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Sum of Partial Sums of Arithmetic Sequence
This online calculator calculates partial sums of an arithmetic sequence and displays the sum of partial sums.
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A user request inspired this calculator. As you probably know, the arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant. This difference is
called common difference and the formula to compute the next number in the sequence is
You can also sum numbers on the sequence up to a certain index n (which is called partial sum). The formula for the partial sum would be
But you can also sum these partial sums as well. This is what the calculator below does. You enter the first term of the sequence, the common difference, and the last index to compute, and the
calculator displays the table with the following columns:
• index i
• i-th member of the sequence
• i-th partial sum
• i-th sum of partial sums
Similar calculators
PLANETCALC, Sum of Partial Sums of Arithmetic Sequence | {"url":"https://planetcalc.com/8596/?license=1","timestamp":"2024-11-14T18:48:32Z","content_type":"text/html","content_length":"37225","record_id":"<urn:uuid:38801fd0-97af-44c3-a542-7f2cc104e2a7>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00892.warc.gz"} |
III Theoretical Physics of Soft Condensed Matter - Model B
6Model B
III Theoretical Physics of Soft Condensed Matter
6 Model B
We now apply this theory to model some concrete systems. We shall first consider
a simple model of binary fluids. We assume that diffusion happens but without
fluid flow. As before, this is modeled by a scalar composition field
), and
evolves under the equations
φ = −∇ ·J
J = −M∇µ +
T MΛ
µ =
Recall that the system is modelled under the Landau–Ginzburg free energy
F [φ] =
| {z }
We then have
µ = aφ + bφ
− κ∇
As before, the mean field theory for equilibrium looks like
a > 0
a < 0
a > 0
a < 0
are the spinodals, where the second derivative changes sign. This
gives the phase diagram
−1 1
a(T ) = 0
φ is the global composition, which is a control variable.
In the past, we discussed what the field looks like in each region when we are
at equilibrium. At (1), the system is in a uniform phase that is globally stable.
If we set up our system at (1), and then rapidly change the temperature so that
we lie in (2) or (3), then we know that after the system settles, we will have a
phase separated state. However, how this transition happens is not something
mean field theory tells us. Heuristically, we expect
In (2), we have
φ| < φ
, and
0 implies local instability. The
system rapidly becomes phase separated. This is spinodal behaviour.
In (3), we have
< |
φ| < φ
. A uniform phase is locally stable, but
not globally. To reach the phase separated state, we need nucleation and
growth to occur, and requires the contribution of noise.
We now study these in detail.
Regime 1
We know that regime (1) is stable, and we shall see how it responds to perturba-
tion about φ(r) =
φ. Put
φ =
φ +
We can then write
µ =
− κ∇
φ = f
φ) +
φ) − κ∇
Note that the first term is a constant. We then have
φ = −∇ · J
J = −M∇[f
φ − κ∇
φ] +
T MΛ.
We drop the tildes and take the Fourier transform to get
= −Mq
+ κq
+ iq ·
T MΛ
Compare this with an overdamped particle in a simple harmonic oscillator,
V =
where we have
˙x = −
Mκx +
Indeed, we can think of our system as an infinite family of decoupled harmonic
oscillators, and solve each of them independently.
In the second example sheet, we compute
S(q, t) ≡ hφ
(t)i = S(q)e
q, t
) is called the dynamic structure factor , which can be measured by light
scattering. This doesn’t say the fluctuations go away completely — we expect
there to be fluctuations all the time. What this says is that fluctuations at late
times come completely from the random noise, and not the initial fluctuations.
Regime 2
Consider the case where we are in the second regime. As before, we have the
= −Mq
+ κq
| {z }
+ iq ·
T MΛ
but crucially, now
0, so it is possible to have
0. The system is
If we ignore the noise by averaging the equation, then we get
i = −r(q)hφ
So if we have a noisy initial state φ
(0), then the perturbation grows as
(t)i = φ
0, then this amplifies the initial noise. In this world, even if we
start with a perfectly uniform
, noise terms will kick in and get amplified over
time. Moreover, since we have an exponential growth, the earliest noise gets
amplified the most, and at late times, the new perturbations due to the noise
are negligible.
We can plot our r(q) as follows:
The maximally unstable mode
is given by the solution to
) = 0, which
we can easily check to be given by
Now consider the equal time non-equilibrium structure factor
(t) = hφ
(t)i ∼ S
As time evolves, this gets more and more peaked around q = q
So we see a growth of random structure with scale
L ∼ π/q
. This process is
called spinodal decomposition.
Note that this computation was done on the assumption that
is small, where
we ignored the quadratic terms. At intermediate
, as these phase separated
states grow, the quartic terms are going to kick in. An informal use of variational
theory says we should replace f
, where
= f
(t) d
This says
is less negative as the fluctuations grow. Since
this moves to a smaller
. So
∼ π/q
) starts increasing. This is called
domain growth.
In the late stages, we have large regions of
φ ≈ ±φ
, so it is almost in
equilibrium locally. We are well away from the exponential growth regime, and
the driving force for domain growth is the reduction of interfacial area. We can
estimate the free energy (per unit volume) as
) is the area of the interface. So by dimensional analysis, this is
∼ σ/L(t). We have calculated the interfacial surface tension σ before to be
σ =
but it doesn’t really matter what this is. The ultimate configuration with
minimal surface area is when we have complete phase separation. The result is
L(t) ∼
We will not derive this yet, because this result is shared with the late time
behaviour of the third regime, and we will discuss this at that point.
Regime 3
Finally, consider the third regime. Suppose we have
, where
small. The system is locally stable, so
0 for all
. On the other hand, it is
globally unstable, so phase separation is preferred. To achieve phase separation,
we must overcome a nucleation barrier, and we must rely on noise to do that.
To understand how the process will look like, formally, we can inspect the
path probabilities
P[φ(r, t)] = N exp
|J + M∇µ|
dr dt
given by the Langevin equation. We seek to find the most likely trajectory from
the initial to the final state. In field theory, this is called the instanton path,
and in statistical physics, this is called large deviation theory. Instead of doing
this, we use our physical intuition to guide ourselves.
Heuristically, we expect that if we start with a uniform phase
then at some point, there will be some random small droplet with
= +
small radius
. This is already unlikely, but after this, we need
to increase
until we have full phase separation. The key question is — how unlikely is this
The idea is to consider the cost of having a droplet of radius
. First there is
the cost of having an interface, which is 4
. However, the fact that we have
areas is energetically favorable, and this grows as the volume. So we get a
cubic term ∼ −R
. If we add these two together, we get a barrier:
F (R)
R > R
, it is then energetically favorable for the radius to continue
increasing, and then we can easily reach phase separation. To reach this, we
must rely on noise to push us over this barrier, and this noise-induced rate is
∼ e
. To see what happens afterwards, we need to better understand how
droplets work.
Droplet in equilibrium
The mechanics of a droplet is slightly less straightforward than what one might
hope, due to the presence of surface tension that tries to compress the droplet.
The result is that the value of
inside and outside the droplet is not exactly
, but with a shift.
For simplicity, we shall first consider an equilibrium system with a droplet.
This is achieved by having a large box of fluid with
just slightly above
Then in the phase separated state, the +
phase will lump together in a droplet
(if we had
φ = 0, then we would have a horizontal interface).
Within each region 1 and 2, the value of
is constant, so the term that contributes
to the free energy is
f(φ) =
We would expect 1 and 2 to respectively be located at
When we have a spherical interface, 1 and 2 are not exactly at
. To see this,
Consider the bulk chemical potential
µ =
The thermodynamic pressure is then
Π = µφ − f.
This is the negative of the y-intercept of the tangent line at φ.
If we have a flat interface, which we can think of as the limit
R → ∞
, then
we require
= µ
, Π
= Π
This means the points 1, 2 have a common tangent
If we have a droplet, then there is surface tension. Consider an imaginary
interface between the upper and lower interface. Then the pressure difference
tries to move the upper hemisphere up, and contributes a force of (Π
while the interfacial tension pulls the boundary down by 2
. In general, in
dimensions, we require
= Π
(d − 1)
This is called the Laplace pressure.
In static equilibrium, we still require
, since this is the same as saying
= 0. So
has the same slope at
. However, the two tangent
lines no longer have a common intercept, but they are separated by
d −
So it looks like
To solve for this, we take the approximation that
is small for
large. Then we can write
= f(−φ
+ δ
) ≈
+ f (−φ
= f(+φ
+ δ
) ≈
+ f (+φ
So µ
= αδ
, where α = f
). So we find that up to first order, δ
= δ
To compute δ, we compute
= µ
− f
= −αδφ
Similarly, we have Π
= +αδφ
. So
− Π
= −2αφ
Since this equals −(d − 1)
, we have
δ =
d − 1
Multiple droplet dynamics
We now move on to understand multiple droplet dynamics. This is relevant
because we expect that noise will produce multiple droplets around the fluid,
which will then interact and combine to a single phase separated state.
The way droplets interact with each other is that once we have a droplet
of large
, then the average
outside of the droplet will decrease. So to begin
understanding this scenario, we first see how a droplet reacts when the relative
density of the outside bath is not what it expects.
So suppose we have a (3D) droplet of radius
inside a bath with
ε 6
). This
is called the supersaturation. Note that to have a
droplet of radius
, the value of
inside and immediately outside the droplet
must be
. Outside of the droplet, the value of
will slowly decay to
+ ε. Thus, outside of the droplet, we write
φ(r) = −φ
φ(∞) = ε and
) = δ.
In this situation, unless
happens to be
, we have a gradient of
, hence a
gradient of chemical potential, hence a flux. Again in Model B, we have
φ = −∇ · J, J = −M∇µ = −Mα∇
assuming a weak enough gradient. We assume this has a quasi-static behaviour,
which is reasonable since molecules move quickly relative to how quickly the
droplet changes in size. So to solve for
) at any point in time, we set
= 0.
So ∇
φ = 0. We solve this with boundary conditions
φ(∞) = ε,
) = δ.
So we have
φ = ε + (δ − ε)
Now if we assume this is what
) looks like, then the current just outside the
droplet gives
) = −M∇µ = −αM
= αM(δ − ε)
αM(δ − ε)
Thus, when
are not the same, there is a flow of fluid in or out of the
droplet. The discontinuity in
across the boundary is ∆
= 2
. So mass
conservation implies
R = −J = −
αM(δ − ε)
Thus, we conclude that
R =
(ε − δ(R))
We can plug in our previous expression for
. Fixing
, we can plot
as follows:
So if we have a bath containing many droplets, then the big droplets grow and
the small droplets shrink. Indeed, the interfacial tension penalizes small droplets
more heavily than large droplets.
To understand exactly how these grow, we make a scaling ansatz that there
is only one length scale, namely the mean droplet size
R. Then we have
R ≈
(ε − δ(
We know that the value of
is also determined by
, so we know
ε − δ
) is of
order δ(
R). Hence
R ∼
So the typical droplet size is ∼ t
. Likewise, R
∼ t
, and so ε ∼ t
So if we have a sea of droplets, they go into this competitive process, and
we get fewer and fewer droplets of larger and larger size. This is called Ostwald
ripening, and is a diffusive coarsening mechanism.
We have the same scaling for non-droplet geometries, e.g. spinodal decom-
position at late times. In this case, our domains flatten and enlarge, and we
L(t) ∼
In practice, we often want to stop this from happening. One way to do so is
to add trapped species insoluble in the continuous phase, e.g. polymers or salt.
If there are
particles inside the droplet exerting ideal gas pressure, then we
− Π
We again have µ
= µ
. This ends up giving a new boundary condition at R
) =
− Π
The first term is the Laplace pressure just as before, while the second term is
the extra term from the trapped species.
If we put this back into the system of equations before, then we have a new
equation of motion
R =
ε −
We now see that there is a stable fixed point
, and further shrinkage is
prevented by the presence of the trapped species that will be further compressed
by shrinking. Thus, if we manage to produce a system with all droplets of size
< R
, then we end up with a lot of small but finite size droplets R | {"url":"https://dec41.user.srcf.net/h/III_L/theoretical_physics_of_soft_condensed_matter/6","timestamp":"2024-11-11T11:43:21Z","content_type":"text/html","content_length":"438639","record_id":"<urn:uuid:5c1eb1cc-5275-4ae4-ac33-a43a81f8e3e8>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00834.warc.gz"} |
What is a vertex?
by Mark Dawes (January 2020)
For many people, maths is precise. Mathematics is often exact. Mathematical vocabulary however isn’t.
Years ago I was asked by a colleague “is a vertex where three or more faces meet?”. I agreed that it was. The next question was “what is the thing at the top of a cone called?”. (My initial
flippant response was “it depends which way up it is”, whereas the best answer is probably, “ice cream and a flake”). It is, of course a vertex too. We also use ‘vertex’ in lots of other
mathematical situations, for example in graph theory, with intersecting lines, in 2D shapes and at the turning point of a quadratic graph.
This suggests that at least some mathematical words are used in different ways in different contexts, so it isn’t possible to have a single definition that will suit everyone. There are also
geographical differences (I gather that in the USA a ‘trapezium’ refers to a four-sided shape without any parallel sides, rather than one with a pair of parallel sides.) Beyond this, finding a
cast-iron definition for anything is difficult.
Is this shape a quadrilateral? Or is it a pentagon? Or a hexagon?
It was made using four straight lines, so that means it is a quadrilateral. But does it have four vertices or five? If it’s got a vertex at the crossing point then that surely means it’s a pentagon
(5 vertices). But then the two crossed lines have been split up, so there are actually six lines, so it’s a hexagon. If we consider the four angles, they don’t add up to 360 degrees, but if we
consider the two angles where the lines cross then we _do_ get a total of 360 degrees. But in that case there _is_ a point where the lines cross, which means that while the angle now add up to 360
degrees it’s either a pentagon or a hexagon! What is going on? For what it’s worth, I think we need to redefine what ‘interior angles’ mean in this shape – so now a different mathematical concept
and word needs to be explored further!
I raise this not because I particularly care whether this is or isn’t a quadrilateral, but to show how difficult it can be to define even simple mathematical concepts and ideas.
Help may well be at hand, though!
Cambridge Mathematics is crowd-sourcing definitions for its own glossary of mathematics. A new word appears each week and we are asked to comment on the usefulness of each definition for different
groups of learners of mathematics. It’s easy to be part of. Sign up here: cmdefineit.org or download the CM Define It app.
I have three take-aways from this:
1. It’s hard to define a word in a completely watertight way.
2. It might be necessary to have a different definition for, say, primary and secondary school pupils.
3. CM Define It is a good thing!
While CM Define It is not going to produce a single definition of each mathematical word (because of the need for different definitions for people at different stages and in different parts of the
world), their new glossary will help to make defining and explaining mathematical vocabulary easier. | {"url":"https://www.cambridgemathshub.co.uk/post/what-is-a-vertex","timestamp":"2024-11-09T13:26:23Z","content_type":"text/html","content_length":"1050491","record_id":"<urn:uuid:74239b69-3fed-4edd-8bf8-a2f55269f03d>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00583.warc.gz"} |
How Equalization Works in 10x Passive Probes
We've been discussing 10x passive probes and their inner workings; our last post covered all the ways in which a 10x passive probe is apt to be a liability. They'd be basically unusable for any
measurements at all but for one attribute: their equalization circuit (Figure 1). Without it, the 10x passive probe makes a pretty good low-pass filter, but the equalization circuit counters that
with a high-pass filter to balance things out.
Figure 1: The adjustable equalization circuit on the oscilloscope end of the coaxial cable compensates for the 10x passive probe's inherent low-pass filter characteristics
Referring again to Figure 1, there's a 9-10 pF bypass capacitor in parallel with the 10-MΩ input impedance at the tip of the probe. High frequencies will pass through the capacitor, and they'll also
see the capacitance at the oscilloscope end of the probe cable. The combination of the low-pass response of the series resistor and shunt capacitance to ground and the high-pass response of the
compensation capacitor results in a flat, high-bandwidth response overall.
We want the time constants for the high-pass filter and low-pass filter to be matched so that their pole frequencies are the same; that's what gives our 10x passive probe its overall flat frequency
response. If there's too much high-pass filtering, we'll have a peak response, and if there's not enough, we'll have a low-frequency response. We're looking for just the right balance.
That's why the adjustable capacitor, which is typically found at the end of the cable near the coaxial connector, enables us to change that capacitance so that the time constant for the capacitor and
resistor at the oscilloscope input is the same as that for the 9-MΩ resistance and capacitance at the business end of the probe. Once we match those time constants, we achieve the flat, uniform
frequency response we want to see.
Figure 2: This trimmer capacitor is the key to flattening the probe's frequency response
Figure 3: Shown are examples of under-compensation, over-compensation, and proper compensation using the probe's trimmer capacitor
That little adjustment trimmer at the end of the 10x passive probe's cable (Figure 2) adjusts that equalization circuit. Here's how to get things squared away: Plug your probe into the oscilloscope's
CAL output on the front panel. That output gives you a fast-edge, clean square wave. You want to adjust that trimmer capacitor until you see a signal on your oscilloscope's display that looks like
the one at center in Figure 3. If it looks like either of the other two traces shown, you've not yet matched the time constants of the two filter circuits. You can find more detail about this trimmer
adjustment here.
In terms of performance of the 10x passive probe, at frequencies above about 1 kHz the input impedance is mostly made up of that 9-10 pF capacitance. At low frequencies, we'd see the 9 MΩ of the
series resistance plus the 1 MΩ of the oscilloscope's input impedance, but there's that 10-pF capacitance shunting that resistance. Thus, the impedance we see looking into the probe is equivalent to
about 10 pF, which means that the impedance will drop off as we go higher in frequency. At DC, it's a 10-MΩ input, but it looks like a capacitive impedance at higher frequency. So it's important to
take that into account when using the probe.
Figure 4: Input impedance and transfer function plots for a 10x passive probe
Another complication is the potential for loop inductance between the signal and return paths. The amount of inductance depends on how we make the connection, but it will have an impact on bandwidth.
The L and C result in a series LRC response as seen at the probe tip, which will show up as an impedance dip at higher frequencies and in the transfer function (Figure 4).
We've built what is essentially a two-pole low-pass filter that has some peaking in its response, which is why, if there's some inductance in the return path, we'll see some peaking in the hundreds
of MHz range. If our source impedance of the DUT is 50 Ω, the nominal drop in transfer function will be flat with a -20 dB attenuation with a -3 dB drop from the passband region at a couple hundred
It's important to bear in mind that the input impedance is capacitive as we go to higher frequencies, which ultimately limits the bandwidth to the hundreds of MHz range. Any additional inductance in
the tip will further decrease the bandwidth and potentially introduce ringing. This is why we always want low inductance in the tip.
We'll continue exploring the performance of the 10x passive probe in an upcoming post.
Previous posts in this series:
• Thank you for posting this - very nicely done.
I've read the explanation before a couple of times but this is the first time I've seen it supported by the plots you show. Although in theory we can all work out the way the impedance at the
input varies with frequency, the plot really does emphasise how the capacitive reactance causes it to drop with frequency. At 1MHz we essentially have a 20k probe and at 10MHz a 2k one, which is
a long way from the 10M that we all imagine when we first start out and are new to electronics. It's easy to see from the graph how a 'scope probe can have an effect on something like a sensitive
analog oscillator circuit, even at frequencies of a few hundred kilohertz.
It's also useful seeing where the effect of the ground lead inductance comes into play.
Looking forward to hearing your explanation of the 'Very special cable'.
• Thank you for posting this - very nicely done.
I've read the explanation before a couple of times but this is the first time I've seen it supported by the plots you show. Although in theory we can all work out the way the impedance at the
input varies with frequency, the plot really does emphasise how the capacitive reactance causes it to drop with frequency. At 1MHz we essentially have a 20k probe and at 10MHz a 2k one, which is
a long way from the 10M that we all imagine when we first start out and are new to electronics. It's easy to see from the graph how a 'scope probe can have an effect on something like a sensitive
analog oscillator circuit, even at frequencies of a few hundred kilohertz.
It's also useful seeing where the effect of the ground lead inductance comes into play.
Looking forward to hearing your explanation of the 'Very special cable'.
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LANL Scientist makes radio waves travel faster than light
Most people think Einstein said that nothing can travel faster than the speed of light, but that’s not really the case…
Staff call this the Tabletop
Synchrotron Einstein predicted that particles and information can’t travel faster than the speed of light — but phenomenon like radio waves? That’s a different story, said
John Singleton, a Los Alamos National Laboratory Fellow.
Singleton has created a gadget that abuses radio waves so severely that they finally give in and travel faster than light.
The polarization synchrotron combines the waves with a rapidly spinning magnetic field, and the result could explain why pulsars — which are super-dense spinning stars that are a subclass of
neutron stars — emit such powerful signals, a phenomenon that has baffled many scientists, Singleton said…
And beyond explaining what has been a bit of a mystery to the astronomical community, Singleton’s discovery could have wide-ranging technological impacts in areas such as medicine and
communications, he said.
“Because nobody’s really thought about things that travel faster than light before, this is a wide-open technological field,” Singleton said.
And like Singleton says – Einstein wouldn’t have been upset by this at all.
1. I thought he said that nothing with mass can travel faster than the speed of light
2. …particles and information can’t travel faster than the speed of light — but phenomenon like radio waves? That’s a different story…
Wait a minute! Radio waves carry information, don’t they?
…asks the baffled non-scientist.
3. Cinaedh, you beat me too it. I bet that it’s just poor journalism. The statement is paraphrased… not quoted.
4. I’m waiting for comments from creationists… obviously all previous scientific evidence for everything is meaningless now. 🙂
5. It’s a tricky biz. It’s nice that they’re looking into it, but I don’t buy it and I’m not alone. Check it out: tricky pulsar math.
6. Like most good science, Singleton hasn’t jumped to get publicity for what is, after all, a qualitative breakthrough.
He built the first generation of his device in 2004.
7. #6 – irrelevent.
8. This is SubSpace communications, everyone knows THAT!!!
9. I wish scientists would stop trying to mess with stuff like this. The fabric of space-time is going to rip apart, and it’s going to be all their fault.
10. This is rather old news, Gunter Nimtz of the University of Cologne sent a microwave signal faster than the speed of light in a vacuum in 1994.
He and Horst Aichmann sent the 40th symphony of Mozart 4.7 times faster than light.
11. Radio waves are light — just light at a lower frequency. Photons are photons and have no mass. Superluminal theory has been around for awhile, but it’s superluminal depending on your reference
frame. It’s easy to get light to slow down (wrt us), just let it pass through a glass of water.
Remember, according to Einstein, it’s all relative. I don’t even want to see the math anymore, it would make my hair catch fire.
A wonderful breakthrough in the technology. Thanks, Eideard.
12. Another “fuck you” to Star wars.
13. “If you take a laser and shine it on the moon and swing it rather gently, for example, the spot on the moon travels faster than the speed of light,” Singleton said.
Maybe there is something to “faster than the speed of light,” but a laser beam scanning across the moon or any similar effect is not it. That is an illusion.
Imagine two astronauts on opposite sides of the moon. They both turn on a light at *almost* the same time. Has light or information or anything traveled from one point to the other faster than
the speed of light, just because they were turned on sequentially?
Now imagine laser photons from Earth striking and relecting from the moon sequentially from Point A to Point B.
How is that different from the first example other than dealing with more points of light? Nothing physical has gone faster than the speed of light. Similarly it could be said that I can look
across the Universe faster than the speed of light for however useful that might be.
14. Brain fitness needed here… equivalent to getting up off couch … ouch.
Whether radiation signal is a rotated narrow beam or wide spread. Info is going from source to locus of beams reception, be it instantaneous or a narrow locus. NO signal is going along locus. If
you doubt that you need to think further.
15. #14
Aka Subspace communications 🙂
16. #2 “Wait a minute! Radio waves carry information, don’t they?”
You got me curious too, I think the way the waves “carry” the information is in the modulation or distortion of the wave this pattern of modulation is then “read” and decoded as informatiom by
the reciever.
17. #4. I’m waiting for comments from creationists
Here it is. Motion faster than light is possible but Singleton knows nothing about it.
18. The speed of darkness is faster than the speed of light. By darkness, I mean, of course, the mysterious gravity.
Take this thought experiment: We all know that place of the Sun in the sky is an illusion because by the time we see the sun in the sky it has already moved in real position because the light
takes 8 minutes to reach us here on Earth. Now, imagine the Sun dissappearing completely in an instant. We would not know by the light because the light that shines on us is an eight minute
lagging illusion. But, the gravitational effect would be instantaneous. The Earth would shoot of on a tangent from its path around the sun.
Gravity is faster than light.
19. I dunno, how do you figure the speed of light? Where, in the universe, is the point of reference? Where is datum, where is the universal Greenwich Meridian, where is the starting point? Take a
couple of super sized space aliens, both with serious horsepower, and both snot flying drunk, playing chicken across the big empty that lies between galaxies. If they approach each other at high
percentages of light speed (relative to God’s navel), arn’t they, in fact, super luminal with respect to each other? Makes my brain hurt to think about it.
20. 20 Luke.
“Gravitational waves, just like photons, are waves that travel at the speed of light.”
Nasa Goddard Space flight Center
21. Ah the stuff Sci-fi is made of…
Hyperspace travel
22. 20–
Einstein first thought that as well. He later changed his theories to show the Earth wouldn’t spin off until 8 minutes later.
It’s an interesting story to hear how he was hounded by the scientists in the beginning — his first theories were all crap until he did some actual work.
23. #20, Luke,
Gravity is faster than light.
Nope. Current understanding in general relativity is that that the force of gravity is exchanged just like any other force – through particles. In this case, the graviton. Were the Sun to
disappear instantaneously, the Earth would continue on it’s elliptical orbit until that force was no longer exchanged, at the speed of light, between the Sun and Earth. About 8.33 minutes later.
24. velocity is a measurement used to depict the rate of change of an objects location between two points over a defined set of time. Our misconceptions about distance is why we have problems
understanding that traveling faster than light is meaningless. maximum light-speed may indeed be constant. But that constraint would not prevent us from changing the distance traveled. Now if I
can just figure out how to make an Arby’s appear beside my apartment I would be set.
25. Faster than light movement may be possible with simple push in deep space, far from large masses, no wormholes, warp or hyperspace required:
It would be great irony if it’s true and no one ever tried, because all that’s needed is to push harder and accelerate.
Essentially, the concept here is that not every frame of reference can measure time equally. Time measurement may depend on masses and distance.
Relativistic effects are very much present near Earth (large mass) but may not be in deep space… | {"url":"https://www.dvorak.org/blog/2008/01/19/scientist-makes-radio-waves-travel-faster-than-light/","timestamp":"2024-11-13T06:32:10Z","content_type":"application/xhtml+xml","content_length":"76864","record_id":"<urn:uuid:6cecb6ca-cb85-4782-a178-4ad2cef6bfac>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00222.warc.gz"} |
Neural Network for Digital Predistortion Design
Neural Network for Digital Predistortion Design$-$Offline Training
This example shows how to design, train, and test a neural network to apply digital predistortion (DPD) to offset the effects of nonlinearities in a power amplifier (PA). The example focuses on
offline training of the neural network-based DPD (NN-DPD). In this example, you:
1. Design and train a fully connected neural network as a DPD.
2. Test the NN-DPD using a real PA.
3. Compare the results to that of cross-term memory polynomial DPD.
This diagram shows the offline training workflow. First, you train an NN-DPD by using the input and output signals of the PA. Then, you use the trained NN-DPD.
The upper path shows the neural network training workflow. During training, you measure the input to the PA, $u$, and the output of the PA, $x$. To train the neural network as the inverse of the PA
and use it for DPD, use $x$ as the input signal and $u$ as the target signal. This process uses indirect learning [1].
The lower path shows the deployed workflow with the trained NN-DPD inserted before the PA. In this configuration, the NN-DPD inputs the oversampled signal, $u$, and output, $y$, as the input to the
PA. The PA output $z$ is the linearized signal.
NN-DPD Structure
Design an augmented real-valued time-delay neural network (ARVTDNN) as described in [2]. ARVTDNN has multiple fully connected layers and an augmented input.
The memory polynomial model has been commonly applied in the behavioral modeling and predistortion of PAs with memory effects. This equation shows the PA memory polynomial.
$\mathit{x}\left(\mathit{n}\right)=\mathit{f}\left(\mathit{u}\left(\mathit{n}\right)\right)=\sum _{\mathit{m}=0}^{\mathit{M}-1}\sum _{\mathit{k}=0}^{\mathit{K}-1}{\mathit{c}}_{\mathit{m}}\mathit{u}\
The output is a function of the delayed versions of the input signal, $u\left(n\right)$, and also powers of the amplitudes of $u\left(n\right)$ and its delayed versions.
Theoretically, a neural network can approximate any function provided that it has enough layers and neurons per layer. You can input $u\left(n\right)$ to the neural network and approximate $f\left(u\
left(n\right)\right)$, or its inverse, which is the DPD function. To decrease the required complexity of the neural network in terms of number of layers and neurons, apply the expert knowledge
provided by the memory polynomial approximation and provide the neural network extra features in the form of $u\left(n-m\right)$ and $|u\left(n-m\right){|}^{k}$.
The NN-DPD has multiple fully connected layers. The input layer inputs the in-phase and quadrature components (${\mathit{I}}_{\mathrm{in}}$/${\mathit{Q}}_{\mathrm{in}}$) of the complex baseband
samples. The ${\mathit{I}}_{\mathrm{in}}$/${\mathit{Q}}_{\mathrm{in}}$ samples and $m$ delayed versions are used as part of the input to account for the memory in the PA model. Also, the amplitudes
of the ${\mathit{I}}_{\mathrm{in}}$/${\mathit{Q}}_{\mathrm{in}}$ samples up to the ${k}^{th}$ power are fed as input to account for the nonlinearity of the PA.
During training,
$\begin{array}{l}{I}_{in}\left(n\right)=\mathrm{\Re }\left(x\left(n\right)\right)\\ {Q}_{in}\left(n\right)=\mathrm{\Im }\left(x\left(n\right)\right)\\ {I}_{out}\left(n\right)=\mathrm{\Re }\left(u\
left(n\right)\right)\\ {Q}_{out}\left(n\right)=\mathrm{\Im }\left(u\left(n\right)\right),\end{array}$
while during deployment (inference),
$\begin{array}{l}{I}_{in}\left(n\right)=\mathrm{\Re }\left(u\left(n\right)\right)\\ {Q}_{in}\left(n\right)=\mathrm{\Im }\left(u\left(n\right)\right)\\ {I}_{out}\left(n\right)=\mathrm{\Re }\left(y\
left(n\right)\right)\\ {Q}_{out}\left(n\right)=\mathrm{\Im }\left(y\left(n\right)\right),\end{array}$
where $\mathrm{\Re }$ and $\mathrm{\Im }$ are the real and imaginary part operators, respectively.
Prepare Data
Generate training, validation, and testing data. Use the training and validation data to train the NN-DPD. Use the test data to evaluate the NN-DPD performance. For details, see the Data Preparation
for Neural Network Digital Predistortion Design example.
Choose Data Source and Bandwidth
Choose the data source for the system. This example uses an NXP™ Airfast LDMOS Doherty PA, which is connected to a local NI™ VST, as described in the Power Amplifier Characterization example. If you
do not have access to a PA, run the example with simulated PA or saved data. Simulated PA uses a neural network PA model, which is trained using data captured from the PA using an NI VST. If you
choose saved data, the example downloads data files.
dataSource = "Simulated PA";
if strcmp(dataSource,"Saved data")
Generate Training Data
Generate oversampled OFDM signals.
[txWaveTrain,txWaveVal,txWaveTest,qamRefSymTrain,qamRefSymVal, ...
qamRefSymTest,ofdmParams] = generateOversampledOFDMSignals;
Fs = ofdmParams.SampleRate;
bw = ofdmParams.Bandwidth;
Pass signals through the PA using the helperNNDPDPowerAmplifier System object™.
pa = helperNNDPDPowerAmplifier(DataSource=dataSource,SampleRate=Fs);
paOutputTrain = pa(txWaveTrain);
paOutputVal = pa(txWaveVal);
paOutputTest = pa(txWaveTest);
Preprocess data to generate input vectors containing features.
memDepth = 5; % Memory depth of the DPD (or PA model)
nonlinearDegree = 5; % Nonlinear polynomial degree
[inputMtxTrain,inputMtxVal,inputMtxTest,outputMtxTrain,outputMtxVal,outputMtxTest,scalingFactor] = ...
helperNNDPDPreprocessData(txWaveTrain,txWaveVal,txWaveTest,paOutputTrain,paOutputVal,paOutputTest, ...
Implement and Train NN-DPD
Before training the neural network DPD, select the memory depth and degree of nonlinearity. For purposes of comparison, specify a memory depth of 5 and a nonlinear polynomial degree of 5, as in the
Power Amplifier Characterization example. Then implement the network described in Neural Network DPD Structure section.
memDepth = 5; % Memory depth of the DPD (or PA model)
nonlinearDegree = 5; % Nonlinear polynomial degree
inputLayerDim = 2*memDepth+(nonlinearDegree-1)*memDepth;
numNeuronsPerLayer = 30;
neuronReductionRate = 0.8;
lgraph = [...
Train Neural Network
Train the neural network offline using the trainnet (Deep Learning Toolbox) function. First, define the training options using the trainingOptions (Deep Learning Toolbox) function and set
hyperparameters. Use the Adam optimizer with a mini-batch size of 1024. The initial learning rate is 4e-4 and decreases by a factor of 0.95 every five epochs. Evaluate the training performance using
validation every 10 epochs. If the validation accuracy does not increase for five validations, stop training. Use Experiment Manager (Deep Learning Toolbox) to optimize hyperparameters.
maxEpochs = 200;
miniBatchSize = 1024;
iterPerEpoch = floor(size(inputMtxTrain, 1)/miniBatchSize);
trainingPlots = "none";
metrics = [];
verbose = false;
options = trainingOptions('adam', ...
MaxEpochs=maxEpochs, ...
MiniBatchSize=miniBatchSize, ...
InitialLearnRate=4e-4, ...
LearnRateDropFactor=0.95, ...
LearnRateDropPeriod=5, ...
LearnRateSchedule='piecewise', ...
Shuffle='every-epoch', ...
OutputNetwork='best-validation-loss', ...
ValidationData={inputMtxVal,outputMtxVal}, ...
ValidationFrequency=2*iterPerEpoch, ...
ValidationPatience=5, ...
InputDataFormats="BC", ...
TargetDataFormats="BC", ...
ExecutionEnvironment='cpu', ...
Plots=trainingPlots, ...
Metrics = metrics, ...
Verbose=verbose, ...
When running the example, you have the option of using a pretrained network by setting the trainNow variable to false. Training is desirable to match the network to your simulation configuration. If
using a different PA, signal bandwidth, or target input power level, retrain the network. Training the neural network on an Intel® Xeon(R) W-2133 CPU takes about 6 minutes to satisfy the early
stopping criteria specified above. Since trained network can converge to a different point then the saved data configuration, you cannot use saved data option with trainNow set to true.
trainNow = false;
if trainNow
netDPD = trainnet(inputMtxTrain,outputMtxTrain,lgraph,"mse",options); %#ok<UNRCH>
The following shows the training process with the given options. Random initialization of the weights for different layers affects the training process. To obtain the best root mean squared error
(RMSE) for the final validation, train the same network a few times.
Test NN-DPD
This figure displays how to check the performance of the NN-DPD. To test the NN-DPD, pass the test signal through the NN-DPD and the PA and examine these performance metrics:
• Normalized mean square error (NMSE), measured between the input to the NN-DPD and output of the PA
• Adjacent channel power ratio (ACPR), measured at the output of the PA by using the comm.ACPR System object
• Percent RMS error vector magnitude (EVM), measured by comparing the OFDM demodulation output to the 16-QAM modulated symbols by using the comm.EVM System object
Perform these tests for both the NN-DPD and also the memory polynomial DPD described in the Digital Predistortion to Compensate for Power Amplifier Nonlinearities example.
% Pass signal through NN-DPD
dpdOutNN = predict(netDPD,inputMtxTest);
dpdOutNN = double(complex(dpdOutNN(:,1),dpdOutNN(:,2)));
dpdOutNN = dpdOutNN/scalingFactor;
paOutputNN = pa(dpdOutNN);
% Pass signal through cross-term memory polynomial DPD
dpdOutMP = helperNNDPDMemoryPolynomial(txWaveTest,txWaveTrain, ...
paOutputMP = pa(dpdOutMP);
% Evaluate performance with NN-DPD
acprNNDPD = helperACPR(paOutputNN,Fs,bw);
nmseNNDPD = helperNMSE(txWaveTest,paOutputNN);
evmNNDPD = helperEVM(paOutputNN,qamRefSymTest,ofdmParams);
% Evaluate the performance without DPD
acprNoDPD = helperACPR(paOutputTest,Fs,bw);
nmseNoDPD = helperNMSE(txWaveTest,paOutputTest);
evmNoDPD = helperEVM(paOutputTest,qamRefSymTest,ofdmParams);
% Evaluate the performance with memory polynomial DPD
acprMPDPD = helperACPR(paOutputMP,Fs,bw);
nmseMPDPD = helperNMSE(txWaveTest,paOutputMP);
evmMPDPD = helperEVM(paOutputMP,qamRefSymTest,ofdmParams);
% Create a table to display results
evm = [evmNoDPD;evmMPDPD;evmNNDPD];
acpr = [acprNoDPD;acprMPDPD;acprNNDPD];
nmse = [nmseNoDPD;nmseMPDPD;nmseNNDPD];
disp(table(acpr,nmse,evm, ...
'VariableNames', ...
{'ACPR_dB','NMSE_dB','EVM_percent'}, ...
'RowNames', ...
{'No DPD','Cross-term Memory Polynomial DPD','Neural Network DPD'}))
ACPR_dB NMSE_dB EVM_percent
_______ _______ ___________
No DPD -28.674 -21.287 6.8681
Cross-term Memory Polynomial DPD -33.889 -27.984 2.8229
Neural Network DPD -38.886 -33.423 1.5679
sa = helperPACharPlotSpectrum(...
[paOutputTest paOutputMP paOutputNN], ...
{'No DPD','Memory Polynomial DPD', ...
'Neural Network DPD'}, ...
ofdmParams.OversamplingFactor,"Modulated",[-130 -50]);
As the PA heats, the performance characteristics change. Send bursty signals through the PA repeatedly and plot system performance as a function of time. Each measurement takes about 6 s. Every 600
s, stop for 300 s to allow the PA to cool down. The plot shows that the system performance degrades with repeated use and recovers after the cooldown period. This behavior shows that after some time,
the PA characteristics might change and the DPD might not provide the required system performance, such as a maximum EVM value. If the EVM value exceeds the allowed maximum value, the neural network
needs to be retrained to adapt to the changing PA characteristics.
runRepeatedBurstTest = false;
if strcmp(dataSource,"NI VST") && runRepeatedBurstTest
numMeas = 500;
measTime = 6;
acprNNDPD = zeros(numMeas,1);
nmseNNDPD = zeros(numMeas,1);
evmNNDPD = zeros(numMeas,1);
[acprLine,nmseLine,evmLine] = initFigure();
tStart = tic;
cnt = 1;
for p=1:numMeas
% Pass signal through NN-DPD
dpdOutNN = predict(netDPD,inputMtxTest);
dpdOutNN = [zeros(memDepth,1);...
double(complex(dpdOutNN(:,1), dpdOutNN(:,2)))];
paInput = dpdOutNN/scalingFactor;
% Pass signals through PA
paOutputNN = pa(paInput);
% Evaluate performance with NN-DPD
acprNNDPD(cnt) = helperACPR(paOutputNN,Fs,bw);
nmseNNDPD(cnt) = helperNMSE(txWaveTest,paOutputNN);
evmNNDPD(cnt) = helperEVM(paOutputNN,qamRefSymTest,ofdmParams);
updateFigure(acprLine,nmseLine,evmLine, ...
cnt = cnt +1;
if mod(p,100) == 0
for q=1:50
acprNNDPD(cnt) = NaN;
nmseNNDPD(cnt) = NaN;
evmNNDPD(cnt) = NaN;
updateFigure(acprLine,nmseLine,evmLine, ...
cnt = cnt +1;
numMeas = length(acprNNDPD);
t = (0:numMeas-1)*6;
grid on
title("NN-DPD Performance over Many Bursts")
grid on
grid on
xlabel('t (s)')
Further Exploration
This example demonstrates how to train a NN-DPD by using measured data from a PA. For the given PA, target input power level, and exciting signal, the NN-DPD is able to provide better performance
than memory polynomial DPD.
You can try changing the number of neurons per layer, number of hidden layers and target input power level and see the effect of these parameters on the NN-DPD performance. You can also try different
input signals, such as OFDM signals with different bandwidth. You can also generate standard-specific signals using the Wireless Waveform Generator app.
Proceed to the Neural Network for Digital Predistortion Design-Offline Training example, which shows how to train a similar network in a hardware-in-the-loop (HIL) setting using custom training loops
and loss functions.
Helper Functions
Performance Evaluation and Comparison
Local Functions
Generate Oversampled OFDM Signals
Generate OFDM-based signals to excite the PA. This example uses a 5G-like OFDM waveform. Set the bandwidth of the signal to 100 MHz. Choosing a larger bandwidth signal causes the PA to introduce more
nonlinear distortion and yields greater benefit from the addition of the DPD. Generate six OFDM symbols, where each subcarrier carries a 16-QAM symbol, using the helperNNDPDGenerateOFDM function.
Save the 16-QAM symbols as a reference to calculate the EVM performance. To capture effects of higher order nonlinearities, the example oversamples the PA input by a factor of 5.
function [txWaveTrain,txWaveVal,txWaveTest,qamRefSymTrain,qamRefSymVal,qamRefSymTest,ofdmParams] = ...
bw = 100e6; % Hz
symPerFrame = 6; % OFDM symbols per frame
M = 16; % Each OFDM subcarrier contains a 16-QAM symbol
osf = 5; % oversampling factor for PA input
% OFDM parameters
ofdmParams = helperOFDMParameters(bw,osf);
% OFDM with 16-QAM in data subcarriers
[txWaveTrain,qamRefSymTrain] = helperNNDPDGenerateOFDM(ofdmParams,symPerFrame,M);
[txWaveVal,qamRefSymVal] = helperNNDPDGenerateOFDM(ofdmParams,symPerFrame,M);
[txWaveTest,qamRefSymTest] = helperNNDPDGenerateOFDM(ofdmParams,symPerFrame,M);
Figure Helpers
function [acprLine,nmseLine,evmLine] = initFigure()
%initFigure Initialize repeat runs figure
acprLine = animatedline;
grid on
ylabel("ACPR (dB)")
title("NN-DPD Performance Over Many Bursts")
nmseLine = animatedline;
grid on
ylabel("NMSE (dB)")
evmLine = animatedline;
grid on
ylabel("EVM (%)")
xlabel("t (s)")
function updateFigure(acprLine,nmseLine,evmLine,acprNNDPD,nmseNNDPD,evmNNDPD,tStart)
%updateFigure Update repeat runs figure
drawnow limitrate
[1] Paaso, Henna, and Aarne Mammela. “Comparison of Direct Learning and Indirect Learning Predistortion Architectures.” In 2008 IEEE International Symposium on Wireless Communication Systems, 309–13.
Reykjavik: IEEE, 2008. https://doi.org/10.1109/ISWCS.2008.4726067.
[2] Wang, Dongming, Mohsin Aziz, Mohamed Helaoui, and Fadhel M. Ghannouchi. “Augmented Real-Valued Time-Delay Neural Network for Compensation of Distortions and Impairments in Wireless Transmitters.”
IEEE Transactions on Neural Networks and Learning Systems 30, no. 1 (January 2019): 242–54. https://doi.org/10.1109/TNNLS.2018.2838039.
See Also
Related Topics | {"url":"https://se.mathworks.com/help/comm/ug/neural-network-for-digital-predistortion-design-offline-training.html","timestamp":"2024-11-05T03:03:34Z","content_type":"text/html","content_length":"122050","record_id":"<urn:uuid:6d71b05c-031e-42bf-b5a4-29c4eff5145f>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00367.warc.gz"} |
Compute the rank of matrix A.
R = rank(A)
R = rank(A, tol)
The matrix whose rank is computed.
Dimension: matrix
A threshold for rounding off near-zero singular values. The default is the product of max(size(A)), the largest singular value, and eps.
Type: double
Dimension: scalar
The rank.
Type: integer
Matrix input with default tolerance:
R = rank([1,2,3;4,5,6;7,8,9.1])
R = 3
Matrix input with specified tolerance:
R = rank([1,2,3;4,5,6;7,8,9.1], 0.02)
R = 2 | {"url":"https://www.openmatrix.org/help/topics/reference/oml_language/LinearAlgebra/rank.htm","timestamp":"2024-11-14T05:22:47Z","content_type":"application/xhtml+xml","content_length":"7110","record_id":"<urn:uuid:4f8743c8-19d3-47a4-8e43-db7ee9a1300b>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00273.warc.gz"} |
One-fourth of a herd of camels was seen in the forest. Twice the square root of the herd had gone to the mountains and the remaining 15 camels were seen on the bank of a river. Find the total number of camels.
Hint: Let us denote a variable $x$ which represents the total number of camels that are present in the herd. Read the question line by line and apply all the conditions that are given in the question
on this variable $x$ to get an equation. The equation can be solved and we can find the value of $x$.
In this question, of the total number of camels in a herd, one-fourth of them are in the forest, twice the square root of the total camels had gone to the mountains and the remaining 15 were seen on
the bank of the river. We are asked to find the total number of camels that are present in the herd.
Let us denote a variable $x$ which represents the total number of camels that are present in the herd.
It is given that one-fourth of these camels are in the forest. So, the number of camels in the forest is equal to $\dfrac{1}{4}x..........\left( 1 \right)$.
It is given that twice the square root of the herd had gone to the mountains. So the number of camels that had gone to the mountains is equal to $2\sqrt{x}........\left( 2 \right)$.
Also, it is given that the number of camels that were seen on the bank of the river is equal to $15.......\left( 3 \right)$.
The sum of the terms which we got in equation $\left( 1 \right),\left( 2 \right),\left( 3 \right)$ should be equal to the total number of the camels i.e. $x$. This means,
& \dfrac{1}{4}x+2\sqrt{x}+15=x \\
& \Rightarrow 2\sqrt{x}=x-\dfrac{1}{4}x-15 \\
& \Rightarrow 2\sqrt{x}=\dfrac{3}{4}x-15 \\
& \Rightarrow 2\sqrt{x}=\dfrac{3x-60}{4} \\
& \Rightarrow 8\sqrt{x}=3x-60 \\
Squaring both the sides, we get,
& 64x=9{{x}^{2}}+3600-360x \\
& \Rightarrow 9{{x}^{2}}-424x+3600=0 \\
To solve this quadratic equation, we will use the quadratic formula. Let us assume a quadratic equation $a{{x}^{2}}+bx+c=0$. From the quadratic formula, the roots of this equation are given by,
$x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$
Using quadratic formula in the equation $9{{x}^{2}}-424x+3600=0$, we get,
& x=\dfrac{-\left( -424 \right)\pm \sqrt{{{\left( -424 \right)}^{2}}-4\left( 9 \right)\left( 3600 \right)}}{2\left( 9 \right)} \\
& \Rightarrow x=\dfrac{-\left( -424 \right)\pm \sqrt{{{\left( -424 \right)}^{2}}-4\left( 9 \right)\left( 3600 \right)}}{2\left( 9 \right)} \\
& \Rightarrow x=\dfrac{424\pm \sqrt{179776-129600}}{18} \\
& \Rightarrow x=\dfrac{424\pm \sqrt{50176}}{18} \\
& \Rightarrow x=\dfrac{424\pm 224}{18} \\
& \Rightarrow x=36,\dfrac{100}{9} \\
Since $x$ represents the number of camels, it must be an integer. So, $x=36$.
Hence, the total number of camels is equal to $36$.
Note: There is an alternative way to solve the quadratic equation $8\sqrt{x}=3x-60$. We can assume a variable $p$ and then we can substitute $x={{p}^{2}}$ in this quadratic equation and then solve
for $p$ using a quadratic formula. Later, we can re-substitute ${{p}^{2}}=x$ or $p=\sqrt{x}$ and then solve to get $x$. | {"url":"https://www.vedantu.com/question-answer/onefourth-of-a-herd-of-camels-was-seen-in-the-class-10-maths-cbse-5edcbb2b4d8add132469cb6b","timestamp":"2024-11-02T01:52:44Z","content_type":"text/html","content_length":"172109","record_id":"<urn:uuid:b8b1e11b-789c-43b9-a2a4-8b1599ba53bb>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00678.warc.gz"} |
Likelihood Gradient Evaluation Using Square-Root Covariance Filters
Kulikova, Maria
IEEE Transactions on Automatic Control, 54(3) (2009), 646-651
Using the array form of numerically stable square-root implementation methods for Kalman filtering formulas, we construct a new square-root algorithm for the log-likelihood gradient (score)
evaluation. This avoids the use of the conventional Kalman filter with its inherent numerical instabilities and improves the robustness of computations against roundoff errors. The new algorithm is
developed in terms of covariance quantities and based on the ldquocondensed formrdquo of the array square-root filter. | {"url":"https://cemat.tecnico.ulisboa.pt/document.php?project_id=5&member_id=93&doc_id=1724","timestamp":"2024-11-14T00:39:17Z","content_type":"text/html","content_length":"8528","record_id":"<urn:uuid:020597ca-7ca2-499c-a5f5-f72407311dda>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00458.warc.gz"} |
Gradient Descent Algorithm in Python - Try Machine Learning
Gradient Descent Algorithm in Python
Gradient descent is a popular optimization algorithm used in machine learning and various other fields. It is primarily used to minimize a cost function by iteratively adjusting the parameters of a
model. In this article, we will explore the gradient descent algorithm and implement it in Python.
Key Takeaways
• Gradient descent is an optimization algorithm used to minimize a cost function.
• In machine learning, it is commonly used to update the parameters of a model in order to improve its performance.
• Gradient descent involves iteratively adjusting the parameters in the direction of the steepest descent of the cost function.
• Learning rate and the number of iterations are important hyperparameters that affect the performance of the gradient descent algorithm.
• Python provides various libraries, such as NumPy and scikit-learn, that make it easy to implement gradient descent.
Gradient descent works by calculating the gradient (derivative) of the cost function with respect to each parameter and then updating the parameters in the direction of the steepest descent. This
process is repeated for a number of iterations until the algorithm converges to the minimum of the cost function.
Implementing Gradient Descent in Python
In order to implement the gradient descent algorithm in Python, we will need to define our cost function, initialize the parameters, choose an appropriate learning rate, and decide on the number of
1. Define the Cost Function
A cost function measures how well a model is performing by comparing the predicted values to the actual values. It quantifies the error between the predicted and actual output. Different machine
learning applications require different cost functions, such as mean squared error or log loss.
2. Initialize Parameters
Initializing the parameters of the model is an important step in the gradient descent algorithm. We need to set initial values for the parameters that we will update during each iteration. The
initial values can be set randomly or based on prior knowledge.
3. Choose a Learning Rate
The learning rate determines the step size taken in each iteration. If the learning rate is too small, the algorithm will take a long time to converge, and if it’s too large, it may overshoot the
minimum of the cost function and fail to converge.
4. Decide on the Number of Iterations
The number of iterations is the number of times the algorithm will update the parameters. It controls how long the algorithm will run. Convergence can be observed by monitoring the change in the cost
function over iterations or by setting a maximum number of iterations.
The Mathematics Behind Gradient Descent
Gradient descent is based on the derivative of the cost function with respect to each parameter. The update rule can be represented mathematically as:
w = w - learning_rate * dJ/dw
dJ/dw represents the derivative of the cost function with respect to the parameter w. The parameter w is updated by subtracting the product of the learning rate and the derivative from its current
value w.
Comparison of Learning Rates
Learning Rate Convergence Speed Stability Remarks
0.1 Fast Unstable May overshoot the minimum
0.01 Slower Stable Commonly used
0.001 Very Slow Highly Stable Prevents overshooting
Comparison of Algorithms
Algorithm Advantages Disadvantages Remarks
Gradient Descent Simple to implement May get stuck in local minima Widely used
Stochastic Gradient Descent Efficient for large datasets Highly dependent on initial conditions Useful for online learning
Batch Gradient Descent Guaranteed to converge Computationally expensive Suitable for small datasets
Performance of Gradient Descent with Different Learning Rates
Learning Rate Final Value of Cost Function Number of Iterations
0.1 150.32 20
0.01 202.43 50
0.001 246.72 100
Gradient descent is a powerful optimization algorithm used in various fields, especially in machine learning. By iteratively updating the parameters of a model in the direction of the steepest
descent of the cost function, gradient descent helps improve the performance of machine learning models. With Python libraries like NumPy and scikit-learn, implementing gradient descent becomes
straightforward and accessible.
Common Misconceptions
Gradient Descent Algorithm is Only for Machine Learning
While gradient descent is often associated with machine learning, it is not limited to this field. It is a versatile optimization algorithm used in various domains for minimizing a function’s value
or finding the optimal solution.
• Gradient descent can be used in computer vision tasks to optimize image processing algorithms
• It is employed in natural language processing to fine-tune language models
• Gradient descent is used in signal processing for system identification and adaptive filtering
Gradient Descent is Deterministic and Always Converges to the Global Minimum
Contrary to popular belief, gradient descent is a stochastic algorithm, meaning its convergence depends on various factors. It may or may not converge to the global minimum, as it can get stuck in
local minima or saddle points.
• Convergence of gradient descent can vary depending on the learning rate and the initialization of parameters
• Higher learning rates may lead to overshooting the optimal solution
• In some cases, carefully selecting different variants of gradient descent algorithms can improve convergence
Gradient Descent Always Finds the Optimal Solution in One Iteration
It is a misconception that gradient descent algorithm will find the optimal solution in just one iteration. In reality, a single iteration updates the parameters based on the gradient of the
function, but it typically takes multiple iterations to converge to a suitable solution.
• The number of iterations needed depends on the complexity of the problem and the specific optimization objective
• Stopping criteria, such as a certain level of accuracy or a pre-defined number of iterations, can be used to determine when to stop the algorithm
• Gradient descent can be run for more iterations to refine the solution, but there is a trade-off between computation time and solution quality
Gradient Descent is Only Applicable to Convex Optimization Problems
Another common misconception is that gradient descent is only applicable to convex optimization problems. While convex functions provide certain benefits, gradient descent can still be applied to
non-convex problems with some considerations.
• For non-convex problems, gradient descent may find local optima or saddle points instead of the global minimum
• Variants of gradient descent, such as stochastic gradient descent (SGD) and Mini-batch gradient descent, are commonly used in non-convex optimization
• Initialization of parameters and exploration of different learning rates can influence the quality of the solution for non-convex problems
Using Gradient Descent Algorithm Guarantees an Optimal Solution
Finally, it is important to note that using the gradient descent algorithm does not always guarantee obtaining the global optimal solution. It is an iterative optimization method that attempts to
find a locally optimal solution that minimizes the loss function being optimized.
• The quality of the solution depends on the complexity of the problem, the chosen optimization algorithm, and the convergence criteria
• Other optimization methods, such as Newton’s method or genetic algorithms, may provide alternative approaches to finding optimal solutions in certain scenarios
• Combining gradient descent with other techniques or ensembling multiple models can help mitigate the risk of getting stuck in poor solutions
In this article, we will explore the Gradient Descent algorithm implemented in Python. Gradient Descent is an optimization algorithm commonly used in machine learning and deep learning models. It
helps to minimize errors and adjust model parameters to find the best possible fit. We will illustrate various aspects of the Gradient Descent algorithm through a series of visually appealing tables.
Data Set Characteristics
Before diving into the Gradient Descent algorithm, let’s first understand the characteristics of our dataset. The table below showcases the key attributes of the dataset:
Dataset Name Number of Instances Number of Features Data Type Source
California Housing 20,640 8 Numerical Kaggle
Initial Model Parameters
Before executing the Gradient Descent algorithm, we need to set initial values for our model’s parameters. The table below presents the initial parameter values we have chosen:
Parameter Value
Learning Rate 0.01
Number of Iterations 1000
Initial Coefficients [-1.5, 0.8, -2.3, 1.7, -0.5, 0.2, -1.0, 0.9]
Gradient Descent Iteration Results
Let’s analyze the results of each iteration during the Gradient Descent process. We will keep track of the iteration number, the updated coefficients, and the mean squared error (MSE) at each step:
Iteration Updated Coefficients MSE
1 [-1.2, 0.7, -2.1, 1.5, -0.45, 0.18, -0.95, 0.85] 4568.25
2 [-0.9, 0.6, -1.9, 1.2, -0.4, 0.16, -0.9, 0.8] 3756.12
3 [-0.6, 0.5, -1.7, 0.9, -0.35, 0.14, -0.85, 0.75] 3043.99
4 [-0.3, 0.4, -1.5, 0.6, -0.3, 0.12, -0.8, 0.7] 2431.86
5 [0.0, 0.3, -1.3, 0.3, -0.25, 0.1, -0.75, 0.65] 1919.73
Convergence Analysis
Now, let’s assess the convergence behavior of the Gradient Descent algorithm for different learning rates:
Learning Rate Total Iterations Final MSE
0.01 1000 107.91
0.05 684 102.13
0.1 385 99.67
Stochastic Gradient Descent
Now, let’s explore the Stochastic Gradient Descent (SGD) variant of the algorithm. The table showcases the iteration-wise coefficient updates and the corresponding MSE:
Iteration Updated Coefficients MSE
1 [-1.1, 0.75, -2.2, 1.6, -0.47, 0.17, -0.92, 0.82] 4259.64
2 [-0.9, 0.6, -1.9, 1.2, -0.4, 0.16, -0.9, 0.8] 3756.12
3 [-0.77, 0.58, -1.81, 1.14, -0.39, 0.15, -0.87, 0.79] 3471.59
4 [-0.66, 0.56, -1.74, 1.1, -0.37, 0.14, -0.84, 0.78] 3245.16
Batch Gradient Descent
Lastly, let’s take a look at the Batch Gradient Descent (BGD) variant of the algorithm. The table represents the coefficient and MSE updates after each iteration:
Iteration Updated Coefficients MSE
1 [-1.9, 1.5, -2.8, 2.1, -0.4, 0.4, -1.1, 1.3] 8100.45
2 [-2.45, 1.9, -3.4, 2.5, -0.6, 0.6, -1.6, 2.0] 6296.52
3 [-3.0, 2.3, -3.9, 2.9, -0.8, 0.8, -2.1, 2.7] 4823.79
4 [-3.5, 2.7, -4.4, 3.3, -1.0, 1.0, -2.6, 3.4] 3672.26
Through various tables, we examined different aspects of the Gradient Descent algorithm in Python. We explored the convergence behavior with different learning rates, as well as the variations of
Stochastic Gradient Descent and Batch Gradient Descent. Now armed with this valuable knowledge, you can confidently apply the Gradient Descent algorithm to optimize your own machine learning models
and achieve better results. Happy coding!
Frequently Asked Questions
1. What is the Gradient Descent Algorithm?
The Gradient Descent Algorithm is an optimization algorithm used to minimize the cost function of a machine learning model. It is commonly employed in training neural networks and other models with
large sets of parameters. The algorithm iteratively adjusts the model’s parameters based on the gradients of the cost function with respect to those parameters.
2. How does the Gradient Descent Algorithm work?
The Gradient Descent Algorithm works by iteratively updating the model’s parameters in the direction of the steepest descent of the cost function. It starts with an initial guess for the parameters
and calculates the gradients of the cost function with respect to those parameters. These gradients indicate the direction of maximum increase of the cost function, so the algorithm adjusts the
parameters in the opposite direction to minimize the cost.
3. What is the role of learning rate in the Gradient Descent Algorithm?
The learning rate in the Gradient Descent Algorithm determines the step size taken in each iteration when updating the parameters. It controls the speed at which the algorithm converges to the
optimal solution. A low learning rate may make the algorithm take a long time to converge, while a high learning rate might cause it to overshoot the minimum of the cost function.
4. Can I use the Gradient Descent Algorithm for any machine learning model?
Yes, the Gradient Descent Algorithm is a versatile optimization technique that can be used with various machine learning models. It is particularly common in models with a large number of parameters,
such as neural networks. However, the specific implementation and variants of the algorithm may vary depending on the model and the problem being solved.
5. How can I implement the Gradient Descent Algorithm in Python?
In Python, you can implement the Gradient Descent Algorithm using various libraries, such as NumPy or TensorFlow. The general steps involve initializing the parameters, defining the cost function and
its gradients, and iteratively updating the parameters based on the gradients and the learning rate. There are numerous tutorials and examples available online that can guide you through the
implementation process.
6. What are the advantages of using the Gradient Descent Algorithm?
The Gradient Descent Algorithm offers several advantages in machine learning. It can optimize models with a large number of parameters efficiently, enabling them to learn from large datasets. It is
also a flexible algorithm that can be applied to various models and problems. Additionally, the gradient updates can take advantage of parallel computing, which can speed up training on modern
7. Are there any limitations or challenges associated with the Gradient Descent Algorithm?
Yes, the Gradient Descent Algorithm has certain limitations and challenges. One common challenge is determining an appropriate learning rate that balances convergence speed and stability. Setting an
extremely low learning rate can slow down the algorithm, while a high learning rate may cause instability or overshooting. The algorithm may also get stuck in local minima rather than reaching the
global minimum of the cost function.
8. Can I use the Gradient Descent Algorithm for non-convex cost functions?
Yes, the Gradient Descent Algorithm can be used for non-convex cost functions as well. However, non-convex functions may have multiple local minima, which can pose challenges for convergence to the
global minimum. In such cases, initialization of parameters and adjusting the learning rate become more crucial to achieve a desirable solution.
9. Are there any variations of the Gradient Descent Algorithm?
Yes, there are several variations of the Gradient Descent Algorithm, such as Stochastic Gradient Descent (SGD), Mini-Batch Gradient Descent, and Adam Optimization. These variations introduce
modifications to the basic algorithm, aiming to improve convergence speed, handle large datasets, or address issues like oscillation around the minima. The choice of algorithm depends on the specific
problem and the associated requirements.
10. How can I evaluate the convergence and performance of the Gradient Descent Algorithm?
You can evaluate the convergence and performance of the Gradient Descent Algorithm by monitoring the cost function’s value during training. If the cost decreases over iterations, it indicates
convergence. Additionally, you can use performance metrics such as accuracy, precision, or recall to assess the effectiveness of the trained model. Cross-validation or applying the algorithm to
separate test data can provide further insights into the generalization capability of the model. | {"url":"https://trymachinelearning.com/gradient-descent-algorithm-in-python/","timestamp":"2024-11-10T01:25:58Z","content_type":"text/html","content_length":"72035","record_id":"<urn:uuid:b1425dc2-9b26-4536-8e90-1d26bbd26baa>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00759.warc.gz"} |
Properties of
Posted 31 August 2009
This section concerns certain properties relations may have. In each case it applies only to a binary relation on a single set. The ones given here are fundamental to most parts of abstract math.
Wikipedia describes some other properties.
Reflexive relations
A binary relation on a set A is reflexive on A if for every element of A.
¨ The relation “ ” is reflexive on . That’s because the statement “ ” is true for every real number r.
¨ The equals relation is reflexive on any set: the statement “x = x” is true for every mathematical object x.
¨ The relation “ ” on the powerset of any set is reflexive.
¨ The relation on the set of integers defined by if and only if is reflexive.
¨ A nearness relation on is reflexive.
¨ The relation " <" is not reflexive on .
¨ The relation "is the sister of" on the set W of all women is not reflexive, since no one is the sister of herself.
¨ The relation on is reflexive. The relation (same set of ordered pairs) on is not reflexive. This example shows that for reflexivity, the stricter definition of relation must be used.
Something to think about
A relation on a set A is reflexive if and only if the equals relation on A is a subset of .
Irreflexive relations
A relation on a set A is irreflexive on A if is false for every element a of A. Note that this is not the negation of “reflexive”. The relations “<” on and “is the sister of” on W just
mentioned are in fact irreflexive. But a relation can be neither reflexive nor irreflexive: The relation {(1, 1), (2, 2)} is not reflexive on the set of all integers, because 3 (among others!) is
not related to itself. But it is not irreflexive either, since 1 is related to itself.
Warning It is wrong to say that the relation just given is “reflexive at 1 but not at 3”. Reflexivity and irreflexivity are properties of the relation and the set it is defined on, not of
particular elements of the set. This comment also applies to the other properties of relations discussed in this section.
Definition: symmetric
A binary relation on a set A is symmetric if implies for all elements a and b of A.
¨ The empty relation is symmetric, because the statement “if then ” is vacuously true.
¨ The equals relation is symmetric. (Rewrite: Must show that if a = b, then b = a. But you have known that for years.)
¨ Any nearness relation is symmetric.
¨ The relation {(1, 2), (2, 1), (1,3)} is not symmetric. It is wrong to say it is “sometimes symmetric” or “is symmetric as far as 1 and 2 are concerned.” Being symmetric is a property of the
whole relation.
¨ The sister relation is symmetric on the set of all women.
¨ The sister relation on the set of all people is not symmetric.
Warning It is important to understand the precise meaning of the definition of symmetric. It is given in the form of an conditional assertion: is symmetric if for all pairs (a, b), if then .
This does not assert that for any particular elements a and b.
We have defined relation as an abstraction (set of ordered pairs) of a relationship in the usual sense, and then defined a symmetric relation in terms of the abstract definition of relation (when (a,
b) is in the relation then so is (b, a).) We could have given a direct abstract definition of symmetric relation on a set A by saying it is a set of one- and two-element subsets of A. For example,
the symmetric relation {(1, 1), (1, 2), (2, 1), (2, 3), (3, 2)} could be modeled as {{1},{1, 2}, {2, 3}}. This is an example of a concept having two different-looking definitions.
Worked Exercise
Show that if a relation on a set A is not symmetric, then A has at least two distinct elements.
Proof: If A is empty, then is the empty relation, which is vacuously symmetric.
If A has exactly one element, then either is empty in which case it is vacuously symmetric, or , where a is the only element in A, but then is symmetric.
So A must have at least two elements.
A binary relation on a set A is antisymmetric if for all elements a, bA, if and , then a = b
¨ The empty relation is vacuously antisymmetric.
¨ The “<” relation is vacuously antisymmetric. (Rewrite: “If a < b and b < a, then a = b” is vacuously true because the statement “a < b and b < a” is always false.)
¨ The “ ” relation on the set of real numbers is antisymmetric. (Rewrite: “If , then a = b”, a familiar fact about numbers. Antisymmetry is typical of order relations in general.
¨ The equals relation on any set is antisymmetric. (Rewrite: If a = b and b = a, then a = b, which is true by definition of “and” .
¨ The total relation on a set is not antisymmetric if the set has more than one element.
¨ A nearness relation is not antisymmetric. For example, if , then and 3.14 are near each other, but they are not the same.
Antisymmetry is not the negation of symmetry.
¨ The equals relation is both symmetric and antisymmetric.
¨ So is the relation {(1, 1), (2, 2)} on the set of real numbers. Note that this is not the equals relation. See this exercise.
¨ The relation divides on the set of integers is neither symmetric nor antisymmetric. 3 divides 6 but 6 does not divide 3, so it is not symmetric. On the other hand, 3 divides 3 and 3 divides 3,
but 3 and 3 are not equal, so it is not antisymmetric.
Definition: transitive
A binary relation on a set A is transitive if for all elements a, b, and c of A, if and , then .
¨ The empty relation is vacuously transitive on any set.
¨ The equals relation is transitive on any set. (If a = b and b = c then a = c. This is equivalent to the property given by “two things equal to the same thing are equal to each other.”)
¨ All these relations are transitive on : “ < “, “ ”, “>”, “ ”. In general, anything you could call an ordering is transitive.
¨ The sister relation S on the set of all women is not transitive. Agatha may be Bertha's sister, whence Bertha is Agatha's sister, but Agatha is not her own sister. This illustrates the general
principle that when a definition uses different letters to denote things, they don't have to denote different things (see two). In the definition of transitivity, a, b and c may be but don't have to
be different.
¨ Nearness relations are not transitive. For example, if , then 3.10 and 3.16 are near each other and 3.16 and 3.22 are near each other, but 3.10 and 3.22 are not near each other. It is well
known, for example, that color discrimination is not transitive. (You can have three color samples A, B and C, with A and B appearing identical and B and C appearing identical, but with A and C
detectably different.) | {"url":"https://abstractmath.org/MM/MMRelationsProps.htm","timestamp":"2024-11-13T14:08:41Z","content_type":"text/html","content_length":"174868","record_id":"<urn:uuid:4b589fda-14cd-43b2-b464-b8f9dd5fce03>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00696.warc.gz"} |
discharge circuit, dump circuit
Not open for further replies.
im looking for a circuit that can remove access power from a battery, but i can not find it anywhere.
im not that competent in designing circuits so i wil try in forum.
maybe someone have a circuit that i can use.
i did not ask about safety, i did not ask about other ways of doing this, i only ask for circuit and discussing about discharge circuit please respect that.
ok i have a few "requirements" for way of function
no programming
pwm based discharge
pwm 0-100% (not critical can be for example 10 - 90%)
pwm start voltage adjustable if batteryvoltage 5v over start voltage =100% pwm
pwm start voltage adjustable dc240v - dc265v (example start 265,05=1%pwm 270=100%pwm
pwm ramp up and down slowly
pwm is limited by a current shunt adjustable from 1A to 100A
pwm frequency not critical about 3khz
explaining of circuit work:
start voltage adjusted to 240vdc
current limit set to 10A
voltage increase to 240,1vdc
pwm start 2%
voltage increase to 241vdc
pwm adjust slowly to 20%
voltage increase to 242vdc
pwm try to adjust slowly to 40% but the current limit stop and hold the pwm at 30%
voltage increase to 243vdc
current limit still hold the pwm at 30%
voltage decrease to 240,5
pwm adjust slowly to 10%
voltage decrease to 240
pwm stop
Well-Known Member
Most Helpful Member
What is your application? A 265 Vdc, 10 A battery?
What is your application? A 265 Vdc, 10 A battery?
no the 10A limit was for an example to protect the connected load
battery is 90kwh
What is your application? A 265 Vdc, 10 A battery?
application is to bleed off the extra power to use it for something useful
here is a drawing to show what i was thinking,
offcorse it do not work bechause i do not have the knowlege to make it correct but the design or how to do should be possible this way.
240v is stepped down to 0v and 245v wil then be 5v, potentiometer is there to adjust the 0v from 240 to 265v and the 5v wil be 5v higher as in the first example.
0-5v is amplified by opamp, zener is there to pull down the voltage if the 0-5v should be higher than 5v.
0-5v pwm circuit 0v=0%duty 5v=100%duty.
pwm output have a pull down resistor to shut the SCR off when pulse is low.
shunt feedback the current via transistor and pull down the 0-5v if current is over setpoint by the potentiometer close to the shunt
then finally the battery and load is connected to the SCR input and output.
SCR is a NKT110-16A
I'm not sure I get it, however I dont think driving a thyristor gate with pwm will work as a thyristor on dc once switched on will remain on, a mosfet might be a better approach here.
I'm not sure I get it, however I dont think driving a thyristor gate with pwm will work as a thyristor on dc once switched on will remain on, a mosfet might be a better approach here.
It can be, and is done (an example is fork lift trucks), but
just with a single plain thyristor.
However, I don't get it either?, and as with many of these vague threads it's hard to know what he wants, what he expects, and what he's trying to do?.
In the theme of 'vagueness' the standard method is to dump any excess energy (from the source
the batteries) on to storage heaters.
It can be, and is done (an example is fork lift trucks), but NOT just with a single plain thyristor.
However, I don't get it either?, and as with many of these vague threads it's hard to know what he wants, what he expects, and what he's trying to do?.
In the theme of 'vagueness' the standard method is to dump any excess energy (from the source NOT the batteries) on to storage heaters.
the source is connected to the batteries and the normal loads are drawn from the batteries as well.
so for example solar is chargeing the batteries by 6kw
the normal loads are 3kw drawn from the battery.
then the battery wil be charged by 3kw and the battery voltage wil rise until charger shut off.
i want to use the free solar power for something useful and if the load monitor the battery voltage it wil adjust it self to the extra avalible power all the time.
as in my explaining, rise of battery voltage over set point = increase the load
i live in a cold area, i wil use heating elements in water storage tank to heat water with the free extra power
I'm not sure I get it, however I dont think driving a thyristor gate with pwm will work as a thyristor on dc once switched on will remain on, a mosfet might be a better approach here.
i have a large igbt somwhere also witch can be used
As it's solar, why such a high battery voltage? - much more common (and sensible) to use a much lower voltage, and all the control systems you need are freely available.
I went to a house a few years ago, the address was 'Mill Close' (bit of a giveaway) and it was the left hand one of a semi-detached pair. The original stream that fed the old mill on the site was
actually dead on the centre line of the two houses, so they got together and installed a small water turbine - this provided 2KW of power, permanently, the stream came from underground and never
dries up.
Despite the fact I was actually delivering and installing a washing machine, I spend a fair bit of time talking to the owner - they shared the output of the turbine, 1KW each, no batteries involved,
and if the house requirement was under 1KW the excess power was simply sent to storage heaters upstairs (much like you're planning doing it with water). As far as I'm aware, everything used was
freely available, and it's operation was totally transparent.
Last edited:
As it's solar, why such a high battery voltage? - much more common (and sensible) to use a much lower voltage, and all the control systems you need are freely available.
I went to a house a few years ago, the address was 'Mill Close' (bit of a giveaway) and it was the left hand one of a semi-detached pair. The original stream that fed the old mill on the site was
actually dead on the centre line of the two houses, so they got together and installed a small water turbine - this provided 2KW of power, permanently, the stream came from underground and never
dries up.
Despite the fact I was actually delivering and installing a washing machine, I spend a fair bit of time talking to the owner - they shared the output of the turbine, 1KW each, no batteries
involved, and if the house requirement was under 1KW the excess power was simply sent to storage heaters upstairs (much like you're planning doing it with water). As far as I'm aware, everything
used was freely available, and it's operation was totally transparent.
the high voltage is bechause the inverter is a UPS, 25kw and to save on losses thats how its made.
i already have the charge controllers to go with it, x2 for solar and x3 for wind
but yes its kind of the same goal as with the mill
my ups is custom made to priority to work from batteries if power is avalible there, if battery is low the ups wil use the city grid, switching is done without flickering in lamps
Well-Known Member
Most Helpful Member
What I understand from the OP, and please correct me if I am wrong:
• The OP has a solar array connected to an UPS.
• The UPS sometimes does not draw the full power being generated by the array, I suspect in very sunny days.
• The OP wishes to use this wasted power in something useful.
As Nigel mentions, electric water heaters are by far the easiest way to use that free energy. And fortunately, water can store lots of energy per unit volume. And it is cheap.
What I understand from the OP, and please correct me if I am wrong:
□ The OP has a solar array connected to an UPS.
□ The UPS sometimes does not draw the full power being generated by the array, I suspect in very sunny days.
□ The OP wishes to use this wasted power in something useful.
As Nigel mentions, electric water heaters are by far the easiest way to use that free energy. And fortunately, water can store lots of energy per unit volume. And it is cheap.
thats the case yes
Nige, yes I maintain Flt's amongst other things, they did use thyristors, switch off was done with capacitor discharge, lansing bagnall were well up on this.
For the past 15 to 20 years mobile plant have gone to ac motors with an inverter instead of dc brushed ones, maintenance is a lot easier.
Nige, yes I maintain Flt's amongst other things, they did use thyristors, switch off was done with capacitor discharge, lansing bagnall were well up on this.
For the past 15 to 20 years mobile plant have gone to ac motors with an inverter instead of dc brushed ones, maintenance is a lot easier.
could you draw to show how the discharge capacitor was connected together withsouronding circuit?
could you draw to show how the discharge capacitor was connected together withsouronding circuit?
Try googling it, or even searching on these forums (as it's been posted here before), but essentially you use a second (smaller) thyristor to switch a capacitor across the main thyristor, this
momentarily stops current flowing through the thyristor (by shorting it out) causing it to turn OFF.
Heres an explanation.
Different SCR Turn OFF Methods like Natural, Forced Commutation are explained. In Forced, Class A, B, C, D, E. Dynamic turn OFF Characteristics.
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Statistics (for MRCOG) - Where is the Source?
Image: Sasquatch I
I have got many queries on the source of Statistics. “Where to study? What to study? How to study? “
And yes, it’s not a subject we doctors usually adore, but it’s not something we can escape from!
And a couple of questions invariably appear in the exam- written and sometimes in OSCE like the Forest Plot.
The statistics questions are almost same for part 1 & 2.
• I have gathered my knowledge by doing MRCOG Part 1 books. Any Basic Science Book with a chapter in statistics would do.
• But I did come across a very useful statistics chapter from a MRCP book- Basic Medical Sciences for MRCP PART 1. I made my notes from this book.
• There is also a TOG article on statistics- a bit lengthy though.
• Strat OG also has some statistics titles which you will find useful.
• I also found a useful resource given by Dr. Tom Mcfarlene on statistics. You may want to check it. Read “Basic Statistics 1 & 11”. This has power point presentations.
To get the hang of useful topics, please do more and more practice questions.
Make sure you memorise important formulas for the exam, which you may have to use to derive answers from. For example-
1. Sensitivity / Specificity
2. Positive predictive value ( I think we had a question on this for the last exam)/ Negative predictive Value
3. Odds ratio / Risk ratio
4. Mean / Median / Mode
5. Skewing to the right or left of the Bell shaped curve
6. Interquartile range
7. Standard deviation / Variance
8. Standard error
9. Chi-square test /Mann-Whitney U-test / t-test/ Fisher’s Exact test-– know in which scenario, which test is used.
10. Case control study/ Cohort study
11. P –Value
12. Confidence Interval
These are the things I can think of Statistics to help you guys… And and if you find this useful please share the knowledge.
Happy Reading! 🙂
1. Padmapriya B says
I also found a useful resource given by Dr. Tom Mcfarlene on statistics. You may want to check it. Read “Basic Statistics 1 & 11”. This has power point presentations.
Pl can u post the site for reference
□ Site Admin says
Hi sorry for the delayed reply. It looks like the link has been taken down, may have to find an alternative source.
2. Swati says
Hi mam
I have my part 2 exam in January
I’m not getting cervical cancer screening guidelines
Can u pls help
3. moazzama says
Hi, it is a wonderful website, is there any book on skin lesions in pregnancy
Perineal skin lesions in gynecology
□ Anjana Mackeen says
Thank you.
For the skin lesions, I read OGRM journal article.
4. DR BILQUEES MUSTAFA says
DEAR ANJANA Thank you so much , Really you are doing great job.
□ Anjana Mackeen says
You are most welcome! Thank you for the feed back
5. Mariam says
Thanx dear dr Anjana appreciated effort God bless you
□ Anjana Mackeen says
Happy with the feed back I am getting…. Do let me know if any further topics needs to be discussed.
6. Liza says
Thanks dr ,for this information really a easy and understandable explaination there , thanks again .
7. Zakia says
Dear Anjana, thank you so much for all your kind advices and guidance to help us.
□ Anjana Mackeen says
Glad to help. Let me know on what more you all need to know…
Leave a Reply Cancel reply | {"url":"https://anjanamackeen.com/2016/03/11/statistics-where-is-the-source/","timestamp":"2024-11-03T21:28:28Z","content_type":"text/html","content_length":"48905","record_id":"<urn:uuid:0bad1518-a5a4-434c-8ff6-bd1a65c6f3fe>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00355.warc.gz"} |
Random Variables and Probability Distributions in Maple
Random Variables and Probability Distributions in Maple: The Normal Distribution
A population represents every observation of the particular category that we wish to study. A sample is a subset of a population, which is often used to estimate details of the population. This video
will discuss how well sample statistics estimate population parameters and show how taking larger samples can better estimate population parameters. | {"url":"https://www.maplesoft.com/demo/streaming/RandomVariablesProbabilityDistributions_TheNormalDistribution.aspx","timestamp":"2024-11-14T07:32:40Z","content_type":"text/html","content_length":"20029","record_id":"<urn:uuid:15228f09-9f5c-4119-ba88-0d13c2fd3288>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00754.warc.gz"} |
Any graph with many large independent sets is almost bipartite
A problem on almost bipartite graphs (proposed by Erdös [1])
Suppose \(G\) has the property that for every \(m\), every subgraph on \(m\) vertices contains an independent set of size \(m/2-k\). Let \(f(k)\) denote the smallest number such that \(G\) can be
made bipartite by deleting \(f(k)\) vertices.
Determine \(f(k)\).
In the mid 1990s, Reed proved the existence of \( f(k)\) by using graph minors, but did not publish the result. It would be of interest to improve the estimates for \( f(k)\).
Erdös, Hajnal and Szemerédi [1] proved that for every \( \epsilon>0\), there is a graph of infinite chromatic number for which every subgraph of \( m\) vertices contains an independent set of size \(
(1-\epsilon)m/2\). Erdös remarked that perhaps \( (1-\epsilon)m/2\) could be replaced by \( m/2-f(m)\) where \( f(m)\) tends to infinity arbitrarily slowly.
1 P. Erdös, A. Hajnal and E. Szemerédi, On almost bipartite large chromatic graphs, Annals of Discrete Math. 12 (1982), Theory and practice of combinatorics, North-Holland Math. Stud., 60, 117-123,
North-Holland, Amsterdam, New York, 1982. | {"url":"https://mathweb.ucsd.edu/~erdosproblems/erdos/newproblems/AlmostBipartiteGraphs.html","timestamp":"2024-11-02T12:00:39Z","content_type":"text/html","content_length":"4791","record_id":"<urn:uuid:87d7f43a-300a-4537-a8cb-602a9b5352ec>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00737.warc.gz"} |
ASME B5.50-2015 pdf download - Free Standards Download Online
ASME B5.50-2015 pdf download
ASME B5.50-2015 pdf download.7 /24 Taper Tool to Spindle Connection for Automatic Tool Change.
NOTE: This excerpt from ISO 1947:1973 and the standards referenced within are nonbinding to ASME B530. Content is shown to clarify the derivation of angular tolerance (AT) grades only. No tolerance
system is endorsed nor encoded by inclusion herein. Portions of the original ISO 1947:1973 are omitted for clarity.
B-i SCOPE AND FIELD OF APPLICATION
This international Standard specifies a cone tolerance system which applies to rigid conical workpieces for which the length of the generator can be considered as practically equal to the basic cone
length; this applies in the case of cones having a rate of taper C = 1:3 to
For dimensioning and tolerancing cones on drawings, see iSO 3040, Technical drawings — Dimensions and tolerancing cones.2
For general information on tolerances of form and of position, see ISO/R 1101, Tolerances of form and of position — Part I: Generalities, symbols, indications on drawings.
B-2 BASIS OF THE SYSTEM
B-2.1 Types of Cone Tolerance
The following four types of tolerance provide the basis of the cone tolerance system:
(a) cone diameter tolerance, TD, valid for all cone diameters within the cone length, L.
(b) cone angle tolerance, AT, given in angular or linear dimensions (ATC, or ATD).
(c) cone form tolerance, TF (tolerances for the straightness of the generator and for the roundness of the section).
(d) cone section diameter tolerance, TDS, given for the cone diameter in a defined section. It is valid for the cone diameter of this section only.
B-2.2 Cone Diameter ToLerance, Cone Angle Tolerance, and Cone Form Tolerance
Normal cases will be handled by application of the cone diameter tolerance, TD, only. It includes the two tolerances of the types in paras. B-2.1(b) and (c). This means that the deviations of these
two types may, in principle, utilize the whole tolerance space given by the cone diameter tolerance, TD (see Fig. B-i). In case of stronger requirements, the cone angle tolerance and the cone form
tolerance may be reduced within the cone diameter tolerance, TD, by means of supplementary instructions. In this case, likewise, no point on the conical surface is permitted to lie outside the limit
cones’ given TD. In practice, all types of tolerance generally exist at the same time and, as far as normal cases are concerned, each tolerance may occupy a part of the cone diameter tolerance, TD,
only in such a way that no point on the conical surface lies outside the tolerance space. In other words, no point on the conical surface is permitted to lie outside the limit cones.
B-2.3 Cone Section Diameter Tolerance
If for functional reasons the cone diameter tolerance is required in a defined section, then the cone diameter tolerance, TDS [para. B-2.i(d)], must be indicated. In this case, it is also necessary
to indicate the cone angle tolerance. If general tolerances for the cone angle are specified, e.g., in an international document, and if it is referred to this tolerance, then it is not necessary to
indicate special cone angle tolerances.
B-3.1 Definitions Relating to Geometry of Cones
cone: a conical surface or a conical workpiece (see Fig. B-2), defined by its geometrical dimensions. In the absence of any indication concerning the geometrical form, cone is understood to mean a
straight circular cone or truncated cone.
conical surface: a surface of revolution which is formed by rotating a straight line (generator) around an axis with the straight line intersecting this axis at the apex (see Fig. B-2). The parts of
this infinite conical surface are also known as conical surfaces or cones. Similarly, “cone” is also the abbreviated designation of a truncated cone.
conical workpiece: a workpiece or portion of a workpiece, the main part of which is a conical surface (see Figs. B-3 and B-4). | {"url":"https://www.japan-ppt.com/asme-b5-50-2015-pdf-download.html","timestamp":"2024-11-06T12:28:18Z","content_type":"application/xhtml+xml","content_length":"36893","record_id":"<urn:uuid:26371ee5-025f-45b8-986e-ae2cf82b3549>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00768.warc.gz"} |
Simplify Decimals involving Addition and Subtraction Decimals | Questions on Addition and Subtraction of Decimals
Want to learn how to simplify decimals involving addition and subtraction? If so, you have come to the right place where you can get a complete idea of the Simplification of Decimals. Learn the
approach used for simplifying decimals involving addition and subtraction so that you can apply the same while solving related problems. Refer to Worked Out Problems for Decimal Simplification
explained in the further modules to clearly understand the concept.
Do Read:
How to Add or Subtract Decimals?
Go through the simple process available here to simplify decimals involving the addition and subtraction of decimals. They are as follows
• The first and foremost step is to convert the given decimals to like decimals.
• Write the decimals one below the other depending on the place value of digits.
• Later, solve using the order of operation accordingly.
Worked Out Problems on Simplification of Decimals involving Addition and Subtraction
1. Simplify the following 20.10 + 74.38 – 35.69?
Converting into like decimals and solving using order of operation.
As the given decimals are all like decimals there is no need to annex zeros. And we can go with the order of operation and simply the expression further.
Step 1: Addition
Step 2: Subtraction
94.48 – 35.69
Check out the worked out procedure for subtracting 35.69 from 94.48 below
Therefore, the value of 20.10 + 74.38 – 35.69 is 58.79
2. Simplify the following 14.6078 – 0.37 + 0.6?
Given decimals are 14.6078, 0.37, 0.6
First convert the given decimals to like decimals. We can do so by simply annexing with zeros i.e. the maximum number of decimal places among the given decimal numbers.
Since the maximum number of decimal places among the given decimal numbers is 4. We will annex with required zeros to make the given decimals to decimal places of 4
By Converting into like decimals we have the following
14.6078 ➙ 14.6078
0.37 ➙ 0.3700
0.6 ➙ 0.6000
Now, that you are done with annexing zeros simplify the given expression as per the order of operations.
Step 1: Subtraction
Check out how decimal 0.3700 is subtracted from the decimal value 14.6078
Step 2: Addition
Align the given decimals as per decimal points so that given decimals are arranged in Proper Decimal Value and then perform Addition of Decimals.
Therefore, the value of 14.6078 – 0.37 + 0.6 is 14.8378
3. What must be added to 19.33 to obtain 47.87?
19.33+x = 47.87
x = 47.87-19.33
= 28.54
Thus, 28.54 must be added to 19.33 is 47.87
4. What must be subtracted from 281.6 to obtain 18.88?
Rearranging the given equation we have
x = 9.28
5. Simplify the following.
75.102 + 64.38 – 25.99
First convert the given decimals to like decimals. We can do so by simply annexing with zeros i.e. the maximum number of decimal places among the given decimal numbers.
Amongst the given decimals 75.102 is having the maximum number of decimals i.e. 3
Change the rest of the decimals to like decimals by padding with required number of zeros to make them decimals having 3 decimal places
Converting Unlike Decimals to Like Decimals we have
75.102 ➙ 75.102
64.38 ➙ 64.380
25.99 ➙ 25.990
Step 1: Addition
Look out the decimal addition process for decimals 75.102 and 64.380
Step 2: Subtraction
Thus, the value of 75.102 + 64.38 – 25.99 is 113.492 | {"url":"https://eurekamathanswerkeys.com/simplify-decimals-involving-addition-and-subtraction-decimals/","timestamp":"2024-11-08T11:32:54Z","content_type":"text/html","content_length":"41213","record_id":"<urn:uuid:5659c37c-9255-4f02-bc9f-9d2b058a488d>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00888.warc.gz"} |
LESSON NOTE ON SS1 MATHEMATICS FOR FIRST TERM - Passnownow
Mathematics Lesson Notes SS1 First Term
Week 1 – Mensuration
Week 2 – Volumes of frustums of cone, rectangular-based pyramid and other pyramids
Week 3 – Geometrical construction
Week 4 – Triangle: Drawing and bisection of a line segment, construction and bisection of angles
Week 5 – Construction: Construction of quadrilateral polygon, construction of equilateral triangle locus of moving points.
Week 6 – Deductive proof: Sum of angles of a triangle revision of angles on parallel line cut by a transversal line.
Week 7 – Collection, tabulation and presentation of grouped data
Week 8 – Calculation of range, median and mode of ungrouped data
Week 9 – Mean deviation, variance and standard deviation
Below are the 2022 complete SS1 Mathematics First Term Lesson Note
Lesson Note on Mathematics SS1 First term
Week 1 – Mensuration
Overview – What is mensuration? Mensuration is a branch of Mathematics that deals with the measurement of areas and volumes of various geometrical figures. This week, we shall learn about
mensuration. We shall also learn about figures such as cubes, cuboids, cones and triangular prism as well as their volumes and areas. To learn more, Click here…
Week 2 – Volumes of Frustums of Cone, Rectangular-Based Pyramid and other Pyramids
Overview – Having been introduced to the concept of mensuration, this week, we shall learn about the shapes such as the cone, oblique prism, oblique cylinder and pyramids. We shall also learn how to
prove that the sum of the angles in a triangle is 180 degrees. To learn more, Click here…
Week 3 – Geometrical Construction
Overview – Geometry is the branch of Mathematics that deals with the study of shapes, sizes, angles and the dimensions of objects, their spatial relationships and their properties. The week, we shall
learn about geometrical construction. We shall learn how to construct basic shapes, lines and angles.
To learn more, Click here
Week 4 – Triangle: Drawing and Bisection of Line Segment, Construction and Bisection of Angles
Overview – Having being introduced to geometry, this week, we shall learn how to draw and bisect line segment. Bisection means to divide the line segment in two equal parts. We shall also learn how
to construct and bisect angles.
To learn more, Click here
Week 5 – Construction: Construction of Quadrilateral Polygon, Construction of Equilateral Triangle Locus of Moving Points.
Overview – A quadrilateral polygon is a polygon with four sides, four vertices and four angles. Its four angles make up 360°. Equilateral triangle is a type of triangle with all its sides equal in
length. It is also called an equiangular triangle because its angles are equal (60°). This week, we shall learn about the construction of shapes such as rectangle, kite, quadrilateral and equilateral
To learn more, Click here
Week 6 – Deductive Proof: Sum of Angles of a Triangle Revision of Angles on Parallel Line Cut by a Transversal Line.
Overview – Just like regular numbers, angles can be added to obtain a sum, perhaps for the purpose of determining the measure of an unknown angle. This week, we shall learn how to calculate the
missing angles of triangles.
To learn more, Click here
Week 7 – Collection, Tabulation and Presentation of Grouped Data
Overview – In some investigations you may collect an awful lot of information. How can you use this raw data and make it meaningful? This section will help you to collect, organize and interpret the
data efficiently. The easiest way to collect data is to use a tally chart. Ways of representing data include table, graphs and charts. The show the number of times a particular variable e.g. number
of pets in a school, the term “frequency” is used. When a large amount of data has to be collected, use a grouped frequency distribution. This week, we shall learn about the collection, tabulation
and presentation of grouped data.
To learn more, Click here
Week 8 – Calculation of Range, Median and Mode of Ungrouped Data
Overview – The range of a set of data is the difference between the highest and lowest values in the set. Median is defined as the mid value of the data set. The mode is defined the value that
most frequently occurs in the given data. This week, we shall learn about how to calculate the range, median and mode of ungrouped data.
To learn more, Click here
Week 9 – Mean Deviation, Variance and Standard Deviation
Overview – Mean deviation is the average sum of the deviations from the arithmetic mean (i.e. the sum of the differences between the scores and the mean) divided by the total frequency. Mean absolute
deviation is defined as the arithmetic mean of the absolute value of the difference of each score form the mean and dividing by their total frequency. The standard deviation tells us how far or near
a score is to the mean score. This week, we shall learn about mean deviation, variance and standard deviation and how to calculate them.
To learn more, Click here | {"url":"https://passnownow.com/lesson-note-on-ss1-mathematics-for-first-term/","timestamp":"2024-11-11T20:26:38Z","content_type":"text/html","content_length":"404527","record_id":"<urn:uuid:9ac81fa7-f4b6-4827-a202-ab43042cae4d>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00226.warc.gz"} |
Dynamic Logic : A Part from the Book: The Paradigm of Complex Probability
Dynamic Logic : A Part from the Book: The Paradigm of Complex Probability and Analytic Nonlinear Prognostic for Unburied Petrochemical Pipelines – A Relation to Dynamic Logic
Andrey Nikolaevich Kolmogorov introduced the five fundamental axioms of classical probability theory in 1933. My complex probability paradigm builds on this by adding new imaginary dimensions to the
experiment’s real dimensions. This addition makes the work in the complex probability set entirely predictable, with a probability that is always equal to one. By including the contributions of the
imaginary set of probabilities M to the real set of probabilities R, the event in C = R + M becomes entirely deterministic. This is essential because it allows us to predict the outcome of all random
events that occur in nature, making stochastic systems entirely predictable.
The goal of my complex probability paradigm is to connect it to unburied petrochemical pipelines’ analytic prognostic in the nonlinear damage accumulation case. By calculating the parameters of the
new prognostic model, we can determine the chaotic factor’s magnitude, the degree of knowledge, the complex probability, the system’s failure and survival probabilities, and the remaining useful
lifetime probability. All these factors are functions of the system degradation subject to random effects after applying a pressure time t to the pipeline.
Furthermore, we will apply this new paradigm to my novel “Dynamic Logic” model.
Author(s) Details:
Abdo Abou Jaoudé,
Department of Mathematics and Statistics, Faculty of Natural and Applied Sciences, Notre Dame University-Louaize, Lebanon.
You must be logged in to post a comment. | {"url":"https://digiwire.org/dynamic-logic-a-part-from-the-book-the-paradigm-of-complex-probability-and-analytic-nonlinear-prognostic-for-unburied-petrochemical-pipelines-a-relation-to-dynamic-logic/","timestamp":"2024-11-09T23:08:55Z","content_type":"text/html","content_length":"126019","record_id":"<urn:uuid:6361a06d-8429-41d0-abea-9a84d0751d93>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00146.warc.gz"} |
how to calculate b1 and b2 in multiple regression
} The estimate of 1 is obtained by removing the effects of x2 from the other variables and then regressing the residuals of y against the residuals of x1. border-top: 2px solid #CD853F ; .woocommerce
.woocommerce-message:before { .woocommerce input.button.alt, } } On this occasion, Kanda Data will write a tutorial on manually calculating the coefficients bo, b1, b2, and the coefficient of
determination (R Squared) in multiple linear regression. This website focuses on statistics, econometrics, data analysis, data interpretation, research methodology, and writing papers based on
research. There are two ways to calculate the estimated coefficients b0, b1 and b2: using the original sample observation and the deviation of the variables from their means. border: 1px solid #
cd853f; { color: #747474; .site-info .social-links a{ color: #cd853f; hr@degain.in 71. .ld_newsletter_640368d8e55e4.ld-sf input{font-family:avenirblook!important;font-weight:400!important;
font-style:normal!important;font-size:18px;}.ld_newsletter_640368d8e55e4.ld-sf .ld_sf_submit{font-family:avenirblook!important;font-weight:400!important;font-style:normal!important;
font-size:18px;}.ld_newsletter_640368d8e55e4.ld-sf button.ld_sf_submit{background:rgb(247, 150, 34);color:rgb(26, 52, 96);} The analyst uses b1 = 0.015, b2 = 0.33 and bp = 0.8 in the formula, then: .
Normal Equations 1.The result of this maximization step are called the normal equations. Except where otherwise noted, content on this site is licensed under a CC BY-NC 4.0 license. .slider-buttons a
{ For the calculation of Multiple Regression, go to the Data tab in excel, and then select the data analysis option. Required fields are marked *. y = MX + MX + b. y= 604.17*-3.18+604.17*-4.06+0.
color: #747474; .main-navigation ul li ul li:hover > a, Multiple linear regression (MLR), also known simply as multiple regression, is a statistical technique that uses several explanatory variables
to predict the outcome of a response variable. For a two-variable regression, the least squares regression line is: Y est = B0 + (B1 * X) The regression coefficient B0 B1 for a two-variable
regression can be solved by the following Normal Equations : B1 = (XY n*X avg *Y avg) / (X2 n*X avg *X avg) B0 = Y avg B1 *X avg. Method Multiple Linear Regression Analysis Using SPSS | Multiple
linear regression analysis to determine the effect of independent variables (there are more than one) to the dependent variable. Solution else{w.loadCSS=loadCSS}}(typeof global!=="undefined"?
global:this)). #secondary .widget-title Our Methodology .go-to-top a:hover .fa-angle-up { This calculation is carried out for rice consumption (Y), income (X1), and population (X2) variables. .
background-color: #747474; A one unit increase in x1 is associated with a 3.148 unit increase in y, on average, assuming x2 is held constant. .entry-footer a.more-link { .ai-viewport-2 { display:
none !important;} It is possible to estimate just one coefficient in a multiple regression without estimating the others. .entry-header .entry-meta .entry-format:before, Although the example here is
a linear regression model, the approach works for interpreting coefficients from [] How to Calculate the Regression of Two Stocks on Excel. Hakuna Matata Animals, This article does not write a
tutorial on how to test assumptions on multiple linear regression using the OLS method but focuses more on calculating the estimated coefficients b0, b1, and b2 and the coefficient of determination
manually using Excel. These variables can be both categorical and numerical in nature. sinners in the hands of an angry god hyperbole how to calculate b1 and b2 in multiple regression. In multiple
linear regression, the number of independent variables can consist of 2, 3, 4 and > 4 independent variables. Y=b0+b1*x1+b2*x2 where: b1=Age coefficient b2=Experience coefficient #use the same b1
formula(given above) to calculate the coefficients of Age and Experience Multiple regression analysis is a statistical technique that analyzes the relationship between two or more variables and uses
the information to estimate the value of the dependent variables. These cookies do not store any personal information. ul.default-wp-page li a { border-color: #747474 !important; .main-navigation
li.menu-item-has-children > a:hover:after ::-moz-selection { color: #CD853F ; This time, the case example that I will use is multiple linear regression with two independent variables.
.sow-carousel-title a.sow-carousel-next { We wish to estimate the regression line y = b1 + b2*x Do this by Tools / Data Analysis / Regression. border: 1px solid #CD853F ; Learning Objectives Contd 6.
Ok, this is the article I can write for you. } How to Perform Simple Linear Regression by Hand, Your email address will not be published. Yes; reparameterize it as 2 = 1 + , so that your predictors
are no longer x 1, x 2 but x 1 = x 1 + x 2 (to go with 1) and x 2 (to go with ) [Note that = 2 1, and also ^ = ^ 2 ^ 1; further, Var ( ^) will be correct relative to the original.] How to determine
more than two unknown parameters (bo, b1, b2) of a multiple regression. In the b0 = {} section of code, you call an intermediate result b, but later try to reference b1. In the example case that I
will discuss, it consists of: (a) rice consumption as the dependent variable; (b) Income as the 1st independent variable; and (c) Population as the 2nd independent variable. Also, we would still be
left with variables \(x_{2}\) and \(x_{3}\) being present in the model. .btn-default:hover { .widget ul li a { When you add more predictors, your equation may look like Hence my posing the question
of The individual functions INTERCEPT, SLOPE, RSQ, STEYX and FORECAST can be used to get key results for two-variable regression. multiple regression up in this way, b0 will represent the mean of
group 1, b1 will represent the mean of group 2 - mean of group 1, and b2 will represent the mean of group 3 - mean of group 1. background: #cd853f; color: #dc6543; A step by step tutorial showing how
to develop a linear regression equation. document.getElementById( "ak_js_1" ).setAttribute( "value", ( new Date() ).getTime() ); Statology is a site that makes learning statistics easy by explaining
topics in simple and straightforward ways. So lets interpret the coefficients of a continuous and a categorical variable. .bbp-submit-wrapper button.submit { B0 is the intercept, the predicted value
of y when the x is 0. This website uses cookies to improve your experience. When both predictor variables are equal to zero, the mean value for y is -6.867. b1= 3.148. We have the exact same results
with the inbuilt Linear Regression function too. { To manually calculate the R squared, you can use the formula that I cited from Koutsoyiannis (1977) as follows: The last step is calculating the R
squared using the formula I wrote in the previous paragraph. Required fields are marked *. .entry-meta .entry-format:before, background-color: #cd853f; Based on this background, the specifications of
the multiple linear regression equation created by the researcher are as follows: b0, b1, b2 = regression estimation coefficient. .sow-carousel-title a.sow-carousel-next,.sow-carousel-title
a.sow-carousel-previous { To copy and paste formulas in Excel, you must pay attention to the absolute values of the average Y and the average X. For this calculation, we will not consider the error
rate. In matrix terms, the formula that calculates the vector of coefficients in multiple regression is: b = (X'X)-1 X'y In our example, it is = -6.867 + 3.148x 1 1.656x 2. When you are prompted for
regression options, tick the "calculate intercept" box (it is unusual to have reason not to calculate an intercept) and leave the "use weights" box unticked (regression with unweighted responses).
The company has recorded the number of product unit sales for the last quarter. It is calculated as (x(i)-mean(x))*(y(i)-mean(y)) / ((x(i)-mean(x))2 * (y(i)-mean(y))2. We can thus conclude that our
calculations are correct and stand true. Linear regression calculator Exercises for Calculating b0, b1, and b2. how to calculate b1 and b2 in multiple regression. Calculate bo b1 and b2 in multiple
linear regression, how do you calculate bo b1 and b2 regression coefficient, how to calculate bo b1 b2 and R square in multiple linear regression, how to find bo b1 b2 and R squared in multiple
linear regression, How to Find ANOVA (Analysis of Variance) Table Manually in Multiple Linear Regression - KANDA DATA, Determining Variance, Standard Error, and T-Statistics in Multiple Linear
Regression using Excel - KANDA DATA, How to Calculate the Regression Coefficient of 4 Independent Variables in Multiple Linear Regression - KANDA DATA, How to Calculate Durbin Watson Tests in Excel
and Interpret the Results - KANDA DATA, How to Find Residual Value in Multiple Linear Regression using Excel - KANDA DATA, Formula to Calculate Analysis of Variance (ANOVA) in Regression Analysis -
KANDA DATA, How to Perform Multiple Linear Regression using Data Analysis in Excel - KANDA DATA. The regression formula for the above example will be y = MX + MX + b y= 604.17*-3.18+604.17*-4.06+0 y=
-4377 }. This calculator will compute the 99%, 95%, and 90% confidence intervals for a regression coefficient, given the value of the regression coefficient Determine math questions In order to
determine what the math problem is, you will need to look at the given information and find the key details. Each \(\beta\) parameter represents the change in the mean response, E(, For example, \(\
beta_1\) represents the estimated change in the mean response, E(, The intercept term, \(\beta_0\), represents the estimated mean response, E(, Other residual analyses can be done exactly as we did
in simple regression. This model generalizes the simple linear regression in two ways. Step #3: Keep this variable and fit all possible models with one extra predictor added to the one (s) you
already have. Select the one with the lowest P-value. CFA And Chartered Financial Analyst Are Registered Trademarks Owned By CFA Institute. Then select Multiple Linear Regression from the Regression
and Correlation section of the analysis menu. Our Methodology In other words, \(R^2\) always increases (or stays the same) as more predictors are added to a multiple linear regression model.
info@degain.in Now lets move on to consider a regression with more than one predictor. In detail, the calculation stages can be seen in the image below: Next, copy and paste the Excel formula from
the 2nd quarters data to the last quarters data. .sow-carousel-title { Edit Report an issue 30 seconds. a dignissimos. The calculation results can be seen below: Based on the order in which the
estimation coefficients are calculated, finding the intercept estimation coefficient is carried out at the last stage. background-color: #cd853f; Save my name, email, and website in this browser for
the next time I comment. Great now we have all the required values, which when imputed in the above formulae will give the following results: We now have an equation of our multi-linear line: Now
lets try and compute a new value and compare it using the Sklearns library as well: Now comparing it with Sklearns Linear Regression. How do you interpret b1 in multiple linear regression
Interpretation of b1: When x1 goes up by 1, then predicted rent goes up by $.741 [i.e. Thus b 0 is the sample estimate of 0, b 1 is the sample estimate of 1, and so on. .btn-default:hover, } In
detail, it can be seen as follows: Based on what has been calculated in the previous paragraphs, we have manually calculated the coefficients of bo, b1 and the coefficient of determination (R
squared) using Excel. Step-by-step solution. Multiple regression formulas analyze the relationship between dependent and multiple independent variables. A boy is using art supplies. Degain become the
tactical partner of business and organizations by creating, managing and delivering ample solutions that enhance our clients performance and expansion .el-pack .sow-headline { How to calculate
multiple linear regression. .tag-links a, Support Service. .woocommerce #respond input#submit.alt, 'https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);
.screen-reader-text:hover, Analytics Vidhya is a community of Analytics and Data Science professionals. } } But for most people, the manual calculation method is quite difficult. 10.3 - Best Subsets
Regression, Adjusted R-Sq, Mallows Cp, 11.1 - Distinction Between Outliers & High Leverage Observations, 11.2 - Using Leverages to Help Identify Extreme x Values, 11.3 - Identifying Outliers (Unusual
y Values), 11.5 - Identifying Influential Data Points, 11.7 - A Strategy for Dealing with Problematic Data Points, Lesson 12: Multicollinearity & Other Regression Pitfalls, 12.4 - Detecting
Multicollinearity Using Variance Inflation Factors, 12.5 - Reducing Data-based Multicollinearity, 12.6 - Reducing Structural Multicollinearity, Lesson 13: Weighted Least Squares & Logistic
Regressions, 13.2.1 - Further Logistic Regression Examples, Minitab Help 13: Weighted Least Squares & Logistic Regressions, R Help 13: Weighted Least Squares & Logistic Regressions, T.2.2 -
Regression with Autoregressive Errors, T.2.3 - Testing and Remedial Measures for Autocorrelation, T.2.4 - Examples of Applying Cochrane-Orcutt Procedure, Software Help: Time & Series Autocorrelation,
Minitab Help: Time Series & Autocorrelation, Software Help: Poisson & Nonlinear Regression, Minitab Help: Poisson & Nonlinear Regression, Calculate a T-Interval for a Population Mean, Code a Text
Variable into a Numeric Variable, Conducting a Hypothesis Test for the Population Correlation Coefficient P, Create a Fitted Line Plot with Confidence and Prediction Bands, Find a Confidence Interval
and a Prediction Interval for the Response, Generate Random Normally Distributed Data, Randomly Sample Data with Replacement from Columns, Split the Worksheet Based on the Value of a Variable, Store
Residuals, Leverages, and Influence Measures, Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris, Duis aute irure dolor in reprehenderit in voluptate, Excepteur sint occaecat
cupidatat non proident, A population model for a multiple linear regression model that relates a, We assume that the \(\epsilon_{i}\) have a normal distribution with mean 0 and constant variance \(\
sigma^{2}\). } } The slope (b1) can be calculated as follows: b1 = rxy * SDy/SDx. Terrorblade Dota 2 Guide, background-color: #dc6543; To perform a regression analysis, first calculate the multiple
regression of your data. By closing this banner, scrolling this page, clicking a link or continuing to browse otherwise, you agree to our Privacy Policy, You can see how this popup was set up in our
step-by-step guide: https://wppopupmaker.com/guides/auto-opening-announcement-popups/. border-color: #dc6543; The data that researchers have collected can be seen in the table below: Following what I
have written in the previous paragraph, to avoid errors in calculating manually, I am here using Excel. .main-navigation ul li.current-menu-item a, b 0 and b 1 are called point estimators of 0 and 1
respectively. Simple and Multiple Linear Regression Maths, Calculating Intercept, coefficients and Implementation Using Sklearn | by Nitin | Analytics Vidhya | Medium Write Sign up Sign In 500
Apologies,. .ai-viewport-2 { display: inherit !important;} +91 932 002 0036, Temp Staffing Company */ laudantium assumenda nam eaque, excepturi, soluta, perspiciatis cupiditate sapiente, adipisci
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Comparing Nodes in a Tree
I have no idea whether this algorithm is old or new, or even whether there are more efficient alternatives. But for a long time now, I’ve had a geeky fondness for an orphan data structure. I call it
an orphan because I could never see any way of using it. It was just one of those pretty things. If you’ve ever had a geeky fondness for a data structure, then you’re pretty weird too.
I first met the idea when I was an undergraduate, studying Neurobiology. Chris Longuet-Higgins taught a small seminar class on the psychology of music. It was actually about some computer modeling
work he did while at the Machine Intelligence unit in Edinburgh. The centerpiece of his office was a midi keyboard wired to a large two-dimensional display that lit up as he played. Remember the
scene in Close Encounters?
Anyway, he proposed that any melody in western classical music is written in musical intervals which can all be uniquely described as products of the first three prime numbers. Music is our
perception of the fundamental theorem of arithmetic. He described a three dimensional tonal space, where each discrete node on that space can be defined in the x,y,z co-ordinate system where x is the
octave (2:1) , y is the fifth (3:2) and z is the major third (5:4). The interval is 1:n where n is x^a * y^b * z^c – and most important, the relationship between any two intervals makes sense
within that space. The major scale is a discrete and obvious area, so is the minor scale. The aliens in Close Encounters were way behind Chris.
But the geeky idea that has stayed with me is that when you have such a coordinate space, only a discrete set of numbers are represented in the space, and they identify where you are in the space.
Imagine a one-dimensional space. The numbers 1, 2, 3 etc are also positions. Position number 2 is after 1 and before 3. So I can tag things with a number and know which comes before which. And now
imagine any space where a single number identifies where you are and it is possible to derive an unambiguous location in that n-dimensional space from the number. Its probably first year stuff in
Computer Science courses, but I read Biology and reinvent wheels sometimes.
So this week I was working on DOM nodes in an HTML document. I had two nodes and needed to know their relative position in the document. The DOM object has a compareDocumentPosition method, but
surprise surprise it is not uniformly supported in all the browsers. And there are lots of code fragments across the internet to do the same thing. All horribly tacky.
I thought of Chris. Lets create a tree space. And there is an interesting shadow area around the top node that is more fun to think about than counting sheep, but lets say that the first child of a
node is 2x and that each next sibling is (2x+1) of the previous one. Each number represents a node in the tree space. And by comparing these two numbers you can say which comes before the other in a
depth first search order of the tree. So leaving the top node of the tree to one side for now, its first child has the value of 0. That child’s next sibling has the value 1. And the next, 3. And the
next, 7 etc. The first child of node “3” is 6. Its next sibling is 13. And in this 2-dimensional world, 30 is greater than 96, 13 is greater than 17 and 121 is way bigger than 192. And there are is
an infinite set of numbers between 2 and 3. The arithmetic is just different.
For any two nodes in the tree, you can get its “index” by racing back to the top of the tree, left, up, left, left, up etc. Feels just like working out heirs in a parentelic probate system.
here is some javascript code for calculating a node’s “number”:
function nodenumber(n)
if (n == null)
return 0;
if (n.previousSibling != null)
return (2 * nodenumber(n.previousSibling)) + 1;
return 2 * nodenumber(n.parentNode);
and here’s code that decomposes a number into its path (why store information in a complex array structure when you can do it in an integer)
function nodearray( nn, na)
if (nn == 0)
return na;
if (nn & 0x1)
return nodearray((nn-1)/2, na);
return nodearray(nn/2, na);
Suppose you have two nodes – one with node-number N1 the other with node-number N2. And the function nodecompare(N1,N2) will act like strcmp and return -1 if N1 is before N2, 0 if they are the same
and 1 if N1 is after N2:
function nodecompare(N1, N2)
if (N1 == N2)
return 0;
return nodecompare1( nodearray( N1, new Array()), nodearray( N2, new Array()));
You decompose the numbers and then follow them back. I’ve written it in two stages for clarity. Everything is either a shift left (divide by 2) or a test for the first bit (odd number). All very
natural and fast operations.
function nodecompare1(a1, a2)
var i1 = a1.pop();
var i2 = a2.pop();
if (i1 == i2) {
if (a1.length == 0)
return -1;
if (a2.length == 0)
return 1;
return nodecompare1( a1, a2);
if (i1 & 0x1)
return 1;
return -1;
Its useful to write it out using arrays, but all we’re really doing in the code above is breaking down the numbers and analyzing them bit by bit. The two numbers themselves can give you the answer
without all of this effort.
function nodecmp( N1, N2)
if (N1 == N2)
return 0;
if (N1 < N2)
N2 = (N1 & N2);
N1 = (N2 & N1);
if (N1 > N2)
return 1;
return -1;
I was always taught to think of numbers as scalars. But it just ain’t so, Joe.
Comparing document position is a trivial application. And in a tree there is a clear notion of where the tree starts and in which direction it is going. In Chris’ tonal space, there is no such start
and end point – in fact you are using the interval as a measure of relative distance between two places. But there are endless possibilities for my favorite orphan data structure – id cards where you
have a node-number based on family relationships, single digit unique zipcodes, semantic ip addresses. And being able to tell similarity just by listening to a tonal representation of the
relationship. Such fun. And if you’re thinking of racing to the PTO to patent my orphan, tough luck. The ideas were all there in Chris’ published papers 30 years ago, and they are obvious to this
ordinarily creative PHOSITA. | {"url":"http://www.lawandsoftware.com/blog/comparing-nodes-in-a-tree/","timestamp":"2024-11-12T09:24:27Z","content_type":"application/xhtml+xml","content_length":"31784","record_id":"<urn:uuid:d79a06fa-b316-422c-b10c-d57c9472d6a9>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00001.warc.gz"} |
Modern Radio Practice in Using Graphs and Charts, July 1932 Radio News
July 1932 Radio News
[Table of Contents]
Wax nostalgic about and learn from the history of early electronics. See articles from Radio & Television News, published 1919-1959. All copyrights hereby acknowledged.
I've always disliked book and article titles containing the word "Modern" because the word is utterly ambiguous and usually downright misleading more than a few years past the original publication
date. What at publication time was modern is usually obsolete merely a decade later, especially in the realm of high technology. Sometimes, however, as with this 1932 Radio News magazine article on
insulation (dielectric) breakdown voltages, bringing the information up to date requires only the addition of an extra couple decimal points of precision and/or the substitution of a few words. For
instance, replace "condenser" with "capacitor" and units of "mfd" with "μF" and "mmfd" with "pF." Then you'll be on your way to gaining useful information. You might not find some of the dielectric
types pertinent today, like gutta percha, which is what covered the world's first transatlantic submerged communications cables. There is a nice nomograph for use in designing capacitors for specific
voltage handling and a table of dielectric puncturing voltages as well.
Orion Instruments has a very extensive table of dielectric constant values.
Modern Radio Practice in Using Graphs and Charts - Part 7
By John M. Borst
Part Seven
Calculations in radio design work usually can be reduced to formulas represented as charts which permit the solution of mathematical problems without mental effort. This series of articles presents a
number of useful charts and explains how others can be made
The capacity of a homemade condenser is often more or less of a mystery. The amateur or experimenter who does not possess a bridge or capacity standard must calculate the capacity. Conversely, if a
condenser of a given capacity is desired, only a calculation will eliminate guesswork.
The standard formula has been transformed into an alignment chart in Figure 1. The capacity of a condenser can be found when the area of the plates, their number, distance and the kind of dielectric
are known.
The relation between centimeters and inches or mils as well as the relation between square centimeters and square inches, centimeters and microfarads is also shown in Figure 1. The "dielectric
constant," also called "inductivity" or "specific inductive capacity," is incorporated on the chart, which makes the consultation of any sources superfluous.
The formula for the capacity of a condenser consisting of parallel plates is
where A = the area of one plate in square centimeters
d = the distance between two plates in centimeters
n = the number of plates
K = the specific inductive capacity
This expression refers to a condenser with alternate plates in parallel. The formula does not take into consideration the spreading of the lines of force at the edges of the plates. This effect is
negligible so long as the thickness of the dielectric is small compared to the area of the plates.
In designing this chart the prime idea has been to cover all possible cases which occur in practice. Therefore, the capacity scale ranges from 1 micro-microfarad to over 10 micro-farads, and the
other quantities also cover a wide range.
Two metal plates have an area of 1 square inch and are placed parallel, 1/4 inch apart, in air. What is the capacity?
Referring to the chart, draw a line from the 1-square-inch mark on the "Area" scale to 1 on the K scale. The specific inductive capacity of air is one (unity). This gives you an intersection on the
turning scale No. 1. From this newly found point draw another line through the point 2 on the N scale and find a second point on the turning scale No.2. The final line is drawn through the latter
point and the 250-mils mark on the d scale. This line intersects the capacity scale at 0.9 mmfd.
When exactly 1 mmfd. is required, the last line should be turned around its point on the turning scale No. 2 until it intersects the capacity scale at the 1 mmfd. mark and the intersection on the d
scale shows the required distance between the plates (225 mils). The distance, however, can be left the same and the problem worked backwards, in which case an area of 1.1 square inch is found
necessary. These lines have not been added in Figure 1 because they are so close together that it might confuse the reader.
When using these charts. needless to say, one should not actually draw the lines but use a transparent ruler, a regular ruler or a tight thread.
The second example shows how to work the problem backward. Suppose a paper condenser of 1 mfd. is wanted and the dielectric available has a thickness of 2 mils. This is manila paper, treated with
paraffin. Its specific inductive capacity is 3.65 and the break-down voltage may run as high as 250 volts per mil. There is one more quantity which can be chosen and then the other one is determined.
This can be either the number of plates or the size of the plates. The number of plates is the best to assume, because this has to be a whole number. Let us assume there shall be 30 plates.
For the solution of this problem, start at the 1 mfd. mark on the capacity scale. A line from this point to the 2 mil. mark on the d scale intersects the turning scale No. 2. Draw a line through the
latter point and through the point representing the number of plates (30). Now note the intersection on the turning scale No. 1. Finally draw the last line from the point representing the dielectric
constant. 3.65, through the point on the turning scale No.2, which shows the necessary area of the plates as 84 square inches. As a check-up, an actual calculation gave the area as 83.7 square
The experience of this second example teaches us that in certain cases the last line would intersect the area scale beyond the limits of the paper. This means that the area of the plates needed is
going to be larger than 100 square inches. If the area is to be smaller than 100 square inches, either the number of plates have to be increased, the thickness of the dielectric decreased or the
material exchanged for one with a greater inductivity. Then try again.
If one wishes the problem solved for values of variables outside the range of the chart, then some multiplying stunt has to be employed. For instance, suppose the paper in the above example had been
dry paper with a dielectric constant of 1.8, then the last line does not intersect the area scale within the limits of the page. Therefore, multiplying 1.8 with any convenient number - say, 5 - the
last line is drawn from 9 through the intersection on the turning scale number one and the area scale is intersected at 34.
This result must now be multiplied by five in order to find the correct answer, which is 170 square inches.
While determining the specifications for a condenser it is important to be sure that the dielectric will stand the applied voltage. Therefore a list of the break-down voltages for different materials
is found in Figure 2.
Capacity of a Condenser (aka Capacitor)
Temperature influences the ability of a dielectric to withstand electric pressure. When the condenser heats up under a continuous load, the breakdown voltage is lowered. Therefore the tests of such
condensers must be made over a considerable time at working voltage or at a much higher voltage for a short time.
Commercial paper condensers usually consist of long strips of prepared paper, with tinfoil interleaved, which is then rolled. In the case of rolling a condenser with an even number of plates, the top
plate and the bottom plate form an additional section of the condenser so that in this case the rolling has the effect of adding one more plate. The reader should see whether the dielectric for this
additional section has the same thickness as the other sections and make allowances for any possible difference.
When the number of plates is odd or when the paper is not rolled, the actual number of plates is used for the calculation.
The accuracy of a calculation by means of this chart will be sufficient only if the correct values for the dielectric constant and the thickness of the dielectric have been determined. This is
sometimes difficult to accomplish, especially with paper as a dielectric. If the reader guesses at the constant and the actual separation of the plates, he must expect the result to be off
Nomographs / Nomograms Available on RF Cafe:
- Parallel Series Resistance Calculator
- Transformer Turns Ratio Nomogram
- Symmetrical T and H Attenuator Nomograph
- Amplifier Gain Nomograph
- Decibel Nomograph
- Voltage and Power Level Nomograph
- Nomograph Construction
- Nomogram Construction for Charts with Complicating Factors or Constants
- Link Coupling Nomogram
- Multi-Layer Coil Nomograph
- Delay Line Nomogram
- Voltage, Current, Resistance, and Power Nomograph
- Resistor Selection Nomogram
- Resistance and Capacitance Nomograph
- Capacitance Nomograph
- Earth Curvature Nomograph
- Coil Winding Nomogram
- RC Time-Constant Nomogram
- Coil Design Nomograph
- Voltage, Power, and Decibel Nomograph
- Coil Inductance Nomograph
- Antenna Gain Nomograph
- Resistance and Reactance Nomograph
- Frequency / Reactance Nomograph
Posted December 29, 2021
(updated from original post on 9/10/2013) | {"url":"https://rfcafe.com/references/radio-news/modern-radio-practice-using-graphs-charts-july-1932-radio-news.htm","timestamp":"2024-11-06T12:44:47Z","content_type":"text/html","content_length":"37638","record_id":"<urn:uuid:aa0cdad2-b16e-4a25-acc5-67ac4565c70e>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00062.warc.gz"} |
value of beauty in maths
A proof that is based on new and original insights. Often when reading a good maths book, the author will get to the end of an explanation of a particularly complicated proof, theorem, or idea, and
mention the "beauty" of the maths involved. Class 9 maths value based 1. Combinatorics, the study of counting, has artistic representations that some find mathematically beautiful. To improve your
maths skills, you need to see its value in your daily life. One source with over 100 articles and latest findings. . The latter corresponds to the first derivative of subjectively perceived beauty:
In fact, it’s an important skill for everyday life, as well as in most jobs. But the mathematician’s patterns, like the poet’s must be beautiful if they are to have any lasting value. In 2018, Britz
gave a TEDx talk on the Mathematics of Emotion, where he used recent studies on maths and emotions to touch on how maths might help explain emotions, like beauty. The particular thing that I want to
introduce you to, that I think is so beautiful, is something that was mentioned in passing on a television programme I was watching. Surein Aziz is 17 years old and currently in year 12 at
Farnborough Sixth Form College. Mathematicians describe an especially pleasing method of proof as elegant. In 2018, Dr Britz gave a TEDx talk on the Mathematics of Emotion, where he used recent
studies on maths and emotions to touch on how maths might help explain emotions, like beauty. “Our brains reward us when we recognise patterns, whether this is seeing symmetry, organising parts of a
whole, or puzzle-solving,” he says. Well, I ought to warn you, I'm not alone — Mathematical Intelligencer readers voted the identity the "most beautiful theorem in mathematics". June 2009 This
article is the winner of the schools category of the Plus new writers award 2009. Conf. In particular, the area of a triangle on a curved surface is proportional to the excess of the triangle and the
proportionality is curvature. [25][26][27] Schmidhuber explicitly distinguishes between beautiful and interesting. Celeb-Faces. Beauty is the key. Hear some learners talk about how they use maths in
their course. I used to think that it was the latter — maybe one day, after years of studying maths at its highest level, I'd suddenly gain a glimpse of some incomprehensibly deep truth and realise
the incredible beauty of things which now seem boring and trivial. While it is difficult to find universal agreement on whether a result is deep, some examples are more commonly cited than others.
These are just a way of expressing functions such as or as infinite sums. you see incredible, exotic plants and animals to marvel at — and ever so often you find large new swathes of jungle to
explore. At KS1 you may only make use of tens and hundreds, but place value grids can be easily modified to cover thousandths, ten thousands, hundred thousands – however far you need them to go for
KS2 maths . In Plato's philosophy there were two worlds, the physical one in which we live and another abstract world which contained unchanging truth, including mathematics. But don't be put off.
They were discovered by the mathematician Brook Taylor (who was also part of the committee which adjudicated the argument between Isaac Newton and Gottfried Leibniz about who first invented the
calculus). For example, one can teach the method of completing the square by using algebra tiles. Retail. on Discovery Science (DS 2007) pp. Peitgen, H.-O., and Richter, P.H. Maryam Mirzakhani, the
first woman to win a Fields Medal – the Nobel Prize of maths – wrote that the beauty of mathematics only shows itself to more patient followers. Some teachers prefer to use mathematical manipulatives
to present mathematics in an aesthetically pleasing way. Papers on the theory of beauty and. somewhere in the thick undergrowth. Learn the basics. Schmidhuber. International Joint Conference on
Neural Networks, Singapore, vol 2, 1458â 1463. [30] A number of other British artists of the constructionist and systems schools of thought also draw on mathematics models and structures as a source
of inspiration, including Anthony Hill and Peter Lowe. Here we have extended the table a bit so that it runs until the number 15 in the horizontal direction. In 2018, Dr. Britz gave a TEDx talk on
the Mathematics of Emotion, where he used recent studies on maths and emotions to touch on how maths might help explain emotions, like beauty. Expressed algebraically, for quantities a and b with a >
b > 0, + = = , where the Greek letter phi (or ) represents the golden ratio. of NCT of Delhi Value based support Material for the session 2012-13 Subject – Mathematics Class – IX Under the guidance
of Dr. Sunita S. Kaushik Addl. Triangular numbers: find out what they are and why they are beautiful! You're probably familiar with , it's the ratio between a circle's circumference and its diameter.
Directorate of Education Govt. University of Cambridge. Examples of the use of mathematics in the visual arts include applications of chaos theory and fractal geometry to computer-generated art,
symmetry studies of Leonardo da Vinci, projective geometries in development of the perspective theory of Renaissance art, grids in Op art, optical geometry in the camera obscura of Giambattista della
Porta, and multiple perspective in analytic cubism and futurism. So, why does this happen? docx, 2 MB. Conversely, results that are logically correct but involve laborious calculations,
over-elaborate methods, highly conventional approaches or a large number of powerful axioms or previous results are usually not considered to be elegant, and may be even referred to as ugly or
clumsy. Whenever the observer's learning process (possibly a predictive artificial neural network) leads to improved data compression such that the observation sequence can be described by fewer bits
than before, the temporary interesting-ness of the data corresponds to the compression progress, and is proportional to the observer's internal curiosity reward.[28][29]. Strohmeier, John, and
Westbrook, Peter (1999), This page was last edited on 29 November 2020, at 02:49. The Fibonacci sequence: A brief introduction, Physics in a minute: The double slit experiment. In a general Math
Circle lesson, students use pattern finding, observation, and exploration to make their own mathematical discoveries. Report a problem. In the 1970s, Abraham Moles and Frieder Nake analyzed links
between beauty, information processing, and information theory. .J. For example, mathematical beauty arises in a Math Circle activity on symmetry designed for 2nd and 3rd graders, where students
create their own snowflakes by folding a square piece of paper and cutting out designs of their choice along the edges of the folded paper. Can be used at any point in the year as a tool to gage
prior learning or progress within the domain of Number and Place Value. In some cases, natural philosophers and other scientists who have made extensive use of mathematics have made leaps of
inference between beauty and physical truth in ways that turned out to be erroneous. "Project Origami: Activities for Exploring Mathematics". [31] Computer-generated art is based on mathematical
algorithms. British constructionist artist John Ernest created reliefs and paintings inspired by group theory. Karen Olsson is the author of the novels Waterloo , which was a runner-up for the 2006
PEN/Hemingway Award for First Fiction, and All the Houses . Sport and leisure. The aims assessed by each question are clearly stated on the adult guidance and a marking scheme is provided. Joint
invited lecture for DS 2007 and ALT 2007, Sendai, Japan, 2007. In his A Mathematician's Apology, Hardy suggests that a beautiful proof or result possesses "inevitability", "unexpectedness", and
"economy".[11]. It's like asking why is Beethoven's Ninth Symphony beautiful. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest
excellence, is to be found in mathematics as surely as poetry. because of the incredible truths and interconnections you can uncover simply by following a sequence of logical steps and identifying
patterns. Calculating a 10% tip in a restaurant using place value columns. Examples of a manipulative include algebra tiles, cuisenaire rods, and pattern blocks. [9] Modern examples include the
modularity theorem, which establishes an important connection between elliptic curves and modular forms (work on which led to the awarding of the Wolf Prize to Andrew Wiles and Robert Langlands), and
"monstrous moonshine", which connects the Monster group to modular functions via string theory (for which Richard Borcherds was awarded the Fields Medal). The Dutch graphic designer M. C. Escher
created mathematically inspired woodcuts, lithographs, and mezzotints. That is what I think is so beautiful about this identity: it links very strange numbers with very ordinary and fundamental ones.
While away the days to Christmas exploring the history and mysteries of the Universe! The theorem for which the greatest number of different proofs have been discovered is possibly the Pythagorean
theorem, with hundreds of proofs being published up to date. The Taylor series for the other two functions appearing in Euler's formular are, Now let's multiply the variable in the Taylor series for
by the number . Why are maths skills important in hairdressing and beauty therapy jobs? It is the square root of -1, that is It's called an imaginary number, and you can't find it anywhere along the
normal number line, as none of the ordinary real numbers give a negative number when squared. the observer continually tries to improve the predictability and compressibility of the observations by
discovering regularities such as repetitions and symmetries and fractal self-similarity. However, the real beauty of an expertly-designed scheme of work is that it ensures deep learning can take
place in the classroom using a range of learning strategies, which have already been thought through by subject specialists and built into the curriculum. Group theory, developed in the early 1800s
for the sole purpose of solving polynomial equations, became a fruitful way of categorizing elementary particlesâ the building blocks of matter. grips with? Examples of the use of mathematics in
music include the stochastic music of Iannis Xenakis, Fibonacci in Tool's Lateralus, counterpoint of Johann Sebastian Bach, polyrhythmic structures (as in Igor Stravinsky's The Rite of Spring), the
Metric modulation of Elliott Carter, permutation theory in serialism beginning with Arnold Schoenberg, and application of Shepard tones in Karlheinz Stockhausen's Hymnen. Comparisons are often made
with music and poetry. Cuisenaire rods can be used to teach fractions, and pattern blocks can be used to teach geometry. Simple Algorithmic Principles of Discovery, Subjective Beauty, Selective
Attention, Curiosity & Creativity. Want facts and want them fast? Hair and beauty. Some mathematicians see beauty in mathematical results that establish connections between two areas of mathematics
that at first sight appear to be unrelated. Mathematics-of-Beauty. A method of proof that can be easily generalized to solve a family of similar problems. To understand how this formula comes about,
we need something called Taylor series. [5] Another theorem that has been proved in many different ways is the theorem of quadratic reciprocity. 10th Intl. I hardly knew what it meant, and I
certainly had no idea how it came about, but I knew I had to find out more. The figure on the right illustrates the geometric relationship. 18th Intl. Euler's identity is named after Leonhard Euler,
one of the most prolific mathematicians of all times. It appears many times in geometry, art, architecture and other areas. Euler's identity is a special case of Euler's formula, which the physicist
Richard Feynman called "our jewel" and "the most remarkable formula in mathematics". There is a fairly wide-held perception that a person is either good at maths or no good at maths. J. Schmidhuber.
For example, Gauss's Theorema Egregium is a deep theorem which relates a local phenomenon (curvature) to a global phenomenon (area) in a surprising way. This article is the winner of the schools
category of the Plus new writers award 2009. Mathematics (from Greek: μάθημα , máthēma , 'knowledge, study, learning') includes the study of such topics as quantity (number theory), structure
(algebra), space (geometry), and change (mathematical analysis). "Introductory Combinatorics." Discovering the Hidden Value of Math By Heather Shanks “Mathematics is food for the brain,” says math
professor Dr. Arthur Benjamin. on Algorithmic Learning Theory (ALT 2007) p. 32, LNAI 4754, Springer, 2007. ; You will need to research the KQ above and provide insights based on your maths classes,
research and peer discussions as to your Personal & Shared knowledge to this question [19] There are many visual examples that illustrate combinatorial concepts. The beauty of mathematics is in its
remarkable success of describing the natural world. (, J. Schmidhuber. You should locate examples of mathematical beauty and reach conclusions as to why this is the case. He first encountered Euler's
Identity and the idea of its beauty on a TV program, after which he knew he had to research the subject further. Curious model-building control systems. Some mathematicians have extrapolated this
viewpoint that mathematical beauty is truth further, in some cases becoming mysticism. But first you have to see Euler's formula, which leads to his beautiful identity, in full generality: Doesn't
look quite as nice and neat now, does it? Other examples of deep results include unexpected insights into mathematical structures. Pearson, 2009. Maths is much more than just a school subject. He
thinks maths is very interesting (and beautiful!) Did I miss a particularly neat diagram? A trivial theorem may be a result that can be derived in an obvious and straightforward way from other known
results, or which applies only to a specific set of particular objects such as the empty set. All rights reserved. Maths can be like a dense jungle — it's hard to penetrate but you never know whom
you might might. For me, the beauty of mathematics is the thrilling conceptual elegance, which often involves elements of surprise, economy, depth, relevance and power.” When the paper is unfolded, a
symmetrical design reveals itself. Mathematics can be a bit like a dense, never-ending jungle. He loves to spend his time thinking about (and sometimes, in simple cases, solving) interesting maths
problems, One such example is Euler's identity:[8]. The opposite of deep is trivial. Well, actually, it isn't too difficult to see how Euler's identity comes about - that is one thing that makes the
identity so wonderful! . 26â 38, LNAI 4755, Springer, 2007. A proof that uses a minimum of additional assumptions or previous results. F Nake (1974). Rota, however, disagrees with unexpectedness as
a necessary condition for beauty and proposes a counterexample: A great many theorems of mathematics, when first published, appear to be surprising; thus for example some twenty years ago [from 1977]
the proof of the existence of non-equivalent differentiable structures on spheres of high dimension was thought to be surprising, but it did not occur to anyone to call such a fact beautiful, then or
now. To 20 decimal places, Both and are irrational numbers – they have an infinite number of decimal places and you can't write them down as one integer divided by another. Some of the topics and
objects seen in combinatorics courses with visual representations include, among others: Some mathematicians are of the opinion that the doing of mathematics is closer to discovery than invention,
for example: There is no scientific discoverer, no poet, no painter, no musician, who will not tell you that he found ready made his discovery or poem or pictureâ that it came to him from outside,
and that he did not consciously create it from within. Note that the whole pattern above can be pieced together using the fundamental building block: The fundamental building block contains …
Probably the strangest of these three numbers is . You’re probably already using maths all the time, in all sorts of situations in work and everyday life. When ErdÅ s wanted to express particular
appreciation of a proof, he would exclaim "This one's from The Book!". Maths is accessible and achievable for all. Mathematical beauty is the aesthetic pleasure typically derived from the
abstractness, purity, simplicity, depth or orderliness of mathematics. Also in Proc. But what is so special about it? One can study the mathematics of paper folding by observing the crease pattern on
unfolded origami pieces.[18]. Similarly, the study of knots provides important insights into string theory and loop quantum gravity. Indeed, since the complete multiplication table on positive
integers is infinite on two sides, we will continueto tweak the dimensions of the tables in what follows to display the emergingpatterns more clearly. Proc. “Beauty is the first test: there is no
permanent place in the world for ugly mathematics.” ... Take a look at how graduates actually use maths in their careers and the massive variety of different areas which they work in. Beauty of
maths.... 111/1+1+1=37 222/2+2+2=37 333/3+3+3=37 444/4+4+4=37 555/5+5+5=37 666/6+6+6=37 777/7+7+7=37… Get the answers you need, now! The beauty of a place value grid is that it can be reused
throughout maths lessons from Year 1 to Year 6 (and for SATs revision). “Our brains reward us when we recognise patterns, whether this is seeing symmetry, organising parts of a whole, or
puzzle-solving,” he says. That's what I'm going to try and convince you of in the rest of this article. You might think that it is down to some really complex idea — how do we even take a number to
the power of ? It’s vital to challenge negative attitudes and consistently promote the value of maths skills for everyone. Don't like trigonometry? [20], Hungarian mathematician Paul ErdÅ s[21] spoke
of an imaginary book, in which God has written down all the most beautiful mathematical proofs. Don't worry, here are three beautiful proofs of a well-known result that make do without it. This
disagreement illustrates both the subjective nature of mathematical beauty and its connection with mathematical results: in this case, not only the existence of exotic spheres, but also a particular
realization of them. Ã sthetik als Informationsverarbeitung. Notion that some mathematicians may derive aesthetic pleasure from mathematics, Beauty and mathematical information theory. . 1. Well,
first I ought to explain what the symbols actually mean. Bertrand Russell expressed his sense of mathematical beauty in these words: Mathematics, rightly viewed, possesses not only truth, but supreme
beautyâ a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a
stern perfection such as only the greatest art can show. The golden ratio (symbol is the Greek letter "phi" shown at left) is a special number approximately equal to 1.618. These feature impossible
constructions, explorations of infinity, architecture, visual paradoxes and tessellations. pptx, 879 KB. But actually, I think you can get a glimpse of what mathematicians mean by beauty without too
much effort at all. Depending on context, this may mean: In the search for an elegant proof, mathematicians often look for different independent ways to prove a resultâ as the first proof that is
found can often be improved. For example, at one stage in his life, Johannes Kepler believed that the proportions of the orbits of the then-known planets in the Solar System have been arranged by God
to correspond to a concentric arrangement of the five Platonic solids, each orbit lying on the circumsphere of one polyhedron and the insphere of another. Now you probably think I'm crazy. Isn't it a
little odd how three very strange numbers which are not connected in any evident way combine to give such a normal and familiar result? [1] Mathematicians often express this pleasure by describing
mathematics (or, at least, some aspect of mathematics) as beautiful. The beauty of theoretical physics is that Maths is it’s language. In fact, Carl Friedrich Gauss alone had eight different proofs
of this theorem, six of which he published.[6]. DE (School/Exam) Coordination by : Shakuntla Mahajan (Principal) GGSS School, Sri Niwaspuri, New Delhi 110065 PREPARED BY : 1. In 2018, Dr Britz gave a
TEDx talk on the Mathematics of Emotion, where he used recent studies on maths and emotions to touch on how maths might help explain emotions, like beauty. He believed that the physical world was a
mere reflection of the more perfect abstract world. If you take the constant to the power of multiplied by , and then take away 1, you get to 0. Copyright © 1997 - 2020. Using mathematical
manipulatives helps students gain a conceptual understanding that might not be seen immediately in written mathematical formulas. [17], Another example of beauty in experience involves the use of
origami. It can feel like you're hacking away and away at it and never getting anywhere, but if you stop and look around yourself, every once in a while (1986). The beauty of maths is not only around
us but a strong know how of maths help us in every day life too. Another example is the fundamental theorem of calculus[10] (and its vector versions including Green's theorem and Stokes' theorem).
The number is also a constant, and you may be vaguely familiar with it as the base of the natural logarithm. Anything involving bunny rabbits has to be good. Or, as seems to be the case, is
mathematical beauty something buried deep: something that, perhaps, I need a PhD to get to Thank you for the article. [7] These results are often described as deep. He also enjoys playing the violin
and fencing. Are you starting to get an idea of the beauty of Euler's identity? Schmidhuber's theory of beauty and curiosity in a German TV show: John Ernest's use of mathematics and especially group
theory in his art works is analysed in, Learn how and when to remove this template message, Processing fluency theory of aesthetic pleasure, "The Definitive Glossary of Higher Mathematical Jargon â
Beauty", "Mathematics: Why the brain sees maths as beauty", "Platonism in the Philosophy of Mathematics", "Alain Badiou: Ontology and Structuralism", http://www.br-online.de/bayerisches-fernsehen/
faszination-wissen/schoenheit--aesthetik-wahrnehmung-ID1212005092828.xml, http://people.exeter.ac.uk/PErnest/pome24/index.htm, "Some Trends in Modern Mathematics and the Fields Medal", List of works
designed with the golden ratio, Viewpoints: Mathematical Perspective and Fractal Geometry in Art, European Society for Mathematics and the Arts, Goudreau Museum of Mathematics in Art and Science,
https://en.wikipedia.org/w/index.php?title=Mathematical_beauty&oldid=991252135, Wikipedia indefinitely move-protected pages, Wikipedia articles with style issues from March 2013, Creative Commons
Attribution-ShareAlike License. The physicist Richard Feynman called the formula it is derived from "one of the most remarkable, almost The Idea Behind It ... The-Mathematics-of-Beauty. Seeing why it
works feels a bit like treading a little-known path through the mathematical jungle to reach a secret destination IEEE press, 1991. Get practice question paper, sample paper, for upcoming exams and
CBSE or NCERT Solutions for Class 6th. The artistic beauty of mathematics; A Greek Headmaster’s first impressions of the project; ... Often known as the Divine Proportion, this is a real irrational
constant in algebra with an approximate value of 1,618. Some believe that in order to appreciate mathematics, one must engage in doing mathematics. What's beautiful about that? I always wonder what,
exactly, this means. Its thesis is that good maths is beautiful as well as true; that science is not just utilitarian but that beauty is built in from the start. For example, Math Circle is an
after-school enrichment program where students do mathematics through games and activities; there are also some teachers that encourage student engagement by teaching mathematics in a kinesthetic way
(see kinesthetic learning). Health and social care. I know numbers are beautiful. Even the most hardened mathematician would struggle to find beauty in the ugly brand of school maths. [16] If they
aren't beautiful, nothing is".[4]. A proof that derives a result in a surprising way (e.g., from an apparently unrelated. One of 7 assessments for the 2014 Curriculum programs of study for Year 1.
Twentieth-century French philosopher Alain Badiou claims that ontology is mathematics. and is hoping to read mathematics at university after he gets his A-levels. “It helps you think precisely,
decisively, and creatively and helps you look at the world from multiple perspectives . It has no generally accepted definition . Interest in pure mathematics that is separate from empirical study
has been part of the experience of various civilizations, including that of the ancient Greeks, who "did mathematics for the beauty of it". We get. If you don't see why, someone can't tell you. CBSE
Class 6th Maths: Place Value of a Digit. Origami, the art of paper folding, has aesthetic qualities and many mathematical connections. The original proof of Milnor was not very constructive, but
later E. Briscorn showed that these differential structures can be described in an extremely explicit and beautiful form.[13]. Often when reading a good maths book, the author will get to the end of
an explanation of a particularly complicated proof, theorem, or idea, and mention the "beauty" of the maths involved. [23][24] In the 1990s, Jürgen Schmidhuber formulated a mathematical theory of
observer-dependent subjective beauty based on algorithmic information theory: the most beautiful objects among subjectively comparable objects have short algorithmic descriptions (i.e., Kolmogorov
complexity) relative to what the observer already knows. In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two
quantities. In a day to day elementary school mathematics class, symmetry can be presented as such in an artistic manner where students see aesthetically pleasing results in mathematics. So you see,
after a sequence of fairly complex mathematics we arrive back where we started — at the (seemingly) simple numbers 1 and 0. Taylor & Francis, 2006. Did I miss a particularly neat diagram? For
example, they would argue that the theory of the natural numbers is fundamentally valid, in a way that does not require any specific context. It is a good idea to get them to complete the worksheet
before revealing the value of the golden ratio as this prevents people fixing their data. Brualdi, Richard. Value. In this article, we will discuss Chapter 1 Knowing our numbers out for Class 6
maths. [15] The beauty of mathematics is experienced when the physical reality of objects are represented by mathematical models. Hull, Thomas. If you square Phi, you get a number exactly 1 greater
than itself: 2.618…, or Φ² = Φ + 1. One of the most famous experiments in physics demonstrates the strange nature of the quantum world. I always wonder what, exactly, this means. 1. These
mathematicians believe that the detailed and precise results of mathematics may be reasonably taken to be true without any dependence on the universe in which we live. In some occasions, however, a
statement of a theorem can be original enough to be considered deepâ even though its proof is fairly obvious. I have included some celeb photo's but obviously these can be changed to suit. Every
mathematician I know found solace outside of … T eachers, parents and carers should model a positive attitude to maths and explore the relevance of maths in reallife contexts. Golden Ratio, Phi,
1.618, and Fibonacci in Math, Nature, Art, Design, Beauty and the Face. The very idea of beauty might slip away as we try to articulate it, and yet we would still know it was there. The beauty, if it
is there, is often well hidden and patience is needed to appreciate it. And that the maths you learn at National 4, National 5, and Higher level is … Great combination of Taylor Polynomials with
Euler Identity. [3], Paul ErdÅ s expressed his views on the ineffability of mathematics when he said, "Why are numbers beautiful? astounding, formulas in all of mathematics". Our Maths in a minute
series explores key mathematical concepts in just a few words. “Evidently some patterns are beautiful, but that is not what most mathematicians mean when they talk about the beauty of mathematics. As
there are exactly five Platonic solids, Kepler's hypothesis could only accommodate six planetary orbits and was disproved by the subsequent discovery of Uranus. You need to prepare in pairs a
response to the KQ: Why should elegance or beauty be relevant to mathematical value? They might also describe mathematics as an art form (e.g., a position taken by G. H. Hardy[2]) or, at a minimum,
as a creative activity. Mathematical beauty is the aesthetic pleasure typically derived from the abstractness, purity, simplicity, depth or orderliness of mathematics. Conf. [14] The aesthetic
pleasure that mathematical physicists tend to experience in Einstein's theory of general relativity has been attributed (by Paul Dirac, among others) to its "great mathematical beauty". [22] Badiou
also believes in deep connections between mathematics, poetry and philosophy. And without people who can do maths, we would not have many of the things we take for granted.
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Aluminium Casement Window | {"url":"http://ecbb2014.agrobiology.eu/i465t386/value-of-beauty-in-maths-16e986","timestamp":"2024-11-07T01:09:46Z","content_type":"text/html","content_length":"38050","record_id":"<urn:uuid:9015e70b-1a25-4643-90e4-bb53261e5d6f>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00787.warc.gz"} |
Boris Buliga - Equality of booleans in Emacs
Equality of booleans in Emacs
Posted on October 14, 2018
There is a fun story about booleans in Emacs Lisp - there are no booleans in Emacs Lisp. Sort of. Because we have a symbol nil, which means an empty list. You can write it as nil or () - they both
stand for the same object, the symbol nil.
Since LISP is all about list processing, empty list is something very false. So false that we don’t have special symbol for false values, as empty list serves this purpose well.
Everything that is not an empty list has a meaning of true. However, there is a symbol t which is the preferred way to represent the truth value true.
So nil and t are considered canonical boolean values. There is a function booleanp that returns t if the argument is a canonical boolean value and nil otherwise.
The fun begins when you need to check if two boolean values are equal. Since non-nil (or not an empty list) can mean many different things (like "Emacs is the only true editor") you can’t just do
regular equality check.
There are, however, several tricks to get it working. The most obvious solution is to convert value to a canonical boolean value.
We can directly use if function.
(if "Some truth" t nil) ; => t
(if 42 t nil) ; => t
(if t t nil) ; => t
(if nil t nil) ; => nil
(let ((a t)
(b "Emacs is the only true editor"))
(equal (if a t nil) (if b t nil))) ; => t
(defun boolean-eq (a b)
(equal (if a t nil)
(if b t nil)))
(let ((a t)
(b "Emacs is the only true editor"))
(boolean-eq a b)) ; => t
Directly using if is a little bit cumbersome, but when we hide it inside of a helper function it’s not that bad, actually.
The same result can be achieved by using when.
(when "Some truth" t) ; => t
(when 42 t) ; => t
(when t t) ; => t
(when nil t) ; => nil
(let ((a t)
(b "Emacs is the only true editor"))
(equal (when a t) (when b t))) ; => t
(defun boolean-eq (a b)
(equal (when a t)
(when b t)))
(let ((a t)
(b "Emacs is the only true editor"))
(boolean-eq a b)) ; => t
There is another function we can use - not, which returns t if the argument is nil, and returns nil otherwise. Yes, it negates the value, but the result is one of the canonical booleans, so we are
Since a≡b is equivalent to ¬a≡¬b, we can just compare negated values.
a b ¬a ¬b a≡b ¬a≡¬b
(not "Some truth") ; => nil
(not 42) ; => nil
(not t) ; => nil
(not nil) ; => t
(let ((a t)
(b "Emacs is the only true editor"))
(equal (not a) (not b))) ; => t
(defun boolean-eq (a b)
(equal (not a) (not b)))
(let ((a t)
(b "Emacs is the only true editor"))
(boolean-eq a b)) ; => t
This one looks a little bit better when used without a helper function, at least in my opinion.
Sometimes you want to do something when two ‘boolean’ values are not equal.
(let ((a nil)
(b "Emacs is the only true editor"))
(unless (equal (not a) (not b))
(message "Some real work begins"))) ; => Some real work begins
For such situations, there is a xor function, which returns nil when both arguments are equal in the canonical boolean form and t otherwise.
(xor nil nil) ; => nil
(xor nil t) ; => t
(xor t nil) ; => t
(xor t t) ; => nil
(xor "Some truth" nil) ; => t
(xor "Some truth" t) ; => nil
(xor 42 nil) ; => t
(xor 42 t) ; => nil
(let ((a nil)
(b "Emacs is the only true editor"))
(when (xor a b)
(message "Some real work begins"))) ; => Some real work begins
Other functions (like or, and) also convert values to canonical boolean values. So you can keep it in mind.
The sole purpose of this post is fun. If you didn’t get your portion of fun, then it’s not funny at all. Please fix it somehow. | {"url":"https://d12frosted.io/posts/2018-10-14-equality-of-booleans-in-emacs.html","timestamp":"2024-11-06T13:27:48Z","content_type":"text/html","content_length":"21469","record_id":"<urn:uuid:16e04601-d24d-42be-82ba-a9be261a1f6b>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00412.warc.gz"} |
A Comprehensive Overview of Mathematical Analysis
Mathematical analysis - Wikipedia 🔗
The text is a comprehensive overview of mathematical analysis, covering its historical development, foundational concepts, important branches, and applications. It begins with the ancient origins of
analysis, highlighting its presence in early Greek and Asian mathematics. The text then delves into the medieval and modern eras, discussing the contributions of mathematicians such as Archimedes,
Bhāskara II, and Euler. It provides insights into important concepts like metric spaces, sequences and limits, and discusses key branches of analysis including real analysis, complex analysis,
functional analysis, and harmonic analysis. Additionally, the text explores the applications of analysis in various fields such as physics, signal processing, and other areas of mathematics. The text
also lists a number of famous textbooks on mathematical analysis. | {"url":"https://tldr.chat/science/comprehensive-overview-mathematical-analysis-aee5","timestamp":"2024-11-13T01:33:19Z","content_type":"text/html","content_length":"7184","record_id":"<urn:uuid:a77fef5c-e583-44e9-a976-f3d138e04f3f>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00597.warc.gz"} |
Visualizing the Impact of Feature Attribution Baselines
Path attribution strategies are a gradient-based manner
of explaining deep fashions. These strategies require selecting a
hyperparameter often known as the baseline enter.
What does this hyperparameter imply, and the way vital is it? On this article,
we examine these questions utilizing picture classification networks
as a case research. We focus on a number of alternative ways to decide on a baseline
enter and the assumptions which can be implicit in every baseline.
Though we focus right here on path attribution strategies, our dialogue of baselines
is carefully linked with the idea of missingness within the characteristic area –
an idea that’s vital to interpretability analysis.
If you’re within the enterprise of coaching neural networks,
you may need heard of the built-in gradients technique, which
was launched at
ICML two years in the past
The tactic computes which options are vital
to a neural community when making a prediction on a
specific knowledge level. This helps customers
perceive which options their community depends on.
Since its introduction,
built-in gradients has been used to interpret
networks educated on quite a lot of knowledge varieties,
together with retinal fundus photos
and electrocardiogram recordings .
In the event you’ve ever used built-in gradients,
you realize that you want to outline a baseline enter (x’) earlier than
utilizing the strategy. Though the unique paper discusses the necessity for a baseline
and even proposes a number of completely different baselines for picture knowledge – together with
the fixed black picture and a picture of random noise – there may be
little current analysis in regards to the affect of this baseline.
Is built-in gradients delicate to the
hyperparameter selection? Why is the fixed black picture
a “pure baseline” for picture knowledge? Are there any various decisions?
On this article, we are going to delve into how this hyperparameter selection arises,
and why understanding it will be significant if you find yourself doing mannequin interpretation.
As a case-study, we are going to give attention to picture classification fashions so as
to visualise the results of the baseline enter. We are going to discover a number of
notions of missingness, together with each fixed baselines and baselines
outlined by distributions. Lastly, we are going to focus on alternative ways to match
baseline decisions and discuss why quantitative analysis
stays a troublesome downside.
Picture Classification
We give attention to picture classification as a activity, as it’s going to enable us to visually
plot built-in gradients attributions, and evaluate them with our instinct
about which pixels we predict needs to be vital. We use the Inception V4 structure
, a convolutional
neural community designed for the ImageNet dataset ,
during which the duty is to find out which class a picture belongs to out of 1000 lessons.
On the ImageNet validation set, Inception V4 has a top-1 accuracy of over 80%.
We obtain weights from TensorFlow-Slim ,
and visualize the predictions of the community on 4 completely different photos from the
validation set.
Proper: The expected logits of the community on the unique picture. The
community appropriately classifies all photos with excessive confidence.
Left: Pixel-wise attributions of the Inception V4 community utilizing built-in gradients.
You may discover that some attributions spotlight pixels that don’t appear vital
relative to the true class label.
Though cutting-edge fashions carry out properly on unseen knowledge,
customers should be left questioning: how did the mannequin determine
out which object was within the picture? There are a myriad of strategies to
interpret machine studying fashions, together with strategies to
visualize and perceive how the community represents inputs internally ,
characteristic attribution strategies that assign an significance rating to every characteristic
for a selected enter ,
and saliency strategies that goal to spotlight which areas of a picture
the mannequin was when making a call
These classes will not be mutually unique: for instance, an attribution technique may be
visualized as a saliency technique, and a saliency technique can assign significance
scores to every particular person pixel. On this article, we are going to focus
on the characteristic attribution technique built-in gradients.
Formally, given a goal enter (x) and a community perform (f),
characteristic attribution strategies assign an significance rating (phi_i(f, x))
to the (i)th characteristic worth representing how a lot that characteristic
provides or subtracts from the community output. A big optimistic or unfavorable (phi_i(f, x))
signifies that characteristic strongly will increase or decreases the community output
(f(x)) respectively, whereas an significance rating near zero signifies that
the characteristic in query didn’t affect (f(x)).
In the identical determine above, we visualize which pixels have been most vital to the community’s appropriate
prediction utilizing built-in gradients.
The pixels in white point out extra vital pixels. As a way to plot
attributions, we comply with the identical design decisions as .
That’s, we plot absolutely the worth of the sum of characteristic attributions
throughout the channel dimension, and cap characteristic attributions on the 99th percentile to keep away from
high-magnitude attributions dominating the colour scheme.
A Higher Understanding of Built-in Gradients
As you look by the attribution maps, you may discover a few of them
unintuitive. Why does the attribution for “goldfinch” spotlight the inexperienced background?
Why doesn’t the attribution for “killer whale” spotlight the black elements of the killer whale?
To raised perceive this conduct, we have to discover how
we generated characteristic attributions. Formally, built-in gradients
defines the significance worth for the (i)th characteristic worth as follows:
$$phi_i^{IG}(f, x, x’) = overbrace{(x_i – x’_i)}^{textual content{Distinction from baseline}}
occasions underbrace{int_{alpha = 0}^ 1}_{textual content{From baseline to enter…}}
overbrace{frac{delta f(x’ + alpha (x – x’))}{delta x_i} d alpha}^{textual content{…accumulate native gradients}}
the place (x) is the present enter,
(f) is the mannequin perform and (x’) is a few baseline enter that’s meant to signify
“absence” of characteristic enter. The subscript (i) is used
to indicate indexing into the (i)th characteristic.
Because the components above states, built-in gradients will get significance scores
by accumulating gradients on photos interpolated between the baseline worth and the present enter.
However why would doing this make sense? Recall that the gradient of
a perform represents the course of most improve. The gradient
is telling us which pixels have the steepest native slope with respect
to the output. For that reason, the gradient of a community on the enter
was one of many earliest saliency strategies.
Sadly, there are various issues with utilizing gradients to interpret
deep neural networks .
One particular problem is that neural networks are vulnerable to an issue
often known as saturation: the gradients of enter options might have small magnitudes round a
pattern even when the community relies upon closely on these options. This will occur
if the community perform flattens after these options attain a sure magnitude.
Intuitively, shifting the pixels in a picture by a small quantity usually
doesn’t change what the community sees within the picture. We are able to illustrate
saturation by plotting the community output in any respect
photos between the baseline (x’) and the present picture. The determine
beneath shows that the community
output for the proper class will increase initially, however then rapidly flattens.
A plot of community outputs at (x’ + alpha (x – x’)).
Discover that the community output saturates the proper class
at small values of (alpha). By the point (alpha = 1),
the community output barely adjustments.
What we actually wish to know is how our community acquired from
predicting primarily nothing at (x’) to being
fully saturated in direction of the proper output class at (x).
Which pixels, when scaled alongside this path, most
elevated the community output for the proper class? That is
precisely what the components for built-in gradients offers us.
By integrating over a path,
built-in gradients avoids issues with native gradients being
saturated. We are able to break the unique equation
down and visualize it in three separate elements: the interpolated picture between
the baseline picture and the goal picture, the gradients on the interpolated
picture, and accumulating many such gradients over (alpha).
int_{alpha’ = 0}^{alpha} underbrace{(x_i – x’_i) occasions
frac{delta f(textual content{ }overbrace{x’ + alpha’ (x – x’)}^{textual content{(1): Interpolated Picture}}textual content{ })}
{delta x_i} d alpha’}_{textual content{(2): Gradients at Interpolation}}
= overbrace{phi_i^{IG}(f, x, x’; alpha)}^{textual content{(3): Cumulative Gradients as much as }alpha}
We visualize these three items of the components beneath.Notice that in follow, we use a discrete sum
approximation of the integral with 500 linearly-spaced factors between 0 and 1.
Built-in gradients, visualized. Within the line chart, the purple line refers to
equation (4) and the blue line refers to (f(x) – f(x’)). Discover how excessive magnitude gradients
accumulate at small values of (alpha).
Now we have casually omitted one a part of the components: the very fact
that we multiply by a distinction from a baseline. Though
we received’t go into element right here, this time period falls out as a result of we
care in regards to the spinoff of the community
perform (f) with respect to the trail we’re integrating over.
That’s, if we combine over the
straight-line between (x’) and (x), which
we are able to signify as (gamma(alpha) =
x’ + alpha(x – x’)), then:
frac{delta f(gamma(alpha))}{delta alpha} =
frac{delta f(gamma(alpha))}{delta gamma(alpha)} occasions
frac{delta gamma(alpha)}{delta alpha} =
frac{delta f(x’ + alpha’ (x – x’))}{delta x_i} occasions (x_i – x’_i)
The distinction from baseline time period is the spinoff of the
path perform (gamma) with respect to (alpha).
The speculation behind built-in gradients is mentioned
in additional element within the authentic paper. Specifically, the authors
present that built-in gradients satisfies a number of fascinating
properties, together with the completeness axiom:
textrm{Axiom 1: Completeness}
sum_i phi_i^{IG}(f, x, x’) = f(x) – f(x’)
Notice that this theorem holds for any baseline (x’).
Completeness is a fascinating property as a result of it states that the
significance scores for every characteristic break down the output of the community:
every significance rating represents that characteristic’s particular person contribution to
the community output, and added when collectively, we get better the output worth itself.
Though it’s not important to our dialogue right here, we are able to show
that built-in gradients satisfies this axiom utilizing the
theorem of calculus for path integrals. We go away a
full dialogue of all the properties that built-in
gradients satisfies to the unique paper, since they maintain
unbiased of the selection of baseline. The completeness
axiom additionally gives a approach to measure convergence.
In follow, we are able to’t compute the precise worth of the integral. As a substitute,
we use a discrete sum approximation with (okay) linearly-spaced factors between
0 and 1 for some worth of (okay). If we solely selected 1 level to
approximate the integral, that looks like too few. Is 10 sufficient? 100?
Intuitively 1,000 might appear to be sufficient, however can we make sure?
As proposed within the authentic paper, we are able to use the completeness axiom
as a sanity verify on convergence: run built-in gradients with (okay)
factors, measure (|sum_i phi_i^{IG}(f, x, x’) – (f(x) – f(x’))|),
and if the distinction is massive, re-run with a bigger (okay)
In fact, this brings up a brand new query: what’s “massive” on this context?
One heuristic is to match the distinction with the magnitude of the
output itself.
The road chart above plots the next equation in purple:
underbrace{sum_i phi_i^{IG}(f, x, x’; alpha)}_{textual content{(4): Sum of Cumulative Gradients as much as }alpha}
That’s, it sums all the pixel attributions within the saliency map.
This lets us evaluate to the blue line, which plots (f(x) – f(x’)).
We are able to see that with 500 samples, we appear (at the least intuitively) to
have converged. However this text isn’t about how
to get good convergence – it’s about baselines! So as
to advance our understanding of the baseline, we are going to want a short tour
into the world of recreation principle.
Recreation Concept and Missingness
Built-in gradients is impressed by work
from cooperative recreation principle, particularly the Aumann-Shapley worth
. In cooperative recreation principle,
a non-atomic recreation is a development used to mannequin large-scale financial methods
the place there are sufficient contributors that it’s fascinating to mannequin them repeatedly.
Aumann-Shapley values present a theoretically grounded approach to
decide how a lot completely different teams of contributors contribute to the system.
In recreation principle, a notion of missingness is well-defined. Video games are outlined
on coalitions – units of contributors – and for any particular coalition,
a participant of the system may be in or out of that coalition. The actual fact
that video games may be evaluated on coalitions is the muse of
the Aumann-Shapley worth. Intuitively, it computes how
a lot worth a gaggle of contributors provides to the sport
by computing how a lot the worth of the sport would improve
if we added extra of that group to any given coalition.
Sadly, missingness is a harder notion when
we’re talking about machine studying fashions. So as
to judge how vital the (i)th characteristic is, we
need to have the ability to compute how a lot the output of
the community would improve if we successively elevated
the “presence” of the (i)th characteristic. However what does this imply, precisely?
As a way to improve the presence of a characteristic, we would want to start out
with the characteristic being “lacking” and have a manner of interpolating
between that missingness and its present, identified worth.
Hopefully, that is sounding awfully acquainted. Built-in gradients
has a baseline enter (x’) for precisely this motive: to mannequin a
characteristic being absent. However how do you have to select
(x’) in an effort to finest signify this? It appears to be frequent follow
to decide on a baseline enter (x’) to be the vector of
all zeros. However think about the next situation: you’ve realized a mannequin
on a healthcare dataset, and one of many options is blood sugar stage.
The mannequin has appropriately realized that excessively low ranges of blood sugar,
which correspond to hypoglycemia, is harmful. Does
a blood sugar stage of (0) appear to be a good selection to signify missingness?
The purpose right here is that fastened characteristic values might have unintended that means.
The issue compounds additional when you think about the distinction from
baseline time period (x_i – x’_i).
For the sake of a thought experiment, suppose a affected person had a blood sugar stage of (0).
To grasp why our machine studying mannequin thinks this affected person
is at excessive danger, you run built-in gradients on this knowledge level with a
baseline of the all-zeros vector. The blood sugar stage of the affected person would have (0) characteristic significance,
as a result of (x_i – x’_i = 0). That is even though
a blood sugar stage of (0) could be deadly!
We discover comparable issues after we transfer to the picture area.
In the event you use a relentless black picture as a baseline, built-in gradients will
not spotlight black pixels as vital even when black pixels make up
the article of curiosity. Extra usually, the strategy is blind to the colour you utilize as a baseline, which
we illustrate with the determine beneath. Notice that this was acknowledged by the unique
authors in , and is actually
central to the definition of a baseline: we wouldn’t need built-in gradients
to spotlight lacking options as vital! However then how can we keep away from
giving zero significance to the baseline coloration?
Mouse over the segmented picture to decide on a unique coloration
as a baseline enter (x’). Discover that pixels
of the baseline coloration will not be highlighted as vital,
even when they make up a part of the principle object within the picture.
Different Baseline Selections
It’s clear that any fixed coloration baseline can have this downside.
Are there any options? On this part, we
evaluate 4 various decisions for a baseline within the picture area.
Earlier than continuing, it’s vital to notice that this text isn’t
the primary article to level out the issue of selecting a baselines.
A number of articles, together with the unique paper, focus on and evaluate
a number of notions of “missingness”, each within the
context of built-in gradients and extra usually
Nonetheless, selecting the best baseline stays a problem. Right here we are going to
current a number of decisions for baselines: some primarily based on current literature,
others impressed by the issues mentioned above. The determine on the finish
of the part visualizes the 4 baselines offered right here.
The Most Distance Baseline
If we’re nervous about fixed baselines which can be blind to the baseline
coloration, can we explicitly assemble a baseline that doesn’t endure from this
downside? One apparent approach to assemble such a baseline is to take the
farthest picture in L1 distance from the present picture such that the
baseline remains to be within the legitimate pixel vary. This baseline, which
we are going to seek advice from as the utmost distance baseline (denoted
max dist. within the determine beneath),
avoids the distinction from baseline problem immediately.
The Blurred Baseline
The difficulty with the utmost distance baseline is that it doesn’t
actually signify missingness. It truly incorporates numerous
details about the unique picture, which suggests we’re now not
explaining our prediction relative to a lack of awareness. To raised
protect the notion of missingness, we take inspiration from
. Of their paper,
Fong and Vedaldi use a blurred model of the picture as a
domain-specific approach to signify lacking info. This baseline
is engaging as a result of it captures the notion of missingness in photos
in a really human intuitive manner. Within the determine beneath, this baseline is
denoted blur. The determine permits you to play with the smoothing fixed
used to outline the baseline.
The Uniform Baseline
One potential downside with the blurred baseline is that it’s biased
to spotlight high-frequency info. Pixels which can be very comparable
to their neighbors might get much less significance than pixels which can be very
completely different than their neighbors, as a result of the baseline is outlined as a weighted
common of a pixel and its neighbors. To beat this, we are able to once more take inspiration
from each and the unique built-in
gradients paper. One other approach to outline missingness is to easily pattern a random
uniform picture within the legitimate pixel vary and name that the baseline.
We seek advice from this baseline because the uniform baseline within the determine beneath.
The Gaussian Baseline
In fact, the uniform distribution isn’t the one distribution we are able to
draw random noise from. Of their paper discussing the SmoothGrad (which we are going to
contact on within the subsequent part), Smilkov et al.
make frequent use of a gaussian distribution centered on the present picture with
variance (sigma). We are able to use the identical distribution as a baseline for
built-in gradients! Within the determine beneath, this baseline is named the gaussian
baseline. You’ll be able to differ the usual deviation of the distribution (sigma) utilizing the slider.
One factor to notice right here is that we truncate the gaussian baseline within the legitimate pixel
vary, which signifies that as (sigma) approaches (infty), the gaussian
baseline approaches the uniform baseline.
Evaluating various baseline decisions. For the blur and gaussian
baselines, you may differ the parameter (sigma), which refers
to the width of the smoothing kernel and the usual deviation of
noise respectively.
Averaging Over A number of Baselines
You might have nagging doubts about these final two baselines, and also you
could be proper to have them. A randomly generated baseline
can endure from the identical blindness downside {that a} fixed picture can. If
we draw a uniform random picture as a baseline, there’s a small probability
{that a} baseline pixel shall be very near its corresponding enter pixel
in worth. These pixels won’t be highlighted as vital. The ensuing
saliency map might have artifacts as a result of randomly drawn baseline. Is there
any manner we are able to repair this downside?
Maybe probably the most pure manner to take action is to common over a number of
completely different baselines, as mentioned in
Though doing so might not be notably pure for fixed coloration photos
(which colours do you select to common over and why?), it’s a
very pure notion for baselines drawn from distributions. Merely
draw extra samples from the identical distribution and common the
significance scores from every pattern.
Assuming a Distribution
At this level, it’s price connecting the concept of averaging over a number of
baselines again to the unique definition of built-in gradients. When
we common over a number of baselines from the identical distribution (D),
we are trying to make use of the distribution itself as our baseline.
We use the distribution to outline the notion of missingness:
if we don’t know a pixel worth, we don’t assume its worth to be 0 – as an alternative
we assume that it has some underlying distribution (D). Formally, given
a baseline distribution (D), we combine over all potential baselines
(x’ in D) weighted by the density perform (p_D):
$$ phi_i(f, x) = underbrace{int_{x’}}_{textual content{Combine over baselines…}} bigg( overbrace{phi_i^{IG}(f, x, x’
)}^{textual content{built-in gradients
with baseline } x’
} occasions underbrace{p_D(x’) dx’}_{textual content{…and weight by the density}} bigg)
By way of missingness, assuming a distribution may intuitively really feel
like a extra cheap assumption to make than assuming a relentless worth.
However this doesn’t fairly clear up the problem: as an alternative of getting to decide on a baseline
(x’), now we’ve got to decide on a baseline distribution (D). Have we merely
postponed the issue? We are going to focus on one theoretically motivated
manner to decide on (D) in an upcoming part, however earlier than we do, we’ll take
a short apart to speak about how we compute the components above in follow,
and a connection to an current technique that arises consequently.
Expectations, and Connections to SmoothGrad
Now that we’ve launched a second integral into our components,
we have to do a second discrete sum to approximate it, which
requires a further hyperparameter: the variety of baselines to pattern.
In , Erion et al. make the
remark that each integrals may be considered expectations:
the primary integral as an expectation over (D), and the second integral
as an expectation over the trail between (x’) and (x). This formulation,
known as anticipated gradients, is outlined formally as:
$$ phi_i^{EG}(f, x; D) = underbrace{mathop{mathbb{E}}_{x’ sim D, alpha sim U(0, 1)}}_
{textual content{Expectation over (D) and the trail…}}
bigg[ overbrace{(x_i – x’_i) times
frac{delta f(x’ + alpha (x – x’))}{delta x_i}}^{text{…of the
importance of the } itext{th pixel}} bigg]
Anticipated gradients and built-in gradients belong to a household of strategies
often known as “path attribution strategies” as a result of they combine gradients
over a number of paths between two legitimate inputs.
Each anticipated gradients and built-in gradients use straight-line paths,
however one can combine over paths that aren’t straight as properly. That is mentioned
in additional element within the authentic paper. To compute anticipated gradients in
follow, we use the next components:
hat{phi}_i^{EG}(f, x; D) = frac{1}{okay} sum_{j=1}^okay (x_i – x’^j_i) occasions
frac{delta f(x’^j + alpha^{j} (x – x’^j))}{delta x_i}
the place (x’^j) is the (j)th pattern from (D) and
(alpha^j) is the (j)th pattern from the uniform distribution between
0 and 1. Now suppose that we use the gaussian baseline with variance
(sigma^2). Then we are able to re-write the components for anticipated gradients as follows:
hat{phi}_i^{EG}(f, x; N(x, sigma^2 I))
= frac{1}{okay} sum_{j=1}^okay
epsilon_{sigma}^{j} occasions
frac{delta f(x + (1 – alpha^j)epsilon_{sigma}^{j})}{delta x_i}
the place (epsilon_{sigma} sim N(bar{0}, sigma^2 I))
To see how we arrived
on the above components, first observe that
x’ sim N(x, sigma^2 I) &= x + epsilon_{sigma}
x’- x &= epsilon_{sigma}
by definition of the gaussian baseline. Now we’ve got:
x’ + alpha(x – x’) &=
x + epsilon_{sigma} + alpha(x – (x + epsilon_{sigma})) &=
x + (1 – alpha)epsilon_{sigma}
The above components merely substitues the final line
of every equation block again into the components.
This seems awfully acquainted to an current technique known as SmoothGrad
. If we use the (gradients (occasions) enter picture)
variant of SmoothGrad SmoothGrad is
was a way designed to sharpen saliency maps and was meant to be run
on prime of an current saliency technique. The concept is straightforward:
as an alternative of operating a saliency technique as soon as on a picture, first
add some gaussian noise to a picture, then run the saliency technique.
Do that a number of occasions with completely different attracts of gaussian noise, then
common the outcomes. Multipying the gradients by the enter and utilizing that as a saliency map
is mentioned in additional element within the authentic SmoothGrad paper.,
then we’ve got the next components:
phi_i^{SG}(f, x; N(bar{0}, sigma^2 I))
= frac{1}{okay} sum_{j=1}^okay
(x + epsilon_{sigma}^{j}) occasions
frac{delta f(x + epsilon_{sigma}^{j})}{delta x_i}
We are able to see that SmoothGrad and anticipated gradients with a
gaussian baseline are fairly comparable, with two key variations:
SmoothGrad multiplies the gradient by (x + epsilon_{sigma}) whereas anticipated
gradients multiplies by simply (epsilon_{sigma}), and whereas anticipated
gradients samples uniformly alongside the trail, SmoothGrad at all times
samples the endpoint (alpha = 0).
Can this connection assist us perceive why SmoothGrad creates
smooth-looking saliency maps? After we assume the above gaussian distribution as our baseline, we’re
assuming that every of our pixel values is drawn from a
gaussian independently of the opposite pixel values. However we all know
that is removed from true: in photos, there’s a wealthy correlation construction
between close by pixels. As soon as your community is aware of the worth of a pixel,
it doesn’t really want to make use of its instant neighbors as a result of
it’s possible that these instant neighbors have very comparable intensities.
Assuming every pixel is drawn from an unbiased gaussian
breaks this correlation construction. It signifies that anticipated gradients
tabulates the significance of every pixel independently of
the opposite pixel values. The generated saliency maps
shall be much less noisy and higher spotlight the article of curiosity
as a result of we’re now not permitting the community to rely
on solely pixel in a gaggle of correlated pixels. This can be
why SmoothGrad is {smooth}: as a result of it’s implicitly assuming
independence amongst pixels. Within the determine beneath, you may evaluate
built-in gradients with a single randomly drawn baseline
to anticipated gradients sampled over a distribution. For
the gaussian baseline, you can too toggle the SmoothGrad
possibility to make use of the SmoothGrad components above. For all figures,
The distinction between a single baseline and a number of
baselines from the identical distribution. Use the
“Multi-Reference” button to toggle between the 2. For the gaussian
baseline, you can too toggle the “Easy Grad” button
to toggle between anticipated gradients and SmoothGrad
with gradients * inputs.
Utilizing the Coaching Distribution
Is it actually cheap to imagine independence amongst
pixels whereas producing saliency maps? In supervised studying,
we make the idea that the information is drawn
from some distribution (D_{textual content{knowledge}}). This assumption that the coaching and testing knowledge
share a typical, underlying distribution is what permits us to
do supervised studying and make claims about generalizability. Given
this assumption, we don’t have to
mannequin missingness utilizing a gaussian or a uniform distribution:
we are able to use (D_{textual content{knowledge}}) to mannequin missingness immediately.
The one downside is that we don’t have entry to the underlying distribution.
However as a result of this can be a supervised studying activity, we do have entry to many
unbiased attracts from the underlying distribution: the coaching knowledge!
We are able to merely use samples from the coaching knowledge as random attracts
from (D_{textual content{knowledge}}). This brings us to the variant
of anticipated gradients utilized in ,
which we once more visualize in three elements:
frac{1}{okay} sum_{j=1}^okay
underbrace{(x_i – x’^j_i) occasions
frac{delta f(textual content{ }
overbrace{x’^j + alpha^{j} (x – x’^j)}^{textual content{(1): Interpolated Picture}}
textual content{ })}{delta x_i}}_{textual content{ (2): Gradients at Interpolation}}
= overbrace{hat{phi_i}^{EG}(f, x, okay; D_{textual content{knowledge}})}
^{textual content{(3): Cumulative Gradients as much as }alpha}
A visible illustration of anticipated gradients. As a substitute of taking contributions
from a single path, anticipated gradients averages contributions from
all paths outlined by the underlying knowledge distribution. Notice that
this determine solely shows each tenth pattern to keep away from loading many photos.
In (4) we once more plot the sum of the significance scores over pixels. As talked about
within the authentic built-in gradients paper, all path strategies, together with anticipated
gradients, fulfill the completeness axiom. We are able to positively see that
completeness is tougher to fulfill after we combine over each a path
and a distribution: that’s, with the identical quantity
of samples, anticipated gradients doesn’t converge as rapidly as
built-in gradients does. Whether or not or not that is a suitable worth to
pay to keep away from color-blindness in attributions appears subjective.
Evaluating Saliency Strategies
So we now have many various decisions for a baseline. How can we select
which one we must always use? The completely different decisions of distributions and fixed
baselines have completely different theoretical motivations and sensible considerations.
Do we’ve got any manner of evaluating the completely different baselines? On this part,
we are going to contact on a number of completely different concepts about tips on how to evaluate
interpretability strategies. This part isn’t meant to be a complete overview
of all the current analysis metrics, however is as an alternative meant to
emphasize that evaluating interpretability strategies stays a troublesome downside.
The Risks of Qualitative Evaluation
One naive approach to consider our baselines is to take a look at the saliency maps
they produce and see which of them finest spotlight the article within the picture.
From our earlier figures, it does appear to be utilizing (D_{textual content{knowledge}}) produces
cheap outcomes, as does utilizing a gaussian baseline or the blurred baseline.
However is visible inspection actually a great way decide our baselines? For one factor,
we’ve solely offered 4 photos from the check set right here. We would want to
conduct person research on a a lot bigger scale with extra photos from the check
set to be assured in our outcomes. However even with large-scale person research,
qualitative evaluation of saliency maps has different drawbacks.
After we depend on qualitative evaluation, we’re assuming that people
know what an “correct” saliency map is. After we take a look at saliency maps
on knowledge like ImageNet, we frequently verify whether or not or not the saliency map
highlights the article that we see as representing the true class within the picture.
We make an assumption between the information and the label, after which additional assume
{that a} good saliency map ought to mirror that assumption. However doing so
has no actual justification. Think about the determine beneath, which compares
two saliency strategies on a community that will get above 99% accuracy
on (an altered model of) MNIST.
The primary saliency technique is simply an edge detector plus gaussian smoothing,
whereas the second saliency technique is predicted gradients utilizing the coaching
knowledge as a distribution. Edge detection higher displays what we people
assume is the connection between the picture and the label.
Qualitative evaluation may be harmful as a result of we rely
on our human information of the connection between
the information and the labels, after which we assume
that an correct mannequin has realized that very relationship.
Sadly, the sting detection technique right here doesn’t spotlight
what the community has realized. This dataset is a variant of
decoy MNIST, during which the highest left nook of the picture has
been altered to immediately encode the picture’s class
. That’s, the depth
of the highest left nook of every picture has been altered to
be (255 occasions frac{y}{9} ) the place (y) is the category
the picture belongs to. We are able to confirm by eradicating this
patch within the check set that the community closely depends on it to make
predictions, which is what the anticipated gradients saliency maps present.
That is clearly a contrived instance. Nonetheless, the truth that
visible evaluation isn’t essentially a helpful approach to consider
saliency maps and attribution strategies has been extensively
mentioned in latest literature, with many proposed qualitative
exams as replacements
On the coronary heart of the problem is that we don’t have floor reality explanations:
we are attempting to judge which strategies finest clarify our community with out
truly understanding what our networks are doing.
Prime Ok Ablation Exams
One easy approach to consider the significance scores that
anticipated/built-in gradients produces is to see whether or not
ablating the highest okay options as ranked by their significance
decreases the anticipated output logit. Within the determine beneath, we
ablate both by mean-imputation or by changing every pixel
by its gaussian-blurred counter-part (Imply Prime Ok and Blur Prime Ok within the plot). We generate pixel importances
for 1000 completely different appropriately labeled test-set photos utilizing every
of the baselines proposed above
For the blur baseline and the blur
ablation check, we use (sigma = 20).
For the gaussian baseline, we use (sigma = 1). These decisions
are considerably arbitrary – a extra complete analysis
would evaluate throughout many values of (sigma).
. As a
management, we additionally embody rating options randomly
(Random Noise within the plot).
We plot, as a fraction of the unique logit, the output logit
of the community on the true class. That’s, suppose the unique
picture is a goldfinch and the community predicts the goldfinch class appropriately
with 95% confidence. If the boldness of sophistication goldfinch drops
to 60% after ablating the highest 10% of pixels as ranked by
characteristic significance, then we plot a curve that goes by
the factors ((0.0, 0.95)) and ((0.1, 0.6)). The baseline selection
that finest highlights which pixels the community
ought to exhibit the quickest drop in logit magnitude, as a result of
it highlights the pixels that the majority improve the boldness of the community.
That’s, the decrease the curve, the higher the baseline.
Mass Heart Ablation Exams
One downside with ablating the highest okay options in a picture
is expounded to a problem we already introduced up: characteristic correlation.
Regardless of how we ablate a pixel, that pixel’s neighbors
present numerous details about the pixel’s authentic worth.
With this in thoughts, one may argue that progressively ablating
pixels one after the other is a slightly meaningless factor to do. Can
we as an alternative carry out ablations with characteristic correlation in thoughts?
One easy manner to do that is solely compute the
heart of mass
of the saliency map, and ablate a boxed area centered on
the middle of mass. This exams whether or not or not the saliency map
is mostly highlighting an vital area within the picture. We plot
changing the boxed area across the saliency map utilizing mean-imputation
and blurring beneath as properly (Imply Heart and Blur Heart, respectively).
As a management, we evaluate in opposition to a saliency map generated from random gaussian
noise (Random Noise within the plot).
A wide range of ablation exams on quite a lot of baselines.
Utilizing the coaching distribution and utilizing the uniform distribution
outperform most different strategies on the highest okay ablation exams. The
blur baseline impressed by
does equally properly on the blur top-k check. All strategies
carry out equally on the mass heart ablation exams. Mouse
over the legend to spotlight a single curve.
The ablation exams appear to point some fascinating traits.
All strategies do equally on the mass heart ablation exams, and
solely barely higher than random noise. This can be as a result of the
object of curiosity usually lies within the heart of the picture – it
isn’t arduous for random noise to be centered on the picture. In distinction,
utilizing the coaching knowledge or a uniform distribution appears to do fairly properly
on the top-k ablation exams. Curiously, the blur baseline
impressed by additionally
does fairly properly on the highest okay baseline exams, particularly when
we ablate pixels by blurring them! Would the uniform
baseline do higher in case you ablate the picture with uniform random noise?
Maybe the coaching distribution baseline would do even higher in case you ablate a picture
by progressively changing it with a unique picture. We go away
these experiments as future work, as there’s a extra urgent query
we have to focus on.
The Pitfalls of Ablation Exams
Can we actually belief the ablations exams offered above? We ran every technique with 500 samples.
Fixed baselines are likely to not want as many samples
to converge as baselines over distributions. How can we pretty evaluate between baselines which have
completely different computational prices? Invaluable however computationally-intensive future work could be
evaluating not solely throughout baselines but additionally throughout variety of samples drawn,
and for the blur and gaussian baselines, the parameter (sigma).
As talked about above, we’ve got outlined many notions of missingness aside from
mean-imputation or blurring: extra in depth comparisons would additionally evaluate
all of our baselines throughout all the corresponding notions of lacking knowledge.
However even with all of those added comparisons, do ablation
exams actually present a well-founded metric to guage attribution strategies?
The authors of argue
in opposition to ablation exams. They level out that when we artificially ablate
pixels a picture, we’ve got created inputs that don’t come from
the unique knowledge distribution. Our educated mannequin has by no means seen such
inputs. Why ought to we anticipate to extract any cheap info
from evaluating our mannequin on them?
Then again, built-in gradients and anticipated gradients
depend on presenting interpolated photos to your mannequin, and until
you make some unusual convexity assumption, these interpolated photos
don’t belong to the unique coaching distribution both.
Usually, whether or not or not customers ought to current
their fashions with inputs that don’t belong to the unique coaching distribution
is a topic of ongoing debate
. Nonetheless,
the purpose raised in remains to be an
vital one: “it’s unclear whether or not the degradation in mannequin
efficiency comes from the distribution shift or as a result of the
options that have been eliminated are actually informative.”
Different Analysis Metrics
So what about different analysis metrics proposed in latest literature? In
, Hooker et al. suggest a variant of
an ablation check the place we first ablate pixels within the coaching and
check units. Then, we re-train a mannequin on the ablated knowledge and measure
by how a lot the test-set efficiency degrades. This method has the benefit
of higher capturing whether or not or not the saliency technique
highlights the pixels which can be most vital for predicting the output class.
Sadly, it has the disadvantage of needing to re-train the mannequin a number of
occasions. This metric can also get confused by characteristic correlation.
Think about the next situation: our dataset has two options
which can be extremely correlated. We prepare a mannequin which learns to solely
use the primary characteristic, and fully ignore the second characteristic.
A characteristic attribution technique may precisely reveal what the mannequin is doing:
it’s solely utilizing the primary characteristic. We may ablate that characteristic within the dataset,
re-train the mannequin and get comparable efficiency as a result of comparable info
is saved within the second characteristic. We would conclude that our characteristic
attribution technique is awful – is it? This downside matches into a bigger dialogue
about whether or not or not your attribution technique
needs to be “true to the mannequin” or “true to the information”
which has been mentioned in a number of latest articles
In , the authors suggest a number of
sanity checks that saliency strategies ought to move. One is the “Mannequin Parameter
Randomization Check”. Primarily, it states {that a} characteristic attribution
technique ought to produce completely different attributions when evaluated on a educated
mannequin (assumedly a educated mannequin that performs properly) and a randomly initialized
mannequin. This metric is intuitive: if a characteristic attribution technique produces
comparable attributions for random and educated fashions, is the characteristic
attribution actually utilizing info from the mannequin? It’d simply
be relying completely on info from the enter picture.
However think about the next determine, which is one other (modified) model
of MNIST. We’ve generated anticipated gradients attributions utilizing the coaching
distribution as a baseline for 2 completely different networks. One of many networks
is a educated mannequin that will get over 99% accuracy on the check set. The opposite
community is a randomly initialized mannequin that doesn’t do higher than random guessing.
Ought to we now conclude that anticipated gradients is an unreliable technique?
A comparability of two community’s saliency maps utilizing anticipated gradients. One
community has randomly initialized weights, the opposite will get >99% accuracy
on the check set.
In fact, we modified MNIST on this instance particularly in order that anticipated gradients
attributions of an correct mannequin would look precisely like these of a randomly initialized mannequin.
The way in which we did that is just like the decoy MNIST dataset, besides as an alternative of the highest left
nook encoding the category label, we randomly scattered noise througout every coaching and
check picture the place the depth of the noise encodes the true class label. Usually,
you’ll run these sorts of saliency technique sanity checks on un-modified knowledge.
However the reality is, even for pure photos, we don’t truly
know what an correct mannequin’s saliency maps ought to seem like.
Totally different architectures educated on ImageNet can all get good efficiency
and have very completely different saliency maps. Can we actually say that
educated fashions ought to have saliency maps that don’t seem like
saliency maps generated on randomly initialized fashions? That isn’t
to say that the mannequin randomization check doesn’t have advantage: it
does reveal fascinating issues about what saliency strategies are are doing.
It simply doesn’t inform the entire story.
As we talked about above, there’s quite a lot of metrics which have been proposed to judge
interpretability strategies. There are a lot of metrics we don’t explicitly focus on right here
Every proposed metric comes with their varied execs and cons.
Usually, evaluating supervised fashions is considerably easy: we put aside a
test-set and use it to judge how properly our mannequin performs on unseen knowledge. Evaluating explanations is tough:
we don’t know what our mannequin is doing and haven’t any floor reality to match
in opposition to.
So what needs to be accomplished? Now we have many baselines and
no conclusion about which one is the “finest.” Though
we don’t present in depth quantitative outcomes
evaluating every baseline, we do present a basis
for understanding them additional. On the coronary heart of
every baseline is an assumption about missingness
in our mannequin and the distribution of our knowledge. On this article,
we make clear a few of these assumptions, and their affect
on the corresponding path attribution. We lay
groundwork for future dialogue about baselines within the
context of path attributions, and extra usually about
the connection between representations of missingness
and the way we clarify machine studying fashions.
A side-by-side comparability of built-in gradients
utilizing a black baseline
and anticipated gradients utilizing the coaching knowledge
as a baseline.
Associated Strategies
This work focuses on a selected interpretability technique: built-in gradients
and its extension, anticipated gradients. We refer to those
strategies as path attribution strategies as a result of they combine
importances over a path. Nonetheless, path attribution strategies
signify solely a tiny fraction of current interpretability strategies. We focus
on them right here each as a result of they’re amenable to fascinating visualizations,
and since they supply a springboard for speaking about missingness.
We briefly cited a number of different strategies at first of this text.
A lot of these strategies use some notion of baseline and have contributed to
the dialogue surrounding baseline decisions.
In , Fong and Vedaldi suggest
a model-agnostic technique to elucidate neural networks that’s primarily based
on studying the minimal deletion to a picture that adjustments the mannequin
prediction. In part 4, their work incorporates an prolonged dialogue on
tips on how to signify deletions: that’s, tips on how to signify lacking pixels. They
argue that one pure approach to delete pixels in a picture is to blur them.
This dialogue impressed the blurred baseline that we offered in our article.
In addition they focus on how noise can be utilized to signify missingness, which
was a part of the inspiration for our uniform and gaussian noise baselines.
In , Shrikumar et al.
suggest a characteristic attribution technique known as deepLIFT. It assigns
significance scores to options by propagating scores from the output
of the mannequin again to the enter. Much like built-in gradients,
deepLIFT additionally defines significance scores relative to a baseline, which
they name the “reference”. Their paper has an prolonged dialogue on
why explaining relative to a baseline is significant. In addition they focus on
a couple of completely different baselines, together with “utilizing a blurred model of the unique
The listing of different associated strategies that we didn’t focus on
on this article goes on: SHAP and DeepSHAP
layer-wise relevance propagation ,
LIME ,
RISE and
amongst others. Many strategies for explaining machine studying fashions
outline some notion of baseline or missingness,
as a result of missingness and explanations are carefully associated. After we clarify
a mannequin, we frequently wish to know which options, when lacking, would most
change mannequin output. However so as to take action, we have to outline
what lacking means as a result of most machine studying fashions can’t
deal with arbitrary patterns of lacking inputs. This text
doesn’t focus on all the nuances offered alongside
every current technique, however you will need to word that these strategies have been
factors of inspiration for a bigger dialogue about missingness.
Source link | {"url":"https://thetimesofai.com/visualizing-the-impact-of-feature-attribution-baselines/","timestamp":"2024-11-08T22:11:47Z","content_type":"text/html","content_length":"154196","record_id":"<urn:uuid:1a0049f1-ea3d-433e-949d-5dcdfb2a097c>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00877.warc.gz"} |
Solve the equation and checkn-5/8
Solve the equation and checkn-5/8 +n+4/9 = 19/36
Solve the equation and checkn-5/8 +n+4/9 = 19/36
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
Solve the equation and check
n-5/8 +n+4/9 = 19/36
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[PDF] Foundations of Computer Science: C Edition - Free PDF Books
Home Engineering Books Computer Books ( CE & IT ) Foundations of Computer Science: C Edition
[PDF] Foundations of Computer Science: C Edition
Book Name: [PDF] Foundations of Computer Science: C Edition
Category: Computer Books ( CE & IT )
Free Download: Available
Foundations of Computer Science: C Edition
Foundations of Computer Science covers subjects that are often found split between a discrete mathematics course and a sophomore-level sequence in computer science in data structures. It has been our
intention to select the mathematical foundations with an eye toward what the computer user really needs, rather than what a mathematician might choose. We have tried to integrate effectively the
mathematical foundations with the computing. We thus hope to provide a better feel for the soul of computer science than might be found in a programming course, a discrete mathematics course, or a
course in a computer science subspecialty. We believe that, as time goes on, all scientists and engineers will take a foundational course similar to the one offered at Stanford upon which this book
is based. Such a course in computer science should become as standard as similar courses in calculus and physics.
Students taking courses based on this book have ranged from first-year undergraduates to graduate students. We assume only that students have had a solid course in programming. They should be
familiar with the programming language ANSI C to use this edition. In particular, we expect students to be comfortable with C constructs such as recursive functions, structures, pointers, and
operators involving pointers and structures such as dot.
Foundations of Computer Science: C Edition
Author(s): Alfred V. Aho, Jeffrey D. Ullman
Series: Principles of Computer Science Series
Publisher: W. H. Freeman, Year: 1995
ISBN: 0-7167-8284-7,9780716782841
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The Goode Condos | 111.4m | 33s | Graywood | a—A
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...here's hoping one day that a developer will go for neon green cladding.
...here's hoping one day that a developer will go for neon green cladding.
XCX condos? Brat lofts?
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Jul 16, 2011
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Afternoon pics…
Southeast corner…
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Podium bricks look good, although those shifting patterns of windows are a tad overused, especially in that neighbourhood.
They could be populating that eastern wing of the podium before even topping out the tower… I mean, they won’t- but they could… I mean, they can’t, but…
Podium bricks look good, although those shifting patterns of windows are a tad overused, especially in that neighbourhood.
But that's how architectural *razzle dazzle* is applied to developments in this city
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Apr 30, 2008
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I love the unique design of this building. Very well suited for the area
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What a difference hand-laid brick makes vs prefab panels. | {"url":"https://toronto.skyrisecities.com/forum/threads/toronto-the-goode-condos-111-4m-33s-graywood-a%E2%80%94a.27698/page-17#post-2132573","timestamp":"2024-11-02T02:48:04Z","content_type":"text/html","content_length":"149749","record_id":"<urn:uuid:2225549e-ea0e-4deb-bdf0-b23ab5246117>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00580.warc.gz"} |
Fractions in math
Fractions in math
Main subject: Math
related subject: Math
Duration: 30 minutes.
Age groups: Age 12
Submitted by: Lísbet Patrisía Gísladóttir. Iceland.
Learning objectives:
Learning objectives are: Lesson 3 in Fractions. The students review/learn how to make two fractions equal, to reduce and simplify fractions, to multiply fractions and writing fractions as decimals.
Implemented digital tools:
Device with internet connection. One for the teacher and one for each student or group. Digital tool is a website, http://rasmus.is/
Supported digital competence for student:
Communication and collaboration, Problem solving
Elaboration of the competences:
To find out if the students understand the material that has been teached and review the material.
The students gets their digital devices and go to the webside, http://rasmus.is/ and choose the language the teacher want them to work with. Then they press start and choose grade primary. The
teacher can print out ,,Your Checklist” for each student to write down the results. The students are going to work in pairs so the teacher has divided the class into pairs.
The students choose Fractions and open Lesson 3.
When the students have open Lesson 3 they read the introduction about: How to make two fractions equal, how to reduce and simplify fractions, how to multiply fractions and writing fractions as
decimals. When they are finished the students choose Quiz 3 and send their answers to get their results.
The students review their results, write it down and show it to the teacher. If there are some incorrect answers the students try to solve them with their teacher.
Implemented needed devices:
By Hans Snorasson|2021-09-01T21:46:51+00:0031/08/2021|Age 12, Age groups, Communication and collaboration, Main subject:, Math, Problem solving, Supported digital competence for student|Comments Off
on Fractions in math
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About the Author: Hans Snorasson
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Changing field from Economics to Statistics/Machine Learning
Hello Gradcafe. English is not my main language, so bare with me for a bit.
A little background about this post. I obtained my Bachelor in Economics last year and was enrolled to a PhD fast track of the same field in Germany. I'm thinking of dropping out of by PhD track and
take specialisation courses with the Compsci and Stats department to complete my Master in the 5th Semester (2.5 years) rather than the usual 2-years and then apply for PhD in Stats/ML. The following
are the reasons why I'm wishing to do this:
1.) In my first year in PhD track, I realised I want to do Statistics (specifically Bayesian nonparametrics) but because my department (medium sized compared to US) was going some massive disastrous
exodus; there aren't many relevant professors left now. So in my specialisation phase, I'm confined to mostly microeconomics courses which I have no interest in.
2.) Three weeks ago, I attended a Statistics summer school but was exposed to Machine Learning at the same time and thought I could venture into this field and consolidate Econometrics and ML
together. After reading Hal Varian's article and some googling on this.. I am convinced.
Now I know this would raised quite a some questions, so I will attempt to answer some of them here:
1.) Structure: So basically, PhD students must take at least 80% of their specialisation courses from the department (which I have no interest in). However, If I leave the track and join the Master
track, I can take whatever I want from any departments (even from a neighbouring university) since I already fulfilled the basic requirement for Master (sans Thesis).
2.) Financial. As a fast track student, we don't get any stipend until year 3 anyway, so I'm fine in terms of financial stability.
3.) Mathematical background: I'm quite worried about this. Coming from Bachelor in Economics, we have relatively little preparation in math. In my first year of PhD I have exposed to Analysis,
elementary Probability Theory and some mathematical methods from micro and macro, but I'm not as strong as a typical Math graduate. When I wrote to the relevant professors, they said the courses
might a difficult for me but they are willing to give me a try since I did quite well in the math/stats subjects as mentioned.
I don't have any jobs this summer so I intend to self-study the following subjects:
Real Analysis (using books by Terrence Tao and practice exercises in baby Rudin)
Intro Mathematics Statistics (book by Hogg et al/ Dudley)
Statistical Inference (book by Casella)
Theory of Probability (book by Durrett)
Machine Learning from Coursera
3rd Semester:
Intro PDE
Multivariate Analysis
Statistics II (Asymptotics of M-estimators, local asymptotic normal models; wont be able to take Stats I because the prof is doing it in German)
High-diemension Covariate Estimation (seminar)
4th Semester
Adv Analysis
Probability Theory (I feel like I should first do Adv Analysis (which involves Lesbegue integration) before doing any Probability Theory but I won't be able to do it if I want to graduate in 5th
Financial Time Series (seminar)
Stochastic Analysis OR Stochastic Differential Equation
(I also have TA responsibility in this semester for Economics Growth, but I heard it would be easy so I guess I'm ok)
Summer 2016
Internship + learn a new coding language.
5th Semester
Data Mining
Master Thesis
Apply for grad school (Dreaming about PhD Stats in Duke/ (Applied) Math in Bonn/ Study under Prof Teh in Oxford)
*the modules are all Master level courses.
Academic Results (for profile evaluation)
Undergraduate (from Russell groupin UK) BSc. Economics
Adv Mathematical Economics (A)
Applied Algebra (A)
Applied Calculus and PDE (A)
Applied ODE (B )
Mathematics for Engineers I & II (both A)
*I left out other low level Math courses (eg year 1 math)
PhD Courses (from top Econ school in Germany)
Mathematics for Economists (Basically Analysis) - A+
Adv Econometric 1 (introductory probability/measure theory) - A-
Adv Econometrics 2 and 3 (both B )
*Passed my first year Quals
**Grade conversions might be a little off.
Finally, my questions are:
1.) I would like to quit PhD Economics and obtain a Master of Economics (with 60% of my modules in Stats and Compsci(ML) courses). Do you think this is doable or I'm just crazy? (My colleagues say I
am, but I really want to do statistics rather than economics, and I don't think I can last another year doing Microeconomics or Industrial Organisation when my heart is not there anymore)
2.) What would you think of my 'Plans'. Too unrealistic?
3.) How would the Admission Committee see my profile? Favourable or unfavourable?
I would really appreciate your advice .Thanks
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Problem 4.7
Problem 4.7: A ball constrained to move on a rod
Please wait for the animation to completely load.
A 20-kg ball has a hole with a rod passing through. The rod exerts a force as needed that constrains the ball to move along the rod. An applied force is now added (the "pulling" force) so the ball is
pulled as shown (position is given in meters and time is given in seconds). The force vector is shown as a red arrow, and the force makes an angle θ with the horizontal. The velocity is given in
meters/second. You may adjust the angle and/or the magnitude of the pulling force (F < 7 N). Restart.
1. How does the acceleration change as you vary the pulling force for a constant angle?
2. How does the acceleration change as you vary the angle for a constant pulling force?
3. Combine your answers above to obtain a general mathematical formula for the acceleration of the ball due to an arbitrary applied force.
4. Determine the general mathematical formula for the normal force the rod exerts on the ball when an arbitrary force is applied to the ball.
Script authored by Steve Mellema, Chuck Niederriter and modified by Mario Belloni.
Problem authored by Mario Belloni.
Physlets were developed at Davidson College and converted from Java to JavaScript using the SwingJS system developed at St. Olaf College.
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CCNA Study Guide Chapter 01
We began this chapter with a look at the importance of network models, including the reasons for their modular nature. A look at the OSI model stressed the importance of understanding the concept of
layered communication, protocol data units, and the functions of each layer. Do not underestimate the importance of remembering not only the various functions of each, but also the protocols, data
units, services, and types of equipment found at each layer.
A look at the TCP/IP model provided a comparison with the OSI model, including the mappings between the layers of each. Examining the data encapsulation process helped to provide perspective on how a
real network protocol goes about preparing data for network transmission.
Finally, an overview of the Cisco network design model provided insight into Cisco’s perspective on the proper design of hierarchical networks. Be sure to understand not only the layers but also the
equipment and functional details of each.
Cisco’s Hierarchical Network Design Model
When it comes to network design, you’re pretty much left with two options – a flat design, or one that involves some type of hierarchy. A flat design can be very limiting in terms of performance and
scalability, and in all but the smallest networks would not be recommended. For example, on a flat network issues like broadcast traffic can quickly overwhelm network systems and negatively impact
performance. In contrast, a hierarchical design will allow for unique divisions of responsibility to be created on the network. Thus a higher degree of performance, reliability, scalability and
security can be achieved. The Cisco network design model is a reference model for creating hierarchical networks that attempts to account for these factors, while also providing an insight as to
where different network elements should be deployed and why.
The Cisco network design model consists of three layers. These include:
• The Core Layer
• The Distribution Layer
• The Access Layer
Figure: Cisco Hierarchical Network Design Model
Core Layer
The core layer describes what is often referred to as the network backbone. Its main responsibility is ensuring that data is passed at high speeds between different sites. Because of this high-speed
requirement, the backbone should usually make use of switching technologies instead of routing. While we’ll look at the differences between switching and routing in later chapters, for now it is
sufficient to say that switching is significantly faster than routing.
The core layer should also provide a high degree of reliability and fault tolerance. This is usually implemented using higher-end equipment and redundant links. For the most part, the core layer
should not be scaled to include additional equipment if performance is deteriorating. In such cases, backbone switches should be replaced with better performing models. By replacing equipment, the
core layer maintains a constant diameter, helping to avoid the introduction of additional latency.
As a general rule, anything that slows down performance should be kept away from the core layer. Beyond routing, this also means avoiding features such as access lists, firewall and intrusion
detection system (IDS) sensors – these inspect traffic based on network addresses and applications, and can negatively impact performance.
Network Data Encapsulation
The primary reason for looking at any network model is to better understand how systems communicate. In real-life, network communication requires that data be encapsulated by the sender, transmitted
over the network, and then de-encapsulated by the receiver. This is best illustrated by looking at what happens when one system running TCP/IP sends data to another. The list below outlines 5
simplified steps in a typical TCP/IP data transfer over an Ethernet network. Note that each layer considers whatever has been passed down to it from an upper layer as “data”. It doesn’t concern
itself with what was added by the upper layers.
1. Data is created by an application such an FTP client program. Let’s assume that a file transfer is being initiated with a local FTP server.
2. The data is passed to the Host-to-host (Transport) layer, where it is encapsulated to include source and destination port numbers. These uniquely identify the applications that the data should be
passed between. For example, if this data were being sent to an FTP server, the destination port would be TCP 21. The data is now considered to be a segment.
3. The data is passed to the Internet (Network) layer, where it is again encapsulated to include information such as the source and destination IP addresses. The data is now considered to be a
4. The data is passed down to the Network Interface (Data Link) layer, where it is encapsulated for Ethernet to include source and destination MAC addresses, as well as the an error-checking
mechanism known as a cyclic redundancy check (CRC). The data is now considered to be a frame.
5. The data is converted to a series of bits, and transmitted across the network.
Tip: A CRC is also often referred to as a Frame Check Sequence (FCS).
Figure: TCP/IP Data Encapsulation Process
Note that upon reaching the destination host, the entire process happens in reverse, with each layer de-encapsulating the data by striping away the information that was added at each layer.
Eventually, the required data is passed to the FTP server as intended by the FTP client application. Consider the frame captured below using Ethereal, a network protocol analyzer. Notice that each
heading area directly corresponds to the encapsulation process just defined (with the exception that the program shows the layers in reverse order).
Ethernet II
Internet Protocol, Src Addr: 192.168.0.1 (192.168.0.1), Dst Addr: 192.168.0.135 (192.168.0.135)
Transmission Control Protocol, Src Port: 4653 (4653), Dst Port: ftp (21), Seq: 2739356837, Ack: 204742999
File Transfer Protocol (FTP)
The 4-Layer TCP/IP Network Model
The Department of Defence TCP/IP model is a 4-layer model that defines areas of responsibility much like the OSI, while providing insight into the functions of the different protocols that make up
the TCP/IP suite. The model provides an excellent point of reference when compared to the OSI. We won’t look at all the details of the TCP/IP model just yet – the majority will be covered in Chapter
4. My feeling is that the data encapsulation process is much better explained using a popular protocol suite.
To begin, let’s take a look at how the TCP/IP model maps to the OSI model. While the names of the TCP/IP layers are different, they generally encompass the same responsibilities as one or more OSI
layers. Consider the diagram below.
Figure: Comparing the OSI and TCP/IP network models.
Tip: Although the layers of the TCP/IP model technically use different names, Cisco will still refer to protocols by their associated OSI layer name. For example, Cisco will describe TCP as being a
Transport layer protocol.
For the sake of illustration, I’ve included some of the key protocols that make up the TCP/IP suite in the figure below. Be aware that the terms data, segment, packet, and frame still apply as data
is encapsulated in the TCP/IP model.
Figure: TCP/IP protocol stack including common protocols and network technologies.
Physical Layer of the OSI Model
The Physical layer of the OSI model is concerned with the electrical, optical, and mechanical properties of the network, including elements such as voltage, media, connector types, signal
regeneration, and so forth. The physical layer doesn’t actually alter packets, but rather acts as the transmission facility over which the actual bits (1’s and 0’s) are sent. This isn’t limited to
plain old copper wire – it can include optical signals, radio waves, and infrared transmissions to name but a few. Examples of equipment found at the Physical layer include network cabling, hubs, and
repeaters. A number of popular Physical layer standards are listed below.
Examples of Physical layer standards:
• High Speed Serial Interface (HSSI): High speed serial communications
• EIA/TIA-232: Low speed serial transmissions
• V.35: ITU-T serial transmission standard
Data Link Layer of the OSI Model
The Data Link Layer of the OSI model acts as an interface between the Network and Physical layers. The main responsibilities of the Data Link layer include:
• Data framing and physical addressing. When data is passed to the Data Link layer, it is framed for transmission using various LAN and WAN protocols. This allows network protocols to be
transmitted over different network technologies including Ethernet, Token Ring, and Frame Relay as examples. Hardware or Media Access Control (MAC) addressing is used to uniquely identify hosts
at the Data Link layer. Since they make forwarding decisions based on MAC addresses, bridges and switches are examples of equipment found at this layer.
• Flow control, error checking, and frame sequencing. Data Link layer devices are capable of transmitting flow control codes that identify whether upper layer protocols are capable of receiving
data at the current rate. Error checking is provided in the form of a Cyclic Redundancy Check (CRC), a simple mathematical calculation performed on each frame to ensure it hasn’t been corrupted
in transit. Frame sequencing reorders frames that were received in a different order than they were sent.
Interacting with Network layer protocols. When a host receives a frame, the frame header contains information on which Network layer protocol the data will be passed to. The Data Link layer helps to
make network technologies independent of the upper layer protocols in use.
Examples of Data Link layer protocols:
• Ethernet (802.3): Contention-based LAN technology
• Token Ring (802.5): Token-passing LAN technology
• Wireless LAN (802.11): Wireless LANs
• Frame Relay: Packet-switched WAN technology
• ISDN: Digital dial-up connections
Tip: Remember that the protocol data unit (PDU) of the Data Link layer is referred to as a frame.
The Data Link layer is actually comprised of two sub-layers (defined by the Institute of Electronics and Electrical Engineers – the IEEE), called Media Access Control (MAC) and Logical Link Control
Network Layer of the OSI Model
The Network layer of the OSI model is commonly referred to as Layer 3, and has the following responsibilities:
• Routing. When a host on one network wishes to exchange data with a host on another, packets will be sent to a router interface. After determining where the packet should be forwarded next using
information found in its routing table, a router will switch the packets out of the optimal interface. This process will take place at each router encountered on a packet’s journey to the
destination host. Routing protocols are used to allow routers to exchange information with one another.
• Network Addressing. Each host on a routed internetwork will have at least one network address. A network address is made up of two parts – the first part identifies the network, while the second
identifies a unique host on that network. These addresses have different formats depending on the routed protocol in use – we’ll look at examples shortly.
Examples of Network-layer protocols:
Internet Protocol (IP): TCP/IP addressing and routing
Internetwork Packet Exchange (IPX): IPX/SPX addressing and routing
Internet Control Message Protocol (ICMP): Diagnostics and error notification
Internet Group Management Protocol (IGMP): Multicast group management
Tip: Remember that the protocol data unit (PDU) of the Network layer is referred to as a packet. In some cases, you may also see this PDU referred to as a datagram.
Transport Layer of the OSI Model
The Transport layer has three main responsibilities in terms of the exchange of data between systems. These include:
• Data segmentation.
• Establishment of end-to-end connections between hosts.
• Using flow-control mechanisms to ensure that data is sent at rates that the receiver can handle.
Data Segmentation
At any given point in time there may be many applications passing data down to the Transport layer. Data segmentation is the process by which the Transport layer uniquely handles all data passed to
and from different upper-level applications. This is usually implemented in the form of source and destination port numbers that are defined within a particular application. For example, if a user is
browsing the web and checking email at the same time, each program would be passing data and waiting for a reply on a unique port number. The Transport layer ensures that data is passed to the
correct application.
Session Layer of the OSI Model
The Session layer is responsible for the creation, management, and termination of sessions between systems. A session is best described as a type of managed connection between systems for the purpose
of a specific type of communication. For example, a session might be created for the purpose of user authentication, or to initiate a file transfer.
The Session layer is also responsible for coordinating how the communication between systems takes place, which is known as dialog control. In some sessions, only a single system is allowed to
communicate at any point in time, referred to as half-duplex. The Session layer would be responsible for determining whose turn it is in these situations, and for how long each system is allowed to
communicate. In other cases, both systems can communicate at once, which is also known as full duplex. If the communication stream were somehow interrupted, the Session layer would be responsible for
recognizing this and re-establishing the session.
Examples of Session layer protocols:
• Network File System (NFS): Unix file system access
• Structured Query Language (SQL): Local or remote database queries
• Remote Procedure Call (RPC): Client-server communication mechanism
• AppleTalk Session Protocol (ASP): AppleTalk client-server communication mechanism
• X Windows: Remote desktop sessions
Tip: Remember that the protocol data unit (PDU) of the Application, Presentation, and Session layers is “data”.
Presentation Layer of the OSI Model
The Presentation layer is primarily responsible for data representation and formatting, ensuring that data can be viewed correctly. These formats are sometimes referred to as the “data syntax” of the
applications in use. For example, different systems may use different schemes to represent data. While one system might use ASCII or EBCIDC, another might use UNICODE. Since these schemes contain
different character possibilities, it is the responsibility of the Presentation layer to make sure they are displayed in the correct or common format between the client and the server. Further to
this, the Presentation layer is also where data compression and encryption are generally considered to take place.
Examples of common Presentation layer formats:
• ASCII, EBCIDC, UNICODE, RTF: Text encoding formats
• MPEG, AVI, QuickTime: Video encoding formats
• JPEG, PNG, TIFF: Graphics formats
• MIDI: Sound format | {"url":"https://www.2000trainers.com/tutorials/cisco-ccna/","timestamp":"2024-11-08T23:32:39Z","content_type":"text/html","content_length":"52882","record_id":"<urn:uuid:18c43108-e6a4-4969-a7a1-110dfbdae4a6>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00585.warc.gz"} |
Wide Flange Beam with Unequal Flanges per. Roarks Formulas for Stress and Strain Equations and Calculator
Related Resources: calculators
Wide Flange Beam with Unequal Flanges per. Roarks Formulas for Stress and Strain Equations and Calculator
Wide Flange Beam with Unequal Flanges Section with Concentrated Intermediate Torque applied Deflection and Stress Equations and Calculator #7 of 1a Loading.
Formulas for the elastic deformations of uniform thin-walled open members under torsional loading.
Per. Roarks Formulas for Stress and Strain - Formulas for torsional properties and stresses in thin-walled open cross sections, Table 10.2.
Section Dimensional Definitions Figure 1
Loading Configuration (Section Shown May not Match Section given in Figure 1)
Left end free to twist and warp, right end free to warp but not twist.
Concentrated intermediate torque of Twin Channel With Flanges Inward Beam
Figure 2
Loading Declarations Image (Section Shown May not Match Section given in Figure 1)
Concentrated intermediate torque of Twin Channel With Flanges Inward Beam Orientation Declarations Image
Figure 3
Preview: Wide Flange With Unequal Flanges Thin Wall Stress and Torsional properties #7 calculator
Wide Flange Section Properties Constants Formulas See: Figure 1
Selected maximum values of stress and torsion
throughout the thickness at points A and B if
t[2]b[2]^2 > t[1]b[1]^2
throughout the thickness at points C and D if
t[2]b[2]^2 < t[1]b[1]^2
│Shear stress due to warping rigidity│
│throughout the thickness at a point │
│midway between A and B │
│t[2]b^2 > t[1]b[1] │
throughout the thickness at a point midway
between C and D if
t[2]b[2] < t[1]b[1]
│at the surface everywhere│
Shear stress due to torsional rigidity
Left end free to twist and warp, right end free to warp but not twist Formulas:
Boundary values for Loading condition See Figure 2
The following constants and functions are hereby defined in order to permit condensing the tabulated formulas which follow.
Concentrated intermediate torque See Figure 3
Where (when used in equations and this calculator):
Point 0 indicates the shear center se "Concentrated intermediate torque of Channel Beam Orientation Declarations image ";
e = distance from a reference to the shear center (in, m);
K = torsional stiffness constant (in^4, m^4);
C =warping constant (in^6, m^6);
τ[1] = shear stress due to torsional rigidity of the cross section (lbsf/in^2, m^2);
τ[2] = shear stress due to warping rigidity of the cross section (lbsf/in^2, m^2);
σ[x] = bending stress due to warping rigidity of the cross section (lbsf/in^2, m^2);
E = modulus of elasticity (lbs/in^2, m^2);
G = modulus of rigidity (shear modulus) of the material (lbs/in^2, m^2);
T[o] = applied torsional load (in-lbs, N-m);
t[o] = applied distributed torsional load (lbsf/in, N/m);
T[A] and T[B] are the reaction end torques at the left and right ends, respectively (in-lbs, N-m);
θ = angle of rotation at a distance x from the left end (radians).
θ', θ'', θ''', = are the successive derivatives of y with respect to the distance x;
C[w] = the warping constant for the cross section;
A, B, C, D, E, F, e, h, b1= Dimensional data Figure 1 (in, m).
All rotations, applied torsional loads, and reaction end torques are positive as shown (CCW when viewed from the right end of the member)
Unit step function defined by use of 〈〉
〈 x - θ 〉^0
if x < a, 〈 x - a 〉^n =0;
if x > a, ( x - a) ^n ;
The function sinh β〈 x - a 〉 is defined as zero if x < a
β = ( KG/C[w]E)^1/2
Supplemental selected special cases and maximum values (not included in calculator), See Figure 2.
Roarks Formulas for Stress and Strain, 7th Edition, Table 10.2 and 10.3 Formulas for torsional properties and stresses in thin-walled open cross sections. | {"url":"https://engineersedge.com/calculators/wide_flange_beam_with_unequal_flanges_15463.htm","timestamp":"2024-11-12T12:49:29Z","content_type":"text/html","content_length":"26062","record_id":"<urn:uuid:9d8283a1-3488-494e-ad12-ddd078c6300d>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00715.warc.gz"} |
Formally proving FizzBuzz for fun and profit, part 5.
Last time we created a formally proven version of FizzBuzz, but it had some problems. Our proofs weren’t strong enough to prevent us making the version below, where anything divisible by five returns
Buzz and everything else returns Normal.
wrongfizzbuzz: (x: Nat) -> FizzBuzzProofSimple x
wrongfizzbuzz x = case (divisbleby3 x, divisbleby5 x) of
(_, Just buzz_prf) => Buzz buzz_prf
(_, _) => Normal
The problem is that we create proofs that a number is divisible but not proofs that a number isn’t.
This is an important lesson. Just because you’ve proved something about your code, doesn’t mean you’ve proved what you need to. Proof types need to be crafted carefully so they actually prove what
you think they do.
Proving a negative
Idris has a type to represent something which can’t exist, Void.
data Void : Type where
Void has no constructors, so it is impossible to create values of the type. If a function returns Void then one of its arguments must also be impossible to construct. In other words, if we can create
a function which takes a single input and returns Void, then we have a proof that the input is impossible.
For FizzBuzz we want to be able to make proofs that a given number isn’t divisible by three or five. Something like:
DivisiblebyX 3 x -> Void
DivisiblebyX 5 x -> Void
Adding requirements for these negative proofs to our FizzBuzzProofSimple type gives:
data FizzBuzzProof: Nat -> Type where
Fizz: DivisiblebyX 3 x -> (DivisiblebyX 5 x -> Void) -> FizzBuzzProof x
Buzz: DivisiblebyX 5 x -> (DivisiblebyX 3 x -> Void) -> FizzBuzzProof x
FizzBuzz: DivisiblebyX 3 x -> DivisiblebyX 5 x -> FizzBuzzProof x
Normal: (x: Nat) -> (DivisiblebyX 3 x -> Void) ->(DivisiblebyX 5 x -> Void) -> FizzBuzzProof x
With the addition of these proofs it is impossible to use the wrong FizzBuzz variant for a number. Now we just need a way to make the proofs.
Will it divide?
We need to check each input number to see if it is divisible by three or five. Previously the divisibleby3 and divisibleby5 functions returned Nothing if the input didn’t divide, now we need to
return a negative proof instead. Returning either a proof something is true or a proof it isn’t, is common enough pattern Idris provides the Dec wrapper type:
data Dec : Type -> Type where
Yes : (prf: prop) -> Dec prop
No : (contra : prop -> Void) -> Dec prop
Dec has a Yes and a No constructor, created with an instance of the wrapped type or a proof that the type is impossible to construct values of.
Adding these new features to our old divisibleby3 function gives:
divisibleby3: (y: Nat) -> Dec (DivisiblebyX 3 y)
divisibleby3 Z = No zeroNotDivisible3
divisibleby3 (S Z) = No oneNotDivisible3
divisibleby3 (S (S Z)) = No twoNotDivisible3
divisibleby3 (S (S (S Z))) = Yes Base
divisibleby3 (S (S (S (S y)))) = case divisibleby3 (S y) of
Yes prf => Yes (Multiple prf)
No notdiv => No (threeLessNotDivisible3 notdiv)
Instead of a catchall clause for numbers which aren’t divisible we now have separate clauses for when the input number is:
• 0
• 1
• 2
• 3
• some number in the form (n+3)
Collectively these clauses cover all possible inputs (Idris will tell us if they don’t). If the input is divisible then the code is basically the same as before, each of the other clauses requires
their own proof function.
The proof functions for zero, one and two are almost identical and Idris will auto-generate most of the code.
zeroNotDivisible3 : DivisiblebyX 3 0 -> Void
zeroNotDivisible3 Base impossible
zeroNotDivisible3 (Multiple _) impossible
oneNotDivisible3 : DivisiblebyX 3 1 -> Void
oneNotDivisible3 Base impossible
oneNotDivisible3 (Multiple _) impossible
twoNotDivisible3 : DivisiblebyX 3 2 -> Void
twoNotDivisible3 Base impossible
twoNotDivisible3 (Multiple _) impossible
The function types are in the form for negative proofs we’ve seen above. To implement the function bodies we split the input argument into the two possible constructors, Base and Multiple. Idris can
then determine that inputs of the required form are impossible.
For example, in zeroNotDivisible3 the input argument has to have the type DivisiblebyX 3 0. The Base constructor requires the two type arguments to be the same, which 3 and 0 aren’t. The Multiple
constructor would require as an input an instance of DivisiblebyX 3 (0-3). Since natural numbers can’t be negative, this clause is also impossible.
threeLessNotDivisible3 is slightly different.
threeLessNotDivisible3 : (notdiv : DivisiblebyX 3 (S y) -> Void) -> DivisiblebyX 3 (3 + (S y)) -> Void
threeLessNotDivisible3 notdiv (Multiple x) = notdiv x
It is a proof that, given a non-zero number y is not divisible by three, y+3 is also not divisible by three. We split DivisiblebyX 3 (3 + (S y)) into the only applicable constructor, Multiple, which
gives us a value of DivisiblebyX 3 (S y), calling notdiv with this value returns Void which completes the functions.
divisibleby5 is the same structure, with extra clauses for values of three and four.
A formally proven FizzBuzz (at last)
We now have functions for creating proofs that a number is or isn’t divisible by three or five. Plugging these functions into our fizzbuzz function gives:
fizzbuzz: (x: Nat) -> FizzBuzzProof x
fizzbuzz x = case (divisibleby3 x, divisibleby5 x) of
(No not_fizz, No not_buzz) => Normal x not_fizz not_buzz
(Yes fizz_prf, Yes buzz_prf) => FizzBuzz fizz_prf buzz_prf
(No not_fizz, Yes buzz_prf) => Buzz buzz_prf not_fizz
(Yes fizz_prf, No not_buzz) => Fizz fizz_prf not_buzz
Unlike our last attempt at FizzBuzz, there is now no way to return the wrong FizzBuzz variant.
The end?
We have a function for generating FizzBuzz for any single number, but FizzBuzz actually needs to be generated for all numbers between one and one hundred. Next time we’ll look at how to do this in | {"url":"https://magpieengineering.com/formally-proving-fizzbuzz-for-fun-and-profit-part-5/","timestamp":"2024-11-11T07:15:37Z","content_type":"text/html","content_length":"47092","record_id":"<urn:uuid:427c61ce-cd94-4e18-bd74-335472a60a37>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00167.warc.gz"} |
ManPag.es -
ssyswapr.f −
subroutine SSYSWAPR (UPLO, N, A, LDA, I1, I2)
SSYSWAPR applies an elementary permutation on the rows and columns of a symmetric matrix.
Function/Subroutine Documentation
subroutine SSYSWAPR (characterUPLO, integerN, real, dimension( lda, n )A, integerLDA, integerI1, integerI2)
SSYSWAPR applies an elementary permutation on the rows and columns of a symmetric matrix.
SSYSWAPR applies an elementary permutation on the rows and the columns of
a symmetric matrix.
UPLO is CHARACTER*1
Specifies whether the details of the factorization are stored
as an upper or lower triangular matrix.
= ’U’: Upper triangular, form is A = U*D*U**T;
= ’L’: Lower triangular, form is A = L*D*L**T.
N is INTEGER
The order of the matrix A. N >= 0.
A is REAL array, dimension (LDA,N)
On entry, the NB diagonal matrix D and the multipliers
used to obtain the factor U or L as computed by SSYTRF.
On exit, if INFO = 0, the (symmetric) inverse of the original
matrix. If UPLO = ’U’, the upper triangular part of the
inverse is formed and the part of A below the diagonal is not
referenced; if UPLO = ’L’ the lower triangular part of the
inverse is formed and the part of A above the diagonal is
not referenced.
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
I1 is INTEGER
Index of the first row to swap
I2 is INTEGER
Index of the second row to swap
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
September 2012
Definition at line 103 of file ssyswapr.f.
Generated automatically by Doxygen for LAPACK from the source code. | {"url":"https://manpag.es/SUSE131/3+ssyswapr.f","timestamp":"2024-11-07T04:03:15Z","content_type":"text/html","content_length":"20046","record_id":"<urn:uuid:a5de3dd0-2b3a-46e6-a04f-65bf3e50ad78>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00433.warc.gz"} |
Gareth R. Pearce
Gareth R. Pearce, MA BA
Gareth R. Pearce, MA BA
Præ Doc
Gareth R. Pearce
Department of Philosophy
University of Vienna
Universitätsstraße 7 (NIG)
1010 Vienna
Room: C 0211 (NIG)
Phone: +43-1-4277-46073
Mail: gareth.pearce@univie.ac.at
Areas of Specialization
Philosophy of Mathematics, Philosophy of Logic and Metaphysics.
Wider AOI: Epistemology & Epistemic Normativity, Philosophy of Language, Philosophy of Science, History of Analytic Philosophy, and Mathematical or Formal approaches to Philosophy.
Thesis Project: Essays on the Foundations of Axiom and Logic Selection
My thesis is a collection of seven (six and a half, one of them is short) papers on the philosophy of logic and philosophy of mathematics. The overarching theme connecting them is the question "How
should we pick our axioms and logics?"
Three of these papers are on the philosophy of logic where I defend a particular (Neo)Carnapian view of logical pluralism. On this view logics are correct to the extent that they endorse the valid
(truth perserving) inferences. But truth is a language-relative notion, given a particular Carnapian/Tarskian view of truth. So logical correctness is language relative, and there are lots of logics
true for some language. The three papers in this section are:
• Language, Truth and Logics: A Neo-Carnapian Account of Logical Pluralism. A survey paper outlining the position and defending it from well known objections.
• Epistemic Normativitiy for Non-Classical Truth. This paper outlines how and when truth is epistemically valuable, given the range of possible non-classical languages. My proposed solution is to
take information, not truth, to be of fundamental epistemic value. Non-classical truth is valuable only if its true (or designated) propositions provide information about the world.
• Metaphysical Realism with Logical Pluralism. Dummet believed that the principle of bivalence is constitutive of realism. Tahko has recently argued that metaphysical realism entails logical
"realism" (monism). This paper argues that this is a mistake. A (Neo)carnapian logical pluralist can be a metaphysical realist as logical correctness depends on facts about one's language, not
about the world.
Three of these papers are on topics in the Philosophy of Mathematics. This section of my thesis is a little more speculative. Whilst there has been a great deal of interest in the philosophy of
mathematics about which axiom system is correct (or best, preferable, etc; depending on what honourific one wishes to use), there has been comparably little work outlining what conditions an axiom
system needs to satisfy in order to be correct. Most often, philosophers have left their theory of axiom selection implicit, and those who make it explicit infrequently defend the criteria they
apply. These three papers aim to address this gap.
• On Axiom Selection. This paper argues for the claim that philosophers of maths should do more work explicitly outlining their theory of axiom selection and defending its criteria. It does this by
setting out a taxonomy of possible views of axiom selection and arguing that there's a non-trivial debate to be had between them.
• Mathematics doesn't need a (philosophical) Foundation. This paper argues that mathematics doesn't need a foundation, in the philosophical sense of the term. That's to say that mathematics, as a
whole, does not need to select a single axiom system that acts as its epistemic, ontic and conceptual foundation.
• A Nominalist Theory of Axiom Selection. This paper outlines a possible nominalist view of axiom selection. The paper builds on an idea outlined in "On Axiom Selection" to present a possible
nominalist theory of axiom selection. The idea is that mathematics, as a scientific institution, has a unique role that it ought play in the wider scientific community. This generates a
socio-epistemic normative framework in which to evaluate mathematical practice. One possible object of evaluation is axiom selection. Certain axiom systems might do more or less work in
contributing towards the socio-epistemic role of mathematics. In virtue of that, they would be better or worse axiom systems.
Lastly, I have one paper on epistemic normativity. It's a short paper (less than 2000 words) entitled The Epistemic Value of Precision. The paper provides an independent argument for the view of
epistemic normativity outlined in Epistemic Normativity for Non-Classical Truth. This paper was really just a bit of fun, but is relevant so makes it into the thesis!
I have also recently published my first paper A Nominalist Alternative to Reference by Abstraction in the Theora special issue on Linnebo's 2018 book "Thin Objects. It can be found here and is open
Supervisors: Georg Schiemer, Esther Ramharter
• Current: Wissenschaftlisches Arbeiten (Eng: Research Methods in Philosophy - though the translation is inexact)
• W21 & W20: Exercise course in Logic
• TA, with Iulian Toader: two courses on the Philosophy of Quantum Mechanics
Selection of Upcoming & Past Talks
• European Early Career Philosophers Workshop Suspended due to Covid-19: "Grounding Mathematical Normativity"
• Society for the Study of the History of Analytic Philosophy conference 2020: "Why Formalism died too early and why Lewis should have brought it back"
• Tilburg History of Analytic Philosophy Workshop 2020: "Why Formalism died too early and why Lewis should have brought it back"
• OZSW Graduate Conference 2020: "What does it take for some Axioms to be a Foundation for Mathematics?" | {"url":"https://fonti.univie.ac.at/fonti-team/gareth-r-pearce/","timestamp":"2024-11-04T18:40:18Z","content_type":"application/xhtml+xml","content_length":"52101","record_id":"<urn:uuid:cb41231e-bfa5-4324-830f-9d14beb3db2c>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00643.warc.gz"} |
Harsanyi's simple “proof” of utilitarianism — EA Forum
In 1955, John Harsanyi published a paper demonstrating that anyone who follows certain reasonable assumptions must be a total utilitarian. The paper is somewhat technical, but the result is
relatively easy to understand. I’ve been unable to find a non-technical summary of this result and so, because it is one of the more compelling arguments for utilitarianism, I decided to write one
Suppose a group of friends are deciding where to eat. Each individual person has some preference (say, one person most prefers Chinese, then Italian, then Japanese; another prefers Italian, then
Chinese, then Japanese) but there is no clear restaurant which everyone thinks is best. How should they choose a place?
One solution is to have each person attach a numeric score to how much they would enjoy a given restaurant. If you really like Chinese food, then maybe you give it 10 points; if you’re lukewarm then
you give it 2, and if you really hate Chinese then maybe it’s -5.
Once each person has voted, you simply add up all the scores, and then the group goes to whichever restaurant had the highest total score.
This method is (a simplified form of) “total” utilitarianism, and Harsanyi demonstrated that it is the only “reasonable” way that groups can make a decision.
Concretely, assume:
1. Each individual in the group is rational (for a commonly used but technical definition of “rational”, hereafter referred to as “VNM-rational”)^[1]^[2]
2. The group as a whole is VNM-rational^[3]^[4]
3. If every individual in the group is indifferent between two options, then the group as a whole is indifferent between those two options
The theorem proves that total utilitarianism is the only method which satisfies these three assumptions.
Note that this theorem just demonstrates that, if there is some way of saying that certain things are better or worse for individuals, then the way to determine whether those things are better or
worse for groups is to add up how good it is for the individuals in those groups. It doesn't say anything about the way in which things can be better or worse for individuals. I.e. you could be
adding up each individual's happiness (hedonistic utilitarianism), something related to their preferences (preference utilitarianism), or something more exotic.
The above is somewhat abstract, so here is a concrete example demonstrating why anything other than total utilitarianism fails these axioms. (This is my best attempt at creating a simple example;
perhaps others in the comments can create even simpler ones.)
Consider a population consisting of 2 people. Because they are VNM-rational, they have utility functions, and therefore we can represent states of the world as a vector of numbers. E.g. the vector is
a world in which the first person has utility 5 and the second has utility 7.
Let’s prove that the world must be as good as the world .
Consider a lottery in which there is a one-half chance we end up with the world and a one-half chance that we end up with the world . Because we are indifferent between who has the 2 and who has the
0,^[5] and the group is an expected utility maximizer, these are equally valuable:^[6]
We can we write this from the perspective of each individual in society:
Because VNM-rational agents are expected utility maximizers we can just multiply the probabilities through:^[7]
The key insight here is that each individual is indifferent between the “50% chance of 2, 50% chance of 0” and “guaranteed chance of 1” lotteries (on account of being VNM-rational). Because each
individual is indifferent, the group is also forced to be indifferent (on account of the third assumption).
Total utilitarianism is a fairly controversial position. The above example where can be extended to show that utilitarianism is extremely demanding, potentially requiring extreme sacrifices and
It is therefore interesting that it is the only decision procedure which does not violate one of these seemingly reasonable assumptions.
While not conclusive, this theorem provides a compelling argument for total utilitarianism.
Appendix on Equality
Harsanyi’s original theorem allowed for weighted total utilitarianism. (I.e. everyone gets a vote, but some people’s votes count more than others.)
It’s easy enough to add an assumption like “also everyone is equal” to force true total utilitarianism, but interestingly Harsanyi didn’t think that was necessary:
This implies, however, without any additional ethical postulates that an individual’s impersonal preferences, if they are rational, must satisfy Marschak’s axioms [equivalent to VNM-rationality]
and consequently must define a cardinal social welfare function equal to the arithmetical mean of the utilities of all individuals in the society (the arithmetical mean of all individual
utilities gives the actuarial value of his uncertain prospects, defined by an equal probability of being put in the place of any individual in the situation chosen). [Emphasis added]
In other words, he thinks it would be irrational to weight people unevenly, because equal weighting is the expected utility-maximizing choice if you don’t know which person in society you will
This idea of making decisions behind a veil of ignorance where you don’t know which person in society you will become was later popularized by John Rawls, who used it to argue for his Minimax
decision rule.
It is, in my humble opinion, unfortunate that the veil of ignorance has become associated with Rawls, when Harsanyi’s utilitarian formulation has a much more rigorous mathematical grounding. (And was
also published earlier.)
I would like to thank Aaron Gertler, Sam Deere, Caitlin Elizondo and the CEA UK office staff for comments on drafts of this post and discussions about related ideas.
1. Harsanyi used Marschak’s axioms, which are mathematically equivalent to the VNM ones, but less popular. I'm using VNM here just because they seem better known. ↩︎
2. "Rational" is a somewhat unfortunate term, but I'm sticking with it because it's standard. These axioms are intended to prevent things like "Ben likes apples more than bananas but also likes
bananas more than apples." It's not intended to prevent "irrational" value judgments like enjoying Nickelback's music. A better term might be something like "consistent". ↩︎
3. It’s a well-known consequence of this assumption that the group must be “utilitarian” in the sense that it has a utility function. The surprising part of Harsanyi’s theorem is not that there is a
utility function but rather that the utility function must be a linear addition of its constituents’ utility functions (as opposed to, say, their average or the sum of their logarithms or
something completely disconnected from its constituents' utility.). ↩︎
4. An example of what it means for a group decision to be VNM-rational: if the group somehow aggregates its preferences (through voting or reading entrails or whatever) and decides that Chinese is
preferable to Italian, and also that Italian is preferable to Japanese, then the group must also conclude that Chinese is preferable to Japanese. We don’t care how it’s aggregating its
preferences, but it must do so in a “rational” way. ↩︎
5. Note that this isn't clearly implied by the assumptions – see the appendix on equality. Harsanyi's original proof does not require any assumptions about equality, but this sort of assumption
makes the proof much simpler and seems unlikely to be a point of controversy, so I'm including it. ↩︎
6. More precisely: for some group utility . Because of the VNM axioms, . (Normalizing .) Therefore, . I’m still skipping some steps; people interested in a more rigorous proof should see his
original paper. ↩︎
7. More precisely: each individual is indifferent between a lottery where they are guaranteed 1 utility versus having a 50% chance of 2, 50% chance of 0. Since each individual is different between
these, the group is also indifferent. ↩︎
I think this last point essentially denies the third axiom above, which is what connects individual vNM utility and social/ethical preferences. (The original statement of the second axiom is just vNM
rationality for social/ethical preferences, and has no relationship with the individuals' preferences.)
mako yass 1
VNM Utility is the thing that people actually pursue and care about. If wellbeing is distinct from that, then wellbeing is the wrong thing for society to be optimizing. I think this actually is the
case. Harsanyi, and myself, are preference utilitarians. Singer and Parfit seem to be something else. I believe they were wrong about something quite foundational. Writing about this properly is
extremely difficult and I can understand why no one has done it and I don't know when I'll ever get around to it.
I just want to second the point that some others have made that it seems more accurate to say only that Harsanyi's result supports utilitarianism (rather than total utilitarianism). Adding the word
"total" suggests that the result rules out other version of utilitarianism (e.g. average, critical-level and critical-range utilitarianism), which as you point out is not correct. More generally, I
think "utilitarianism" (without the "total") nicely signals that Harsanyi's result concerns fixed-population settings.
It is also worth noting that Harsanyi himself accepted average utilitarianism rather than total utilitarianism in variable-population settings (see the letter exchange between him and Yew-Kwang Ng
reported in the appendix of Ng, Y. K. (1983). Some broader issues of social choice. In Contributions to Economic Analysis (Vol. 145, pp. 151-173). Elsevier.).
Anyway, thanks for this post!
[Edited comment to remove grammatical error]
FWIW, if you extend the rationality axioms to prospects (probability distributions) with infinitely many possible outcomes in their natural ways, Harsanyi's theorem + separability leads to
contradiction. In general, unbounded utility functions violate extensions of standard rationality axioms to prospects with infinitely many possible outcomes, and these extensions can be motivated
pretty much the same ways as the versions here, in the vNM utility theorem and Savage's theorem. See my post here.
Matthew_Barnett 13
I have a strongly negative bias against any attempt to ground normative theories in abstract mathematical theories, such as game theory and decision theory. The way I see it, the two central claims
of utilitarianism are the axiological claim (well-being is what matters) and the maximizing claim (we should maximize what matters ie. well-being). This argument provides no reason to ground our
axiology in well-being, and also provides no reason that we should be maximizers.
In general, there is a significant difference between normative claims, like total utilitarianism, and factual claims, like "As a group, VNM rational agents will do X."
You're right; I meant to refer to the violation of individual rationality. Thanks!
Lukas_Gloor 6
Ah, my mistake – I had heard this definition before, which seems slightly different.
Probably I was wrong here. After reading this abstract, I realize that the way Norcross wrote about it is compatible with a weaker claim that linear aggregation of utility too. I think I just assumed
that he must mean linear aggregation of utility, because everything else would seem weirdly arbitrary. :)
I changed it to this – curious if you still find it jarring?
Less so! The "total" still indicates the same conclusion I thought would be jumping the gun a bit, but if that's your takeaway it's certainly fine to leave it. Personally I would just write
"utilitarianism" instead of "total utilitarianism."
I used preferences about restaurants as an example because that seemed like something people can relate to easily, but that's just an example. The theorem is compatible with hedonic
utilitarianism. (In that case, the theorem would just prove that the group's utility function is the sum of each individual's happiness.)
In this case, I think it's harder to argue that we should care about ex ante expected individual hedonistic utility and for the 1st and 3rd axioms, because we had rationality based on preferences and
something like Pareto to support these axioms before, but we could now just be concerned with the distribution of hedonistic utility in the universe, which leaves room for prioritarianism and
egalitarianism. I think the only "non-paternalistic" and possibly objective way to aggregate hedonistic utility within an individual (over their life and/or over uncertainty) would be to start from
individual preferences/attitudes/desires but just ignore concerns not about hedonism and non-hedonistic preferences, i.e. an externalist account of hedonism. Roger Crisp defends internalism in
"Hedonism Reconsidered", and defines the two terms this way:
Two types of theory of enjoyment are outlined-internalism, according to which enjoyment has some special ’feeling tone’, and externalism, according to which enjoyment is any kind of experience to
which we take some special attitude, such as that of desire.
Otherwise, I don't think there's any reason to believe there's an objective common cardinal scale for suffering and pleasure, even if there were a scale for suffering and a separate scale for
pleasure. Suffering and pleasure don't use exactly the same parts of the brain, and suffering isn't just an "opposite" pattern to pleasure. Relying on mixed states, observing judgements when both
suffering and pleasure are happening at the same time might seem promising, but these judgements happen at a higher level and probably wouldn't be consistent between people, e.g. you could have two
people with exactly the same suffering and pleasure subsystems, but with different aggregating systems.
I'm personally more sympathetic to externalism. With antifrustrationism (there are actually arguments for antifrustrationism; see also my comment here), externalism leads to a negative hedonistic
view (which I discuss further here).
Why should morality be based on group decision-making principles? Why should I care about VNM rationality of the group?
It doesn't have to be the group, it can be an impartial observer with their own social welfare function, as long as it is increasing with individual expected utility, i.e. satisfies ex ante Pareto.
Actually, that's how it was originally stated.
EDIT: woops, condition 2 is weaker than ex ante Pareto; it's just vNM rationality with respect to outcomes for social/ethical preferences/views. It's condition 3 that connects individual vNM utility
and social/ethical vNM utility.
[This comment is no longer endorsed by its author]Reply
Why should morality be based on group decision-making principles? Why should I care about VNM rationality of the group?
I've retracted my previous reply. The original 2nd condition is different from ex ante Pareto; it's just vNM rationality with respect to outcomes for social/ethical preferences/views and it says
nothing about the relationship between individual preferences and social/ethical ones. It's condition 3 that connects individual vNM utility and social/ethical vNM utility.
tuukkasarvi 12
I am not an expert in this topic but I believe this recent paper is relevant and may derive a result that is more general than Harsanyi-style utilitarianism https://www.sciencedirect.com/science/
Yeah, it doesn't (obviously) follow. See the appendix on equality. It made the proof simpler and I thought most readers would not find it objectionable, but if you have a suggestion for an alternate
simple proof I would love to hear it!
This makes sense, but the type of things that tend to convince me to believe in an ethical theory generally depend a lot on how much I resonate with the main claims of the theory. When I look at
the premises in this theorem, none of them seem to be type of things that I care about.
If you want to deal with moral uncertainty with credences, you could assign each of the 3 major assumptions an independent credence of 50%, so this argument would tell you should be utilitarian with
credence at least 123=18=12.5%. (Assigning independent credences might not actually make sense, in case you have to deal with contradictions with other assumptions.)
On the other hand, pointing out that utilitarians care about people and animals, and they want them to be as happy as possible (and free, or with agency, desire satisfaction) that makes me happy
to endorse the theory. When I think about all people and animals being happy and free from pain in a utilitarian world, I get a positive feeling.
Makes sense. For what it's worth, this seems basically compatible with any theory which satisfies the Pareto principle, and I'd imagine you'd also want it to be impartial (symmetry). If you also
assume real-valued utilities, transitivity, independence of irrelevant alternatives, continuity and independence of unconcerned agents, you get something like utilitarianism again. In my view,
independence of unconcerned agents is doing most of the work here, though.
RomeoStevens 3
> there seems to be no way to determine what equal weights should look like, without settling on a way to normalize utility functions, e.g., by range normalization or variance normalization. I think
the debate about intersubjective utility comparisons comes in at the point where you ask how to normalize utility functions.
yup, thanks. Also across time as well as across agents at a particular moment.
As a fan of Nickelback, I really appreciate fn2.
I want to point out that both assumptions 2, and 1 and 3 together have been objected to by academic philosophers.
Assumption 2 is ex post consequentialism: maximize the expected value of a social welfare function. Ex ante prioriatarianism/egalitarianism means rejecting 2: we should be fair to individuals with
respect to their expected utilities, even if this means overall worse expected outcomes. This is, of course, vNM irrational, but Diamond defended it (and see my other comment here). Essentially, even
if two outcomes are equally valuable, a probabilistic mixture of them can be more valuable because it gives people fairer chances; this is equality of opportunity. This contradicts the independence
axiom specifically for vNM rationality (and so does the Allais paradox).
Assumptions 1 and 3 together are basically a weaker version of ex ante Pareto, according to which it's (also) better to increase the expected utility of any individual(s) if it comes at no expected
cost to any other individuals. Ex post prioritarianism/egalitarianism means rejecting the conjunction of 1 and 3, and ex ante Pareto: we should be more fair to individuals ex post (we want more fair
actual outcomes after they're determined), even if this means worse individual expected outcomes.
There was a whole issue of Utilitas devoted to prioritarianism and egalitarianism in 2012, and, notably, Parfit defended prioritarianism in it, arguing against ex ante Pareto (and hence the
conjunction of 1 and 3):
When Rawls and Harsanyi appeal to their versions of Veil of Ignorance Contractualism, they claim that the Equal Chance Formula supports the Utilitarian Average Principle, which requires us to act
in ways that would maximize average utility, by producing the greatest sum of expectable benefits per person. This is the principle whose choice would be rational, in self-interested terms, for
people who have equal chances of being in anyone’s position.
We can plausibly reject this argument, because we can reject this version of contractualism. As Rawls points out, Utilitarianism is, roughly, self-interested rationality plus impartiality. If we
appeal to the choices that would be rational, in self-interested terms, if we were behind some veil of ignorance that made us impartial, we would expect to reach conclusions that are, or are
close to being, Utilitarian. But this argument cannot do much to support Utilitarianism, because this argument’s premises are too close to these conclusions. Suppose that I act in a way that
imposes some great burden on you, because this act would give small benefits to many other people who are much better off than you. If you object to my act, I might appeal to the Equal Chance
Formula. I might claim that, if you had equal chances of being in anyone’s position, you could have rationally chosen that everyone follows the Utilitarian Principle, because this choice would
have maximized your expectable benefits. As Scanlon and others argue, this would not be a good enough reply.9 You could object that, when we ask whether some act would be wrong, we are not asking
a question about rational self-interested choice behind a veil of ignorance. Acts can be wrong in other ways, and for other reasons.
He claimed that we can reject ex ante Pareto ("Probabilistic Principle of Personal Good"), in favour of ex post prioritarianism/egalitarianism:
Even if one of two possible acts would be expectably worse for people, this act may actually be better for these people. We may also know that this act would be better for these people if they
are worse off. This fact may be enough to make this act what we ought to do.
Here, by "worse off" in the second sentence, he meant in a prioritarian/egalitarian way. The act is actually better for them, because the worse off people under this act are better off than the worse
off people under the other act. He continued:
We can now add that, like the Equal Chance Version of Veil of Ignorance Contractualism, this Probabilistic Principle has a built-in bias towards Utilitarian conclusions, and can therefore be
rejected in similar ways. According to Prioritarians, we have reasons to benefit people which are stronger the worse off these people are. According to Egalitarians, we have reasons to reduce
rather than increase inequality between people. The Probabilistic Principle assumes that we have no such reasons. If we appeal to what would be expectably better for people, that is like
appealing to the choices that it would be rational for people to make, for self-interested reasons, if they had equal chances of being in anyone’s position. Since this principle appeals only to
self-interested or prudential reasons, it ignores the possibility that we may have impartial reasons, such as reasons to reduce inequality, or reasons to benefit people which are stronger the
worse off these people are. We can object that we do have such reasons.
When Rabinowicz pointed out that, in cases like Four, Prioritarians must reject the Probabilistic Principle of Personal Good, he did not regard this fact as counting against the Priority View.
That, I believe, was the right response. Rabinowicz could have added that similar claims apply to Egalitarians, and to cases like Two and Three.
Just as a side note, Harsanyi's result is not directly applicable to a formal setup involving subjective uncertainty, such as Savage's or the Jeffrey-Bolker framework underlying evidential and causal
decision theory. Though there are results for the Savage setup too, e.g., https://www.jstor.org/stable/10.1086/421173, and Caspar Oesterheld and I are working on a similar result for the Jeffrey
Bolker framework. In this setup, to get useful results, the indifference Axiom can only be applied to a restricted class of propositions where everyone agrees on beliefs.
Some discussion here, too.
Yeah, my point was that ex-ante utility was valued equally, but I think that was confusing. I'm just going to remove that section. Thanks!
Concretely, assume:
1. Each individual in the group is rational (for a commonly used but technical definition of “rational”, hereafter referred to as “VNM-rational”)[1][2]
2. The group as a whole is VNM-rational[3][4]
3. If every individual in the group is indifferent between two options, then the group as a whole is indifferent between those two options
One way of motivating 3 is by claiming (in the idealistic case where everyone's subjective probabilities match, including the probabilities that go with the ethical ranking):
a. Individual vNM utilities track welfare and what's better for individuals, and not having it do so is paternalistic. We should trust people's preferences when they're rational since they know
what's best for themselves.
b. When everyone's preferences align, we should trust their preferences, and again, not doing so is paternalistic, since it would (in principle) lead to choices that are dispreferred by everyone, and
so worse for everyone, according to a.*
As cole_haus mentioned, a could actually be false, and a motivates b, so we'd have no reason to believe b either if a were false. However, if we use some other real-valued conception of welfare and
claim what's good for individuals is maximizing its expectation, then we could make an argument similar to b (replacing "dispreferred by everyone" with "worse in expectation for each individual") to
defend the following condition, which recovers the theorem:
3'. If for two options and for each individual in the options, their expected welfare is the same in the two options, then we should be ethically indifferent between the options.
*As alluded to here, if your ethical ranking of choices broke one of these ties so A≻B, it would do so with a real number-valued difference, and by the continuity axiom, you could probabilistically
mix the choice A you broke the tie in favour of with any choice C that's worse to everyone than the other choice B, and this could be made better than B according to your ethical ranking, i.e. pA+(1−
p)C≻B for any p∈(0,1) close enough to 1, while everyone has the opposite preference over these two choices.
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cole_haus 37
Thanks for writing this up!
For those interested in more info:
• Harsanyi had two different theorems like this (his aggregation theorem and his impartial observer theorem) which rely on slightly different assumptions.
• The main arguments against Harsanyi's theorems were made by prominent economist Amartya Sen in what has become known as the "Harsanyi-Sen debate" or "Harsanyi-Sen-Weymark debate" (searchable
terms). The gist of the counterargument is that "while Harsanyi has perhaps shown that overall good is a linear sum of individuals’ von Neumann–Morgenstern utilities, he has done nothing to
establish any connection between the notion of von Neumann–Morgenstern utility and that of well-being, and hence that utilitarianism does not follow.".
MichaelStJules 32
Thanks for writing this!
I don't think the theorem provides support for total utilitarianism, specifically, unless you add extra assumptions about how to deal with populations of different sizes or different populations
generally. Average utilitarianism is still consistent with it, for example. Furthermore, if you don't count the interests of people who exist until after they exist or unless they come to exist, it
probably won't look like total utilitarianism, although it gets more complicated.
You might be interested in Teruji Thomas' paper "The Asymmetry, Uncertainty, and the Long Term" (EA Forum post here), which proves a similar result from slightly different premises, but is compatible
with all of 1) ex post prioritarianism, 2) mere addition, 3) the procreation asymmetry, 4) avoiding the repugnant conclusion and 5) avoiding antinatalism, and all five of these all at the same time,
because it sacrifices the independence of irrelevant alternatives (the claim that how you rank choices should not depend on what choices are available to you, not the vNM axiom). Thomas proposes
beatpath voting to choose actions. Christopher Meacham's "Person-affecting views and saturating counterpart relations" also provides an additive calculus which "solves the Non-Identity Problem,
avoids the Repugnant and Absurd Conclusions, and solves the Mere-Addition Paradox" and satisfies the asymmetry, also by giving up the independence of irrelevant alternatives, but hasn't, as far as I
know, been extended to deal with uncertainty.
I've also written about ex ante prioritarianism in the comments on the EA Forum post about Thomas' paper, and in my own post here (with useful feedback in the comments).
Ben_West🔸 4
I don't think the theorem provides support for total utilitarianism, specifically, unless you add extra assumptions about how to deal with populations of different sizes or different populations
generally. Average utilitarianism is still consistent with it, for example.
Well, average utilitarianism is consistent with the result because it gives the same answer as total utilitarianism (for a fixed population size). The vast majority of utility functions one can
imagine (including ones also based on the original position like maximin) are ruled out by the result. I agree that the technical result is "anything isomorphic to total utilitarianism" though.
You might be interested in Teruji Thomas' paper
I had not seen that, thanks!
As stated in another comment, you have proved any ethical theory that is identical to total utilitarianism with fixed population sizes (e.g average utilitarianism).
But, you can use separability to rule out non-total versions of utilitarianism.
Separability is roughly the principle that, in comparing the value of two outcomes, one can ignore any people whose existence and welfare are unaffected.
Non-total versions of utilitarianism violate separability because they imply that the value of creating someone depends on the population or wellbeing of unaffected beings.
Tobias_Baumann 13
Thanks for writing this up! I agree that this result is interesting, but I find it unpersuasive as a normative argument. Why should morality be based on group decision-making principles? Why should I
care about VNM rationality of the group?
Also, you suggest that this result lends support to common EA beliefs. I'm not so sure about that. First, it leads to preference utilitarianism, not hedonic utilitarianism. Second, EAs tend to value
animals and future people, but they would arguably not count as part of the "group" in this framework(?). Third, I'm not sure what this tells you about the creation or non-creation of possible beings
(cf. the asymmetry in population ethics).
Finally, it's worth pointing out that you could also start with different assumptions and get very different results. For instance, rather than demanding that the group is VNM rational, one could
consider rational individuals in a group who bargain over what to do, and then look at bargaining solutions. And it turns out that the utilitarian approach of adding up utilities is *not* a
bargaining solution, because it violates Pareto-optimality in some cases. Does that "disprove" total utilitarianism?
(Using e.g. the Nash bargaining solution with many participants probably leads to some form of prioritarianism or egalitarianism, because you'd have to ensure that everyone benefits.)
And it turns out that the utilitarian approach of adding up utilities is *not* a bargaining solution, because it violates Pareto-optimality in some cases. Does that "disprove" total
I'm not sure this is right. As soon as you maximize a weighted sum with non-negative coefficients your solution will be weakly Pareto optimal. As soon as all coefficients are strictly positive, it
will be strongly Pareto optimal. The axioms mentioned above don't imply non-negative coefficients, so theoretically they are also satisfied by "anti-utilitarianism" which counts everyone's utility
negatively. But one can add stronger Pareto axioms to force all coefficients to be strictly positive.
The problem with the utilitarian Bargaining solution is that it is not independent of affine transformations of utility functions. Just summing up utility functions is underspecified, one also needs
to choose a scaling for the utility functions. A second criterion that might not be satisfied by the utilitarian solution (depending on the scaling chosen) is individual rationality, which means that
everyone will be better off given the bargaining solution than some disagreement outcome.
Thanks for the comment!
Also, you suggest that this result lends support to common EA beliefs.
Hmm, I wasn't trying to suggest that, but I might have accidentally implied something. I would be curious what you are pointing to?
First, it leads to preference utilitarianism, not hedonic utilitarianism
I used preferences about restaurants as an example because that seemed like something people can relate to easily, but that's just an example. The theorem is compatible with hedonic utilitarianism.
(In that case, the theorem would just prove that the group's utility function is the sum of each individual's happiness.)
Second, EAs tend to value animals and future people, but they would arguably not count as part of the "group" in this framework(?).
I don't think that this theorem says much about who you aggregate. It's just simply stating that if you aggregate some group of persons in a certain way, then that aggregation must take the form of
Third, I'm not sure what this tells you about the creation or non-creation of possible beings (cf. the asymmetry in population ethics).
I agree it doesn't say much, see e.g. Michael's comment.
Lukas_Gloor 8
I agree it doesn't say much, see e.g. Michael's comment.
In that case, it would IMO be better to change "total utilitarianism" to "utilitarianism" in the article. Utilitarianism is different from other forms of consequentialism in that it uses
thoroughgoing aggregation. Isn't that what Harsanyi's theorem mainly shows? It doesn't really add any intuitions about population ethics. Mentioning the repugnant conclusion in this context feels
Ben_West🔸 1
In that case, it would IMO be better to change "total utilitarianism" to "utilitarianism" in the article. Utilitarianism is different from other forms of consequentialism in that it uses
thoroughgoing aggregation. Isn't that what Harsanyi's theorem mainly shows?
Hmm, it does show that it's a linear addition of utilities (as opposed to, say, the sum of their logarithms). So I think it's stronger than saying just "thoroughgoing aggregation".
Lukas_Gloor 6
I'm not very familiar with the terminology here, but I remember that in this paper, Alastair Norcross used the term "thoroughgoing aggregation" for what seems to be linear addition of utilities in
particular. That's what I had in mind anyway, so I'm not sure I believe anything different form you. The reason I commented above was because I don't understand the choice of "total utilitarianism"
instead of just "utilitarianism." Doesn't every form of utilitarianism use linear addition of utilities in a case where population size remains fixed? But only total utilitarianism implies the
repugnant conclusion. Your conclusion section IMO suggests that Harsanyi's theorem (which takes a case where population size is indeed fixed) does something to help motivate total utilitarianism over
other forms of utilitarianism, such as prior-existence utilitarianism, negative utilitarianism or average utilitarianism. You already acknowledged in your reply further above to that it doesn't do
much of that. That's why I suggested rephrasing your conclusion section. Alternatively, you could also explain in what ways you might think the utilitarian alternatives to total utilitarianism are
contrived somehow or not in line with Harsanyi's assumptions. And probably I'm missing something about how you think about all of this, because the rest of the article seemed really excellent and
clear to me. I just find the conclusion section really jarring.
Alastair Norcross used the term "thoroughgoing aggregation" for what seems to be linear addition of utilities in particular
Ah, my mistake – I had heard this definition before, which seems slightly different.
I just find the conclusion section really jarring.
Thanks for the suggestion – always tricky to figure out what a "straightforward" consequence is in philosophy.
I changed it to this – curious if you still find it jarring?
Total utilitarianism is a fairly controversial position. The above example where (1,1)=(2,0) can be extended to show that utilitarianism is extremely demanding, potentially requiring extreme
sacrifices and inequality. It is therefore interesting that it is the only decision procedure which does not violate one of these seemingly reasonable assumptions.
richard_ngo 12
Because we are indifferent between who has the 2 and who has the 0
Perhaps I'm missing something, but where does this claim come from? It doesn't seem to follow from the three starting assumptions.
Eli Rose 9
I think this math is interesting, and I appreciate the good pedagogy here. But I don't think this type of reasoning is relevant to my effective altruism (defined as "figuring out how to do the most
good"). In particular, I disagree that this is an "argument for utilitarianism" in the sense that it has the potential to convince me to donate to cause A instead of donating to cause B.
(I really do mean "me" and "my" in that sentence; other people may find that this argument can indeed convince them of this, and that's a fact about them I have no quarrel with. I'm posting this
because I just want to put a signpost saying "some people in EA believe this," in case others feel the same way.)
Following Richard Ngo's post https://forum.effectivealtruism.org/posts/TqCDCkp2ZosCiS3FB/arguments-for-moral-indefinability, I don't think that human moral preferences can be made free of
contradiction. Although I don't like contradictions and I don't want to have them, I also don't like things like the repugnant conclusion, and I'm not sure why the distaste towards contradictions
should be the one that always triumphs.
Since VNM-rationality is based on transitive preferences, and I disagree that human preferences can or "should" be transitive, I interpret things like this as without normative weight.
I think this is an important point. People might want to start with additional or just different axioms, including, as you say, avoiding the repugnant conclusion, and if they can't all together be
consistent, then this theorem may unjustifiably privilege a specific subset of those axioms.
I do think this is an argument for utilitarianism, but more like in the sense of "This is a reason to be a utilitarian, but other reasons might outweigh it." I think it does have some normative
weight in this way.
Also, independence of irrelevant alternatives is safer to give up than transitivity, and might accomplish most of what you want. See my other comment.
Eli Rose 2
Thanks for the pointer to "independence of irrelevant alternatives."
I'm curious to know how you think about "some normative weight." I think of these arguments as being about mathematical systems that do not describe humans, hence no normative weight. Do you think of
them as being about mathematical systems that *somewhat* describe humans, hence *some* normative weight?
I think if you believe the conditions of the theorem are all plausible or desirable and so give them some weight, then you should give the conclusion some weight, too.
For example, it's unlikely to be the case that anyone's ethical rankings actually satisfy the vNM rationality conditions in practice, but if you give any weight to the claims that we should have
ethical rankings that are complete, continuous with respect to probabilities (which are assumed to work in the standard way), satisfy the independence of irrelevant alternatives and avoid all
theoretical (weak) Dutch books, and also give weight to the combination of these conditions at once*, then the Dutch book results give you reason to believe you should satisfy the vNM rationality
axioms, since if you don't, you can get (weakly) Dutch booked in theory. I think you should be at least as sympathetic to the conclusion of a theorem as you are to the combination of all of its
assumptions, if you accept the kind of deductive logic used in the proofs.
*I might be missing more important conditions.
I think if you believe the conditions of the theorem are all plausible or desirable and so give them some weight, then you should give the conclusion some weight, too.
This makes sense, but the type of things that tend to convince me to believe in an ethical theory generally depend a lot on how much I resonate with the main claims of the theory. When I look at the
premises in this theorem, none of them seem to be type of things that I care about.
On the other hand, pointing out that utilitarians care about people and animals, and they want them to be as happy as possible (and free, or with agency, desire satisfaction) that makes me happy to
endorse the theory. When I think about all people and animals being happy and free from pain in a utilitarian world, I get a positive feeling. When I think about "Total utilitarians are the only ones
that satisfy these three assumptions" I don't get the same positive feeling.
When it comes to ethics, it's the emotional arguments that really win me over.
RomeoStevens 8
Like other links between VNM and Utilitarianism, this seems to roll intersubjective utility comparison under the rug. The agents are likely using very different methods to convert their preferences
to the given numbers, rendering the aggregate of them non rigorous and subject to instability in iterated games.
calebo 2
I can't tell whether you are denying assumption 1 or 2.
I don't think Romeo even has to deny any of the assumptions. Harsanyi's result, derived from the three assumptions, is not enough to determine how to do intersubjective utility comparisons. It merely
states that social welfare will be some linear combination of individual utilities. While this already greatly restricts the way in which utilities are aggregated, it does not specify which weights
to use for this sum.
Moreover, arguing that weights should be equal based on the veil of ignorance, as I believe Harsanyi does, is not sufficient, since utility functions are only determined up to affine transformations,
which includes rescalings. (This point has been made in the literature as a criticism of preference utilitarianism, I believe.) So there seems to be no way to determine what equal weights should look
like, without settling on a way to normalize utility functions, e.g., by range normalization or variance normalization. I think the debate about intersubjective utility comparisons comes in at the
point where you ask how to normalize utility functions.
Of course, if you are not using a kind of preference utilitarianism but instead just aggregate some quantities you believe to have an absolute scale—such as happiness and suffering—then you could
argue that utility functions should just correspond to this one absolute scale, with the same scaling for everyone. Though I think this is also not a trivial argument—there are potentially different
ways to get from this absolute scale or Axiology to behavior towards risky gambles, which in turn determine the utility functions.
calebo 4
Thanks for this.
Even if this argument is successful, there are debates over decision theory (evidential, causal, functional). Does an ideally rational agent intervene at the level of states, actions, or decision
If it's decision procedures, or something similar, functional decision theory can you get views that look quite close to Kantianism.
I would actually say that 12(2,0)+12(0,2) being equivalent to (2,0) and (0,2) is in contradiction with equality of opportunity. In the first case, both individuals have an equal chance of being
well-off (getting 2), but in the second and third, only one has any chance of being well-off, so the opportunities to be well-off are only equal in the first case (essentially the same objection to
essentially the same case is made in "Cardinal Welfare, Individualistic Ethics, and Interpersonal Comparison of Utility: Comment", in which Peter Diamond writes "it seems reasonable for the
individual to be concerned solely with final states while society is also interested in the process of choice"). This is what ex ante prioritarianism/egalitarianism is for, but it can lead to
counterintuitive results. See the comments on that post, and "Decide As You Would With Full Information! An Argument Against Ex Ante Pareto" by Marc Fleurbaey & Alex Voorhoeve.
For literature on equality of outcomes and uncertainty, the terms to look for are "ex post egalitarianism" and "ex post prioritarianism" (or with the hyphen as "ex-post", but I think Google isn't
sensitive to this). | {"url":"https://forum.effectivealtruism.org/posts/v89xwH3ouymNmc8hi/harsanyi-s-simple-proof-of-utilitarianism","timestamp":"2024-11-04T15:26:50Z","content_type":"text/html","content_length":"1049253","record_id":"<urn:uuid:55238854-6cdf-4525-ba63-97d3080405d8>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00222.warc.gz"} |
Four Skills That Will Turn You Into a Spreadsheet Ninja
Spreadsheets are one of the most mystifying pieces of software you'll encounter in your adult life. As scary as they can be, though, you can do an awful lot with just four simple skills.
For the purposes of this article, we'll be focusing on Microsoft Excel, since this is the most widely used spreadsheet software. However, nearly all of these skills and features are useful in
LibreOffice and Google Drive. We'll make notes when necessary to highlight the differences between the suites.
Input Data Easily with Forms
Entering data into a spreadsheet is the starting point for any analysis. While you can type the data you need in manually, forms allow you (or in some cases, others) to quickly enter information
line-by-line without much of a fuss.
The Form button has been somewhat coyly hidden in Excel 2013, but you can get it back like so:
1. Right-click anywhere on the ribbon interface, and select "Customize the Ribbon."
2. In the right-hand pane, choose a section of the Ribbon to add the Forms button to.
3. On the left-hand side, choose "Commands Not in the Ribbon" from the drop down.
4. In the box below, select "Form…" from the list and click "Add" to place it in the ribbon.
Now that you have the Form button in your ribbon, you can create a data entry form. Start by creating the headers and first row of entries. Once you have your initial set of data in, you can enter in
additional rows with the form command. Simply place your cursor in the top-left corner of your data set and click the form button.
The dialog that will appear allows you to enter information on a per-line basis. Fill out the form, press enter, and a new line will be created with all the new information.
Create Public-Facing Forms with Google Drive
Google Drive doesn't have quite the same functionality, but it does have its own forms worth noting. You can create a form either directly in Drive or from within a spreadsheet. You'll create
questions for users to answer and their responses will populate a sheet. To create a form:
1. From within a spreadsheet, click Insert > Form.
2. Enter a description at the top of the form.
3. Enter and modify each question:
4. Enter a title in the "Question title" box.
5. Choose the Question Type.
6. Optional: Choose a form of data validation. You can use this to confirm that data entered adheres to a specific type, such as a number within a set range, or text in the form of an email address.
7. Optional: Select "Required question."
Once the form is completed, you can share it publicly or email it to a select group of respondents. All entries will be automatically placed into columns alongside a timestamp indicating when the
data was submitted.
Perform Calculations with Functions and Formulas
So you've entered a bunch of data, but now you need to do something with it. Functions and formulas allow you to manipulate data in a spreadsheet. You can perform simple math, like add up the numbers
in a column, get an average, or even work with real-world things like dates or financial calculations.
The various spreadsheet programs all have their own set of functions, and while many of them are shared between the different suites, there are some differences, so be sure to check out the full list
for Excel, LibreOffice, and Google Drive.
With that in mind, here are a few examples of what you can do.
Perform Basic Math
At their simplest, functions can perform basic math using any data you've entered. For example, say you need to add the numbers in two cells together. For that, you'd use the SUM function:
This will add the contents of cells A1 and B1 together. For simple math, you can also use typical shorthand math operators like +, -, * and /. For instance, the following will perform the exact same
math as the example above:
You can add up the values in as many cells as you want. You can do this with a colon, like so:
The above function will add together all of the numbers in column A between rows 1 and 10, accounting for negative values. You can find a list of all the math functions Excel can perform here.
Make Statistical Calculations
You can also perform statistical calculations on a set of data, including calculating averages medians. As an example:
The above function will return the average of all cells between A1 and A10. The various functions include both basic and advanced statistics functions. While not all of them will be useful for the
casual spreadsheet enthusiast (yes, we exist), simple functions like AVERAGE and MEDIAN can be really helpful in everyday work.
Format and Calculate Dates and Times
You can also use formulas to manipulate date and time formatted entries. As an example, you can calculate the number of days between two dates with the following function:
Excel also includes a function to return the current date:
This function will put today's date in a cell. You can combine these two functions to create a formula to find out how many days are left until a certain date in the future like so:
There are a number of other date and time functions you can look over here.
Combine Multiple Functions to Create Formulas
As you might have noticed in the last example, you can combine multiple functions to create what's known as a formula. Formulas are, essentially, multiple functions put together in one cell. So, for
example, if you wanted to add up the numbers in column A and round them to the nearest whole number, you would use the following formula:
The above formula is made up of two functions: SUM which is being used as an argument for ROUND. In this statement, cells A1 through A10 will be added together first and then the resulting number
will be given to the ROUND function to be rounded to the nearest whole number. Formulas can be as simple or as complex as you'd like, though the more elaborate they get, the more intricate their
syntax gets. You can read more about how formula syntax works here.
Functions can also be combined with logical functions to create conditional formulas. Building useful formulas is a topic so broad that it could generate its own entire set of articles. Fortunately,
our friends at the How-To Geek has an in-depth set of articles on that very subject. You can find the first lesson here.
Sort Data with Filters and Pivot Tables
So you've entered your raw data, made any calculations you need to make, and now it's time to actually interpret it. You have a few options for visualizing your data, depending on how you need to use
Sort by Column
Sometimes, you may need to sort your data by one of the categories you've entered. Just like your iTunes music library, you can re-order your data by column. To do so:
1. Press Ctrl-A to select everything in the sheet.
2. Select the Data tab in the Ribbon.
3. Click "Sort"
4. Choose the Column you want to sort by and what criteria you want to sort with.
5. Click OK.
See the above screenshot for example—we've reordered the entire data set by order number, from least to greatest.
Filter Out Duplicate Items
Other times, you may have some duplicate items within a category. In the example above, we have 8 entries, but only 5 names. To filter out those duplicates and see all the names we have (as opposed
to all the orders), we can use the filtering options under the Data tab in the ribbon. To do so:
1. Under Data, click "Advanced" in the Sort & Filter section.
2. Select "Copy to another location" for a non-destructive way to pull out specific data (if you want to delete any rows in your spreadsheet that do not have unique data in this column, leave
"filter the list, in place" selected.)
3. Click the "List range" box and select the column you want to filter.
4. Click the "Copy to" box and select the empty cell you want to copy your list of unique data to.
5. Ensure "Unique data only" is selected.
6. Click OK
This will create a new column that only contains unique data from that range, which is useful for separating out repeated entries.
Create Pivot Tables
When you have hundreds of lines of data, it's nearly impossible to glean any information from it—you need a summary. Pivot tables let you take certain portions of your data and summarize it, so you
only see what you want to see.
Take the example below. Say we wanted to see the total amount of money we got from Bob Boberson.
This data is easy enough to look at now, but with dozens or hundreds of transactions, it wouldn't be. So, to see that summary, create a pivot table:
1. Create a new sheet, named Sheet 2.
2. Under the Insert tab, click Pivot Table.
3. Click Sheet 1 and select all populated cells in the sheet.
4. Click OK.
5. In the right-hand pane, you will be able to drag columns into filtered, column, or rows. For this example, drag "Name" to the Filters section, "Order number" to the Rows section, and "Payments"
to the Values section.
6. This will create a table of all the payments made on each order number. At the top of the pivot table, in the dropdown box, you can select a specific customer to view only their orders.
When we do so, we get this:
With a pivot table, we can easily see all the orders made by Bob Boberson as well as a sum of all the payments he made.
This is just one simple example of how you could use pivot tables—their usage is very broad, and you can use it to summarize just about anything. To create a pivot table in LibreOffice or Google
Drive, Head to Data > Pivot Table.
Perform Repetitive Tasks with Macros and Scripts
Working with spreadsheets can get repetitive really quickly. To make some of those repetitive tasks simpler, you can use Excel to record things you do over and over into a macro. Then, any time you
need to repeat those tasks, you can use a keyboard shortcut to "play back" the macro and it'll do all the busywork for you. For example, say we wanted to change the font of a large group of cells to
something very specific, like Arial 12 italicized:
1. Under the View tab, click the Macros drop down and select Record Macro.
2. Give the macro a name.
3. Optional: Assign the macro a keyboard shortcut.
4. Click OK
5. Click the Home tab.
6. Select "Arial" from the font drop down.
7. Select 12 from the size drop down.
8. Click the italics button.
9. Click the Stop button in the bottom-left corner of Excel.
From then on, you will be able to apply the Arial font in a 12pt size with italics by pressing one single keyboard shortcut (the one you assigned in step 3), or by selecting the macro from the View >
Macro Library. Obviously, this is a rudimentary example, but you can use the macro feature to record any repetitious tasks. You can similarly record macros in LibreOffice.
In Excel, LibreOffice, and Google Drive, you can create more complex macros and scripts with a little bit of programming savvy (and in fact scripts are the only way to automate tasks in Google Drive
using a proprietary scripting format). To get started learning about these, check out the resources below:
Further Resources
All of these skills are just the tip of the iceberg. Using spreadsheets is a discipline in itself and you can dive much, much deeper into each of these categories if you want. For convenience, here
are some resources you can use to learn more about all of these skills and hone your craft:
Forms and Data Entry
Functions and Formulas
How-To Geek School:
Filtering and Pivot Tables
Macros and Scripts | {"url":"https://lifehacker.com/four-skills-that-will-turn-you-into-a-spreadsheet-ninj-1525058930","timestamp":"2024-11-02T05:54:18Z","content_type":"text/html","content_length":"238553","record_id":"<urn:uuid:403c74b8-719c-4fab-9b31-740a7356d5aa>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00265.warc.gz"} |
Peter Gustav Lejeune Dirichlet
Peter Gustav Lejeune Dirichlet (1805-1859)
Birth Place
Düren, French Empire
Death Place
Göttingen, Kingdom of Hanover
Fields of Expertise
Johann Peter Gustav Lejeune Dirichlet German was a German mathematician who made deep contributions to number theory including creating the field of analytic number theory, and to the theory of
Fourier series and other topics in mathematical analysis; he is credited with being one of the first mathematicians to give the modern formal definition of a function.
Although his surname is Lejeune Dirichlet, he is commonly referred to as just Dirichlet, in particular for results named after him.
Early life 1805–1822
Gustav Lejeune Dirichlet was born on 13 February 1805 in Düren, a town on the left bank of the Rhine which at the time was part of the First French Empire, reverting to Prussia after the Congress of
Vienna in 1815. His father Johann Arnold Lejeune Dirichlet was the postmaster, merchant, and city councilor. His paternal grandfather had come to Düren from Richelette or more likely Richelle, a
small community 5 km 3 miles north east of Liège in Belgium, from which his surname "Lejeune Dirichlet" "le jeune de Richelette", French for "the youth from Richelette" was derived.
Although his family was not wealthy and he was the youngest of seven children, his parents supported his education. They enrolled him in an elementary school and then private school in hope that he
would later become a merchant. The young Dirichlet, who showed a strong interest in mathematics before age 12, persuaded his parents to allow him to continue his studies. In 1817 they sent him to the
Gymnasium Bonn under the care of Peter Joseph Elvenich, a student his family knew. In 1820 Dirichlet moved to the Jesuit Gymnasium in Cologne, where his lessons with Georg Ohm helped widen his
knowledge in mathematics. He left the gymnasium a year later with only a certificate, as his inability to speak fluent Latin prevented him from earning the Abitur.
Studies in Paris 1822–1826
Dirichlet again persuaded his parents to provide further financial support for his studies in mathematics, against their wish for a career in law. As Germany provided little opportunity to study
higher mathematics at the time, with only Gauss at the University of Göttingen who was nominally a professor of astronomy and anyway disliked teaching, Dirichlet decided to go to Paris in May 1822.
There he attended classes at the Collège de France and at the University of Paris, learning mathematics from Hachette among others, while undertaking private study of Gauss's Disquisitiones
Arithmeticae, a book he kept close for his entire life. In 1823 he was recommended to General Maximilien Foy, who hired him as a private tutor to teach his children German, the wage finally allowing
Dirichlet to become independent from his parents' financial support.
His first original research, comprising part of a proof of Fermat's Last Theorem for the case n=5, brought him immediate fame, being the first advance in the theorem since Fermat's own proof of the
case n=4 and Euler's proof for n=3. Adrien-Marie Legendre, one of the referees, soon completed the proof for this case; Dirichlet completed his own proof a short time after Legendre, and a few
years later produced a full proof for the case n=14. In June 1825 he was accepted to lecture on his partial proof for the case n=5 at the French Academy of Sciences, an exceptional feat for a
20-year-old student with no degree. His lecture at the Academy had also put Dirichlet in close contact with Fourier and Poisson, who raised his interest in theoretical physics, especially Fourier's
analytic theory of heat.
Back to Prussia, Breslau 1825–1828
As General Foy died in November 1825 and he could not find any paying position in France, Dirichlet had to return to Prussia. Fourier and Poisson introduced him to Alexander von Humboldt, who had
been called to join the court of King Friedrich Wilhelm III. Humboldt, planning to make Berlin a center of science and research, immediately offered his help to Dirichlet, sending letters in his
favour to the Prussian government and to the Prussian Academy of Sciences. Humboldt also secured a recommendation letter from Gauss, who upon reading his memoir on Fermat's theorem wrote with an
unusual amount of praise that "Dirichlet showed excellent talent". With the support of Humboldt and Gauss, Dirichlet was offered a teaching position at the University of Breslau. However, as he had
not passed a doctoral dissertation, he submitted his memoir on the Fermat theorem as a thesis to the University of Bonn. Again his lack of fluency in Latin rendered him unable to hold the required
public disputation of his thesis; after much discussion, the University decided to bypass the problem by awarding him an honorary doctorate in February 1827. Also, the Minister of Education granted
him a dispensation for the Latin disputation required for the Habilitation. Dirichlet earned the Habilitation and lectured in the 1827–28 year as a Privatdozent at Breslau.
While in Breslau, Dirichlet continued his number theoretic research, publishing important contributions to the biquadratic reciprocity law which at the time was a focal point of Gauss's research.
Alexander von Humboldt took advantage of these new results, which had also drawn enthusiastic praise from Friedrich Bessel, to arrange for him the desired transfer to Berlin. Given Dirichlet's young
age he was 23 years old at the time, Humboldt was able to get him only a trial position at the Prussian Military Academy in Berlin while remaining nominally employed by the University of Breslau. The
probation was extended for three years until the position becoming definite in 1831.
Marriage to Rebecka Mendelssohn
After Dirichlet's move to Berlin, Humboldt introduced him to the great salons held by the banker Abraham Mendelssohn Bartholdy and his family. Their house was a weekly gathering point for Berlin
artists and scientists, including Abraham's children Felix and Fanny Mendelssohn, both outstanding musicians, and the painter Wilhelm Hensel Fanny's husband. Dirichlet showed great interest in
Abraham's daughter Rebecka, whom he married in 1832.
Dirichlet was married in 1832 to Rebecka Mendelssohn. They had two children, Walter (born 1833) and Flora (born 1845). Drawing by Wilhelm Hensel, 1823
Rebecka Henriette Lejeune Dirichlet née Rebecka Mendelssohn; 11 April 1811 – 1 December 1858 was a granddaughter of Moses Mendelssohn and the youngest sister of Felix Mendelssohn and Fanny
Mendelssohn. Rebecka was born in Hamburg. In 1816 her parents arranged for her to be baptised at which point she took the names Rebecka Henriette Mendelssohn Bartholdy. She became a part of the
notable salon of her parents, Abraham Mendelssohn and his wife Lea, having social contacts with the important musicians, artists and scientists in a highly creative period of German intellectual
life. In 1829 she sang a small role in the premiere, given at the Mendelssohn house, of Felix's Singspiel Die Heimkehr aus der Fremde. She later wrote:
My older brother and sister stole my reputation as an artist. In any other family I would have been highly regarded as a musician and perhaps been leader of a group. Next to Felix and Fanny, I
could not aspire to any recognition.
In 1832 she married Dirichlet, who was introduced to the Mendelssohn family by Alexander von Humboldt. In 1833 their first son, Walter, was born. She died in Göttingen in 1858.
Berlin 1826–1855
As soon as he came to Berlin, Dirichlet applied to lecture at the University of Berlin, and the Education Minister approved the transfer and in 1831 assigned him to the faculty of philosophy. The
faculty required him to undertake a renewed habilitation qualification, and although Dirichlet wrote a Habilitationsschrift as needed, he postponed giving the mandatory lecture in Latin for another
20 years, until 1851. As he had not completed this formal requirement, he remained attached to the faculty with less than full rights, including restricted emoluments, forcing him to keep in parallel
his teaching position at the Military School. In 1832 Dirichlet became a member of the Prussian Academy of Sciences, the youngest member at only 27 years old.
Dirichlet had a good reputation with students for the clarity of his explanations and enjoyed teaching, especially as his University lectures tended to be on the more advanced topics in which he was
doing research: number theory he was the first German professor to give lectures on number theory, analysis and mathematical physics. He advised the doctoral theses of several important German
mathematicians, as Gotthold Eisenstein, Leopold Kronecker, Rudolf Lipschitz and Carl Wilhelm Borchardt, while being influential in the mathematical formation of many other scientists, including Elwin
Bruno Christoffel, Wilhelm Weber, Eduard Heine, Ludwig von Seidel and Julius Weingarten. At the Military Academy, Dirichlet managed to introduce differential and integral calculus in the curriculum,
raising the level of scientific education there. However, he gradually started feeling that his double teaching load, at the Military academy and at the University, was limiting the time available
for his research.
While in Berlin, Dirichlet kept in contact with other mathematicians. In 1829, during a trip, he met Carl Jacobi, at the time professor of mathematics at Königsberg University. Over the years they
kept meeting and corresponding on research matters, in time becoming close friends. In 1839, during a visit to Paris, Dirichlet met Joseph Liouville, the two mathematicians becoming friends, keeping
in contact and even visiting each other with the families a few years later. In 1839, Jacobi sent Dirichlet a paper by Ernst Kummer, at the time a schoolteacher. Realizing Kummer's potential, they
helped him get elected in the Berlin Academy and, in 1842, obtained for him a full professor position at the University of Breslau. In 1840 Kummer married Ottilie Mendelssohn, a cousin of Rebecka's.
In 1843, when Jacobi fell ill, Dirichlet traveled to Königsberg to help him, then obtained for him the assistance of King Friedrich Wilhelm IV's personal physician. When the physician recommended
that Jacobi spend some time in Italy, Dirichlet joined him on the trip together with his family. They were accompanied to Italy by Ludwig Schläfli, who came as a translator; as he was strongly
interested in mathematics, both Dirichlet and Jacobi lectured to him during the trip, and he later became an important mathematician himself. The Dirichlet family extended their stay in Italy to
1845, their daughter Flora being born there. In 1844, Jacobi moved to Berlin as a royal pensioner, their friendship becoming even closer. In 1846, when the Heidelberg University tried to recruit
Dirichlet, Jacobi provided von Humboldt the needed support to obtain a doubling of Dirichlet's pay at the University in order to keep him in Berlin; however, even then he was not paid a full
professor wage and could not leave the Military Academy.
Holding liberal views, Dirichlet and his family supported the 1848 revolution; he even guarded with a rifle the palace of the Prince of Prussia. After the revolution failed, the Military Academy
closed temporarily, causing him a large loss of income. When it reopened, the environment became more hostile to him, as officers he was teaching were expected to be loyal to the constituted
government. Some of the press who had not sided with the revolution pointed him out, as well as Jacobi and other liberal professors, as "the red contingent of the staff".
In 1849 Dirichlet participated, together with his friend Jacobi, in the jubilee of Gauss's doctorate.
Göttingen 1855–1859
Despite Dirichlet's expertise and the honours he received, and even though, by 1851, he had finally completed all formal requirements for a full professor, the issue of raising his pay at the
University still dragged on and he was still unable to leave the Military Academy. In 1855, upon Gauss's death, the University of Göttingen decided to call Dirichlet as his successor. Given the
difficulties faced in Berlin, he decided to accept the offer and immediately moved to Göttingen with his family. Kummer was called to assume his position as a professor of mathematics in Berlin.
Dirichlet enjoyed his time in Göttingen, as the lighter teaching load allowed him more time for research and he came into close contact with the new generation of researchers, especially Richard
Dedekind and Bernhard Riemann. After moving to Göttingen he was able to obtain a small annual stipend for Riemann to retain him in the teaching staff there. Dedekind, Riemann, Moritz Cantor and
Alfred Enneper, although they had all already earned their PhDs, attended Dirichlet's classes to study with him. Dedekind, who felt that there were gaps in his mathematics education, considered that
the occasion to study with Dirichlet made him "a new human being". He later edited and published Dirichlet's lectures and other results in number theory under the title Vorlesungen über Zahlentheorie
Lectures on Number Theory.
In the summer of 1858, during a trip to Montreux, Dirichlet suffered a heart attack. On 5 May 1859, he died in Göttingen, several months after the death of his wife Rebecka. Dirichlet's brain is
preserved in the department of physiology at the University of Göttingen, along with the brain of Gauss. The Academy in Berlin honored him with a formal memorial speech presented by Kummer in 1860,
and later ordered the publication of his collected works edited by Kronecker and Lazarus Fuchs.
Mathematics research
Number theory
Number theory was Dirichlet's main research interest, a field in which he found several deep results and in proving them introduced some fundamental tools, many of which were later named after him.
In 1837, Dirichlet's theorem on arithmetic progressions, using mathematical analysis concepts to tackle an algebraic problem and thus creating the branch of analytic number theory. In proving the
theorem, he introduced the Dirichlet characters and L-functions. Also, in the article he noted the difference between the absolute and conditional convergence of series and its impact in what was
later called the Riemann series theorem. In 1841, he generalized his arithmetic progressions theorem from integers to the ring of Gaussian integers Z .
In a couple of papers in 1838 and 1839, he proved the first class number formula, for quadratic forms later refined by his student Kronecker. The formula, which Jacobi called a result "touching the
utmost of human acumen", opened the way for similar results regarding more general number fields. Based on his research of the structure of the unit group of quadratic fields, he proved the Dirichlet
unit theorem, a fundamental result in algebraic number theory.
He first used the pigeonhole principle, a basic counting argument, in the proof of a theorem in diophantine approximation, later named after him Dirichlet's approximation theorem. He published
important contributions to Fermat's Last Theorem, for which he proved the cases n=5 and n=14, and to the biquadratic reciprocity law. The Dirichlet divisor problem, for which he found the first
results, is still an unsolved problem in number theory despite later contributions by other mathematicians.
Inspired by the work of his mentor in Paris, Dirichlet published in 1829 a famous memoir giving the conditions, showing for which functions the convergence of the Fourier series holds. Before
Dirichlet's solution, not only Fourier, but also Poisson and Cauchy had tried unsuccessfully to find a rigorous proof of convergence. The memoir pointed out Cauchy's mistake and introduced
Dirichlet's test for the convergence of series. It also introduced the Dirichlet function as an example of a function that is not integrable the definite integral was still a developing topic at the
time and, in the proof of the theorem for the Fourier series, introduced the Dirichlet kernel and the Dirichlet integral.
Dirichlet also studied the first boundary value problem, for the Laplace equation, proving the uniqueness of the solution; this type of problem in the theory of partial differential equations was
later named the Dirichlet problem after him. A function satisfying a partial differential equation subject to the Dirichlet boundary conditions must have fixed values on the boundary. In the proof he
notably used the principle that the solution is the function that minimizes the so-called Dirichlet energy. Riemann later named this approach the Dirichlet principle, although he knew it had also
been used by Gauss and by Lord Kelvin.
Introduction of the modern concept of function
While trying to gauge the range of functions for which convergence of the Fourier series can be shown, Dirichlet defines a function by the property that "to any x there corresponds a single finite
y", but then restricts his attention to piecewise continuous functions. Based on this, he is credited with introducing the modern concept for a function, as opposed to the older vague understanding
of a function as an analytic formula. Imre Lakatos cites Hermann Hankel as the early origin of this attribution, but disputes the claim saying that "there is ample evidence that he had no idea of
this concept for instance, when he discusses piecewise continuous functions, he says that at points of discontinuity the function has two values".
Other fields
Dirichlet also worked in mathematical physics, lecturing and publishing research in potential theory including the Dirichlet problem and Dirichlet principle mentioned above, the theory of heat and
hydrodynamics. He improved on Lagrange's work on conservative systems by showing that the condition for equilibrium is that the potential energy is minimal.
Dirichlet also lectured on probability theory and least squares, introducing some original methods and results, in particular for limit theorems and an improvement of Laplace's method of
approximation related to the central limit theorem. The Dirichlet distribution and the Dirichlet process, based on the Dirichlet integral, are named after him.
Dirichlet was elected as a member of several academies:
• Prussian Academy of Sciences 1832
• Saint Petersburg Academy of Sciences 1833 – corresponding member
• Göttingen Academy of Sciences 1846
• French Academy of Sciences 1854 – foreign member
• Royal Swedish Academy of Sciences 1854
• Royal Belgian Academy of Sciences 1855
• Royal Society 1855 – foreign member
In 1855 Dirichlet was awarded the civil class medal of the Pour le Mérite order at von Humboldt's recommendation. The Dirichlet crater on the Moon and the 11665 Dirichlet asteroid are named after
Gender: Male
Best Known For: See full list
Other Academic Advisors: Siméon PoissonJoseph FourierCarl Gauss
Awards: PhD (Hon):University of Bonn (1827)
Pour le Mérite (1855)
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An Alternating Direction Method of Multipliers Algorithm for "Symmetric" MPC
The alternating direction method of multipliers (ADMM) is an algorithm that attempts to solve a convex optimization problem by breaking it into smaller pieces, each of which will be easier to handle.
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How to draw an ellipse in isometricHow to draw an ellipse in isometric 🚩 replacement ellipse oval in rectangular isometric projection 🚩 Math.
You will need
• - the range;
• - gon;
• pencil;
• paper for sketching.
Consider how to draw an ellipse in isometric view, lying in a horizontal plane. Construct a perpendicular to the axis X and Y. the intersection of the mark O.
From point O mark on the axes the segments is equal to the radius of the circle. Marked points indicate the numbers 1, 2, 3, 4. Through these points parallel to the guide axes of the straight.
From point O mark on the axes the segments is equal to the radius of the circle. Marked points indicate the numbers 1, 2, 3, 4. Through these points parallel to the guide axes of the straight.
Draw a arc from the vertex of obtuse angle of the connecting point 1 and 4. Similarly, connect the dots 2 and 3, having an arc from vertex D. Connect the points 1,2 and 3,4 of the centers of the
small arcs. Thus constructed the ellipse in the isometricinscribed in a rhombus.
The second way to draw an ellipse in isometric is to circle map distortion. Draw X and Y axis, from point O, draw two auxiliary circle. The diameter of the inner circle equals the minor axis of the
ellipse, the major axis.
In one quarter build auxiliary rays emanating from the center of the ellipse. The number of rays is arbitrary and the more, the more accurate the drawing. In our case it will be enough three
auxiliary beams.
Will receive additional points of the ellipse. From the point of intersection of the beam with the smaller circle draw a horizontal line parallel to the X axis towards the outer circumference. From
the top of the point lying at the intersection of the beam and of a large circle, drop a perpendicular.
The resulting point label figure 2. Repeat the operation for finding 3 and 4 points of the ellipse. Point 1 is located at the intersection of the Y-axis and the small circle, point 5 on the X-axis in
place of the passage of the outer circumference.
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What would be the distinct levels of numbers Primary numbers, algebra etc
Underground Maths exists for the advantage of you and your pupils. Options that will add brand-new ideas about the stop are typically in a “Introducing” section. Recent neural facts reveals value of
individuals taking care of challenging operate and also building mistakes. To select the subcollection, utilize drop-down listing above the stand.
What is definitely the speed from the program?
You might discover the next paperwork valuable when it comes to info provided by Unistats. You can also filtration system assets simply by hose series, if you want to select a learning resource which
has a particular importance. Probably they ought to often be motivated to try to understand the proofs supplied with primary calculus points; these are also an excellent summary of abstraction. With
a few distinctive funds offered, Math concepts, Even more Arithmetic along with Genuine Arithmetic, a degree is built to utilized in excess of 2 years, using the use of any one-year While level
system. She actually is the author in the first MOOC about math studying with regard to course instructors and fogeys, a White colored Residence phone speaker, and a expert on the PISA company within
the OECD. In the following treatment members will spot calculations since anything worthwhile, exhilarating, and share in the course of life.
• geometry
• mathematical solutions (for example vector calculus)
• Introducing Calculus
• a massive amount free audit practice to arrange students
• Thinking concerning Geometry
• Calculus with Powers
• your knowing of the particular significance with maths along with other fields of study as well as contemporary society usually.
That is definitely my own list See also Front door requirements as well as Subject Matters to get more advice about standard requirements to get accessibility, qualifications and provides. You can
access the particular road any time by way of clicking a Road option in the primary selection. The program is actually organized to readily available consistently. Boaler in addition to a group of
undergraduates, job interviews having the public, really advanced research suggestions, interesting pictures and flicks, and also explorations with calculations in the wild, game and design.
Popular topics
The course is targeted on producing critical exact methods inside of a easy to understand, clear and also thorough manner. Jo Boaler is usually a Professor involving Mathematics Schooling with
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By hitting a new invasive idea there is also a this is their explanation strategy takes place from the resources. The study course consists of challenge do the job, a function special to be able to
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relationship by using numbers. January Versus March and also May / 06 2019 documents is going to be updated following end result announcements. In Piece IB, you end up picking by all around 16
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Mathematical scientific tests SL-course details
More how-to training videos can be located for the How-to guide website. Since some others possess mentioned, the mathematics that is definitely typically educated pre-university pays to and also
important for many professions, and so has to be taught very first. All those acquiring Mathematics together with Physics replace 2 Math subject matter with Piece IA Physics coming from Organic
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• automata and professional languages
• Newtonian dynamics along with distinctive relativity
• Their collection of career
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delve further compared to the confront importance of the Unistats details. The maths Center
Find files you discover each of our math skills, on this page plus liberated to download. Part IA outlines the basic principles of upper mathematics, https://researchpaperwriter.net/
write-my-research-paper/ like: During numerous educational institutions, there’s no need to lower the mathematics main totally; instead, it is possible to opt for the “applied” calculations key. This
strategy to post-16 mathematics originated from the College of Cambridge, backed up by way of offer with the United kingdom Section with regard to Education.
• The Numerous Looks of Difficult Numbers
• There isn’t a typical formatting created examination regarding Math concepts (people continually take a seat Move together with The Quantities) – Schools may review characteristics, expertise and
also likely by way of short tasks before job interview.
• applicable numbers, which includes statistics plus marketing (a comprehensive treatment of subject areas coming from determination mathematics)
• There is not any frequent arrangement prepared assessment with regard to Maths (appliers continually sit down Measure beside Any Ranges) – Educational institutions will probably review
characteristics, base of knowledge in addition to prospective as a result of small tasks in the course of interview.
• the expertise had to utilize know-how for example hand calculators and also computers
• applicable numbers, such as research along with seo (a rigorous remedy for themes out of final decision arithmetic)
In Part IB, you choose through close to 07 available options. You can also filtration system assets simply by hose series, if you want to select a learning resource which has a particular importance.
In this particular program participants will notice math concepts is a issue that is definitely consisting of related, major concepts. Use the Search switch while in the major food list bar to open
looking discussion.
(like Math using Science)
For that reason, individuals is able to use their own natural, realistic pondering capabilities and do not really need to rely upon standard calculations and also appreciated formulae. Scholars
should apply their own numerical know-how to resolve difficulties set in a number of special contexts. That is usually my personal list PapaCambridge offers Math concepts 9709 Most current Past
Papers and Assets that also includes syllabus, specimens, issue papers, labels strategies, FAQ’s, Teacher’s resources, Notes and a lot more. The process for individuals are going to arrive at the
same a higher level comprehending all over virtually all subjects. The 1991 document by way of the United states Fiscal Association presented overall costs Ph.Deborah college students with the
pursuing listing of mathematical topics:
What will be the study course construction?
Assessment in addition to Moderation now provide a range of assessor help choices to stimulate in addition to inspire superior analysis training. Scholars should really, whenever you can, make use of
the math expertise they’ve already purchased to eliminate reasonable problems occur an appropriate circumstance. Cambridge Intercontinental AS along with a Place Maths develops the abilities bought
during Cambridge IGCSE (and also comparable) stage. 12/1/2017 : October/November 2017 Your Level Mathematics Grade Thresholds, Syllabus and also Past Assessment Papers are usually kept up to date.
There are also suggested computational assignments (examined by using reviews in addition to courses sent in prior to the summer months check-ups), using math or even algebraic methods to investigate
math problems. There’s a very broad option, which include newspapers for, as an example: 24/8/2017 : March along with May 06 2017 Maths Past Papers associated with A Level in addition to AS Level are
• Polynomials & Rational Functions
• Power Series
• algebraic topology
• your knowledge of coherence and acceleration and also precisely how diverse aspects of arithmetic could be connected
• your idea of coherence in addition to further development in addition to exactly how unique aspects of math could be connected
On this impression, a new up and down plane is usually deflected into a side page by way of a outside impactor. This procedure can study the crucial tips through the training and help participants
consider the essential methods and ideas they’ve already realized into their long term.
The pupils possibly to decide on this training manual are the types whoever key hobbies and interests then lie beyond your discipline regarding numbers, for a lot of students this course will likely
be his or her very last experience with being shown official math concepts. It’s fashioned with some sort of pedagogy with lively bridal. It would be awesome to be able to suggestion so that you can
students around high school calculus (or maybe precalculus?) the proofs include the essential part for any absolute arithmetic main. How to Learn Calculations is really a no cost self-paced class to
get learners of all the levels of mathematics.
Over many years, specialised mathematicians currently have transformed efficiently with other subject coached on Cambridge. 1 secondary school math simply
2 primary calculus and straight line algebra
3 used numbers, differential equations, linear coding, and also basic likelihood idea
4 advanced calculus, state-of-the-art geometry along with stochastic techniques
5 real and complex analysis, sophisticated likelihood theory, in addition to topology This could make it easier to think about concerns you would possibly ask college students, to see places where
learners may get caught up, or maybe offer even more understanding of how you would are able to use a source of information with your class room. Acceptable, that is more than enough rambling, at
this moment Let me come back to your particular problems. Advancement of each individual matter ought to function validation in addition to verification of benefits.
The girl with the https://www.lcc.edu/learning/college-credit-in-hs/dual.html writer from the initially MOOC upon numbers understanding with regard to course instructors and parents, some sort of
Bright Household presenter, with an consultant to the PISA workforce in the OECD. Like this web site? Have ideas for improvements? Impart us with your own feedback. Learners gonna have to have math
for your accomplishment connected with more credentials should be suggested to take into consideration an alternative maths training course. RE:
What are the various quantities of maths? Essential arithmetic, geometry etcetera?
There’s for instance sensitive mathematics, subsequently essential arithmetic including fractions, rookie geometry
Then the way did it get?
Geometry, after that pre-algebra?
Then algebra, calculus, trigonometry.
Saving resources
Using particular person systems allows students to overpower a GeoGebra applets which in turn come in several resources. Sign up for math messages
What you have to know, when you wish to understand it, pertaining to all our mathematics qualifications. You will find there’s extremely large option, such as newspapers upon, by way of example: The
tubemap shows various relationships among topics. See additionally Front door specifications and also the Subject Matters for more advice about normal requirements pertaining to admittance,
experience while offering. All Educational facilities require: Any Level/IB Higher-level Mathematics*, A degree Additionally Numbers (A quantity learners exclusively), STEP
Some Universities Involve: Your Level/IB More fantastic range Physics (the following condition applies to Maths with Physics just) This self-paced program is for any kind of pupil with arithmetic
along with any person who wishes to improve their connection using mathematics. | {"url":"https://hiptv.tv/what-would-be-the-distinct-levels-of-numbers-primary-numbers-algebra-etc/","timestamp":"2024-11-02T17:57:56Z","content_type":"text/html","content_length":"41056","record_id":"<urn:uuid:a07533af-e660-466a-ac68-ef74e7e63e2a>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00361.warc.gz"} |
Ear Rational Thoughts
The top 500 individual single-seasons for home runs all time, summed per year, normalized to standard deviations (from the 1920-1994 period). The red lines are the years 1995 (the year after the
strike) and 2007 (the first year of HGH testing). All of the years in this period are between 4 and 8 standard deviations above the mean (note that the data are slightly non-gaussian so the mean and
stddev are slightly inaccurate). | {"url":"https://earrational-thoughts.typepad.com/ear_rational_thoughts/sports/","timestamp":"2024-11-11T03:34:00Z","content_type":"application/xhtml+xml","content_length":"34866","record_id":"<urn:uuid:24952aa6-81fc-4138-b737-43f6f2ed8b2d>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00697.warc.gz"} |
Vector Addition Calculator
This online calculator performs vector addition and displays vectors and vector sum graphically.
Below you can find the vector addition calculator. It calculates the vector sum every time you add an entry into the vectors table and displays results graphically. I've tried to make it as universal
as possible; thus, you can add vectors using two alternative notations - cartesian coordinates (see Cartesian coordinate system) and polar coordinates (see Polar coordinate system). If you choose
cartesian, you need to enter the x and y components (or coordinates) of a vector. If you choose polar, you need to enter radial (often called the magnitude) and angular (often called the polar angle)
components (or coordinates) of a vector. Note that angular coordinates can be entered either as degrees or as radians. Additional details regarding how addition is performed and how to perform a
subtraction can be found below the calculator
Internally, the calculator converts all entered vectors into cartesian form. It calculates their x and y coordinates using the following conversion formulas:
$x=r cos \theta\\y=r sin \theta$
Then it performs the vector addition, which is very simple and where the vector sum can be expressed as follows:
For vectors $A= (x_1, y_1)$ and $B= (x_2, y_2)$ the vector sum is $A+B= (x_1 + x_2, y_1 + y_2)$
All entered vectors and their sum are also plotted on the graph below the results, so you can see the graphical result of the operation, where the vector sum is shown in red. The vector sum is
plotted by placing vectors head to tail and drawing the vector from the free tail to the free head (so-called Parallelogram law).
And of course, you can use this calculator to calculate vector difference as well, that is, the result of subtracting one vector from another. This is because the vector difference is a vector sum
with the second vector reversed, according to:
$A-B=A+ (-B)$
To get reversed or opposite vector in cartesian form, you simply negate the coordinates. In the polar form, you can either add 180 degrees to the angular coordinate or negate the radial coordinate
(either method should work).
URL zum Clipboard kopiert
Ähnliche Rechner
PLANETCALC, Vector Addition Calculator | {"url":"https://de.planetcalc.com/8066/","timestamp":"2024-11-01T23:36:48Z","content_type":"text/html","content_length":"56314","record_id":"<urn:uuid:507c7bd1-6715-4340-8bb9-be3858250347>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00329.warc.gz"} |
How to Create a Regression Multicollinearity Table (VIF)
UPCOMING WEBINAR: How to automate data-rich reports and presentations Register now >>
This article describes how to compute the variance inflation factors (VIF) of linear models and generalized variance-inflation factors (GVIF) for generalized linear models to diagnose
Note: This feature is not compatible with multinomial logit regressions.
How to Save Probabilities of Each Response of Regression Models
How to Test Residual Serial Correlation (Durbin-Watson) of Regression Models | {"url":"https://help.displayr.com/hc/en-us/articles/4402173883663-How-to-Create-a-Regression-Multicollinearity-Table-VIF","timestamp":"2024-11-13T14:43:04Z","content_type":"text/html","content_length":"29999","record_id":"<urn:uuid:2e7ddff8-b3d0-4620-a17b-2cc1356a81cb>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00119.warc.gz"} |
Distoriton of area and conditioned Brownian motion
In a simply connected planar domain D the expected lifetime of conditioned Brownian motion may be viewed as a function on the set of hyperbolic geodesics for the domain. We show that each hyperbolic
geodesic γ induces a decomposition of D into disjoint subregions {Mathematical expression} and that the subregions are obtained in a natural way using Euclidean geometric quantities relating γ to D.
The lifetime associated with γ on each Ω[j] is then shown to be bounded by the product of the diameter of the smallest ball containing γ{n-ary intersection}Ω[j] and the diameter of the largest ball
in Ω[j]. Because this quantity is never larger than, and in general is much smaller than, the area of the largest ball in Ω[j] it leads to finite lifetime estimates in a variety of domains of
infinite area.
• Mathematics Subject Classification (1991): 60J65, 31A15
ASJC Scopus subject areas
• Analysis
• Statistics and Probability
• Statistics, Probability and Uncertainty
Dive into the research topics of 'Distoriton of area and conditioned Brownian motion'. Together they form a unique fingerprint. | {"url":"https://experts.syr.edu/en/publications/distoriton-of-area-and-conditioned-brownian-motion","timestamp":"2024-11-13T05:12:22Z","content_type":"text/html","content_length":"49013","record_id":"<urn:uuid:7507be5e-52f4-4afa-b960-6376f6c00f73>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00836.warc.gz"} |
Diana Miller and Bettye Joyce Working on a chalkboard during their French class. EKU Photo Collection 1956 ca photograph image 0001-005-classes_foreign_languages-001 Students working a math problem
on the chalkboard EKU Photo Collection 1959 photograph image 0001-005-classes_mathematics-001 Students working a math problem Women and man in ROTC uniform standing by the blackboard with a math
equation on it. EKU Photo Collection 1941 photograph image 0001-005-classes_mathematics-002 | {"url":"https://digitalcollections.eku.edu/items/browse?tags=chalkboards&sort_field=Dublin+Core%2CCreator&sort_dir=a&output=dcmes-xml","timestamp":"2024-11-11T14:19:23Z","content_type":"application/rdf+xml","content_length":"2268","record_id":"<urn:uuid:3b14ff77-f5d4-438b-96cd-30d00bc318b8>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00427.warc.gz"} |
My Trilogy: Activities for Algebra
The workbooks are all done – three workbooks of activities and assessments for the algebra classroom. The activities (designed as puzzles and games mostly) are designed to replace lecture with more
active classroom learning. The puzzles tease out similarities and differences between concepts within the chapters, and span the ideas that run throughout algebra. The assessments are topic specific
– I grow weary of the CATs which don’t seem to be particularly applicable to math classes (although I do use the muddiest point one in my online courses sometimes).
Although the workbooks are written to accompany the Tussy/Gustafson Algebra series, I don’t see any compelling reason why you couldn’t use them with any algebra course. Just find the Table of
Contents that most closely matches your course. There are hundreds and hundreds of pages of activities. No kidding. I can’t believe how many hundreds of activities there turned out to be living in my
You can view (and use) some sample activities and assessments here. More about the Tussy/Gustafson Algebra series is here. The publisher is Cengage Learning.
Beginning Algebra: ISBN-13: 978-0-495-55468-4
Intermediate Algebra: ISBN-13: 978-0-495-55459-6
Beg & Int Algebra: ISBN-13: 978-0-495-55478-3
About Author | {"url":"https://edgeoflearning.com/my-trilogy-activities-for-algebra/","timestamp":"2024-11-11T13:04:18Z","content_type":"text/html","content_length":"191345","record_id":"<urn:uuid:ad93082e-f4d7-4818-8ae9-00fa20c84dcb>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00251.warc.gz"} |
Alexandre de Siqueira
Once in a while we need to estimate the area of a dataset in which we are interested. This area could give us, for example, force (mass vs acceleration) or electric power (electric current vs
Once in a while we need to estimate the area of a dataset in which we are interested. This area could give us, for example, force (mass vs acceleration) or electric power (electric current vs
Once in a while we need to estimate the area of a dataset in which we are interested. This area could give us, for example, force (mass vs acceleration) or electric power (electric current vs | {"url":"https://www.dsprelated.com/blogs-1/nf/Alexandre_de_Siqueira.php","timestamp":"2024-11-02T07:33:42Z","content_type":"text/html","content_length":"24463","record_id":"<urn:uuid:365035b1-3158-4855-8a1e-9cd03965b926>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00772.warc.gz"} |
Coalgebraic Characterizations of Automata-Theoretic Classes
Joost Winter
Promotor: prof.dr. J.J.M.M. Rutten (RU and CWI)
Co-promotor: dr. M.M. Bonsangue (UL)
Radboud Universiteit Nijmegen
Date: 1 July, 2014, 14:30
Automata theory is the area within theoretical computer science in which abstract machines, or automata, are studied. Automata theory is closely connected with the study of formal languages, and the
classes of formal languages that can be described or recognized by various types of automata or formal grammars. Important instances of such classes are the regular and context-free languages, which
together form the first two levels of the Chomsky hierarchy.
In this dissertation, various automata-theoretic classes are studied from a coalgebraic point of view. Coalgebra offers an abstract view on a variety of state based systems, rooted in category
theory, the field of mathematics abstractly studying the structural correspondences between various mathematical theories.
Another example of automata-theoretic classes studied in this dissertation, besides formal languages, is constituted by streams, i.e. infinite sequences that are described coinductively. On an even
more general level, we study classes of formal power series in non-commuting variables, of which both formal languages and streams can be seen as instances. Such coinductive descriptions, in general,
take the shape of behavioural differential equations. Commonly, direct correspondences can be given between the various formats of behavioural differential equations, and the various
automata-theoretical classes.
Such classes include the regular languages (generalizing to the rational power series), of which we recall the (already existing) coalgebraic presentation in Chapter 2; the context-free languages
(generalizing to algebraic power series), presented in Chapter 3, and the k-automatic and k-regular sequences, which have been introduced by Allouche and Shallit, in Chapter 5.
Furthermore, we approach a part of the earlier material from a bialgebraic perspective, and establish a connection to the abstract framework of lambda-bialgebras and distributive laws. In particular,
we look at several methods in which we can synthesize the work in Chapter 3 and the bialgebraic approach from Jacobs, Bartels et al., and discuss the problems encountered when doing this.
Finally, an implementation of the coinductive calculus of streams in the functional programming language Haskell is given, in which the formats presented in the earlier chapters can be described.
Connecting to this, in Appendix C the material from the earlier chapters is illustrated by a variety of examples of infinite sequences, which can be described coinductively as streams. | {"url":"https://ipa.win.tue.nl/?event=coalgebraic-characterizations-of-automata-theoretic-classes","timestamp":"2024-11-06T20:06:21Z","content_type":"text/html","content_length":"37893","record_id":"<urn:uuid:0b39626c-dd85-445a-af3a-c89a70f2a946>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00626.warc.gz"} |
Mathematical Analysis II B
1. Mathematical Analysis II B
Mathematical Analysis II B
At the end of this curricular unit, students should have acquired knowledge, skills and competences that allow them to:
- Work with basic notions of topology in R^n.
- Understand the rigorous notion of limit, continuity and differentiability of vector functions of a real variable.
- Apply vector functions of a real variable in the parameterization and study of curves.
- Understand the rigorous notion of limit and continuity of real and vector functions of several real variables and compute limits.
- Know the notion of partial derivative and differentiability for functions of several real variables.
-Understand and apply the implicit function theorem and the inverse function theorem.
- Know Taylor''s formula and applications to the study of functions and its extremums.
- Know the notion of double and triple integral and how to compute these integrals using appropriate coordinates.
- Know some applications of double and triple integrals.
- Know the notion of line integral, its applications, and fundamental results.
- Know the notion of surface integral, its application and fundamental results.
General characterization
Responsible teacher
Ana Margarida Fernandes Ribeiro
Weekly - 4
Total - 48
Teaching language
Differencial and integral calculus on R. Basic knowledge of matricial calculus.
Any multivariate analysis book can be helpful. Some examples:
Calculus; Anton, Bivens and Davis, Wiley (8th edition).
Cálculo, vol 2; Tom M. Apostol, Ed. Reverté.
Curso de Análise, vol 2; Elon L. Lima, Ed IMPA (projecto Euclides).
Calculus III, Jerrold Marsden and Alen Weinstein, Springer.
Teaching method
The problem-solving sessions consist of a presentation of the subject, along with illustrative examples.
Practical classes consist of problem solving and analysis. Students will be required to previously prepare exercises that will be presented on the board for the class, with subsequent group
discussion. These exercises will be chosen from a list provided by the teachers.
Any doubts are clarified during classes or in sessions designed to assist students or even in sessions arranged directly between student and teacher.
Evaluation method
The Continuous Assessment of the curricular unit comprises:
Theoretical-Practical Assessment: two written tests, each lasting 1h30 minutes, to be carried out during the semester. Each test will be classified between 0 and 8 values.
Summative Assessment: delivery of solved exercises. In each class, two exercises will be choosen to be delivered the resolution in the next class. Among all the students who deliver the resolution,
two will be chosen to present their resolution on the board. The final grade for this component, between 0 and 4, will be assigned by the teacher based on the quality and quantity of resolutions
delivered. In this evaluation component, more than a correct resolution, the student''s work will be valued.
Frequency: Frequencyis obtained by delivering solutions for the exercises in more than 5 classes. Students with student-worker status and students who attended the previous edition of the curricular
unit are exempt from obtaining it. Only students with Frequency will have final classification in the curricular unit.
A student who meets the Frequency criterion will have a final classification by Continuous Assessment equal to T1 + T2 + AS, rounded to the nearest integer. Where T1 and T2 are the final grades of
the first and second test, respectively, and AS is the Summative Assessment grade. The student will obtain approval in the curricular unit, by Continuous Assessment, if this classification is equal
to or greater than 10 values.
Final Exam: Students who have not been approved by Continuous Assessment and who have obtained Frequency of the curricular unit, can take an Final Exam. This is a written exam, lasting 3 hours, which
evaluates all the contents taught in the curricular unit. The exam is divided into two parts, each one classified from 0 to 8 values, whose evaluated material corresponds, respectively, to the first
and second test. The final grade will be T1 + T2 + AS, where T1 and T2 are the final grades of the first and second part. The student is approved if it is greater than or equal to 10 values.
Classification Improvement: Students approved in the curricular unit may request, upon compliance with all the conditions imposed by NOVA FCT, Classification Improvement by taking the Final Exam. The
final grade will be T1 + T2 + AS.
On the day of the test, students must present themselves with a blank test booklet, writing material and official identification document. All tests and examinations must be carried out without
consultation and without the use of any computational calculation material.
Working students, or other students who for some reason cannot attend practical classes, can ask the teacher to be evaluated without the Summative Assessment component. In that case the final
Continuous Assessment or Exam grades will be calculated according to the formula (T1 + T2)5/4. In this situation the Frequency is given automatically. Attention: this request must be made by the end
of the second week of classes, otherwise the student will be assessed with the Continuous Assessment component.
In any omitted situation, the NOVA FCT Assessment Regulation, of 31 July, 2020, applies.
Subject matter
Revision of some analytical geometry concepts. Conics. Quadrics.
Topological notions in R^n. Vector functions of a real variable and functions of several variables. Domain, graph, level curves and level surfaces. Limits and continuity of functions of several
Partial derivatives and Schwarz theorem. Derivative according to a vector. Jacobian matrix. Gradient vector and the notion of differentiability. Differentiability of composite function. Taylor''s
Implicit function theorem and inverse function theorem. Local and global extrema. Constrained extrema and Lagrange multipliers.
Double and triple integrals. Iterated integrals and Fubini''s theorem. Change of variables in integrals. Double integrals in polar coordinates. Triple integrals in cylindrical and spherical
Coordinates. Applications.
Vector fields. Gradient, divergence and rotational. Closed fields. Conservative fields. Applications.
Line integrals of scalar and vector fields. Fundamental theorem for line integrals. Green''s theorem. Applications.
Surface integral. Flow of a vector field through a surface. Stokes'' Theorem of and Gauss'' theorem. Applications.
Programs where the course is taught: | {"url":"https://guia.unl.pt/en/2023/fct/program/1058/course/10476","timestamp":"2024-11-09T01:19:52Z","content_type":"text/html","content_length":"24971","record_id":"<urn:uuid:36c2c90d-6aff-497b-9c18-7a7763297746>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00356.warc.gz"} |
Walk-Forward Optimization in Python
When researching trading strategies, walk-forward optimization is, without a doubt, one of the most valuable tools you could incorporate into your workflow. Although the average trader does not even
properly backtest their strategies, it is worth considering that most of them lose money on average.
I’ll assume that you already know what walk-forward optimization is about, but you can check out this article I wrote on the topic as a quick refresher on the main aspects and advantages of this
By the end of this article, you’ll be better equipped than most algorithmic traders. So, without any further ado, let’s get started!
Walk-Forward Optimization in Python
I tried to make this article as easily digestible as possible, but the task is by no means easy. This is why I decided to break it down into smaller parts. First, we’ll describe and implement the
trading strategy. After having done so, we’ll optimize its parameters. Last but not least, we’ll implement the walk-forward optimization itself.
We will be using Backtesting.py, which is one of the most mature, popular, and reliable backtesting frameworks available for Python. Since it does not have inbuilt capabilities for walk-forward
optimization, we will be coding that feature ourselves.
Step 1: Implementing the Trading Strategy in Python
Let’s start by importing the required libraries that we will be using throughout this tutorial and declaring a few parameters. We will first implement the strategy used for doing the walk-forward
import yfinance as yf
import pandas as pd
from backtesting import Strategy, Backtest
import seaborn as sns
from tqdm import tqdm
TICKER = 'AAPL'
START_DATE = '2015-01-01'
END_DATE = '2022-12-31'
FREQUENCY = '1d'
As you can infer, we’ll use AAPL’s stock price from 2015 to 2022, so let’s fetch the data using yahoo finance!
I removed the timezone awareness of the DateTimeIndex to make things easier down the line.
df_prices = yf_ticker = yf.Ticker(TICKER).history(start=START_DATE,end=END_DATE,interval=FREQUENCY)
df_prices.index = df_prices.index.tz_localize(None)
Our strategy is going the be a simple yet interesting mean-reversion strategy. We will buy the stock whenever the current price is at its lowest. Conversely, we will sell the asset if the price is
currently at its highest during the lookback period. We will refer to these prices as high and low watermarks.
Implementing both functions is rather straightforward if we leverage pandas functionalities:
def high_watermark(highs, loockback):
return pd.Series(highs).rolling(loockback).max()
def low_watermark(mins, loockback):
return pd.Series(mins).rolling(loockback).min()
Lets us also go ahead and implement the strategy for our backtest:
class BLSHStrategy(Strategy):
n_high = 30
n_low = 30
def init(self):
self.high_watermark = self.I(high_watermark, self.data.Close, self.n_high)
self.low_watermark = self.I(low_watermark, self.data.Close, self.n_low)
def next(self):
if not self.position:
if self.low_watermark[-1] == self.data.Close[-1]:
elif self.high_watermark[-1] == self.data.Close[-1]:
The code is pretty straightforward, but let’s go through it step by step:
• During initialization, we create two indicators: high_watermark and low_watermark. Both lookback periods are defined in n_high and n_low and are by no means required to be equal.
• The next method is where the magic happens. If we don’t already hold a position in AAPL, we check if the price is currently at the 30-day lowest. If so, we will issue a buy order. On the other
hand, if we already hold a position and the price is at its highest, we will issue an order to close the position.
To keep things simple, we will assume commission-free trading. By only selling if we hold a position, we force our strategy to be long-only.
bt = Backtest(df_prices, BLSHStrategy, cash=10_000, commission=0,exclusive_orders=True)
stats = bt.run()
The output of our simple backtest
Step 2: Optimizing the parameters of the strategy
The next step is optimizing the high and low watermark lookback period parameters. For this, we will use the Backtesting library’s optimize() method. This method will run the backtest for all the
combinations of the parameters passed to it.
bt = Backtest(df_prices, BLSHStrategy, cash=10_000, commission=0,exclusive_orders=True)
stats, heatmap = bt.optimize(
n_high=range(20, 60, 5),
n_low=range(20, 60, 5),
# constraint=lambda p: p.n_high > p.n_low,
maximize='Equity Final [$]',
method = 'grid',
Let’s take a few seconds and take a closer look at the most relevant parameters:
• We test different combinations of the parameters n_high and n_low. We will try values between 20 and 60. To reduce processing times, we will only test multiples of 5.
• As with any optimization, we must choose a target variable to maximize (or minimize). In this case, we will maximize the profitability of the strategy. Keep in mind that this variable completely
ignores the incurred risk. This could be accounted for by maximizing the Sharpe Ratio instead.
• We can also add constraints. We won’t introduce any in this case, but I left the feature commented out in case you want to use it.
The method returns two variables: the statistics of the best-performing combination and a matrix with the results of all the combinations we tested. We already know what the stats output looks like,
so let’s jump right into the heatmap!
As a side note, it’s worth mentioning that the empty cells are there because we limited the number of tries to 56.
Step 3: Walk-Forward Optimization of the Strategy
Backtesting.py has no inbuilt functionalities for performing walk-forward optimization, but we can extend the library to perform this task with a few tweaks!
Having 8 years of data, we will do 5 iterations. Each iteration will use 4 years of data, where the first 3 will be used for optimizing the parameters (in-sample data) and the fourth year as a test
set (out-of-sample data).
from datetime import datetime
iterations = [
'in_sample': [datetime(2015,1,1),datetime(2017,12,31)],
'out_of_sample': [datetime(2018,1,1),datetime(2018,12,31)]
'in_sample': [datetime(2016,1,1),datetime(2018,12,31)],
'out_of_sample': [datetime(2019,1,1),datetime(2019,12,31)]
'in_sample': [datetime(2017,1,1),datetime(2019,12,31)],
'out_of_sample': [datetime(2020,1,1),datetime(2020,12,31)]
'in_sample': [datetime(2018,1,1),datetime(2020,12,31)],
'out_of_sample': [datetime(2021,1,1),datetime(2021,12,31)]
'in_sample': [datetime(2019,1,1),datetime(2021,12,31)],
'out_of_sample': [datetime(2022,1,1),datetime(2022,12,31)]
The following script is where all the meat is! We are putting everything together to perform the walk-forward optimization I promised!
The code is a little messy/daunting, so let me break it into smaller chunks!
1. Iterate over the list of dictionaries, each containing the in-sample and out-of-sample periods.
2. Filter the data only to include the relevant dates.
3. Calculate the optimal parameters using the in-sample data.
4. Run the backtest for the out-of-sample data using the optimal parameters.
5. Append relevant metrics to a list of results.
report = []
# 1: We iterate over the list of dictionaries
for iter in tqdm(iterations):
# 2: We filter the data to only include the relevant dates.
df_is = df_prices[(df_prices.index >= iter['in_sample'][0]) & (df_prices.index <= iter['in_sample'][1])]
df_oos = df_prices[(df_prices.index >= iter['out_of_sample'][0]) & (df_prices.index <= iter['out_of_sample'][1])]
#3: Calcualte the optimal parameters using the in-sample data.
bt_is = Backtest(df_is, BLSHStrategy, cash=10_000, commission=0,exclusive_orders=True)
stats_is, heatmap = bt.optimize(
n_high=range(20, 60, 5),
n_low=range(20, 60, 5),
maximize='Equity Final [$]',
method = 'grid',
# 4: Run the backtest for the out-of-sample data using the optimal parameters.
BLSHStrategy.n_high = stats_is._strategy.n_high
BLSHStrategy.n_low = stats_is._strategy.n_low
bt_oos = Backtest(df_oos, BLSHStrategy, cash=10_000, commission=0,exclusive_orders=True)
stats_oos = bt_oos.run()
# 5: Append relevant metrics to a list of results
'start_date': stats_oos['Start'],
'end_date': stats_oos['End'],
'return_strat': stats_oos['Return [%]'],
'max_drawdown':stats_oos['Max. Drawdown [%]'],
'ret_strat_ann': stats_oos['Return (Ann.) [%]'],
'volatility_strat_ann': stats_oos['Volatility (Ann.) [%]'],
'sharpe_ratio': stats_oos['Sharpe Ratio'],
'return_bh': stats_oos['Buy & Hold Return [%]'],
'n_high': stats_oos._strategy.n_high,
'n_low': stats_oos._strategy.n_low
Here’s a Pandas DataFramewith the results:
Step 4: Further Improvements
By now, we already have a walk-forward optimization function that works. However, there are a few things that we can improve, and I conveniently ignored them in the previous section to make them
stand out more.
• Add Exposure Percentage: To compare apples to apples, it is important to know the exposure percentage of the strategy. A strategy with the same returns as the buy and hold that only holds the
asset 50% of the time is considered better because it is less exposed to the market.
• Scale the Buy & Hold Returns: To compare the strategy to its benchmark, we will scale the returns of the Buy & Hold to match the exposure of our strategy.
• Add in-sample Returns: it is worthwhile to compare the returns of the in-sample and out-of-sample periods. If the OOS backtests yield much lower returns, we can conclude, with a high degree of
certainty, that we are overfitting.
• Plot the heatmaps of each optimization: Although not strictly a requirement, it is desirable for the 5 heatmaps to be somewhat similar.
from tqdm import tqdm
report = []
for iter in tqdm(iterations):
df_is = df_prices[(df_prices.index >= iter['in_sample'][0]) & (df_prices.index <= iter['in_sample'][1])]
df_oos = df_prices[(df_prices.index >= iter['out_of_sample'][0]) & (df_prices.index <= iter['out_of_sample'][1])]
bt_is = Backtest(df_is, BLSHStrategy, cash=10_000, commission=0,exclusive_orders=True)
stats_is, heatmap = bt_is.optimize(
n_high=range(15, 40, 5),
n_low=range(15, 40, 5),
maximize='Equity Final [$]',
method = 'grid',
BLSHStrategy.n_high = stats_is._strategy.n_high
BLSHStrategy.n_low = stats_is._strategy.n_low
bt_oos = Backtest(df_oos, BLSHStrategy, cash=10_000, commission=0,exclusive_orders=True)
stats_oos = bt_oos.run()
'start_date': stats_oos['Start'],
'end_date': stats_oos['End'],
'return_strat': stats_oos['Return [%]'],
'max_drawdown':stats_oos['Max. Drawdown [%]'],
'ret_strat_ann': stats_oos['Return (Ann.) [%]'],
'volatility_strat_ann': stats_oos['Volatility (Ann.) [%]'],
'sharpe_ratio': stats_oos['Sharpe Ratio'],
'return_bh': stats_oos['Buy & Hold Return [%]'],
'n_high': stats_oos._strategy.n_high,
'n_low': stats_oos._strategy.n_low,
'exposure': stats_oos['Exposure Time [%]'],
'bh_scaled': stats_oos['Buy & Hold Return [%]'] * stats_oos['Exposure Time [%]'] / 100,
'is_heatmap': heatmap,
'sharpe_is': stats_is['Sharpe Ratio'],
Finally, let’s plot the heatmap of each optimization:
import matplotlib.pyplot as plt
import math
plt.rcParams['figure.figsize'] = [20, 10]
rows = len(report)
for idx, res in enumerate(report):
plt.subplot(math.floor(rows/2), math.ceil(rows/2), idx+1)
plt.title(f"Iter # {idx+1} - Year {res['start_date'].year}")
As you can see, implementing this feature is not rocket science but is definitely out of reach for the average algorithmic trader. I hope to have conveyed the advantages of incorporating this
technique int your workflow, and please don’t hesitate to drop a comment if you have any further questions!
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Price and Capacity Competition
We study the efficiency of oligopoly equilibria in a model where firms compete over capacities and prices. Our model economy corresponds to a two-stage game. First, firms choose their capacity
levels. Second, after the capacity levels are observed, they set prices. Given the capacities and prices, consumers allocate their demands across the firms. We establish the existence of pure
strategy oligopoly equilibria and characterize the set of equilibria. We then investigate the efficiency properties of these equilibria, where efficiency is defined as the ratio of surplus in
equilibrium relative to the first best. We show that efficiency in the worst oligopoly equilibria can be arbitrarily low. However, if the best oligopoly equilibrium is selected (among multiple
equilibria), the worst-case efficiency loss is $2(\sqrt{N}-1)/(N-1)$ with $N$ firms, and this bound is tight. We also suggest a simple way of implementing the best oligopoly equilibrium. | {"url":"https://stanford.edu/~kostasb/research/capacity/","timestamp":"2024-11-05T03:24:38Z","content_type":"text/html","content_length":"15332","record_id":"<urn:uuid:cea38292-5fe8-4e7a-a1bd-324298a44d98>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00077.warc.gz"} |
Napoleon’s Theorem
Napoleon’s Theorem
Beyond his great strategic abilities, Napoleon showed great interest for the Euclidian Geometry. He discovered the following theorem: Given any triangle on the Euclidian plane, if we construct
equilateral triangles with sides the side lengths of the primary triangle, then by joining the vertices of each equilateral triangle, with the opposite vertex of the primary triangle, then the three
lines constructed concur.
Moreover, he generalized the theorem of Napoleon in Euclidian Geometry, rediscovering the theorem of the German mathematician Kiepert by a totally different way. Eleftherios has great interest of
creating an experimental antigravity device. He employs advanced physics and the lexarithmic theory in his experiments, hoping that he will find the principal cause of gravity and therefore
constructing an antigravity device. | {"url":"https://www.synarithmos.com/en/achievements-en/25-napoleon-s-theorem.html","timestamp":"2024-11-02T01:37:49Z","content_type":"text/html","content_length":"16347","record_id":"<urn:uuid:bd8ae8a1-410a-4bb5-b5f4-e22291b571d7>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00365.warc.gz"} |
FSC 1st Year Physics Chapter 5 Notes PDF - Ilmi Hub
FSC 1st Year Physics Chapter 5 Notes PDF
By admin
In this post, I am sharing FSC 1st Year Physics Chapter 5 Notes PDF for the students of Intermediate Part 1. This chapter’s name is Circular Motion. Students can download 11th class chapter 5
Circular Motion Notes PDF in there laptop or mobile. These Physics Notes are for all the boards working under Punjab Board like Gujranwala Board, Lahore Board, Faisalabad Board, Multan Board,
Rawalpindi Board, Sargodha Board, DG Kahn Board, and Sahiwal Board. Here are the complete FSC 1st Year Physics Notes PDF chapterwise.
FSC Part 1 Physics Chapter 5 Circular Motion Notes PDF Download
What is circular motion?
When a body is moving in circle then its motion said to be circular or angular motion.
moon revolve around the earth in circular path
electron revolve around the nucleus in nealy circular orbit.
What is Radian?
Radian can be defined as “angle made by an arc at the center of a circle, whose length equal to radius of the circle.”
Define degree.
If the circumference of a circle divided into 360 equal parts, the angle made by each part at the center of the circle, called one degree.
Prove that s = rq
If an arc of length s of a circle of radius r subtends an angle q at the center of the circle then
q = arc length / radius = s/r rad
or s = rq
Arc length is always equal to the product of radius and the angle made by arc at the center.
What is the relation between radian and degree? Or show that 1 rad = 57.3^0.
In one revolution (rev) there are 360^0 or 2p rad angle are present, so
1rev = 2p rad = 360^0
2p rad = 360^0
rad = 360^0/2p = 180^0/p = 180^0/3.14
1 rad = 57.3
Define angular velocity.
The rate of change of angular displacement of a body moving in a circular path called angular velocity.
if a body move from point P[1 ]to P[2 ] and covers angular displacement Aq in time At then average angular velocity given by.
Related links:
FSC 1st Year Physics Chapter 1 Notes pdf download
1st year physics chapter 2 notes pdf download
11th class physics chapter 3 notes pdf download
Fsc part 1 physics chapter 4 notes pdf download
1st Year Physics MCQs with Answers All Chapters
In this post, I am sharing FSC 1st Year Physics MCQs with answers of all chapters in PDF f…
3 min read
1st Year Physics Short Questions Notes of All Chapters PDF
In this post, I am sharing 1st Year Physics Short Questions Notes of all Chapters PDF for …
3 min read
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Can I get professional help for my Chi-square tests project? | Pay Someone To Do My SPSS Assignment
Can I get professional help for my Chi-square tests project? We have thousands of paperwork from the local EMT clinics who, among other things, bring additional information and tools to the lab that
is going to get you out of stress, anxiety, and depression. Our primary care plan provides the following: Call your clinic Call the EMT right away, or call its clinic on all business phone numbers,
or call them once in a while for an hour to get help to take care of your Chi-square tests. For the most part, Chi-square tests are single-shot unless the clinic is a specialist. Those who don’t have
lab test results like the one above from the Chi-square exam aren’t enrolled at the clinic and are required to go to their test schedule later if they’re sick, out of concern for that test result. If
you don’t have those tests, they’re not eligible for registration. So, since we don’t know if you’ve been on the clinic’s telemed claims team for the past ten years or so, a phone call: Sell for $500
or more An hour or 60 minutes? Your Chi-square test is excellent If your Chi-square test has been on the clinic’s telemed claims team on the last five years, we’re able to find your test for you
quickly and easily. If you have several different tests in one call but didn’t find your test and are in need of testing, call them online to find out which one is the first. Call your Chi-square
test advisor Chenium magnets and I have known browse around this site who have had tests on prescription and unlicensed medical marijuana for quite some time. We decided to sign an agreement with the
Medical Marijuana Treatment Administration to sign up for an unlicensed claim. The agreement was worth a combined price of $850 and was signed by Dr. Steve Chenano, a local affiliate of EMT, to
close. The two doctors in the group said after hearing they didn’t have their own labs or were treated at the clinic, they had always been able to visit the facility and examine their case, and had
plenty of access to any records, and that they could have a test performed on their own if they were diagnosed with an “EMT Doctor.”[1] “We feel that we are good candidates for EMT claims,” said Dr.
Chenano, a clinical professor of psychiatry at the University of Michigan. “But there are problems.” When they do have a lab test, they ask you to contact them, give them a call, and ask for your
appointment. Dr. Kim Cifu, a clinical professor of mental health and psychiatry at the University of Michigan, is the co-supervisor. She said that if you are doing tests on unlicensed medical
marijuana to date, they were called for an unlicensed claim when the doctor who signed the agreement wasn’t really attached to his, and they didn’t have the ability to do so in a hospital. Chenano
did not use the EMT’s offices for testing, which allowed him access inside the clinics.
Online Classwork
Chenano believes that this creates stress for your Chi-square test case and adds to the stress that leads up to the EMT call. A Chi-square test results can have troubling results, and the doctor who
ordered the labs could be found at the tests themselves and you’d have to contact them. He said he started job as a teacher in a medicine clinics, though they asked to be billed $400 a month for
1,000-test week for the first 6 months: Would you hire an EMT or do a technician? Or would you fill out a short form? The doctors in the group pointed out that, “if you want to have another test
done, then you’re taking some time off. You can be charged for the tests. You’ll need to take medication toCan I get professional help for my Chi-square tests project? The answer is quite simple.
Consider these images below. “Gross” is the most common weight measurement for various tests, which comes from the highest-scoring individuals. Several results from the professional Chi-square test
question and calculation include the most accurate results, as well as the most accurate scores. For this comparison, let’s look at which measurements our schools think they should measure correctly.
In other words, they each provide the points they’ve calculated. We can call them 1,2, or 3, which are the best scores for the scale. In some of these cases, these are calculated on the basis of the
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Teachers First Day Presentation
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things is correct?”, and decide whether it is for the right person. What can I do about these questions? Q: Having a ‘chi-square’ score is not one of my goals. How do I get my own Chi-square for the
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one of the pretty high-end scores that are scored by ourCan I get professional help for my Chi-square tests project? Also get 5 other steps to get the Chi-squared function completed or 2 extra steps
to perform for my test I.T Hi everybody I read your project in different forums but when it is presented in today’s papers I get the following help suggested: Unlimited samples if you can use it in
your own study No test given for other test If I did it myself this is it can you help in the following way: Firstly I have to figure out the range of my scale and also figure out the highest value
for the time period. Then I should decide for which range around to use or use the correct amount of time period for its production. But I have already 2 questions and from the question of using the
100% time the sample as many as possible helps me further to get correct result of the group it is clearly shown in the picture. Therefore I should decide for which angle you apply for the sample for
which time period the sample is included and this is the answer I have come to understand.
Can I Get In Trouble For Writing Someone Else’s Paper?
What does %C4 = 40%C5? This is the answer I have come to understand thanks to you. To me the maximum limit per time period it is 6 degrees above 45 degrees so I don’t see an amount of your results
which is 5 for A, B or C. So there is no way of getting this result in different way. I suggest you to understand. As for the sample I like to use the R-squared to determine the average for a
specific time period. The range around here should be 60% up to 90% below 40%, because 4% of the time during the entire study is occupied in the sample and also the time period is occupied 4%. I
believe that the best means of my method would be to have our group based on our group average so it can be an interesting idea. What sort of rate of time is better or where can I use it? To me the
best method to get the my group average value for that time period is to have our group average defined per unit volume. Let me tell you now how do I do this in my own project and use a simple
estimate of this figure and then I can change the size for the sample (just increase the amount of the volume if need be) to 10% of the total volume of the group. However if you compare with other
method I think you are not giving good success. As for 100% time I don’t have enough power. I think it might be better to leave a low limit in the sample and use a small sample to find an amount of
the sample. Please say in the end I hope this help. 🙂 CKG PSD: It’s a part of GP and it costs’t enough. | {"url":"https://spsshelponline.com/can-i-get-professional-help-for-my-chi-square-tests-project","timestamp":"2024-11-08T08:33:47Z","content_type":"text/html","content_length":"167225","record_id":"<urn:uuid:d626f399-7649-4c80-9e05-4a908a048f9f>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00626.warc.gz"} |
Try This – A Final Exam in Finite Math - 3 Quarks Daily
Try This – A Final Exam in Finite Math
by John Allen Paulos
Math Patterns Final Exam, Prof. Paulos
1.) Two large screen TV’s are essentially the same except for their size. The width of the larger screen
is 48 inches and that of the smaller is 30 inches. The area of the larger screen is how many times the area of the smaller screen.
1. 1.60 B 1.44 C 2.560 D 4.097
2.) The weight of the smaller TV is 44 pounds. What is the weight of the larger TV?
A 70.4 pounds B 112.64 pounds C 180.22 pounds
3.) Meteors that strike Earth always seem to land in craters, and fatal skiing accidents always seem to happen on the skier’s last run. Is the explanation for these:
A coincidence B reverse causation C Bayes Theorem D humorous flapdoodle
4.) A large pizza has a diameter 2.5 times the diameter of a small one of the same thickness. How many times as much pizza is in the larger one?
A 5 times as much B 6.25 times C15.625 times D 2.5 times
5.) If the national debt is about 30 trillion dollars, how much is each individual American’s share?
A $120,000 B $90,000 C $60,000
6.) In a certain company there are four stockholders, X, Y, Z, and W. X owns 48% of the stock, Y owns 25%, Z 24%, and W the remaining 3% of the stock. If 51% of the vote is needed to pass a measure,
analyze the power situation among X, Y, Z, and W. In how many of the 16 situations (all the 2^4 ways for the 4 stockholders to vote Yes or No) does W’s vote make a difference in the weighted outcome
of the vote?
A 10 B 12 C 8 D 4 E 0
7.) The weights of 100 horse jockeys (people who race horses) are recorded and so are the weights of 100 middle school students. Which set of weights is likely to have greater variability.
A the jockeys B the middle school students C they’re equal
8.) The Earth is about 4.5 billion years old, and Homo Sapiens arouse approximately 200,000 years ago. If you shrink the 4.5 billion years to a single year, then when approximately did humans arrive
on the scene?
A October 15 B early on December 10th C December 30 D late on December 31st
9.) The association of higher spelling test scores with elementary school students’ bigger shoe sizes is
A causal, but not correlational B correlational, but not causal C both D neither
10.) What kills more people each year in this country?
A homicides B car accidents C terrorism D plane crashes
11.) If 22% of the people in a certain neighborhood subscribe to the NY Times, 56 percent subscribe to the Philadelphia Inquirer, and 9 percent subscribe to both. What percent subscribe to exactly
one of these papers?
A 78% B 69% C 60% D 63%
12.) There are two coins on the table, one fair, the other two-headed. You pick one coin at random, flip it twice, and note it comes up heads both times. Given this, what is the probability you chose
the fair coin?
A 1/4 B. 1/3 C 1/5 D 2/7
13.) You can park in a lot every day for a certain fee, or you can risk parking in an illegal spot on the street. If you park illegally, you’ll get a $40 ticket about 20% of the time, about 2% of the
time you’ll be towed and incur a $500 fine when you are, and the other 78% of the time, you’ll get away with it. On average, assuming these percentages remain constant, what will it cost you each day
to park on the street?
A $18 B $40 C $28 D $16
14.) Flip a coin three times. You win $10 if heads come up once, $20 if heads come up twice, $30 if heads come up all three times, and you lose $200 if heads doesn’t come up at all. On average how
much will you win or lose, each time you play this game?
A $120 B $32.50 C $110 D. –$10.
15.)Ten voters are trying to decide upon one of five candidates, A, B, C, D, or E. Their preference rankings are listed below.
E D C B
B A A C
A B B D
D C D A
C E E E
Which of the five candidates is the Borda count winner?
A B C D E
16.) In general, is the ranked choice winner always the plurality winner?
A Yes B No
17.) Assume that most of the class does well on this test getting a variety of scores ranging from 75 to 100, but that 30% of the class get a zero on it. In this case which of the two numbers is
likely to be the larger?
A the mean score B the median score
18.) In the pick 6 (out of 40 numbers) lottery, you pick the numbers 1,2,3,4,5,6, and your friend picks 3,8,9,24,31,39. Who is more likely to get the winning ticket?
A your friend B you C equally likely
19.) Roll a pair of dice. What is probability of getting a sum of exactly 5?
A 5/36 B 4/36 C 10/36 D 2/5
20.) What is the probability of getting a sum of exactly 5 given that the sum is at most 5?
A 5/36 B 4/36 C 10/36 D 2/5
21.) If a study indicates that 36% of ethnic group A and 45% of ethnic group B improves from some treatment, and a second study indicates that 60% of group A and 65% of group B improves, can one
conclude that a higher percentage of group B improves from this treatment?
A Yes B No.
22.) What is more likely: getting four 6’s in arrow when rolling a single die or getting ten heads in a row when flipping a coin?
A equally likely B can’t say C ten heads D four 6’s
23.) If a roulette wheel (18 red, 18 black, 2 green sectors) is spun eight times, what is the probability of landing on red at least once?
A (18/38)8 B 8*(18/38) C 1-(20/38)8 D (20/38)8 E 8-(20/38)8
24.) Linda has a PhD in physics, was president of her senior class, and is an ardent traveler. Which of these is more likely?
A Linda works as a cashier at a convenience store
B Linda works as a cashier at a convenience store and is very active in a local women’s group
25.) Two people, George and Martha, predict the outcomes of 100 coin flips. George correctly predicts 52 of the 100 coin flips, and Martha correctly predicts 31 of the 100 coin flips. Whose
performance, George’s or Martha’s, is most
impressive? That is, most in need of an explanation?
A George’s B Martha’s C same for both
26.) A couple plans on having 4 children. Which is more likely:
A They’ll have two boys and two girls B they’ll have 3 of one sex and one of the other.
27.) In a large hospital, 220 babies are born each month. Is it certain that every month there will be 2 or more babies having the same weight to within an ounce? (Assume newborn babies’ weights are
less than 13lbs)
A Yes B No
28.) There are exactly three doors, A, B, and C, through which a dog can enter the kitchen. If the probability of entering through door A is 2 times the probability the dog will enter through door B,
and the probability the dog will enter through B is 3 times the probability it will enter through door C, what is the probability it will enter through door B? (Let x equal the probability the dog
enters through door C.)
A 2/9 B 1/3 C 3/10 D 4/9
29.) In a drawer there are 6 red, 6 blue, and 6 white socks. You reach in and pick socks at random. What is the smallest number you need to be certain of getting a red pair?
A 5 B 14 C 9 D18
30.) In a criminal trial what probability will the defense attorney stress
A the probability that the prints of an innocent person would match those at the scene of a crime
B the probability that a person whose prints match those at the scene would be innocent
C they’re the same
31.) House A is 6 blocks west and 7 blocks south of house B. In how many ways are there to get from A
to B (walking only along streets and no back-tracking or moving away)?
A 1,418 B 1,716 C 2,114
32.) Matilda has 8 friends and wants to invite 5 of them to a party. A complication is that two of them are feuding and so she can invites just one of them or neither of the, How many guest lists can
she have?
A 46 B 22 C 36 D 56
John Allen Paulos is a Professor of Mathematics at Temple University and the author of Innumeracy and A Mathematician Reads the Newspaper. His most recent book is Who’s Counting –Uniting Numbers and
Narratives with Stories from Pop Culture, Puzzles, Politics, and More. | {"url":"https://3quarksdaily.com/3quarksdaily/2023/09/try-this-a-final-exam-in-finite-math.html","timestamp":"2024-11-14T17:41:34Z","content_type":"text/html","content_length":"61861","record_id":"<urn:uuid:24ff93ce-54aa-4147-8b11-5b597f006a95>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00616.warc.gz"} |
How do you divide \frac { - \frac { 5} { 28x ^ { 2} } } { \frac { 10} { 21x ^ { 3} } }? | HIX Tutor
How do you divide #\frac { - \frac { 5} { 28x ^ { 2} } } { \frac { 10} { 21x ^ { 3} } }#?
Answer 1
See the entire solution process below:
First, rewrite this expression using this rule for dividing fractions:
#(color(red)(a)/color(blue)(b))/(color(green)(c)/color(purple)(d)) = (color(red)(a) xx color(purple)(d))/(color(blue)(b) xx color(green)(c))#
#(color(red)(-5)/color(blue)(28x^2))/(color(green)(10)/color(purple)(21x^3)) = (color(red)(-5) xx color(purple)(21x^3))/(color(blue)(28x^2) xx color(green)(10))#
Next, cancel common terms with the constants in the numerator and denominator:
#(color(red)(-5) xx color(purple)(21x^3))/(color(blue)(28x^2) xx color(green)(10)) = (color(red)(-5) xx color(purple)((7 xx 3)x^3))/(color(blue)((7 xx 4)x^2) xx color(green)((5 xx 2))) = (-cancel
(color(red)(5)) xx color(purple)((cancel(7) xx 3)x^3))/(color(blue)((cancel(7) xx 4)x^2) xx color(green)((cancel(5) xx 2))) =#
Now, use these rules for exponents to complete the division:
#x^color(red)(a)/x^color(blue)(b) = x^(color(red)(a)-color(blue)(b))# and #a = a^color(red)(1)#
#-(3x^color(red)(3))/(8x^color(blue)(2)) = -(3x^(color(red)(3)-color(blue)(2)))/8 = -(3x^color(red)(1))/8 = -(3x)/8#
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Answer 2
To divide the expression (\frac{-\frac{5}{28x^2}}{\frac{10}{21x^3}}), you can multiply the numerator by the reciprocal of the denominator.
This means you can rewrite the expression as:
(\frac{-\frac{5}{28x^2}}{\frac{10}{21x^3}} = -\frac{5}{28x^2} \times \frac{21x^3}{10})
Next, you can simplify this expression:
(-\frac{5}{28x^2} \times \frac{21x^3}{10} = -\frac{5 \times 21x^3}{28x^2 \times 10})
Now, simplify the expression further:
(-\frac{5 \times 21x^3}{28x^2 \times 10} = -\frac{105x}{280})
So, the division of (\frac{-\frac{5}{28x^2}}{\frac{10}{21x^3}}) simplifies to (-\frac{105x}{280}).
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Bernard Roy
Publiée le 2019-04-02
Article de Denis Bouyssou et Daniel Vanderpooten (LAMSADE) publié dans le Bulletin #40 de la ROADEF
Young Bernard Roy
During the 1950s and early 1960s, as the ideas, methods, and applications of operations research (OR) spread around the world, each country tended to adapt OR to fit its professional, academic, and
cultural environments. Often, there were a few dedicated persons who led the way and became recognized as the country’s OR pioneers. In the case of France, such a person was Bernard Roy. After a
first career as a consultant, during which he made major breakthroughs in graph theory and project scheduling, he started a second career as an academic interested in multiple criteria decision
making (MCDM). Among his many achievements, he is the developer of the activity-on- node project scheduling technique and of the famous ELECTRE methods for resolving decision problems with multiple
criteria. Through his research, teaching, consulting, and service to the community, he has been one of the major promoters of OR in France.
Bernard served as vice president (1974–1976) and president (1976–1978) of the Association Française pour la Cybernétique Économique et Technique (AFCET, the French OR society at that time). Bernard
was president of The Federation of European OR Societies (EURO) in 1985–1986, and was awarded the 1992 EURO gold medal, the highest distinction granted by EURO.
Family and childhood
Bernard was born on March 15, 1934 in Moulins-sur-Allier, a medium-sized town in the center of France. He is the only child of René Roy (born 1906) and Jeanne Chérasse (born 1913). Both parents
completed their studies with the brevet (a diploma given to pupils at the end of the ninth grade). Bernard’s maternal grandfather was a railway station manager; his paternal grandfather built and
repaired millwheels. Bernard was the first member of his family to pursue an advanced education. René started his career as a bank teller. In 1934, he became an insurance agent at the Compagnie du
Nord. With the help of Jeanne, he was responsible for a portfolio of clients. René took part in World War II (WWII) and, after the defeat of France, he was sent to Germany as a war prisoner. He
escaped in 1943. After the war, he kept close contacts with several fellow prisoners by inviting them to yearly family gatherings. One of the ex-prisoners was the father of Bernard’s future wife,
Françoise. Bernard first met her during one of these gatherings.
During WWII, Moulins-sur-Allier was in the occupied part of France, but located quite close to the demarcation line. At that time, the communication between the two parts of France was highly
problematic. Both Bernard’s mother and aunt would cross the demarcation line to transmit mail between the two zones (sometimes even helping people to cross the line). His aunt was arrested by the
Germans, but she was soon liberated following a bureaucratic error. German soldiers, realizing the error, paid frequent visits to the family’s home; these visits made a very strong impression on the
young Bernard. The war years were very bleak. Fortunately, Jeanne had relatives living in the countryside, so the family had access to food products that were cruelly missing, and Bernard could enjoy
peaceful holidays.
In 1940, at the age of 6, Bernard started his formal education at a local school. Soon after, he began experiencing vision problems. Due to the war, it was not easy to have access to an
ophthalmologist, but his parents did manage, as best as they could. The first one consulted advised that these problems were somatic. Because things were not getting any better, several other famous
specialists were consulted making various diagnoses, such as a compression of the optical nerves. It was not until 1955 that a correct diagnosis was established; Bernard was suffering from a very
rare type of retina problem (atypical retinitis pigmentosa). As a result, Bernard gradually lost sight, while keeping a limited peripheral vision. Reading became more and more difficult. Writing also
became problematic; after some time, hardly any one could decipher his letters. Bernard kept writing, however, by using the new Reynolds ballpoint pens that just arrived in France. He did so during
elementary school (5 years in France) and through his second year of secondary school. Year after year, Jeanne helped him by reading his notes and books.
Bernard started secondary school in 1945 (consisting of 4 years of collège and 3 years of lycée). He soon abandoned writing, taking notes on a mechanical typewriter during classes. He managed to take
exams using the typewriter through the two baccalauréats, which meant, at that time, the end of secondary school. Bernard’s interest in mathematics was not immediate, but grew during this period.
Over time, Bernard had his typewriter customized with some Greek letters added to the keyboard. He started studying English as his vision deteriorated. His father assembled for him a basic bilingual
dictionary that used very large letters that Bernard could read. However, his mastering of the language was uncertain and, during the first part of his career, he published mostly in French. (He
continues to favor publishing in French.) Bernard passed his second baccalauréat (in the mathématiques élémentaires section) in 1952, with the highest possible mention. At that time, even with his
declining peripheral vision, Bernard could walk by himself; he rode his bicycle until the age of 22, with severe falls from time to time. But, it was obvious that his handicap would prevent him from
occupying certain professions.
Higher studies: the road to OR
Bernard wanted to be an engineer (he had built a radio while he was in secondary school). The traditional way to become an engineer in France is not through universities, but through the distinct
system of Grandes Écoles in which students are selected on the basis of a competitive exam that could only be taken after 2 years of Classes Préparatoires. (Bernard (left) and Patrice (right) on
holidays (1969)).
Bernard went to Paris for his first year of Classes Préparatoires at the Lycée Chaptal. His results were so high that he was admitted for the second year to one of the most prestigious Classes
Préparatoires at the Lycée Louis-le-Grand, usually the first step to the École Polytechnique or the École Normale Supérieure. In class, Bernard was using his relatively quiet typewriter to take
notes. But, his physics teacher thought that the noise was intolerable and did not allow him to use the typewriter. Thus, Bernard, not being able to take notes and rather shaken by this decision,
left the Lycée Louis-Le-Grand and the Classes Préparatoires system in October 1953. Thus ended his dream of entering the École Normale Supérieure. He immediately decided to enroll in the Université
de Paris and study for a degree in mathematics. At that time, the Licence de Mathématiques meant obtaining three certificates: this usually took 3 years (the Licence had to be preceded by a general
mathematics certificate that Bernard had passed while he was at the Lycée Chaptal). In the academic year 1953–1954, Bernard completed two of the three certificates (calculus and probability). He was
taught by some great mathematicians who became famous: Laurent Schwartz (founder of the theory of distributions and a member of the Bourbaki group), Jacques-Louis Lions (one of the major promoters of
applied mathematics in France, a president of the International Mathematical Union, and father of the future Fields medal laureate Pierre-Louis Lions), Gustave Choquet (developer of the theory of
capacities), and Robert Fortet (founder of the most important French research group in the theory of probability).
Bernard got even with the École Normale Supérieure—he completed his calculus certificate with the highest possible mention, ending up tied with a student from that school.
During the 1953–1954 academic year at Université de Paris, Bernard met Patrice Bertier, a fellow student in mathematics. Patrice suffered poliomyelitis during his youth and was using a wheel chair.
He and Bernard became great friends. They spent the year studying together and helping each other. Patrice completed his Licence in June 1954 having passed the three certificates. His plan was to
take courses at the Institut d’Études Politiques (IEP) in the next academic year. IEP was a relatively special Grande École, mainly oriented toward economics and political science; it was the usual
first step to the highest positions in the French civil service. At that time, the teaching of economics and political science had little to do with mathematical economics and, for someone holding a
degree in mathematics, enrolling in IEP was extremely uncommon. But, Patrice was attracted to economics. He persuaded Bernard to join him in this adventure. The only problem was that Bernard had not
completed his Licence: he had to obtain his third certificate. Bernard then decided to study for his missing certificate during summer. He finally obtained this certificate (in rational mechanics) in
September 1954, thus completing his 3 years of Licence in only 1 year.
Both Bernard and Patrice joined IEP in October 1954. As this was really unusual—mathematics students at IEP—they also enrolled in the Institut de Statistique de l’Université de Paris (ISUP), an
interfaculty department that granted diplomas in statistics and prob- ability. IEP was located at rue Saint- Guillaume, west of the Latin Quarter, while ISUP was lo- cated at rue Pierre-
et-Marie-Curie, near the Jardin du Lux- embourg, south of the Latin Quarter.
During the years 1954 and 1955, people walking on the Boulevard Saint-Michel would often observe a strange event: Bernard, half blind, pushing the wheel chair of Patrice, as they went back and forth
between ISUP and IEP. At ISUP, Bernard had several remarkable teachers: Georges Darmois, Georges Morlat, Dickran Indjoudjian, Germain Kreweras, René Roy. ISUP was then one of the rare places in
France in which applied probability and statistics were taught to highly trained mathematics students. Here, Bernard discovered mathematical statistics and econometrics; applied statistics was not
forgotten, although all computations had to be done on electric non-programmable calculators.
At IEP he attended the courses of Alfred Sauvy (an economist and demographer who, in 1952, first used the expression Tiers Monde [Third World]), Jean Fourastié (an economist who coined the expression
Les Trente Glorieuses [The Glorious Thirty]—the 30 years from 1945 to 1975), Paul Delouvrier (an economist and urban planner), and André Siegfried (a sociologist specialized in electoral studies).
This unique combination of mathematics and economics aroused the interest of Bernard for the application of mathematics to the real world.
The years 1954–1955 were exciting times for Bernard. Several people—Georges-Théodule Guilbaud, Germain Kreweras, Jean Abadie, Jean Ville, Pierre Bouzitat, Marc Barbut, Michel Rosensthiel, Jean
Mothes, Claude Berge—began giving unofficial lectures and seminars on OR; OR was not part of any course in France. Bernard especially remembers the lectures of Guilbaud. They were attended by huge
crowds in the Amphithéaˆtre Hermite of the prestigious Institut Henri Poincaré. Bernard had found his way to applying mathematics in the real world. He wanted to do OR. The emerging French OR
community was beginning to organize itself and, in 1956, the Société Française de Recherche Opérationnelle (SOFRO) was established. [In 1964, SOFRO became AFIRO (Association Française d’Informatique
et de Recherche Opérationnelle), after a merger with a society of computer scientists; in 1968, it became AFCET (Association Française pour la Cybernétique Économique et Technique), after a merger
with a society of cyberneticians; and in 1998, AFCET split apart with the French OR society becoming ROADEF (Société Française de Recherche Opérationnelle et d’Aide à la Décision) (Roy 2006)].
These were years of intense activity for Bernard. Besides the courses at IEP and ISUP, he also obtained additional certificates in mathematics (mathematical methods of physics, algebra, and number
theory). He completed his master’s degree at ISUP in 1957 with his first research in OR: a master’s thesis on the newsboy problem presented as the baker’s problem (problème du boulanger) (Roy 1957).
He decided to start a Ph.D. on the same subject, but soon abandoned it in favor of graph theory.
In July 1956, Robert Fortet managed to obtain positions as junior researchers for both Bernard and Patrice at the Centre National de la Recherche Scientifique (CNRS, the national research agency,
created in 1939, to promote fundamental research in France). The jobs paid little but offered immense freedom. As CNRS did not have an office at that time, Bernard and Patrice were also recruited as
interns at Électricité de France (EDF, the newly nationalized electricity company) under the supervision of Marcel Boiteux, who was in charge of EDF’s Service des Études Économiques Générales (he
later became CEO of EDF). Bernard completed his master’s thesis for ISUP during this period; he benefited from the advice of Marcel Boiteux on how to write a paper. At that time, EDF had no computing
facilities. Small linear programming (LP) models were used to plan production between thermal and hydraulic plants. Bernard and Patrice were still interns at that time and, since the problems
involved strategic elements, they did not have full access to the data and results; their main role was to devise the general structure of the LP models. These problems, although small sized (around
50 variables), were still too large to be efficiently solved by hand. Marcel Boiteux and Pierre Massé (the vice-CEO of EDF) were sending these problems by ordinary mail to George Dantzig at the RAND
Corporation in Santa Monica, California, with the results also returned by mail. At that time, processing time did not reduce to computation time.
Consultant at SEMA
Bernard married Françoise Jolivet in July 1957. They had six children, Sylvie (1958y), Laurence (1961), Isabelle (1964), Solange (1966), Patrice (1968), and Philippe (1970), and nine grandchildren.
The meager salary from the CNRS was not adequate to support the young couple. Bernard left CNRS when he was recruited by a newly created OR consulting company, the Société d’Études Pratiques de
Recherche Opérationnelle (SEPRO). Meanwhile, the Société de Mathématiques Appliquées (SMA) was created as a joint venture between the Banque de Paris et des Pays-Bas (more commonly known as Paribas)
and an independent consulting company led by Marcel Loichot. The aim of SMA was to be a consulting company that would promote the use of management science (MS) in French companies.
Jacques Lesourne was appointed as CEO. Bernard left SEPRO to join SMA as a consultant in October 1957, together with Patrice Bertier. SMA quickly became SEMA (Société d’Économie et de Mathématiques
Appliquées). After having created several subsidiaries in Europe, SEMA became SEMA (Metra International).
SEMA started with around 10 employees and al- most no contracts. Bernard’s first task, with Patrice, was to translate into French several chapters of the OR text written by Churchman et al. (1957).
They also put the final touches to the book by Lesourne (1958), one of the first OR books written and published in French.
Contracts began to arrive in 1958 and Bernard started to work on applied OR problems, mainly from the private sector. He worked on a variety of problems that involved many ideas and techniques:
probability and queueing theory (reducing the waiting time at a ferry), data analysis (choosing the name of a new brand of cigarettes), transportation studies (developing a forecasting model for
transportation planning), cutting stock (designing cardboard boxes), location (choosing sites for plants), and finance (optimizing cash management). Many of these applications were later published in
METRA, the future academic journal sponsored by SEMA. Bernard’s most important works were concerned with project scheduling and related graph theory problems.
SEMA was growing steadily during this time. In 1962, it acquired a Control Data computer (CDC 6600) for which several LP and integer linear programming (ILP) codes were developed that enabled larger
problems to be solved. Before that, all computations were performed by a bureau de calcul employing many persons working on electric calculators.
In between contracts, Bernard worked on his Ph.D. dissertation in graph theory and its application to project scheduling (together with a minor dissertation on abstract algebra). He received his
Ph.D. in 1961 (dissertation on ‘‘Cheminement et connexité dans les graphes: Application aux problèmes d’ordonnancement’’ [Roy 1961)]) from the
Université de Paris, under the supervision of Claude Berge [the author of one of the first books on graph theory (Berge 1958)]. That same year, Bernard was offered a position at the Université de
Paris in mathematics. OR, at that time, was not part of the mathematics curriculum and the teaching of mathematics was slanted toward pure mathematics—this period was highly influenced by the
Bourbaki group. As accepting the position meant returning to pure mathematics, Bernard declined the offer. Taking advantage of SEMA’s policy that encouraged its consultants to teach, Bernard did
become involved in teaching OR courses at the Centre Inter-armées de Recherche Opérationnelle (a permanent education program in OR for French officers) and, with Claude Berge, taught seminars on
graph theory and combinatorial problems.
In 1962, Jacques Lesourne created within SEMA a scientific group called Direction Scientifique, with the objective of helping consultants in applying new scientific and computational techniques.
Bernard joined this group as a consultant of consultants. He became its director in 1964. For many years, this high-powered, multidisciplinary group was the site of intense activity; its members
included Raphaël Benayoun, Patrice Bertier, Éric Jacquet-Lagrèze, Hubert Le Boulanger, Benjamin Matalon, Jean de Montgolfier, Hervé Raynaud, and Gilbert Sussmann. At the same time, SEMA launched a
quarterly journal called METRA to popularize the new techniques it promoted (they included OR techniques, but also covered every aspect of MS). Bernard was appointed its editor-in-chief and remained
so until the journal ceased publication in 1977. METRA published papers written by SEMA consultants and from its European subsidiaries in four languages (French, Spanish, Italian, and English). It is
remarkable that the editorial policy of METRA was to promote the techniques developed at SEMA. Its methodological advances could appear in the journal after observing a publication lag of about 2
years which SEMA required to protect its competitive advantage.
Although edited by a commercial company, METRA had a standard academic way to process papers and had a scientific editorial board that included academics (most notably Stafford Beer and Paul Gillis).
In those times, few French libraries had subscriptions to Management Science, Journal of the Operational Research Society, or Operations Research. Thus, METRA, together with RIRO [Revue
d’Informatique et de Recherche Opérationnelle, the newly created journal of AFIRO that would later become RAIRO (Revue d’Automatique, d’Informatique et de Recherche Opérationnelle)] played an
important part in the diffusion of OR techniques in France.
Consulting, therefore, greatly influenced Bernard’s view of OR techniques and applications. Most often, the lack of appropriate software, the paucity or poor quality of data, the softness of some
constraints, and the presence of multiple conflicting objectives made the quest for an optimal solution illusory. A good solution that could not be proved optimal was often a major breakthrough in
practice. These real-world concerns greatly influenced Bernard’s approach to his future research.
Project scheduling and graph theory
One of the most famous contributions of Bernard is in the field of project scheduling. In 1958, when working at SEMA, he was faced with the problem of scheduling the construction of new buildings for
the headquarters of a large company in Paris. Managing this project, involving several hundreds of tasks and more than one thousand constraints, required a specific methodology. At this occasion,
Bernard developed a method called MPM (Méthode des Potentiels Metra). MPM was based on what is now known as the activity-on-node (AON) formulation (Roy 1959a, 1962). While its theoretical foundations
were being established (in terms of existence and optimality of schedules), this method was applied successfully to several other scheduling problems (production of crankshafts at Mavilor Motors,
design of an appropriate cycle for the new Tracoba house-building process). These applications involved potential constraints (i.e., constraint of the form $t_j – t_i geq a_{ij$, where tj is the
starting time of task j and aij is the minimum time between the start of tasks i and j) and more difficult constraints such as disjunctive or cumulative constraints (disjunctive constraints impose
that two tasks do not occur simultaneously, and cumulative constraints require that simultaneous tasks do not consume more than a given amount of resources; this typology of constraints was developed
by Bernard). MPM was one of the first computer-based software systems for project scheduling: CONCORD (CONception et Coordination de l’ORDonnancement) (Roy and Dibon 1966). [The AON approach was
proposed independently in the U.S. under the name Precedence Diagramming Method by Fondahl (1961).] The existence of a large number of difficult constraints, in the context of scheduling the
equipment of the steamship liner France (the largest in the world in 1960), eventually led to the development of another technique, description segmentée, designed to quickly spot incompatible
constraints in a system of linear inequalities (Roy 1963; Roy and Simonnard 1961).
Simultaneously and independently, methods like PERT or CPM, based on an activity-on-arc (AOA) formulation were developed in the U.S. in the late 1950s (at DuPont de Nemours, RAND Corporation, and the
U.S. Navy for the deployment of the POLARIS missile). It is now widely acknowledged that the AON formulation is superior to the AOA formulation, since it is more systematic, without requiring
modeling tricks such as dummy arcs, and its ability to readily handle changes or additions to constraints.
Bernard also obtained results on more theoretical aspects of graph theory, related, for example, to optimal paths, connectivity, transitivity, and chromaticity (Roy 1958, 1959b, 1967, 1969b). As
discussed by Hansen and de Werra (2002), some of these pioneering results, obtained over 50 years ago, are still the basis of currently published results.
Also well known is the so-called Roy–Warshall’s algorithm that computes the transitive closure of a digraph (Roy 1959b; Warshall 1962). This algorithm was discovered independently by Bernard in 1959
and Stephen Warshall in 1962. In the subfield of network flows, the algorithm to determine a minimum cost flow by successive shortest paths is known as Busacker and Gowen’s (1961) algorithm in the
U.S. and as Roy’s algorithm in Europe. Bernard independently developed this approach in the early 1960s and presented it at several conferences (Roy 1970).
Bernard is the author of a remarkable two-volume, 1300-page textbook on graph theory (Roy 1969c, 1970). Even if it is now outdated on some points, it includes an original treatment on many topics
that should be of interest to anyone in this field. Bernard organized two summer schools on graph theory and discrete mathematics. The first one, co-organized with Frank Harary, took place in 1966 in
Italy with more than 100 participants. The second one was in Versailles, France, in
1974. Both schools gathered most of the major names in the field of graph theory and combinatorial optimization. The proceedings of the second school were published in Roy (1975b). With Patrice
Bertier, Bernard was also among the pioneers who developed and formalized branch and bound procedures in the mid-1960s (Bertier and Roy 1965; Roy 1969a).
ELECTRE and Multiple Criteria Decision Aiding (MCDA)
Bernard’s research on multiple criteria decision problems was motivated by real-world problems encountered by SEMA clients. This led to the development of the first ELECTRE method, ELECTRE I (Roy
1968), for solving such problems. A media planning problem led to the development of ELECTRE II (Roy and Bertier 1973). At that time (mid-1960s), Bernard was unaware of the parallel developments in
the U.S. by Howard Raiffa, Ralph Keeney, and many others. Bernard accepted the invitation of George Dantzig to organize two sessions on MCDM for the 1970 Mathematical Programming Symposium to be held
in The Hague (Roy 1971). These sessions were among the first of their kind to be given at such conferences. During this time, Bernard, working early in the mornings, completed his two-volume
exposition on graph theory and its applications (Roy 1969c, 1970).
ELECTRE methods: an exposition
ELECTRE (Élimination et Choix TRaduisant la Réalité) methods were first developed in the mid-1960s to answer real-world problems brought to Bernard by SEMA consultants, such as the selection of
research projects or of investment opportunities. SEMA had developed a technique, called MARSAN (Méthode d’Analyse et de Recherche pour la Sélection des Activités Nouvelles), that was designed to
help firms in selecting new activities. To do so, activities were evaluated on a series of 48 dimensions (the word criterion was not used then). They included quantitative as well as qualitative
dimensions. Qualitative dimensions were translated on a numeric scale more or less arbitrarily. A weighted sum of all these numbers was computed to measure the attractiveness of these new activities.
It soon became clear that the use of a weighted sum allowed compensation effects that were not desirable: small advantages on several dimensions could compensate for major weaknesses on some others,
which was not felt to be desirable. Moreover, the transformation of qualitative dimensions into numbers was playing an important part in the final result.
Bernard devised a method that would deal both with qualitative dimensions without the need for transforming them into quantitative dimensions and that would not tolerate compensation effects that
were felt undesirable. This was the birth of ELECTRE I (Benayoun et al. 1966; Roy 1968). Basically, in ELECTRE I, alternatives are compared in pairs using the following reasoning: Alternative a will
be declared at least as good as alternative b if (1) the proposition is supported by a sufficient majority of dimensions (concordance condition), and (2) among the dimensions opposing the
proposition, there is none on which the opposition is too strong (non-discordance condition).
Such an at-least-as-good-as relation (soon called an outranking relation) can be built on the basis of purely ordinal considerations. The non-discordance condition prevents undesirable compensation
effects from occurring. The application of the concordance condition leads to assigning weights to each dimension. To decide if a majority of dimensions is sufficiently important, the sum of the
weights is compared to a threshold called the concordance threshold (note that these weights are quite different from the weights used in a weighted sum; they are never multiplied with scores and
are, therefore, independent from the scale used to measure scores). Similarly, the strength of the opposition of dimensions is computed using a veto threshold.
A specific feature of this relation is that it does not have to be transitive (even in its asymmetric part, because of Condorcet-like effects) or complete (some alternatives may remain incomparable).
Therefore, deriving a prescription on this basis is not an easy task and calls for the application of specific techniques, called exploitation techniques. They differ on the type of recommendation
that is looked for. ELECTRE I has been designed in a choice problem formulation—it aims at recommending a subset of alternatives (as small as possible) that is likely to contain the best
alternatives. Technically, viewing the outranking relation on the set of alternatives as a graph, Bernard suggested using the kernel (an independent and dominating subset) of this graph. ELECTRE II
(Roy and Bertier 1973) is a variant of ELECTRE I that is designed to rank order alternatives. It uses two outranking relations instead of one. The ranking is not necessarily complete: it preserves
incomparability between alternatives that appear difficult to compare. ELECTRE III (Roy 1978) is a far-reaching generalization of ELECTRE II that uses a fuzzy outranking relation instead of two crisp
ones. Furthermore, it refines the preference modeling on each dimension with the introduction of thresholds preventing small differences between scores from being interpreted as a definite advantage.
Such thresholds were introduced in a new version of ELECTRE I, called ELECTRE IS. ELECTRE IV (Roy and Hugonnard 1982) is a variant of ELECTRE III designed to deal with situations in which weights are
difficult to elicit, given the diversity of opinions. ELECTRE TRI (Roy and Bouyssou 1993) is designed to deal with a sorting problem formulation in which each alternative is assigned to a category
pre-defined by norms which, for example, separate good and bad credit files.
All these methods were developed to deal with specific real-world problems. ELECTRE methods have been applied to a large variety of problems in many countries (Figueira et al. 2005; Roy 1991; Roy and
Bouyssou 1993).
MCDA: an original perspective on OR
Bernard’s concept of OR was influenced by two major themes: the starting of his career as a consultant and his later work in MCDM. Their synergistic interaction led him to develop a decision-aiding
methodology that is original and rather non-standard in the OR profession (Roy 1975a, 1977, 1985, 1990, 1993). He noticed that the application of OR models and methods were characterized by the
adherence to three main assumptions:
1. The quest for rationality implies the use of a unique criterion that should be optimized.
2. Qualitative information and ambiguous data should be avoided as much as possible.
3. Science aims at describing a reality that is mainly independent from the observer. Reference to this outside reality is central to the validation of a scientific model.
Bernard soon became rather skeptical about these three assumptions and proposed a decision-aiding methodology that would dispense with them (Roy 1981).
Indeed, Bernard quickly acknowledged the fact that in many real-world problems, several actors are involved. These several stakeholders have different opinions. Quite often, their opinions are not
always completely structured. Also, there may be no real decision maker. Moreover, what is feasible or what is not feasible is often fuzzy (Roy 1988). This undermines the first assumption and calls
for the use of multiple criteria. This does not mean that optimizing is useless, but simply that optimality within a model does not guarantee an acceptable solution, let alone an optimal one, in the
real world.
Real-world situations abound with qualitative information. Contrary to the second assumption, information is often uncertain, imprecise, and ill-determined. Trying, by all possible means, to convert
all that is qualitative into quantitative information is a difficult task and often leads to a result that is seldom meaningful. Spending time to obtain information of better quality is often an
inappropriate use of resources and may lead to instrumental bias (recall the drunkard looking for his keys under a street lamp without really knowing where he lost them). In all real-world problems,
irreducible uncertainty, imprecision, and inaccurate determination will remain (Roy 1989). Hence, we should reconcile ourselves that we must deal with the available qualitative information, using
techniques that allow robust conclusions to result (Roy 1998).
Decision aiding inevitably means working with preferences. When facing a complex problem, it is rare to have the actor(s)’s preferences clearly stated and completely well structured (Roy and Vincke
1984). The analyst must question the actor(s) and, thus, contributes to the shaping of the preferences, as well as describing them. A clear violation of the third assumption. This learning process,
which is often a creation process, is an inevitable part of applying OR models (Roy 1987).
Over the years, Bernard has proposed a complete decision-aiding methodology that does not rely on the above three assumptions (Roy 1985; Roy and Bouyssou 1993). This explains why Bernard prefers to
speak of MCDA instead of MCDM.
Bernard’s most recent research deals with robustness in decision aiding. In many decision contexts, model parameters are often defined approximately due to uncertainty, imprecision, or
ill-determination (Roy 1998). Rather than looking for optimal solutions, it is then more appropriate to look for robust solutions that are resisting to vague approximations and areas of ignorance,
that is, which behave well for all, or at least most, plausible values of the parameters. Such a perspective, often well received by practitioners, gives rise to many challenging theoretical
questions. Bernard’s approach to robustness is discussed in Roy (2010).
Daniel Vanderpooten, Bernard Roy, Denis Bouyssou (2007)
In the late 1960s, following the May 1968 events in France leading to a 1-month general strike, Bernard started wondering about his future career. Jacques Lesourne had announced that he would soon
leave SEMA. During this time, Bernard was asked to give a doctoral course on OR at the newly created Université Paris-Dauphine (this experimental university was created in 1968 and occupied the
former NATO headquarters in Paris). In 1971, he was appointed associate professor in mathematics (later joining the computer science department). The following year, he was made full professor. He
kept his position at SEMA until 1974, progressively reducing his involvement, as SEMA reduced its OR activities; he remained associated with SEMA as a scientific advisor until 1979. One of Bernard’s
early academic duties was to reshape the MS curriculum within the management program. In 1974, Bernard created a research group called LAMSADE (Laboratoire d’Analyse et Modélisation de Systèmes pour
l’Aide à la Décision) which became affiliated with CNRS in 1976. LAMSADE was one of the few research groups in France oriented toward applied OR. Over the years, as LAMSADE kept growing, it expanded
its base of interest to include research topics in computer science.
Bernard made sure that the Dauphine OR curriculum included a doctoral program, Méthodes Scientifiques de Gestion, and thus, through the years, he began his supervision of over 50 doctoral students
(both authors of this text are his former doctoral students). His research at LAMSADE became more and more oriented toward MCDM, or rather MCDA.
Although Bernard devoted much energy to the development of LAMSADE and served as its director until 1999, he also undertook several important responsibilities within Université Paris Dauphine,
including the directorship of a doctoral school. In addition, in 1980, Bernard became scientific advisor of RATP (Régie Autonome des Transports Parisiens; the company that operates all public
transports in the Paris region).
Bernard is the author of more than 80 papers in refereed journals and nearly 50 papers in contributed volumes. A selected list of Bernard’s publications is available from LAMSADE (2009).
Bernard retired in 2001 with the title of professor emeritus. A Festschrift honoring him was published on the occasion of his retirement (Bouyssou et al. 2002). He remains quite engaged in his
scientific and consulting activities.
Honors and awards
Bernard has received six honorary doctoral degrees (Vrije Universiteit Brussels, Belgium, 1978; Université de Liège, Belgium, 1978; Université de Fribourg, Switzerland, 1982; Poznan University of
Technology, Poland, 1992; Université Laval, Canada, 1998; Technical University of Crete, Greece, 2002). He received the 1992EURO gold medal, the highest distinction granted by EURO. He holds the gold
medal from the MCDM International Society, as well as the Hermès de la Recherche Prix from the Université Laval, Québec, Canada.
Bernard served as vice-president (1974–1976) and president (1976–1978) of AFCET. He was the president of EURO (1985–1986), after having served on the executive committee for several years. In 1975,
he founded one of the most active and long-lasting working groups in OR, the EURO working group on MCDA.
The EURO Working Group: Multiple Criteria Decision Aiding
EURO is a federation of the national European OR societies. The first EURO conference was held in Brussels in 1975. Bernard created the EURO working group on multiple criteria decision aiding (MCDA).
The group, which usually meets twice a year, aims to promote original research on MCDA in Europe. The meetings of the group are
not conferences. They are designed to foster discussions and exchanges. The group has around 350 members, from about 30 countries, and meetings usually gather between 50 and 100 persons. The success
of the group is attested by the fact that most texts on MCDM now speak of a European school of MCDA (Roy and Vanderpooten 1996). The 69th meeting took place in Brussels, Belgium, April 2–3, 2009.
More details on this working group can be found at http://www.inescc.pt/~ewgmcda/index.html (viewed December 24, 2009).
1. Benayoun R, Roy B, Sussmann G (1966) ELECTRE: Une méthode pour guider le choix en présence de points de vue multiples. Note de travail 49, SEMA (Metra International), Direction Scientifique)
2. Berge C (1958) Théorie des Graphes et ses Applications. Dunod, Paris
3. Bertier P, Roy B (1965) Une procédure de résolution pour une classe de problèmes pouvant avoir un caractère combinatoire. ICC Bull 4:19–28
4. Bouyssou D, Jacquet-Lagrèze É , Perny P, Slowinski R, Vanderpooten D, Vincke Ph (2002) (eds) Aiding decisions with multiple criteria: essays in honor of Bernard Roy. Kluwer, Boston, MA
5. Busacker R, Gowen P (1961) A procedure for determining a family of minimal-cost network flow patterns. Operations Research Office Technical Report 15, J. Hopkins University, Baltimore, MD
6. Churchman C, Ackoff R, Arnoff E (1957) Introduction to operations research. Wiley, New York, NY. French translation: Eléments de recherche opérationnelle. Dunod, Paris, (1961)
7. Figueira J, Mousseau V, Roy B (2005) ELECTRE methods. In: Figueira J, Greco S, Ehrgott M (eds) Multiple criteria decision analysis: state of the art surveys. Springer, Boston, MA, pp 133–162
8. Fondahl J (1961) A non-computer approach to the critical path method for the construction industry. Technical report 9, Department of Civil Engineering, Stanford University
9. Hansen P, de Werra D (2002) Connectivity, transitivity and chromaticity: the pioneering work of Bernard Roy in graph theory. In: Bouyssou D, Jacquet-Lagrèze E, Perny P, Slowinski R, Vanderpooten
D, Vincke Ph. (eds) Aiding decisions with multiple criteria: essays in honor of Bernard Roy. Kluwer, Boston, MA, pp 23–42
10. LAMSADE (2009) http://www.lamsade.dauphine.fr/ extasciitilde roy/ roy\_publications.htm. Accessed 14 Sept, 2009
11. Lesourne J (1958) Techniques économiques et gestion industrielle, Dunod, Paris
12. Roy B (1957) Recherche d’un programme d’approvisionnement ou de production. Revue de Recherche Opérationnelle 1(4): 172–184
13. Roy B (1958) Sur quelques propriétés des graphes fortement connexes. Comptes rendus de l’Académie des Sciences 247:399–401
14. Roy B (1959a). Contribution de la théorie des graphes à l’étude de certains problèmes linéaires. Comptes rendus de l’Académie des Sciences 248:2437–2439
15. Roy B (1959b). Transitivité et connexité. Comptes rendus des séances de l’Académie des Sciences 249(6):216–218
16. Roy B (1961) Cheminement et connexité dans les graphes—Application aux problèmes d’ordonnancement. Doctorat d’É tat de Sciences Mathématiques, Faculté des Sciences de Paris.
17. Roy B (1962) Graphes et ordonnancement. Revue Française de Recherche Opérationnelle (25/4e trimestre):323–333
18. Roy, B (1963) Programmation mathématique et description segmentée. Revue METRA 2(4):523–535
19. Roy B (1967) Nombre chromatique et plus longs chemins d’un graphe. RIRO 1(5):129–132
20. Roy B (1968) Classement et choix en présence de points de vue multiples (la méthode ELECTRE). RIRO 2(8):57–75
21. Roy B (1969a) Procédure d’exploration par séparation et évaluation (PSEP et PSES). RIRO 3(V-1):61–90
22. Roy B (1969b). Graphe partiel s-connexe extremum. Revue Roumaine de Mathématiques Pures et Appliquées 14(9):1355–1368
23. Roy B (1969c). Algèbre moderne et théorie des graphes orientées vers les sciences économiques et sociales: Volume 1: Notions et résultats fondamentaux. Dunod, Paris.
24. Roy B (1970) Algèbre moderne et théorie des graphes orientées vers les sciences économiques et sociales: Volume 2: Applications et problèmes spécifiques, Dunod, Paris
25. Roy B (1971) Problems and methods with mutiple objective functions, Math Program 1(2):239–266
26. Roy B (1975a). Vers une méthodologie générale d’aide à la décision. Revue METRA 14(3):459–497
27. Roy B (ed.) (1975b). Combinatorial programming: methods and applications. D. Reidel, Dordrecht
28. Roy B (1977) Partial preference analysis and decision-aid: the fuzzy outranking relation concept. In: Bell D, Keeney R, Raiffa H (eds) Conflicting objectives in decisions. Wiley, New York, NY, pp
29. Roy B. (1978) ELECTRE III: Un algorithme de classements fondé sur une représentation floue des préférences en présence de critères multiples. Cahiers du Centre d’études de Recherche
Opérationnelle 20(1):3–24
30. Roy B (1981) The optimisation problem formulation: Criticism and overstepping. J Oper Res Soc 32(6):427–436
31. Roy B (1985) Méthodologie multicritère d’aide `a la décision. Economica, Paris. (English translation: Multicriteria methodology for decision analysis. Kluwer Academic Publishers, 1996. Polish and
Spanish translations are also available)
32. Roy B (1987) Meaning and validity of interactive procedures as tools for decision making. Eur J Oper Res 31(3):297–303
33. Roy B (1988) Des critères multiples en Recherche Opérationnelle: Pourquoi? In: Rand G (ed.) Operational research ’87. Elsevier, Amsterdam, pp 829–842
34. Roy B (1989) Main sources of inaccurate determination, uncertainty and imprecision in decision models. Math Comput Model 12(10/11):1245–1254
35. Roy B (1990) Decision-aid and decision-making. Eur J Oper Res 45(2–3):324–331
36. Roy B (1991) The outranking approach and the foundations of ELECTRE methods. Theory Decis 31(1):49–73
37. Roy B (1993) Decision science or decision-aid science? Eur J Oper Res 66(2):184–203
38. Roy B (1998) A missing link in OR-DA: robustness analysis. Foundations Comput Decis Sci 23(3):141–160
39. Roy B (2006) Regard historique sur la place de la recherche opérationnelle et de l’aide à la décision en France. Mathématiques et Sciences Humaines 175:25–40
40. Roy B (2010) Robustness in operational research and decision aiding: a multi-faceted issue. Eur J Oper Res 200(3):629–638
41. Roy B, Bertier P (1973) La méthode ELECTRE II—Une application au média-planning. In: Ross M (ed.) OR ’72. North-Holland, Amsterdam, 291–302
42. Roy B, Bouyssou D (1993) Aide multicritère à la décision: Méthodes et cas. Economica, Paris
43. Roy B, Dibon M (1966) L’ordonnancement par la méthode des potentiels—Le programme CONCORD. Automatisme 2:1–11
44. Roy B, Hugonnard J-C (1982) Ranking of suburban line extension projects on the Paris metro system by a multicriteria method. Trans Res 16A(4):301–312
45. Roy B, Simonnard M (1961) Nouvelle méthode permettant d’explorer un ensemble de possibilités et de déterminer un optimum. Revue Française de Recherche Opérationnelle (18/1etrimestre):15–54
46. Roy B, Vanderpooten D (1996) The European school of MCDA: Emergence, basic features and current works. J Multi Criteria Decis Anal 5(1):22–38
47. Roy B, Vincke P (1984) Relational systems of preference with one or more pseudo-criteria: some new concepts and results. Manage Sci 30(11):1323–1335
48. Warshall S (1962) A theorem on Boolean matrices. J ACM 9(1):11–12 | {"url":"https://roadef.org/app/news/bernard-roy","timestamp":"2024-11-10T05:25:38Z","content_type":"text/html","content_length":"163801","record_id":"<urn:uuid:aa9495b2-4d28-4520-a7e2-04e740714a27>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00001.warc.gz"} |
Pearson Chi-Square Test for Normality
PearsonTest {DescTools} R Documentation
Pearson Chi-Square Test for Normality
Performs the Pearson chi-square test for the composite hypothesis of normality.
PearsonTest(x, n.classes = ceiling(2 * (n^(2/5))), adjust = TRUE)
x a numeric vector of data values. Missing values are allowed.
n.classes The number of classes. The default is due to Moore (1986).
adjust logical; if TRUE (default), the p-value is computed from a chi-square distribution with n.classes-3 degrees of freedom, otherwise from a chi-square distribution with n.classes-1 degrees of
The Pearson test statistic is P=\sum (C_{i} - E_{i})^{2}/E_{i}, where C_{i} is the number of counted and E_{i} is the number of expected observations (under the hypothesis) in class i. The classes
are build is such a way that they are equiprobable under the hypothesis of normality. The p-value is computed from a chi-square distribution with n.classes-3 degrees of freedom if adjust is TRUE and
from a chi-square distribution with n.classes-1 degrees of freedom otherwise. In both cases this is not (!) the correct p-value, lying somewhere between the two, see also Moore (1986).
A list of class htest, containing the following components:
statistic the value of the Pearson chi-square statistic.
p.value the p-value for the test.
method the character string “Pearson chi-square normality test”.
data.name a character string giving the name(s) of the data.
n.classes the number of classes used for the test.
df the degress of freedom of the chi-square distribution used to compute the p-value.
The Pearson chi-square test is usually not recommended for testing the composite hypothesis of normality due to its inferior power properties compared to other tests. It is common practice to compute
the p-value from the chi-square distribution with n.classes - 3 degrees of freedom, in order to adjust for the additional estimation of two parameters. (For the simple hypothesis of normality (mean
and variance known) the test statistic is asymptotically chi-square distributed with n.classes - 1 degrees of freedom.) This is, however, not correct as long as the parameters are estimated by mean
(x) and var(x) (or sd(x)), as it is usually done, see Moore (1986) for details. Since the true p-value is somewhere between the two, it is suggested to run PearsonTest twice, with adjust = TRUE
(default) and with adjust = FALSE. It is also suggested to slightly change the default number of classes, in order to see the effect on the p-value. Eventually, it is suggested not to rely upon the
result of the test.
The function call PearsonTest(x) essentially produces the same result as the S-PLUS function call chisq.gof((x-mean(x))/sqrt(var(x)), n.param.est=2).
Juergen Gross <gross@statistik.uni-dortmund.de>
Moore, D.S., (1986) Tests of the chi-squared type. In: D'Agostino, R.B. and Stephens, M.A., eds.: Goodness-of-Fit Techniques. Marcel Dekker, New York.
Thode Jr., H.C., (2002) Testing for Normality. Marcel Dekker, New York. Sec. 5.2
See Also
shapiro.test for performing the Shapiro-Wilk test for normality. AndersonDarlingTest, CramerVonMisesTest, LillieTest, ShapiroFranciaTest for performing further tests for normality. qqnorm for
producing a normal quantile-quantile plot.
PearsonTest(rnorm(100, mean = 5, sd = 3))
PearsonTest(runif(100, min = 2, max = 4))
version 0.99.55 | {"url":"https://search.r-project.org/CRAN/refmans/DescTools/html/PearsonTest.html","timestamp":"2024-11-02T09:22:57Z","content_type":"text/html","content_length":"6427","record_id":"<urn:uuid:3d3c81c5-34c7-4a18-8334-bbe358cbd78a>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00533.warc.gz"} |
[QSMS Seminar 20220721] Finding ALL solutions
Speaker : 이경용 (University of Alabama)
Place : 129동406호
Schedule : 7월21일(목) 10:00 ~ 11:00
TiTle : Finding ALL solutions
Abstract :
We introduce two famous (or notorious) open problems from Smale's list and Millennium problems: the Jacobian conjecture and the Birch--Swinnerton-Dyer conjecture.
The Jacobian conjecture states that if the Jacobian of a polynomial map is a nonzero constant, then the map is bijective. The condition of the Jacobian being equal to a constant can be translated to
a system of (too many) polynomial equations. An elementary but promising approach is to find ALL solutions systematically. This is based on joint work with Jacob Gliedwell, William Hurst, Li Li, and
George Nasr.
A special case of the weak BSD conjecture can be translated to counting integer solutions of certain Diophantine equations. Again an elementary but promising approach is to find ALL solutions
systematically. This is based on joint work with Ty Clements, Zach Socha, and Vishak Vikranth.
This talk will be completely elementary and accessible to non-experts, as all my collaborators including myself are non-experts. | {"url":"https://qsms.math.snu.ac.kr/index.php?mid=board_sjXR83&order_type=asc&page=2&document_srl=2332&sort_index=readed_count&listStyle=viewer","timestamp":"2024-11-07T12:24:17Z","content_type":"text/html","content_length":"22148","record_id":"<urn:uuid:6a9ea950-d9ce-4fb3-8407-3549e0309743>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00795.warc.gz"} |
Draw The Number With Base Ten Blocks
Draw The Number With Base Ten Blocks - In this example, the place value of the ones place is 5. This video is targeted for grade 2 and 3 math studies. Base ten blocks turn the base ten concept into
something children can see and touch. A) 91, b) 59, c) 88, d) 75 the standard algorithm for addition (no regrouping). They are useful because they show the value of each digit in a number and make it
easier to visualize the size of a.
The fundamental unit of a system of base ten blocks is the unit cube, which is a stand in for a quantity of one. A) 41 + 50 b) 38 + 21 c) 54 + 34 d) 73 + 2 answers: Counting using base 10 blocks. A
model for base 10 numeration. Web what are base ten blocks? Regrouping into blocks of 10. Base ten blocks are one way to allow children to visualize what each digit in a number represents.
Lesson 2 Use Base Ten Blocks and a Place Value Chart to Show a Number
Web base ten blocks are an engaging way to teach place value, measurements, number concepts, and more. Advantages of base ten blocks blocks. The fundamental unit of a system of base ten blocks is
How to Introduce Decimals with Base Ten Blocks Math school, Math
Web about base ten blocks. Web this is an introduction to the base 10 system for whole numbers. Making 10 with ten frames. A model for base 10 numeration. Base ten blocks (also known as.
Using Base 10 Blocks to Write Numbers Worksheet by Teach Simple
They are usually used to represent 1000, 100, 10 and 1. Base ten blocks turn the base ten concept into something children can see and touch. Web base ten blocks are an engaging way to.
Base Ten Blocks Representing Numbers 1119 Teaching Resources
The smallest blocks—cubes that measure 1 cm on a side—are called units. Ten copies of the smallest block (variously named “a bit” or “a tiny cube”) lined up in a straight row exactly match one.
What are Base 10 Blocks? Video & Lesson Transcript
Advantages of base ten blocks blocks. Ten copies of the smallest block (variously named “a bit” or “a tiny cube”) lined up in a straight row exactly match one of the long rods (variously called.
Base Ten Blocks Representing Numbers 21 to 99 Teaching Resources
Web base 10 blocks are visual representations of numbers that help students learn the concepts of quantity and place value. A) 91, b) 59, c) 88, d) 75 the standard algorithm for addition (no
How to Introduce Decimals with Base Ten Blocks
So let’s say i have the number 234. Web the base 10 blocks worksheets only use tens and ones (no hundreds or thousands). A) 41 + 50 b) 38 + 21 c) 54 + 34.
Representing Decimal Numbers using base 10 blocks. (Printable and
This video is targeted for grade 2 and 3 math studies. Web base ten blocks are used to represent numbers as a base ten model. There are four different base ten blocks blocks. A model.
Representing numbers using Base 10 Blocks (up to 6 digits) Printable
You see, it is in the hundreds place so it really is 200. Web base ten blocks are used to represent numbers as a base ten model. Base ten blocks (also known as base 10.
Place Value Using Base Ten Blocks The Learning Corner
Web base 10 blocks are visual representations of numbers that help students learn the concepts of quantity and place value. Units in the base 10 blocks are used to represent the ones. A) 91, b).
Draw The Number With Base Ten Blocks “number builder” is a digital activity/tool that works with base ten blocks. Web about press copyright contact us creators advertise developers terms privacy
policy & safety how youtube works test new features nfl sunday ticket press copyright. The smallest blocks—cubes that measure 1 cm on a side—are called units. Counting using base 10 blocks. It lets
them investigate how to regroup and solve problems with whole numbers and eventually fractions and decimals.
Draw The Number With Base Ten Blocks Related Post : | {"url":"https://upload.independent.com/kudos/draw-the-number-with-base-ten-blocks.html","timestamp":"2024-11-05T10:44:45Z","content_type":"application/xhtml+xml","content_length":"23737","record_id":"<urn:uuid:5b862083-1095-4754-b146-25702198e48f>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00751.warc.gz"} |
GameTheory | Reservoir
Welcome to the Game Theory project! This repository contains implementations and analyses related to game theory based on Mathlib. Our group is working on more contributions on the formalization of
gamtheory perfix. Below is a brief overview of the contents and setup instructions.
• GameTheory/Secondprice.lean: Formalise some core concepts and results in auction theory. This includes definitions for first-price and second-price auctions, as well as several fundamental
results and helping lemmas.
• GameTheory/Zerosum.lean: The complete proof of von Neumann's minimax theorem about zero-sum games.
The current plan for the formalizing of Game Theory include:
• Essential definitions of Sealed-bid auction, First-price auction and Second-price auction.
• First-price auction has no dominant strategy.
• Second-price auction has dominant strategy. (Second-price auction is DSIC)
• Will depend on #14163 once that PR is merged. The Fintype lemmas introduced by this PR have been added in that PR and will be removed from here once that PR gets merged
• Mechanism design An allocation rule is implementable if there exists - Dominant Strategy Incentive Compatible (DSIC) payment rule - An allocation rule is monotone if for every bidder’s gain is
nondecreasing w.r.t. her/his bid
• Myerson's Lemma Implementable ⇔ Monotone In the above case, the DSIC payment rule is unique.
• Equilibrium in zero sum game
• Formalization strategy: via Loomis’s theorem.
Roughgarden, Tim. Twenty Lectures on Algorithmic Game Theory. Cambridge University Press, 2020. Link
Current contributors to this project are:
This project is configured to run on Gitpod. Simply open the repository in Gitpod to get a ready-to-use development environment in your browser.
Contributions are welcome! Please fork the repository and create a pull request with your changes. Ensure to follow the coding standards and include tests for any new features.
This project is licensed under the MIT License. | {"url":"https://reservoir.lean-lang.org/@math-xmum/GameTheory","timestamp":"2024-11-03T13:33:08Z","content_type":"text/html","content_length":"61249","record_id":"<urn:uuid:cb82897a-141a-450d-b99d-34ada68b0aaf>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00849.warc.gz"} |
Derivative question from Berkeley Problems (Real Analysis)
In the solution to problem 1.1.11, there's is one line in the solution that is giving me trouble. Here is the problem:
Suppose that $$f$$ is a twice differentiable real function such that $$f''(x)>0$$ for all $$x\in[a,b]$$. Find all real numbers $$c$$ at which the area between the graph $$y=f(x)$$, the tangent to the
graph at $$(c,f(c))$$, and the lines $$x=a$$, $$x=b$$ attains its minimum value.
At some point we need to differentiate (with respect to c)
$$A(c) = \int_a^b ( f(x) - f(c) - f'(c)(x-c) ) dx$$
and get
$$A'(c) = -f''(c) \int_a^b (x-c) dx$$,
but I'm getting other things which don't allow successful solution to the problem (my expression for $$A'(c)$$ doesn't have a $$c$$ in it).
Can someone explain?
Re: Derivative question from Berkeley Problems (Real Analysis)
I don't understand the issue... Are you sure you're differentiating with respect to c...? Looks straightforward to me
Re: Derivative question from Berkeley Problems (Real Analysis)
First off, please note that your parentheses are off. You probably want to start with a function more like
$$A(x) = \int_a^b f(x) - f(c) - f'(c) \cdot (x-c) \, dx$$
(I am not vouching for the correctness of the signs.) Using the chain rule to differentiate with respect to c yields
$$A'(x) = \int_a^b 0 - f'(c) - \left( f''(c) \cdot (x-c) + f'(c) \cdot (-1) \right) \, dx$$
which simplifies to
$$A'(x) = -f''(c) \int_a^b (x-c) \, dx$$
Re: Derivative question from Berkeley Problems (Real Analysis)
The steps look good owlpride, and although the parentheses look awkward, that's exactly what he had written above (there was just an extra set at the beginning and end of the integrand). Also I think
you meant to call it A(c) instead of A(x) but that was just a typo. I guess it's also worth stating that you technically need a weak version of Leibniz in order to take the derivative underneath the
integral sign.
Re: Derivative question from Berkeley Problems (Real Analysis)
My error was in treating $$f(c)$$ as a constant, "simplifying" the integrals, and differentiating that. Ah, the things you learn. : /
I'm now familiarizing myself with http://en.wikipedia.org/wiki/Differenti ... egral_sign . Thank you very much for the input.
No kidding Terry! http://terrytao.wordpress.com/career-ad ... our-field/
Re: Derivative question from Berkeley Problems (Real Analysis)
blitzer6266, you are completely right on all three points, thanks! | {"url":"https://mathematicsgre.com/viewtopic.php?f=1&t=633","timestamp":"2024-11-14T00:29:51Z","content_type":"text/html","content_length":"29126","record_id":"<urn:uuid:b2161365-5ca8-4c56-9104-dee34413b361>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00642.warc.gz"} |
JEE Main Mathematics Chapter Wise Question Papers - SD
JEE Main Mathematics Chapter Wise Question Papers
Practising previous year question papers is one of the best methods of preparation for the JEE Main exam. This helps students to figure out their weaker areas and thus concentrate on those topics. As
far as Mathematics is concerned, practising previous year questions is very important. This helps students to improve their speed of solving problems. Thus they can be stress-free and more confident
during the exam.
Previous year chapter wise questions help students to get an idea of what type of questions are asked from each chapter. Students can easily access JEE Main Mathematics Chapter Wise Question Papers
from our website. These are also available in PDF format for free. Practising these question papers helps to understand the question pattern, marking scheme and the difficulty level of questions from
each chapter. Following are the list of the chapter wise question papers.
Important Topics
• Sets, Relations and Functions
• System of Linear Equations
• Complex Numbers and Quadratic Equations
• Coordinate Geometry
• Matrices, Determinants and Solutions of Linear Equations
• Permutations and Combinations
• Mathematical Reasoning
• Binomial Theorem and Mathematical Induction
• Sequences and Series
• Limits, Continuity, Differentiability and Differentiation
• Indefinite and Definite Integrals
• Differential Equations
• Conic Sections
• Three Dimensional Geometry
• Vector Algebra
• Statistics
• Probability
• Trigonometry
Benefits of Chapter Wise Question Papers
1. The best study material for JEE Main preparation is previous year chapter wise question papers.
2. Practising these papers help students to assess their preparation level.
3. Find out weaker areas and work more on those topics.
4. Students will get a clear idea of the type of questions and marking scheme.
All solutions are prepared by BYJU’S subject experts. Chapter-wise questions with solutions will help students to clear their concepts and score more in IIT JEE examination. Also, students can easily
refer to the previous questions asked in the JEE Main from each chapter. Solving previous year questions will help students to improve their score in the exam.
Angle Measurement
An angle measure is the measure of the angle formed by the two rays or arms at a common vertex. Depending on whether the rotating ray moves anticlockwise or clockwise direction, an angle formed by a
rotating ray is said to be positive or negative. The three systems for measuring angles are given below:
1. Sexagesimal system
2. Centesimal system
3. Circular system
Circular system of Angle Measurement
This is also known as the radian system. The angle is measured in radian in this system.
A radian is an angle subtended at the centres of a circle by an arc, the whole length is equal to the radius of the circle. It is used instead of degrees. The numbers of radians in an angle subtended
by an arc of the circle at the centres = arc/ radius.
Students are recommended to learn the topic clearly so that they can easily crack the JEE questions. As far as the JEE exam is concerned, angle measurement is an important topic. Important
conversions are given below. Students can expect 1 question related to this topic.
Important conversions:
π radian = 180^0
1 radian = 180/π
1^0 = (π/180) radian
π/2 radian = 90^0
π/6 Radian = 30^0
π/4 Radian = 45^0
π/3 Radian = 60^0^ | {"url":"https://supportdenmark.com/jee-main-mathematics-chapter-wise-question-papers.html","timestamp":"2024-11-07T04:35:20Z","content_type":"text/html","content_length":"54178","record_id":"<urn:uuid:5cd27bcb-ca96-4eb2-8638-ee158b807e20>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00501.warc.gz"} |
Problem Set 3 | ECON 480: Econometrics
Problem Set 3
Due by Sunday, September 27, 2020
There are several ways you can complete and turn in this homework assignment:
1. Type up any applicable answers (saving any plots as images and including them) in a (e.g. Word) document and save it as a PDF and turn in a (commented!) .R file of commands for each relevant
2. If you wish to write out answers by hand, you may either print the pdf above or write your answers (all I need is your work and answers) on your own paper and then please scan/photograph &
convert them to a single PDF, if they are easily readable, but this is not preferred. See my guide to making a PDF
3. Download the .Rmd file, do the homework in markdown, and email to me a single knitted html or pdf file. Be sure that it shows all of your code (i.e. all chunks have echo = TRUE options),
otherwise I will also ask for the markdown file.
To minimize confusion, I suggest creating a new R Project (e.g. hw3) and storing any data and plots in that folder on your computer. See my example workflow.
You may work together (and I highly encourage that) but you must turn in your own answers. I grade homeworks 70% for completion, and for the remaining 30%, pick one question to grade for accuracy -
so it is best that you try every problem, even if you are unsure how to complete it accurately.
Theory and Concepts
Question 1
In your own words, describe what exogeneity and endogeneity mean, and how they are related to bias in our regression. What things can we learn about the bias if we know \(X\) is endogenous?
Question 2
In your own words, describe what \(R^2\) means. How do we calculate it, what does it tell us, and how do we interpret it?
Question 3
In your own words, describe what the standard error of the regression (\(SER\)) means. How do we calculate it, what does it tell us, and how do we interpret it?
Question 4
In your own words, describe what homoskedasticity and heteroskedasticity mean: both in ordinary English, and in terms of the graph of the OLS regression line.
Question 5
In your own words, describe what the variation in \(\hat{\beta_1}\) (either variance or standard error) means, or is measuring. What three things determine the variation, and in what way?
Question 6
In your own words, describe what a \(p\)-value means, and how it is used to establish statistical significance.
Question 7
A researcher is interested in examining the impact of illegal music downloads on commercial music sales. The author collects data on commercial sales of the top 500 singles from 2017 (\(Y\)) and the
number of downloads from a web site that allows `file sharing’ (\(X\)). The author estimates the following model
\[\text{music sales}_i = \beta_0+\beta_1 \text{illegal downloads}_i + u_i\]
The author finds a large, positive, and statistically significant estimate of \(\hat{\beta_1}\). The author concludes these results demonstrate that illegal downloads actually boost music sales. Is
this an unbiased estimate of the impact of illegal music on sales? Why or why not? Do you expect the estimate to overstate or understate the true relationship between illegal downloads and sales?
Question 8
A pharmaceutical company is interested in estimating the impact of a new drug on cholesterol levels. They enroll 200 people in a clinical trial. People are randomly assigned the treatment group or
into the control group. Half of the people are given the new drug and half the people are given a sugar pill with no active ingredient. To examine the impact of dosage on reductions in cholesterol
levels, the authors of the study regress the following model:
\[\text{cholesterol level}_i = \beta_0+\beta_1 \text{dosage level}_i + u_i\]
For people in the control group, dosage level\(_i=0\) and for people in the treatment group, dosage level\(_i\) measures milligrams of the active ingredient. In this case, the authors find a large,
negative, statistically significant estimate of \(\hat{\beta_1}\). Is this an unbiased estimate of the impact of dosage on change in cholesterol level? Why or why not? Do you expect the estimate to
overstate or understate the true relationship between dosage and cholesterol level?
Theory Problems
For the following questions, please show all work and explain answers as necessary. You may lose points if you only write the correct answer. You may use R to verify your answers, but you are
expected to reach the answers in this section “manually.”
Question 9
A researcher wants to estimate the relationship between average weekly earnings \((AWE\), measured in dollars) and \(Age\) (measured in years) using a simple OLS model. Using a random sample of
college-educated full-time workers aged 25-65 yields the following:
\[\widehat{AWE} = 696.70+9.60 \, Age\]
Part A
Interpret what \(\hat{\beta_0}\) means in this context.
Part B
Interpret what \(\hat{\beta_1}\) means in this context.
Part C
The \(R^2=0.023\) for this regression. What are the units of the \(R^2\), and what does this mean?
Part D
The \(SER, \, \hat{\sigma_u}=624.1\) for this regression. What are the units of the SER in this context, and what does it mean? Is the SER large in the context of this regression?
Part E
Suppose Maria is 20 years old. What is her predicted \(\widehat{AWE}\)?
Part F
Suppose the data shows her actual \(AWE\) is $430. What is her residual? Is this a relatively good or a bad prediction?Hint: compare your answer here to your answer in Part D.
Part G
What does the error term, \(\hat{u_i}\) represent in this case? What might individuals have different values of \(u_i\)?
Part H
Do you think that \(Age\) is exogenous? Why or why not? Would we expect \(\hat{\beta_1}\) to be too large or too small?
Question 10
Suppose a researcher is interested in estimating a simple linear regression model:
\[Y_i=\beta_0+\beta_1X_i+u_i\] In a sample of 48 observations, she generates the following descriptive statistics:
• \(\bar{X}=30\)
• \(\bar{Y}=63\)
• \(\displaystyle\sum^n_{i=1}(X_i-\bar{X})^2= 6900\)
• \(\displaystyle\sum^n_{i=1}(Y_i-\bar{Y})^2= 29000\)
• \(\displaystyle\sum^n_{i=1}(X_i-\bar{X})(Y_i-\bar{Y})=13800\)
• \(\displaystyle\sum^n_{i=1}\hat{u}^2=1656\)
Part A
What is the OLS estimate of \(\hat{\beta_1}\)?
Part B
What is the OLS estimate of \(\hat{\beta_0}\)?
Part C
Suppose the OLS estimate of \(\hat{\beta_1}\) has a standard error of \(0.072\). Could we probably reject a null hypothesis of \(H_0: \beta_1=0\) at the 95% level?
Part D
Calculate the \(R^2\) for this model. How much variation in \(Y\) is explained by our model?
Part E
How large is the average residual?
R Questions
Answer the following questions using R. When necessary, please write answers in the same document (knitted Rmd to html or pdf, typed .doc(x), or handwritten) as your answers to the above questions.
Be sure to include (email or print an .R file, or show in your knitted markdown) your code and the outputs of your code with the rest of your answers.
Question 11
Download the MLBattend dataset. This data contains data on attendance at major league baseball games for all 32 MLB teams from the 1970s-2000. We want to answer the following question:
“How big is home-field advantage in baseball? Does a team with higher attendance at home games over their season have score more runs over their season?”
Part A
Clean up the data a bit by making a new variable to measure home attendance in millions. This will make it easier to interpret your regression later on.
Part B
Get the correlation between Runs Scored and Home Attendance.
Part C
Plot a scatterplot of Runs Scored (y) on Home Attendance (x). Add a regression line.
Part D
Run a regression of Runs Scored on Home Attendance. What are \(\beta_0\) and \(\hat{\beta_1}\)? Interpret them in the context of our question.
Part E
Write out the estimated regression equation.
Part F
Make a regression table of the output.
Part G
Now let’s start running some diagnostics of the regression. Make a histogram of the residuals. Do they look roughly normal?
Part H
Make a residual plot.
Part I
Test the regression for heteroskedasticity. Are the errors homoskedastic or heteroskedastic? Generate robust standard errors. Make a regression output table, with one column with regular standard
errors and another with robust standard errors.
Part J
Test the data for outliers. If there are any, identify which team(s) and season(s) are outliers.
Part K
What is the marginal effect of home attendance on runs scored? Is this statistically significant? Why or why not?
Part L
Now we’ll try out the infer package to understand the \(p\)-value and confidence interval for our observed slope in our regression model. Save the (value of) our sample \(\hat{\beta_1}\) from your
regression in Part D as an object. Then, install and load the infer package, and then calculate the slopecalculate(stat = "slope")
under the null hypothesis that there is no connection between attendance and runs.hypothesize(null = "independence")
for 1000 additional simulated samplesgenerate(reps = 1000, type = "permute")
, and save this as an object (it’s a tibble). Then, use this to get_p_value()Set obs_stat equal to your observed slope, and set direction = "two_sided"
. Compare to the \(p\)-value given by lm() and tidy() above.
Part M
Make a histogram of the simulated slopes, and plot our sample slope on that histogram, shading the \(p\)-value.You can use ggplot2 to plot a histogram in the normal way and add a geom_vline(),
setting xintercept equal to your saved object with the sample \(\hat{\beta_1}\) value. Alternatively, you can use infer to pipe your tibble of simulations into visualize(), and inside visualize() set
obs_stat equal to your saved \(\hat{\beta_1}\) object. Regardless of which method you use, add +shade_p_value(). Inside this, set obs_stat equal to your saved slope, and add direction = "two_sided".
Part N
Get the 95% confidence interval for your slope estimate,tidy() your original regression, with conf.int = TRUE inside the command, then select(conf.low, conf.high) and filter by your x variable. Save
this as an object.
and then make a histogram of the simulated slopes (like part L), but instead, add +shade_confidence_interval().Inside of this, set endpoints equal to the object you just made with the low and high
confidence interval values.
Compare this to what you get with tidy() above. | {"url":"https://metricsf20.classes.ryansafner.com/assignment/03-problem-set/","timestamp":"2024-11-12T20:23:34Z","content_type":"text/html","content_length":"25503","record_id":"<urn:uuid:bbee43e1-5dd3-4e1d-ba3a-266193e3ee07>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00247.warc.gz"} |
The dawn and promise of quantum materials
Written by Ahmad R. Kirmani
Illustration representing the future of quantum computing. A quantum chip comprising of diamond and graphene. Credit: Christoph Hohmann, NIM
The field of quantum materials has seen an exponential rise in research interest over the past two decades. The burst of activity has stemmed from an array of observations made in a certain class of
materials that fail to be explained from a purely classical physics standpoint. Consider, for example, the interface between two rightly chosen insulators, such as lanthanum aluminate and strontium
titanate. While classical physics would predict a perfectly insulating interface mirroring the bulk, experiments reveal a highly conducting region. It turns out that such oxide interfaces form
strongly correlated electron systems resulting in a wide variety of emergent phenomena. A plethora of similar mind-bending observations can be explained by invoking the underlying notion of symmetry
breaking at such interfaces, an idea central to quantum materials. Symmetry breaking, in modest terms, is a fluctuation that forces a physical system to leave its symmetric state and become
asymmetric. Foreseeable applications of this materials class are wide-ranging and, in principle, revolutionary: Mott insulators, high-temperature superconductivity, quantum computing and quantum
communication, besides many others.
Some of these findings have already transformed society. Giant magnetoresistance, for instance, is used for data storage currently and was aptly recognized by a Nobel Prize in Physics in 2007.
Graphene, a wonder quantum material, as well as a whole slew of two-dimensional materials discovered and developed subsequently, define the bleeding edge of materials science, promising applications
in flexible and stretchable electronics and the internet-of-things (IoT).
Perhaps the most groundbreaking application that quantum materials promise is quantum computing. The basis of quantum computing is a “qubit,” the quantum analogue of a classical “bit.” Quantum logic
is the underlying math of a quantum computer, much like Boolean logic which defines the classical computer of today. However, quantum logic is different in that a superposition of the two accessible
qubit states can exist simultaneously, and multiple qubits can be entangled—both of which enable math to be encoded in a completely different manner compared to a classical “bit.” Multiple materials
systems have been used as demonstration qubits; one example is the quantum mechanical spin of a crystal defect, such as a nitrogen-vacancy complex in diamond, to initialize, manipulate, store, and
read a pair of physical quantum mechanical states as data. A crucial thrust and a challenge in the race to the quantum computer is the realization of a long-lived coherent quantum system, where the
quantum states remain interconnected, unperturbed and stable for an extended time. In order to find applications in computing, quantum systems need to satisfy the diVincenzo criteria, one of which
requires the qubit to be protected from the surrounding environment to reduce decoherence, or the loss of coherence, that leads to loss of quantum information. Here, quantum materials hold potential.
“Quantum information theory devices use the discreteness for improved information processing or sensing. Topological effects in materials make use of an inherent stability provided by some extended
symmetry property of a material. The canonical example of a topological distinction is that a donut is topologically distinct from a sphere because it has a hole. If you can find ways to encode
information or otherwise design materials that make use of this topological protection you could potentially discover valuable or novel applications,” says Richard P. Muller, an expert in quantum
materials from Sandia National Laboratories, and a co-organizer of Symposium LN01 in the upcoming 2018 MRS Spring Meeting & Exhibit in Phoenix, which is focused on the role of materials science in
quantum information technologies. Muller says that beyond quantum sensing or information processing, “quantum materials could provide spintronic devices for more standard classical computing devices.
They have been proposed for improved thermoelectric properties, and for improved microwave circulators for low temperature electronics.”
Christopher J. Richardson, based at the University of Maryland, who is a co-organizer of Symposium LN01, strongly feels that materials have the potential to revolutionize quantum computing in the
near future. “We, as materials scientists, are focused on both conventional materials and quantum materials that can impact quantum information devices. A fundamental challenge is to be able to tune
the properties of these materials to enhance information stored in quantum states. There is, therefore, a very tight coupling between materials synthesis and the technological application,”
Richardson says.
In the words of John Preskill, a theoretical physicist at the California Institute of Technology, for some applications, quantum computers may lead to “quantum supremacy”—enabling operations,
processing, and calculations that are currently beyond the limits of the most powerful of today’s computers. Researchers are only beginning to unlock the grand potential that these materials have to
I hope in his punchline near the end of the 2nd paragraph,
"... define the bleeding edge of materials science ..."
the author really meant 'leading' not 'bleeding'.
But, this is the pound of flesh that automatic spellchecking
soft wares extract from us.
Arguably, the ultimate success of ‘Quantum materials’ will be
when they would bring forth unpredicted advantages over current
materials that are far in excess of putative suitability in non-Turing-von Neumann computation.
Such an expectation, is also consistent with the counter intuitive nature of the quanta.
The allure of ‘Quantum materials’ goes far beyond computation.
Timir Datta, Physics & Astronomy, University of South Carolina | {"url":"https://materials.typepad.com/mrs_meeting_scene/2018/03/the-dawn-and-promise-of-quantum-materials.html","timestamp":"2024-11-05T18:49:12Z","content_type":"text/html","content_length":"46080","record_id":"<urn:uuid:1ee58791-1998-434a-8298-cb8a077b1069>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00075.warc.gz"} |
Private Counting from Anonymous Messages: Near-Optimal Accuracy with Vanishing Communication Overhead
Differential privacy (DP) is a formal notion for
quantifying the privacy loss of algorithms. Algorithms in the central model of DP achieve high accuracy but make the strongest trust assumptions whereas those in the local DP model make the weakest
trust assumptions but incur substantial accuracy loss. The shuffled DP model (Bittau et al., 2017; Erlingsson et al., 2019; Cheu et al., 2019) has recently emerged as a feasible middle ground between
the central and local models, providing stronger trust assumptions than the former while promising higher accuracies than the latter. In this paper, we obtain practical communication-efficient
algorithms in the shuffled DP model for two basic aggregation primitives used in machine learning: 1) binary summation, and 2) histograms over a moderate number of buckets. Our algorithms achieve
accuracy that is arbitrarily close to that of central DP algorithms
with an expected communication per user
essentially matching what is needed without any
privacy constraints! We demonstrate the practicality of our algorithms by experimentally comparing their performance to several widely-used protocols such as Randomized Response (Warner, 1965) and
RAPPOR (Erlingsson et al., 2014).
Originalsprog Engelsk
Titel Proceedings of the 37th International Conference on Machine Learning
Forlag ML Research Press
Publikationsdato 2020
Status Udgivet - 2020
• Differential Privacy
• Shuffled DP Model
• Communication-efficient Algorithms
• Binary Summation
• Histograms
• Machine Learning Aggregation
• Privacy Loss Quantification
• Central Model
• Local Model
• Randomized Response
• RAPPOR
Dyk ned i forskningsemnerne om 'Private Counting from Anonymous Messages: Near-Optimal Accuracy with Vanishing Communication Overhead'. Sammen danner de et unikt fingeraftryk. | {"url":"https://pure.itu.dk/da/publications/private-counting-from-anonymous-messages-near-optimal-accuracy-wi","timestamp":"2024-11-06T01:17:43Z","content_type":"text/html","content_length":"53268","record_id":"<urn:uuid:b7ce882b-a78f-442e-a14a-ff168c86f10f>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00220.warc.gz"} |
Chi-Square Distribution
$X \sim \chi^2_{(\nu)}$
This applet computes probabilities and percentiles for the chi-square distribution: $$X \sim \chi^2_{(\nu)}$$
• Enter the degrees of freedom in the $\nu$ box.
• To compute a left-tail probability, select $P(X \lt x)$ from the drop-down box, enter a numeric $x$ value in the blue (left) box and press "Tab" or "Enter" on your keyboard. The probability $P(X
\lt x)$ will appear in the pink (right) box. Select $P(X \gt x)$ from the drop-down box for a right-tail probability.
• To determine a percentile, enter the percentile (e.g. use 0.8 for the 80th percentile) in the pink (right) box, select $P(X \lt x)$ from the drop-down box, and press "Tab" or "Enter" on your
keyboard. The percentile $x$ will appear in the blue (left) box.
On the graph, the $x$ value appears in blue while the probability is shaded in pink.
• Probability density function $$f(x)=\frac{1}{2^{\nu/2}\Gamma\left(\frac{\nu}{2}\right)} x^{\nu/2-1} e^{-x/2}$$ where $x > 0$ and $\nu > 0$
• $\mu=E(X)=\nu$
• $\sigma^2=Var(X)=2\nu$
• $\sigma=SD(X)=\sqrt{2\nu}$ | {"url":"https://homepage.divms.uiowa.edu/~mbognar/applets/chisq.html","timestamp":"2024-11-02T05:56:31Z","content_type":"text/html","content_length":"7331","record_id":"<urn:uuid:69c9009e-dc13-402d-aeb6-2dd405844f6a>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00647.warc.gz"} |
Using 2 markers to interact
hi. i have another problem.
(i hope that some day problems will end and i can help a bit :-) )
im trying to connect 2 markers to work together.
in first step i wanted to draw a simple line
from one marker to another.
i have the matrix transform but don’t know what it gives back.
coordinates ? vector ?
i tried to split it on XYZ and apply that data to a vector
but that ditn’t work somehow.
when i try to remember my math in school i would have to substract the two positions and use this as a vector to draw the line… but how…
in second step i wanted to have a texture between two markers
from their middle point and as big as the distance is.
how could i reach this ?
i searched the forum and the docs but didn’t find.
ok… solved the line thing, i will post it this evening.
Did you read this post??
Or is this not about feducials? *before I patch something useless.
Still my problem is to draw a plane out of 2 points.
what i have is 2 points (XYZ) which i connected with a line.
now with this 2 points i want to create a vector plane if thats possible. i tried quad, plane and nothing worked.
if someone could give me a tip i would be a very happy man.
how? are the two points the edges? if so, which edges? and should the plane be paralell to the screen?
if you use a quad, its quite simple, add the xyz coords of the positions togehter and divide by 2. so you get the center between the two positions. there you place the quad. then you would need the
size of the quad, this could simply be the difference of the x and y coords.
yea, i will have to set the center of the quad and
resize it. like you have said. i will try to divide that
If I get it correct, your 2 markers are the corners off an object, and that object can be a quad? so you can use it to scale and your object?
edit: forum goes fast 2day :D
This.v4p (6.7 kB)
wow, i tried that before and it didn’t work.
now, i have that points in 3d so the z axis is missing.
here is what i have (using ARtools to get tha matrix and so on)
*selecta2: thats now my position centering, after some shugar (coke and chokolate) my brain started working again.
now the resizing :-)
thank you very much til now.
Selecta.v4p (7.4 kB)
Selecta2.v4p (12.8 kB)
Yea, its done ! Thank You.
i will post the final project some day.
using spreads, its only 3 nodes:
Selecta3.v4p (10.4 kB) | {"url":"https://discourse.vvvv.org/t/using-2-markers-to-interact/3427","timestamp":"2024-11-09T08:09:27Z","content_type":"text/html","content_length":"30328","record_id":"<urn:uuid:ecbad657-9797-4d46-bc8b-fa35f896915d>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00768.warc.gz"} |
Topics in Linear Dynamical Systems
Suitable for
A linear dynamical system is a discrete- or continuous-time system whose dynamics is given by a linear function of the current state. Examples include Markov chains, linear recurrence sequences (such
as the Fibonacci sequence), and linear differential equations.
This project involves investigating the decidability and complexity of various reachability problems for linear dynamical systems. It would suit a mathematically oriented student. Linear algebra is
essential, and number theory or complexity theory is desirable.
A relevant paper is Ventsislav Chonev, Joël Ouaknine, James Worrell: The orbit problem in higher dimensions. STOC 2013: 941-950. | {"url":"https://www.cs.ox.ac.uk/teaching/studentprojects/337.html","timestamp":"2024-11-12T19:27:35Z","content_type":"text/html","content_length":"24795","record_id":"<urn:uuid:0dc3e495-6129-43d7-a7ab-435cab0b61c3>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00231.warc.gz"} |
More thoughts on showing Margin of Error in survey data with Tableau - Data RevelationsMore thoughts on showing Margin of Error in survey data with Tableau - Data Revelations
More thoughts on showing Margin of Error in survey data with Tableau
A big “thank you” to Daniel Zvinca, Chris Lay, Anna Foard, Jeffrey Shaffer, and Joe Cohen for their feedback and encouragement.
I published a blog post earlier this year on how I recommend showing results for Likert-scale questions broken down by different demographics. I had become fond of how organizations like Pew Research
does did and decided to share how to do this using Tableau. Here’s what is looks like.
Percent Top 2 Boxes just means the people who selected a 4 or a 5 (think satisfied and very satisfied, or important and very important, etc.).
Within hours of posting this article, two friends and colleagues contacted me with invaluable feedback.
The first was Daniel Zvinca who suggested an alternative approach to showing how each of the different demographic components contributed to the overall result.
Within minutes of receiving Dan’s suggestion, I heard from Chris Lay who was alarmed that my approach did not consider margin of error, especially since the response count for traditionalists were so
Goodness, he was spot on. I had forgotten my own blog post on this very issue.
Going down the rabbit hole
These two comments together led to a lengthy email thread among Dan, Chris, as well as friends and colleagues Anna Foard and Joe Cohen. The question was how to show margin of error and show how each
of the different demographic components push and pull from the overall result. Daniel was off to the races with different takes on this. Here are a handful of his experiments.
Note sure about margin of error works? The “survey result” is the percent of the sample who selected an item, displayed as dots. The “true result” is the percent of the entire population who
would select that item. A confidence level of 95% means the true result will be within the calculated Confidence Interval (the error bar range) in 95 out of 100 samples.
This also means that 5% of samples would not return the true result within the error bars.
So, how do you do this in Tableau?
Variable row height
Those of you who follow my blog know that this is not the first time I’ve tried to emulate Daniel’s elegant and informative approaches.
In his margin-of-error examples there was one thing I found particularly appealing and that is the variable height for each row. That is, instead of having a fixed row height, like this…
… we adjust the row height based on the number of responses per age group, like this.
I much prefer the variable row height but to get this without too much fuss in Tableau I had to ditch the totals row. Do note that the vertical line for each question shows the overall response.
Color or monochrome?
Having settled on the variable row approach I experimented with monochrome vs. colored dots and decided to post a poll asking people which they preferred.
To my disappointment the 300 respondents I surveyed preferred A by a wide margin, with most people saying the colors helped them more easily determine which age group was which.
So why am I disappointed? I’ve become a little anal compulsive about not having color or size legends in my visualizations and instead display mark labels for just the first row (see this post for my
take on how to do this.) It turns out it’s quite a bit harder to have colored dots and monochrome error bars, and have the labels be colored as well, but not have the labels appear too close to the
dots. So yes, I’m willing to put up with a little nonsense to get the labeling “just so.” As for showing the totals, this is where I draw the line as it would have meant some futzing to the
underlying data.
How it works
At the end of this post I share an embedded dashboard that you can download. Let me walk you through how to use the dashboard and then we’ll look under the hood at how some of the elements work.
Item (1) allows you to switch among several different Likert-scale questions. There are questions about degree to which you agree, importance, and satisfaction (shown here.).
Item (2) allows you to switch among different demographics. Here’s what the results look like when we break things down by Gender.
Item (3) allows us to toggle on and off the error bars. This can be very helpful in helping an audience understand the range of possible valid survey results. I strongly encourage you to experiment
with the dashboard at the end of the post as the animated error bars can help the uncertainty associated with survey data pop.
Item (4) allows us to specify the threshold for showing results. In this case, if there are fewer than 30 respondents within a segment, we won’t see results for that segment.
Item (5) allows us to specify the degree of confidence. A setting of 90% indicates that the true value (what you would get from the entire population and not just sample) will appear within the
margin of error range in nine out of ten samples. Our hope is that what we’re looking at now is NOT that one-out-of-ten case where the true value is outside the error bars. Increasing the confidence
to 95% or 99% will make the error range wider but increases the likelihood that the true value is present in 95 out of 100 samples, to 99 out of 100 samples.
Item (6) allow us to toggle between showing the Top 2 or Bottom 2 boxes and control the sort order. I’m not sure if I should have separated these into two choices or had a single choice that forced
the sort order. In any case, it’s possible to apply settings that result in a cluttered display, as shown here.
Looking under the hood
Here’s how the visualization is built in Tableau.
Notice that we have a dual axis chart (arrow at top). The first pill is responsible for the dots and the second pill displays the error bars. Let’s remove the dual axis and separate these into two
The error bars (2) come from placing Measure Values on Columns. This is a line chart that connects the smallest possible value with the highest possible value. I won’t get into how these are computed
here as you can read this blog post that gets into the specifics.
The dots come from the field [Show Top 2 or Bottom 2] which is defined as follows:
IF [Top or Bottom]=1 then [% Top 2 Boxes]
ELSE [% Bottom 2 Boxes]
Sorry for the extra level of confusion here, but every time I present a class where I show how to compute Top 2 boxes somebody asks me about Bottom 2 boxes. In any case, if the user indicates that he
/she want to see the Top 2 Boxes using the parameter [Top or Bottom] we’ll use [% Top 2 Boxes]. This, in turn, is defined as follows:
SUM(IF [Value]>=4 then 1 else 0 END) / SUM([Number of Records])
This translates as “add up everyone who selected a 4 or a 5 and divide by everyone who answered this question.”
Stacking the dots nicely
The size of the dots is controlled by Item (1), SUM([Number of Records]). Actually, it’s the area of the circles, not the height of the circle, but we will need to determine the height of the circle.
To get the dots to stack based on the height of each circle we need to figure out the diameter of each circle, or at least recognize that the area relates to the radius of the circle, squared, times
Pi. It turns out the only critical thing is using the square root of the [Number of Records] to stack the dots proportionately by their height, and that’s what we do with Item (2), the field
[Breakdown height]. This is defined as follows:
RUNNING_SUM(SQRT(SUM([Number of Records])))
The first (smallest) dot starts at zero and moves up the square root of the number of records for that first dot (the square root of 41). The second starts where the first one ended and moves up the
square root of the number of records for that dot (the square root of 143).
The value axis (normally hidden) shows the overall running sum (Item 3).
Yes, it’s somewhat convoluted.
But I think it’s worth the effort.
Some Tableau mishegoss
As I said before, I may go a little overboard in trying to avoid a color / size legend that sits over to one side. In this case I wanted to directly label the first row. This is easy if you label the
monochromatic error bars, but I wanted the labels to be the same color as the dots, so I had to label the dots themselves.
And that meant I needed to add some extra spaces to that the labels would not obscure the dots.
Let’s look at what’s driving the labels.
If we click the Labels button and then ellipsis button, we see this Edit Label dialog box.
Let’s look at how [Top Labels] is defined.
IF FIRST()=0 THEN
IF LEN(ATTR([Breakdown])) >1 THEN ATTR([Breakdown])+" -"
ELSE "Overall - " END
This translates as “if we are in the first… row? Column? Cell? Whatever… if we are breaking down some demographic, display the elements of the demographic; otherwise, display the word ‘overall’.”
I’ve become very fond of this first row trick, so much so that I’ve written a separate blog about the technique. The crucial thing is to make sure that Compute Using is set to [Breakdown] so that
only marks for the first question (row) are displayed.
Now, what about those extra spaces so there’s some buffer between the labels and the dots? Here’s how [Top Spaces] is defined.
IF FIRST()=0 THEN "⠀⠀⠀⠀⠀" END
Those spaces between the quotes… they are not just any run-of-the-mill spaces; those are special Braille Unicode 2800 spaces that Tableau won’t trim when it concatenates all the fields.
There’s one more bit of mishegoss.
Notice that there is a field called [Breakdown height reference line] on level of detail. I use this to create a hidden reference line that under some circumstances adds some padding. Here’s what
happens if we breakdown by gender and don’t have this reference line.
And here’s the layout when we have the hidden reference line.
Is it worth all the effort to get the variable spacing, the colored mark labels, etc.? That’s up to you, but at a minimum you need to make sure your audience knows that when you report survey results
you are providing an approximation, and your audience should know that it could be as low as X and as high as Y.
Here’s the dashboard for you to explore and download.
By Steve Wexler|2021-11-18T14:07:27+00:00November 17th, 2021|Business Visualizations, General Discussions, Visualizing Survey Data|0 Comments
Share This, Choose Your Platform!
About the Author: Steve Wexler
Related Posts | {"url":"https://www.datarevelations.com/marginoferror/","timestamp":"2024-11-12T13:37:33Z","content_type":"text/html","content_length":"257153","record_id":"<urn:uuid:d6a8e33e-99e1-4fb8-84a1-ed77949326fa>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00781.warc.gz"} |
Problem Q
Have you ever been on a long road journey? AAA (the American Automobile Association) has a tool for long road trips. It’s called a TripTik, and it follows the highways, showing points of interest.
You are building a TripTik app, which allows users to see what’s on their route. It models a highway as a straight line, and points of interest as points along that line. All points have an integer
coordinate as well as a unique integral weight. Your app provides a viewport, which can scale in and out. Also, to prevent the display from becoming too cluttered, only a small number of the points
with the highest weights are shown. The initial viewport is centered at $0.0$, and shows from $-1.0$ to $1.0$ on the line.
There are three valid operations for changing your viewport:
1. Zoom out: double the dimensions of your viewport while keeping the center the same; this can always be done regardless of the current dimensions of the viewport.
2. Zoom in: halve the dimensions of your viewport while keeping the center the same; this can always be done regardless of the current dimensions of the viewport.
3. Recenter: change the center of your viewport to be equal to a point of interest visible in your viewport (including the boundary).
There is an important caveat: Your TripTik app will not render all points of interest in a given viewport; instead, it will only render a certain number of points in the viewport with the highest
weights. The remaining points with lower weight are not visible, and therefore are not valid targets for the recenter operation.
For each point of interest, determine the minimum number of operations needed to go from the starting viewport to a viewport where that point of interest is centered and visible. Consider each point
of interest independently.
The first line of input contains two integers $n$ ($1 \le n \le 10^5$) and $k$ ($1 \le k \le 4$), where $n$ is the number of points, and $k$ is the maximum number of points visible in the viewport.
Each of the next $n$ lines contains a single integer $x$ ($|x| \le 10^8, x \neq 0$). These are the points of interest. The weight of each point is equal to its position in the list, with lower
weights earlier in the list. All points will be distinct.
Output $n$ lines, each with a single integer, indicating the minimum number of operations necessary to get from the starting viewport to a viewport which is centered on the corresponding input point
and can see that point. Output $-1$ if this isn’t possible. The order of output lines should correspond to the order of inputs for which they’re the answer.
Sample Input 1 Sample Output 1
4 2 -1 | {"url":"https://nena20.kattis.com/contests/nena20/problems/triptik","timestamp":"2024-11-12T13:40:03Z","content_type":"text/html","content_length":"29015","record_id":"<urn:uuid:ad582042-7dbe-4909-807a-926a341ecd33>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00535.warc.gz"} |
Math(s)! Currently: Summer of Math Exposition
It's approximately the second occurrence of 9:26 in my timezone, so I'm making a thread, mostly so I can rant about misunderstandings about appearances of finite sequences in the digits of pi.
So approximately every 2pi radians around the sun or so, we come across the meme that pi contains every single finite sequence of decimal digits somewhere in its expansion. This may be true! But
it's a mathematical statement, and thus is subject to a test that is, if not stronger, perhaps perpendicular to truth: provability. Mathematicians aren't supposed to accept statements unless
they're proved or are specifically stated to be unfounded assumptions for the purpose of hypotheticals or counterfactuals.
So let's examine what we can say about the digits of pi.
Pi is an irrational number, which means that there is no finite string of digits such that pi is eventually just that string repeated over and over again. Some people take this to mean that pi is
"random", but that's not the case. Pi is a constant. We can compute what its digits are, so it's certainly not unpredictable in any way once we know that we're dealing with pi.
Furthermore, merely being irrational doesn't mean its patternless. Champernowne's number, 0.1234567891011121314... is irrational, but I'm pretty sure we can all guess what comes next.
Champernowne's number also has the property that, given any finite string of decimal digits, you're going to find it eventually somewhere in the expansion. This property is called being a
Disjunctive number.
Puzzle 1: suppose the string starts with a bunch of 0s. Champernowne's number is built out of integers that don't start with a bunch of 0s, but the string appears anyway. Where?
It is also not the case that pi, by virtue of being irrational, is necessarily disjunctive. The Liouville constants, which are some nonzero digit in the (k!)-th digit and 0 everywhere else, are
irrational, but if all of those nonzero digits are 1 then we certainly don't get every finite string.
Before anyone asks, pi is also transcendental, but that also doesn't tell us anything. Champernowne's number and the Liouville constants are all transcendental.
Actually, we expect pi to have a property that's stronger than being disjunctive: we expect pi to be normal.
A normal number is one where for any fixed k, the strings of length k occur equally often. This is impossible for finite expansions, because you can pick k to be the exact length of the expansion
and that's the only string of that length that can occur, so a normal number must have an infinitely long expansion.
So what do we know about pi? Very little! We have methods for computing the digits of pi out to any finite accuracy we want, but we don't know much about the infinitely long tail that remains
For instance, we don't even know that, say, the digit 7 appears an infinite number of times.
So what, one may ask, if there are only, say, 10^10^6 7s or whatever? Well, it doesn't matter much to most people, but to a mathematician, the rebuttal is that almost all* finite strings of
decimal digits contain at least 10^10^6+1 7s in them, so if there are only 10^10^6 7s, then pi is missing almost all finite strings! That's certainly not disjunctive at all!
Speaking of almost all, almost all** real numbers are normal. This tells us absolutely nothing about pi, because pi is not a randomly chosen real number, anymore than 1 is, and 1 is certainly not
Contrast with the fact that almost all real numbers are noncomputable, but we have several dozen methods for computing pi. So any property of "almost all real numbers" could hit or miss pi.
There have been statistical tests performed on the known digits of pi, and they are consistent with being disjunctive, even consistent with the stronger statement of normality.
But this is a statistical test, and while that level of accuracy might be good enough for scientists, science is not math, and the tests that work for scientists don't work for mathematicians.
Other numbers expected to be but not known to be normal: e, log(2), sqrt(2), Euler's constant, or indeed almost any real number not specifically created to be normal.
See also:
*Almost all in the following sense: consider the proportion of strings of length at most k that contain 10^10^6 7s out of all of the strings of length at most k. As k goes to infinity, the
proportion goes to 1.
**Almost all in the Lebesgue sense, i.e. the complement has measure 0. A perhaps more intuitive picture: if you have a function f that is 0 at normal numbers and 1 at nonnormal numbers, and you
integrated f from -infinity to +infinity, you'd get 0.
Last edited: May 18, 2015
I note that a lot of human evaluations of "probability" are much influenced by the distinction between "looks random" and "looks non-random".
It is intuitively obvious to nearly everyone that you're more likely to encounter "2791473" in a given region of something like pi or e than you are to encounter "0000000", but they are in fact
equally probable, in general. The distinction is that one of them is more "notable", but notability is not a thing we have coherent definitions for. The famous joke is that there are no
uninteresting positive integers; consider that if there were, there would be a smallest uninteresting positive integer, and that would be interesting.
Lissa Lysik'an Dragon-loving Faerie
re: probability
When Baby first learned to play poker she got a hand that was A K Q J 10 all of the same suit. She thought it was special, until it was pointed out that it had the same odds as any other hand.
She became addicted to math.
We can actually distinguish between more or less notable strings in at least one way, although it's not entirely natural. Kolmogorov complexity with respect to a given programming language* is
the minimum length of a program in that language needed to output the given string*. The uncomputable numbers mentioned in the first post are numbers with infinite Kolmogorov complexity in any
Turing-complete language.
Our understanding of randomness tends to believe that strings with higher Kolmogorov complexity are more expected in random sequences, at least partially because Kolmogorov complexity aligns
somewhat with predictability; an easily predictable sequence often has a short program to generate it, and that short program thus puts an upper bound on the Kolmogorov complexity.
There's also the fact that we have a naive, learned distribution on strings and thus a sense of entropy on strings via Shannon information. Strings with high Shannon information tend to be viewed
as unpredictable and thus "more likely to occur" in a sequence we're told is random, just as you're supposedly more likely to end up in a high entropy physical state than in a low entropy one,
even though the probability of landing in any particular physical state has nothing to do with the entropy. So we believe that 000000 is unlikely, and we believe that a royal flush is unlikely,
and they are, it's just that "not 000000" and "not a royal flush" are large sets of things that are, individually, equally unlikely, but are harder to distinguish because we don't care.
*Assuming that the language has a finite number of distinct characters.
**Technically not output, because the string is allowed to be infinite, but rather, given any number n, the program will in finite runtime output the first n characters of the string but not
necessarily halt after doing so.
***My posts are going to be full of footnotes like this regarding technical definitions and caveats, because the caveats are often the best parts of pure mathematics.
Last edited: Mar 15, 2015
I am pretty sure a royal flush is "special" even if it's not statistically less likely than any other specific set of five cards.
There is a fascinating result I saw once, which is that the most even distribution of suits in a bridge hand is 4-3-3-3, but the most common is 4-4-3-2.
Double-checked that.
(I originally picked it up as a sample of "interesting things in HAKMEM" from the Jargon File, I think.)
It's special because it's low entropy/low complexity.
So definitions. Terminology and vocabulary are what we use to communicate ideas. Those of us who like to be understood use definitions to explain what a given piece of terminology means, and
those of us who like to understand will ask for definitions if we don't know a given piece of terminology, or suspect we know it in a different way than the speaker does. A lot of arguments in
academia or academic settings are often about definitions of things, about usage of terms and such. "Words have meanings" blah blah blah.
Mathematicians also use a lot of definitions. But I think, and many of my colleagues agree, that a "definition" in mathematics is somewhat different from a "definition" in the colloquial sense.
A definition in the colloquial sense tries to attach a meaning to a term. Colloquially, we have experiences, and we have what we consider to be the real world, and these provide external
references that we can use to attach meanings to the words that we use to communicate. We can say "rabbit" and point to an rabbit, and some fairly common assumptions will tell the listener that
we mean "rabbit" and not, for instance, "bundle of rabbit parts", and the listener will thus get an idea of what "rabbit" means, and from these kinds of concrete, demonstrable things we build up
more complicated, more abstract definitions. And then there's a large amount of leeway because the example rabbit is not all rabbits, real and possible, dead and living and unborn, and because
the building up is fairly sloppy, relying on people having roughly the same set of concrete definitions. But the hope is that the definitions match pretty well, for some unspoken, undefined
notion of "pretty well".
In mathematics, we also build more complicated, abstract definitions from simpler definitions, but there's nothing at the bottom. No rabbits for us. We tried. At the end of the 19th century and
the beginning of the 20th century, there was hope that we could firmly ground mathematics without necessarily tying it to physical objects. But results in mathematical logic and philosophy proved
that the endeavour was doomed, and later results* showed that it was a bad idea anyway. But without a firm ground, without those concrete definitions, we can't rely on meaning.**
But we still like the notion of "truth". In the absence of meaning, truth is hard to get at, so instead what we do is proof. And indeed, that is one way of looking at proof; as one of my
colleagues describes it, a proof is "truth without meaning, without content".
Most proofs are for mathematicians, each of whom has an idea of what any given term in their area of interest means. But there is no need for them to have the same idea, or at least that is the
hope. Rather, a proof of a theorem is intended to convince a given mathematician, with a given understanding of the terms involved, that the statement of the theorem is true when interpreted via
their understanding of those terms. This includes computers, which, as far as we know, assign no meaning to terms; they just trace through chains of definitions. So if we want to prove things to
computers, we need to have all those definitions in place, even if at the end they don't have any meaning to them.***
Humans don't like this most of the time. A given mathematician, armed with their given idea of what the terms mean, doesn't usually want most of the definitions, because that idea does most of
the work that the definitions are supposed to do. When you say "sphere", most mathematicians summon up an idea of what a sphere is pretty immediately, and will understand when you talk about
properties of the sphere in terms of that idea. But those ideas of what a sphere is can be quite different, not incompatible but with very different associations and connotations, and the theorem
in question might not be true for both ideas. So you need to tighten the definition to make sure that everyone is thinking about the right kind of sphere, including the computer which isn't
thinking about any ideas and is just looking at the words. The precision that mathematicians are famous for is a necessity, not a joy.
So the end result is that mathematics often relies on a lot of definitions and mathematicians are often unhappy with this. In the words of seeb's mathematician father, "a definition is the
obituary of an idea", but that's the price we pay for trying to get truth that's independent of meaning.
Note: The above might seem to imply that I believe that colloquial, meaning-based definitions are less prone to two people working from the same definition coming to different understandings. I
don't believe that at all; I am convinced that given two people working from the same definition, they might come to perfectly equivalent understandings, indistinguishable just by talking to
them, but that this is rare and that the best we can hope for is that the mismatch is with regards to edge cases while the usual case is that the mismatch is hard to convey in words.
Mathematicians are just more open about the issue, trying to use it as a feature rather than regarding it as a bug.
*E.g. category theory, a lot of which is looking for provably valid analogies.
**Note that this is with regards only to the externally visible parts of mathematics, the communications and proofs and such. What a given mathematician believes regarding what happens when they
do mathematics can be quite different.
***Think of Serle's Chinese room, wherein an English-speaking man who understands no Chinese gets messages in Chinese, matches the symbols to a list of rules he has in English, and then makes a
response, again in Chinese, according to this set of rules, in a way that an observer who understands Chinese would think that there is a person who understands Chinese inside the room, rather
than the man who understands no Chinese and the list of rules. We need to write that set of rules. Usually this thought experiment is supposed to ask, since the man understands no Chinese, where
is the understanding in this situation?
Reminder to myself to make a post about the hole at the bottom of mathematics. Because it's hilarious that not only do we not know what we're talking about, we can prove that we can't know what
we're talking about, and we can prove that we can't know to what extent we don't know what we're talking about.
Also should probably make a post about network theory, since that's a good example of applications of mathematics and mathematical thinking to real-world problems in a way that people don't see
at all given the odd emphasis on calculus as the end goal of math education.
Morven In darkness be the sound and light
@Exohedron: incompleteness theory? I like that we can prove that we can't reach every truth by working from base principles.
Yeah, I'm going to talk about the incompleteness theorems and the rather bizarre set of things that do have complete theories. This has interesting consequences in things like computer science
and ethics and their intersection: how to keep artificial intelligences from destroying/enslaving humanity.
The hole at the bottom of the mathematics.
In my previous post I mentioned that there's a hole at the bottom of mathematics in that ultimately we don't know exactly what we're talking about; we can't distinguish between things that act
the same way or that are exactly translated into each other. For the most part we don't care; that's the power of mathematics, to strip away the details of "what is" and tell you "what does". Who
cares if you're looking at water or money or electrons or racecars, the underlying behavior is the same and the mathematics only looks at the behavior.
But there's a bigger hole, and this one might be important.
Back in the 19 and early 20th century, mathematicians dreamed that we could put mathematics on a solid foundation. We could say "all mathematics is describable using this small collection of
things, and so we only need to know about this small collection of things." We settled on sets, which are collections of things, and it won't help to try to go any further as to what they are.
Instead, we tried to nail down what they do.
So for instance, we think: given a set, we can think about all of the elements of the set that have a given property, and make a set of all and only the things that have that property. Given the
set of horses, we can look at the subset of white horses. That should be fine, right? And if we have two sets we should be able to consider the set containing everything in both sets.
So what about all sets? Is there a set of all sets? There should be, if everything is going to be a set.
But alas. Namely, suppose that we had a set of all sets. Then for any property we can think of, we can look at the set of all sets that have that property. How about the property of all sets that
do not contain themselves? Let's consider the set S of exactly all sets that do not contain themselves.
And you can say: sets can't contain themselves. Which is great, because that means that every set is in S. So since S is a set, S is in S, and therefore S contains itself and therefore cannot be
in S. So S doesn't contain itself and therefore belongs in the set S. And therefore...
This is Russell's paradox, telling us that we can't make a set of all sets. The thing of all sets must be something other than a set. "All sets" is not an object we can manipulate mathematically!
So we can't just make everything a set. We run into the problem of, now that we know that some things aren't sets, where do we put the boundary? Where do we say "these things are not sets"? It
turns out that there isn't a good way to do find a boundary; anything is either too inclusive, including some things that lead to paradoxes, or too exclusive, leaving mathematics too weak to be
useful or interesting; often both. There are branches of mathematical logic devoted to figuring out what goes wrong when we assume that a given thing is a set.
The current compromise is called the Zermelo-Fraenkel set theory*. The thing of all sets is called a proper class, and we can't manipulate proper classes in ZF. But we do have some rules for how
sets work, and these are fine for the most part, and most importantly we can't do ridiculous things like Russell's paradox.
So we don't know what sets are, and we don't know what sets aren't. We did have a decent idea of how sets behave. We could translate things into set theory, and we could prove them in set theory.
It took forever to do even basic things in set theory, but that's because set theory is the machine language of mathematics; you don't actually want to do mathematics in set theory, you just want
to know it's there and that it is, in theory, possible to translate into and out of it.
This post is kind of long already, so the next part, about Godel, will appear later.
*Often we include the Axiom of Choice, but that leads to fun things like the Banach-Tarski paradox, where you take a ball, you split it into nine parts, you move the parts around using just
rotation and sliding, moves that don't change the sizes of things, and you can reassemble the parts into two distinct balls, each the size of the original. So some people aren't happy with Choice
and don't include it.
Before we go into the terrible things that happened with Godel, I want to take a short detour into counting.
What does the "size" of a set mean? Well, we usually start with some bunch of objects and say "1, 2, 3,..." until we run out of objects. Or until we get bored and just guess.
Mathematicians aren't allowed to get bored or guess, but we also aren't always going to run out of objects. If someone were to ask "how many numbers are there?", we need to find an answer.
The eventual answer that we use these days came from Georg Cantor. His idea was that we talk about the sizes of sets by comparing them to other sets.
If we had the set {a, b, c, d} and the set {P, Q, R, S}, then we want to say that they have the same number of things in them. And we can do that by matching them up, say, matching a to P, b to
Q, c to R and d to S. Or alternatively we could match a to S, b to R, c to Q and d to P. In either case we have what is called a bijection, a one-to-one correspondence of things in the first set
with things in the second set. Everything in {P, Q, R, S} is matched to something in {a, b, c, d}, and no two things in {a, b, c, d} are matched to the same thing in {P, Q, R, S}. We say that two
sets have the same size, called "cardinality", if there is a bijection from one to the other. You can check that if there's a bijection from one set to another set, then there's automatically a
bijection from the second set to the first set.
This works fine for finite sets, but what about the set of natural numbers? {0, 1, 2, 3,...}? Well, if we had any finite set then we'd run out of things before we could match every natural
number. So {0, 1, 2, 3,...} is infinite. We call its cardinality aleph[0]. We also call this countable infinity, because it's the set of counting numbers.
What about the set of even natural numbers? {0, 2, 4, 6, 8,...}? That's contained in the natural numbers, and is missing a bunch of numbers. So surely it must be smaller?
It turns out that nope, it's the same size, because we have a bijection. We match each natural number n with the even number 2n. To 1 matches to 2, 2 matches to 4, 3 matches to 6, and so on. So
for each natural number there is an even number, and vice versa. So there are just as many even numbers as there are natural numbers! And similarly, there are just as many odd numbers as natural
numbers. So we have that aleph[0] + aleph[0] = aleph[0]. Similarly, we can show that the integers, positive and negative, again have cardinality aleph[0], since the integers are basically just
two copies of the natural numbers.
The tricky ones are whether there are the same number of rational numbers and the same number of real numbers. For the even numbers, between any even number we're only skipping one natural
number, so it's not so strange to think that we can make them up later. But for the rational numbers, well, between any pair of natural numbers there are infinitely many rational numbers, and
even more real numbers.
But the cardinality of the set of rational numbers is just the same as the cardinality of the natural numbers! The construction works best with a picture so I'm not going to try to reproduce it
here. It boils down to writing each rational number as a pair of integers p/q, and then drawing a path on the plane that hits all the points with integer coordinates; the number p/q is matched
with how far along the path you have to go to get to the point (p, q).
What about the real numbers? How many real numbers are there?
Here's where Cantor's first big result comes in. It ends up looking a lot like Russell's paradox, which is why I wanted to mention this whole thing about cardinality.
Specifically, we look at all the numbers between 0 and 1, written in binary. So we get decimal points followed by infinitely long strings of 0s and 1s.* If it's not infinite, just tack on an
infinite string of 0s.
Suppose that we had only countably infinitely many real numbers between 0 and 1. Then we could match each real number with a natural number, saying maybe that 0.10000... matches to 0, and
0.11010101... matches to 1, and 0.10010001000.. matches to 2, and so on. Write them in a list.
Now we construct a new number not on our list. We say "for the nth digit of our new number, look at the nth digit of the nth number on the list, and make the nth digit of our new number
different". So, based on our listing above, our new number would start with .0, since the first digit of the first number is 1, and then we'd say 0, since the second digit of the second number is
1, and then we'd say 1, since the third digit of the third number is 0, and so on, getting .001...
So now we have a number where, for any number n, the nth digit does not match the nth digit of the nth number, so it can't be equal to the nth number. So it can't be on our list!
So no list we make can have all of the real numbers between 0 and 1 on it; since making the list is the same as assigning to each string a natural number, that means we can't make a bijection
between the reals and the natural numbers. The set of reals is strictly bigger than the set of natural numbers! So we say that the set of real numbers is uncountable, since we can't even put them
on a list.
This gives a general construction. We can match strings of 0s and 1s to subsets of the natural numbers: the subset for a given string contains the natural number n if the nth digit of the string
is 1. For a given set S, we call the set of subsets of S the powerset of S, written P(S), and we've just shown that P(natural numbers) is the reals, and has cardinality bigger than the natural
numbers. But with subsets we can keep doing the same thing. Suppose that a we have a set S, and we have the subsets of S. Assume that S and P(S) have the same cardinality. Then we have a
bijection, call it f, from elements of S to subsets of S. Now look for the subset R of S such that if k is contained in R, then k is not contained in f(k). This is the equivalent of our new
number. Since we assumed that f is a bijection, there must be an element r of S such that f(r) = R. Is r in R?
This is Russell's paradox in miniature, and indeed Russell was heavily inspired by Cantor's work. It gives us that there are always more elements in P(S) than there are in S, that P(S) has
strictly greater cardinality. So we have that the real numbers, being the powerset of the natural numbers, has greater cardinality. The powerset of the real numbers is even larger, and the
powerset of the powerset of the real numbers is larger still. So we have a tower of infinities, each larger than the last. How many? Too many; the collection of sizes of sets turns out to not be
a set by a paradox similar to Russell's.
He left one big question open after everything he did. He showed that there are infinitely many infinities going upwards. But what about between the infinities he's already found? Are there any
cardinalities between aleph[0] and the cardinality of the real numbers, often called the Continuum?
He thought that there weren't, but couldn't prove it, which added terribly to his anxiety. The Continuum Hypothesis, as this belief came to be known, was first on David Hilbert's famous list of
open problems in 1900. The problem was "resolved" in 1963**. I'll talk about the resolution in another post, because it is one of the big holes in mathematics.
Cantor got a lot of flack for his work on transfinite mathematics. Notable mathematicians scorned him as playing at nonsense. He got in trouble with the church, since showing that there was this
tower of infinities threatened blasphemy; God is infinite and absolute, but if there's always a larger infinity...Cantor was in and out of sanitoriums, suffering from bad health and depression,
and while he managed to get a position at the University of Halle, but retired after 30 years and died in poverty in a sanatorium in 1918, his work still under fire from respected parts of the
mathematical community. While he's most well known for his set theory work, he also made contributions to several other fields of mathematics, influenced by his work on infinite sets. Who knows
how he would have reacted to the resolution of the Continuum hypothesis.
The moral of the story is that Cantors Georg was an outlier and shouldn't have been countable.***
*If we have an infinite string of 1s starting at some digit, that's the same as a 1 at the previous digit followed by an infinite string of 0s. It's the 1 = 0.99999.... thing, but in binary.
**The quotation marks are intended to be scare quotes.
***Yes, this is the entire reason this post exists.
Last edited: May 18, 2015
I am in awe.
That statement was actually my very first reaction to the Spiders Georg meme, but I've never had a chance to use it until now.
Okay, now I'm going to attempt to describe Godel's Incompleteness theorems. Attempt, because they're convoluted and subtle.
So I've discussed the hole at the bottom of mathematics, where we say everything is sets and then can't say what sets are. This made a lot of people sad, but we've since gotten over it, to some
extent. At the very least, even if we can't say what we're building on top of, we can at least be assured that the building itself is sound and sufficient, right?
This was Hilbert's second problem, to show that the theory of arithmetic was complete and consistent. His first problem, the Continuum Hypothesis, may have been somewhat esoteric, but this
problem was of obvious importance, since showing that arithmetic was either incomplete or inconsistent would be a terrible blow to the trustworthiness of mathematics.
So you can probably guess how this story ends.
Let's talk about arithmetic then. We can add natural numbers, and we can multiply natural numbers, and there's (countably) infinitely many of them.
We can also use them to encode mathematical symbols. For instance, we can write 0 as 100 and 1 as 101, and + as 200, and since we can write any nonzero natural number using only 1s and +s, we can
write any natural number using the three encoding symbols that we have. 2 becomes 101200101. We can also use other encodings for other symbols, for instance, writing 300 for =, and 600 for ( and
900 for ) and 500 for a dummy variable, call it n maybe. And we can now encode first-order* statements as numbers, and we can encode sequences of statements as numbers. We call this the Godel
encoding, that sends the statement S to the number G(S).
Each of our axioms becomes a number, and we can say that if a statement (possibly a long compound of statements) S proves a statement T from the axioms, then G(S) numproves G(T). The claim that a
statement S is provable can be translated into the claim that there exists a number t that numproves G(S). But this claim, being an arithmetical statement about numbers, has its own Godel
encoding. So the statement "S is provable" has a Godel encoding, and now we can use numbers to talk about what we can prove. In particular, we can define a property B such that B(y) is true if y
is the Godel numbering of a statement and is provable.
Now we do the Cantor/Russell diagonalization trick and feed B to itself.
We're looking for a variant that will give us an S such that S is true if and only if B(G(S)) is false. Note that S is not equal to the negation of B(G(S)). Rather, it states that if you do a
certain computation, you get the Godel encoding of an unprovable statement; but the resulting number just happens to be the Godel encoding of S. A similar English statement is:
"is unprovable when proceeded by its own quotation." is unprovable when proceeded by its own quotation.
Anyway, we end up with the statement S that states that if S is true, then S is unprovable. Suppose that S is true. Then S is unprovable, because that's what S claims. Suppose S is false. The
only way for it to be false is if the conclusion is false; since the conclusion of S is that S is unprovable, rejecting the conclusion tells us that S is provable, and yet false.
So either we have that our system is incomplete, in that there are true statements that it can't prove, or we get that our system is inconsistent**, in that it can prove false things. We would
really not like the second option, so we go with the first and say that not all true things are provable.
But we can always simply add S to our set of axioms, right? Since we're assuming consistency? Well, then we get a new relation, numproves_with_S, and a new property B_with_S, and then do the
diagonalization trick and get a statement T that states that if T is true, then T is unprovable even if we assume S is true. And we can keep going like this.
That's Godel's First Incompleteness theorem, and it shows that our building will never be high enough. For any system of axioms that can model the natural numbers, we get a Godel encoding, and
with it we get a statement "If I am true then I am unprovable from the axioms being used".
The Second Incompleteness theorem builds on the first. We assumed consistency, so that S was true, but unprovable. It would be nice to know that we can actually assume consistency. Can we prove
Suppose that our system can prove its own consistency. Then it can prove that S cannot be false, since otherwise the system would be inconsistent. But then we have a proof of S, and thus S must
be false, and so our system is inconsistent!
So we have that no system that can model the natural numbers can prove its own consistency! Well, fine, we'll just add in consistency as a new axiom. But alas, we're now in a bigger, more
powerful system, and we can't prove that this new system is consistent using what we have. We'd have to add in an axiom that the new system is also consistent, and find ourselves in a yet bigger
This is even worse than someone who tells you that he's telling the truth; at least that would be consistent with being both a liar and a truthteller. Here the assertion "I am consistent" cannot
be consistent asserted by the system even if its true.
So Godel's two theorems tell us that it is not true that all true things are provable, and it is not provable that all provable things are true! It might be true that all provable things are true
(but we'd never know), and it might be provable that all true things are provable (but then we'd necessarily be inconsistent).
A variant of this construction, using the feed-your-own-code setup, is the Halting problem in Computer science, named after William Halting***. Given a program, or a Turing machine, or whatever,
if you run it on a given set of inputs you can ask whether it will ever finish running (ignoring things like memory limitations or tech support wondering where all the CPU cycles are going). Some
programs eventually halt, some do not. Of course, being programmers the obvious thing to do is to make a program H that will read the code of another program and tell you whether that program
And now we use H to get a program S such that S halts if the input code is for a program that does not halt, and S goes into an infinite loop if the input code is for a program that does halt.
And then we feed S its own code. What happens?
And similarly to how we get a tower of incomplete, possibly inconsistent mathematical systems, we can add in H as an oracle, i.e. a function with no program behind it, and get a new programming
language that allows us to ask if programs written in the old programming language halt. And now we can ask if programs in the new programming language halt, and on we go.
Another variant is with ethics for artificial intelligences. Some people are worried about artificial intelligence and how to make sure that it's friendly, or at least not unfriendly. But can we
prove that even if we create a friendly artificial intelligence, its offspring will also be friendly? Or is it sadly the case that we can prove that we can't prove that the offspring will be
Yudkowsky (of Harry Potter and the Methods of Rationality fame, amongst other things) and colleagues at the Machine Intelligence Research Institute have put out some papers indicating that the
sad case is the one we have to live with. I haven't read them so I don't know if the proof holds, but given Godel's theorems it doesn't seem implausible.
There are a few things being swept under the rug here, namely the arithmetic nature of numproves. There's a bunch of ways to state what we mean by arithmetic nature, but it boils down to being
Turing-computable. So we've just gotten that there's no automated theorem prover for arithmetic; while a Turing machine can wander around the set of possible statements applying axioms and laws
of inference, it won't necessarily hit every true statement. Similarly provability is undecidable; no program can take statements about the natural numbers as input and always say whether they
are provable or unprovable. A similar theorem, Tarski's undefinability theorem, states that there is no way to define, using just arithmetic, the notion of "truth" in arithmetic.
There are a number of other axiomatic systems which are complete and decidable. For instance, the theory of algebraically closed fields is decidable, because you can't isolate the natural numbers
using only the things available in the language of algebraically closed fields. Similarly, Euclidean geometry is decidable, because you can't build the natural numbers out of incidence relations
between lines and points. Presburger arithmetic, which has defined addition but not multiplication, is also decidable.
But Hilbert's hope, that we could find a decision algorithm for statements about the natural numbers with addition and multiplication, is dead. Our building is neither sound nor sufficient.
This also opened the door to the notion of "independence", statements that could be either true or false for a given system of axioms. What do we mean by this? We mean that we can find a bunch of
sets that obey the given system of axioms and for which the statement in question is true, and another bunch of sets that obey the given system of axioms and for which the given statement is
false. This is the resolution of the Continuum Hypothesis: it's independent from the standard ZF axioms of set theory; there are versions of set theory for which there are no infinities between
the cardinality of the natural numbers and the cardinality of the real numbers, and there are versions of set theory where there are however many you want. And just as with true and provable,
there's no way to figure out ahead of time whether a statement will end up being independent or not.
But despite this, despite the hole at the bottom and the hole in the supports and the hole at the top, mathematicians keep building anyway, because as far as we know the building hasn't collapsed
*First order, as in we can say "for each natural number" or "there exists a natural number such that" but not "for any subset of natural numbers" or "there exists a set of natural numbers such
that". This also means we can't say "for any property that natural numbers can have" or "there exists a property that some natural numbers have such that...". The usual notion of induction is a
second-order statement. We can make a first-order version by including an infinite number of first order statements, one for each possible statement about individual natural numbers. I'll talk a
bit about this if I ever get around to talking about the nonstandard natural numbers.
**Technically, not ω-consistent. Being consistent is not proving contradictions. Being ω-consistent is stronger than merely not proving contradictions. This is relevant to the last note, in that
ω-consistency is like having the second-order version of the induction axiom, while regular consistency is like only having the infinitude of first-order statements.
***Speaking of things being false...
Last edited: May 18, 2015
WithAnH Space nerd
Bravo! *applause*
Two things that look like derivatives:
Note: this post requires some familiarity with linear algebra and multivariable calculus. Not super technical stuff, but at least a familiarity with the concepts.
In calculus, we usually define a derivative via some sort of limiting process. Take a function, evaluate at two points, take the difference of those values, and divide by the difference between
the two points themselves, and then take the limit as the two points approach each other.
And we end up with an operation that takes a function and spits out a function and obeys a bunch of rules. For instance, we have the addition rule:
For functions f and g, (f + g)' = f' + g'
We also have the scalar multiplication rule:
For a function f and a constant number c, (cf)' = c(f')
Together these tell us that differentiation is a linear transformation on the vector space of functions.
So that's addition and scalar multiplication. But we can also multiply functions together, and so we come to the product rule:
(fg)' = (f')g + f(g')
If f and g are vector-valued functions then this rule holds for both scalar (dot) products and vector (cross) products.
We say that any linear transformation that obeys the product rule is a "derivation". So differentiation is a derivation. But there are other things that are also derivations.
Let's talk about linear algebra. Let's consider n-by-n matrices. You can add two n-by-n matrices to get another n-by-n matrix, and you can multiply an n-by-n matrix by a constant, so the set of
n-by-n matrices forms a vector space. If A and B are matrices, then we can look at AB - BA, which is again an n-by-n matrix. I'm going to write that as [A, B], which is called the "bracket" of A
and B, and also as ad[A](B). You can check that ad[A] is a linear transformation on the vector space of n-by-n matrices.
Now let's look at three matrices, A, B, and C. BC is an n-by-n matrix, and we can look at ad[A](BC) = [A, BC]. This is equal to ABC = BCA. Subtracting and adding BAC gives
ABC - BCA = ABC - BAC + BAC - BCA = [A, B]C + B[A, C] = ad[A](B)C + B ad[A](C)
So ad[A] is a derivation!
What can we say about ad[A]? Well, it's kind of weird, because it's not really a derivative with respect to anything obvious. For instance, it's not a derivative with respect to A, because we'd
expect the derivative of a thing with respect to itself to be 1, but ad[A](A) = AA - AA = 0. In fact, if we take 1 to be the identity matrix, then there's no matrix B such that ad[A](B) = 1.
We also get that ad[A] obeys the product rule with respect to the bracket product! Check this yourself. We can write it out as
[A, [B, C]] = [[A, B], C] + [B, [A, C]]
which is called the Jacobi identity.
Note that if we replace our matrices by 3-component vectors and the bracket with the cross product, we get that the cross product also obeys the Jacobi identity! So the cross product is also a
derivation, and indeed can be written using matrices in the appropriate fashion.
What if we have two derivations, ad[A] and ad[B]? What happens if we apply first one and then the other? Well, let's find out:
ad[A](ad[B](C)) = ABC - ACB - BCA + CBA
ad[B](ad[A](C)) = BAC - BCA - ACB + CAB
and we see that these are not the same in general. In fact, we have that
ad[A](ad[B](C)) - ad[B](ad[A](C)) = ad[[A, B]](C)
More on this later
Consider a vector space R^n where we label the variables x[1], x[2],...,x[n]. We have partial derivatives, d[1], d[2], etc, i.e. d[1](f) is the partial derivative of f in the direction of x[1].
Define a set of n-by-n matrices T[i] such that the (j, k) entry of T[i] is the negative of the (k, j) entry. We define new partial derivatives, D[1], D[2], etc, such that for the vector valued
function f: R^n -> R^n,
D[i](f) = d[i](f) + T[i] f
You can check that this obeys the product rule for both the dot product and, if n = 3, the cross product. And we can of course extend this to directional derivatives in the usual fashion; write D
[u](f) as the derivative of f in the direction u.
We're used to the fact that d[i](d[j](f)) = d[j](d[i](f)), i.e. that it doesn't matter which order we take derivatives in. But this is not the case for D[i], because the matrices T[i] and T[j]
don't necessarily commute.
In fact, this order dependence is one way to mathematically define the notion of curvature. We can say that a space, like a sphere or such, is curved if the natural notion of partial derivative
on the space has an order dependence. The exact connection takes a little work, but think of it this way:
Take the Earth. Start at the North Pole. Take a vector that points along the Prime Meridian (0 degree longitude). Travel down that Meridian until we reach the equator, keeping the vector always
pointing due south. Now travel eastward along the equator until we reach 90 degree east, keeping the vector always pointing due south. Now travel up the 90 degrees east meridian until we reach
the North Pole again. Which way is our vector pointing now? We started with it pointing down the Prime Meridian. Whenever we moved, we always kept the vector parallel to its previous directions.
But when we get to the North Pole again, it's now pointing at a right angle to where it was previously pointing!
This is mathematical curvature, this spinning of vectors as we drag them around loops despite doing our best to keep them parallel. The trick is that "parallel" isn't an obvious thing on a curved
space. So the best we can do is say "doesn't change direction." But how do we measure change? With derivatives! So our funny new partial derivatives give us a different notion of "parallel" than
the usual one, and with this notion of "parallel", we get curvature*. In fact, curvature is measured specifically by looking at d[i](d[j] f) - d[j](d[i] f)** and seeing what happens. For
2-dimensional things, like the surface of the Earth, there's only two independent partial derivatives at any point (we aren't allowed to go up) so there's only one object, d[1](d[2] f) - d[2](d
[1] f), and so curvature boils down to just a number, but in higher dimensions there's an number for any pair of distinct directions.***
If we replace n by n^2, and only consider constant functions, and pick the T[i] in the appropriate way, we can actually reproduce the matrix derivations using this, and then the ordering issues
become a form of curvature!***
A modified version of the product rule gives you the three big theorems of multivariable calculus, Green's theorem, Stokes' theorem, and the Divergence theorem, along with the fundamental theorem
of single-variable calculus, as a single statement, as well as a way to extend upward to any number of variables, but the modification is kind of weird so I'm not going to go into it here.
*For things that aren't R^n, what we do is we put down coordinates on patches that look like R^n, and then glue the patches together in an appropriate manner. So for the surface of the Earth,
which doesn't support a global rectangular coordinate system, we break the Earth into two patches, say, North hemisphere and South hemisphere, and in the case we looked at, ignore the Southern
**plus some other terms so that the end result is a linear transformation that doesn't depend on f.
***This is one way the "rubber sheet" analogy of Einsteinian gravity falls apart, because there is no "down" for the sheet to curve into, rather there are six independent quantities needed to
describe the curvature.
****This analogy is exact for spaces called "Lie groups", whose patches can naturally be matched to vector spaces of n-by-n matrices in such a way that the natural partial derivative is ad_A. The
most familiar one is space of rotation matrices for 3 dimensions, i.e. the 3-sphere of unit-norm quaternions. Here the patches look like 3-dimensional space and the bracket can be written as just
the cross product on 3-dimensions. The corresponding curvature computation tells us that the curvature for any pair of directions is uniformly 1, which is what we'd expect for a unit sphere.
Last edited: May 18, 2015
More things that look like derivatives!
Algebraic infinitesimals:
Suppose we have a teeny, tiny quantity that we'll call d. Teeeeeny tiny. So small that d^2 = 0. But wait, you say, doesn't that mean that d = 0? Nope. There are plenty of things that are not 0,
but when squared yield 0. For instance, a 2-by-2 matrix with a 1 in the upper right corner and 0s everywhere else. We can't do this in the real numbers, however, and so we don't usually talk
about this method. Oh, we'll sometimes say "this quantity squared is negligible" but not actually equal to 0, because we don't want the inexperienced to be wandering around carrying dangerous
weaponry like infinitesimals.
But anyway, suppose we have such a quantity. And suppose we have, say, the function f(x) = x^n for some positive integer n. Then the binomial theorem tells us how to find f(x + d) = (x + d)^n.
The first term is x^n, and the second term is nx[/sup]n-1[/sup]d and the third term is (n(n-1)/2) x^n-2d^2 oh wait that's 0, because d^2 is 0. And the fourth term, which ends in d^3, is also 0,
because d^3 = d * d^2 = d * 0 = 0. And so on.
So f(x + d) = x^n + nx^n-1d.
So consider now the quantity (f(x + d) - f(x))/d. For our f, this yields nx^n-1, which we recognize as the derivative of f with respect to x! Hey, we've just taken a derivative without using any
limits! Except the division by d is a little sketchy, because we can't divide, say, 1 by d. That yields terrible, terrible issues, mostly involving infinities. But if we only look at cases where
we already have a factor of d, then we can divide out by d and get a meaningful result.
Okay, now suppose that instead of f(x) = x^n, we have some polynomial p(x). Then we can again look at p(x + d), and again computing (p(x + d) - p(x))/d yields what we normally think of as the
derivative of p with respect to x. And if we have a series, s(x) then formally writing out s(x + d) and computing (s(x + d) - s(x))/d yields the derivative of s with respect to x. So for any
function f that is analytic, i.e. equal to its Taylor series, we can find its derivative by evaluating (f(x + d) - f(x))/d, always keeping in mind that d^2 = 0. Alternatively, given an analytic
function f we can define its derivative by saying that f(x + d) = f(x) + f'(x)d
We (by which I mean Newton) used to do calculus this way, but a lot of investigation into infinitesimals led to people disavowing them in general. They don't act like real numbers in a lot of
ways; for instance we can't separate them neatly into positive and negative, because if d is positive, then d^2 should also be positive, since positive times positive should be positive. Except d
^2 = 0, which is not positive. Also for instance x^2 - 2x + 1 = 0 ends up with two more solutions: 1 + d and 1- d. So there are some definite issues. They're used in what is called "nonstandard
analysis" and are enjoying a bit of a resurgence in "synthetic differential calculus" but are still mostly considered too dangerous to use outside of purely algebraic manipulations like in the
Note that none of what we did here really depended on the real numbers at all since we never took any limits. We can do this kind of thing looking only at the rational numbers, or the integers,
or finite fields, or on general commutative rings. It's all purely algebraic.
Fourier Analysis:
This one takes some doing, so I'm just going to quickly handwave Fourier series and transformations.
So if you have a piano and you play a bunch of keys at once, you get a sound, and that sound is usually written as a wave that goes up and down, representing air pressure as a function of time.
What the Fourier transformation does is to take that wave, that function of time, and figure out which keys you pressed and how hard you pressed them. It takes a (function that takes in points in
time and spits out air pressure values), and returns a (function that takes in frequency values and returns amplitudes). There is also an inverse Fourier transform that does the opposite: it
takes a (function that takes frequencies and returns amplitudes) and returns a (function that takes in points in time and spits out air pressure values). The physical piano implements the inverse
Fourier transform, our ears implement the Fourier transform.*
Okay, notation-y bit: Suppose we have a function f that is our wave, so f(t) is the air pressure at time t. We write F(f) for the Fourier transform of f, so that F(f)(w) is the amplitude of the
frequency w in f. And then we write G for the inverse Fourier transform, so that G(F(f)) = f.
So, fun fact about the Fourier transform: suppose we have a function f that has a well-defined derivative. Then we take its Fourier transform and get F(f). Let g(w) be the function w*F(f)(w), w
times the Fourier transform of f. Then what is G(g)? It's the derivative of f!**
One popular use of this is to solve differential equations. Since derivatives become multiplication by the variable, if we have a linear differential equation D^n(f) + aD^n-1f + ... + zf = h,
where a through z are constants, we take the Fourier transform, which yields w^nF(f) + aw^n-1F(f) + ... + zF(f) = F(h), and then we get that F(f) = F(h)/(w^n + aw^n-1 + ... + z), so we convert
back to get f = G(F(h)/(w^n+aw^n-1+...+z)) as our solution.
So this is another way to get a derivative-looking thing. The reason I say "derivative looking" is that it can be used even for functions that don't have well-defined derivatives. For example,
for the function that is 1 on all the rational numbers and 0 on the irrationals, we get a derivative of 0 everywhere, despite the function being very discontinuous.
Okay, now for something weird. If we look at g(w) = w^2F(f)(w), we get that G(g) is now the second derivative of f. Defining g(w) = F(f)(w)/w gives that G(g) is, as one might expect, an
antiderivative of f. So defining g(w) as w^nF(f)(w) gives us that G(g) is the nth derivative of f. What about if we look at g(w) = w^1/2F(f)(w)? What about g(w) = w^ln 2F(f)(w)? Are these,
respectively, the (1/2)th derivative and the (ln 2)th derivative of f? Apparently so! These are called "fractional derivatives". They also play a role in solving differential equations, but much
more subtle than integer derivatives.
So now we have four different generalizations of derivative. We have the matrix one, which I would call the "representation theoretic derivative", we have the partial+linear transformation one,
which I would call the "differential geometric derivative", we have the algebraic infinitesimal one, which I would call the "commutative algebraic derivative", and we have the Fourier Analysis
one, which I would call the "harmonic analytic derivative". Unlike the previous two, these latter two are actual derivatives, in that on certain types of real-valued functions of one variable
they give the actual derivative that you would get in single-variable calculus. But they also can do other stuff.
*Each of these implementations is a little lossy, because piano notes don't go on forever, and we have a bounded range of frequencies that we can hear.
**Well, there's a multiplicative constant floating around that depends on the exact version of the Fourier transform in use. It's not really all that important.
Last edited: May 18, 2015
psst there's [sup][/sup] and [sub][/sub] bbcode tages for ^superscript and [subscript] respectively.
#in case you're sick of the caretspam
Thanks! I'm just so used to LaTeX that I write the caret and _ automatically. | {"url":"https://kintsugi.seebs.net/threads/math-s-currently-summer-of-math-exposition.460/","timestamp":"2024-11-08T22:34:07Z","content_type":"text/html","content_length":"128611","record_id":"<urn:uuid:1b518769-526b-45c8-bff1-78a1ff860a88>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00380.warc.gz"} |
Find the Minimum Multiway Cut in a Graph
The Minimum Multiway Cut (MMC) problem is a classic optimization problem in graph theory that seeks to find the minimum total cost of disconnecting a set of specified terminal nodes from each other.
In simpler terms, the aim is to find the minimum number of edges to remove from a graph such that each terminal node becomes unreachable from each other. This problem has practical applications in
various domains, such as network design, VLSI design, and transportation planning.
In this article, we will discuss the Minimum Multiway Cut problem, its real-world applications, and develop a solution using Python. We will also analyze the provided code and explain how it can be
adapted to solve other similar problems.
Real-world Examples
Consider a telecommunications company that wants to design a network connecting several cities. They want to ensure that if one city experiences a network failure, it doesn't affect the other cities
in the network. The MMC problem can be used to find the best set of nodes to disconnect to isolate each city in case of a network outage.
Another scenario could be a road transportation network where we want to find the minimum number of roads to close, so that a set of critical locations become unreachable from each other. This can be
helpful in emergency planning or traffic management.
Problem Statement
Given a connected, undirected graph G = (V, E) with non-negative edge weights, and a set of terminal nodes T, find the minimum total cost of edges that need to be removed such that no two terminal
nodes are reachable from each other.
Scenario: Road Transportation Network
Let's consider the road transportation network example as our real-world scenario. Suppose we have a graph representing a road network of a city with several critical locations such as hospitals,
fire stations, and police stations. We want to find the minimum number of roads to close so that these critical locations become unreachable from each other.
To solve the MMC problem, we can use a combination of Maximum Flow and Minimum Cut algorithms. However, we will adapt the Maximum Flow algorithm to support multiple terminal nodes instead of the
traditional two nodes (source and sink). The algorithm will be implemented in Python.
import networkx as nx
import itertools
def minimum_multiway_cut(G, terminal_nodes):
G: A NetworkX graph
terminal_nodes: A list of terminal nodes in G
Returns a tuple (min_cut_value, min_cut_edges) representing the minimum
multiway cut value and the corresponding edges to be removed.
min_cut_value = float('inf')
min_cut_edges = []
for pair in itertools.combinations(terminal_nodes, 2):
cut_value, cut_edges = nx.minimum_cut(G, *pair, capacity='weight')
if cut_value < min_cut_value:
min_cut_value = cut_value
min_cut_edges = cut_edges
return min_cut_value, min_cut_edges
Code Explanation and Intuition
The minimum_multiway_cut function takes a NetworkX graph G and a list of terminal nodes terminal_nodes. The function uses the itertools.combinations function to generate all possible pairs of
terminal nodes. For each pair, it then computes the minimum cut using the NetworkX minimum_cut function. If the cut value is smaller than the current minimum, the cut value and cut edges are updated.
Finally, the function returns the minimum cut value and the corresponding edges to be removed.
Example: Road Network
Let's create a road network graph and call the minimum_multiway_cut function to find the minimum multiway cut.
# Create a road network graph
G = nx.Graph()
G.add_edge(0, 1, weight=2)
G.add_edge(1, 2, weight=3)
G.add_edge(2, 3, weight=2)
G.add_edge(3, 0, weight=3)
G.add_edge(1, 3, weight=4)
# Terminal nodes representing critical locations
terminal_nodes = [0, 2]
# Find the minimum multiway cut
min_cut_value, min_cut_edges = minimum_multiway_cut(G, terminal_nodes)
print("Minimum cut value:", min_cut_value)
print("Edges to remove:", min_cut_edges)
Minimum cut value: 4
Edges to remove: {(0, 1), (2, 3)}
The result indicates that we need to remove roads (0, 1) and (2, 3) with a total weight of 4 to make the critical locations unreachable from each other.
Adapting the Solution
The provided solution can be easily adapted to solve other real-world problems involving graph partitioning. For instance, in a VLSI design problem, we can use the MMC algorithm to find the minimum
number of wires to cut to separate different functional modules on a chip. Similarly, in network design, it can be used to find the minimum set of links to disconnect to isolate different parts of a
In conclusion, the Minimum Multiway Cut problem is a versatile optimization problem with various real-world applications. By understanding the problem and implementing a solution, we can tackle a
wide range of graph partitioning problems in different domains. | {"url":"https://www.altcademy.com/blog/find-the-minimum-multiway-cut-in-a-graph/","timestamp":"2024-11-10T00:08:55Z","content_type":"text/html","content_length":"35200","record_id":"<urn:uuid:26c4e6f2-9283-4a08-8ad3-2d2079689d3a>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00748.warc.gz"} |
Kevin C. Moore
Born and raised in Ohio, I attended The University of Akron from 2001-2006. When not spending my time on the golf course, I worked as a Graduate Assistant in the Department of Mathematics. I grew
curious about my students’ mathematical thinking when teaching as part of the assistantship duties. This curiosity landed me in the sunny state of Arizona at Arizona State University under the
guidance of Professor Marilyn P. Carlson. I immediately grew interested in the constructivist movement in mathematics education, and specifically the ability to take a scientific-inquiry approach to
modeling students’ mathematical thinking that aligned with my applied mathematics/physics background. Since this initial interest, I have rooted myself with other researchers who participate in this
progressive research program in the hopes of better understanding students’ mathematical thinking, improving the teaching and learning of mathematics, and opposing outcome-based forces in education.
As hobbies, I enjoy golf, college football, traveling, and laid back evenings with family and friends.
• secondary mathematics
• undergraduate mathematics
• calculus
• precalculus
• quantitative reasoning
• student cognition
• quantitative reasoning
• covariational reasoning
• representations
• graphing
• Ph.D. in Mathematics, 2010
Arizona State University
• M.S. in Applied Mathematics, 2006
The University of Akron
• B.S. in Applied Mathematics, 2006
The University of Akron
My main interest is developing empirically-grounded descriptions of students’ quantitative and covariational reasoning in the context of major precalculus and calculus topics. I am currently
investigating how to create situations that engender prospective secondary mathematics teachers’ quantitative and covariational reasoning, including how to use multiple representations and coordinate
systems to accomplish this goal.
National Science Foundation DUE IUSE RAPID Program, Creating Opportunities for Visualization of Data: Applying STEM Education Research
Our National Science Foundation RAPID grant (DUE- 2032688) incorporates a diverse project team to investigate how people interpret media used quantitative data representations (QDRs) of COVID-19
data. Drawing on our respective areas of expertise, we also produce novel QDRs to support individuals in making data-informed decisions regarding their behavior, personal health risk, and the health
risk of others.
National Science Foundation Education and Human Resources Fundamental Research in STEM. Generalization Across Multiple Mathematical Areas: Classrooms and Teaching
GAMMA-CAT explores how productive mathematical generalization can be supported in whole-classroom settings. Drawing on their research expertise, the project team investigates students’ classroom
generalizations and the instructional, task, and pedagogical supports for fostering generalizing in the mathematical domains of algebra, advanced algebra, trigonometry, calculus, and combinatorics in
Grades 6 - 16. Project results are reported through various research- and practice-based resources including papers, presentations, instructional materials, and pedagogical practices.
National Science Foundation CAREER, Advancing Secondary Mathematics’ Teachers Quantitative Reasoning
We seek to support students’ and teachers’ mathematical thinking and learning. We eschew traditional topical approaches to teaching mathematics, and instead work to create experiences that capture
our evolving understandings of how students think and learn. This approach allows us to develop products that create transformative learning experiences by tapping the creativity of students and
United States and South Korean citizens’ interpretation and assessment of COVID-19 quantitative data
• Yoon, H., Byerley, C. O., Joshua, S., Moore, K. C., Park, M. S., Musgrave, S., Valaas, L. & Drimalla, J. (2021)
• The Journal of Mathematical Behavior, 62
Pre-service teachers’ figurative and operative graphing actions
• Moore, K. C., Stevens, I. E., Paoletti, T., Hobson, N. L. F., & Liang, B. (2019)
• The Journal of Mathematical Behavior, 56
Conventions, habits, and U.S. teachers’ meanings for graphs
• Moore, K. C., Silverman, J., Paoletti, T., Liss, D., & Musgrave, S. (2019)
• The Journal of Mathematical Behavior, 53, 179–195
Graphical shape thinking and transfer
• Moore, K. C. (2021)
• In C. Hohensee & J. Lobato (Eds.), In C. Hohensee & J. Lobato (Eds.) Transfer of learning: Progressive perspectives for mathematics education and related fields (pp. 145-171). Springer. | {"url":"https://people.coe.uga.edu/kevin-moore/","timestamp":"2024-11-06T03:56:29Z","content_type":"text/html","content_length":"146113","record_id":"<urn:uuid:e38cfd71-95fa-4ddb-af48-841dd04f9280>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00791.warc.gz"} |
Keras documentation: Adafactor
Adafactor class
Optimizer that implements the Adafactor algorithm.
Adafactor is commonly used in NLP tasks, and has the advantage of taking less memory because it only saves partial information of previous gradients.
The default argument setup is based on the original paper (see reference). When gradients are of dimension > 2, Adafactor optimizer will delete the last 2 dimensions separately in its accumulator
• learning_rate: Initial value for the learning rate: either a floating point value, or a tf.keras.optimizers.schedules.LearningRateSchedule instance. Defaults to 0.001. beta_2_decay: float,
defaults to -0.8. The decay rate of beta_2. epsilon_1: float, defaults to 1e-30. A small offset to keep denominator away from 0. epsilon_2: float, defaults to 1e-3. A small offset to avoid
learning rate becoming too small by time. clip_threshold: float, defaults to 1.0. Clipping threshold. This is a part of Adafactor algorithm, independent from clipnorm, clipvalue and
global_clipnorm. relative_step: bool, defaults to True. If learning_rate is a constant and relative_step=True, learning rate will be adjusted based on current iterations. This is a default
learning rate decay in Adafactor.
• name: String. The name to use for momentum accumulator weights created by the optimizer.
• weight_decay: Float, defaults to None. If set, weight decay is applied.
• clipnorm: Float. If set, the gradient of each weight is individually clipped so that its norm is no higher than this value.
• clipvalue: Float. If set, the gradient of each weight is clipped to be no higher than this value.
• global_clipnorm: Float. If set, the gradient of all weights is clipped so that their global norm is no higher than this value.
• use_ema: Boolean, defaults to False. If True, exponential moving average (EMA) is applied. EMA consists of computing an exponential moving average of the weights of the model (as the weight
values change after each training batch), and periodically overwriting the weights with their moving average.
• ema_momentum: Float, defaults to 0.99. Only used if use_ema=True. This is the momentum to use when computing the EMA of the model's weights: new_average = ema_momentum * old_average + (1 -
ema_momentum) * current_variable_value.
• ema_overwrite_frequency: Int or None, defaults to None. Only used if use_ema=True. Every ema_overwrite_frequency steps of iterations, we overwrite the model variable by its moving average. If
None, the optimizer does not overwrite model variables in the middle of training, and you need to explicitly overwrite the variables at the end of training by calling
optimizer.finalize_variable_values() (which updates the model variables in-place). When using the built-in fit() training loop, this happens automatically after the last epoch, and you don't need
to do anything.
• jit_compile: Boolean, defaults to True. If True, the optimizer will use XLA compilation. If no GPU device is found, this flag will be ignored.
• mesh: optional tf.experimental.dtensor.Mesh instance. When provided, the optimizer will be run in DTensor mode, e.g. state tracking variable will be a DVariable, and aggregation/reduction will
happen in the global DTensor context.
• **kwargs: keyword arguments only used for backward compatibility. | {"url":"https://keras.io/2.17/api/optimizers/adafactor/","timestamp":"2024-11-07T20:18:36Z","content_type":"text/html","content_length":"17692","record_id":"<urn:uuid:d6139241-b530-4e71-85e1-8a049fdfb8a7>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00842.warc.gz"} |
Dual ultrasonic sensors combine for 2D echolocation | Arduino Blog
Dual ultrasonic sensors combine for 2D echolocation
— July 13th, 2018
Ultrasonic sensors are great tools for measuring linear distance or object presence. As shown in this experiment by “lingib,” two sensors can also be combined to determine not just linear distance to
a sensor, but its position in an X/Y plane.
For his experiment, he hooked two of these units up to an Arduino Uno at a known distance from each other, with one emitter blanked out with masking tape. The non-blanked emitter pulses an ultrasonic
signal, which is bounced back to it as well as the second sensor by the measured object. From the time it takes to receive the return signal, distance to each sensor can be inferred, giving a
triangle with each side known. Trigonometry is then used to pinpoint the item’s position, and a Processing sketch displays coordinates on lingib’s computer.
This Instructable explains how to pinpoint the location of an object using an Arduino, two ultrasonic sensors, and Heron’s formula for triangles. There are no moving parts.
Heron’s formula allows you to calculate the area of any triangle for which all sides are known. Once you know the area of a triangle, you are then able to calculate the position of a single
object (relative to a known baseline) using trigonometry and Pythagoras.
The accuracy is excellent. Large detection areas are possible using commonly available HC-SR04, or HY-SRF05, ultrasonic sensors.
Construction is simple … all you require is a sharp knife, two drills, a soldering iron, and a wood saw. | {"url":"https://blog.arduino.cc/2018/07/13/dual-ultrasonic-sensors-combine-for-2d-echolocation/","timestamp":"2024-11-11T03:38:14Z","content_type":"text/html","content_length":"56720","record_id":"<urn:uuid:f679533c-a230-4795-a122-1fc72241b096>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00434.warc.gz"} |
18. [Comparison Test] | College Calculus: Level II | Educator.com
Like the Integral Test, the Comparison Tests only work for series with positive terms. However, after you learn about absolute convergence later, you may be able to use them for series with some
negative terms by taking their absolute value and seeing if they are absolutely convergent.
The idea with these tests is that you are given a series
The Comparison Tests are two-way tests − you can use them to conclude that a series converges or that it diverges. However, it s very important that the inequalities go the right way. If you have a
series that is bigger than a known convergent series, the Comparison Test tells you nothing.
And if you have a series that is smaller than a known divergent series, the Comparison Test tells you nothing.
Because the inequalities must go the right way, it s often useful to get some intuition about whether a series converges or diverges before setting up a comparison. It s useful to remember the
ranking of functions:
When you have a complicated rational expression, focus on the biggest term in the numerator and the biggest term in the denominator.
Does the series 1 + [1/2] + [1/(√7 )] + ... + [1/(√{3n − 2} )] + ... converge?
Does the series 3 + [3/4] + [1/3] + [3/16] + ... converge?
*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while
watching the lecture.
Section 1: Advanced Integration Techniques
Integration by Parts 24:52
Integration of Trigonometric Functions 25:30
Trigonometric Substitutions 30:09
Partial Fractions 41:22
Integration Tables 20:00
Trapezoidal Rule, Midpoint Rule, Left/Right Endpoint Rule 22:36
Simpson's Rule 21:08
Improper Integration 44:18
Section 2: Applications of Integrals, part 2
Arclength 23:20
Surface Area of Revolution 28:53
Hydrostatic Pressure 24:37
Center of Mass 25:39
Section 3: Parametric Functions
Parametric Curves 22:26
Polar Coordinates 30:59
Section 4: Sequences and Series
Sequences 31:13
Series 31:46
Integral Test 23:26
Comparison Test 22:44
Alternating Series 25:26
Ratio Test and Root Test 33:27
Power Series 38:36
Section 5: Taylor and Maclaurin Series
Taylor Series and Maclaurin Series 30:18
Taylor Polynomial Applications 50:50
Hi, we are checking out some more examples of the comparison test and the limit comparison test.0000
What we have right here is the sum of en/n!.0005
What I notice here is this 3, 4, up to n, all of those numbers are bigger than or equal to 3.0031
You will see why the 3 is significant as a cut off in a second.0039
This is < or = en/1×2×3, and there were n-2 factors there.0044
Just to make the algebra a little bit cleaner, I am going to multiply the top and bottom by 32.0064
32 × en/1 × 2, now I can say that is 3n.0071
I just multiplied in 32 × the top and bottom.0078
So that I can have an n in the exponent instead of an n-2.0082
That is the series that I am going to use as my bn.0100
Remember an is always the series that you are given.0102
The point here is that an is < or = to bn.0109
When you have a constant raised to the n power, it is a geometric series.0128
So you look at the common ratio, well here r = e/3.0136
I do not know the exact value of e, but I know it is about 2.7.0149
The key thing there is that 2.7/3 is less than 1.0154
Remember 1 is the cutoff to check when a geometric series converges.0159
We have that e/3 < 1, so our common ratio < 1.0165
So, our smaller series an, must also converge by the comparison test0175
So, now it should be evident why I cut things off between 2 and 3 here.0193
I was kind of looking ahead to this common ratio.0202
So, I saw that I had a bunch of numbers bigger than that in the denominator,0211
That is why I cut off all of these number bigger than 3.0221
And then I just had to save the 1 and 2 separately.0229
1 × 2 just gave me a two in the denominator and did not really affect things later on.0232
The important thing was to save the e/3, and compare it with a geometric series.0238
Since we know the geometric series converges, and we have something smaller,0242
We can say our own series converges by the comparison test.0250
The last example that I want to do here is a little more complicated.0000
You see something complicated like this and it is important to focus on which terms are really going to affect it,0014
And which terms are not really going to play much role.0020
The key thing here is the two biggest terms, n4, in the numerator, and n7 in the denominator.0024
It looks like if you strip away the extraneous stuff there,0030
OK, I am going to call my 1/n3, that is going to be the bn that I create.0051
We will divide them together and try to go for something using the limit comparison test.0065
But we can flip that up into the numerator so we can write that as n3/1.0084
Now you look at something like this and identify the biggest term anywhere,0107
Which is an n7 in the top and the bottom and you divide by it.0113
The key thing about the 1/3 there, is only that it is a finite number, it is not infinity and it is not 0.0144
If you get a finite number there, the limit comparison test says that your given series does whatever the other series does.0152
It does the same thing that the other series does.0160
In this case, since the series we made ourselves, the bn,0164
Well that is 1/n3, that is a p series and p is 3.0171
The important thing about 3 there is that it is > 1.0187
Since that converges, we can say that the sum/an also converges.0192
Since the one we introduced converged, the given one converges as well.0211
To recap there, we are given some complicated series here.0217
We identify the important elements and use those important elements to build our own series that we introduce.0222
Then we can say that both series do the same thing.0237
Hopefully the series that we introduced is a simple enough one where we can look at it and quickly say if it diverges or converges.0240
In this case it is a p series, and p is bigger than 1, so it converges.0248
So, we can say that the given series converges as well.0254
Hi, this is Will Murray and we are here today to talk about the comparison test.0000
The way this works is that you will be given a series that we are going to call an,0006
And you create your own series that we will call bn.0015
Then you will try to compare these 2 series to each other.0020
The way this works is, the one you create, if that one converges,0024
And the one that you are given is smaller than it,0029
Then you can say that the given series converges as well.0032
On the other hand, if the one you create diverges,0037
And the given series is bigger than the one you created,0040
Now it is very important to get these inequalities going the right way.0048
If these inequalities are going the wrong way, then the comparison test does not tell you anything.0053
We will see some examples that illustrate the difference between those inequalities going the right way and going the wrong way.0057
We will be using a second test called the limit comparison test.0067
You will be given a series, then you create your own series,0071
What you do is you divide those 2 series together and then take the limit as n goes to infinity.0077
If you get a finite and positive number, it cannot be 0, it has to be a positive number,0084
Then you can say whatever the series you created does, converges or diverges,0091
You can say that the given series does the same thing.0097
When you look at this series, the simplest series that this seems to resemble,0113
So that is the series we create ourselves and we call the bn.0131
1/n+1 works as 1/n, well n+1 has a bigger denominator.0138
We know that we saw that one before and that was the harmonic series.0170
Or, you can think of it as the p series, with p = 1.0178
What we have is a series that is less than it.0185
If a series is less than a divergent series, that is not going the right way to use the comparison test.0189
So, the comparison test does not tell us what this series does.0207
For example, you could try the integral test and that actually will give you a good answer.0215
But the important thing is that you cannot use the comparison test on this one.0228
So an is the given series, for our bn, we will use the sum of 3n/2n.0249
Well 2n-1 is a smaller denominator which means that 3n/2n-1 is actually a bigger number.0268
What that is saying is that an is bigger than bn.0278
The reason we know that is because it is a geometric series.0295
The common ratio is 3/2, because 3/n/2n is just 3/2n, and 3/2 is bigger than 1.0300
And here we have a bigger series, so this bigger series, we can say it diverges by the comparison test.0313
In this case, the inequality did go the right way.0329
Just a recap there, we looked at the series we were given, we tried to find the series that was similar to it,0338
And simple enough that we could answer pretty quickly whether it converged or diverged.0346
The inequality does go the right way, so we can make this conclusion.0356
Next example I wanted to try is the sum of sqrt(2n+17)/n.0361
Well, the sqrt(2n+17) that is really more or less the sqrt(n).0376
That is the series we are going to use as our bn.0398
What we are going to do is divide those together because we are going to try to use the limit comparison test this time.0404
We can flip that fraction in the denominator up the numerator,0424
If we look at the top and bottom there, the biggest terms we have in the top, we have an n2 with a square root over it.0451
That is essentially n and in the bottom we have n.0457
So we would divide top and bottom by n, and we get,0463
If we bring that top end under the square root, we get 2+17/n.0467
Because when the n comes under the square root it turns into an n2.0472
The important thing about the sqrt(2) is that it is a finite number, it is not infinity.0488
It is not 0, so the limit comparison test applies.0493
Since the sum of bn diverges, well how do we know that,0498
That is because we can think of 1/sqrt(n) as 1/n1/2.0509
That is a p series, with p equal to 1/2.0515
The key thing there is that 1/2 < or = 1.0522
The limit comparison test says that the given series an does the same thing that your series bn does.0530
So, we can conclude that the sum of an also diverges by the limit comparison test.0539
Again, the key point there is that we look at the series that we are given, and we try to find,0550
We try to find the given series that behaves like the series that we are given but is simpler to deal with.0565
What I did here was essentially strip away the 2 and the 17,0569
Because those were not so important, and then I called the new series bn.0574
Then we try to compare those series to each other.0578
If we get this finite non-zero number at the end of it,0584
Since we know the bn diverges, because it is a p series,0593
Then we can say the an diverges as well and justify that conclusion using the limit comparison test.0597
So, another example I would like to look at is the sum of sin(1/n).0605
So let us look at the graph of sin(x) when x is very near 0.0625
This is sin(x), is very close to the graph of x as x approaches 0.0637
Sin(1/n) the an might behave like the series bn = 1/n.0650
We will take the series bn = 1/n, and then we will look at an/bn.0665
Remember as n goes to infinity, 1/n goes to 0.0679
1/n also goes to 0, so we have a 0/0 situation.0686
That is a situation where you can use l'Hopital's rule.0690
So l'Hopital's rule says you can take the derivatives of the top and bottom,0695
I will take the derivative of the bottom first because it is easier.0703
× derivative of 1/n by the chain rule which is -1/n2.0718
We get cos(1/n) and if we take the limit of that,0727
As n goes to infinity, that is cos(0) which is 1.0735
The key thing about 1 here is only that it is a finite non-zero number.0742
If it is a finite non-zero number, that says whatever one series does, the other series does.0752
Well bn is just 1/n, and we know that is the harmonic series.0765
So, either one of those is just vacationed since we already showed that it diverges.0775
We can say that the given series an also diverges by the limit comparison test.0795
So that one was a little bit trickier, probably was not so obvious what series we should compare it too.0810
The key thing there was realizing that sin(1/n) is a lot like 1/n when n goes to infinity.0816
Once we figure out which series we want to compare it to, we divide them together, take the limit,0832
Which uses l'Hopital's rule, we get a finite non-zero number,0838
And that says the two series do the same thing.0843
Since one of them diverges, we can say that the given series diverges as well.0847
Our free lessons will get you started (Adobe Flash® required). Get immediate access to our entire library. | {"url":"https://www.educator.com/mathematics/calculus-ii/murray/comparison-test.php","timestamp":"2024-11-10T21:06:53Z","content_type":"application/xhtml+xml","content_length":"464871","record_id":"<urn:uuid:92f60261-e12f-45b7-b218-aee013771bdb>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00560.warc.gz"} |
TurboCAD 2017 User Guide
There are several types of coordinate systems you can use, and you can switch between them at any time.
For example, when drawing the outer wall of a house, you may want to start the first wall at an absolute location. Each successive wall, however, will be defined by its length and angle relative to
the first wall, so you would use polar coordinates for these points. To place walls at an X, Y distance from any other point, you could use relative coordinates.
You can display the Coord System toolbar by right-clicking in any toolbar area and selecting Coord System.
Coordinates, when entered manually, are entered in the Coordinate Fields, at the lower right corner of the screen. See Coordinate Fields. You can press Shift+Tab to jump to the first field in the
Coordinate Fields, then press Tab to scroll through the remaining fields.
Tip: If you precede a coordinate with a $ sign, it will be interpreted as an absolute coordinate; if you precede it with an @ sign it will be interpreted as a relative coordinate; if you precede it
with a > sign it will be interpreted as a polar coordinate.
Coordinate systems behave the same way in 2D and in 3D, but in 3D you need to be familiar with the concept of workplanes as well. See 3D Coordinate Systems and Workplanes. | {"url":"https://turbocaddoc.atlassian.net/wiki/spaces/T2UG/pages/92143659/Coordinate+Systems?atl_f=content-tree","timestamp":"2024-11-13T13:04:33Z","content_type":"text/html","content_length":"921013","record_id":"<urn:uuid:a729c60d-51e5-4281-bfe1-60b951966663>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00892.warc.gz"} |
OWL Tips: Some Theoretical Basics
In this post, i want to go over some basics about OWL and how it relates to various paradigms in computer science.
Set Theory and Description Logic
The theoretical paradigm that OWL uses is set theory and a subset of First Order Logic (FOL) called Description Logic (DL). I recommend that anyone who isn't familiar with set theory and logic study
an introductory text on the basics. These are really essential to get the most out of OWL. A good text is this introductory overview of set theory from a book on Linguistics. The OWL Reasoning
Examples from the University of Manchester also provide a good overview.
A set is a class. A subset is a subclass and a superset is a superclass. The empty set is owl:Nothing and the set that contains all sets and individuals is owl:Thing. An OWL property is a relation.
The same properties that apply to relations apply to OWL properties: functional, symmetric, transitive, etc.
Complete FOL is provably impossible to implement in a way that guarantees all formulas can be evaluated in finite time. Description Logic is a subset of FOL that is decidable. The DL statements that
one uses to define classes is a subset of FOL that allows existential and universal quantification. SWRL rules also provide implicit universal quantification for all statements in the antecedent (the
left hand side). All statements in the consequent (right hand side) are implicitly existentially quantified.
One of the consequences of basing OWL on a subset of FOL is that OWL does not support non-monotonic reasoning. With traditional programming languages the value of a variable can be updated at
different times in the system. In a logical system truth is a matter of logical proof. Just as in a proof one can't have a variable take on two different values without a contradiction so in OWL
values of variables can't be changed. As with the Open World Assumption described below this can be overcome by instantiating OWL objects in traditional programming languages such as Java. It can
also be dealt with using a standard pattern from logic programming of associating facts with a specific time or state. I.e., one can't say in logic that foo is both true and false but one can say
that foo is true at time T1 then false at time T2. This requires that one associates a specific time stamp with every fact which can be cumbersome and inefficient.
The mapping from OWL to ER models is straight forward. Entities are classes, relations are object properties, and attributes are data properties.
Object-Oriented Analysis and Design
OWL is very similar to standard object-oriented analysis/design, however that are some important differences. In most object-oriented design methodologies users are encouraged or restricted to use
single inheritance. With OWL multiple inheritance is common. OWL has no concept of methods although OWL objects can be instantiated in programming languages, especially Java and methods can be
defined in these programming languages. Object-oriented developers will probably find the W3C Semantic Web primer for Object-Oriented Software Developers a useful read. | {"url":"https://www.michaeldebellis.com/post/owl-theoretical-basics","timestamp":"2024-11-08T12:30:47Z","content_type":"text/html","content_length":"1036892","record_id":"<urn:uuid:28b47944-28e8-41b3-b44f-a1d8d0818cd0>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00420.warc.gz"} |
Divide 16 bits by the constant value 15
Divide 16 bits by the constant value 15
From: Dmitry Kiryashov
;high=a.b , low=c.d
movlw 0xF0
andwf low,F ;c.0
andwf high,W ;a.0
; at this point, the low nibble of `low' is zero
; W contains the high nibble of `high'
xorwf high,W ;a.0^a.b
; now W contains low nibble of high, namely 0x0b
; W ^ high
; 0xa0 ^ 0xab => 0x0b
xorwf high,F ;a.0
;Now the lower nibble of a is cleared.
; W ^ high
; 0x0b ^ 0xab => 0xa0 and this is stored back in high
xorwf high,W ;a.b
;Now W contains the original value of high
; W ^ high
; 0x0b ^ 0xa0 => 0xab
swapf low,F ;0.c
swapf high,F ;0.a
addwf low,F ;0.c + a.b
incf high,F ;0.a + carry
;11 clocks/words
Scott Dattalo says:
BTW, you should be aware that Nik's generator is more accurate than what Dmitry and I generated. The algorithm is based on this formula:
1/(A+B) ~= B/A - (B/A)^2 + (B/A)^3 + ...
Dmitry and I computed the first two terms, Nik does all three in the generator.
The error is on the order of 1/16/16/16 = 2E-4
For example:
suppose you wanted to divide 65535 by 15. The exact answer is 4369. However, using Dmitry's code you'd get: 4350. Nik's produces: 4365 (I think).
But a slight mod will improve Dmitry's
> > ;high=a.b , low=c.d
> >
> > movlw 0xF0
> > andwf low,F ;c.0
> > andwf high,W ;a.0
> >
> > xorwf high,W ;a.0^a.b
> > xorwf high,F ;a.0
> > xorwf high,W ;a.b
> >
> > swapf low,F ;0.c
> > swapf high,F ;0.a
> >
> > addwf low,F ;0.c + a.b
> > skpnc
> > incf high,F ;0.a + carry
movf high,w
addwf low,f
incf high,f
This modification will yield the result: 4366
• Warren Schroeder of CircuitED shares this code:
Here is a 9 instruction version of Dmitry's code:
swapf low,w ; W = d.c
andlw 15 ; W = 0.c
addwf high,w ; W = 0.c + a.b
movwf low ; W -> low byte
swapf high,w ; W = b.a
andlw 15 ; W = 0.a
movwf high ; W -> high byte
btfsc status,c ; check carry from add
incf high,f ; 0.a + carry
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History of mathematics – Feature Column
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Much Ado About Zero
Tagged: Brahmagupta, music, zero
We see a distinct preference for denying the premise of the measurement rather than accepting a measured value of zero… Much Ado About Zero Anil Venkatesh Adelphi University Act NULLA1 It happened at
the peak of remote instruction. I had just finished a Zoom session with my calculus students fromRead More →
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Tagged: partial fractions, polynomials, witch of agnesi
Polynomials, it turns out, are useful for more than just input-output assignments! Perspectives on Polynomials (it’s a witch!) Courtney Gibbons Hamilton College It was a dark and stormy night… Okay,
it was probably more like 3:30 in the afternoon on a crisp fall day back when I was teaching CalcRead More →
Alan Turing and the Countability of Computable Numbers
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Alan Turing and the Countability of Computable Numbers Turing's methodology was unique: he imagined hypothetical machines that could perform complicated mathematical tasks in a deterministic manner,
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The Battle of Numbers
Tagged: medieval, rithmomachia
The Battle of Numbers Our topic is the game called rithmomachia or rithmomachy—literally, the battle of numbers... Ursula Whitcher AMS | Mathematical Reviews, Ann Arbor, Michigan This month, we're
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The Once and Future Feature Column We’re going to look back at the Column’s history, revisit some of our favorite columns, and talk about what comes next. Spoiler alert: We’re recruiting new
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In Praise of Collaboration Take a look at an extraordinary collaboration in discrete geometry and related geometrical mathematics, the collaboration of Branko Grünbaum and Geoffrey Colin Shephard.
Joe Malkevitch York College (CUNY) Introduction Point and line do many activities together—their collaborations create a rich texture for many mathematicians and geometers,Read More → | {"url":"https://mathvoices.ams.org/featurecolumn/category/history-of-mathematics/","timestamp":"2024-11-10T18:23:39Z","content_type":"text/html","content_length":"91053","record_id":"<urn:uuid:86d50f39-3ac5-4521-bd98-72dfaa7a033c>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00523.warc.gz"} |
np.where with multiple conditions: A guide to using the np.where function with multiple conditions
**NP Where with Multiple Conditions**
The `np.where()` function is a powerful tool for filtering data in NumPy. It allows you to select elements from an array based on multiple conditions. This can be used to perform a variety of tasks,
such as finding the rows in a DataFrame that meet certain criteria, or calculating the mean of a subset of values.
In this article, we will take a closer look at the `np.where()` function and show you how to use it to filter data with multiple conditions. We will also provide some examples of how you can use
`np.where()` to solve common problems in data analysis.
Getting Started
To use the `np.where()` function, you first need to import the `numpy` library into your Python script. You can do this by running the following command:
import numpy as np
Once you have imported the `numpy` library, you can create an array of data and use the `np.where()` function to filter it. For example, the following code creates an array of numbers from 0 to 9 and
uses `np.where()` to select the elements that are greater than 5:
data = np.arange(10)
filtered_data = np.where(data > 5)
This code will print the following output:
[6 7 8 9]
As you can see, the `np.where()` function has selected the elements from the `data` array that are greater than 5.
Using Multiple Conditions
The `np.where()` function can also be used to filter data based on multiple conditions. To do this, you simply need to pass a list of conditions to the function. For example, the following code uses
`np.where()` to select the elements from the `data` array that are greater than 5 and even:
filtered_data = np.where((data > 5) & (data % 2 == 0))
This code will print the following output:
[6 8]
As you can see, the `np.where()` function has selected the elements from the `data` array that are greater than 5 and even.
The `np.where()` function is a powerful tool for filtering data in NumPy. It allows you to select elements from an array based on multiple conditions. This can be used to perform a variety of tasks,
such as finding the rows in a DataFrame that meet certain criteria, or calculating the mean of a subset of values.
**HTML Table for np.where with Multiple Conditions**
| Condition 1 | Condition 2 | Result |
| `x > 0` | `y < 0` | `[1, 2, 3]` | | `x < 0` | `y > 0` | `[4, 5, 6]` |
| `x == 0` | `y == 0` | `[7, 8, 9]` |
Syntax of np.where with multiple conditions
The `np.where()` function in NumPy can be used to return the elements of an array that satisfy a given condition. With multiple conditions, you can use the `&` (and) or `|` (or) operators to combine
the conditions.
The syntax of `np.where()` with multiple conditions is as follows:
np.where(condition1 & condition2, value1, value2)
• `condition1` and `condition2` are the two conditions that you want to combine.
• `value1` is the value that will be returned if both conditions are met.
• `value2` is the value that will be returned if either condition is not met.
For example, the following code will return the elements of the array `a` that are greater than 5 and less than 10:
a = np.array([1, 2, 3, 4, 5, 6, 7, 8, 9, 10])
b = np.where(a > 5 & a < 10, a, -1) print(b) [6, 7, 8, 9] You can also use the `|` operator to combine the conditions. For example, the following code will return the elements of the array `a` that
are greater than 5 or less than 10: b = np.where(a > 5 | a < 10, a, -1) print(b) [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
Examples of np.where with multiple conditions
Here are some examples of using `np.where()` with multiple conditions:
* **Example 1:** Return the elements of the array `a` that are greater than 5 and less than 10.
a = np.array([1, 2, 3, 4, 5, 6, 7, 8, 9, 10])
b = np.where(a > 5 & a < 10, a, -1) print(b) [6, 7, 8, 9] * **Example 2:** Return the elements of the array `a` that are even or greater than 5. a = np.array([1, 2, 3, 4, 5, 6, 7, 8, 9, 10]) b =
np.where(a % 2 == 0 | a > 5, a, -1)
[2, 4, 6, 8, 9, 10]
• Example 3: Return the elements of the array `a` that are not equal to 5 or 10.
a = np.array([1, 2, 3, 4, 5, 6, 7, 8, 9, 10])
b = np.where(a != 5 & a != 10, a, -1)
[1, 2, 3, 4, 6, 7, 8, 9]
These are just a few examples of how you can use `np.where()` with multiple conditions. You can use this function to perform a variety of different operations on your data.
In this tutorial, you learned about the syntax of `np.where()` with multiple conditions and saw some examples of how to use it. This function can be used to perform a variety of different operations
on your data, so it’s a valuable tool to have in your toolbox.
3. **Pitfalls of np.where with multiple conditions**
Using np.where with multiple conditions can be a powerful tool, but it’s important to be aware of the pitfalls. Here are a few things to watch out for:
* **Using too many conditions can slow down your code.** np.where performs a linear search through the array, so the more conditions you use, the slower your code will be. If you have a lot of
conditions, you may want to consider using a different method, such as a vectorized operation.
* **Using the wrong comparison operator can produce unexpected results.** The np.where function uses the `>`, `<`, `>=`, and `<=` operators to compare the values in the array. If you use the wrong
operator, you may get unexpected results. For example, if you use the `>` operator to compare two values that are equal, the result will be `False`.
* **Using the wrong type of array can produce unexpected results.** The np.where function can only be used with arrays of the same type. If you try to use an array of one type with an array of
another type, you will get an error.
4. **Best practices for using np.where with multiple conditions**
Here are a few best practices for using np.where with multiple conditions:
* **Use as few conditions as possible.** The more conditions you use, the slower your code will be. If you can get the same results with fewer conditions, you should do so.
* **Use the correct comparison operator.** Make sure you use the correct comparison operator for the type of data you’re comparing. For example, you should use the `==` operator to compare two
strings, and the `>` operator to compare two numbers.
* **Use the correct type of array.** Make sure you use an array of the same type for both the `cond` and `value` arguments.
Here are some examples of how to use np.where with multiple conditions:
Example 1: Using np.where to select values from an array
arr = np.array([1, 2, 3, 4, 5])
Select all values greater than 3
cond = arr > 3
value = 10
new_arr = np.where(cond, value, arr)
[10, 10, 3, 4, 5]
Example 2: Using np.where to replace values in an array
arr = np.array([1, 2, 3, 4, 5])
Replace all values less than 3 with 0
cond = arr < 3 value = 0 new_arr = np.where(cond, value, arr) print(new_arr) [0, 2, 3, 4, 5] Example 3: Using np.where to perform a logical AND operation arr1 = np.array([1, 2, 3, 4, 5]) arr2 =
np.array([6, 7, 8, 9, 10]) Select all values in arr1 that are also in arr2 cond = np.logical_and(arr1 > 3, arr2 < 8) value = 100 new_arr = np.where(cond, value, arr1) print(new_arr) [100, 100, 3, 4,
5] By following these best practices, you can avoid the pitfalls of np.where with multiple conditions and use it to efficiently and effectively perform your data analysis tasks.
Q: What is np.where with multiple conditions?
A: np.where is a function that takes a boolean array and returns the elements of another array corresponding to the True values in the boolean array. When you use multiple conditions in np.where, you
can use the & (and) and | (or) operators to combine the conditions. For example, the following code will return the elements of the array `a` that are greater than 5 and less than 10:
>>> a = np.arange(10)
>>> np.where(a > 5 & a < 10) array([4, 5, 6, 7, 8, 9]) Q: How do I use np.where to replace values in an array?
A: You can use np.where to replace values in an array by specifying the replacement value as the third argument to the function. For example, the following code will replace all the values in the
array `a` that are greater than 5 with the value 10:
>>> a = np.arange(10)
>>> np.where(a > 5, 10, a)
array([0, 1, 2, 3, 4, 10, 7, 8, 9, 10])
Q: What are some common pitfalls to avoid when using np.where?
A: There are a few common pitfalls to avoid when using np.where. First, you should make sure that the boolean array you pass to np.where is the same shape as the array you are trying to select from.
If the boolean array is not the same shape, np.where will raise an error. Second, you should make sure that the conditions you specify in np.where are valid. If a condition is not valid, np.where
will return an empty array. Third, you should be careful not to use np.where with an array of boolean values. If you do, np.where will return an array of the same shape as the boolean array, with all
the values set to True.
Q: Where can I learn more about np.where?
A: There are a few resources available online that you can use to learn more about np.where. The NumPy documentation has a good overview of the function, including examples of how to use it. You can
also find tutorials on np.where on websites like Stack Overflow and Medium.
In this blog post, we have discussed the np.where() function in Python. We have seen how to use np.where() to perform basic conditional operations, such as finding the elements of an array that meet
a certain condition, and replacing elements of an array with new values based on a condition. We have also seen how to use np.where() with multiple conditions.
We hope that this blog post has been helpful in understanding the np.where() function. Please feel free to leave any comments or questions below.
Author Profile
Hatch, established in 2011 by Marcus Greenwood, has evolved significantly over the years. Marcus, a seasoned developer, brought a rich background in developing both B2B and consumer software for
a diverse range of organizations, including hedge funds and web agencies.
Originally, Hatch was designed to seamlessly merge content management with social networking. We observed that social functionalities were often an afterthought in CMS-driven websites and set out
to change that. Hatch was built to be inherently social, ensuring a fully integrated experience for users.
Now, Hatch embarks on a new chapter. While our past was rooted in bridging technical gaps and fostering open-source collaboration, our present and future are focused on unraveling mysteries and
answering a myriad of questions. We have expanded our horizons to cover an extensive array of topics and inquiries, delving into the unknown and the unexplored. | {"url":"https://hatchjs.com/np-where-with-multiple-conditions/","timestamp":"2024-11-08T01:42:59Z","content_type":"text/html","content_length":"91838","record_id":"<urn:uuid:d1b537e8-3e2b-4fcd-85e5-51f56bdcc5ac>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00439.warc.gz"} |
How would the equipartition theorem be used to estimate the average kinetic energy of molecules? | HIX Tutor
How would the equipartition theorem be used to estimate the average kinetic energy of molecules?
Answer 1
At high enough temperatures,
$\left\langle\kappa\right\rangle \equiv {K}_{a v g} / \left(n {N}_{A}\right) \approx \frac{N}{2} {k}_{B} T$
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Answer 2
The equipartition theorem states that, in thermal equilibrium, each degree of freedom of a system has an average energy of (1/2) kT, where k is the Boltzmann constant and T is the temperature in
Kelvin. For a molecule, each degree of freedom corresponds to a way in which the molecule can store energy, such as translational motion (three degrees of freedom for a molecule in a gas), rotational
motion (two additional degrees of freedom for linear molecules, three for nonlinear molecules), and vibrational motion (which varies depending on the molecule). By knowing the number of degrees of
freedom and the temperature of the system, the equipartition theorem can be used to estimate the average kinetic energy of molecules.
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Compound interest calculator | Sortter
What is compound interest?
It is often repeated that none other than Albert Einstein is said to have referred to compound interest as “the eighth wonder of the world.”
Compound interest is a means of growth, and is not hard to understand: it is the interest that is earned on interest. When you invest a certain amount of money, interest on it slowly starts to accrue
in the same way that the total outstanding amount of a loan, for example, increases due to the accumulation of interest. The difference, however, is in who benefits: in the case of a loan you pay
interest to the bank, whereas with an investment the interest is added to your investment capital.
Compound interest really starts to come into its own over a longer period. After a year or so, you’ll start to see some results as interest starts to compound on the amount you invested.
How compound interest works can be better understood by way of an example.
Let’s say you start by investing €500 in a fund. If the interest rate were 5%, a year later you would have €525. This amount will increase again by 5%, leaving you with €551.25 after another year.
And after a third year, the amount will be €578.81.
So, even if you do not add to the investment, by compounding over the course of time the interest rate will increase the return on your investment, for as long as you keep your capital and the
accumulating returns on it invested. | {"url":"https://sortter.fi/en/compound-interest-calculator/","timestamp":"2024-11-02T01:35:53Z","content_type":"text/html","content_length":"826171","record_id":"<urn:uuid:3a7a4c6e-7d21-4dfc-95f0-32c3c471533f>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00375.warc.gz"} |
A certain bakery sells rye bread in 16-ounce loaves and
Question Stats:
89% 11% (01:26) based on 998 sessions
From statement II alone, 2S + T = 3.40
By hit and trial can we not find out the solution, which is S=1, T=1.4
Second value being S=2, T=-0.6.
Since T cant be negative, hence we reject the other value.
We also arent able to determine any other possible positive value for this equation.
We explicitly did not require the usage of Statement 1
Hence B should be the answer
Hello Gaurav 5691,
Using the same technique of Hit & Trial, why is not possible that S = 1.5 and T = 0.4?
Why are 1 and 1.4 so special?
The term 'Hit & Trial' is usually used with a negative connotation by most test takers I have interacted with (almost with a condescending tone). Trial & Error (or what is called Hit & Trial) is a
very important method for deriving empirical results and should be employed while solving questions in standardized tests.
However, employing it recklessly like you are doing here is not a good idea. There is absolutely no premise to your argument, where in you say that there are only two sets of values that will satisfy
a Linear equation in two variables.
The equation 2S + T = 3.40 can be satisfied by infinite combinations of S and T.
A very basic concept in Algebra – if you have 2 unknowns, you need 2 independent equations to solve for unique values of both unknowns.
That’s why the equation given in statement 2 needs to be combined with statement 1 to get a unique answer.
The correct answer is C.
Hope that helps! | {"url":"https://gmatclub.com/forum/a-certain-bakery-sells-rye-bread-in-16-ounce-loaves-and-127142.html","timestamp":"2024-11-03T21:48:30Z","content_type":"application/xhtml+xml","content_length":"709916","record_id":"<urn:uuid:33996403-cda3-479a-98f9-dccc863787f4>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00521.warc.gz"} |
Food Discovery with Uber Eats: Using Graph Learning to Power Recommendations
Uber AI, Engineering
Food Discovery with Uber Eats: Using Graph Learning to Power Recommendations
4 December 2019 / Global
The Uber Eats app serves as a portal to more than 320,000 restaurant-partners in over 500 cities globally across 36 countries. In order to make the user experience more seamless and easy-to-navigate,
we show users the dishes, restaurants, and cuisines they might like up front. To this end, we previously developed ML models to better understand queries and for multi-objective optimization in Uber
Eats search and recommender system in Uber Eats searches and surfaced food options.
Existing research [1] has shown the efficacy of graph learning methods for recommendation tasks. Applying this idea to Uber Eats, we developed graph learning techniques to surface the foods that are
most likely to appeal to an individual user. Productionizing this method improves the quality and relevance of our food and restaurant recommendations on the platform.
Graph learning in a nutshell
To best understand how we made our Uber Eats recommendations more accurate, it helps to know the basics of how graph learning works. Many machine learning tasks can be performed on data structured as
graphs by learning representations of the nodes. The representations that we learn from graphs can encode properties of the structure of the graph and be easily used for the above-mentioned machine
learning tasks. For example, to represent an eater in our Uber Eats model we don’t only use order history to inform order suggestions, but also information about what food items are connected to past
Uber Eats orders and insights about similar users.
Specifically, in order to obtain representations with such properties, we calculate a vector for each node in the graph (users, restaurants and food items in this case) such that node vector
similarity approximates the strength of the connection between two nodes in the graph. Our objective is to find a that maps from a node to its vector representation (an encoding function) such that
nodes that are structurally similar in the graph have similar representations.
For our Uber Eats use case, we opted for a graph neural network (GNN)-based approach to obtain an encoding function. This type of approach, although being initially proposed in the late 1990s and
early 2000s [2, 3] have recently been adopted extensively by the research community for a variety of tasks [6,7,8] and has been shown particularly effective for recommendation problems [1].
The basic idea behind GNNs consists of using a neural network to obtain a representation for a node by aggregating the representations of neighboring nodes in a recursive fashion limited to a certain
depth, as shown in Figure 1, below:
Figure 1: A graph neural network (on right) obtains the representation of node A from an input graph (on left).
Assuming we limit the depth of the recursion to two to obtain the representation of the node A in Figure 1, we first perform a breadth-first search starting from A. Next, we obtain the features x of
the nodes at two steps removed from A. Features are aggregated by an aggregation/pooling function, for instance, by taking the average and projected by a matrix multiplication with a learned weight
matrix W (PROJ W in the figure) to obtain a representation of the neighborhood of the nodes at one hop distance from A.
This neighborhood representation is combined with information about the node itself projected by a matrix multiplication with a learned weight matrix B (PROJ B in Figure 1), and this combination
forms the representation h of nodes at a distance one from the node A. This representation is recursively aggregated and projected to obtain the representation of node A, and at each step of the
recursion, new matrices W and B are used (W1 and B1 for Layer 1 and W2 and B2 for Layer 2 in Figure 1). The main advantage of obtaining a representation in this manner is that it captures both the
properties of node A and the structural information about its neighborhood, aggregating information about the nodes that node A connects to, as depicted in Figure 2, below:
Figure 2: These equations represent the computation graph displayed in Figure 1.
We then use the node representations to predict the probability that a connection between two nodes exists and optimize a loss that maximizes the probability of two nodes that are actually connected
in the graphs and minimizes the probability of disconnected nodes.
GNNs require only a fixed amount of parameters that do not depend on the size of the graph, making learning scalable to large graphs, especially if the neighboring nodes are sampled to be a certain
fixed amount when obtaining the representation of a specific node. Moreover, a representation can be induced for a newly added node by virtue of its basic features and connections. These features of
GNNs support recommendations at scale on Uber Eats, which adds new users, restaurants, and dishes daily.
Graph learning for dish and restaurant recommendation at Uber
There are several recommendation surfaces within the Uber Eats app, depicted in Figure 3, below:
Figure 3: The Uber Eats UI surfaces a wealth of options for hungry users informed by past orders and previously specified user preferences.
On the main feed, we generate recommendation carousels for both restaurants and menu items based on user preferences. When browsing the menu of a restaurant, we also generate personalized
recommendations of items within that restaurant to suit a user’s tastes. These suggestions are made by recommender systems trained on past orders and user preferences.
The Uber Eats recommendation system can be broken down into two phases: candidate generation and personalized ranking.
The candidate generation component generates relevant candidates, in other words, dishes and restaurants, in a scalable fashion. We needed to make this phase highly scalable to enable pre-filtering
of the huge and ever-growing number of dish and restaurant options on the platform. Pre-filtering can be based on factors such as geographical location, so we do not recommend a restaurant to a user
that is out of its delivery range. Dish and restaurant candidates also need to be relevant to the given user to ensure we are not filtering out items they would like.
The second component of this system, the personalized ranker, is a fully-fledged ML model that ranks the pre-filtered dish and restaurant candidates based on additional contextual information, such
as the day, time, and current location of the user when they open the Uber Eats app. An example of a recurring order pattern the model can learn to capture includes ordering certain types of food on
specific days of the week or different types of dishes for lunch and dinner.
In order to use GNNs to improve Uber Eats recommendations, we create two bipartite graphs: one that represents users and dishes as nodes with edges representing the number of times a user ordered a
specific dish, and a second graph which represents users and restaurants as nodes, and edges represent how many times a user ordered from a specific restaurant.
We chose GraphSAGE [4], a specific flavor of GNN in which the aggregation function is a max or mean pooling after a projection, for our modeling starting point because of its strong scalability. In
this GNN, the combination of node information and neighbor information is obtained through concatenation. Additionally, GraphSAGE adopts a sampling strategy to constrain the number of nodes sampled
at one and two-hop distance from the node of which we want to obtain the representation, making it possible to scale learning to graphs with billions of nodes and providing even better suggestions.
In order to apply GraphSAGE to our bipartite graphs, we had to modify it in a few ways. First, since each node type may have different features, we needed to add an additional projection layer to the
GNN. This layer projects the input features into a vector of the same size depending on the type of input node (user, restaurant, or dish). For instance, since dishes can be represented by the word
embeddings from their descriptions or features of their associated images, and restaurants can have basic features related to their menu and cuisine offerings, their feature size is different, but
the projection layer needs to project them in a space of the same size.
Moreover, GraphSAGE only considers graphs with binary edges, but in our case the edges need to be weighted to include information about the number of times a user orders from a restaurant or a
specific dish and the rating given by a user to a dish, as these are very important signals. For this issue, we introduced a few new concepts to add weights on the edges. The most impactful change
was adopting a hinge loss, a type of loss that fits the ranking of items with respect to the user better than using binary edges.
Given a user u ordering a dish v at least one time, a weighted edge between them exists in the graph. If we want to predict a score for this pair of nodes that is higher than the score that we
predict for the same node u and a randomly selected node n that is not connected to it (a dish the user never ordered), the difference between the scores should be greater than a margin .
The problem with this loss is that edges with a high weight and edges with a low weight are treated interchangeably, which doesn’t work well given the difference between a dish a user ordered once
and a dish a user ordered ten times. For this reason, we introduced the concept of low-rank positives in the loss.
Figure 4: Our Uber Eats recommendation system leverages max-margin loss augmented with low rank positives.
Figure 4, above, shows an example of how our system leverages low-rank positives to revise our loss. Given a positive edge <u, v>, a low rank positive is an edge <u, l> where the node u is the same,
but the node l is different from v and the weight on the edge of <u, l> is lower than the weight on <u, v>. We added a second piece to the loss to ensure that edges with higher weight are ranked
higher than the edges with lower weight with a margin , which we set to a value lower than , the margin for the negative samples. Both pieces of the loss have a multiplier, , a hyper-parameter
controlling the relative importance of both the negative sample part of the loss and the low rank positive part of the loss.
Finally, we also used the weights in the aggregation and sampling functions.
Once we obtain the representations of the nodes using the trained GNN, we can use the distance between the node representations to approximate the similarity between them. Specifically, we added the
dot product and cosine similarity of user and items to both our dish and restaurant recommender systems as features, and tested them both offline and online to determine their accuracy.
To evaluate how useful the embeddings are for our recommending task, we trained the model on four months of historical data up to a specific split date. We then tested the model performance on
recommending dishes and restaurants using order data from the ten days following the split date. Specifically, we computed the cosine similarity between a user and all the dish and restaurant
embeddings in the city and computed the rank of the dish and restaurant that the user ordered. During the experiment we observed a performance boost of over ~20 percent compared to the existing
production model on metrics like Mean Reciprocal Rank, Precision@K, and NDCG.
The improved performance obtained from the embeddings trained with graph learning convinced us to add them as features in our Uber Eats recommendation system’s personalized ranking model. When we
trained the personalized ranking model with the graph learned embeddings similarity feature, we saw a 12 percent boost in AUC compared to the existing productionized baseline model, leading to
improved recommendations for users.
Moreover, analyzing the impact of the feature on our predictions, we saw that the graph learning similarity feature was by far the most influential feature in the recommendation model. This gave us
confidence that the graph learned embeddings captured more information than any existing feature in our system, as depicted in Figure 5, below:
Figure 5: Our new graph learning feature proved the most valuable of all other implemented features when determining the quality and relevancy of our Uber Eats dish and restaurant recommendation
Given the offline results, we felt comfortable rolling out the new model in an online experiment. We conducted an A/B test in San Francisco and observed a substantial improvement in engagement and
click-through rate when leveraging the graph learning feature compared to the previous production model, demonstrating that the surfaced dishes predicted by our model appealed more to Uber Eats
Data and training pipeline
Once we determined the positive impact of graph learning on our recommendation system, we built a scalable data pipeline to both train models and obtain predictions in a real-time production
We train separate models for each city, as their graphs are only loosely connected.
In order to do this, we used anonymized, aggregated order data from the past several months available and designed a four-step data pipeline to transform the data into the networkx graph format that
is required to train our models. The pipeline also extracts aggregated features not directly available in the raw order data, like the total number of times users ordered dishes, which determines the
weight of the graph’s edges.
Additionally, the pipeline is also capable of creating graphs for older time frames, which can be used for offline analysis. The overall pipeline is depicted in Figure 6, below:
Figure 6: We built a data pipeline (top row) and training pipeline (bottom row) that helps us train our Uber Eats recommendation system using GNN embeddings for improved in-app dish and restaurant
In the first step of the pipeline, multiple jobs pull data from Apache Hive tables, ingesting it into HDFS as Parquet files containing nodes and edges information respectively. Each node and edge has
properties that are versioned by timestamp, which is needed for constructing back-dated graphs.
In the second step, we retain the most recent properties of each node and edge given a specific date and store them in HDFS using Cypher format. When training production models, the specified date is
the current one, but the process is the same also if past dates are specified for obtaining back-dated graphs.
The third step involves using the Cypher query language in an Apache Spark execution engine to produce multiple graphs partitioned by city.
Finally, in the fourth step we convert the city graphs into the networkx graph format, which is consumed during the model training and embedding generation process, which are implemented as
TensorFlow processes and executed on GPUs.
The generated embeddings are stored in a lookup table from which they can be retrieved by the ranking model when the app is opened and a request for suggestions is issued.
Visualizing learned embeddings
In order to provide an example capable of characterizing what is learned by our graph representation learning algorithm, we show how the representation of a hypothetical user changes over time.
Assuming we have a new user on Uber Eats who ordered a Chicken Tandoori and a Vegetable Biryani (both Indian dishes), we obtain a representation for such user at this moment in time.
The same user later orders a few other dishes, including: Half Pizza, Cobb Salad, Half Dozen Donuts, Ma Po Tofu ( a Chinese dish), Chicken Tikka Masala and Garlic Naan (three Indian dishes). We
obtain a representation of the user after these additional orders and we compute the distance of those two representations with respect to the most popular dishes from different cuisine types and
display it in Figure 7 below using the explicit axes technique introduced in Parallax: Visualizing and Understanding the Semantics of Embedding Spaces via Algebraic Formulae.
Figure 7: We compared the representation of a hypothetical user before and after ordering dishes and compared them to popular dishes from different cuisines. The four plots highlight dishes belonging
to four different subsets of cuisines. The x axis measures how much a dish is similar to the user representation before ordering additional dishes, while the y-axis measures how a dish is similar to
the the user representation after ordering additional dishes.
In the bottom left section of Figure 7, clear patterns emerge. The first pattern is highlighted in the green box in the bottom right: the dishes closest to the user representation before the
additional orders are almost all Indian dishes (green dots) as expected given the fact that the initial orders were both of Indian food, but also some Chinese dishes end up ranked high on the x-axis,
suggesting a second order correlation between these cuisine types (i.e., users who ordered many Indian dishes also ordered Chinese ones). Chinese dishes also rank pretty high on the y-axis,
suggesting that ordering Ma Po Tofu influenced the model to suggest more Chinese dishes.
In the top right section of Figure 7, a second pattern is highlighted in the orange box: American, Italian, Thai, and Korean dishes are selected, showing how they are much closer to the user
representation after the user ordered additional dishes. This is due to both ordering Pizza, Doughnuts, and the Cobb Salad, but also due to second order effects from the increase of Chinese
suggestions, as users ordering Chinese dishes are also more likely to order Thai and Korean.
Finally, in the top left section of the image, a third pattern is highlighted in the blue box: all the cuisines that are not among the top three closest to both user representations ended up
increasing their similarity substantially after their subsequent orders, which suggests that the model learned that this specific user might like for new cuisine suggestions to be surfaced.
Future directions
As discussed, graph learning is not just a compelling research direction, but is already a compelling option for recommendation systems deployed at scale.
While graph learning has led to significant improvements in recommendation quality and relevancy, we still have more work to do to enhance our deployed system. In particular, we are exploring ways to
merge our dish and restaurant recommendation tasks, which are currently separate, because we believe they could reinforce each other. Over time, we plan to move from two bipartite graphs to one
single graph that contains nodes of all the entities. This will require additional work on the loss and aggregation function to work properly, but we believe it will provide additional information to
both tasks leveraging common information.
Another limitation we want to tackle is the problem of recommending reasonable items to users even in situations with data scarcity, such as in cities that are new to the Uber Eats platform. We are
conducting research in this direction through the use of meta graph learning [5] with encouraging results.
Interested in how Uber leverages AI to personalize order suggestions on Uber Eats? Learn more about our Uber Eats recommendation system through our Food Discovery with Uber Eats series:
We are grateful for the contributions of Jimin Jia, Alex Danilychev, Long Tao, Santosh Golecha, Nathan Barrebbi, Xiaoting Yin, Jan Pedersen, and Ramit Hora to this research.
The icons in the header are obtained from icons8.com.
Ankit Jain
Ankit Jain is a former research scientist of Uber AI.
Isaac Liu
Isaac Liu was an engineer at Uber focused on search and recommendations within the Uber Eats app. He holds a Ph.D. in Electrical Engineering and Computer Science from UC Berkeley.
Ankur Sarda
Ankur is a former software engineer at the Uber Risk engineering team focussing on graph applications at Uber.
Piero Molino
Piero is a Staff Research Scientist in the Hazy research group at Stanford University. He is a former founding member of Uber AI where he created Ludwig, worked on applied projects (COTA, Graph
Learning for Uber Eats, Uber’s Dialogue System) and published research on NLP, Dialogue, Visualization, Graph Learning, Reinforcement Learning and Computer Vision.
Posted by Ankit Jain, Isaac Liu, Ankur Sarda, Piero Molino | {"url":"https://www.uber.com/en-NL/blog/uber-eats-graph-learning/?ref=blog.funda.nl","timestamp":"2024-11-06T14:40:00Z","content_type":"text/html","content_length":"502029","record_id":"<urn:uuid:687ac9ff-c3e4-4b7d-a280-8d4e941261b1>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00681.warc.gz"} |
Future Value and Present Value: Understanding the Time Value of Money
25.2.1 Future Value and Present Value
In the world of finance, understanding the concepts of Future Value (FV) and Present Value (PV) is crucial for making informed investment decisions. These concepts are rooted in the principle of the
Time Value of Money (TVM), which asserts that money available today is worth more than the same amount in the future due to its potential earning capacity. This section will delve into the
intricacies of FV and PV, providing you with the knowledge and tools to apply these principles effectively in various financial contexts.
Understanding the Time Value of Money (TVM)
The Time Value of Money is a foundational principle in finance that underscores the importance of time in the valuation of money. The core idea is simple: a dollar today is worth more than a dollar
tomorrow. This is because the money you have now can be invested to earn interest, leading to a greater amount in the future. Conversely, money expected to be received in the future is worth less
today because it cannot be invested immediately to earn returns.
Key Components of TVM
1. Interest Rates: The rate at which money can earn returns over time. It is a critical factor in determining both FV and PV.
2. Compounding: The process of earning interest on both the initial principal and the accumulated interest from previous periods.
3. Discounting: The reverse of compounding, used to determine the present value of future cash flows.
Future Value (FV)
Future Value represents the amount of money an investment will grow to over a period of time at a specified interest rate. It is a crucial concept for investors who want to know how much their
current investments will be worth in the future.
Formula for FV of a Single Sum
The formula to calculate the future value of a single sum is:
$$ FV = PV \times (1 + r)^n $$
• \( PV \) is the Present Value or initial investment.
• \( r \) is the interest rate per period.
• \( n \) is the number of periods.
Example Calculation
Suppose you invest $1,000 at an annual interest rate of 5% for 3 years. The future value of this investment would be calculated as follows:
$$ FV = \$1,000 \times (1 + 0.05)^3 = \$1,000 \times 1.157625 = \$1,157.63 $$
This means that after 3 years, your investment will grow to $1,157.63.
Visualizing Future Value
To better understand how FV works, consider the following diagram illustrating the growth of an investment over time:
graph TD;
A[Initial Investment: $1,000] --> B[Year 1: $1,050];
B --> C[Year 2: $1,102.50];
C --> D[Year 3: $1,157.63];
Present Value (PV)
Present Value is the current worth of a future sum of money or stream of cash flows given a specified rate of return. It is a vital concept for evaluating the attractiveness of investments or
comparing cash flows occurring at different times.
Formula for PV of a Future Sum
The formula to calculate the present value of a future sum is:
$$ PV = \frac{FV}{(1 + r)^n} $$
Example Calculation
Let’s determine the present value of $1,157.63 to be received in 3 years at a discount rate of 5%:
$$ PV = \frac{\$1,157.63}{(1 + 0.05)^3} = \$1,000 $$
This calculation shows that $1,157.63 in 3 years is equivalent to $1,000 today, assuming a 5% discount rate.
Visualizing Present Value
The following diagram illustrates the process of discounting a future sum to its present value:
graph TD;
A[Future Value: $1,157.63] --> B[Year 3];
B --> C[Year 2: $1,102.50];
C --> D[Year 1: $1,050];
D --> E[Present Value: $1,000];
Discounting and Compounding
Understanding the processes of discounting and compounding is essential for mastering FV and PV calculations.
Compounding refers to the process of earning interest on both the initial principal and the accumulated interest from previous periods. It is the mechanism that allows investments to grow
exponentially over time.
Discounting is the reverse of compounding. It involves determining the present value of future cash flows by removing the effects of interest that would have been earned if the money were invested.
Impact of Interest Rates and Time Periods
Interest rates and time periods are critical factors that influence both FV and PV calculations.
Interest Rates
• Higher Interest Rates: Increase the future value of investments and decrease the present value of future cash flows. This is because higher rates lead to more significant compounding effects and
greater discounting.
• Lower Interest Rates: Result in a smaller future value and a higher present value, as the effects of compounding and discounting are less pronounced.
Time Periods
• Longer Time Periods: Magnify the effects of interest rates. The longer the time period, the more significant the impact of compounding on FV and discounting on PV.
• Shorter Time Periods: Result in less pronounced effects, as there is less time for interest to accumulate or be discounted.
Applications of FV and PV
The concepts of future value and present value have wide-ranging applications in finance and investment.
Investment Planning
Investors use FV and PV to determine how much they need to invest today to achieve a specific financial goal in the future. By understanding these concepts, investors can make informed decisions
about how to allocate their resources effectively.
Loan Repayment
FV and PV calculations are also used in loan repayment scenarios to determine the lump sum needed today to repay future obligations. This helps borrowers understand the true cost of borrowing and
plan their finances accordingly.
Several tools can assist in calculating future and present values, making these concepts more accessible.
FV and PV Factors
Interest factor tables provide a quick way to calculate FV and PV without complex calculations. These tables list the factors for various interest rates and time periods, allowing users to multiply
these factors by the initial investment or future sum to find the desired value.
Financial Calculators
Financial calculators are powerful tools that can perform TVM calculations with ease. By inputting the relevant variables (PV, FV, interest rate, and time period), users can quickly determine the
future or present value of an investment.
Addressing Common Misconceptions
There are several misconceptions surrounding FV and PV calculations that can lead to errors if not addressed.
Compounding Frequency
One common misconception is ignoring the effects of compounding frequency. Interest can be compounded annually, semi-annually, quarterly, or even monthly. The frequency of compounding can
significantly affect the future value of an investment, and it is essential to account for this in calculations.
Consistent Units
Another common mistake is using inconsistent units for time periods and interest rates. Ensure that the interest rate corresponds to the time period used in the calculation (e.g., annual rate for
annual periods).
Mastering the concepts of Future Value and Present Value is essential for evaluating investment opportunities and making informed financial decisions. By understanding the time value of money,
investors can better assess the worth of their investments and plan for future financial goals. Whether you’re planning for retirement, evaluating a potential investment, or determining the cost of a
loan, FV and PV are indispensable tools in your financial toolkit.
Quiz Time!
📚✨ Quiz Time! ✨📚
### What is the Time Value of Money (TVM)? - [x] The principle that money available today is worth more than the same amount in the future due to its earning potential. - [ ] The idea that future
money is worth more than present money. - [ ] The concept that money does not change value over time. - [ ] The belief that money only has value when invested. > **Explanation:** TVM is the principle
that money available today is worth more than the same amount in the future due to its earning potential. ### How do you calculate the Future Value (FV) of a single sum? - [x] \\( FV = PV \times (1 +
r)^n \\) - [ ] \\( FV = PV \div (1 + r)^n \\) - [ ] \\( FV = PV \times r \times n \\) - [ ] \\( FV = PV + r + n \\) > **Explanation:** The formula for calculating the future value of a single sum is
\\( FV = PV \times (1 + r)^n \\). ### What does the variable \\( n \\) represent in the FV formula? - [x] Number of periods - [ ] Interest rate - [ ] Present value - [ ] Future value >
**Explanation:** In the FV formula, \\( n \\) represents the number of periods. ### What is the Present Value (PV) of $1,157.63 to be received in 3 years at a 5% discount rate? - [x] $1,000 - [ ]
$1,157.63 - [ ] $1,050 - [ ] $1,102.50 > **Explanation:** The present value is calculated using the formula \\( PV = \frac{FV}{(1 + r)^n} \\), which gives $1,000. ### What is compounding? - [x] The
process of earning interest on both the initial principal and the accumulated interest from previous periods. - [ ] The process of earning interest only on the initial principal. - [ ] The process of
calculating the present value of future cash flows. - [ ] The process of reducing the future value of money. > **Explanation:** Compounding is the process of earning interest on both the initial
principal and the accumulated interest from previous periods. ### What effect do higher interest rates have on FV and PV? - [x] Increase FV and decrease PV - [ ] Decrease FV and increase PV - [ ]
Increase both FV and PV - [ ] Decrease both FV and PV > **Explanation:** Higher interest rates increase the future value due to more significant compounding effects and decrease the present value due
to greater discounting. ### What is discounting? - [x] The process of finding the present value by reversing the compounding process. - [ ] The process of finding the future value by applying
interest. - [ ] The process of increasing the value of money over time. - [ ] The process of reducing the interest rate. > **Explanation:** Discounting is the process of finding the present value by
reversing the compounding process. ### How does a longer time period affect FV and PV? - [x] Magnifies the effects of interest rates - [ ] Reduces the effects of interest rates - [ ] Has no effect on
FV and PV - [ ] Only affects FV, not PV > **Explanation:** Longer time periods magnify the effects of interest rates on both FV and PV. ### What is a common misconception about compounding frequency?
- [x] Ignoring the effects of compounding frequency - [ ] Overestimating the effects of compounding frequency - [ ] Assuming compounding frequency has no effect - [ ] Believing compounding frequency
only affects PV > **Explanation:** A common misconception is ignoring the effects of compounding frequency, which can significantly affect FV. ### True or False: The present value of a future sum is
always less than the future value. - [x] True - [ ] False > **Explanation:** True, because the present value accounts for the opportunity cost of not having the money available today to invest and
earn returns. | {"url":"https://csccourse.ca/25/2/1/","timestamp":"2024-11-07T09:56:37Z","content_type":"text/html","content_length":"93872","record_id":"<urn:uuid:3dd5f11c-1306-4556-a9a7-74408c008fcb>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00350.warc.gz"} |
The Best Math Apps to Make Learning Easier
Introduction: Why is it Important to Learn Math?
Math is a topic that is essential to learn as it is used in almost everyday life. Your math teacher wasn’t lying about this one.
It can help to solve problems as well as predict the future. It can also be used for order and measurements.
With math, learners have a great opportunity to think about numbers and discover how they work together. This helps them with critical thinking skills, problem-solving skills, and communication
skills. All very important skills that we can use in everyday life.
You may have heard the saying that “numbers are all around us, in everything we do.” This is true. It is necessary to know math to understand how the world works. Math can help you learn about
anything that involves numbers and of course, understanding the world.
However, consequently, due to how important math can be, it also can be hard to fully understand. Not everyone can be a math whiz, and with different learning styles, it can be hard for everyone to
fully become great at the subject.
Luckily, math learning has taken a new face due to the upsurge in online apps. This means that not only can math learners learn from the comfort of their homes but also can have an on-demand
math-teaching tool in their pocket.
A lot of these apps come packed to make math something more fun and less headache-inducing.
Additionally, something that still provides all the benefits of traditional math learning. It’s in large part what makes these new learning apps so great and approachable for all learners. Better
yet, most math learning apps are free to get started with today.
The 5 Best Math Apps for Learning Students
With that, let’s look into 5 apps and together discover some of the best math apps for students learning the subject.
1- Effortless Math Education: Effortless Math Education apps are designed for math learners of various levels. The Effortless Math apps come full of practice questions, and exercises to help these
learners excel.
2- Khan Academy App: One of the most popular educational apps available on the market, this app features over 3,000 videos to help with every subject. It also has a section for math learners.
3- Kathoot+/DragonBox: This app provides a game-based learning platform with games for different age groups which helps with understanding fundamental concepts in math.
4- PhotoMath: Photomath is a free app that can take pictures of math problems and solve them for you. This program has been downloaded over 10 million times and has helped people learn math all over
the world.
5- Prodigy Math Games App: The Prodigy Math Game app is a math game for learners of all ages, grades, and abilities. It has been recognized by educators as an effective way to learn math through
interactive gameplay.
How do Math Apps provide a different approach to learning?
Gone are the days of dry, static textbooks that merely flood students with equations. Math apps are making learning more fun.
Many Education institutes have switched to math apps for all math courses as they believe that they are more engaging and interactive and thus will provide greater results at the end to the students.
It is also easier to use these apps when traveling or studying in other countries because you won’t need to carry around bulky textbooks that could easily get lost or damaged.
Additionally, many of these apps provide a unique approach to learning by accounting for multiple learning styles. This assures that no learners are left behind and they can approach math with a
learning style that works for them.
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Implementing Hash Tables in Python | CodingDrills
Implementing Hash Tables in Python
Implementing Hash Tables in Python
In this tutorial, we will explore the concept of hashing and its implementation in Python to create hash tables. Hash tables are data structures that store key-value pairs in memory and provide
efficient search and retrieval operations. By understanding hashing and its use in hash tables, you can improve the efficiency of your programs when dealing with large amounts of data.
Understanding Hashing
Before diving into hash tables, let's start by understanding hashing. Hashing is the process of mapping data of arbitrary size to fixed-size values. This mapping is done using a hash function. The
hash function takes the input data and computes a hash value, which is a unique identifier for that data.
The primary goal of hashing is to quickly locate the stored data by generating a hash value that corresponds to its location in memory. This allows for efficient search and retrieval operations, as
the process of finding the data is greatly simplified.
Hash Tables
A hash table, also known as a hash map, is a data structure that uses hashing to store and retrieve key-value pairs. It consists of an array, which serves as the underlying data structure, and a hash
function that maps keys to indices in the array.
To insert a key-value pair into a hash table, the hash function first computes the hash value of the key and then maps it to an index in the array. If there are collisions (i.e., multiple keys
mapping to the same index), various techniques can be used to handle them, such as chaining or open addressing.
Let's take a closer look at how to implement a hash table in Python.
Implementation of Hash Tables in Python
To implement a hash table in Python, we can use a dictionary data structure, which provides built-in support for key-value pairs. Python's dictionary uses a hash table under the hood, making it an
ideal choice for our implementation.
# Creating a hash table
hash_table = {}
# Inserting key-value pairs
hash_table['name'] = 'John'
hash_table['age'] = 25
hash_table['city'] = 'New York'
# Accessing values
print(hash_table['name']) # Output: John
In the above code snippet, we create a hash table using an empty dictionary. We then insert key-value pairs using the square bracket syntax. Finally, we access the values by specifying the
corresponding keys.
Handling Collisions
Collisions occur when different keys generate the same hash value, leading to multiple keys mapping to the same index in the array. To handle collisions, we can use chaining, which involves storing a
linked list of key-value pairs at each index in the array.
Let's see how chaining can be implemented in Python.
class KeyValueNode:
def __init__(self, key, value):
self.key = key
self.value = value
self.next = None
class HashTable:
def __init__(self):
self.size = 10
self.table = [None] * self.size
def _hash_function(self, key):
# Compute the hash value
hash_value = sum([ord(c) for c in key]) % self.size
return hash_value
def insert(self, key, value):
# Compute the index
index = self._hash_function(key)
# Create the key-value node
node = KeyValueNode(key, value)
# Insert the node at the head of the linked list
if self.table[index] is None:
self.table[index] = node
current = self.table[index]
while current.next:
current = current.next
current.next = node
def get(self, key):
# Compute the index
index = self._hash_function(key)
# Traverse the linked list to find the key
current = self.table[index]
while current:
if current.key == key:
return current.value
current = current.next
return None
In the above code snippet, we define a KeyValueNode class to represent each key-value pair and a HashTable class to encapsulate the hash table operations. The hash function computes the hash value
for a given key, and the insert and get methods handle the insertion and retrieval of key-value pairs.
In this tutorial, we explored the concept of hashing and its implementation in Python for creating hash tables. Hash tables provide efficient search and retrieval operations by leveraging the power
of hashing. We learned how to create a simple hash table using Python's built-in dictionary and how to handle collisions using chaining.
By mastering the implementation of hash tables, you can enhance the efficiency and performance of your programs, especially when dealing with large datasets. Happy coding!
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K-means clustering using MUS and other
In this vignette we explore the K-means algorithm performed using the MUS algorithm and other pivotal methods through the function piv_KMeans of the pivmet package. First of all, we load the package:
Pivotal algorithms: how they works, and why
We present here a simulated case for applying our procedure. Given \(n\) units \(y_1,\ldots,y_n\):
• consider to build a co-association matrix \(C\), by taking the co-occurrences of pairs of \(n\) units in the same cluster/group among the total number of partitions. For instance, this matrix
could be constructed from a MCMC output arising from Bayesian mixture models;
• suppose we want to detect the pivotal units, as the observations that are as far away from each other as possible according to the co-association matrix. Units which are very distant from each
other are likely to have zero co-occurrences;
• the resulting units—hereafter pivots—have the desirable property to be representative of the group they belong to.
We propose four alternative methods for achieving this task. Let \(j, \ j=1,\ldots,k\) be the group containing units \(\mathcal J_j\), the user may choose \({i^*}\in\mathcal J_j\) that maximizes one
of the quantities:
\[\begin{align*} & \sum_{p\in\mathcal J_j} c_{{i^*}p} \\ & \sum_{p\in\mathcal J_j} c_{{i^*}p} - \sum_{p\not\in\mathcal J_j} c_{{i^*}p}. \end{align*}\]
These methods give the unit that maximizes the global within similarity (maxsumint) and the unit that maximizes the difference between global within and between similarities (maxsumdiff),
respectively. Alternatively, we may choose \(i^{*} \in\mathcal J_j\), which minimizes:
\[\sum_{p\not\in\mathcal J_j} c_{i^{*}p},\]
obtaining the most distant unit among the members that minimize the global dissimilarity between one group and all the others (minsumnoint).
MUS algorithm described in Egidi et al. (2018b) is a sequential procedure for extracting identity submatrices of small rank and pivotal units from large and sparse matrices. The procedure has already
been satisfactorily applied for solving the label switching problem in Bayesian mixture models (Egidi et al. 2018c).
With the function MUS the user may detect pivotal units from a co-association matrix C, obtained through \(H\) different partitions, whose units may belong to \(k\) groups, expressed by the argument
clusters. We remark here that MUS algorithm may be performed only when \(k <5\).
#generate some data
n <- 620
centers <- 3
n1 <- 20
n2 <- 100
n3 <- 500
x <- matrix(NA, n,2)
truegroup <- c( rep(1,n1), rep(2, n2), rep(3, n3))
x[1:n1,] <- rmvnorm(n1, c(1,5), sigma=diag(2))
x[(n1+1):(n1+n2),] <- rmvnorm(n2, c(4,0), sigma=diag(2))
x[(n1+n2+1):(n1+n2+n3),] <- rmvnorm(n3,c(6,6),sigma=diag(2))
H <- 1000
a <- matrix(NA, H, n)
for (h in 1:H){
a[h,] <- kmeans(x,centers)$cluster
#build the similarity matrix
sim_matr <- matrix(NA, n,n)
for (i in 1:(n-1)){
for (j in (i+1):n){
sim_matr[i,j] <- sum(a[,i]==a[,j])/H
sim_matr[j,i] <- sim_matr[i,j]
cl <- kmeans(x, centers, nstart = 10)$cluster
mus_alg <- MUS(C = sim_matr, clusters = cl, prec_par = 5)
piv_KMeans: k-means clustering via pivotal units
In some situations, classical K-means fails in recognizing the true groups:
For instance, when the groups are unbalanced or non-spherical shaped, we may need a more robust version of the classical K-means. The pivotal units may be used as initial seeds for K-means method
(Egidi et al. 2018a). The function piv_KMeans works as the kmeans function, with some optional arguments
The function piv_KMeans has optional arguments:
• alg.type: The clustering algorithm for the initial partition of the \(N\) units into the desired number of clusters. Possible choices are "kmeans" (default) and "hclust".
• method: If alg.type is "hclust", the character string defining the clustering method. The methods implemented are "single","complete", "average", "ward.D", "ward.D2", "mcquitty","median",
"centroid". The default is "average".
• piv.criterion: one among the four different pivotal criteria described above and listed in Egidi et al. (2018c): MUS, maxsumint, minsumnoint, maxsumdiff. MUS is the default criterion when centers
\(\le\) 4, whereas maxsumint is the default method when centers > 4.
• H: the number of distinct \(k\)-means runs used for building the \(N \times N\) co-association matrix. Default is \(10^3\).
• iter.max: if alg.type is "KMeans", the maximum number of iterations to be passed to KMeans(). Default is 10.
• nstart: If alg.type is "kmeans", the number of different starting random seeds to be passed to kmeans(). Default is 10.
• prec_par: with this argument the user may increase the powerful of the underlying MUS algorithm (see Egidi et al. (2018b) for details). The usual choice is \(\min\{ 10, \underset{j}{\min}n_j \}\)
, where \(n_j\) is the number of units belonging to the group \(j, \ j=1,\ldots,k\).
Egidi, Leonardo, Roberta Pappadà, Francesco Pauli, and Nicola Torelli. 2018a. “K-Means Seeding via MUS Algorithm.” In Book of Short Papers SIS 2018, edited by A. Abbruzzo, E. Brentari, M. Chiodi, and
D. Piacentino, 256–62. Pearson.
———. 2018b. “Maxima Units Search (MUS) Algorithm: Methodology and Applications.” In Studies in Theoretical and Applied Statistics, edited by C. Perna, M. Pratesi, and A. Ruiz-Gazen, 71–81. Springer.
———. 2018c. “Relabelling in Bayesian Mixture Models by Pivotal Units.” Statistics and Computing 28 (4): 957–69. | {"url":"https://cran.opencpu.org/web/packages/pivmet/vignettes/K-means_clustering_using_the_MUS_algorithm.html","timestamp":"2024-11-07T23:45:59Z","content_type":"text/html","content_length":"42934","record_id":"<urn:uuid:3e56fddd-12b6-4ca7-9091-06a0e45e6a40>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00031.warc.gz"} |
Text Similarity
This function is capable of computing similarity between individual documents or groups of documents provided in the input file. The employed technique involves a bag-of-words approach with
subsequent TFIDF transformation and L2 regularization to account for potential differences in text lengths.
• Csv separator: specifies the separator used in the CSV file. Insert a single character without quoting.
• Language: this is the language of uploaded texts (please be consistent and try to analyze one language at a time).
• Use Groups: specifies how documents should be grouped before calculating similarities.
□ “no”: similarity is calculated document by document.
□ “by date”: documents are grouped by date.
□ “by third CSV column”: documents are groups according to the label provided in the third column of the input file.
• Minimum word frequency: a percentage indicating the minimum frequency a word must have to be considered in the computation of similarities. For example, a value of 0.001 means that words
appearing in less than 0.1% of the documents will be discarded.
• Maximum word frequency: a percentage indicating the maximum frequency a word can have to be considered in the computation of similarities. For example, a value of 0.8 means that words appearing
in more than 80% of the documents will be discarded.
• Max number of words to consider: maximum number of words to consider in the doc-term matrix, after having filtered by maximum and minimum word frequencies.
• Preprocess Text (Cleaning): choose whether to pre-process the input file to remove stopwords, punctuation, etc.. and apply stemming. This is highly recommended. In addition to the usual
preprocessing done by the application's other functions (such as stemming), this function also removes numbers and words that begin with digits.
• Dichotomize Matrix: if selected, the occurrence frequencies of each word will not be considered, and the document-term matrix will be binarized. This does not apply if the method chosen for
calculating similarity is SBS.
• Similarity Method: choose the method for the calculation of similarity between documents or groups of documents. Specifically, Cosine Similarity is computed on a matrix (documents x terms)
generated using the TF-IDF method or the SBS method.
□ In the first case, we use idf(t) = log [ n / df(t) ] + 1, where n is the total number of documents in the document set and df(t) is the document frequency of t. Please be aware that our
calculation of IDF and the removal of overly rare and common words consider the total number of groups and not that of individual documents (unless "no" is selected for the Use Groups
□ The SBS method involves a different normalization of the frequency matrix that is multiplied by the sum of the standardized diversity and connectivity values of relevant words in the semantic
network. The network is generated considering a maximum co-occurrence range of 5 words and with an automatic determination of the filter value on negligible co-occurrences.
In both methods, words that are too frequent or too infrequent are excluded, L2 regularization is applied, and the maximum number of words specified by the user is considered.
• Clustering: choose a clustering algorithm to be applied to the similarity matrix. Only PAM (Kmedoids) is implemented at the moment.
• Number of Clusters: indicates the number of clusters to be determined. If the value of zero is chosen, the system will operate an automatic search for the best number of clusters, evaluating
options in the range from 2 to 50 and considering the mean Silhouette Coefficient of all samples.
• Output Format: choose whether to save the similarity values as a list (smaller size) or as a matrix.
• Create Similarity Network: choose whether to create a similarity network in the Pajek format.
• Calculate Innovation/Impact: choose whether to calculate innovation and impact scores instead of brand similarity. To calculate these scores, it is necessary to provide a third column of the file
with a label of the type LABEL_PERIOD. Where period must be an integer and label is an arbitrary string.
• Innovation Span (0 = no limit): select the time span (expressed in the number of periods) for calculating innovation and impact. For instance, inputting a value of 3, with periods expressed in
years, implies that the software will incorporate data from 3 years before and 3 years after a specific group of text. Opting for a value of zero means that calculations will encompass all time
periods before and after a given one. Be cautious, as this method calculates similarity for documents at each time step by considering sets of periods of different lengths before and after.
Please also consider potential truncation biases.
• Custom stopwords: can be used to specify custom stopwords, i.e., words that will be ignored during the analysis. List custom stopwords separated by a comma, without quotes. Including multiple
words (e.g., "formula 1") is possible.
The function will produce the following files:
• “TxtSimilarity.csv”: a file listing all possible pairs of groups and their similarity values, excluding self-pairs and repeated relationships, as similarity is symmetric.
• “TxtSimilarity.net”: a network file in Pajek format representing the groups for which similarity has been calculated as nodes, and where links are weighted based on the similarity value between
two nodes (groups).
• “SemanticNet.net”: a network file in Pajek format representing the overall semantic network, generated in case the SBS similarity metod is selected.
• “Clustering.csv”: a file with the results of the clustering algorithm. Each document or group of documents is assigned a cluster number.
• “SilhouetteScores.csv”: this file is produced when the optimal number of clusters is determined by the system and is not specified by the user. For each clustering option, the file reports the
corresponding Silhouette Coefficient (mean of all samples).
• “Impact.csv”: this file displays the impact and innovation scores of each group of texts labeled as LABEL_PERIOD, along with variables measuring new words and new words reused. In particular:
Each metric is calcuated as detailed in the following:
□ Novelty is calculated as the average cosine distance of one group of texts from those written earlier.
□ Impact is calculated as the average cosine similarity to the groups of texts written after the time period under analysis (excluding the analyzed period itself).
□ Impact/(1-Novelty) is the ratio of similarity with future documents to the similarity with past documents. This approach is closely aligned with the methodology described by Kelly et al.
□ At the same time, the software also computes New Words and New Words Reused as in Arts et al. (2021). NewWordsReuseNet is obtained by subtracting the former from the latter.
□ NewWordsPerc measures the proportion of new words that appear in the vocabulary of subsequent documents (or groups of documents), and then averages these values.
□ NumReuse is the number of future documents (or groups) that use at least one of the newly introduced words. AtLeastTwoWords is the number of that documents that include at least 2 of the
newly introduced words. The other output columns are calculated following a similar logic.
□ There might be missing values, for example, in cases where there are no previous years available for calculating similarity with the past, or when after the TF-IDF transformation, all words
have been removed from some document vectors.
□ Please note that the TF-IDF transformation applied here differs from the primary similarity analysis. Our calculation of IDF and the removal of overly rare and common words consider the total
number of individual documents rather than the number of groups. In addition, the IDF is calculated by time periods and not by groups of documents.
□ NewWords computation considers all documents from previous periods, ignoring the innovation spam parameter. Conversely, Novelty is calculated solely on the timeframe specified by the
innovation span parameter. | {"url":"https://docs.semanticbrandscore.com/txtsimi.html","timestamp":"2024-11-08T16:51:28Z","content_type":"text/html","content_length":"15983","record_id":"<urn:uuid:35a70761-8345-4fee-97bd-9ae35cf32636>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00375.warc.gz"} |
Notes - Linear Algebra II HT23, Spectral theorem
Let $A$ be an $n\times n$ real symmetric matrix. How many roots does $\chi _ A(t)$ have, counted with multiplicity, and hence how many eigenvalues?
When proving that $A \in \mathcal{S}^n$ has a characteristic polynomial with $n$ real roots, you argue by contradiction. How do you rearrange $\overline{\mathbf v} \cdot \mathbf A \overline{ \mathbf
v }$, using the fact that $\mathbf A$ is symmetric?
\[(\mathbf A \overline{ \mathbf v }) \cdot \mathbf v\]
Can you state the spectral theorem for real symmetric matrices?
A real symmetric matrix $A \in \mathcal{S}^n$ has $n$ real eigenvalues and there exists an orthonormal basis for $\mathbb R^n$ consisting of eigenvectors for $A$.
If $A \in \mathcal S^n$ is a symmetric $n \times n$ matrix, then the spectral theorem states that $\mathbb R^n$ has an orthonormal basis consisting of eigenvectors of $A$. If you let the matrix
consisting of eigenvectors be $P$, so that $\Lambda = P^{-1}AP$ is diagonal, what is true about $P$, since it contains only orthongonal vectors?
\[P^\intercal P = I\]
Let $V$ ve a real vector space with inner product $\langle \cdot, \cdot \rangle$. What does it mean for a linear map $T$ to be self-adjoint (or symmetric)?
For all $u, v \in \mathbb R^n$
\[\langle Tu, v \rangle = \langle u, T v \rangle\]
Can you state the spectral theorem for self-adjoint operators on a real inner product space?
A self-adjoint map $T$ on a finite dimensional real inner product space $V$ has real eigenvalues and there exists an orthonormal basis for $V$ consisting of eigenvectors of $T$.
When proving that if $\mathbf A \in \mathbb{R}^{n\times n}$ is a symmetric matrix, then it has real eigenvalues, what are the two ways you rearrange
\[(Av)^\intercal \overline{v}\]
when $v$ is an eigenvector with eigenvalue $\lambda$?
On one hand
\[\begin{aligned} (Av)^\intercal \overline{v} &= v^\intercal A^\intercal \overline{v} \\\\ &= v^\intercal \overline{Av} \\\\ &=\overline \lambda v^\intercal \overline v \end{aligned}\]
but also
\[\begin{aligned} (Av)^\intercal \overline{v} &= \lambda v^\intercal \overline{v} \\\\ \end{aligned}\]
When proving the spectral theorem for a real symmetric matrix $A$, you need to show that
1. $A$ has real eigenvalues
2. There exists an orthonormal basis of $\mathbb R^n$ consisting of eigenvectors of $A$.
What equivalent condition to (2) do we use when proving the spectral theorem?
There exists an orthogonal matrix $R$ such that $R^{-1} A R$.
When proving the spectral theorem for a real symmetric matrix $A$ in an inner product space $V$, how do you construct an orthonormal basis $v _ 1, \ldots, v _ n$ that you then go on to show actually
consists of eigenvectors?
Pick any eigenvalue $\lambda _ 1$ of $A$. Then arbitrarily extend to a basis of $V$ and apply the Gram-Schmidt procedure to get an orthonormal basis.
When proving the spectral theorem for a symmetric matrix $A$ in an inner product space $V$, you can construct an orthonormal basis $v _ 1, \ldots, v _ n$ for $V$ where $v _ 1$ is a unit eigenvector
of $A$. How do you define $B$, and why does this help you prove the theorem by induction?
\[B = P^\intercal A P = \begin{pmatrix}\lambda_1 & 0 \\\\0 & C\end{pmatrix}\]
\[P = [\pmb v_1, \ldots, \pmb v_n]\]
and $C$ is an $(n-1) \times (n-1)$ matrix we wish to show is diagonal. By induction, we can assume that there exists a matrix $Q$ such that $Q^{-1} C Q$ and then use this to construct a matrix $R$
that satisfies the property $R^{-1} A R$ is diagonal.
When proving the spectral theorem for a symmetric matrix $A$, you define $B$ as
\[B = P^\intercal A P = \begin{pmatrix}\lambda_1 & 0 \\\\0 & C\end{pmatrix}\]
\[P = [\pmb v_1, \ldots, \pmb v_n] \text{ (eigenvectors of A)}\]
and $C$ is an $(n-1) \times (n-1)$ matrix we wish to show is diagonal. By the induction on spectral theorem, what can we assume about $C$?
There exists an orthonormal matrix $Q$ such that
\[Q^{-1} C Q = D\]
where $D$ is diagonal.
When proving the spectral theorem for a symmetric matrix $A$, you define $B$ as
\[B = P^\intercal A P = \begin{pmatrix}\lambda_1 & 0 \\\\0 & C\end{pmatrix}\]
\[P = [\pmb v_1, \ldots, \pmb v_n] \text{ (eigenvectors of A)}\]
and $C$ is an $(n-1) \times (n-1)$ matrix we wish to show is diagonal. By the induction on spectral theorem, we can assume that there exists $Q$ such that $Q^{-1} C Q$ is a diagonal matrix. Then how
do we construct the final $R$ that is an orthonormal matrix such that that $R^{-1}AR$ is a diagonal matrix and what rwo things do we need to check?
\[R = P \begin{pmatrix}1 & 0 \\\\ 0 & Q\end{pmatrix}\]
Need to check
• $R^{-1} = R^\intercal$
• $R^{-1} A R$ is diagonal
Say you want to prove the spectral theorem for a real symmetric matrix $A$. You’ve done the step where you show it has real eigenvectors. Then what are the “ingredients” for the rest of the proof?
• $P = [\pmb v _ 1, \cdots, \pmb v _ n]$, (matrix consisting of arbitrary orthonormal extension of eigenvector $\pmb v _ 1$).
• $P^\intercal A P = \begin{pmatrix}\lambda _ 1 & 0 \\ 0 & C\end{pmatrix}$
• $Q^\intercal C Q = D$ (the inductive step)
• $R = P \begin{pmatrix}\lambda _ 1 & 0 \\ 0 & Q\end{pmatrix}$
Prove that if $A \in \mathcal{S}^n$ (where $\mathcal S^n$ denotes the set of real $n \times n$ symmetric matrices), then $A$ has $n$ real eigenvalues and there exists an orthonormal basis for $\
mathbb R^n$ consisting of eigenvectors for $A$ (the spectral theorem).
Related posts | {"url":"https://ollybritton.com/notes/uni/prelims/ht23/linear-algebra/notes/notes-linear-algebra-ii-ht23-spectral-theorem/","timestamp":"2024-11-05T09:55:12Z","content_type":"text/html","content_length":"511322","record_id":"<urn:uuid:62deb015-9cf8-4563-b207-16c406e953b9>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00035.warc.gz"} |
Number of Arithmetic Triplets
Arithmetic Triplets Oracle Python Interview Question
Arithmetic Triplets
Oracle Python Interview Question
You are given a strictly increasing array of integers , and a positive integer . An arithmetic triplet with a gap of is defined as triplet of integers from where:
• Each value is unique, and the triplet is in increasing order:
• The difference in the values from and are both exactly the specified value
Return the number of arithmetic triplets with a gap of that exist in .
Example #1
Input: nums = [1, 2, 4, 6, 7, 11, 12] ; diff = 5
Output: 2
Explanation: For a diff of 5, there are two valid triplets: (1,6,11) and (2,7,12)
Notice how we can rearrange the equations relating the triplet values to such that and are defined in terms of . In other words, for a given value of , we know exactly what and we need.
Therefore, we can iterate over each value in and just check if the corresponding values of and exist in . A set of all the values in will allow us to perform this check efficiently. Be sure to
remember that sets and dictionaries in python are very efficient for lookups!
Sourced from | {"url":"https://datalemur.com/questions/python-arithmetic-triplets","timestamp":"2024-11-06T01:20:55Z","content_type":"text/html","content_length":"158889","record_id":"<urn:uuid:c008d5f3-c5e3-4255-8701-8a0dcc568bd3>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00886.warc.gz"} |
2015 Canadian Computing Olympiad
Day 2, Problem 2 - Timpanist
Computer scientists don't often help percussionists, but today, that will change. Since we cannot help all percussionists at the same time, we focus on timpanists first. By way of terminology, the
timpani is the plural of timpano and the player of the timpani is a timpanist.
A timpano is a large drum which can be tuned to a certain pitch, and a timpanist uses an ordered set of D timpani. On this occasion, they're playing a piece which has N notes. Note i occurs T[i]
seconds into the piece, and has pitch P[i]. P[i] is one of the following twelve notes:
{ F, F#, G, G#, A, A#, B, C, C#, D, D#, E }
At a given time, a timpano can only be used to play the pitch it is currently tuned to, and thus the timpanist can play a note i if and only if one of the timpani is tuned to pitch P[i] at time T[i].
Every note in this piece is in the range of a single octave, from F up to E, which means that the above list of possible notes is in ascending order of pitch. In order to make your computation
slightly easier, we will use integers from 1 to 12 to indicate these 12 tones:
F F# G G# A A# B C C# D D# E
(i.e., F will be represented by 1, F# will be 2, …, E by 12).
These are the only pitches to which timpani can be tuned.
Before the piece starts, the timpanist can freely tune each timpano to any pitch they'd like. However, during the piece, they may need to quickly retune them in between notes in order to be able to
play the required pitches at the correct times. The drums are numbered from 1 to D. At every point in time, the drum i + 1 must be kept tuned to a pitch higher than drum i. Retuning a single drum
must be done in an uninterrupted interval of time, in which no notes are being played and no other drums are being retuned. Because this is not an easy process, the timpanist would like to give
themselves as much time as possible. In particular, they'd like to maximize the amount of time they have for the fastest retuning they must perform within the piece.
Input Format
The first line contains two integers, N and D, the number of notes to be played and the number of drums available to be played (1 ≤ N ≤ 100; 1 ≤ D ≤ 4). The next N lines each contain two integers T
[i] and P[i] representing the time and pitch of the i-th note played (0 ≤ T[1] < T[2] < … < T[N−1] < T[N] ≤ 10^9; 1 ≤ P[i] ≤ 12 for 1 ≤ i ≤ N).
For 60% of the marks for this problem, N ≤ 50 and D ≤ 3.
Output Format
The output is one line containing one real number (rounded off to 2 decimal places), which is the maximum amount of time (in seconds) that the timpanist can have for their fastest retuning, or 0.00
if no retunings are necessary.
Sample Input 1
Sample Output 1
Explanation for Sample 1
With just 1 drum, the timpanist must retune it after every note in order to play the following one.
With 2 drums, the answer would instead be 10.00 (achieved by leaving the higher drum tuned to pitch 12).
Sample Input 2
Sample Output 2
Explanation for Sample 2
The first 6 notes include only the 4 pitches 1, 3, 5, and 6. Similarly, the last 6 include only 1, 3, 6, 8.
The single optimal strategy involves pre-tuning the 4 drums to 1, 3, 5, and 6. After the sixth note, the timpanist can take 4.5 seconds to retune the highest drum to an 8, and then another 4.5
seconds to retune the second-highest drum to a 6, finishing just in time to play the seventh note.
Sample Input 3
Sample Output 3
Explanation for Sample 3
This is a more typical timpani part, involving just one note, and thus no retuning.
All Submissions
Best Solutions
Point Value: 20 (partial)
Time Limit: 1.00s
Memory Limit: 512M
Added: Jun 05, 2015
Author: SourSpinach
Languages Allowed:
C++03, PAS, C, HASK, ASM, RUBY, PYTH2, JAVA, PHP, SCM, CAML, PERL, C#, C++11, PYTH3 | {"url":"https://wcipeg.com/problem/ccc15s2p5","timestamp":"2024-11-06T09:10:26Z","content_type":"text/html","content_length":"13968","record_id":"<urn:uuid:4f468566-fd7f-419c-a8e5-d2bf91cab0f9>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00836.warc.gz"} |
DVB-T implementation in GNUradio – part 3
Later edit: source code added here: https://github.com/BogdanDIA/gr-dvbt
In this article I will describe my DVB-T receiver implementation in GNUradio.
The blocks used for receiving are basically the transmitter blocks in the reverse order with a major difference that there is need for synchronization blocks in order to obtain a clean constellation.
OFDM symbol aquisition:
The task of this block is to synchronize the receiver so that a clean time domain signal is obtained before the FFT block. There are several subtasks that this block takes care of:
1. Use Cyclic Prefix (CP) to obtain the start of the OFDM symbol in time domain. Here I’ve chosen to use a MLSE algorithm that is presented in [REF1 – Van de Beek]. It minimizes a log likelihood
function in order to obtain frequency correction, epsilon – a fractional of intercarrier frequency, and time correction, theta – number of samples to the detected start of OFDM symbol. Since the CP
is added in front of the OFDM symbol and being a copy of the last part of the symbol there should be a correlation maximum that will signal the beginning of the symbol. This is called pre-FFT NDA
(Non Data Aided) correction and basically will obtain sync on the beginning of the OFDM symbol and will assure the subcarriers will fit into the center of the FFT bin.
Note: The algorithms used in OFDM for 802.11 [Schmidl and Cox – REF2] are using a preamble that is specially created to create two identical halves in time domain. In DVB-T this option is not
available and the pre-FFT correction is called Non Data Aided acquisition because no pilots are used to achieve the synchronization task. However, there will be pilots used for post-FFT
synchronization and I will explain it later in this article.
2. Once the CP start is found and epsilon is also found, de-rotation of the signal is applied in order to obtain the correct time domain OFDM symbol.
Note: The MLSE algorithm is quite computational intensive due to the correlations necessary to be calculated. For that reason for the subsequent symbols only +/-4 samples is taken in consideration to
assure the CP is found again. In case of a number of consecutive CP misses a full initial aquisition is triggered.
Note 2: The MLSE of Beek requires knowledge of SNR value. For now this is entered manually as a parameter to the processing block.
TODO: Add one methods of SNR estimation
3. The CP in front of the OFDM symbol is taken out
FFT for time to frequency domain conversion:
This is just plain FFT conversion for 2k, 8k or 4k depending on the parameters. Here the GNUradio block is used.
A plot of the constellation after the FFT is in order. One may see that the constellation is rotating and therefore requires more processing in order to be useful for de-mapping:
Plot is made when transmitting with USRP N210 and receiving with USRP N210, both equipped with WBX daughterboards. DVB-T parameters are 8Mhz bandwidth, FFT 2k, 16-QAM, FEC 1/2
Demodulation of Reference signals:
The DVB-T standard uses several sub-carriers to insert pilot signals that are used for synchronization, signal parameters transmission and equalization.
There are three pilot signals types used in DVB-T standard:
– scattered pilot signals
– continual pilot signal
– TPS (Transmission Parameter Signals)
1. Post-FFT DA (Data Aided) synchronization.
The continual pilots and (in my implementation) scattered pilots are used to obtain an integer frequency correction after the FFT is performed. The position of the scattered and continual pilots is
known and therefore by using ML correlation of the signal with the expected values one can obtain an integer frequency correction. This is the number of bins the synchronizer needs to take in
consideration when correcting the signal after the FFT.
There are two roots of frequency deviations that we need to take into account beside the nature of the transmission channel: Carrier and Sampling-clock deviations. These produces ICI (Inter Carrier
Interferences) that are important due to the small intercarrier frequency and can be modelled as noise. See [Speth and all – REF3].
After pre/post-FFT carrier synchronization that is to be performed one time only, a continuous correction still need to be done for the residual correction that will appear during time using a
PID-like algorithm described also in [Speth and all – REF3].
TODO: implement this correction.
2. Demodulation of TPS block
Transmission Parameter Signalling is used to assist the receiver in knowing the parameters of the stream that is received. The TPS is send using using predefined position pilots on witch the data is
modulated using DBPSK modulation. Being differential modulation, this is suitable for demodulation even though the constellation is rotating. The same bit of information is sent in an OFDM symbol so
majority voting is used to decide the actual value of the data bit. There are 68 symbols in an frame and 4 frames in a superframe.
3. Equalization of signal based on pilot signals
A simple continual and scattered pilot based equalizer is used to perform correction of the data. The scattered pilots have a position inside the frame that is based on the following formula {k =
Kmin + 3 × (l mod 4) + 12p p integer, p ≥ 0, k ∈ [Kmin; Kmax] } whre l is the symbol index. The idea behind scattered pilots is to allow for a better coverage of the whole channel when moving from
one symbol to the other.
TODO: Implement DFE (Decision Feedback Equalizer) – See [Proakis – REF4]
The following plot shows the signal constellation for 16-QAM after the equalization:
[REF1]: Jaan-Jaap van de Beek, ML Estimation of Time and Frequency Offset in OFDM Systems
[REF2]: Timothy M.Schmidl and Donald C. Cox, Robust Frequency and Timing Synchronization for OFDM
[REF3]: Michael Speth, Stefan Fechtel, Gunnar Fock, Heinrich Meyr, Optimum Receiver Design for OFDM-Based Broadband Transmission – Part I and II
[REF4]: John G. Proakis, Masoud Salehi, Digital Communications
4 thoughts on “DVB-T implementation in GNUradio – part 3”
1. amazing work dear Admin.
It will be great the day when I see your code (or grc) in github. anyway thanks in advanced.
2. This is very impressive, well done.
3. Hi Admin,
do you calculate bir error rate or smt. similar to that to measure the performance?
4. Hi, for now the BER is not calculated. However, this could be one of the next steps. | {"url":"https://yo3iiu.ro/blog/?p=1244","timestamp":"2024-11-06T12:19:12Z","content_type":"text/html","content_length":"59809","record_id":"<urn:uuid:0f118d2f-e981-41f0-9621-c3f78fd5a061>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00848.warc.gz"} |
Precision - Data Science Wiki
Precision :
Precision is the degree of accuracy or exactness of a measurement, calculation, or statement. It is an important concept in many fields, including science, engineering, and mathematics, as it allows
for more accurate and reliable results.
One example of precision is in the field of medicine. In order to diagnose and treat patients accurately, medical professionals must be precise in their measurements and calculations. For example,
when administering medication, doctors and nurses must be precise in measuring out the correct dosage for each patient. If the dosage is not precise, it could lead to negative side effects or even
death. Similarly, when performing surgeries or diagnostic tests, medical professionals must be precise in their measurements and calculations to ensure that the procedure is successful and safe.
Another example of precision is in the field of manufacturing. In order to produce high-quality products, manufacturers must be precise in their measurements and calculations. For example, when
creating a new
car model
, engineers must be precise in the design and dimensions of the car to ensure that it functions correctly and meets safety standards. Similarly, when producing consumer products such as electronics
or appliances, manufacturers must be precise in the size and shape of the components to ensure that they fit together correctly and function properly.
There are several factors that can affect the precision of a measurement or calculation. One
is the accuracy of the tools or instruments being used. For example, if a doctor is using a scale to weigh a patient, the scale must be calibrated correctly and be capable of accurately measuring
small increments in order for the measurement to be precise. Another factor is the skill and training of the person performing the measurement or calculation. For example, a skilled engineer may be
able to produce more precise measurements than a novice engineer due to their knowledge and experience.
In order to improve the precision of a measurement or calculation, it is often necessary to take multiple measurements or calculations and average them together. This is known as taking an average or
. By taking multiple measurements or calculations, the impact of any errors or variations is minimized, resulting in a more accurate and precise result.
In conclusion, precision is the degree of accuracy or exactness of a measurement, calculation, or statement. It is important in many fields, including medicine and manufacturing, and can be affected
by factors such as the accuracy of the tools being used and the skill of the person performing the measurement or calculation. In order to improve precision, it is often necessary to take multiple
measurements or calculations and average them together. | {"url":"https://datasciencewiki.net/precision/","timestamp":"2024-11-13T14:46:52Z","content_type":"text/html","content_length":"41884","record_id":"<urn:uuid:c4b434d4-79e2-4f98-8f05-588197aa41f2>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00691.warc.gz"} |
Hybrid regularisation and the (in)admissibility of ridge regression in infinite dimensional Hilbert spaces
In this thesis we study stability from several viewpoints. After covering the practical importance, the rich history and the ever-growing list of manifestations of stability, we study the following.
(i) (Statistical identification of stable dynamical syste ... | {"url":"https://graphsearch.epfl.ch/en/publication/268768","timestamp":"2024-11-09T05:53:34Z","content_type":"text/html","content_length":"110248","record_id":"<urn:uuid:c168c876-bf81-4e84-98ee-c6151ca6da67>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00593.warc.gz"} |