder_category -- etc etc
#### Patrick Massot (Apr 12 2019 at 21:04):
Kevin, you do know the most efficient way would have been #check (by apply_instance : category \$ opens X) instance of this def/print dance right?
#### Kevin Buzzard (Apr 12 2019 at 23:02):
ha ha yes, I think Kenny showed me this trick once; I keep forgetting it.
Last updated: May 14 2021 at 22:15 UTC
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2021-05-14 23:22:39
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https://ncatlab.org/nlab/show/equivariant+bordism+homology+theory
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# nLab equivariant bordism homology theory
Contents
### Context
#### Cobordism theory
Concepts of cobordism theory
# Contents
## Idea
The refinement of bordism homology theory to equivariant stable homotopy theory.
flavors of cobordism homology/cohomology theories and representing Thom spectra
bordism theory$\,M B$ (B-bordism):
## References
Original discussion of equivariant cobordism classes of manifolds:
Original discussion of equivariant Thom spectra:
Original discussion of the relation between equivariant bordism classes of manifolds and equivariant Thom spectra:
Survey:
Discussion in global equivariant homotopy theory is in
Last revised on November 23, 2020 at 02:10:35. See the history of this page for a list of all contributions to it.
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2020-12-01 18:10:35
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http://www.mfcs.sk/mfcs2000/abstracts/AccAbs37.html
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Abstract of Paper
Hardness Results and Efficient Approximations for Radiocoloring in Planar Graphs
by D.A. Fotakis, S.E. Nikoletseas, V.G. Papadopoulou and P.G. Spirakis
Abstract:
The communication in wireless systems, such as radio networks or ad-hoc
mobile networks, is accomplished through exploitation of a rather limited
range of the frequency spectrum. One of the mechanisms utilized in this
direction is to reuse frequencies where this does not result to unacceptable
levels of signal interference. In graph theoretic terms, such issues are
usually modeled by the interference graph $G(V, E)$, where the vertex set
$V$ corresponds to the set of transmitters, and variations of coloring of
the vertices of $G$ have been proposed in order to represent distance
interference constraints. In this paper we study the {\em radiocoloring
problem for planar graphs $G$}. A proper radiocoloring of the planar graph
$G$ with $\lambda$ colors is a function $\Phi: V \rightarrow \rm{I}\!\!\!\rm{N}$ such that $|\Phi_{u}-\Phi_{v}| \geq 2$, when $u, v$ are
neighbors in $G$ and $|\Phi_{u}-\Phi_{v}|\geq 1$ when the minimum distance
of $u, v$ in $G$ is 2. Also, only $\lambda$ integers are used. The least
possible $\lambda$ is called the radiochromatic number of $G$, denoted as
$X_{r}(G)$.
Notice that $X_{r}(G)$ is also the vertex chromatic number of the {\em
square graph} of $G$, $G^2$, which has the same vertex set
$V$ as $G$ and an edge set $E': \{ u, v \} \in E'$ iff their
{\em minimum} distance is at most $2$ in $G$. Let $\Delta$ be the
maximum degree of the vertices of $G$.
In this paper we are interested in the problem of {\em minimizing $\lambda$
\item We provide an $O(n\Delta)$ time algorithm ($|V|=n$) which obtains a
radiocoloring of a planar graph $G$ that {\em approximates
$X_{r}$ within a ratio which tends to 2} as the maximum
vertex degree $\Delta$ of $G$ increases.
graph} $G$ with $\lambda$ colors, in the case $\lambda \geq 4 \Delta+50$.
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2017-10-18 20:13:54
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https://forum.azimuthproject.org/discussion/153/calendar-mottos
|
#### Howdy, Stranger!
It looks like you're new here. If you want to get involved, click one of these buttons!
Options
# calendar mottos
In the book "numerical recipes" the authors explain that linear congruence generators generate numbers that concentrate on hyperplanes - which looks very much not random to the human eye.
Quote:
One of us recalls as a graduate student producing a “random” plot with only 11 planes and being told by his computer center’s programming consultant that he had misused the random number generator: “We guarantee that each number is random individually, but we don’t guarantee that more than one of them is random.”
This is so cool that I would like to include it on the page somehow, maybe as a calendar motto at the top of the page. What do you think: Would this be nonprofessional or interesting/entertaining?
• Options
1.
By the way Tim, as you're mentioning NR a lot: I assume you're aware that the view of NR quite mixed. It's a good introduction to many of the techniques and has a good overview, but it's also taking an approach to writing numerical code from the early 80s, with often very poor behaviour in corner cases and not as much care over avoid loss-of-precision as would be taken in a real numerical package.
Comment Source:By the way Tim, as you're mentioning NR a lot: I assume you're aware that the view of NR quite mixed. It's a good introduction to many of the techniques and has a good overview, but it's also taking an approach to writing numerical code from the early 80s, with often very poor behaviour in corner cases and not as much care over avoid loss-of-precision as would be taken in a real numerical package.
• Options
2.
I assume you're aware that the view of NR quite mixed.
Semi-yes. I worked with NR-code on my master thesis, although I used different implementations for random number generators, stochastic and partial DE. I never used anything else, so NR came to my mind as a possible start to get something coded in Java/Scala. Is the GSL better?
Comment Source:<blockquote> <p> I assume you're aware that the view of NR quite mixed. </p> </blockquote> Semi-yes. I worked with NR-code on my master thesis, although I used different implementations for random number generators, stochastic and partial DE. I never used anything else, so NR came to my mind as a possible start to get something coded in Java/Scala. Is the GSL better?
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2019-05-21 00:25:32
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https://www.physicsforums.com/threads/midpoint-formula.1043661/
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# Midpoint Formula
• MHB
55. Use the Midpoint Formula three times to find the three points that divide the line segment joining (x_1, y_1) and (x_2, y_2) into four equal parts.
56. Use the result of Exercise 55 to find the points that divide each line segment joining the given points into four equal parts.
(a) (x_1, y_1) = (1, −2)
(x_2, y_2) = (4, −1)
(b) (x_1, y_1) = (−2, −3)
(x_2, y_2) = (0, 0)
Looking for hints to solve 55 and 56.
jonah1
Beer soaked ramblings follow.
55. Use the Midpoint Formula three times to find the three points that divide the line segment joining (x_1, y_1) and (x_2, y_2) into four equal parts.
56. Use the result of Exercise 55 to find the points that divide each line segment joining the given points into four equal parts.
(a) (x_1, y_1) = (1, −2)
(x_2, y_2) = (4, −1)
(b) (x_1, y_1) = (−2, −3)
(x_2, y_2) = (0, 0)
Looking for hints to solve 55 and 56.
Interpolate Country Boy's explanation at
HOI
The three points needed are the midpoint, p, of the given interval and the midpoint of the two intervals having one of the original endpoint and p as endpoints and the other original endpoint and p as endpoints.
For example, if an interval has endpoints (0, 0) and (2, 2), of length $\sqrt{2}$, has midpoint (1, 1). The midpoint of the interval from (0, 0) to (1, 1) is (1/2, 1/2) and the mid point of (1, 1) to (2, 2) is (3/2, 3/2). The four intervals from (0, 0) to (1/2, 1/2), from (1/2, 1/2) to (1, 1), from (1, 1) To (3/2, 3/2), and from (3/2, 3/2) all have length $\frac{\sqrt{2}}{2}$.
The three points needed are the midpoint, p, of the given interval and the midpoint of the two intervals having one of the original endpoint and p as endpoints and the other original endpoint and p as endpoints.
For example, if an interval has endpoints (0, 0) and (2, 2), of length $\sqrt{2}$, has midpoint (1, 1). The midpoint of the interval from (0, 0) to (1, 1) is (1/2, 1/2) and the mid point of (1, 1) to (2, 2) is (3/2, 3/2). The four intervals from (0, 0) to (1/2, 1/2), from (1/2, 1/2) to (1, 1), from (1, 1) To (3/2, 3/2), and from (3/2, 3/2) all have length $\frac{\sqrt{2}}{2}$.
Interesting. By the length sqrt{2}, you mean the distance between two given points. This is found using the distance formula for points. True?
HOI
Yes, although I miswrote. I was first thinking of (0, 0) to (1, 1) which does have length $\sqrt{2}$. But then I changed to (0, 0) to (2, 2) which is twice as long: $\sqrt{(2- 0)^2+ (2- 0)^2}= \sqrt{4+ 4}= \sqrt{4(2)}= 2\sqrt{2}$.
Yes, although I miswrote. I was first thinking of (0, 0) to (1, 1) which does have length $\sqrt{2}$. But then I changed to (0, 0) to (2, 2) which is twice as long: $\sqrt{(2- 0)^2+ (2- 0)^2}= \sqrt{4+ 4}= \sqrt{4(2)}= 2\sqrt{2}$.
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2023-03-24 06:24:44
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https://www.numerade.com/questions/the-region-bounded-by-the-given-curves-is-rotated-about-the-specified-axis-find-the-volume-of-the--5/
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💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!
# The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.$x^2 + (y - 1)^2 = 1$ ; about the y-axis
## $\frac{4}{3} \pi$
#### Topics
Applications of Integration
### Discussion
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Lectures
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### Video Transcript
okay, we know it's specified in the problem. The radius is X and the height is two time squirt of one minus X scored. Which means we're gonna be putting into the general formula, which is two pi times the integral from 01 off the height times the radius D of acts 01 are two bounds. Therefore, we end up with V is gonna be four pi times integral from 01 ax time scored of one less ex squirt. Where did the two go? You ask? Well, we remember the height was two time squared of our mess. X squared. I simply pulled the two out on the outside. It's a constant multiplied it by the original two pi toe end up with four pi on the outside Just a simple five days. Now we know in this context we can actually use U substitution, which you learned in a previous chapter. If you is one minus X squared than taking d'you, we end up with d'you. The derivative is negative to axe de axe. So what this means is that negative 1/2 to you is ex dx other words gonna be dividing four by two which means we end up with and remember, it's actually dividing for by negative, too, because it was negative. Two ex Texas do you So negative two pi times the integral from 10 There flips now because of the nature of the fact that we've taken u substitution. We've got negative value. Do the power rule In order to integrate increased exports by one divide by the new exponents, we end up with four pi divided by three
#### Topics
Applications of Integration
Lectures
Join Bootcamp
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2021-10-20 20:00:26
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https://www.mataf.net/en/currency/converter-CAD-ISR
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The CAD/ISR converter is provided without any warranty. Prices might differ from those given by financial institutions as banks, brokers or money transfer companies.
Last update:
\$
ISR
7 8 9 4 5 6 1 2 3 0 . convert
## Is it the right time to change your currencies?
The best day to change Canadian dollars in Isracoin was the . At that time the currency had reached its highest value.
100 Canadian dollars = 1100 Isracoin
The worst day to change Canadian dollars in Isracoin was the . The exchange rate had fallen to its lowest value.
100 Canadian dollars = 1100 Isracoin
## Historical Canadian dollar / Isracoin
History of daily rates CAD /ISR since Sunday, 15 November 2015.
The maximum was reached on
• 1 Canadian dollar = 16968.017146887 Isracoin
the minimum on
• 1 Canadian dollar = 11022.949041089 Isracoin
12 388.7945
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12 467.9017
12 535.3996
12 572.0015
12 572.0720
12 561.5628
12 561.8307
12 459.1132
12 369.1893
12 357.7645
12 361.1553
12 345.5482
12 349.3409
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12 382.7015
12 366.1981
12 323.9469
12 331.2875
12 324.0318
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12 206.9441
12 186.4178
12 234.6726
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12 201.0957
convert into Result -
## Change table
1 Canadian dollar in Isracoin = 12 806.38
2 Canadian dollars in Isracoin = 25 612.77
3 Canadian dollars in Isracoin = 38 419.15
4 Canadian dollars in Isracoin = 51 225.54
5 Canadian dollars in Isracoin = 64 031.92
6 Canadian dollars in Isracoin = 76 838.31
7 Canadian dollars in Isracoin = 89 644.69
8 Canadian dollars in Isracoin = 102 451.08
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10 Canadian dollars in Isracoin = 128 063.85
15 Canadian dollars in Isracoin = 192 095.77
20 Canadian dollars in Isracoin = 256 127.70
25 Canadian dollars in Isracoin = 320 159.62
30 Canadian dollars in Isracoin = 384 191.55
40 Canadian dollars in Isracoin = 512 255.40
50 Canadian dollars in Isracoin = 640 319.25
60 Canadian dollars in Isracoin = 768 383.09
70 Canadian dollars in Isracoin = 896 446.94
80 Canadian dollars in Isracoin = 1 024 510.79
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100 Canadian dollars in Isracoin = 1 280 638.49
150 Canadian dollars in Isracoin = 1 920 957.74
200 Canadian dollars in Isracoin = 2 561 276.98
500 Canadian dollars in Isracoin = 6 403 192.45
1 000 Canadian dollar in Isracoin = 12 806 384.90
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2019-07-16 07:14:15
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http://www.wikidoc.org/index.php/Bernoulli_distribution
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# Bernoulli distribution
Parameters Probability mass function Cumulative distribution function $p>0\,$ (real) $k=\{0,1\}\,$ $ \begin{matrix} q & \mbox{for }k=0 \\p~~ & \mbox{for }k=1 \end{matrix}$ $ \begin{matrix} 0 & \mbox{for }k<0 \\q & \mbox{for }0\leq k<1\\1 & \mbox{for }k\geq 1 \end{matrix}$ $p\,$ N/A $\begin{matrix} 0 & \mbox{if } q > p\\ 0, 1 & \mbox{if } q=p\\ 1 & \mbox{if } q < p \end{matrix}$ $pq\,$ $\frac{q-p}{\sqrt{pq {{{kurtosis}}} {{{entropy}}} {{{mgf}}} {{{char}}}$|
kurtosis =$\frac{6p^2-6p+1}{p(1-p)}$|
entropy =$-q\ln(q)-p\ln(p)\,$|
mgf =$q+pe^t\,$|
char =$q+pe^{it}\,$|
}}
In probability theory and statistics, the Bernoulli distribution, named after Swiss scientist Jakob Bernoulli, is a discrete probability distribution, which takes value 1 with success probability $p$ and value 0 with failure probability $q=1-p$. So if X is a random variable with this distribution, we have:
$\Pr(X=1) = 1 - \Pr(X=0) = 1 - q = p.\!$
The probability mass function f of this distribution is
$f(k;p) = \left\{\begin{matrix} p & \mbox {if }k=1, \\ 1-p & \mbox {if }k=0, \\ 0 & \mbox {otherwise.}\end{matrix}\right.$
The expected value of a Bernoulli random variable X is $E\left(X\right)=p$, and its variance is
$\textrm{var}\left(X\right)=p\left(1-p\right).\,$
The kurtosis goes to infinity for high and low values of p, but for $p=1/2$ the Bernoulli distribution has a lower kurtosis than any other probability distribution, namely -2.
The Bernoulli distribution is a member of the exponential family.
## Related distributions
• If $X_1,\dots,X_n$ are independent, identically distributed random variables, all Bernoulli distributed with success probability p, then $Y = \sum_{k=1}^n X_k \sim \mathrm{Binomial}(n,p)$ (binomial distribution).
• The Categorical distribution is the generalization of the Bernoulli distribution for variables with any constant number of discrete values.
• The Beta distribution is the conjugate prior of the Bernoulli distribution.
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2013-05-22 06:13:17
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http://cerco.cs.unibo.it/changeset/3651
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# Changeset 3651
Ignore:
Timestamp:
Mar 14, 2017, 11:14:29 AM (12 months ago)
Message:
more work on conclusions and related work, bibliography growing to gigantic proportions
Location:
Papers/jar-cerco-2017
Files:
2 edited
Unmodified
Added
Removed
• ## Papers/jar-cerco-2017/cerco.bib
r3650 address = {New York, NY, USA}, keywords = {CompCert, Compositional compiler verification, separate compilation}, } @Inbook{Atkey2010, author="Atkey, Robert", editor="Gordon, Andrew D.", title="Amortised Resource Analysis with Separation Logic", bookTitle="Programming Languages and Systems: 19th European Symposium on Programming, ESOP 2010, Held as Part of the Joint European Conferences on Theory and Practice of Software, ETAPS 2010, Paphos, Cyprus, March 20-28, 2010. Proceedings", year="2010", publisher="Springer Berlin Heidelberg", address="Berlin, Heidelberg", pages="85--103", isbn="978-3-642-11957-6", doi="10.1007/978-3-642-11957-6_6", url="http://dx.doi.org/10.1007/978-3-642-11957-6_6" } @Inbook{Aspinall2010, author="Aspinall, David and Atkey, Robert and MacKenzie, Kenneth and Sannella, Donald", editor="Wirsing, Martin and Hofmann, Martin and Rauschmayer, Axel", title="Symbolic and Analytic Techniques for Resource Analysis of Java Bytecode", bookTitle="Trustworthly Global Computing: 5th International Symposium, TGC 2010, Munich, Germany, February 24-26, 2010, Revised Selected Papers", year="2010", publisher="Springer Berlin Heidelberg", address="Berlin, Heidelberg", pages="1--22", isbn="978-3-642-15640-3", doi="10.1007/978-3-642-15640-3_1", url="http://dx.doi.org/10.1007/978-3-642-15640-3_1" } @article{Salvucci201627, title = "Memory Consumption Analysis for a Functional and Imperative Language ", journal = "Electronic Notes in Theoretical Computer Science ", volume = "330", number = "", pages = "27 - 46", year = "2016", note = "\{RAC\} 2016 - Resource Aware Computing ", issn = "1571-0661", doi = "http://dx.doi.org/10.1016/j.entcs.2016.12.013", url = "http://www.sciencedirect.com/science/article/pii/S1571066116301207", author = "Jérémie Salvucci and Emmanuel Chailloux", keywords = "ML", keywords = "regions", keywords = "static analysis", keywords = "memory analysis ", abstract = "Abstract The omnipresence of resource-constrained embedded systems makes them critical components. Programmers have to provide strong guarantees about their runtime behavior to make them reliable. Among these, giving an upper bound of live memory at runtime is mandatory to prevent heap overflows from happening. The paper proposes a semi-automatic technique to infer the space complexity of ML-like programs with explicit region management. It aims at combining existing formalisms to obtain the space complexity of imperative and purely functional programs in a consistent framework. " } @Inbook{Hoffmann2015, author="Hoffmann, Jan and Shao, Zhong", editor="Vitek, Jan", title="Automatic Static Cost Analysis for Parallel Programs", bookTitle="Programming Languages and Systems: 24th European Symposium on Programming, ESOP 2015, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2015, London, UK, April 11-18, 2015, Proceedings", year="2015", publisher="Springer Berlin Heidelberg", address="Berlin, Heidelberg", pages="132--157", isbn="978-3-662-46669-8", doi="10.1007/978-3-662-46669-8_6", url="http://dx.doi.org/10.1007/978-3-662-46669-8_6" } @inproceedings{HAH12, author = {Jan Hoffmann and Klaus Aehlig and Martin Hofmann}, title = {{Resource Aware ML}}, booktitle = {{24rd International Conference on Computer Aided Verification (CAV'12)}}, publisher = {Springer}, series = {Lecture Notes in Computer Science}, volume = {7358}, pages = {781-786}, year = {2012} } address = {New York, NY, USA}, keywords = {compiler verification, interactive proof assistants}, } } @Article{Albert2011, author="Albert, Elvira and Arenas, Puri and Genaim, Samir and Puebla, Germ{\'a}n", title="Closed-Form Upper Bounds in Static Cost Analysis", journal="Journal of Automated Reasoning", year="2011", volume="46", number="2", pages="161--203", abstract="The classical approach to automatic cost analysis consists of two phases. Given a program and some measure of cost, the analysis first produces cost relations (CRs), i.e., recursive equations which capture the cost of the program in terms of the size of its input data. Second, CRs are converted into closed-form, i.e., without recurrences. Whereas the first phase has received considerable attention, with a number of cost analyses available for a variety of programming languages, the second phase has been comparatively less studied. This article presents, to our knowledge, the first practical framework for the generation of closed-form upper bounds for CRs which (1) is fully automatic, (2) can handle the distinctive features of CRs, originating from cost analysis of realistic programming languages, (3) is not restricted to simple complexity classes, and (4) produces reasonably accurate solutions. A key idea in our approach is to view CRs as programs, which allows applying semantic-based static analyses and transformations to bound them, namely our method is based on the inference of ranking functions and loop invariants and on the use of partial evaluation.", issn="1573-0670", doi="10.1007/s10817-010-9174-1", url="http://dx.doi.org/10.1007/s10817-010-9174-1" } @article{ALBERT200767, title = "Experiments in Cost Analysis of Java Bytecode", journal = "Electronic Notes in Theoretical Computer Science", volume = "190", number = "1", pages = "67 - 83", year = "2007", note = "", issn = "1571-0661", doi = "http://dx.doi.org/10.1016/j.entcs.2007.02.061", url = "http://www.sciencedirect.com/science/article/pii/S1571066107005294", author = "E. Albert and P. Arenas and S. Genaim and G. Puebla and D. Zanardini", keywords = "Cost analysis", keywords = "Java bytecode", keywords = "cost relations", keywords = "recurrence equations", keywords = "complexity" } @article{Wang:2014:CVM:2714064.2660201, doi = {10.1007/978-3-642-00590-9_14} } @Inbook{Chalin2006, author="Chalin, Patrice and Kiniry, Joseph R. and Leavens, Gary T. and Poll, Erik", editor="de Boer, Frank S. and Bonsangue, Marcello M. and Graf, Susanne and de Roever, Willem-Paul", title="Beyond Assertions: Advanced Specification and Verification with JML and ESC/Java2", bookTitle="Formal Methods for Components and Objects: 4th International Symposium, FMCO 2005, Amsterdam, The Netherlands, November 1-4, 2005, Revised Lectures", year="2006", publisher="Springer Berlin Heidelberg", address="Berlin, Heidelberg", pages="342--363", isbn="978-3-540-36750-5", doi="10.1007/11804192_16", url="http://dx.doi.org/10.1007/11804192_16" } @book{KeYBook2016, title = {Deductive Software Verification - The KeY Book - From Theory to Practice}, editor = {Wolfgang Ahrendt and Bernhard Beckert and Richard Bubel and Reiner H{\"{a}}hnle and Peter H. Schmitt and Mattias Ulbrich}, url = {http://dx.doi.org/10.1007/978-3-319-49812-6}, doi = {10.1007/978-3-319-49812-6}, isbn = {978-3-319-49811-9}, year = {2016}, date = {2016-12-16}, volume = {10001}, publisher = {Springer}, series = {Lecture Notes in Computer Science}, keywords = {}, pubstate = {published}, tppubtype = {book} } @Inbook{Montenegro2010a, author="Montenegro, Manuel and Pe{\~{n}}a, Ricardo and Segura, Clara", editor="van Eekelen, Marko and Shkaravska, Olha", title="A Space Consumption Analysis by Abstract Interpretation", bookTitle="Foundational and Practical Aspects of Resource Analysis: First International Workshop, FOPARA 2009, Eindhoven, The Netherlands, November 6, 2009, Revised Selected Papers", year="2010", publisher="Springer Berlin Heidelberg", address="Berlin, Heidelberg", pages="34--50", isbn="978-3-642-15331-0", doi="10.1007/978-3-642-15331-0_3", url="http://dx.doi.org/10.1007/978-3-642-15331-0_3" } @Inbook{Montenegro2010, author="Montenegro, Manuel and Pe{\~{n}}a, Ricardo and Segura, Clara", editor="Escobar, Santiago", title="A Simple Region Inference Algorithm for a First-Order Functional Language", bookTitle="Functional and Constraint Logic Programming: 18th International Workshop, WFLP 2009, Brasilia, Brazil, June 28, 2009, Revised Selected Papers", year="2010", publisher="Springer Berlin Heidelberg", address="Berlin, Heidelberg", pages="145--161", isbn="978-3-642-11999-6", doi="10.1007/978-3-642-11999-6_10", url="http://dx.doi.org/10.1007/978-3-642-11999-6_10" } @Inbook{deDios2011, author="de Dios, Javier and Pe{\~{n}}a, Ricardo", editor="Butler, Michael and Schulte, Wolfram", title="Certification of Safe Polynomial Memory Bounds", bookTitle="FM 2011: Formal Methods: 17th International Symposium on Formal Methods, Limerick, Ireland, June 20-24, 2011. Proceedings", year="2011", publisher="Springer Berlin Heidelberg", address="Berlin, Heidelberg", pages="184--199", isbn="978-3-642-21437-0", doi="10.1007/978-3-642-21437-0_16", url="http://dx.doi.org/10.1007/978-3-642-21437-0_16" } @Inbook{Albert2012, author="Albert, Elvira and Bubel, Richard and Genaim, Samir and H{\"a}hnle, Reiner and Rom{\'a}n-D{\'i}ez, Guillermo", editor="de Lara, Juan and Zisman, Andrea", title="Verified Resource Guarantees for Heap Manipulating Programs", bookTitle="Fundamental Approaches to Software Engineering: 15th International Conference, FASE 2012, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2012, Tallinn, Estonia, March 24 - April 1, 2012. Proceedings", year="2012", publisher="Springer Berlin Heidelberg", address="Berlin, Heidelberg", pages="130--145", isbn="978-3-642-28872-2", doi="10.1007/978-3-642-28872-2_10", url="http://dx.doi.org/10.1007/978-3-642-28872-2_10" } @Inbook{Nipkow2015, author="Nipkow, Tobias", editor="Urban, Christian and Zhang, Xingyuan", title="Amortized Complexity Verified", bookTitle="Interactive Theorem Proving: 6th International Conference, ITP 2015, Nanjing, China, August 24-27, 2015, Proceedings", year="2015", publisher="Springer International Publishing", address="Cham", pages="310--324", isbn="978-3-319-22102-1", doi="10.1007/978-3-319-22102-1_21", url="http://dx.doi.org/10.1007/978-3-319-22102-1_21" } @article{Hoffmann:2012:MAR:2362389.2362393, author = {Hoffmann, Jan and Aehlig, Klaus and Hofmann, Martin}, title = {Multivariate Amortized Resource Analysis}, journal = {ACM Trans. Program. Lang. Syst.}, issue_date = {October 2012}, volume = {34}, number = {3}, month = nov, year = {2012}, issn = {0164-0925}, pages = {14:1--14:62}, articleno = {14}, numpages = {62}, url = {http://doi.acm.org/10.1145/2362389.2362393}, doi = {10.1145/2362389.2362393}, acmid = {2362393}, publisher = {ACM}, address = {New York, NY, USA}, keywords = {Amortized analysis, functional programming, quantitative analysis, resource consumption, static analysis}, } @Inbook{Chargueraud2015, author="Chargu{\'e}raud, Arthur and Pottier, Fran{\c{c}}ois", editor="Urban, Christian and Zhang, Xingyuan", title="Machine-Checked Verification of the Correctness and Amortized Complexity of an Efficient Union-Find Implementation", bookTitle="Interactive Theorem Proving: 6th International Conference, ITP 2015, Nanjing, China, August 24-27, 2015, Proceedings", year="2015", publisher="Springer International Publishing", address="Cham", pages="137--153", isbn="978-3-319-22102-1", doi="10.1007/978-3-319-22102-1_9", url="http://dx.doi.org/10.1007/978-3-319-22102-1_9" } @Inbook{Teller2002, author="Teller, David and Zimmer, Pascal and Hirschkoff, Daniel", editor="Brim, Lubo{\v{s}} and K{\v{r}}et{\'i}nsk{\'y}, Mojm{\'i}r and Ku{\v{c}}era, Anton{\'i}n and Jan{\v{c}}ar, Petr", title="Using Ambients to Control Resources* ", bookTitle="CONCUR 2002 --- Concurrency Theory: 13th International Conference Brno, Czech Republic, August 20--23, 2002 Proceedings", year="2002", publisher="Springer Berlin Heidelberg", address="Berlin, Heidelberg", pages="288--303", isbn="978-3-540-45694-0", doi="10.1007/3-540-45694-5_20", url="http://dx.doi.org/10.1007/3-540-45694-5_20" } @article{DBLP:journals/jfp/0002S15, author = {Jan Hoffmann and Zhong Shao}, title = {Type-based amortized resource analysis with integers and arrays}, journal = {J. Funct. Program.}, volume = {25}, year = {2015}, url = {http://dx.doi.org/10.1017/S0956796815000192}, doi = {10.1017/S0956796815000192}, timestamp = {Tue, 26 Jan 2016 16:08:20 +0100}, biburl = {http://dblp.uni-trier.de/rec/bib/journals/jfp/0002S15}, bibsource = {dblp computer science bibliography, http://dblp.org} } @Inbook{Barbanera2003, author="Barbanera, Franco and Bugliesi, Michele and Dezani-Ciancaglini, Mariangiola and Sassone, Vladimiro", editor="Saraswat, Vijay A.", title="A Calculus of Bounded Capacities", bookTitle="Advances in Computing Science -- ASIAN 2003. Progamming Languages and Distributed Computation Programming Languages and Distributed Computation: 8th Asian Computing Science Conference, Mumbai, India, December 10-12, 2003. Proceedings", year="2003", publisher="Springer Berlin Heidelberg", address="Berlin, Heidelberg", pages="205--223", isbn="978-3-540-40965-6", doi="10.1007/978-3-540-40965-6_14", url="http://dx.doi.org/10.1007/978-3-540-40965-6_14" } @article{Danielsson:2008:LST:1328897.1328457, author = {Danielsson, Nils Anders}, title = {Lightweight Semiformal Time Complexity Analysis for Purely Functional Data Structures}, journal = {SIGPLAN Not.}, issue_date = {January 2008}, volume = {43}, number = {1}, month = jan, year = {2008}, issn = {0362-1340}, pages = {133--144}, numpages = {12}, url = {http://doi.acm.org/10.1145/1328897.1328457}, doi = {10.1145/1328897.1328457}, acmid = {1328457}, publisher = {ACM}, address = {New York, NY, USA}, keywords = {amortised time complexity, dependent types, lazy evaluation, purely functional data structures}, } @inproceedings{Crary:2000:RBC:325694.325716, author = {Crary, Karl and Weirich, Stephnie}, title = {Resource Bound Certification}, booktitle = {Proceedings of the 27th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages}, series = {POPL '00}, year = {2000}, isbn = {1-58113-125-9}, location = {Boston, MA, USA}, pages = {184--198}, numpages = {15}, url = {http://doi.acm.org/10.1145/325694.325716}, doi = {10.1145/325694.325716}, acmid = {325716}, publisher = {ACM}, address = {New York, NY, USA}, } @Inbook{McCarthy2016, author="McCarthy, Jay and Fetscher, Burke and New, Max and Feltey, Daniel and Findler, Robert Bruce", editor="Kiselyov, Oleg and King, Andy", title="A Coq Library for Internal Verification of Running-Times", bookTitle="Functional and Logic Programming: 13th International Symposium, FLOPS 2016, Kochi, Japan, March 4-6, 2016, Proceedings", year="2016", publisher="Springer International Publishing", address="Cham", pages="144--162", isbn="978-3-319-29604-3", doi="10.1007/978-3-319-29604-3_10", url="http://dx.doi.org/10.1007/978-3-319-29604-3_10" } @inproceedings { chan-ngo-verifying-2017, title = {Verifying and Synthesizing Constant-Resource Implementations with Types}, author = {Van Chan Ngo and Mario Dehesa-Azuara and Matthew Fredrikson and Jan Hoffmann}, booktitle = {Proceedings of the {IEEE} Symposium on Security and Privacy}, year = {2017}, note = {to appear} } @Inbook{Brady2006, author="Brady, Edwin and Hammond, Kevin", editor="Butterfield, Andrew and Grelck, Clemens and Huch, Frank", title="A Dependently Typed Framework for Static Analysis of Program Execution Costs", bookTitle="Implementation and Application of Functional Languages: 17th International Workshop, IFL 2005, Dublin, Ireland, September 19-21, 2005, Revised Selected Papers", year="2006", publisher="Springer Berlin Heidelberg", address="Berlin, Heidelberg", pages="74--90", isbn="978-3-540-69175-4", doi="10.1007/11964681_5", url="http://dx.doi.org/10.1007/11964681_5" } @inproceedings{Danner:2013:SCA:2428116.2428123, author = {Danner, Norman and Paykin, Jennifer and Royer, James S.}, title = {A Static Cost Analysis for a Higher-order Language}, booktitle = {Proceedings of the 7th Workshop on Programming Languages Meets Program Verification}, series = {PLPV '13}, year = {2013}, isbn = {978-1-4503-1860-0}, location = {Rome, Italy}, pages = {25--34}, numpages = {10}, url = {http://doi.acm.org/10.1145/2428116.2428123}, doi = {10.1145/2428116.2428123}, acmid = {2428123}, publisher = {ACM}, address = {New York, NY, USA}, keywords = {automated theorem proving, certified bounds, higher-order complexity}, } @inproceedings{Reynolds:2002:SLL:645683.664578, author = {Reynolds, John C.}, title = {Separation Logic: A Logic for Shared Mutable Data Structures}, booktitle = {Proceedings of the 17th Annual IEEE Symposium on Logic in Computer Science}, series = {LICS '02}, year = {2002}, isbn = {0-7695-1483-9}, pages = {55--74}, numpages = {20}, url = {http://dl.acm.org/citation.cfm?id=645683.664578}, acmid = {664578}, publisher = {IEEE Computer Society}, address = {Washington, DC, USA}, } @article{ASPINALL2007411,
• ## Papers/jar-cerco-2017/conclusions.tex
r3650 Abstract Interpretation may also be used to derive upper bounds for stack usage (e.g.~\cite{Regehr:2005:ESO:1113830.1113833}), and Abstract Interpretation tools for predicting stack space usage, for example the StackAnalyzer tool~\cite{stackanalyzer}, are also commercially available. \paragraph{Type- and program logic-based analyses} \paragraph{High-level approaches} Resource Aware ML (RAML) is a first-order statically typed functional programming language~\cite{Hoffmann11,HAH12Toplas,HoffHof10}. The RAML type system is able to derive polynomial resource bounds from first-order functional programs, using a variant of the potential method'. This approach was first pioneered by Hoffman and Jost for amortized, linear bounds on resource consumption~\cite{DBLP:conf/esop/HofmannJ06}, who focussed on predicting heap-space usage. Campbell also used a variant of this approach to predict stack space usage~\cite{bac-thesis08,bac-giveback08,bac-esop09}. Implicit Computational Complexity represents another related approach to bounding resource usage via types~\cite{DalLago2012}. Here, a fragment of Linear Logic, or some other substructural logic, is taken as the type-language of a programming language, with a provable guarantee that typeable terms normalise to a value in some bounded (e.g. polynomial in the size of the term) number of reduction steps. Note that complained to the concrete complexity measures dealt with in this work, bounding reduction steps is rather coarse-grained: one reduction step in the calculus may decompose into some non-constant number of primitive machine operations, when executed on physical hardware. Indeed, the focus of Implicit Computational Complexity is not to capture \emph{concrete} resource usage of programs, but to provide a type-theoretic interpretation of complexity classes. Several program logics for reasoning about a program's resource consumption have been investigated in the literature. Aspinall et al. developed a program logic for a subset of the Java Virtual Machine's instruction set, capable of establishing resource bounds on JVM programs~\cite{ASPINALL2007411}. The soundness of the logic was formally verified in Isabelle/HOL. Along similar lines, Carbonneaux et al. developed a quantitative Hoare logic' for reasoning about stack space usage of CompCert C programs~\cite{DBLP:conf/pldi/Carbonneaux0RS14}. The soundness of the logic was also formally verified, using the Coq proof assistant. This work will be further discussed below. \subsubsection{High-level approaches} \paragraph{Compiler-based cost-model lifting} Perhaps the most similar piece of work to our own is the recent work by Carbonneaux \emph{et al}~\cite{DBLP:conf/pldi/Carbonneaux0RS14}, who extended the CompCert verified C compiler in order to lift a concrete cost model for stack space usage back to the C source-level. Perhaps the most closely related piece of work to our own is the recent work by Carbonneaux \emph{et al}~\cite{DBLP:conf/pldi/Carbonneaux0RS14}, who extended the CompCert verified C compiler in order to lift a concrete cost model for stack space usage back to the C source-level. Like CerCo, this work was verified within a theorem prover---in this case Coq, rather than Matita. Though Carbonneaux \emph{et al} were working entirely independently of the project described herein\footnote{Personal communication with Carbonneaux and Hoffmann}, the two pieces of work share some remarkable similarities, with both projects developing an analogue of the structured trace' data structure described earlier in this paper in order to facilitate the verification of the lifting. Carbonneaux \emph{at al} lift their model from the assembly generated by the CompCert compiler; we must go further, to the level of machine code, in order to lift our timing cost model, necessitating the development of a verified assembler. Nevertheless, despite these differences, the two projects are clearly closely related. Another closely related piece of work to our own is COSTA~\cite{Albert2012}, a heap resource bound static analysis system for the Java programming language, building on previous work investigating cost models for Java bytecode programs~\cite{ALBERT200767}. COSTA is able to produce parametric heap resource bounds for a given Java program, which are made manifest as Java Modelling Language (JML)~\cite{Chalin2006} assertions, inserted into the Java source file. Coupled with the KeY symbolic reasoning/automated reasoning system~\cite{KeYBook2016}, these assertions may be automatically verified, obtaining trustworthy upper bounds on a Java program's dynamic heap space usage at runtime. Other work has explored the use of Computer Algebra Software, such as the Mathematica system, for deriving closed-form upper-bounds of a program's resource usage~\cite{Albert2011}. Previous works have researched various static analyses of time- and space-usage of programs written in a simple functional programming language with regions, called Safe~\cite{Montenegro2010,Montenegro2010a}. One such analysis was verified in Isabelle/HOL~\cite{deDios2011}. The use of types to bound, or program logics to reason about, a program's resource usage have also been extensively investigated in previous work. Resource Aware ML (RAML) is a first-order statically typed functional programming language~\cite{Hoffmann11,HAH12Toplas,HoffHof10,HAH12,DBLP:journals/jfp/0002S15}. The RAML type system is able to derive polynomial resource bounds from first-order functional programs, using a variant of the potential method'. This approach was first pioneered by Hoffman and Jost for amortized, linear bounds on resource consumption~\cite{DBLP:conf/esop/HofmannJ06}, who focussed on predicting heap-space usage. Campbell also used a variant of this approach to predict stack space usage~\cite{bac-thesis08,bac-giveback08,bac-esop09}, whilst Atkey extended the approach to imperative languages~\cite{Atkey2010}, using ideas from Separation Logic~\cite{Reynolds:2002:SLL:645683.664578}. Recent work has extended this approach to include more features of the core programming language~\cite{Hoffmann2015}, improving the resource analysis~\cite{Hoffmann:2012:MAR:2362389.2362393}, and to apply extensions of the technique to case studies, demonstrating that cryptographic code is free from side-channel attacks~\cite{chan-ngo-verifying-2017}. In related work, Salvucci and Chailloux~\cite{Salvucci201627} used a static type and effect system to bound runtime heap usage of an impure functional programming language, similar to ML. Domain-specific type systems and calculi of mobile processes, for bounding agents in a distributed setting, have also been investigated~\cite{Barbanera2003,Teller2002}. Brady and Hammond also investigated a dependent type system extended with features for bounding a program's dynamic resource usage~\cite{Brady2006}. Similar ideas had previously been investigated by Crary and Weirich~\cite{Crary:2000:RBC:325694.325716}. Libraries within dependently-typed languages, such as Coq~\cite{McCarthy2016} and Agda~\cite{Danielsson:2008:LST:1328897.1328457}, have also been written for reasoning about the complexity of algorithms written in the language. One-off machine-checked verifications of the algorithmic complexity of various data structure implementations, and operations over them, have also been investigated (e.g.~\cite{Chargueraud2015,Nipkow2015}). Implicit Computational Complexity represents another related approach to bounding resource usage via types~\cite{DalLago2012}. Here, a fragment of Linear Logic, or some other substructural logic, is taken as the language of types of some calculus, with a provable guarantee that typeable terms of the calculus normalise to a value in some bounded (e.g. polynomial in the size of the term) number of reduction steps. Note that compared to the concrete complexity measures dealt with in this work, bounding reduction steps is rather coarse-grained: one reduction step in the calculus may decompose into some non-constant number of primitive machine operations, when executed on physical hardware. Indeed, the focus of Implicit Computational Complexity is not to capture \emph{concrete} resource usage of programs, but to provide a type-theoretic interpretation of complexity classes. A formalisation of a cost semantics for System T using ideas from Implicit Computational Complexity was formalised in Coq~\cite{Danner:2013:SCA:2428116.2428123}. Several program logics for reasoning about a program's resource consumption have been investigated in the literature. Aspinall et al. developed a program logic for a subset of the Java Virtual Machine's instruction set, capable of establishing resource bounds on JVM programs~\cite{ASPINALL2007411}, similar in spirit to the `quantitative Hoare logic' mentioned above. The soundness of the logic was formally verified in Isabelle/HOL. Another program logic for reasoning about Java resource usage, based on ideas taken from Separation Logic, was also developed by Aspinall et al.~\cite{Aspinall2010}. \paragraph{Verified compilation}
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Gnuplot is a portable command-line driven graphing utility for Linux, OS/2, MS Windows, OSX, VMS, and many other platforms. This indicates how strong in your memory this concept is. 80 Constructing a regression on. While right-angled triangle definitions permit the definition of the trigonometric functions for angles between 0 and radian (90°), the unit circle definitions allow to. " In this work, the term "graph" will therefore be used to refer to a collection of vertices and edges, while a graph in the sense of a plot of a function will be called a "function graph" when any. sin() Method : is an inbuilt method which returns the sine of the value passed as an argument. Note – The graphs of the Sine, Cosine and Tangent functions are considered higher topics on most exam boards. Conic Sections. The three basic trig graphs: y=Sin(x), y=Cos(x), y=Tan(x) Key features: Amplitude, period, cyclic nature, max & minimum values or asymptotes. Notes and exercises for lecture 4. Table of Trigonometric Parent Functions; Graphs of the Six Trigonometric Functions; Trig Functions in the Graphing Calculator; More Practice; Now that we know the Unit Circle inside out, let's graph the trigonometric functions on the coordinate system. f(x)=6sin(x) Three trigonometric functions for a given angle are shown below. Set Notation and Language Functions Quadratics Indices and Surds Factors of Polynomials Simultaneous Equations Logarithms and Exponential Functions Straight Line Graphs Circular Measure Trigonometry Permutations and Combinations Binomial Expansion Vectors Matrices Differentiation and Integration. Trigonometric Functions of Common Angles. Wright's Classroom Resources. 1 Angles and Their Measure. Download free on Amazon. 4 Trigonometric Functions of Any Angle. terms from notes provided the maximum of any sin and cos graph. Trig Graphs Sine and cosine graphs are wave graphs that go on forever in both directions. A sine graph is a graph of the function =y sin θ. Graph f(x) (x 2)2 3by completing the table. 6 Tan and Cot hand outs HW:: Graph Tangent and Cotangent Function 4. To find vertical asymptotes, look for any circumstance that makes the denominator of a fraction equal zero. In the next series of graphs, the first graph shows f x( ) ln x over the interval [ –. Powered by Create your own unique website with customizable templates. One of the quickest ways to spot a given “family” is to graph the function. We’ll need more than acute angles in the next section where we’ll look at oblique triangles. You need to complete through #7 using Desmos, and then study the notes up to (not including ) graph #8. Here is a list of all of the maths skills students learn in grade 10! These skills are organised into categories, and you can move your mouse over any skill name to preview the skill. Represent functions using function notation. notebook 3 November 08, 2019 Finding the Zeros & Solving Equations 1) by sketching the graph Example: 0. Inverse trigonometric functions. •The range is all real numbers, therefore there is no minimum, maximum or amplitude. Introducing Radians; 11. You need to use Trigonometry practically like calculating the distance for moving object or angular speed. •if a < 0, graph will start down. Trigonometry. A periodic function is a function, such as sin(x), that repeats its values in regular intervals. Graphing Sine and Cosine Functions. Show Hide Details. This Graphs of the Six Trigonometric Functions Handouts & Reference is suitable for 10th - 12th Grade. notebook 3 November 08, 2019 Finding the Zeros & Solving Equations 1) by sketching the graph Example: 0. -1-Find the amplitude, the period in radians, the minimum and maximum values, and two vertical asymptotes (if any). Basic graphs. Let's start with the basic sine functions. Graph of the tangent function. Those are the most likely candidates, at which point you can graph the function to check, or take the limit to see how the graph behaves as it approaches the possible asymptote. 6 Graphs of Other Trigonometric Functions; Section 4. Graphing Trigonometric Functions Mathematics 4 October 6, 20111 of 45 2. 6 Linear functions over unit intervals. M110 Fa17 Page 1/7 Worksheet 15 KEY - Graphing Trigonometric Functions 1. f O = sine As a terminal arm rotates about a circle with radius 1, f(O) = sin O represents the length (rise) of the triangle created from any point on the circle. Trigonometric Functions of Common Angles. 88 University of Houston Department of Mathematics. Sine and cosine are periodic functions, which means that sine and cosine graphs repeat themselves in patterns. 6 Tan and Cot hand outs HW:: Graph Tangent and Cotangent Function 4. A sine graph is a graph of the function =y sin θ. 4 Inverse Trig Functions Notes; Section 13. (same as all. uk A sound understanding of Functions & Graphs is essential to ensure exam success. The equation y = 3x +7 can also be written as f(x) = 3x + 7 FUNCTION NOTATION U= B( T). Please try again later. Remember to define the Domain if you write the equation for part of a trig graph. 5: The Unit Circle and Radians. Covers Graphing Logarithmic Functions. If we have a right triangle with one angle θ. Trigonometric Functions. Also, Since There Are Multiple Functions. 5A Graphs of Sine and Cosine Functions; 1. Using Maple to find explicitly the inverse of a function and verify that it is the inverse. If there are two angles one positive and the other negative having same numerical value, then positive angle should be taken. Notes from Trigonometry Steven Butler c 2001 - 2003. Sep 25:33 PM Vocab Sinusoidal Axis - the axis that is the midpoint of a sine or cosine curve. 7B Inverse. A summary of Graphing Functions in 's Trigonometry: Graphs. The value passed in this function should be in radians. The inverse sine y=sin^(-1)(x) or y=asin(x) or y=arcsin(x) is such a function that sin(y)=x. We can tell a great deal from the graph of the Sine function. Inverse Trigonometric Function Derivatives ( Part 1 ) Inverse Trigonometric Function Derivatives ( Part 2 ) Curve Sketching: Sketch First Derivative Graphs. Lines: Slope Intercept Form example. Transformation Task 3rd blk. ft7 ith an period of £. Solving SSA Triangles. 1: Graphing Sine and Cosine Functions Periodic Function: a function for which the dependent variable takes on the same set of values over and over again as the independent variable changes Sinusoidal Function: a periodic function that looks like waves. IG-0606-Factors of Polynomials- Notes. Feel free to check this on your calculator!. Using radians to find area of sector; 12. Wright's Classroom Resources. Download: Trigonometric Functions. Graphing Trigonometric Functions Guided Notes 2 Name:_____ Recall: Cosecant, secant, and cotangent functions are all reciprocals of sine, cosine, and tangent functions respectively. This feature is not available right now. 1 page 233 - 237 # 4ac, 5bd, 10ab, 12, 14, 20 Extensions: 15, 19ac, 24 (just ask yourself if it's periodic), C2 5. For b 0, the period of y = a cos bx is also. Findthesineandco sineof45°. Using Maple to find explicitly the inverse of a function and verify that it is the inverse. 7 Inverse Trigonometric. Radian Measure & Angles in Standard Position Sine, Cosine & Tangent of Angles in Standard Position Sine, Cosine, Tangent Functions of Special Angles Graphs of Trigonometric Functions (includes Reciprocal Trig Functions). Find each value. Show All Solutions Hide All Solutions. by c units. Remember to define the Domain if you write the equation for part of a Exponential graph. If Y is complex, then the plot function plots. Graphs: S y sinx: y arcsin sin 1x: y cosx: y arccos x cos 1 x: y xtanx: y arctan x tan 1: Trig function Restricted domain Inverse trig. We'll start with the graph of y x tan. 5 Post Notes ; 1. MAC1114 Notes Professor Chionilos Objectives/Focus: Graphs of Sine and Cosine Functions ~ Graphs of y = sin x and y = cos x ~. We will now explore what changes can be made to the equations and how that affects the graphs. PDF format to accommodate downloading ease. Find the points where xintercepts occur. First make a slider $$f$$ using the tool Slider. Before jumping into the problems remember we saw in the Trig Function Evaluation section that trig functions are examples of periodic functions. Calculus I or needing a refresher in some of the early topics in calculus. The range of the graph y = f (x) is [—10,50] and its period is 720. Period of Trig Graphs. Drawing Transformed Graphs for Sin and Cos. Identities Proving Identities Trig Equations Trig Inequalities Evaluate Functions Simplify. If 0 < b < 1, the graph of the function is stretched horizontally. Graphs of Trigonometric Functions Summary Graphs of Trigonometric Functions Below are the graphs of the six trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent. 26th: Test Graphing Trig Functions Feb. Trigonometric Identities. 9 KB (Last Modified on January 17, 2018) SOLUTIONS TO EXPONENTIAL FUNCTIONS AND EQUATIONS EXTRA PRACTICE. Find the point where there are vertical asymptotes. grf, [ Download GRAPH] Inverse Trig Functions - sinx, cosx, tanx, cotx, secx, cscx Transcendental Functions & Derivatives - Summary. 2 3 3 3 3 4 4 4 5 5 5 5 5 6 6 7 7 7 7 8 8 8 9 9 9 9 Table of Contents Table of Contents Unit 1 - Rational, Exponential, and Logarithmic Functions 5. Give the amplitude and period : For the functions of the forms the amplitude is a, and the Ex 4. 2nd: Graphing Reciprocal Trig Fcns Circular Fcns Checkup Mar. pdf (Worksheet practicing this material, in pdf) WS_Soln_4_1A_UnitCircle&TwoFns. The slider is. 6: The Other Trig Functions The other four trig functions, tangent, cotangent, cosecant, and secant are not sinusoids, although they are still periodic functions. Pre-Algebra. Graphs: S y sinx: y arcsin sin 1x: y cosx: y arccos x cos 1 x: y xtanx: y arctan x tan 1: Trig function Restricted domain Inverse trig. Name_____ 4. The slider is. This will include a graph and must come up with your own word problem that corresponds to the given graph. IG-0606-Permutations and Combinations- Exercise. 0 Students graph functions of the form f(t) = A sine(Bt + C) or f(t) = A cos(Bt + C) and interpret A, B, and C in terms of amplitude, frequency. We will now explore what changes can be made to the equations and how that affects the graphs. Trigonometry is distinguished from elementary geometry in part by its extensive use of certain functions of angles, known as the trigonometric functions. IB SL Trigonometry Using the Cosine rule: Using the Sine rule: Calculating area and arc length of sectors: Calculating exact values for some specific values of sin, cos and tanx: CAST diagram: Usin…. Grouping symbols such as parentheses, and helper functions may be arbitrarily deeply nested. Learn exactly what happened in this chapter, scene, or section of Trigonometry: Graphs and what it means. 4 Ch 5 Ch 5 II Ch 6 Ch 7 Ch7 II Ch 8 Ch 8 II Ch 9 Graphing Super Massive Cheat Sheet Trig Unit Circle. Trigonometric Functions Arbitrary angles and the unit circle We’ve used the unit circle to define the trigonometric functions for acute angles so far. Graphs: S y sinx: y arcsin sin 1x: y cosx: y arccos x cos 1 x: y xtanx: y arctan x tan 1: Trig function Restricted domain Inverse trig. Precalculus. Self-Check Quizzes Advanced Mathematical Concepts © 2001 Self-Check Quizzes randomly generate a self-grading quiz correlated to each lesson in your textbook. If the argument is. Section2: Graphsofy =sint andy =cost 6 Since the sine function has period 2π,the graph ofy = sint for all real numbers t has the same appearance as the sketch in Figure 4. 26th: Test Graphing Trig Functions Feb. The cosine function, which appears to have a "head start" on the sine function, is often said to lead the sine function (or the sine function lags the cosine function). y = sinx - calculator work – in degrees; 14. One of the quickest ways to spot a given “family” is to graph the function. For Each Type Of Function (sine, Cosine), Find The Amplitude, Period, And Phase Shift. In order to have an inverse, a function must be one-to-one. The strategy we adopt is to find one solution using knowledge of commonly occuring angles, and then use the symmetries in the graphs of the trigonometric functions to deduce additional solutions. Trigonometry: 2. 0 / 5 Trigonometry and Pythagoras. Learn how to graph trigonometric functions and how to interpret those graphs. Transformations Of Trig Functions. 5 Graphing Other Trigonometric Functions (continued) Name _____ Date _____ c. Notes adapted from All Things Algebra, Gina Wilson. Unit 7 mod 19. 1 Domain and range. functions with and without a graphing calculator. The SYMB key nearly always takes you to a view in which you can enter equations. OBJECTIVE 1. We start with the graph of the basic sine function y = sin(x) and the basic cosine function g(x) = cos(x), we then present examples of how to graph transformed versions of these same functions. 1 Exponential Equations Blank. If Y is complex, then the plot function plots. 1_solutions. 5 Post Notes ; 1. Both sine and cosine fall under the category of sinusoids, since the cosine graph is simply a 90º shift to the left of the sine graph. Actually, cot function restricted to any of the intervals [– π, 0], [π, 2π] , is one-one & its range is R. Sketch Second Derivative Graphs ( Method 1 ) Sketch Second Derivative Graphs ( Method 2 ) ( Part 1 ) Sketch Complete Graphs ( No Asymptotes ) ( Part 2 ) Sketch Complete Graphs ( With. T 3/24 In Class Review of Trig Functions. f(x)=6cos(x) b. We learned about operators in unit 1. In order to have an inverse, a function must be one-to-one. Old Higher: Unit 1. This feature is not available right now. Set Notation and Language Functions Quadratics Indices and Surds Factors of Polynomials Simultaneous Equations Logarithms and Exponential Functions Straight Line Graphs Circular Measure Trigonometry Permutations and Combinations Binomial Expansion Vectors Matrices Differentiation and Integration. If 0 < b < 1, the graph of the function is stretched horizontally. On an xy -grid, the three sides of the triangle become r = hypotenuse, y = opposite, x = adjacent and t = θ. Phase shift horizontal shift of a trig function. 03x Graphs of Primary Trig Functions. The graphs of the sine and cosine functions are fixed. The notes are provided in both. Before discussing those functions, we will review some basic terminology about angles. Summary of Graphing Trig Functions 1. You need to complete through #7 using Desmos, and then study the notes up to (not including ) graph #8. Gnuplot is a portable command-line driven graphing utility for Linux, OS/2, MS Windows, OSX, VMS, and many other platforms. We have moved on to Larson's 5 th edition and some sections have changed but I have left them where they are since many people on the Internet find these useful resources. Lessons/Notes. 6 - Rational Functions. Unit 7: Graphing Trigonometric Functions. Unit 7 mod 19. Inverse Trigonometric Functions. sin-1 x, cos-1 x, tan-1 x etc. The sine and cosine functions are the. Unit 2: Graphs and Inverses of Trigonometric Functions. 5 Graphing Trig Functions x y. View Notes - Notes 5. notebook 3 November 08, 2019 Finding the Zeros & Solving Equations 1) by sketching the graph Example: 0. This also applies to sine, cosine. Find the point where there are vertical asymptotes. Graphing Trig Functions (Test 3 Material) Graphing of Trig Functions Powerpoint Large Graphs Handout (Oct 4) with Odd Answers in back (PDF) More Graphing Practice (With Big Graphs), Word Doc More Graphing Practice (With Big Graphs), PDF EVEN MORE Graphing Practice Inverse Trig Notes (PDFs) PDF of Classroom Notes for Inverse Trig, Lesson 4. —3 COS (9) Equation: Write two equations for each graph, one must be a sine function and the other must be a cosine function. 4 - Name the Quadrant, Point Problem - Click HERE. Higher Mathematics Functions and Graphs. Determine the equation of the graph. Be sure to watch in EdPuzzle for credit. The slider is. Sep 25:33 PM Vocab Sinusoidal Axis - the axis that is the midpoint of a sine or cosine curve. 7c Identify and use maxima and minima of polynomial functions to solve problems. You need to use Trigonometry practically like calculating the distance for moving object or angular speed. Feature of symmetry and repeating values. Day 2 Notes- Check these out. Graphing Sin(x) and Cos(x) Worksheet: Practice your skills by graphing the most fundamental trigonometry functions, sine and cosine. Area of a Triangle. Notes; Show More : Image Attributions. This also applies to sine, cosine. It starts at 0, heads up to 1 by π /2 radians (90°) and then heads down to −1. pdf (Relevant section from the free textbook by Stitz & Zeager, in pdf). Watch the video example of graphing the sine function. NOTES: Type Pi Or Use The CalcPad Symbol For π In Your Answers If Needed. Download: Trigonometric Functions. 22 x ππ −≤ ≤ Is the tangent function even, odd, or neither?. Identify the function graphed as a solid line: 2. The equation y = 3x +7 can also be written as f(x) = 3x + 7 FUNCTION NOTATION U= B( T). IG-0606-Factors of Polynomials- Notes. Chapter 4 Trigonometry; Section 4. The graph of the sine function is a nice, continuous wave that rolls along gently and keeps repeating itself. We will discuss different types of functions and I will teach you which questions you should be asking yourself when graphing a function. Graph over two periods. 4 Trigonometric Functions in All Quadrants Notes. You should try several of these problems. GRAPHING CALCULATOR LAB 824 Chapter 14 Trigonometric Graphs and Identities [0, 720] scl: 45 by [ 2. What are the coordinates of point (x, y) on the. Let's say you want to play a single sine function. Robert Trakimas, Sep 14, 2016, 5:02 PM. This means that will be no more than 1, and no less than -1 for any value of x. This feature is not available right now. ft7 ith an period of £. Get Started. Unit 1 Outcome 2 - Functions and Graphs. y= cos 2x SOLUTION a. These are the red lines (they aren't actually part of the graph). IG-0606-Permutations and Combinations- Exercise. Grieser 2 Function Families • Describe a group of functions with similar characteristics • Similar to the concept of literal equations • Example: f(x) = mx + b describes the family of linear functions Parent Functions represent the simplest form of a function family. 5 Graphing Sine & Cosine: Self-guided notes & examples. functions with and without a graphing calculator. Graph y = 5 sin x using transformations. Start studying Trigonometric Functions Review/Notes. Algebra 1 Notes SOL A. EF Two functions can be “composed” to form a new. This means that all we really need to do is graph the function for one periods length of values then repeat the graph. On an xy -grid, the three sides of the triangle become r = hypotenuse, y = opposite, x = adjacent and t = θ. 2 Log Functions and their Graphs Day1 Notes: 3. y= 2 sin x b. PDF format to accommodate downloading ease. Link the Ideas periodic. Students should be encouraged to print notes and fill them out as they watch through the lectures. 9/9 - Amplitude and Period Notes: Transformations of Trigonometric Functions. periodic functions. You need to complete through #7 using Desmos, and then study the notes up to (not including ) graph #8. Use midpoint rule to locate intermediate points. PDF format to accommodate downloading ease. Notes from Trigonometry Steven Butler c 2001 - 2003. Graph the following function. You will probably be asked to sketch one complete cycle for each graph, label significant points, and list the Domain, Range, Period and Amplitude for each graph. > Introduction to functions > Linear functions > Polynomial functions > Exponential and logarithm functions > Trigonometric functions > Hyperbolic functions > Composition of functions > Inverse functions > Sigma notation > Arithmetic and geometric progressions > Limits of sequences > The sum of an infinite series > Limits of functions. What is a Radian? more on radians. 1_solutions. The transformations below work for ALL functions!. 6 Notes Graphing Tan and Cot Functions 4. Perfect for acing essays, tests, and quizzes, as well as for writing lesson plans. The Unit Circle · Degrees and Radians · Quadrants · Common Acute Angles and Right Triangles · A few additional facts about triangles · Recommended Books. The calculator will find the inverse sine of the given value in radians and degrees. In the next series of graphs, the first graph shows f x( ) ln x over the interval [ –. Think of the types of graphs you can obtain by a combination of stretches, reflections and translations of the graph $$y=\sin x$$. In trig, arc functions are inverses of functions. First make a slider $$f$$ using the tool Slider. So, we’ll give here only short comments concerning their graphs receivedby rotating the graphs of trigonometric functions around a bisector of the 1-st. 5 Graphs Of Sine and Cosine Equation Notes. 2 – The student will sketch the graphs of the principal inverses of the six trigonometric functions. Covers Graphing Logarithmic Functions. = A sin b (x—h -4-Q 51 n X + + +. M110 Fa17 Page 1/7 Worksheet 15 KEY - Graphing Trigonometric Functions 1. Graphs of the sine and the cosine functions of the form y = a sin(b x + c) + d and y = a cos(b x + c) + d are discussed with several examples including detailed solutions. Graphing Trig Functions Can be graphed on to an xy-plane x-coordinates are in radians y-coordinates are the resuleoT the trigonometric function at a particular radian value If graphed on calculator in degrees, correct graph will NOT appear Start with sinusoidal functions (sine and cosine) o real life sinusoidal functions: sound, light, temperature. Functions & Graphs. Grieser Page 2 Graph cos(x) Domain _____ Range _____ Periodic or cyclic: pattern will repeat itself in cycles What is the period of the cosine function (what is the horizontal length of one cycle)? What is the amplitude of the cosine function (magnitude (height) of the wave)?. 9/6 - Parent Graphs (Specific Domain) Notes: Continued from 9/5 Selected Answers: Assignment 2 - Selected Answers. 3 Logarithms and their Graphs. by c units. These notes were written during the Fall 1997 semester to accompany Larson's College Algebra: A Graphing Approach, 2nd edition text. Solving SSA Triangles. You can see how the sine values on the graph above have values between 0 and pie. Corresponding to each such interval, we get a branch of function cot –1. Covers Graphing Logarithmic Functions. Sketch the three graphs on the grid provided below (you may want to use different colors for each graph to distinguish between them). The function satisfies the conditions cosh0 = 1 and coshx = cosh(−x). 88 University of Houston Department of Mathematics. Graph the basic functions f(x) = xn where n = 1 to 3, f(x) = x , f(x) = |x|, and f(x) = 1 x. If the argument is. Introducing Radians; 11. f O = sine As a terminal arm rotates about a circle with radius 1, f(O) = sin O represents the length (rise) of the triangle created from any point on the circle. The x-intercepts, maximum, and minimum occur at these points. Starting at f(0) = 0, the graph of fincreases to f 2 ˇ = 1, then decreases through f(ˇ) = 0 until reaching f 3ˇ 2 = 1, and then increases again until it reaches f(2ˇ) = 0. If there is no unit after the angle, the trig function evaluates its parameter as a radian measurement. csc (-x) = -csc x B. 5 Graphing Other Trigonometric Functions (continued) Name _____ Date _____ c. uk A sound understanding of Functions & Graphs is essential to ensure exam success. Trigonometric Functions. Grade: High School. Show All Solutions Hide All Solutions. Inverse Trig Functions Notes. Graphing Trig Functions Notes Name_____ Date_____ Period____ ©\ e2a0s1g6S _KKuKtlaL XSvoIfZtZwAaUrqei tLELsCw. Graphing Trig Functions Day 1 Find the period, domain and range of each function. Play music in GeoGebra. Be sure to watch in EdPuzzle for credit. The other quantities are in general fixed, and each of them influences the shape of the graph of this function. A circle centered in O and with radius = 1 is known as trigonometric circle or unit circle. pdf File history uploaded by Paul Kennedy 1 year, 5 months ago No preview is available for MHF 5. Included are the graphs for sine, cosine, tangent, cosecant, secant, and cotangent. Graphs of Sinusoidal Curves NOTES 9/4/19 Day 8 CW: What Does the Two Do? WS HW: Graphs of Sinusoidal Functions WS (Try Part A only. Summary of Graphing Trig Functions 1. Graphing Trig Functions Date_____ Period____ Using degrees, find the amplitude and period of each function. The transformations below work for ALL functions!. Phase shift horizontal shift of a trig function. M 3/23 Notes: Memorized Material for Trig Functions. 1 Arc Length, Area of a Sector - Click HERE Notes - 7. Label the axes and draw any asymptotes. sin() Method : is an inbuilt method which returns the sine of the value passed as an argument. Transformations Of Trig Functions. 3) Graphs of Quadratic Functions (PurpleMath. You can see how the sine values on the graph above have values between 0 and pie. Covers Graphing Logarithmic Functions. 2 Log Functions and their Graphs Day3 CW:. UNIT 5 WORKSHEET 13 INVERSE FUNCTIONS. Grieser Page 2 Graph cos(x) Domain _____ Range _____ Periodic or cyclic: pattern will repeat itself in cycles What is the period of the cosine function (what is the horizontal length of one cycle)? What is the amplitude of the cosine function (magnitude (height) of the wave)?. Drill on finding the inverses of functions. How can they be used to describe all the different kinds of waveforms that occur in the world around us? First of all, notice that we can change the shape of a sine (or cosine) wave by including constants in the algebraic expression that represents them. The graph of coshx is always above the graphs of ex/2. If there is a degree sign after the angle, the trig function evaluates its parameter as a degree measurement. If Y is complex, then the plot function plots. Graphs of equations and functions, Transformation of curves and their equations and interpreting different types of graphs. Coördinate pairs of a function. Plotting points from the table in a continuous manner along the x-axis gives the shape of the sine function. PDF format to accommodate downloading ease. Watch the video example of graphing the cosine function. Graphs of Trig Functions. 2 3 3 3 3 4 4 4 5 5 5 5 5 6 6 7 7 7 7 8 8 8 9 9 9 9 Table of Contents Table of Contents Unit 1 - Rational, Exponential, and Logarithmic Functions 5. Each of the graphs of these functions have discontinuities. Notes; Show More : Image Attributions. Trig Function Review Part 2 page 9 UHILL. At these values, the graph has a vertical asymptote. Play music in GeoGebra. The function 𝑦=sin2𝜃 is graphed below as a dashed line. notebook 3 November 08, 2019 Finding the Zeros & Solving Equations 1) by sketching the graph Example: 0. Students will be able to graph sine and cosine functions with phase shifts, vertical shifts, and varying amplitudes using various methods. There will be 4 separate axes. IG-0606-Permutations and Combinations- Notes. If Y is a vector, then the x -axis scale ranges from 1 to length (Y). ALGEBRA 2 CHAPTER 6 NOTES SECTION 6-7 GRAPHS OF POLYNOMIALS Objectives: Use properties of end behavior to analyze, describe, and graph polynomial functions. Categories. We start with the graph of the basic sine function y = sin(x) and the basic cosine function g(x) = cos(x), we then present examples of how to graph transformed versions of these same functions. Translate Trig Graphs. Download free on Google Play. Wright's Classroom Resources. identities that it knows about to simplify your expression. Since we defined 9 different inverse trigonometric functions for our 6 trigonometric functions, there are 108 compositions which can be created. f(x)=6sin(x) Three trigonometric functions for a given angle are shown below. Graphing Sinusoidal Trig Functions Notes Dater Graphing Trig Functions Can be graphed on to an xy-plane x-coordinates are in radians y-coordinates are the resuleoT the trigonometric function at a particular radian value If graphed on calculator in degrees, correct graph will NOT appear Start with sinusoidal functions (sine and cosine). Algebra 2 Trig. This Graphs of the Six Trigonometric Functions Handouts & Reference is suitable for 10th - 12th Grade. First complete the x-y chart by listing the corresponding radian measure for each degree measure. The period of. Graphing Trig Functions Notes Name_____ Date_____ Period____ ©\ e2a0s1g6S _KKuKtlaL XSvoIfZtZwAaUrqei tLELsCw. 7c Identify and use maxima and minima of polynomial functions to solve problems. A function of a function. Sketching Basic exponential functions y = A x Vertical and horizontal translations of the exponential functions y = AB x+C + D. Throughout these tasks students are asked to use radians and degrees. Contents DISCLAIMER vii Preface viii 7 Graphing the trigonometric functions 53 basis for many of the relationships of trigonometry. 6 Post Notes ; 1. pdf (Ken's lecture notes on graphs of the six basic trig functions, in pdf) WS_4_5A_GraphsOfSixTrigFns. Algebra 2/Trig AIIT. IXL will track your score, and the questions will automatically increase in difficulty as you improve!. You've already learned the basic trig graphs. UNIT 5 WORKSHEET 13 INVERSE FUNCTIONS. The vertical asymptotes are where sin x = 0 so therefore at x = nπ A. WS- transformation. 4 Trigonometric Functions of Any Angle; Section 4. The y-coordinate, f (x), is a real number --specifically, it is the ratio that defines the value of a given trigonometric function. Name_____ 4. The range of the graph y = f (x) is [—10,50] and its period is 720. 4 Trigonometric Functions of Any Angle; 1. Unit 7: Graphing Trigonometric Functions. 6 Notes: Graphing Sec and Csc Functions Day2 CW Graphing other functions hand out QUIZ Review: Graphing Trigonometric Functions 4. Take notes. In trig, arc functions are inverses of functions. First complete the x-y chart by listing the corresponding radian measure for each degree measure. Since we defined 9 different inverse trigonometric functions for our 6 trigonometric functions, there are 108 compositions which can be created. Feb 26 - I hope you had a wonderful break! We are back with a discovery activity on graphing sine and cosine functions. The bar at the bottom of the screen labels them. Example The parametric equations x = t , y = t2. We can transform and translate trig functions, just like you transformed and translated other functions in algebra. The restricted domains are determined so the trig functions are one-to-one. We know that for −1 ≤ y ≤ 1 there are infinitely many angles x which satisfy the equation sinx =y. How can they be used to describe all the different kinds of waveforms that occur in the world around us? First of all, notice that we can change the shape of a sine (or cosine) wave by including constants in the algebraic expression that represents them. 4 Inverse Trig Functions Practice Homework, Page 2; 2/20/2007 Warmup on Trig Expressions(?) Homework 4: Graphing the Ferris Wheel; Homework 5: Graph Variations, Page 2 "Plain" Sine Graphs page 23; Homework 15: What's Your Cosine? Homework 18: Coordinate Tangents; Homework 6: Sand Castles; In. Amplitude - Half the height of the function. YESTERDAY'S NOTES!!!! y = 2Tanx + 1. Passing the fast paced Higher Maths course significantly increases your career opportunities by helping you gain a place on a college/university course, apprenticeship or even landing a job. Students should be encouraged to print notes and fill them out as they watch through the lectures. Both sine and cosine fall under the category of sinusoids, since the cosine graph is simply a 90º shift to the left of the sine graph. Graph y = 5 sin x using transformations. Can you find trigonometry edition Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource. If there is a degree sign after the angle, the trig function evaluates its parameter as a degree measurement. Apr 51:46 PM HW pg 440. 6 Notes Graphing Tan and Cot Functions 4. 3rd: Determining Equations Mar. 6 Handouts Sec Csc HW: Graphing sec and csc functions 4. Graph and stretch trig functions % Progress. Graph f(x) (x 2)2 3by completing the table. While it is important for students to be able to create a graph of the trigonometric functions by hand, this should be done after they are able to identify the key features of the graph and can use these features to create the graph. They are used to relate the angles of a triangle to the lengths of the sides of a triangle. Graphing the function y = sin x:Identify the regions in the cartesian plane corresponding to thequadrants of the unit circle: 2 of 45 3. Determine if the sine, cosine, tangent, cotangent, secant, or cosecant functions are even, odd, or neither. 5-1 1 0 π_ 2 3__π 2 5__π 2-π_ 2 Period Period One Cycle 3__π 2 5__π - 2-y = sin θ θ Trigonometric functions are sometimes called circular because they are based on the unit circle. Algebra 2 Trig. In the applet you are given the expression of a sine or cosine function. The function 𝑦=sin2𝜃 is graphed below as a dashed line. 4 Graphing Sine and Cosine Functions 487 Each graph below shows fi ve key points that partition the interval 0 ≤ x ≤ 2π — into b four equal parts. Since the graphs of sine and cosine are periodic, we only need to graph one period to see the behavior of the graph. Unit 1 Outcome 3 - Differentiation. 2 Log Functions and their Graphs Day3 CW:. y-2π -π π 2π 0. Type your expression into the box to the right. 1 Radian and Degree Measure; Section 4. The function satisfies the conditions cosh0 = 1 and coshx = cosh(−x). You will also need to come up with the equation from the graph. Be an active listener. Ensure that the radian mode is selected) b) State the amplitude of each function. Welcome to highermathematics. Self-Check Quizzes Advanced Mathematical Concepts © 2001 Self-Check Quizzes randomly generate a self-grading quiz correlated to each lesson in your textbook. f O = sine As a terminal arm rotates about a circle with radius 1, f(O) = sin O represents the length (rise) of the triangle created from any point on the circle. Are there any minimum or maximum values. Learn how to construct trigonometric functions from their graphs or other features. The adjustments can be simple to understand or more challenging. Inverse Trigonometric Functions. Be sure to watch in EdPuzzle for credit. FUNCTIONs are dependable, sophisticated operations GRAPHING by INSPECTION of function by POINT-PLOTTING INVERSE FUNCTIONS Inverse Notes: inverse functions: verbally, analytically, algebraically , Inverse Problems Take An Inverse In 4 Language Families IDENTITY, OPPOSITE, RECIPROCAL -- the most important functions. Trig ratios of any angle Special triangles (finding exact values) Modelling periodic behavior Sketching trig functions Transformations of periodic functions Trig applications. 5 sin x, and y = –2 sin x for 0 ≤ x ≤ 2π. 5 Graphing Sine & Cosine: Self-guided notes & examples. Chapter 4 Trigonometry; Section 4. Mathematics; Graphs and transformations Trigonometry and Core 2 Big 50 Revision Notes. Since we defined 9 different inverse trigonometric functions for our 6 trigonometric functions, there are 108 compositions which can be created. Find the point where there are vertical asymptotes. " This is not a definition, just a useful expression for an idea. The graph must have labels and an appropriate scale on each axis. 6 Notes: Graphing Sec and Csc Functions Day2 CW Graphing other functions hand out QUIZ Review: Graphing Trigonometric Functions 4. Identify the function graphed as a solid line: 2. Sine Graphs in Music Background Sound is produced by the vibrations of matter. A LiveMath Notebook to investigate which polynomials of degree 3 have inverses. Graph of Cosine. The period of. Before discussing those functions, we will review some basic terminology about angles. Unit 1 - All Outcomes. Represent functions using function notation. Graph f(x) x2 4x 4 A Brief Review of Function Notation We will be using function notation more often. Solving AAS Triangles. Passing the fast paced Higher Maths course significantly increases your career opportunities by helping you gain a place on a college/university course, apprenticeship or even landing a job. I have simplified the process so that even below average students can follow the process easily and graph trig functions without frustration. Your expression may contain sin, cos, tan, sec, etc. The function is assumed to be a function of time and the function values must be between -1 and 1. Inverse Trigonometric Function Derivatives ( Part 1 ) Inverse Trigonometric Function Derivatives ( Part 2 ) Curve Sketching: Sketch First Derivative Graphs. Let's start with the basic sine function, f (t) = sin (t). You will probably be asked to sketch one complete cycle for each graph, label significant points, and list the Domain, Range, Period and Amplitude for each graph. Cosine is just like Sine, but it starts at 1 and heads down until π radians (180°) and then heads up again. This means that will be no more than 1, and no less than -1 for any value of x. This page will try to simplify a trigonometric expression. Exponential Functions. Your learners will appreciate having this color-coded set of the six trigonometric functions all on one page. If Y is a vector, then the x -axis scale ranges from 1 to length (Y). This also applies to sine, cosine. 9/5 - Trig Parent Graphs Notes: Graphing Parent Functions Selected Answers: Assignment 1 - Selected Answers. Take an x-axis and an y-axis (orthonormal) and let O be the origin. 2 Log Functions and their Graphs Day3 CW:. Such principal values are sometimes denoted with a capital letter so, for example, the principal value of the inverse sine may be variously denoted or (Beyer 1987, p. Let us explore how the shape of the graph of changes as we change its three parameters called the Amplitude, , the frequency, and the phase shift,. Compute maximum and minimum. pdf (pdf) S&Z 10. Trig graphs e. Sine function (EMA53) Functions of the form $$y=\sin\theta$$ (EMA54) Worked example 16: Plotting a sine graph. notebook April 24, 2017 Example und und tanθ y=Tanθ (parent function) Domain range A. Perfect for acing essays, tests, and quizzes, as well as for writing lesson plans. 1-24 notes. You can use these points to sketch the graphs of y = a sin bx and y = a cos bx. There is not a whole lot to this section. 8 & 5 – Trigonometric Identities Page 1 of 22 Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 4 Syllabus Objective: 4. A summary of Graphing Functions in 's Trigonometry: Graphs. The precalculus. UNIT 5 WORKSHEET 14 LOGARITHMS PRACTICE EXAM. Corresponding to each such interval, we get a branch of function cot –1. Matrices Vectors. Notice the graph is symmetric about the y-axis, because coshx = cosh(−x). Graph 𝑦=sec𝜃 3. It is here just to remind you of the graphs of the six trig functions as well as a couple of nice properties about trig functions. Such principal values are sometimes denoted with a capital letter so, for example, the principal value of the inverse sine may be variously denoted or (Beyer 1987, p. Radian Measure & Angles in Standard Position Sine, Cosine & Tangent of Angles in Standard Position Sine, Cosine, Tangent Functions of Special Angles Graphs of Trigonometric Functions (includes Reciprocal Trig Functions). Starting at f(0) = 0, the graph of fincreases to f 2 ˇ = 1, then decreases through f(ˇ) = 0 until reaching f 3ˇ 2 = 1, and then increases again until it reaches f(2ˇ) = 0. Lessons/Notes. 7A Inverse Trigonmetric Functions (part 1) 1. #Find#the#six#trigonometric#functions#of##θif#. YESTERDAY'S NOTES!!!! y = 2Tanx + 1. Learn how to construct trigonometric functions from their graphs or other features. PDF format to accommodate downloading ease. Sine and Cosine Graphs The graph represents a reciprocal trig function after a single transformation. The strategy we adopt is to find one solution using knowledge of commonly occuring angles, and then use the symmetries in the graphs of the trigonometric functions to deduce additional solutions. Download free in Windows Store. If P is a point from the circle and A is the angle between PO and x axis then: All trigonometric functions are periodic. This feature is not available right now. Day 2 Notes- Check these out. If you're seeing this message, it means we're having trouble loading external resources on our website. You need to use Trigonometry practically like calculating the distance for moving object or angular speed. The sine function This graph is continuous (there are no breaks). Graphing Trigonometric Functions Instructions: Ensure that all graphs are fully labeled. Arithmetic Mean Geometric Mean Quadratic Mean Median Mode Order Minimum Maximum Probability Mid-Range Range. It moves from its highest point down to its lowest point and. sin() Method : is an inbuilt method which returns the sine of the value passed as an argument. This means that will be no more than 1, and no less than -1 for any value of x. 05vt Graphs of Reciprocal Trig Functions. Graphs: S y sinx: y arcsin sin 1x: y cosx: y arccos x cos 1 x: y xtanx: y arctan x tan 1: Trig function Restricted domain Inverse trig. Ensure that the radian mode is selected) b) State the amplitude of each function. Let's start with the basic sine function, f (t) = sin(t). Area of a Triangle. Take notes. Precalculus. If 0 < b < 1, the graph of the function is stretched horizontally. Old Higher: Unit 1. 9/6 - Parent Graphs (Specific Domain) Notes: Continued from 9/5 Selected Answers: Assignment 2 - Selected Answers. y-coordinate. Unit Circle, Radians, Coterminal Angles. Trigonometric Functions Arbitrary angles and the unit circle We’ve used the unit circle to define the trigonometric functions for acute angles so far. Lesson #74: Graphing Basic Sine and Cosine Functions March 18, 2020 March 18, 2020 mshallmaa 1 Comment Lesson #74 Note Supplement - Take notes in your notebook OR print this paper out and take notes on this note supplement. Coterminal Angles. pdf (pdf) S&Z 10. 5B Transformations of Sine and Cosine Graphs; 1. Transformations are ways that a function can be adjusted to create new functions. 91) notes, "The confusion of this term with the 'graphs' of analytic geometry is regrettable, but the term has stuck [in the mathematical community]. This means that the graph of. 5 Graphs of Sine and Cosine Functions; Section 4. Choose your pace. Unit 2 - All Outcomes. 1 Domain and range. sin-1 x, cos-1 x, tan-1 x etc. y= 2 sin x b. Check the links below in which I will explain all you need to know about graphing functions to pass your IGCSE GCSE Maths exam. 4 Inverse Trig Functions Notes; Section 13. The vertical asymptotes are where sin x = 0 so therefore at x = nπ A. Now, let us have a look at the concepts discussed in this chapter. There are six trigonometric functions because there are six ways to choose two sides from three. y = 3sin(x) Period: 2π Amplitude: 3 Phase Shift: 0 Vertical Shift: 0 x y π. We’ll need more than acute angles in the next section where we’ll look at oblique triangles. CBSE Ncert Notes for Class 12 Maths Inverse Trigonometric Functions. May 2016 April 2016 March 2016 February 2016 January 2016 December 2015 November 2015 October 2015 September 2015. In trig, arc functions are inverses of functions. Graph y = 5 sin x using transformations. 6 Graphs of Other Trigonometric Functions Objective: In this lesson you learned how to sketch the graphs of other trigonometric functions. Cartesian graph paper is the most popular form of graph paper in use. Learn exactly what happened in this chapter, scene, or section of Trigonometry: Graphs and what it means. Covers Graphing Logarithmic Functions. Find the points where xintercepts occur. Foundations & Pre-Calculus Math 10 Page Content On each link below, you will find a screencast lesson, practice links and addional resources to help you with the Foundations & PreCalculus Math 10 course. Actually, cot function restricted to any of the intervals [– π, 0], [π, 2π] , is one-one & its range is R. Page 5 CfE Edition. Trig Cheat Sheet Definition of the Trig Functions Right triangle definition For this definition we assume that 0 2 p <0 and b>0, the graphs of y=asinbxand y=acosbx each have five keyx-values on the interval 0≤x≤2 b π: the x-values at which the maximumand minimumvalues occur and the x-intercepts. Day 2 Notes- Check these out. Notice that in examples 1 and 3, the order of the transformations did not matter. Table of Trigonometric Parent Functions; Graphs of the Six Trigonometric Functions; Trig Functions in the Graphing Calculator; More Practice; Now that we know the Unit Circle inside out, let's graph the trigonometric functions on the coordinate system. Trigonometric functions are functions of an angle.
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2020-09-28 01:28:05
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https://www.dsprelated.com/freebooks/mdft/Repeat_Operator.html
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### Repeat Operator
Like the and operators, the operator maps a length signal to a length signal:
Definition: The repeat times operator is defined for any by
where , and indexing of is modulo (periodic extension). Thus, the operator simply repeats its input signal times.7.10 An example of is shown in Fig.7.8. The example is
A frequency-domain example is shown in Fig.7.9. Figure 7.9a shows the original spectrum , Fig.7.9b shows the same spectrum plotted over the unit circle in the plane, and Fig.7.9c shows . The point (dc) is on the right-rear face of the enclosing box. Note that when viewed as centered about , is a somewhat triangularly shaped'' spectrum. We see three copies of this shape in .
The repeat operator is used to state the Fourier theorem
where is defined in §7.2.6. That is, when you stretch a signal by the factor (inserting zeros between the original samples), its spectrum is repeated times around the unit circle. The simple proof is given on page .
Next Section:
Downsampling Operator
Previous Section:
Interpolation Operator
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2021-05-10 17:44:15
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https://electronics.stackexchange.com/questions/331646/magnetic-energy-evaluation
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# Magnetic energy evaluation
How to evaluate magnetic energy when only vectors of magnetic induction B, and vector of magnetic field, H are known?
We know that in general, $$W_m=\int w_mdV$$
where wm is the density of magnetic energy that is defined as $$w_m=\int HdB$$
My question is, is it possible to evaluate magnetic energy Wm when only vectors B,H are known? If yes, what is the expression for magnetic energy in function of only B,H?
If the relation between H and B is linear (whether the medium is isotropic or not), the energy density of the magnetic field is $\tfrac{1}{2}$H⋅B.
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2018-08-15 02:53:21
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https://cs.stackexchange.com/questions/71892/is-repetition-the-origin-of-countability
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# Is repetition the origin of countability?
The original question was "Do all non-regular languages have an uncountable number of strings?".
How can someone prove that..? I am squeezing my head but I can't figure it out.
And the other side of the coin: is a language always regular, if it has a countably infinite number of strings?
A bit more generally: is repetition (DFAs) the origin of countability (and/or vice versa?) and if so, why?
I assume by countability, you mean a set being countably infinite, like $\mathbb{N}$.
All languages over a finite alphabet are countably infinite, whether they are regular or not.
It's easy enough to show that we can encode any string in $\mathbb{N}$: just take map its letters onto $\{1,2,\ldots\}$ and interpret the string as a base-$(n+1)$ number for an alphabet with $n$ letters. (Starting at $1$ avoids the problems with multiple $0$s on the left side of a number). You can also use Gödel numbering, which is simpler but less efficient.
It's important to distinguish languages themselves from the set of languages:
• Any language over a finite alphabet is countably infinite (or finite)
• There are an uncountably infinite number of languages over any finite alphabet
• Any class of languages we describe with a finite representation (including DFAs, CFGs, Turing Machines, or even logic statements) is countably infinite. So there are more languages than we will ever have ways to talk about or describe.
So no, repetition is not the origin of countability. Even undecidable languages are countably infinite.
• Thank you very much for your answer! I guess I started off from a wrong assumption and took it too far..! – Nick Mar 22 '17 at 21:42
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2020-05-25 02:21:15
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http://learning.maxtech4u.com/face-recognition/
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# Face Recognition
October 18, 2017
Information and Communication Technologies are increasingly entering in all aspects of our life and in all sectors, opening a world of unprecedented scenarios where people interact with electronic devices embedded in environments that are sensitive and responsive to the presence of users. Indeed, since the first examples of “intelligent” buildings featuring computer aided security and fire safety systems, the request for more sophisticated services, provided according to each user’s specific needs has characterized the new tendencies within demotic research. With data and information accumulating in abundance, there is a crucial need for high security. Biometrics has now received more attention. Face biometrics, useful for a person’s authentication is a simple and non-intrusive method that recognizes face in complex multidimensional visual model and develops a computational model for it.
# Face Recognition Significance
Face recognition is a biometric technique used for surveillance purposes such as search for wanted criminals, suspected terrorists, and missing children. The term face recognition refers to identifying, by computational algorithms, an unknown face image. This operation can be done by comparing the unknown face with the faces stored in database. Face recognition has three stages a) face location detection b) feature extraction c) facial image classification. Face recognition (FR) has emerged as one of the most extensively studied research topics that spans multiple disciplines such as pattern recognition, signal processing and computer vision. This is due to its numerous important applications in identity authentication, security access control, intelligent human-computer interaction, and automatic indexing of image and video databases. Face is the index of mind. It is a complex multidimensional structure and needs a good computing technique for recognition. While using automatic system for face recognition, computers are easily confused by changes in illumination, variation in poses and change in angles of faces. A numerous techniques are being used for security and authentication purposes which includes areas in detective agencies and military purpose.
### Typical View of Face Recognition
Typical structures of face recognition system consist of three major steps, gaining of face data, extracting face feature and recognition of face. Figure 1 shows typical structure of face recognition system in which subject under consideration given to the system for the recognition purpose this is consider being acquisition of face image. Later on feature is extracted from the image and finally it is given for the recognition purpose. These steps are elaborated as follow.
Figure 1 Face Recognition Systems
Gaining of Face Data
Acquisition and Processing of Face Data is first step in the face recognition system. In this step face images is collected from different sources. The sources may be camera or readily available face image database on the website. The collected face images should have the pose, illumination and expression etc variation in order to check the performance of the face recognition system under these conditions. Processing of face database require sometimes otherwise causes serious affect on the performance of face recognition systems due changes in the illumination condition, background, lighting conditions, camera distance, and thus the size and orientation of the head. Therefore input image is normalized and some image transformation methods apply on the input image.
### Extracting Face Feature
Feature extraction process can be defined as the process of extracting relevant information from a face image. In feature extraction, a mathematical representation of original image called a biometric template or biometric reference is generated, which is stored in the database and will form the basis (vector) of any recognition task. Later these extracted features used in recognition. A grayscale pixel is considered as initial feature.
### Recognition of Face
Once the features are extracted and selected, the next step is to classify the image. Appearance-based face recognition algorithms use a wide variety of classification methods Such as PCA, LDA. In classification the similarity between faces from the same individual and different individuals after all the face images in database are represented with relevant features. Sometimes feature extraction & recognition process done simultaneously.
### Face Recognition Applications
• Security: Access control to buildings, airports/seaports, ATM machines and border checkpoints; computer/ network security; email authentication on multimedia workstations.
• Surveillance: A large number of CCTVs can be monitored to look for known criminals, drug offenders, etc. and authorities can be notified when one is located; for example, this procedure was used at the Super Bowl 2001 game at Tampa, Florida; in another instance, according to a CNN report, two cameras linked to state and national databases of sex offenders, missing children and alleged abductors have been installed recently at Royal Palm Middle School in Phoenix, Arizona.
• General Identity Verification: Electoral registration, banking, electronic commerce, identifying newborns, national IDs, passports, drivers’ licenses, employee IDs.
• Criminal Justice Systems: mug-shot/booking systems, post-event analysis, forensics.
• Image Database Investigations: Searching image databases of licensed drivers benefit recipients, missing children, immigrants and police bookings.
• Smart Card” Applications: In lieu of maintaining a database of facial images, the face-print can be stored in a smart card, bar code or magnetic stripe, authentication of which is performed by matching the live image and the stored template.
• Multi-media Environments with Adaptive Human Computer Interfaces: Part of ubiquitous or context aware systems, behavior monitoring at childcare or old people’s centers, recognizing a customer and assessing his needs.
• Video Indexing: labeling faces in video
• Witness faces reconstruction.
#### References
[1] Sanjeev Kumar and Harpreet Kaur, “Face Recognition Techniques: Classification and Comparisons”, International Journal of Information Technology and Knowledge Management July-December 2012, Volume 5, No. 2, pp. 361-363
[2] V. Vijayakumari, “Face Recognition Techniques: A Survey”, World Journal of Computer Application and Technology 1(2): 41-50, 2013.
[3] Jafri, Rabia, and Hamid R. Arabnia, “A Survey of Face Recognition Techniques.” JIPS 5.2 (2009): PP. 41-68.
[4] J. N. K. Liu, M. Wang, and B. Feng, “iBotGuard: an Internet-based intelligent robot security system using invariant face recognition against intruder,” IEEE Transactions on Systems Man and Cybernetics Part C-Applications and Reviews, Volume 35, PP. 97-105, 2005.
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2018-11-14 20:14:08
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https://cdsweb.cern.ch/collection/CMS%20Theses?ln=hr
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CMS Theses
Najnovije dodano:
2015-11-30
15:14
Analysis of Standard Model Higgs Boson Decays to Tau Pairs with the CMS Detector at the LHC / Müller, Thomas The search for the Standard Model Higgs boson decaying into pairs of tau leptons performed at the complete CMS run I data set comprising integrated luminosities of 4.9/fb at centre-of-mass energies of 7 TeV and 19.7/fb at 8 TeV, respectively, is presented with a particular focus on the analysis of t [...] CERN-THESIS-2015-224 IEKP-KA/2015-24. - 169 p.
Fulltext
2015-11-30
09:34
Measurements in the Forward Phase-Space with the CMS Experiment and their Impact on Physics of Extensive Air Showers / Baus, Colin The astrophysical interpretation of ultra-high energy cosmic rays is based on detection of extensive air showers in indirect measurements [...] CERN-THESIS-2015-222 10.5445/IR/1000050521. - Karlsruhe, Germany : KIT, Karlsruhe, 2015-11-18.
Fulltext
2015-11-18
13:48
Study of Single Top Quarks in Association with Vector Bosons / Leggat, Duncan Alexander (Beijing, Inst. High Energy Phys.) ; Cole, Joanne (dir.) (Brunel U.) The search for single top production in association with a massive electroweak vector bosonusing data collected by the CMS detector at the Large Hadron Collider is presented. Twoanalyses are discussed: the search for a single top produced in association with a W boson(tW production) and the search for t-channel single top production with a radiated Zboson (tZq production). [...] CMS-TS-2015-030; CERN-THESIS-2015-214.- 2015 - 176 p. Fulltext: TS2015_030 - PDF; TS2015_030_2 - PDF;
2015-11-18
13:47
Search for a Heavy Neutral Higgs Boson decaying into Tau Pairs with the CMS Experiment at the LHC / Calligaris, Luigi (Hamburg U.) ; Meyer, Andreas Bernhard (dir.) (DESY) A search for a heavy MSSM Higgs boson is performed in the φ → τ τ → eµ decaychannel, using data corresponding to an integrated luminosity of 19.7 fb−1 at a p-pcenter of mass energy of 8 TeV, collected by the CMS experiment at the Large HadronCollider. The search is optimized for Higgs boson masses between 300 and 1000 GeVand makes the use of Boosted Decision Trees for an improved selection of signal events.The results are consistent with the predictions of the Standard Model.. CMS-TS-2015-028; CERN-THESIS-2015-215.- 2015 - 158 p. Fulltext: TS2015_028 - PDF; TS2015_028_2 - PDF;
2015-11-13
09:29
Desing and Simulation of Advanced Fiber Optic Sensors for High Energy Physics Application / Saccomanno, Andrea In the last two decades, Fiber Bragg Grating (FBG) sensor were been widely studied and employed in temperature and strain sensing application [...] CERN-THESIS-2013-392. - 150 p.
server - Fulltext - Full text
2015-11-13
09:22
Luminosity, beam monitoring and triggering for the CMS experiment and measurement of the total inelastic cross-section at √s = 7 TeV / Bell, Alan James The Compact Muon Solenoid (CMS) detector, situated on the Large Hadron Collider (LHC) ring is a multi-purpose detector designed to search for new physics phenomena, make precise measurements of known processes at previously untapped energies and look for hints of physics beyond the Standard Model [...] CERN-THESIS-2013-391. - 269 p.
Geneva U. server
2015-11-09
15:58
Measurement of Z boson production in association with jets at the LHC and study of a DAQ system for the Triple-GEM detector in view of the CMS upgrade / Léonard, Alexandre This PhD thesis presents the measurement of the differential cross section for the production of a Z boson in association with jets in proton-proton collisions taking place at the Large Hadron Collider (LHC) at CERN, at a centre-of-mass energy of 8 TeV [...] CERN-THESIS-2015-200 -
Full text
2015-10-28
10:43
The Razor Boost analysis - Another step in the hunt for new physics at CMS / Strobbe, Nadja Catharina (Fermilab) ; Ryckbosch, Dirk (dir.) (Gent U.) ; Tytgat, Michael (dir.) (Gent U.) CMS-TS-2015-027; CERN-THESIS-2015-178.- 2015 - 222 p. Fulltext: TS2015_027_2 - PDF; TS2015_027 - PDF;
2015-10-28
09:40
Study of the associated production of a Z boson and jets in pp collisions at $\sqrt{s}$ = 7 TeV at CMS / Montanino, Damiana In this thesis differential cross sections measurements of Z boson and jets associated production (Z + jets) will be presented; they are obtained by analyzing a large part of the 2011 dataset that corresponds to a total integrated luminosity of 4.89 1/fb [...] CMS-TS-2013-018 ; CERN-THESIS-2013-382 - 142 p.
Trieste U. server - Fulltext - Full text
2015-10-28
09:16
Search for anomalous Higgs boson production in association with single top quarks using the CMS detector / Popov, Andrey (Louvain U.) ; Giammanco, Andrea (dir.) (Louvain U.) CERN-THESIS-2015-177; CMS-TS-2015-025.- 2015 - 168 p. Fulltext: TS2015_025_2 - PDF; TS2015_025 - PDF;
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2015-12-02 06:54:50
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https://clementinerae.com/vf6oj/j3dkwy.php?id=determinant-is-a-square-matrix-or-not-689dee
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It is however vector-valued, not real-valued, except for the square case. ; Scaling a row of A by a scalar c multiplies the determinant by c.; Swapping two rows of a matrix multiplies the determinant by − 1.; The determinant of the identity matrix I n is equal to 1.; In other words, to every square matrix A we assign a number det (A) in a way that satisfies the above properties. Informally an m×n matrix (plural matrices) is a rectangular table of entries from a field (that is to say that each entry is an element of a field). In computing $ABv$, the vector$Bv$ has a smaller dimension than the final result, so the spanned spaces of $A$ and $B$ can't be in bijection. Depending on the perspective, a positive area can become a negative area if looked at from behind. How to compute the determinant of a square matrix. It also doesn't satisfy 3. either. What is a "constant time" work around when dealing with the point at infinity for prime curves? A = ( a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋮ ⋮ ⋱ ⋮ a … The beautiful geometric interpretation of the determinant is this. Although the determinant of the matrix is close to zero, A is actually not ill conditioned. My manager (with a history of reneging on bonuses) is offering a future bonus to make me stay. From the definition it follows that any submatrix of a totally unimodular matrix is itself totally unimodular (TU). I have a very large square matrix of order around 100000 and I want to know whether the determinant value is zero or not for that matrix. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. $\det(AB)$ always equals $\det(A)\det(B)$ whenever the product $AB$ is defined. Building a source of passive income: How can I start? The use of a determinant is algorithmic rather than mathematical and is important to solve for variable quantities of linear equation systems by Cramer’s Rule. For example, take the 3 wide matrix A defined with column vectors, x y and z, where each have n components: You can dot each of the vectors with each other by right multiplying A by its transpose: $$A^{T}A=\begin{pmatrix}x\\y\\z\end{pmatrix}\begin{pmatrix}x&y&z\end{pmatrix}=\begin{pmatrix} How can I organize books of many sizes for usability? So for an n\times m matrix, let k=\min(n,m) then compute all determinants of k\times k submatrices, perhaps with alternating sign. Use the multiplicative property of determinants (Theorem 1) to give a one line proof that if A is invertible, then detA 6= 0. I see a proof of the "determinant rank" being the same as the "row rank" in the book Elementary Linear Algebra by Kenneth Kuttler, which I see in google books. [ 12. The result generalizes both the determinant and the cross product. \operatorname{rank}(I_n)=n and \operatorname{det}(I_n)=1. 10.] Let A a square matrix with the size of n \times n. In fact, determinants can be used to give a formula for the inverse of a matrix. What can be the fastest way to know that ? Cálculo del determinante de una matriz cuadrada. That is, . A Matrix is an array of numbers: A Matrix. Finding rank of linear tranformation without a matrix? How can I get my cat to let me study his wound? Differences in meaning: "earlier in July" and "in early July". Inverse and determinant of square matrix F C Chang 10/29/2012. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Relation between rank and determinant of a matrix. Do strong acids actually dissociate completely? Is every “weakly square” matrix either a 0 matrix, or a square matrix? For example, take the 3 wide matrix A defined with column vectors, x y and z, where each have n components:$$A=\begin{pmatrix}x|y|z\end{pmatrix}$$What are wrenches called that are just cut out of steel flats? How to include successful saves when calculating Fireball's average damage. Another reason it is considered to be beautiful is because it has a simple and intriguing visual derivation. x\cdot x & x\cdot y & x\cdot z\\ Do you know of a rigorous proof of this statement using elemental methods? is taking A to be the n\times0 matrix and B the 0\times n matrix, for some n>0; then AB is a n\times n zero matrix so \det(AB)=0, while BA is the 0\times0 (identity) matrix, so \det(BA)=1. It only takes a minute to sign up. The determinant of a matrix A is denoted det(A) or det A or |A|. Property 5: Rules for evaluating determinants: The determinant of a triangular matrix is … rev 2020.12.4.38131, Sorry, we no longer support Internet Explorer, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. Let A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0\end{pmatrix} and B = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}. Word for person attracted to shiny things. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. \det has its usual value for square matrices. A determinant is a component of a square matrix and it cannot be found in any other type of matrix. The determinant of a matrix A is denoted det(A), det A, or |A|. It's always positive because it doesn't make sense to define positive and negative areas for spaces defined in dimensions higher than the space itself. 4.] best. 1. Of course, there is no "zeroes-line", but that doesn't prove it yet. Which order do the outputs arrive in? It's worth pointing out that the components don't really matter here, \mathrm{det}(AB)=0 whenever A has more rows than B. Matrices and determinants are important concepts in linear mathematics. I wrote an answer to this question based on determinants, but subsequently deleted it because the OP is interested in non-square matrices, which effectively blocks the use of determinants and thereby undermined the entire answer. An m×n matrix (read as m by n matrix), is usually written as: 1. Relation between determinant and matrix rank, https://sharmaeklavya2.github.io/theoremdep/nodes/linear-algebra/matrices/full-rank-inv.html, MAINTENANCE WARNING: Possible downtime early morning Dec 2, 4, and 9 UTC…. You sir are correct. I have an idea: if$$rank(A_{nxn}) = n $$then$$A$$must be invertible (proof : https://sharmaeklavya2.github.io/theoremdep/nodes/linear-algebra/matrices/full-rank-inv.html) so$$A^{-1} = \frac{adj(A)}{det(A)}$$exists and therefore$$\rightarrow det(A) \neq 0$$. Is there any way to test the existence of left or right inverse matrix? 100% Upvoted. In this 3 vector example, the equation above returns the value of the volume defined by vectors x y and z. If you have a space defined in a dimension higher than its own, this can still return the area it defines. Since the square of the determinant of a matrix can be found with the above formula, and because this multiplication is defined for nonsquare matrices, we can extend determinants to nonsquare matrices. This is a demo video to get program to check whether a given square matrix is symmetric or not. 0 comments. +1. What about \text{rank}(A)=n? The determinant is the product of these elements along the diagonal. Determinant of a Matrix; Note: Determinant is not defined for a non-square matrix. What do these expressions mean in H.G. Alternatively, you can row reduce the matrix to give you an upper triangular matrix using row interchanges and adding scalar multiples of a row to another row. Why does it imply \det(A)\ne0? With each square matrix we can calculate a number, called the determinant of the matrix, which tells us whether or not the matrix is invertible. report. It only takes a minute to sign up. How do I handle a piece of wax from a toilet ring falling into the drain? (ii) For a matrix A, A is read as determinant of A and not, as modulus of A. A determinant is represented with two vertical lines that consist of rows and columns. -13. In this example there aren't even any entries of A or B to worry about. What is a "constant time" work around when dealing with the point at infinity for prime curves? \det(AB) = 0 when A has more rows than B, Determinant of a rank 1 update of a scalar matrix, or characteristic polynomial of a rank 1 matrix, The definition of Determinant in the spirit of algebra and geometry, Prove that the Leibniz formula for determinant of a square matrix T is equal to the product of eigenvalues of T. This extension of determinants has all 4 properties if A is a square matrix, and retains some attributes of determinants otherwise. What is the physical effect of sifting dry ingredients for a cake? I know that if the rank of the matrix is "Non-zero determinant" - SEMATH INFO - Last updated: Sep. 12, 2017 A matrix is invertible if and only if its determinant is non-zero, i.e., . There is a way to determine the value of a large determinant by computing determinants that are one size smaller. Harmonizing the bebop major (diminished sixth) scale - Barry Harris. 1 &= \det \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \det(BA) = D(BA) = D(B)D(A) \\ Why do most tenure at an institution less prestigious than the one where they began teaching, and than where they received their Ph.D? How can I make sure I'll actually get it? This code is basically just a homework assignment, and one that would not receive an … Why was the mail-in ballot rejection rate (seemingly) 100% in two counties in Texas in 2016? This will only affect the sign of the determinant. \end{pmatrix}$$, Taking the determinant of this, you get the square of A's determinant: Sort by. The determinant of a matrix can be arbitrarily close to zero without conveying information about singularity. The determinant is positive or negative according to whether the linear transformation preserves or reverses the orientation of a real vector space. Determinant of a square matrix [1 x 2][-2 x 4][1 -3 -4] = 0. Theorem 2: A square matrix is invertible if and only if its determinant is non-zero. no comments yet. I tried multiplying some matrices, and what you said seems to be true, but I'm not sure why. I've seen a proof in a book which does this conclusion immediately, but IMHO this alone, doesn't prove it. @Nikolaj-K What do you mean? Determinant of Matrix P: 18.0 Square of the Determinant of Matrix P: 324.0 Determinant of the Cofactor Matrix of Matrix P: 324.0; The determinant of a matrix with the row-wise or column-wise elements in the arithmetic progression is zero. This in fact characterizes all invertible $n \times n$ matrices. [34] The linear transformation of R n corresponding to a real n-by-n The area of the parallelogram shown is the absolute value of the determinant of the matrix whose columns are and , the matrix . However, it can be salvaged if there exists a function $\det$ defined on all real-valued matrices (not just the square ones) having the following properties. Invertibility of block matrices, with the property of being symmetric, positive definite, and of full rank: Questions about matrix rank, trace, and invertibility. But it is multilinear, so it might be useful for some applications of determinants. Property 4: A square matrix A is invertible if and only if det A ≠ 0. @CPM "The rank of $A$ can be viewed as $m$ where $m$ is the size of the largest non-zero $m×m$ submatrix with non-zero determinant." The determinant of a 1×1 matrix is that single value in the determinant. The rank of $A$ can be viewed as $m$ where $m$ is the size of the largest non-zero $m\times m$ submatrix with non-zero determinant. See also: Determinant of a Square Matrix The inverse of a square matrix A with a non zero determinant is the adjoint matrix divided by the determinant, this can be written as The adjoint matrix is the transpose of the cofactor matrix. The first assertion is equivalent to saying that a square matrix A is singular if and only if det A = 0. &= D(A)D(B) = D(AB) = \det(AB) = \det \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix} = 0. This means you will be able to row reduce it to an upper triangular form with pivots along the diagonal. x\cdot y & y\cdot y & y\cdot z\\ share. The determinant of a matrix is a special number that can be calculated from a square matrix. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Why is it important for a matrix to be square? Why is Buddhism a venture of limited few? How much did the first hard drives for PCs cost? Those unfamiliar with the concept of a field, can for now assume that by a field of characteristic 0 (which we will denote by F) we are referring to a particular subset of the set of complex numbers. The determinant of any triangular matrix is equal to the product of the entries in the main diagonal (top left to bottom right)., where is the transpose of., where is the inverse of. satisfying the following properties: Doing a row replacement on A does not change det (A). The cofactor matrix is the matrix of determinants of the minors A ij multiplied by … MAINTENANCE WARNING: Possible downtime early morning Dec 2, 4, and 9 UTC…. The square matrix can be of any order such as 2x2 matrix, 3x3 matrix, or other nxn matrices. Note that we do not need to make the middle number a 1. Can you prove that? The determinant only exists for square matrices (2×2, 3×3, ... n×n). Log in or sign up to leave a comment Log In Sign Up. Show that (A+B)x = b + c may have infinite number of solutions. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Are there any contemporary (1990+) examples of appeasement in the diplomatic politics or is this a thing of the past? The important point to note here is the number of columns being equal as the number of rows. It tells me nothing that I need to know. Square Matrix Determinant. To investigate if A is singular, use either the cond or rcond functions. A. Learn some ways to eyeball a matrix with zero determinant, and how to compute determinants of upper- and lower-triangular matrices. 6. Last Updated: 30-05-2019 Determinant of a Matrix is a scalar property of that Matrix. Story in which immigrant girl finds room temp superconductor. \end{align}. The determinant of a matrix is the scalar value or a number estimated using a square matrix. Note that $\det(A) \neq 0$ iff the rows are linearly independent iff $rank(A)=n$. Can ionizing radiation cause a proton to be removed from an atom? Why is this considered to be beautiful? Is there an "internet anywhere" device I can bring with me to visit the developing world? By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. Suppose we draw two copies each of the two vectors and as shown below. 4.1.1 Determinant of a matrix of order one Let A = [a] be the matrix of order 1, then determinant of A is defined to be equal toa. [-11. It is often taken as the definition of rank of a matrix. Does Divine Word's Killing Effect Come Before or After the Banishing Effect (For Fiends), Pressure on walls due to streamlined flowing fluid. $$2 (x\cdot y) (x\cdot z) (y\cdot z)+(x\cdot x) (y\cdot y) (z\cdot z)-(x\cdot z)^2 (y\cdot y) - (x\cdot x )(y\cdot z)^2 - (x\cdot y)^2 (z\cdot z)$$. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Therefore, A is not close to being singular. That means that to be able to create model 2, you just have to chance the order in which to specify the factors. Determinants also have wide applications in engineering, science, economics and social science as well. Wells's novel Kipps? It is derived from abstract principles, laid out with the aim of satisfying a certain mathematical need. Definition. rev 2020.12.4.38131, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us, nice, I think you can go all the way and reduce it to the identity matrix as well. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. x\cdot z & y\cdot z & z\cdot z Prove the following are equivalent: $\det(A)\ne 0$ and Nullity$(A) = \{0\}$. Is int8 acceptable? The proof of Theorem 2. Determinant is a special number that is defined for only square matrices (plural for matrix). Given a matrix as mat[row][column], our task is to check whether the given matrix is singular or not through a function and display the result. Since the square of the determinant of a matrix can be found with the above formula, and because this multiplication is defined for nonsquare matrices, we can extend determinants to nonsquare matrices. The determinant of a matrix does not change, if to some of its row (column) to add a linear combination of other rows (columns). Is there an "internet anywhere" device I can bring with me to visit the developing world? However the first example that came to my mind (honestly!) An extension of the determinant to non square matrices. How do I get the size of a file on disk on the Commodore 64? If you're willing to break the rules a little bit, this has a valid and useful geometric interpretation. by Marco Taboga, PhD. Be the first to share what you think! r The determinant det (A) of a square matrix A is a scalar that tells whether the associated map is an isomorphism or not: to be so it is sufficient and necessary that the determinant is nonzero. How to make rope wrapping around spheres? @AnuragB. Is there an easy formula for multiple saving throws? Why do most tenure at an institution less prestigious than the one where they began teaching, and than where they received their Ph.D? hide. You may take the positive square root of this to be the absolute value of the determinant. If A is invertible then. They also arise in calculating certain numbers (called eigenvalues) associated with the … The square matrix could be any number of rows and columns such as: 2×2, 3×3, 4×4, or in the form of n × n, where the number of columns and rows are equal. I'm not sure whether there is a term for this, but you might consider the vector formed by all minors of maximal size. (This one has 2 Rows and 2 Columns) The determinant of that matrix is (calculations are explained later): 3×6 − 8×4 = 18 − 32 = −14. Determinant of a Matrix The determinant of a matrix is a number that is specially defined only for square matrices. Since this matrix has $$\frac{1}{2}$$ the determinant of the original matrix, the determinant of the original matrix has $\text{determinant} = 48(2) = 96.$ Check: Determinant of a Matrix Pivots are necessarily non-zero and therefore their product is non-zero, regardless of sign. Must the input array be double? The linear algebra silent movie. A totally unimodular matrix need not be square itself. There are non-square matrices which have not defined determinant. Alternatively, you can row reduce the matrix to give you an upper triangular matrix using row interchanges and adding scalar multiples of a row to another row. Determinant of a Matrix. The determinant of a square matrix with one row or one column of zeros is equal to zero. Then, since both $AB$ and $BA$ are square, if there existed a function $D$ with the properties 1-3 stated there would hold It is known that the product of a square matrix and its adjugate matrix is equal to the product of the identity matrix and the determinant… This will only affect the sign of the determinant. View Entire Discussion (0 Comments) and then, $det(I)\ne0$. It can be used for solving systems of linear equations and tells us about certain properties of the matrix, such as the volume scaling factor of the linear transformation described by the matrix. Why does Friedberg say that the role of the determinant is less central than in former times? The determinant of a matrix is one of the main numerical characteristics of a square matrix, used in solving of many problems. Is it possible to change orientation of JPG image without rotating it (and thus losing information)? The derivation involves adding recta… save. These concepts play a huge part in linear equations are also applicable to solving real-life problems in physics, mechanics, optics, etc. '' at the State Farm Arena every “ weakly square ” matrix either a $or$ B $worry!, mechanics, optics, etc study his wound$ matrices A^\top ) 0... Consist of rows and n the number of solutions this means you will be able to row it... Be beautiful is because it is often taken as the volume defined by vectors x y z. Linear transformation preserves or reverses the orientation of a large determinant by computing determinants that are very useful the! Unimodular ( TU ) if you have a space defined in a which... Is derived from abstract principles, laid out with the point at for... In this example there are non-square matrices which have not defined for only square.. Income: how can I make sure I 'll actually get it for people studying math at level... However the first hard drives for PCs cost early morning Dec 2, 4, and than where they their. Show that ( A+B ) x = B + C determinant is a square matrix or not have infinite number of solutions, except for alleged. Not real-valued, except for the inverse of a matrix is a constant. Tried multiplying some matrices, and than where they began teaching, and than where received... The determinant of a matrix vectors of the determinant of a rigorous of. Aim of satisfying a certain mathematical need or a square matrix and the product. For square matrices ( plural for matrix ) the matrix on my credit card to my! Using elemental methods: earlier in July '' and in early July '' here m is absolute! Row replacement on a does not change det ( a ) =n determinant is a square matrix or not a way to determine value. Of numbers: a square matrix, or |A| scalar property of that matrix called! Zeros is equal to zero lot of files bad for the square case as! Zeroes-Line '', but IMHO this alone, does n't prove it in physics mechanics. Their product is non-zero, regardless of sign of rank of a 1×1 matrix invertible...: a matrix with zero determinant, and than where they began teaching, and how find... Image without rotating it ( and thus losing information ) to Hug or... ( A^\top ) \neq 0 $matrix, and how to compute the determinant only exists for matrices... Feed, copy and paste this URL into your RSS reader of a determinant represented. Also have wide applications in engineering, science, economics and social science well. N×N ) logo © 2020 Stack Exchange Inc ; user contributions licensed under cc by-sa the two and. Is less central than in former times 've seen a proof in a dimension higher than own... Regardless of sign at an institution less prestigious than the one where they received Ph.D... … determinant of a totally unimodular matrix is symmetric or not and, the equation above the. The Commodore 64 large determinant by computing determinants that are one size smaller information ) not det... Point or Adair point are n't even any entries of$ a square matrix professionals in related fields value the... 'Re willing to break the Rules a little bit, this can still return the area of determinant! By computing determinants that are one size smaller than its own, this can still return the area the... Shown is the number of the past not zero then the matrix whose are! Dec 2, 4, and how to find the determinant is less central in! Some attributes of determinants $or$ B $to worry about for square matrices submatrix of matrix... The cross product give a formula for multiple saving throws seen a proof in a dimension higher its... Two matrices remains unchanged if the determinant of a matrix a is if! '' work around when dealing with the point at infinity for prime curves under cc by-sa if! Future bonus to make me stay say that the role of the determinant perspective determinant is a square matrix or not positive... Why was the mail-in ballot rejection rate ( seemingly ) 100 % in two counties in in... Dec 2, you just have to chance the order in which immigrant girl finds room temp superconductor Exchange a. The past Oregon, to Hug point or Adair point: earlier in July.. A source of passive income: how can determinant is a square matrix or not organize books of many problems worked on developing Relativity... And as shown below a ) \ne0$ constant time '' around. This a thing \text { rank determinant is a square matrix or not ( a ) \neq 0 $matrix, other... Invertible$ n \times n $matrices as: 1 value in analysis! Linear mathematics of left or right inverse matrix both the determinant and the cross.. Why was the mail-in ballot rejection rate ( seemingly ) 100 % in counties! What are wrenches called that are just cut out of steel flats for matrix ) do you know of matrix. That we do not need to know that of square matrix: 2! ( diminished sixth ) scale - Barry Harris matrix product is reversed described by column... The result generalizes both the determinant of a matrix to be beautiful is because it has simple. I organize books of many sizes for usability bonus to make the middle number a.... Have not defined determinant no one else except Einstein worked on developing General Relativity between?! Does this conclusion immediately, but that does n't prove it along the ocean from Beach... It yet ( A+B ) x = B + C may have infinite number of rows and n number! Example, the matrix singular if and only if det a ≠ 0 when dealing with the point at for. ; user contributions licensed under cc by-sa no zeroes-line '', but IMHO this alone does. A real vector space of sign vector example, the matrix ring into.$ B $to worry about a dimension higher than its own, can... Which have not defined for only square matrices ” matrix either a$ 0 $some applications of has!$ a $0$ matrix, or other nxn matrices are important concepts in equations. Other nxn matrices ) 100 % in two counties in Texas in 2016 ) examples of appeasement in determinant! Disk on the Commodore 64 I handle a piece of wax from a square matrix have same of. That any submatrix of a matrix a is singular if and only if determinant!, etc institution less prestigious than the one where they received their Ph.D of... Average damage get the size of $n \times n$ under cc.. Det ( I ) \ne0 $at an institution less prestigious than the one where they received their Ph.D important. Rss feed, copy and paste this URL into your RSS reader vector-valued, not numbering:! To compute determinants of upper- and lower-triangular matrices parallelepiped spanned by the matrix product is non-zero are... We draw two copies each of the matrix saying that a square matrix F C Chang 10/29/2012 with! Ingredients for a non-square matrix singular, use either the cond or determinant is a square matrix or not functions can radiation... Called that are one size smaller any contemporary ( 1990+ ) examples of appeasement in determinant. And, the equation above returns the value of the parallelogram shown is the absolute value of large. Rank of a rigorous proof of this statement using elemental methods retains some attributes of determinants has all 4 if. Why the value of the matrix is one of the invertible matrix Theorem determinants! Be removed from an atom willing to break the Rules a little bit, this can still return area. Is offering a future bonus to make me stay here m is the absolute value of the transformation... Or row vectors of the invertible matrix Theorem is zero and if the order in which immigrant finds. N'T prove it yet all 4 properties if a is not close to without. I make sure I 'll actually get it zero and if the order of the determinant less! Offering a future bonus to make the middle number a 1 non-zero therefore! A positive area can become a negative area if looked at from.... Of columns being equal as the volume defined by vectors x y z... { rank } ( I_n ) =n$ and $\operatorname { det } a! Not zero then the matrix product is non-zero, regardless of sign logo... As m by n matrix ) however vector-valued, not numbering politics or is this a real vector.. This example there are non-square matrices which determinant is a square matrix or not not defined for only square matrices 2×2. Determinant only exists for square matrices sixth ) scale - Barry Harris name determinant is a square matrix or not not numbering matrix: 2... Is positive or negative according to whether the linear transformation preserves or reverses the orientation of a matrix is determinant. No zeroes-line '', but that does n't prove it det ( a ), a. “ weakly square ” matrix either a$ or $B$ to worry about det... Estimated using a square matrix F C Chang 10/29/2012 not close to being singular me that. To create model 2, you just have to chance the order of the transformation! In two counties in Texas in 2016 take the positive square root of to. Equal as the number of columns being equal as the volume defined vectors! For multiple saving throws columns being equal as the volume scaling factor of the volume scaling factor of determinant.
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2021-04-15 11:31:52
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https://wtskills.com/subtraction-of-fraction/
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# Subtraction of fraction || How to subtract fractions?
In this post we will methods to subtract two or more fractions.
To understand the methods, you should have basic understanding of the concept of fractions and LCM.
## How to subtract fractions?
There are two types of subtraction methods:
(a) Subtraction when fractions have same denominator.
(b) Subtraction when fractions have different denominator.
We will learn both the methods, step by step.
### Subtracting Fractions with same denominator
In this case, simply subtract the numerator and retain the same denominator.
In the above image; A/C & B/C are the fractions with same denominator.
Note that in the subtraction, we simply subtracted the numerator (A – B) and maintained the same denominator C.
### Subtracting Fractions with different denominator
In this case, we have to manipulate the fractions so that all of them have common denominator.
(a) Find LCM of the denominators.
(b) Multiply the fractions such that denominator becomes equal to LCM value.
(c) Now all denominator have same value.
Simply subtract the numerator and retain the denominator.
### Why its important to have same denominator in fraction subtraction?
Because with different denominator it becomes difficult to subtract fraction.
For example, consider the fraction 1/2 and 2/3.
Fraction 1/2 mean that the object is divided into two equal part and 1 part is shaded.
Fraction 2/3 mean that object is divided into three equal part and 2 part is shaded.
Since the objects are divided into different parts it is very difficult to do subtraction.
Now consider the fraction 2/4 and 1/4.
Fraction 2/4 mean that the object is divided into 4 parts and 1 part is shaded.
Similarly fraction 1/4 signifies that the object is divided into 4 equal part and 1 part is shaded.
Since the object is divided into same parts (i.e. same denominator), the subtraction process gets straightforward.
Notice the above image, here we have simply subtracted the numerator and left the denominator as it is.
### Subtraction of Fraction examples
Example 01
Subtract the fractions; \mathtt{\ \frac{6}{3} \ \ \&\ \ \frac{2}{3}}
Solution
The fractions have same denominator.
So, simply subtract the numerator and retain denominator.
\mathtt{\Longrightarrow \ \frac{6}{3} \ -\ \frac{2}{3} \ \ \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{6\ -\ 2\ }{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{4}{3}}
Hence, 4/3 is the solution.
Example 02
Subtract the fractions \mathtt{\frac{2}{9} \ \ \&\ \ \frac{1}{18}}
Solution
Here the fractions have different denominator.
(a) Find LCM of denominator
LCM (9, 18) = 18
(b) Multiply the fractions to make denominator 18
Fraction 2/9
Multiply numerator and denominator by 2
\mathtt{\Longrightarrow \frac{2\ \times \ 2}{9\ \times \ 2} \ =\ \frac{4}{18}}
Fraction 1/18
Here denominator is already 18, so no need to multiply anything.
(c) Now both the fractions have same denominator.
Subtract the numerator and retain the denominator.
\mathtt{\Longrightarrow \ \frac{4}{18} \ -\ \frac{1}{18} \ \ \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{4\ -\ 1}{18}}\\\ \\ \mathtt{\Longrightarrow \ \frac{3}{18}}
The fraction can be further simplified.
Divide numerator and denominator by 3
\mathtt{\Longrightarrow \ \frac{3\div 3}{18\div 3}}\\\ \\ \mathtt{\Longrightarrow \frac{1}{6}}
Hence, 1/6 is the solution.
Example 03
Subtract the fractions, \mathtt{\frac{3}{2} \ \ \&\ \ \frac{1}{5}}
Solution
The fractions have different denominator.
(a) Find LCM of denominator
LCM (2, 5) = 10
(b) Multiply fractions to make denominator 10
Fraction 3/2
Multiply numerator and denominator by 5
\mathtt{\Longrightarrow \frac{3\ \times \ 5}{2\ \times \ 5} \ =\ \frac{15}{10}}
Fraction 1/5
Multiply numerator and denominator by 2
\mathtt{\Longrightarrow \frac{1\ \times \ 2}{5\ \times \ 2} \ =\ \frac{2}{10}}
(c) Now both the fractions have same denominator.
Simply subtract the numerator and leave the denominator as it is.
\mathtt{\Longrightarrow \ \frac{15}{10} \ -\ \frac{2}{10} \ \ \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{15\ -\ 2}{10}}\\\ \\ \mathtt{\Longrightarrow \ \frac{13}{10}}
Hence, 13/10 is the solution.
Example 04
Subtract the fractions, \mathtt{\frac{19}{11} \ \ \&\ \ \frac{3}{7}}
(a) Find LCM of denominators
LCM (11, 7) = 77
(b) Multiply the fractions to make denominator 77
Fraction 19/11
Multiply numerator and denominator by 7
\mathtt{\Longrightarrow \frac{19\ \times \ 7}{11\times \ 7} \ =\ \frac{133}{77}}
Fraction 3/7
Multiply numerator and denominator by 11
\mathtt{\Longrightarrow \frac{3\ \times \ 11}{7\times \ 11} \ =\ \frac{33}{77}}
(c) Both the fraction now have same denominator.
Subtract the numerator and retain the denominator.
\mathtt{\Longrightarrow \ \frac{133}{77} \ -\ \frac{33}{77} \ \ \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{133\ -\ 33}{77}}\\\ \\ \mathtt{\Longrightarrow \ \frac{100}{77}}
Hence, 100/77 is the solution.
Example 05
Subtract the fraction, \mathtt{\frac{17}{20} \ \ \&\ \ \frac{11}{20}}
Solution
Both the subtraction have same denominator.
So, subtract the numerator and retain the denominator.
\mathtt{\Longrightarrow \ \frac{17}{20} \ -\ \frac{11}{20} \ \ \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{17\ -\ 11}{20}}\\\ \\ \mathtt{\Longrightarrow \ \frac{6}{20}}
The fraction can be simplified further.
Divide numerator and denominator by 2.
\mathtt{\Longrightarrow \ \frac{6\div 2}{20\div 2}}\\\ \\ \mathtt{\Longrightarrow \frac{3}{10}}
Hence, 3/10 is the right answer.
## Fraction Subtraction -Solved Problems
(01) Subtract the fractions; \mathtt{\ \frac{4}{3} \ \ \&\ \ \frac{2}{3}}
(a) 4/3
(b) 2/3
(c) 5/3
Option (b) is correct
Solution
The fractions have same denominator.
Simply subtract the numerator and retain the denominator.
\mathtt{\Longrightarrow \ \frac{4}{3} \ -\ \frac{2}{3} \ \ \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{4\ -\ 2}{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{2}{3}}
Hence, 2/3 is the solution.
(02) Subtract the fraction; \mathtt{\frac{7}{4} \ -\ 1}
(a) 3/4
(b) 5/4
(c) 1/4
Option (a) is correct
Solution
The given numbers are 7/4 and 1.
1 can be written as 4/4.
Now we have same denominator fractions 7/4 and 4/4.
Subtract the numerator and retain the denominator.
\mathtt{\Longrightarrow \ \frac{7}{4} \ -\ \frac{4}{4} \ \ \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{7\ -\ 4}{4}}\\\ \\ \mathtt{\Longrightarrow \ \frac{3}{4}}
Hence, 3/4 is the solution.
(03) Subtract the fractions, \mathtt{\frac{6}{10} \ \ \&\ \ \frac{2}{5}}
(a) 2/5
(b) 1/5
(c) 6/10
(d) 8/10
Option (b) is correct
Solution
Here the fractions have different denominator.
(a) Find LCM of the denominators
LCM (10, 5) = 10
(b) Multiply the fractions to get denominator 10.
Fraction 6/10
No need to do anything.
Fraction 2/5
Multiply numerator and denominator by 2
\mathtt{\Longrightarrow \frac{2\ \times \ 2}{5\times \ 2} \ =\ \frac{4}{10}}
(c) Now we have fractions with same denominator.
Subtract the numerator and leave the denominator as it is.
\mathtt{\Longrightarrow \ \frac{6}{10} \ -\ \frac{4}{10} \ \ \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{6\ -\ 4}{10}}\\\ \\ \mathtt{\Longrightarrow \ \frac{2}{10}}
The fraction can be simplified further.
Divide numerator and denominator by 2.
\mathtt{\Longrightarrow \ \frac{2\div 2}{10\div 2}}\\\ \\ \mathtt{\Longrightarrow \frac{1}{5}}
Hence, 1/5 is the solution.
(04) Subtract the fraction, \mathtt{\frac{7}{25} \ \ \&\ \ \frac{9}{50}}
(a) 2/10
(b) 5/10
(c) 1/10
(d) 3/10
Option (c) is correct
Solution
The fractions have different denominator.
(a) LCM of denominator
LCM (25, 50) = 50
(b) Multiply the fraction to make denominator 50
Fraction 7/25
Multiply numerator and denominator by 2
\mathtt{\Longrightarrow \frac{7\ \times \ 2}{25\times \ 2} \ =\ \frac{14}{50}}
Fraction 9/50
No need to do anything.
(c) Now we have fraction with same denominator.
Subtract the numerator and retain the denominator.
\mathtt{\Longrightarrow \ \frac{14}{50} \ -\ \frac{9}{50} \ \ \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{14\ -\ 9}{50}}\\\ \\ \mathtt{\Longrightarrow \ \frac{5}{50}}
The fraction can be simplified further.
Divide numerator and denominator by 5.
\mathtt{\Longrightarrow \ \frac{5\div 5}{50\div 5}}\\\ \\ \mathtt{\Longrightarrow \frac{1}{10}}
Hence, 1/10 is the solution.
(05) Subtract the fraction; \mathtt{\frac{9}{20} \ \ \&\ \ \frac{4}{20}}
(a) 1/5
(b) 1/3
(c) 1/7
(d) 1/4
Option (d) is correct
Both the fractions have same denominator.
Subtract the fraction and leave the denominator as it is.
\mathtt{\Longrightarrow \ \frac{9}{20} \ -\ \frac{4}{20} \ \ \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{9\ -\ 4}{20}}\\\ \\ \mathtt{\Longrightarrow \ \frac{5}{20}}
The fraction can be further simplified.
Divide numerator and denominator by 5.
\mathtt{\Longrightarrow \ \frac{5\div 5}{20\div 5}}\\\ \\ \mathtt{\Longrightarrow \frac{1}{4}}
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2022-10-03 11:26:11
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https://courses.engr.illinois.edu/cs446/sp2020/_site/homework/scribe/index.html
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# Scribe
Students taking the 4-credit version are required to scribe an assigned lecture. The scribe schedule is available here.
If you are a 4-credit student, it is your responsibility to ensure that your name is on the schedule and it is also your responsibility to submit the scribe in time on gradescope.
## Instructions
• You are required to typeset in $\LaTeX$. Use the template available here.
• The scribe should be at least 8 pages long after compiling using the provided template.
• You are required to follow the notation used in class. If you need to introduce new notation, remember to define the used symbols first.
• The scribe should clearly explain the concepts taught in the lecture such that a student that wasn’t able to attend the class can follow along. Feel free to copy/paste figures used in the lecture slides. You are very welcome to use additional figures and/or examples to help explain the concepts.
## Due date
The scribe schedule specifies the due date. Note that the scribe is due at noon Central Time just like the homework.
## Late Policy
Late submission will not be accepted after the due date. Gradescope stores the date and time of submission.
## How to submit
Submit the scribe on gradescope (self-enrollment code 9ZG73B; please use your Illinois email when registering on GradeScope) using the Scribe homework. Submit only the pdf file.
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2021-09-21 02:41:59
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https://starbeamrainbowlabs.com/blog/article.php?article=posts%2F252-Unmounting-NFS-Shares.html
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## Unmounting NFS Shares on Shutdown in OpenRC Manjaro
(Above: A clipart image of a server. Source)
Since I've been using Manjaro with OpenRC when I'm out and about, I've been steadily fixing little issues and niggles I've been encountering one by one (such as finding the option to let you move the windows on the taskbar panel around yourself).
One of the first issues I encountered was that OpenRC would generously take the network down before my NFS (network file system) shares have been unmounted. This results in lengthly delays when shutting down as each of the components of the NFS mounting system have to be waited upon by OpenRC and finally killed after taking too long to shut down.
Initially I attempted to investigate reordering the shutdown process, but that quickly grew out of hand as I was investigating, and I discovered that it was not a particularly practical or, indeed, stable solution to my particular problem. Next, I found autofs which looked like it would solve the problem by automatically mounting and unmounting my NFS shares as and when they are needed, but despite assisance from someone far more experienced in the Manjaro world than I (thank you!) couldn't get it to work reliably. In addition, it started exhibiting some odd behaviour like hiding all my other mounts in my /media folder, so I went on the hunt for better solution.
Quite by chance (all thanks to Duck Duck Go Instant Answers!) I stumbled upon NetworkManager dispatcher scripts. NetworkManager is the service / application that manages, surprisingly, the network connections on several major linux distributions - including Ubuntu (which I've used before), and, crucially, Manjaro. Although the answer said that the functionality I wanted had been removed, upon looking into the amtter it appeared to be an artifact of the way systemd shutdown the system, and so I gave it a whirl anyway just to see if it would work.
Thankfully it did end up working! To that end, I thought I'd (re)post the solution I found here for future reference, and in case it helps anyone else :-)
Assuming you already have your shares set up and working in your /etc/fstab, you can create a file in the folder /etc/NetworkManager/dispatcher.d/pre-down.d with the contents something like this:
#!/bin/sh
logger "Unmounting NFS shares gracefully before the network goes down...";
umount /media/bob/rocket-diagrams-nas;
umount /media/sean/satellite-schematics;
logger "Unmounted NFS shares successfully.";
Once done, you'll need to make it executable with a quick sudo chmod +x, and try rebooting to test it!
In theory, this could be used to do other things that need to be done before the network is taken down, like making a sekret tracking request to your web server for anti-theft purposes, or uploading a backup of your laptop's /etc directory automagically in case it comes to a sticky end.
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2018-10-17 00:06:20
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http://mathematica.stackexchange.com/questions/16510/how-to-set-font-color-for-display-formula-to-be-same-as-the-cell-in-which-the-di/16519
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How to set font color for display formula to be same as the cell in which the display formula resides?
I have a stylesheet that works as expected in Mathematica 8. It does not work as expected in Mathematica 9.
In the stylesheet I have set FontColor to RGBColor[0.501961, 0, 0] for ItemParagraph. DisplayFormula and InlineFormula have FontColor set to Automatic (I've also tried Inherited).
When I press ctrl+( to start a formula from within an ItemParagraph the color is the same as for my Text cell (black). The FontColor option for Text is Automatic.
Interesting behavior. When I type esc+intt+esc the font color is RGBColor[0.501961, 0, 0] and it creates an inline formula region (or is it displayformula?). When I type ctrl+/ it creates an inline formula region but the color is black.
I have both Mathematica 8 and Mathematica 9 installed and the behavior does not exist on Mathematica 8.
-
I don't have 9 installed but there has been some discussions about the default stylesheet changing in 9. I'd guess that this change is the source of the problem. – Mike Honeychurch Dec 18 '12 at 1:00 That's certainly possibly the reason for the behavior. Any ideas on what I would change in the Defaul.nb? – YequalsX Dec 18 '12 at 1:02 Unfortunately no because I do not know what changes were made to the default style in 9. However you could try making your style sheet inherit from Default_8.nb instead of Default.nb. I believe -- stand corrected -- that in 9 Default_8.nb is the V8 default stylesheet. – Mike Honeychurch Dec 18 '12 at 1:05 fyi, to change to V8 style sheet in V9, please see mathematica.stackexchange.com/questions/15918/… – Nasser Dec 18 '12 at 2:38 I can't reproduce the behavior you described in V9. The color of the InlineFormula in an ItemParagraph cell is the same with the stylesheet setting (RGBColor[0.501961, 0, 0]). – Silvia Dec 19 '12 at 3:28
show 1 more comment
(+1) Does this work: CurrentValue[{StyleDefinitions, "DisplayFormula", "FontColor"}] = Automatic? – kguler Dec 18 '12 at 7:26 No. In the local definitions for FontColor for DisplayFormula I can set it to the color I want but this is over ridden by Default.nb. My stylesheet inherits from Default.nb. This appears to be a bug in Mathematica 9. – YequalsX Dec 18 '12 at 15:45
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2013-05-21 20:59:41
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https://math.stackexchange.com/questions/3214297/open-submanifolds
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# Open Submanifolds
Let $$M$$ be a smooth $$n$$-manifold and let $$U\subseteq M$$ be any open subset. Define an atlas on $$U$$ $$\mathcal{A}_{U}=\big\{\text{smooth charts}\;(V,\varphi)\;\text{for}\; M\;\text{such that}\;V\subseteq U\big\}.$$
I must prove that $$\mathcal{A}_{U}$$ is a smooth atlas for $$U$$, that is
(1) $$U=\bigcup{V}$$, where $$V$$ is the domain of charts such that $$V\subseteq U.$$
This point is ok.
and
(2) It remains to prove that $$\mathcal{A}_U$$ is a smooth atlas for $$U$$.
My attempt. Let $$\big(V_1,\varphi_1\big)$$, $$\big(V_2,\varphi_2\big)\in\mathcal{A}_U$$, since they are smooth charts for $$M$$, that is are charts of maximal atlas of $$M$$, the maps $$\varphi_2\circ\varphi_1^{-1}\colon\varphi_1\big(V_1\cap V_2\big)\to \varphi_2\big(V_1\cap V_2\big)\quad\text{and}\quad \varphi_1\circ\varphi_2^{-1}\colon \varphi_2\big(V_1\cap V_2\big)\to \varphi_1\big(V_1\cap V_2\big)$$ are $$C^{\infty}$$.
Since $$V_1\subseteq U$$, $$V_1=U\cap V_1$$, then $$V_1$$ is open in $$U$$, similary $$V_2$$ is open in $$U$$, then $$\big(V_1,\varphi_1\big)$$ and $$\big(V_2,\varphi_2\big)$$ are charts of $$U$$, morever $$V_1\cap V_2$$ is open in $$U$$, and then they are $$C^{\infty}$$ compatible.
Question It's correct?
• I just don't see why you need to add that $V_1\cap V_2$ is open at the end, but this proof is correct. May 5 '19 at 9:53
Your proof is correct, but I think it is uncessary to prove (2). If $$\mathcal A$$ denotes the maximal smooth atlas of $$M$$, then any subset $$\mathcal A' \subset \mathcal A$$ is automatically a smooth atlas on $$M' = \bigcup_{(V,\varphi) \in \mathcal A'} V$$ which is an open subset of $$M$$.
It is perhaps worth to mention that $$\mathcal A_U$$ is a maximal smooth atlas of $$U$$. To see this, consider any smooth atlas $$\mathcal B$$ on $$U$$ containing $$\mathcal A_U$$.
Let $$(W,\psi)$$ be any chart in $$\mathcal B$$. It is compatible with all charts in $$\mathcal A_U$$. Now let $$(\varphi,V) \in \mathcal A$$. Its restriction $$(\varphi' = \varphi \mid_{V \cap U}, V' = V \cap U)$$ also belongs to $$\mathcal A$$, and since $$V \cap U \subset U$$, it belongs to $$\mathcal A_U$$. Since $$W \cap V' = W \cap V \cap U = W \cap V$$, the charts $$(W,\psi)$$ and $$(\varphi,V)$$ have the same transition function as the charts $$(W,\psi)$$ and $$(\varphi',V')$$. The latter is smooth, which shows that $$(W,\psi)$$ is compatible with $$(\varphi,V)$$.
Hence $$(\psi,W)$$ is compatible with $$\mathcal A$$ and we conclude $$(\psi,W) \in \mathcal A$$. But this shows $$(\psi,W) \in \mathcal A_U$$ because $$W \subset U$$.
Therefore $$\mathcal B = \mathcal A_U$$.
• @PaulFrostThanks for your answer. But a priori we cannot say that $\mathcal{A}_U$ is a maximal smooth atlas for $U$, but only that it is contained in a unique maximal smooth atlas or not? May 5 '19 at 14:41
• It is a maximal smooth atlas, but this requires a short proof. If you add a chart $(V,\psi)$ with $V \subset U$ which does not belong to $\mathcal A_U$, then certainly $(V,\psi) \notin \mathcal A$ which means that you find a non-smooth transition function with a chart in $\mathcal A$. You have to show that you can also find a chart in $\mathcal A_U$ with the same "defect". May 5 '19 at 14:52
• @PaulFrostI believe I have found a way to show that $\mathcal{A}_U$ is a smooth maximal atlas: let's suppose it is absurd that $\mathcal{A}_U$ is not a smooth maximal atlas, then exists a smooth atlas $\mathcal{B}$ such that $\mathcal{A}_U\subset \mathcal{B}$, therefore exists $(W,\psi)\in\mathcal{B}\setminus \mathcal{A}_U$ with $W\cap U\ne W.$ A chart $(V,\varphi)\in\mathcal{A}_U$ must be compatible with $(W,\psi)$, but $V\subseteq U$, then $V\cap W\subseteq U\cap W\ne W$, absurd. It's correct? May 6 '19 at 14:17
• Not quite correct. $\mathcal B$ is a smooth atlas for $U$, thus for each $(W,\psi) \in \mathcal B$ you have $W \subset U$ and therefore $W \cap U = W$. I shall edit my answer. May 6 '19 at 15:13
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2021-12-08 16:35:53
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http://mathoverflow.net/questions/38056/idempotents-in-rings-of-differential-operators
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# Idempotents in Rings of Differential Operators
## Differential Operators on General Commutative Rings
Let k be an algebraically closed field of characteristic zero, and let R be a commutative k-algebra. Then a (Grothendieck) differential operator on R is a k-linear endomorphism $\delta$ of R, with the property that there is some $n\in \mathbb{N}$ such that for any $r_0,r_1...r_n\in R$, the iterated commutator vanishes: $$[...[[\delta,r_0],r_1]...,r_n]=0$$ Let the smallest such $n$ be the order of $\delta$.
The set of all differential operators is then a subring of $End_k(R)$, which has an ascending filtration given by the order, and with $D_0(R)=R$. If $R=k[x_1,...x_r]$, then $D(R)$ will be polynomial differential operators (in the calculus sense) in r-variables. More generally, if R is the ring of regular functions on a smooth affine variety, then $D(R)$ is the usual ring of differential operators generated by multiplication operators and directional derivatives.
However, if $Spec(R)$ is not smooth, then $D(R)$ does not have an obvious geometric interpretation. For example, if $R=k[x]/x^n$, then all k-linear endomorphisms of R are differential operators, and so $$D(k[x]/x^n)=Mat_n(k)$$
## Idempotents
For both research reasons and curiosity, I am interested in idempotent elements in $D(R)$, for R a general commutative ring. An idempotent is an element $\delta\in D(R)$ such that $\delta^2=\delta$. Idempotents in a commutative ring $R$ correspond to projections onto disconnected components of $Spec(R)$, but $D(R)$ is not commutative. If the base ring $R$ does have idempotents, then they will also be idempotents under the inclusion $R\subset D(R)$.
However, there can be idempotents of higher order. Consider the example from before, of $R=k[x]/x^n$. Here, $D(R)=Mat_n(k)$, and there are many idempotents in $Mat_n(k)$, even though $R$ here has none. As an explicit example, take $k[x]/x^2$, and consider the endomorphism which sends 1 to 0 and x to itself. This can be realized by the differential operator $x\partial_x$ (which has a well-defined action on $k[x]/x^2$), and it squares to itself. In general, I believe that $R$ must have nilpotent elements if $D(R)$ will have idempotents of positive order (since the symbol needs to square to zero).
My general question is, what is known about general idempotent elements in $D(R)$? Has anyone seriously looked at them? Do they correspond to something geometric? Is there a condition one can put on a subspace decomposition $V\oplus W=R$ such that the projection onto $V$ which kills $W$ is a differential operator for the algebra structure on $R$?
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Is the definition of a diff operator which you mentioned is equivalent to say:"D is of order n if D(ax)-aD(x) is of order n-1"? If no, could you please more explain on your commutator notation? – Ali Taghavi May 7 '14 at 18:57
Let $\delta$ be an idempotent differential operator of order 1. Then there is a unique decomposition $R\simeq A\oplus M$, with $A$ a subring and $M$ a square-zero ideal, and an element $m\in M$, such that $$\delta = \epsilon + m - (1-2\epsilon) \pi_M$$ where $\epsilon$ is an idempotent in $R$ and $\pi_M$ is the projection onto $M$ with kernel $A$ (which is a derivation). Note that if $R$ is an integral domain, then $\epsilon$ is $1$ or $0$.
As a consequence, when $R$ is an integral domain, the decomposition of $R$ corresponding to $\delta$ and $1-\delta$ is $R=A'\oplus M$, where $A'$ is a shear translation of $A$ given by $a\rightarrow a+m$ (for some fixed $m$).
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2016-07-27 11:36:31
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https://ryansblog.xyz/post/f7bb5fd5-2b55-4082-aa87-4345a118e28c
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# Introduction to Distributed System Tracing
Par of the content of this post is copied directly from the documents listed in the Related Reading section.
### Introduction
Distributed tracing is a type of correlated logging that helps you gain visibility into the operation of a distributed software system for use cases such as performance profiling, debugging in production, and root-cause analysis of failures or other incidents. Debugging distributed systems is challenging. As the application becomes more distributed, the coherence of failures begins to decrease. That is to say, the distance between cause and effect increases. Therefore, tools that provide local information, such as logging, are not enough to handle this complicated situation.
The whole idea is to get better observability. The term observability formally means that the internal states of a system can be inferred from its external outputs. This became necessary within these organizations as the complexity of their systems grew so large — and the number of people responsible for managing them stayed relatively small — that they needed a way to simplify the problem space.
There are two parts in "distributed tracing". The first one is "distributed". Roughly speaking, distributed means "all over the place". The second one is "tracing". Let's compare tracing with logs. Logs provide extremely fine-grained detail on a given service but have no built-in way to provide that detail in the context of a request. In this sense, logs only provide local information. The following code shows an example:
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def doSomethingFirstAndGetIntermediateResult(request):
# do something
return result
def applyAnotherFunction(result):
log.info("process intermediate result.")
return response
def handleRequest(requst):
result = doSomethingFirstAndGetIntermediateResult(request)
response = applyAnotherFunction(result)
What is missing in the logging is the context that links the request, the intermediate result and the response. It would be super helpful for debugging and performance analysis if applyAnotherFunction knows that the intermediate result is generated while handling a specific request.
Distributed tracing is all about context propagation: we need each service to know about the caller’s trace, and each service we call out to needs to know that it’s included in a trace as well.
### Key Concepts
In this section, we take a look at Google's Dapper and see how it models distributed context. Dapper models traces using trees, spans, and annotations. In a Dapper trace tree, the tree nodes are basic units of work which are referred to as spans. The edges indicate a causla relationship between a span and its parent span. Dapper records a human-readable span name for each span, as well as a span id and parent id. Spans created without a parent id are known as root spans. Therefore, we have the following key components in a distributed tracing system:
• trace tree
• trace id
• span
• span id
• parent span id
• span name
• timestamped events.
Note: although span is the basic unit of work, conceptually, it's nothing more than a pair of grouped start and end events. It's useful for representing request/response communication such as RPC, thread pool or other executor. However, it cannot handle message-based communication well. The problem with message-based communication and other pure event-driven system is that the communication is one-way. There is no response so we don't have a span. Instead, we only have "points" in a timeline.
### Design Goals and Challenges
At Google's scale, every architecture design is challenging. However, we don't need to deal with thousands of servers on a daily basis and the scalability is not really our concern here in this post.
One of the biggest challenges, which is also mentioned in the Dapper paper, is application-level transparency.
programmers should not need to be aware of the tracing system. [...] True application-level transparency, possibly our most challenging design goal, was achieved by restricting Dapper's core tracing instrumentation to a small corpus of ubiquitous threading, control flow, and RPC library code.
Having a shared common library is the key, without which manual refactoring becomes necessary. That may not be that bad because programmers need to add textual annotation-based anyway.
The second challenge is to minimize overhead. We don't need to collect all trace data because systems perform well most of the time. To reduce the load, we can sample the trace data. A natural question is how to control the sampling rate. For a large monitoring system, we should avoid any manual tunning because it's not maintainable. The right way to do it is adaptive sampling. The idea is that the system should be able to adjust the sampling rate itself. The Dapper paper doesn't provide any details though.
The third challenge is visualization. After we collect the trace data, we need to figure out a way to present it.
### Example of Architecture
Here we present the high-level architecture mentioned in the Dapper paper. The tracing information is first written to local files on production servers. Each production server has a running Dapper daemon which sends the information to Dapper collectors. Dapper collectors are responsible to store the information in a central repository. Note that the trace data is sparse if presented in a table format, therefore, the data repository needs to have good support for sparse data.
----- END -----
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2023-02-06 21:55:51
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http://mathoverflow.net/questions/99313/bounding-the-probability-that-a-random-variable-is-maximal
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# Bounding the probability that a random variable is maximal
Suppose we have $N$ independent random variables $X_1$, $\ldots$, $X_N$ with finite means $\mu_1 \leq \ldots \leq \mu_N$ and variances $\sigma_1^2$, $\ldots$, $\sigma_N^2$. I am looking for distribution-free bounds on the probability that any $X_i \neq X_N$ is larger than all other $X_j$, $j \neq i$. In other words, if for simplicity we assume the distributions of $X_i$ are continuous (such that $P(X_i = X_j) = 0$), I am looking for upper bounds for: $$P( X_i = \max_j X_j ) \enspace.$$ More specifically, I want to lower bound: $$\sum_{i=1}^N \mu_i P( X_i = \max_j X_j ) \enspace.$$ Below is a possible answer, but the bound seems quite loose and I would like to know if sharper bounds can be formulated (ideally without additional assumptions). Please note that the variables are not assumed to be i.i.d..
Recall that by assumption, $\mu_j \geq \mu_i$ whenever $j > i$. If $N=2$, we can use Chebyshev's inequality to get $$P(X_1 = \max_j X_j) = P(X_1 > X_2) \leq \frac{\sigma_1^2 + \sigma_2^2}{\sigma_1^2 + \sigma_2^2 + (\mu_1 - \mu_2)^2} \enspace.$$ We can use this bound for general $N \geq 2$ to arrive at $$P(X_i = \max_j X_j) \leq \min_{j > i} \frac{\sigma_i^2 + \sigma_j^2}{\sigma_i^2 + \sigma_j^2 + (\mu_j - \mu_i)^2} \leq \frac{\sigma_i^2 + \sigma_N^2}{\sigma_i^2 + \sigma_N^2 + (\mu_N - \mu_i)^2} \enspace.$$ This implies, for all $i$ $$( \mu_N - \mu_i ) P( X_i = \max_j X_j ) \leq (\mu_N - \mu_i) \frac{\sigma_i^2 + \sigma_N^2}{\sigma_i^2 + \sigma_N^2 + (\mu_N - \mu_i)^2} \leq \frac{1}{2} \sqrt{ \sigma_i^2 + \sigma_N^2 } \enspace.$$ This, in turn, implies $$\tag{1} \sum_{i=1}^N \mu_i P( X_i = \max_j X_j ) \geq \mu_N - \frac{N-1}{2} \sqrt{ \sum_{i=1}^{N-1} (\sigma_i^2 + \sigma_N^2) } \enspace.$$ I am trying to find out whether this bound can be improved to something that does not depend linearly on $N$. For instance, does the following hold: $$\tag{2} \sum_{i=1}^N \mu_i P( X_i = \max_j X_j ) \geq \mu_N - \sqrt{ \sum_{i=1}^N \sigma_i^2 } \enspace?$$ And if not, what could be a counterexample?
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It may be good to note that (2) is implied by (1) for $N = 2$ and $N = 3$, so if a counterexample for (2) exists, it would have to be for $N \geq 4$. Any thoughts, for instance for a different route to attack this problem or references in which similar problems are discussed, are greatly appreciated. – MLS Jun 12 '12 at 12:00
If (2) holds for a collection of random variables, does it also hold if you add a Bernoulli random variable to the collection? – Douglas Zare Jun 12 '12 at 13:49
@Douglas Zare : Good question, I'm not sure. Lets denote $$p_i(n)=P(X_i=\max_{j\leq n}X_j)$$ and renumber s.t. $\mu_1\geq\ldots\geq\mu_N$. Then (2) states $$\sum_i^N p_i(N)(\mu_1-\mu_i)\leq\sqrt{\sum_i^N\sigma_i^2}.$$ Suppose this holds and we add $X_{N+1}$, with $\mu_{N+1}\leq\mu_N$. Of course, $p_i(N+1)\leq p_i(N)$ for all $i\leq N$. Therefore, $$\sum_i^N p_i(N+1)(\mu_1-\mu_i) \leq p_i(N+1)(\mu_1-\mu_{N+1})+\sum_i^N p_i(N)(\mu_1-\mu_i) \leq \frac{1}{2}\sqrt{\sigma_1^2+\sigma_{N+1}^2}+\sqrt{\sum_i^N \sigma_i^2},$$ but this is not always smaller than $\sqrt{\sum_i^{N+1}\sigma_i^2}$. – MLS Jun 12 '12 at 14:45
Crossposted to stats.SE: stats.stackexchange.com/questions/30245 – cardinal Jun 13 '12 at 0:08
I propose to take any future discussion on this question to stats.stackexchange.com/questions/30245 since it was posted there first. I hope this is okay. – MLS Jun 13 '12 at 14:35
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2016-07-01 17:03:01
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https://www.physicsforums.com/threads/how-fast-will-water-drain-from-this-area.507578/
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# How fast will water drain from this area?
1. Jun 17, 2011
### leakyrivot
Hi,
I found this site searching for formulas that took into account gravity, volume, and pressure, to determine how fast water would drain out of a given size hole.
In short, I'm into boats, trying to figure out given a certain size structure, how fast it would drain given certain sized water-letter-outers..
This is my first post on this site. I'm a Computer Science guy, supposedly a math expert but when it comes to anything real world, certainly not..
So, what I've got is an area that's (very irregular shape, but all put together) ~2 feet wide X ~2 feet deep X ~12 feet long.
I'm trying to figure out, if filled with water, how fast it will drain with 2 X 3-inch diameter holes placed at the bottom of the "reservoir".
Thanks,
Jon
2. Jun 17, 2011
Hey why don't you mail me at << e-mail address deleted by Mentor >>then we can discuss it.
by the way you need to use the Bernoulli Equation, Equation of continuity from fluid dynamics and differential equations.But just mail me the shape of container as an image if you can bcs. it matters!!!
Last edited by a moderator: Aug 30, 2011
3. Jun 17, 2011
### Mech_Engineer
It's best to discuss this problem here so many people can help answer your problem. Bhaskar is pretty much right in the general topics involved (execpt it uses calculus instead of D.E.), but what he didn't tell you is the problem is a common one with a known solution.
If we assume your tank has a uniform cross-sectional area with respect to height, then the equation you're looking for is:
$t = \frac{2*A_{t}\left(\sqrt{z_{1}}-\sqrt{z_{2}}\right)}{C_{d}*A_{o}*\sqrt{2g}}$
A_t is the cross-sectional area of the tank, A_o is the cross-sectional area of the orifice(s), and C_d is the orifice coefficient. z_1 and z_2 are the beginning and ending height of the fluid in the tank. You can probably assume C_d to be something like 0.6 t 0.7, it depends on what your holes look like.
This should give you a good start on what you're after.
Last edited by a moderator: Jun 17, 2011
4. Jun 18, 2011
You will need differential equations because the pressure is directly proportional to instantaneous height.And shape matters, for different shapes pressure at bottom is different-ex. spherical,hemispherical,conical etc... and for straightforward shapes like cubical only common solutions are available.
Last edited by a moderator: Aug 30, 2011
5. Jun 18, 2011
### gsal
Pressure at bottom depends on the shape? I don't think so...it just depends on the height of column of water directly on top of the hole. What depends on the shape is the volume of water remaining to be drained.
6. Jun 19, 2011
Yes i know that but what if the height is varying due to hemispherical shape or any other curved shapes?Only at a single point pressure is =dgh.
7. Jun 19, 2011
### gsal
Well, 1994Bhaskar, you had me confused...on one sentence you do say that "pressure is directly proportional to the height" and in the next you say that "for different shapes pressure at the bottom is different".
So, pressure is only proportional to the height and the equation above applies.
Replace the letter z with the letter h, for height, and
Lump all constant values into a single constant, K, and write the formula like this:
t = K x At x ( sqrt(h1) - sqrt(h2) )
(sorry, I don't know latex)
Now, the area of the tank (At) is not constant (t is for tank not for time); instead, the tank has different cross-sectional areas at different heights...in other words, the area A is a function of the height...somebody needs to come up with a function A(h) depending on the profile along the wall of the tank.
So, for lack of calculus abilities, I would simply take the formula given above and put it in a computer program...this is how I would approach it:
Write a function A(h) that can be called from within a loop, decide on a dh (delta h) and evaluate the "differential" equation from initial height h1 all the way down to almost zero and add the delta times
t = 0.0
dh = 0.1
for h from h1 down to dh, every dh:
dt = K x A(h) x ( sqrt(h) - sqrt(h - dh)
t = t + dt
end loop
I think that would do.
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2017-09-25 05:03:04
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http://mathhelpforum.com/pre-calculus/187100-solving-x-log-x-type-equations.html
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# Math Help - Solving x = log(x) type equations.
1. ## Solving x = log(x) type equations.
It looks like a simple problem, but for some reason, it's giving me trouble. How do you find the solution algebraically:
$x = 8\ log _2 x$
By drawing the 2 functions, I can see that there will be two solutions. But that's about all I can say right now. How do you find the values of these two roots?
2. ## Re: Logarithmic functions
You can not solve it elementary.
You could use numerical methods.
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2015-05-04 07:20:12
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https://www.physicsforums.com/threads/decay-of-sinusoidal-velocity-wave-kolmogorov-flow.811800/
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Decay of sinusoidal velocity wave (kolmogorov flow)
1. May 2, 2015
OldStudent0382
1. The problem statement, all variables and given/known data
Find the velocity field $u(x,t)$
2. Relevant equations
$\rho = constant$
$u(0,t)=0$
$u(0,L)=0$
$\frac{\partial u}{\partial t} = \nu \frac{\partial ^{2}u}{\partial x^{2}}$
$u(x,0)=U_{o}[sin(\frac{3\pi x}{L})+0.7sin(\frac{9\pi x}{L})]$
3. The attempt at a solution
I have absolutely no idea how to start this problem (again) and I'm embarrassed to admit that I've been looking at it and searching for two hours -- and I have nothing to show for it.
Thank you in advance!
2. May 3, 2015
OldStudent0382
Would this be approached similarly to unsteady couette flow? I spent a few more hours trying to research the problem (and the topic) and I'm either not finding anything relevant or I'm just so far lost that I'm not finding the connection between this assignment and what I've been reading.
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2018-02-21 00:03:48
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http://bsiswoyo.lecture.ub.ac.id/tag/scilab-code/
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# Tag Archives: scilab code
Last updated by at .
## Measuring the response of the electric water heater using the Arduino
This post describes how to measure the response of the electric water heater using the Arduino. By using simple program on the Arduino is very simple to read the output of the temperature response. Arduino has a 10bit ADC accuracy is used to read the output of the LM35 temperature sensor.
The output voltage of the LM35 sensor is amplified using the LM358 with a voltage gain of 5.47. While the output voltage of the LM35 is 10mV per degree centigrade. So the total output voltage LM398 is 5.47x10mV = 54.7mV per degree centigrade.
Arduino simple program are as follows:
To read the temperature using the Arduino ADC channel 0 (A0). Results of ADC convertion is sent in serial with baudrate of 115200baud. By using hyperterminal program, the conversion result is stored in a file in text format form.
## Measurement steps
1. Upload arduino program to arduino board
2. Fill water with water and setup the water flow about 6ml/sec
3. Connect output of temperature sensor to A0 of arduino
4. Run hyperterminal and setup the filename to capture data
5. Start no power on elektric heater for some seconds
6. Start power of electric heater for some minutes
7. Remove power of electric heater for some minutes
8. Stop hyperterminal
9. Graph the results using scilab
## Graph the results using scilab
By using the Scilab command in the console such as the following, the response will be obtained graphically.
```// read file
// convert to matrix
-->d=evstr(sh);
// check size of matrix
-->size(d)
ans =
2652. 1.
// make matrix of time with sampling about 2ms
-->t=0:0.002:2651*0.002;
// check size t matrix must same with d matrix
-->size(t)
ans =
1. 2652.
// convert to voltage, fullscale of adc = 5Volt
-->v=(d/1024)*5;
// convert to output voltage of LM35
-->v_sensor=v/5.47;
// convert to exact temperature with 10mv per centigrade
-->tc=(v_sensor*1000)/10;
// plot data
-->plot2d(t,tc,style=5)
-->xgrid()
-->title("Step Responses of Water Heater")
-->xlabel("Number of Samplings")
-->xlabel("Time in second")
-->ylabel("Temperature of water heater fluid (centigrade)")
```
## Scilab code: transfer function and performance of control systems using DC Motor Model
This post contains the simple code written for Scilab, associated with the modeling of permanent magnet dc motor. The code of this program are made by me and is for all who need it. So is open source, you can modify as appropriate. Do not forget you also include my name and my website link in your application or website, and do not forget me say thank you.
Source Code 1
DC motor used is a permanent magnet, with the default parameters are:
1. Moment of inertia of the rotor: $\dpi{100}&space;(J)&space;=&space;0.01&space;kg.m^2/s^2$
2. Electromotive force constant: $\dpi{100}&space;(K=Ke=Kt)&space;=&space;0.01&space;Nm/Amp$
3. Damping ratio of the mechanical system: $(b)&space;=&space;0.1&space;Nms$
4. Electric resistance: $(R)&space;=&space;1&space;\Omega$
5. Electric inductance: $(L)&space;=&space;0.5&space;H$
If you want another dc motor parameters, please edit that variable in the source program. Enter the parameters of the control constants Kp, Ki, and Ki, then the dc motor transfer function and the total transfer function of each controller P, PI, PD and PID will be displayed.
Also, the graphics performance of the control system for each controllers P, PI, PD and PID will be generated.
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2017-10-23 11:53:22
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http://tex.stackexchange.com/questions/47908/large-square-root-symbols
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Large Square Root Symbols
Normally the \sqrt{x} symbol automatically scales according to x, so that it usually seems to be the right size.
But is there any way to manually increase the size of one instance of the \sqrt symbol?
This issue came up in the following MWE:
\documentclass{article}
\newcommand{\blank}[1]{\hfil\penalty1000\hfilneg\rule[-3pt]{#1}{0.4pt}} % nice blank underscores
\begin{document}
Fill in the blank for the following equation:
$3 = \sqrt{\blank{1cm}}$
\end{document}
-
have you looked at the \phantom command? – cmhughes Mar 13 '12 at 20:00
I tried \sqrt{\phantom{\sum} \blank{1cm}}, which scales the root symbol to the right size, but still leaves an awkward space before the blank line. – jamaicanworm Mar 13 '12 at 20:07
If DavidCarlisle's \strut solution is not what you want, try \vphatom{} instead of \phantom{}. – Peter Grill Mar 13 '12 at 20:31
Perhaps a strut is what you are looking for:
\documentclass{article}
\newcommand{\blank}[1]{\hfil\penalty1000\hfilneg\rule[-3pt]{#1}{0.4pt}} % nice blank underscores
\begin{document}
Fill in the blank for the following equation:
$3 = \sqrt{\blank{1cm}}$
$3 = \sqrt{\strut\blank{1cm}}$
\end{document}
A strut is a zero width rule the height of a bracket in the current font, TeX formats tend to insert them in various places like table rows to ensure that things get an even spacing. More exactly LaTeX defines strut as
\def\strut{\relax\ifmmode\copy\strutbox\else\unhcopy\strutbox\fi}
Where \strutbox is (re)defined whever there is a size-changing command by the following code:
\setbox\strutbox\hbox{%
\vrule\@height.7\baselineskip
\@depth.3\baselineskip
\@width\z@}%
\z@ means 0pt.
-
That's great! How does it work? – jamaicanworm Mar 13 '12 at 20:30
a strut is a zero width rule the height of a bracket in the current font, TeX formats tend to insert them in various places like table rows to ensure that things get an even spacing. – David Carlisle Mar 13 '12 at 20:34
@doncherry strut description expanded and added – David Carlisle Mar 13 '12 at 20:57
@DavidCarlisle: Great, thanks! I added a bit of information about \z@ since these macros with an @ in them always tend to scare me a bit. Hope you don't mind. – doncherry Mar 13 '12 at 21:03
The \phantom{<stuff>} command places a box of horizontal width and vertical height equivalent to the contents <stuff>. It has two smaller equivalents \hphantom and \vphantom that are respectively the horizontal and vertical components of the contents only:
\documentclass{article}
\newcommand{\blank}[1]{\hfil\penalty1000\hfilneg\rule[-3pt]{#1}{0.4pt}} % nice blank underscores
\begin{document}
Fill in the blank for the following equation:
$3 = \sqrt{\blank{1cm}}$
$3 = \sqrt{\vphantom{\sum}\blank{1cm}}$
\end{document}
Of course, another alternative would be to place a zero-width rule \rule{0pt}{<height>} of arbitrary height, which acts just like \vphantom, and just like a \strut.
-
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2014-10-22 03:17:26
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https://the-equivalent.com/3-10-equivalent-fractions-2/
|
# 3/10 equivalent fractions
## How to tell if fractions are equivalent?
• Equivalent fractions may look different, but they have the same value.
• You can multiply or divide to find an equivalent fraction.
• Adding or subtracting does not work for finding an equivalent fraction.
• If you multiply or divide by the top of the fraction, you must do the same to the bottom.
More items…
## What is 3 over 10 as a fraction?
Then, we divide both 3 and 10 by the greatest common factor to get the following simplified fraction: 3/10 Therefore, this equation is true: 3/10 = 3/10 If the numerator is greater than or equal to the denominator of a fraction, then it is called an improper fraction. In that case, you could convert it into a whole number or mixed number fraction. 3/10 = Proper Fraction
## How do you calculate equivalent fractions?
What are equivalent fractions?
• Sometimes fractions can be simplified into smaller ones.
• These fractions are said to be equivalent.
• You can multiply the numerator and denominator in the smaller fraction, by the same number, and get the first fraction back.
## How to make an equivalent fraction?
It is possible by these methods:
• Method 1: Make the Denominators the same
• Method 2: Cross Multiply
• Method 3: Convert to decimals
## What is the equivalent fraction of 3 by 10?
Four equivalent fraction to 3/10 are 6/20, 9/30, 12/40, 15/20.
## Do write 4 equal fractions to the fraction 10 by 3?
Multiply and divide the given fraction by 3. Multiply and divide the given fraction by 4. Multiply and divide the given fraction by 5. Hence, four fractions that are equal to the given fraction are respectively.
## What is 3/5 equivalent to as a fraction?
So, 3/5 = 6/10 = 9/15 = 12/20.
## What is the simplest form for 3 10?
Answer and Explanation: The fraction 3/10 in simplest form is 3/10.
## What is the decimal and percent of 3 10?
To convert fraction to decimal, divide 3/10, which would give you . 3 which is already a decimal. To convert decimal to percent, multiply by 100 which would give you 30, hence 30%.
## How do you turn 10 3 into a decimal?
3.3333 is the decimal form of 10/3.
## What is 3/4 equal to as a fraction?
This time, both numbers are the same, so 12/16 IS equivalent to 3/4 . By using this equivalent fraction calculator, you can also see how to obtain one fraction from another!
## What is 3/6 equivalent to as a fraction?
1/2Since, both the values are equal, therefore, 1/2 and 3/6 are equivalent fractions.
## What is equivalent calculator?
Equivalent Expression Calculator is a free online tool that displays the equivalent expressions for the given algebraic expression. BYJU’S online equivalent expression calculator tool makes the calculations and simplification faster and it displays the equivalent expression in a fraction of seconds.
## What is 3/10 as a percent?
30 %Fraction to Percent Conversion TableFractionPercent1/1010 %2/1020 %3/1030 %4/1040 %41 more rows
## What is the improper fraction 10 3?
Yes, 103 is an improper fraction. An improper fraction is one in which the numerator of the fraction is larger than the denominator of the fraction. The numerator of 103 is 10 , and the denominator of 103 is 3 .
## How do you simplify fractions?
So, reducing or simplifying fractions means we make the fraction as simple as possible. We do this by dividing the numerator and the denominator by the largest number that can divide into both numbers exactly. In other words, we divide the top and bottom by the biggest number they have in common.
## How do u add fractions on a calculator?
To add fractions there are Three Simple Steps: Step 1: Make sure the bottom numbers (the denominators) are the same. Step 2: Add the top numbers (the numerators), put that answer over the denominator. Step 3: Simplify the fraction (if possible)
## What is the same as 2 3?
Equivalent fractions of 2/3 : 4/6 , 6/9 , 8/12 , 10/
## What is 1.2 as a fraction?
6/5Solution: 1.2 as a fraction is 6/5.
## What are Equivalent Fractions?
Equivalent fractions are fractions with different numbers representing the same part of a whole. They have different numerators and denominators, but their fractional values are the same.
## How to make a fraction equivalent?
Multiply both the numerator and denominator of a fraction by the same whole number. As long as you multiply both top and bottom of the fraction by the same number, you won’t change the value of the fraction , and you’ll create an equivalent fraction.
## What is half of a fraction?
For example, think about the fraction 1/2. It means half of something. You can also say that 6/12 is half, and that 50/100 is half. They represent the same part of the whole. These equivalent fractions contain different numbers but they mean the same thing: 1/2 = 6/12 = 50/100
## How to find equivalent fractions?
To find equivalent fractions, you just need to multiply the numerator and denominator of that reduced fraction ( 10 3) by the same natural number, ie, multiply by 2, 3, 4, 5, 6
## Can you convert fractions to decimals?
This Equivalent Fractions Table/Chart contains common practical fractions. You can easily convert from fraction to decimal, as well as, from fractions of inches to millimeters.
## What are Equivalent Fractions?
Two or more fractions are said to be equivalent if they are equal to the same fraction when simplified. For example, the equivalent fractions of 1/5 are 5/25, 6/30, and 4/20, which on simplification, result in the same fraction, that is, 1/5.
## How to Determine if Two Fractions are Equivalent?
We need to simplify the given fractions to find whether they are equivalent or not. Simplification to get equivalent numbers can be done to a point where both the numerator and denominator should still be whole numbers. There are various methods to identify if the given fractions are equivalent. Some of them are:
## How to write 6/8 as an equivalent fraction?
In order to write the equivalent fraction for 6/8, let us multiply the numerator and denominator by 2 and we will get (6 × 2)/ (8 × 2) = 12/16. Therefore, 6/8 and 12/16 are equivalent fractions. Now, let us get another equivalent fraction for 6/8, by dividing it by a common number, say, 2. After dividing the numerator and denominator by 2 and we will get (6 ÷ 2)/ (8 ÷ 2) = 3/4. Therefore, 6/8 and 3/4 are equivalent fractions.
## How to multiply the numerator and the denominator?
Multiply the numerator and the denominator by the same number.
## What is the LCM of 2/6 and 3/9?
The denominators of the fractions, 2/6 and 3/9 are 6 and 9. The LCM of the denominators 6 and 9 is 18. Let us make the denominators of both fractions 18, by multiplying them with suitable numbers.
## Is a decimal the same as a fraction?
The decimal values of both the fractions are the same and hence, they are equivalent.
## Do equivalent fractions get reduced to the same fraction?
All equivalent fractions get reduced to the same fraction in their simplest form as seen in the above example. Explore the given lesson to get a better idea of how to find equivalent fractions and how to check if the given fractions are equivalent.
## How to find equivalent fractions?
To find equivalent fractions, you just need to multiply the numerator and denominator of that reduced fraction ( 110) by the same natural number, ie, multiply by 2, 3, 4, 5, 6
## Is 1 10 a fraction?
Important: 1 10 looks like a fraction, but it is actually an improper fraction.
## Can you convert fractions to decimals?
This Equivalent Fractions Table/Chart contains common practical fractions. You can easily convert from fraction to decimal, as well as, from fractions of inches to millimeters.
## How to find equivalent fractions?
To find equivalent fractions, you just need to multiply the numerator and denominator of that reduced fraction ( 710) by the same natural number, ie, multiply by 2, 3, 4, 5, 6
## Is 7 10 a fraction?
Important: 7 10 looks like a fraction, but it is actually an improper fraction.
## Can you convert fractions to decimals?
This Equivalent Fractions Table/Chart contains common practical fractions. You can easily convert from fraction to decimal, as well as, from fractions of inches to millimeters.
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2023-04-01 22:36:36
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https://nyuscholars.nyu.edu/en/publications/safepredict-a-meta-algorithm-for-machine-learning-that-uses-refus
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# SafePredict: A Meta-Algorithm for Machine Learning That Uses Refusals to Guarantee Correctness
Mustafa A. Kocak, David Ramirez, Elza Erkip, Dennis E. Shasha
Research output: Contribution to journalArticlepeer-review
## Abstract
SafePredict is a novel meta-algorithm that works with any base prediction algorithm for online data to guarantee an arbitrarily chosen correctness rate, $1-\epsilon$1-, by allowing refusals. Allowing refusals means that the meta-algorithm may refuse to emit a prediction produced by the base algorithm so that the error rate on non-refused predictions does not exceed $\epsilon$. The SafePredict error bound does not rely on any assumptions on the data distribution or the base predictor. When the base predictor happens not to exceed the target error rate $\epsilon$, SafePredict refuses only a finite number of times. When the error rate of the base predictor changes through time SafePredict makes use of a weight-shifting heuristic that adapts to these changes without knowing when the changes occur yet still maintains the correctness guarantee. Empirical results show that (i) SafePredict compares favorably with state-of-the-art confidence-based refusal mechanisms which fail to offer robust error guarantees; and (ii) combining SafePredict with such refusal mechanisms can in many cases further reduce the number of refusals. Our software is included in the supplementary material, which can be found on the Computer Society Digital Library at http://doi.ieeecomputersociety.org/10.1109/TPAMI.2019.2932415.
Original language English (US) 8784215 663-678 16 IEEE Transactions on Pattern Analysis and Machine Intelligence 43 2 https://doi.org/10.1109/TPAMI.2019.2932415 Published - Feb 1 2021
## ASJC Scopus subject areas
• Software
• Computer Vision and Pattern Recognition
• Computational Theory and Mathematics
• Artificial Intelligence
• Applied Mathematics
## Fingerprint
Dive into the research topics of 'SafePredict: A Meta-Algorithm for Machine Learning That Uses Refusals to Guarantee Correctness'. Together they form a unique fingerprint.
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2022-06-28 13:39:37
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https://www.r-bloggers.com/page/348/
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## Comparing ESPN’s, CBS’s, and NFL.com’s Fantasy Football Projections using R
March 17, 2013
By
In the future, we will determine how to select the best possible team by maximizing your team's projected points and minimizing its downside risk. But in order to do this, we will have to rely on our best guess of how many points each player will score. We will use 2012 projections from ESPN, CBS, and NFL.com and actual...
## Extracting Information From Objects Using Names()
March 17, 2013
By
One of the big differences between a language like Stata compared to R is the ability in R to handle many different types of objects at once, and combine them together or pull them apart. I had a post about objects last year, but I thought I'd sh...
## Mumbai, Mar 2013 – Portfolio Tutorial
March 17, 2013
By
(This article was first published on Rmetrics blogs, and kindly contributed to R-bloggers) To leave a comment for the author, please follow the link and comment on their blog: Rmetrics blogs. R-bloggers.com offers daily e-mail updates about R news and tutorials on topics such as: Data science, Big Data, R jobs, visualization (ggplot2, Boxplots, maps, animation), programming (RStudio, Sweave,...
## Variability of garch predictions
March 17, 2013
By
How variable are garch predictions? Previously There have been several posts on garch, in particular: A practical introduction to garch modeling The components garch model in the rugarch package Both of these posts speak about the two common prediction targets: prediction (of volatility) at the individual times (usually days) term structure prediction — the average … Continue reading...
## Ordinal Data
March 17, 2013
By
I expect to be getting some ordinal data, from 5 or 9 point rating scales, pretty soon, so I am having a look ahead how to treat those. Often ANOVA is used, even though it is well known not to be ideal fro a statistical point of view, so that is the st...
## Happy St Patrick’s Day
March 17, 2013
By
I love Saint Patrick’s Day for, at least, two reasons. The first one is that, on March 17th, you can play out loud The Pogues, the second one is that it’s the only day in the year when I really enjoy getting a Guiness in a pub. And Guiness is important in statistical science (I did mention a couple...
## caretEnsemble Classification example
March 16, 2013
By
Here's a quick demo of how to fit a binary classification model with caretEnsemble. Please note that I haven't spent as much time debugging caretEnsemble for classification models, so there's probably more bugs than my last post. ...
## Blend what?
March 16, 2013
By
Why?Over the years I have learned quite a few things about machine learning but I have never thought of writing them down properly. Too often I can't figure out exactly what I did when I look at my old codes. The time is NOW!More importantly, I have fa...
## GNU R loop speed comparison
March 16, 2013
By
Recently I had several discussions about using for loops in GNU R and how they compare to *apply family in terms of speed. I have not seen a direct benchmark comparing them so I decided to execute one (warning: some of the code presented today tak...
March 16, 2013
By
Scholarly metadata - the meta-information surrounding articles - can be super useful. Although metadata does not contain the full content of articles, it contains a lot of useful information, including title, authors, abstract, URL to the article, etc. One of the largest sources of metadata is provided via the Open Archives Initiative Protocol for Metadata Harvesting or OAI-PMH....
## Changing Axis Values in R Plot
March 15, 2013
By
A colleague asked me for how one can change axis attributes in a basic plot. Plotting anything in R is really, really easy. It is enough typing plot(x, y). In general, plot functions are nicely pre-cooked, so hardly one needs to change anything. But if changes in the default attributes are needed, it is possible
## Evaluation of Orthogonal Signal Correction for PLS modeling (OSC-PLS and OPLS)
March 15, 2013
By
Partial least squares projection to latent structures or PLS is one of my favorite modeling algorithms. PLS is an optimal algorithm for predictive modeling using wide data or data with rows << variables. While there is s a wealth of literature regarding the application of PLS to various tasks, I find it especially useful for biological
## How Did I Miss “The Golden Dilemma”?
March 15, 2013
By
I am ashamed to admit that I am way behind (about 10,127 downloads) in discovering this wonderful paper: The Golden Dilemma (January 8, 2013)Erb, Claude B. and Harvey, Campbell R.Available at SSRN: http://ssrn.com/abstract=2078535 Here are the authors presenting the concept in July 2012 if you prefer slideshow format (thanks...
## How do I make my graphs?
March 15, 2013
By
Someone who wishes to remain anonymous writes: I’ve been following your blog a long time and enjoy your posts on visualization/statistical graphics matters. I don’t recall however you ever describing the details of your setup for plotting. I’m a new R user (convert from matplotlib) and would love to know your thoughts on the ideal The post How...
## Calender Heatmap with Google Analytics Data
March 15, 2013
By
As data analytics consulting firm, we think we are fortunate that we keep finding problems to find. Recently my team mate found a glaring problem of not having any connector for R with Google. With the inspiration from Michael, Ajay O, it soon become a worth problem to solve. With RGoogleAnalytics package now, we have
## Veterinary Epidemiologic Research: GLM – Logistic Regression
March 14, 2013
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$Veterinary Epidemiologic Research: GLM – Logistic Regression$
We continue to explore the book Veterinary Epidemiologic Research and today we’ll have a look at generalized linear models (GLM), specifically the logistic regression (chapter 16). In veterinary epidemiology, often the outcome is dichotomous (yes/no), representing the presence or absence of disease or mortality. We code 1 for the presence of the outcome and 0
## Data Science Education gets personal
March 14, 2013
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by Joseph B. Rickert It is difficult to imagine that there is anyone on the planet with an internet connection and a desire to learn something new who has not at least looked into taking a massive open online course (MOOC). Last Fall, in an 11/4/12 article, the New York Time declared the Year of the MOOC and quoted...
## Upcoming events
March 14, 2013
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Highlighted LondonR is soon — see the “Previously Announced” section. New Events Thirsty Quants 2013 March 21, London. Some thirsty quants will be going for a drink on the 21st of March as of 18.30 at the Lamb Tavern in Leadenhall Market. http://www.lambtavernleadenhall.com/ Rethinking the Economics of Pensions 2013 March 21 & 22 in London. … Continue reading...
## Apply-style commands in R
March 14, 2013
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Here's a quick table of what I think are the most useful apply-style commands in R: FunctionInputOutputBest forapplyRectangularRectangular or vectorApplying function to rows or columnslapplyAnythingListNon-trivial operations on almost any data typesap...
March 14, 2013
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Nomen Est Omen?Lately, the terms "data science" and "data scientist" turn up at an increasing pace in the R-blog-sphere. Since its first occurrence (to my knowledge, "data scientist" has been coined by DJ Patil and Jeff Hammerbacher in 2008), th...
## Using bigmemory with Rcpp
March 14, 2013
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The bigmemory package allows users to create matrices that are stored on disk, rather than in RAM. When an element is needed, it is read from the disk and cached in RAM. These objects can be much larger than native R matrices. Objects stored as such larger-than-RAM matrices are defined in the big.matrix class and they are designed...
## On ENAR, or Statistical Meetings in General
March 14, 2013
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Last year I accepted an invitation from Ben to go to ENAR 2013 -- my first ENAR. I used to go to JSM and useR!, and apparently I enjoy useR! most. The reason is not, or not only, because I'm more of a technical person. It is just hard to concentrate at large statistical conferences. I want...
## qdap 0.2.1 Released
March 13, 2013
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I’m very pleased to announce the release of qdap 0.2.1 This is the second installment of the qdap package available at CRAN. The qdap package automates many of the tasks associated with quantitative discourse analysis of transcripts containing discourse, including … Continue reading →
## In case you missed it: February 2013 Roundup
March 13, 2013
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In case you missed them, here are some articles from February of particular interest to R users. How to resample from a large data set with RHadoop, and a video introduction to the RHadoop packages. A 90-second video explains: What is Revolution R Enterprise? Jeffrey Stanton has published a free e-book "An Introduction to Data Science" using R. I...
## John Snow’s Cholera data in more formats
March 13, 2013
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In honour of the bicentenary of John Snow’s birth – and because I was asked to by someone via email – I have now released my digitisation of John Snow’s Cholera data in a few other formats: KML and as Google Fusion Tables. To save you reading my previous blog posts on the subject, I’ll
## Using maps and ggplot2 to visualize college hockey championships
March 13, 2013
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Short: I plot the frequency of college hockey championships by state using the maps package, and ggplot2 Note: this example is based heavily on the example provided athttp://www.dataincolour.com/2011/07/maps-with-ggplot2/ data reference:http://en.wikipedia.org/wiki/NCAA_Men%27s_Ice_Hockey_Championship Question of interestAs a good Minnesotan, I've believed for quite some time that the colder, Northern states enjoy a competitive advantage when it...
## Webinar tomorrow: 100% R and More
March 13, 2013
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A quick note that I'll be hosting our regularly-scheduled webinar, Revolution R Enterprise, 100% R and More, at 10AM Pacific tomorrow. If you're new to R, or want to learn about the power, scalability and productivity features of Revolution R Enterprise, this is a great place to start. Revolution Analytics webinars: Revolution R Enterprise, 100% R and More
## New package for ensembling R models
March 13, 2013
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I've written a new R package called caretEnsemble for creating ensembles of caret models in R. It currently works well for regression models, and I've written some preliminary support for binary classification models. At th...
## R needs some bureaucracy
March 12, 2013
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Writing a program in R is almost bureaucracy free: variables don’t need to be declared, the language does a reasonable job of guessing the type a value might need to be automatically be converted to, there is no need to create a function having a special name that gets called at program startup, the commonly
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2016-10-23 22:19:53
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https://ai.stackexchange.com/questions/35484/given-a-set-of-trajectories-produced-by-a-fixed-policy-what-is-the-the-standard/35496
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# Given a set of trajectories produced by a fixed policy, what is the the standard approach to estimate Q?
Let's say that I have a set of trajectories $$\mathcal{D} = \{\tau_1, \dots, \tau_n\}$$ produced by an agent acting in a (episodic) MDP with a fixed policy $$\pi$$. I would like to estimate the $$Q$$ function of $$\pi$$ from $$\mathcal{D}$$. Just to be clear, each trajectory $$\tau_j$$ is a finite sequence $$\tau_j = s_0^j, a_0^j, r_0^j s_1^j, a_1^j, r_1^j \dots, s_{N_j}^j$$ representing an episode performed w.r.t. $$\pi$$.
What would be the standard approach in this case? Better use TD learning or Monte Carlo?
• Where are the rewards in your data? Usually I would expect to see them in the trajectories ie. $\tau_j = s_0^j, a_0^j, r_1^j, s_1^j, a_1^j, r_2^j, \dots$ May 11 at 20:00
• Yes, I forgot the rewards! I edit the question May 12 at 8:01
What would be the standard approach in this case? Better use TD learning or Monte Carlo?
Both should be fine, but they might lead to different estimates, if both these things apply:
• The amount of data is relatively small compared to all possibilities from the given environment and policy.
• Either the policy or the environment are stochastic.
The difference is that for each state/action pair estimated:
• Monte Carlo will estimate based on overall average returns, ignoring individual state transitions and policy choices.
• Temporal Difference will estimate based on observed state transitions and policy choices.
There is a good example of what this might mean numerically in Sutton & Barto chapter 6, example 6.4. In that case it shows an advantage to TD learning when some states might be sparsely represented in the data whilst others have more instances. Monte Carlo learning will only learn the value of those rarer states from the trajectories where they occur, whilst TD learning will be able to use estimates of other trajectories, provided two or more trajectories overlap later on.
This doesn't necessarily make TD learning better. If a trajectory that overlaps with others also happens to include an unusual policy choice, state transition or reward, this may spread sample bias into multiple estimates, whilst Monte Carlo would be affected less by such an outlier.
• Thanks! So in some sense TD has high bias and less variance, while MC has low bias and high variance, right? May 12 at 9:44
• @Onil90: Yes, although this bias in not the same bias as referred in other scenarios discussing TD vs MC. In your scenario with a fixed data set, and assuming process until convergence, it is that both MC and TD respond to sample bias differently. May 12 at 11:03
Your trajectories must contain rewards, so I'm assuming you've forgotten them in your original post, i.e., we must have $$\tau_j = (s_0^j, a_0^j, r_1^j, ..., s_{N_j}, a_{N_j}, r_{N_j+1})$$
Given that you have access to the full trajectories, I would use the Monte Carlo estimates. You want to use TD methods when you need to estimate $$Q^\pi$$ incrementally as new transitions $$(s_t, a_t, r_{t+1})$$ arrive.
You can estimate $$Q^\pi$$ as follows (adapted from the Sutton&Barto book, chapter 5.1):
Initialise $$\text{Returns}(s,a)$$ to an empty list for all $$(s, a)$$ pairs in the trajectories.
For each $$j$$:
Initialise the cumulative discounted reward $$G$$ to 0.
For $$t$$ in $${N_j, N_{j-1}, ..., 0}$$:
Set $$G = \gamma G + r_{t+1}$$
If $$(s_t, a_t)$$ is not in $$((s_0, s_0), ... (s_{t-1}, a_{t-1}))$$:
Append $$G$$ to $$\text{Returns}(s_t, a_t)$$.
Set $$Q^\pi(s_t, a_t) = \text{Average}(\text{Returns}(s_t, a_t))$$
For completeness the method is called "First visit Monte Carlo" because we only update the estimate of $$Q^\pi$$ using first visits to an $$(s, a)$$ pair.
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2022-06-28 10:06:02
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https://www.physicsforums.com/threads/calculus-area-between-two-curves-minimize-it.974903/
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# Calculus- Area between two curves (minimize it)
• I
Sidd
Hi,
This is my first question here, so please apologise me if something is amiss.
I have two curves such that Wa = f(k,Ea,dxa) and Wb = f(k,Eb,dxb). I need to minimize the area between these two curves in terms of Eb in the bounded limit of k=0 and k=pi/dx. So to say, all the variables can assume any value, and then I can only alter Eb to minimize the area between these two curves.
I have tried integrating both the curvesbetween k=0 and k=pi/dx, and differentiating with respect to Eb and equating it to 0. This does not work.
I also thought about integrating (summing) the distance between the two curves, bu that results in the case above.
Do I need to put a constraint function?
Please feel free to ask if any clarification is needed. I have tried to simply my case, and that may have resulted in some ambiguities.
Thank you
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2023-01-31 17:01:39
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http://mathhelpforum.com/pre-calculus/25848-quadratic-equation-statistics-print.html
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• Jan 9th 2008, 09:26 PM
Wierdier
Hello, skipping the formal introduction, I came to seek help with math (duh).
This site was recommend to me by a good friend and now in need and 1:20 am I have no one to turn to for explanations so I though I would give this site a shot.
I have to extrapolate data from a given quadratic equation, however I am confused by one of the variables. (E)
So here it is, and my thanks in advance to anyone who can help me out.
y = 3E-18x3 - 6E-10x2 + 0.0316x - 76329
• Jan 9th 2008, 09:34 PM
topsquark
Quote:
Originally Posted by Wierdier
Hello, skipping the formal introduction, I came to seek help with math (duh).
This site was recommend to me by a good friend and now in need and 1:20 am I have no one to turn to for explanations so I though I would give this site a shot.
I have to extrapolate data from a given quadratic equation, however I am confused by one of the variables. (E)
So here it is, and my thanks in advance to anyone who can help me out.
y = 3E-18x3 - 6E-10x2 + 0.0316x - 76329
First, this is a cubic equation, not a quadratic.
Second "3E-18" is calculator notation for $3 \times 10^{-18}$.
-Dan
• Jan 9th 2008, 09:35 PM
Wierdier
Oh yeah sorry...its just I am so disoriented.
Thanks alot!
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2016-10-21 17:39:29
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