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Pulse Width Modulation (PWM)

What is pulse-width modulation, anyway?

If you're working with a switching power converter for motor control, it probably has a set of half-bridge output stages. Each of these has two switches connected in series across a DC link that carries some voltage, VDC.

At any given instant, we have three choices for controlling the half bridge: we can pull the output phase low to the negative side of the DC link (let's call this the "0" state), we can pull the output phase high to the positive side of the DC link (let's call this the "1" state), and we can leave the output phase open-circuited. For the moment let's ignore this last choice, and any transients involved going between the 0 and 1 states. We can output 0 or 1, and switch back and forth any way we like over time.

Here's the 0 state:

and here's the 1 state:

PWM switches between the 0 and 1 state at a fixed frequency. In each period of time T, the switch is at the 1 state some fraction of the time, and at the 0 state the rest of the time. The average fraction of time is D (the "duty cycle"), and this creates an average voltage on the half-bridge output of ${V}_{out} = D \times {V}_{dc}$.

# A bit of history

It may help to get some historical perspective on PWM for motor control.

Early switching converters (1960s-1980s) for three-phase motors generally used an analog circuit technique for generating the PWM waveforms, called "sine-triangle" PWM. The idea is that if you want to generate a duty cycle with an average value that approximates a sine wave, create a triangle waveform and compare it to the desired output waveform (sine wave), using the results of an analog comparator to determine whether or not to be in the 0 or 1 states.

# Three-phase PWM, and zero-sequence voltage

Sine-triangle PWM will work for 3 sine waves that are 120° out of phase:

The maximum line-to-line voltage you can get in this case, without creating distorted waveforms, is $\frac{\sqrt{3}}{2}{V}_{dc}$, or 86.6% of the DC link voltage. But we can do better than this.

In 1974, K. G. King of Westinghouse Brake & Signal Co. Ltd. received U.S. Patent #3839667 on adding harmonics of the operating frequency to increase the line-to-line voltage of an inverter. In particular he described adding 3rd, 6th, 9th, etc. harmonics, also known as "triplens", to each of the phase voltages. This affects the line-to-ground voltage, but the line-to-line voltages are unaffected, because the triplen harmonics on each phase line up. The resulting line-to-ground voltages appear squashed (figure from this patent):

This general idea involves adding what is known as a zero-sequence component, namely the same voltage added identically to each phase. (The term "zero-sequence" is from C. L. Fortescue's 1918 paper on symmetrical components to analyze three-phase power systems: positive sequence represents equal-amplitude sine waves in the normal phase order, e.g. A leading B leading C; negative sequence represents equal-amplitude sine waves in the reverse phase order e.g. A leading C leading B; and zero sequence represents equal-amplitude sine waves that are aligned in phase. Fortescue showed that any set of 3 phase sinusoidal voltages can be expressed as the sum of a positive, negative, and zero-sequence component. Only the positive and negative sequence components affect the voltage measured between any two phases.)

In 1984, Houldsworth and Grant published a paper demonstrating that a 3rd-harmonic component, with 1/6 the amplitude of the fundamental, can be added to each phase, to produce waveforms that have line-to-line voltages reaching the full DC link voltage (15.5% higher than without any triplen harmonics).


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OR
(ii). The volume, terms of pi, of the smaller cone that was removed.
ii). Calculate, in terms of pi, the surface area of the remaining solid after the smaller cone is removed.

format: plain


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# The Angles of Depression of the Top and Bottom of a Tower as Seen from the Top of a 60 Sqrt(3) M High Cliff Are 45° and 60° Respectively. Find the Height of the Tower. - Mathematics

The angles of depression of the top and bottom of a tower as seen from the top of a 60  sqrt(3) m high cliff are 45° and 60° respectively. Find the height of the tower.

#### Solution

Let AD be the tower and BC be the cliff.
We have,
BC = 60 sqrt(3) , ∠ CDE = 45° and ∠BAC = 60°
⇒ BE = AD = h
⇒ CE =  BC - BE= 60 sqrt(3)  -  h
In  ΔCDE,

 tan 45° = (CE)/(DE)

⇒ 1 = (60 sqrt(3) -h)/(DE)

⇒ DE = 60 sqrt(3) - h

⇒ AB = DE = 60 sqrt(3) - h                      ............(1)

Now, in ΔABC

tan 60° = (BC)/(AB)

⇒ sqrt(3)= (60 sqrt(3) )/ (60 sqrt(3) -h)                 [ Using (1)]

⇒ 180 - h sqrt(3) = 60 sqrt(3)

⇒ h sqrt(3) = 180- 60 sqrt(3)

⇒ h = (108 -60sqrt(3) )/sqrt(3) xx sqrt(3)/sqrt(3)

⇒  h = ( 180 sqrt(3)-180) /3

⇒ h = (180 sqrt(3)-1) /3

∴ h = 60 ( sqrt(3)-1)

= 60 (1.732 -1)

= 60 (0.7.32)

Also, h = 43.92m

So, the height of the tower is 43. 92 m.

Concept: Heights and Distances
Is there an error in this question or solution?

#### APPEARS IN

RS Aggarwal Secondary School Class 10 Maths
Chapter 14 Height and Distance
Exercises | Q 22

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# Why do the discriminant and primality of the group order of an elliptic curve affect security?

In a book about cryptography and elliptic curves, there was a mention that not all curves are secure, and a statement than in order to pick a secure curve the curve must satisfy 3 requirements.

1. The curve's equation is $$y^2 = x^3 + ax + b\quad\pmod p$$
2. with $$4a^3 + 27b^2 \neq 0\quad\pmod p$$
3. and the order $$\#E(\mathbb F_p)$$ of the Elliptic Curve group is prime.

The third point is weird since in DLP like Diffie-Hellman modulo $$p$$, the order $$p-1$$ can be any number not necessarily a prime (maybe it can be extra secure if it's a prime I don't know).

I understand the first statement of course but I'm completely lost on why the other two statements must be true for an elliptic to be secure. So if someone can explain why statements 2/3 must be true for a good elliptic curve and what can happen if these two conditions aren't true. Any small example with small $$p, a, b$$ will be really appreciated so I can understand the mathematics behind it.

1. $$y^2 = x^3 + ax + b\bmod p$$ is the Short Weierstrass equation. The theory behind it is here

Using Bezout’s Theorem, it can be shown that every irreducible cubic has a flex (a point where the tangent intersects the curve with multiplicity three) or a singular point (a point where there is no tangent because both partial derivatives are zero). [Reducible cubics consist of a line and a conic, which are easy to study.] An irreducible cubic with a flex can be affinely transformed into a Weierstrass equation: $$Y^2 + a_1 XY + a_3 Y = X^3 + a_2 X^2 + a_4 X + a_6$$

And this can be converted into a short Weierstrass equation if $$p$$ is larger than $$3$$. The transformation can be achieved by

Change $$Y\longrightarrow y-(a_1x+a_3)/2$$ and $$X \longrightarrow x$$

$$y^2 = x^3 - (1/4 a_1^2 +a_2 )x^2 + (1/2 a_1 a_3)x + (a_4+ 1/4 a_3^2 + a_6)$$

with renaming

\begin{align} A & = - (1/4 a_1^2 +a_2 )\\ B & = 1/2 a_1 a_3\\ C & = a_4+ 1/4 a_3^2 + a_6 \end{align}

now the new equation has of the form $$y^2=x^3+Ax^2+Bx+C.$$

And now change $$x\longrightarrow x-A/3$$, $$y^2 = x^3 - \left(1/3A^2x + Bx \right)+ \left( 2/27 A^3 - 1/3AB + C \right)$$ with renaming;

\begin{align} a &= 1/3A^2 + B\\ b & = 2/27 A^3 - 1/3AB + C\\ \end{align} now the new equation has of the form $$y^2=x^3+ax+b.$$

The reason for $$p \neq 2,3$$ can be seen from the divisions. Also one should note that the two changes of variables are invertible, therefore their composition is too. This concludes that the rational points on both equations are in bijective correspondence.

For $$p=2$$ and $$p=3$$, there are other short forms. In Cryptography we work in either binary extension fields ($$\mathbb{F}_{2^m}$$), which are no more secure, large prime fields ($$\mathbb{F}_{p}$$), and their extensions ($$\mathbb{F}_{p^m}$$).

2. If the discriminant is zero ($$\Delta = 0$$), then the curve is singular and the DLP is easy. Geometrically, this means that the graph has cusps, self-intersections (nodes), or isolated points.

Cusp; when $$\Delta=0$$ and $$a=0$$ ( $$y^2 = x^3$$);

Intersection or Node; when $$\Delta=0$$ and $$a\neq 0$$ ( $$y^2 = x^3 +3x +1$$);

Acnode (isolated point); $$y^2 + x^3 + x^2$$, at $$(0,0)$$

Note these plots are not in finite fields where we cannot see such figures easily. A curve over a finite field is just points when we draw on the plane as below, and we cannot see the above singular points; however, we can still calculate them.

The above is the point of the curve over $$\operatorname{GF}(307)$$ with the curve equation $$y^2 = x^3 + x^2 + x$$ and this curve's group of points is isomorphic to $$\mathbb Z_{160} \oplus \mathbb Z_2$$

And, Wolfram has a nice page about those singular points.

1. This is not completely true. Some curves have prime order (i.e., $$\#E(\mathbb F_p)$$ is a prime), and they are called prime curves. All of the elements other than the identity can be a generator. This is the direct result of Lagrange's theorem on the group theory, i.e., the order of the subgroup generated by an element must divide the order of the group. To see what kind of groups can be formed by the points of a curve over a finite field see the theorem on the bottom.

Some curves don't have prime order, like the Curve25519 and Curve448. Curve25519 has a co-factor $$h=8$$ i.e. $$h = \frac{\#E(\mathbb F_p)}{n}$$ where $$n$$ is the order of the generator point $$G$$. In this case, we chose the largest prime to form the subgroup and select a generator for it.

There is also twist security related to the order that one must consider. Although the NIST P-224 has co-factor 1 ( i.e. it has prime order), it doesn't have the twisted security.

Theorem: Let $$E$$ be an elliptic curve group over the finite field $$\mathbb{F_q}$$. Then $$E(\mathbb{F_q}) \simeq \mathbb{Z_p} \text{ or } \mathbb{Z_{n_1}} \oplus \mathbb{Z_{n_2}}$$ for some integer $$n \geq 1$$, or for some integers $$n_1,n_2 \geq 1$$ with $$n_1|n_2$$.

If the order of the curve is prime then it must be $$E(\mathbb{F_q}) \simeq \mathbb{Z_p}$$.

SageMath code for the transformation. First part;

var('x y z a b c')
f = y^2 - x^3- a*x^2-b*x-c
print(f)
g = f.subs(x == x-a/3)
print(g)
g.expand()

The second part;

var('X Y Z x y z a1 a2 a3 a4 a5 a6')
f = Y^2 + a1 * X*Y + a3*Y - X^3 - a2*X^2 - a4*X - a6
f
g = f.subs(Y == Y - (a1*X +a3)/2)
print(g)
g.expand()

• Thank's a lot this helped me out.
– KMG
Dec 13, 2020 at 13:54


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# Again Computing limits

Calculus Level pending

Let $$\{a_n \}$$ be a sequence of real numbers such that $$e^{a_{n}}+ na_{n}=2$$ for all positive integers $$n$$. Compute the limit

$\lim_{n \to \infty} n(1-na_{n})$


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# How do you find the slope and intercept of y = –6x + 2?

To find x-intercept, make y = 0 --> $x = \frac{2}{6} = \frac{1}{3}$


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Viewpoint

# A New Way to Laser-Cool Molecules

Physics 13, 89
Researchers exploit the peculiar structure of yttrium monoxide to cool the gas to ultralow temperatures and record-breaking densities.

The study of ultracold atoms has driven major advances in fundamental and applied quantum science for decades—from observing new phases of matter to searching for new fundamental particles and forces. Recently, the quantum community has grown excited about what might be learned from studying controlled quantum interactions between ultracold molecules. With their additional complexity, molecules have more degrees of freedom, offering the possibility to investigate new exotic phenomena. Unfortunately, this extra complexity means that molecules cannot be cooled to ultralow temperatures by simply applying the same methods that produce ultracold atoms. Now, Jun Ye and colleagues at the University of Colorado, Boulder, demonstrate a novel laser-cooling technique that exploits the structural peculiarities of molecules with a nuclear spin and an electron residing on the same species—in their case, the yttrium atom in yttrium monoxide (YO) [1]. By leveraging the unusual structure of YO, the team reaches ultralow temperatures and record-breaking high densities in a laser-cooled molecular gas.

Cooling molecules to the quantum regime—the temperatures at which delicate quantum states are protected from random thermal effects—has many applications within quantum chemistry, many-body physics, and fundamental physics. For example, polar molecules offer a promising platform for quantum simulation, with their electric dipole moments providing strong and tunable interactions. They are also sensitive to the effects of high-energy physics: their large internal electromagnetic fields would amplify hypothetical electromagnetic interactions that are predicted to resolve the Universe’s matter-antimatter asymmetry. Polar molecules’ chemical bonds, meanwhile, offer the ability to study controlled quantum chemistry, in which well-defined quantum states are observed as a chemical reaction takes place.

Currently, in nearly all quantum experiments with atoms, the necessary ultracold temperatures are achieved using laser cooling. This technique works by repeatedly scattering laser photons from the atomic gas to provide a viscous damping force. This force—very much like friction—pushes against the motion of the atoms, and therefore cools the gas. The reason the same cannot be done easily with molecules is that molecules have internal structure. This means that scattering a photon can (and often does) make the molecule rotate or vibrate. In other words, it injects energy into the molecule and stifles cooling.

In the past decade, several groups (including Ye’s) have overcome these challenges and achieved laser cooling and trapping of several molecules: strontium fluoride (SrF) [2], calcium fluoride (CaF) [3, 4], and yttrium oxide (YO) [5]—all of which have an electronic structure particularly suited to laser cooling. These molecules feature a valence electron that acts as though it is bound only to the metal atom. A consequence of this electron localization is that the electron is largely decoupled from the molecular motion, and the electronic transition excited by the laser is unlikely to generate unwanted internal modes. Unlike the metal atoms in SrF and CaF, however, the Y atom in YO has a nuclear spin, which results in a strong electron-nucleus interaction. This “Fermi contact” interaction produces a rather different electronic structure compared to SrF and CaF, and it changes the magnetic properties of the entire molecule. Ye and colleagues use these altered magnetic properties to their advantage to design a new cooling scheme.

Molecules that have been cooled by interacting with photons warm up again when their excited states decay spontaneously, and the absorbed photons are re-emitted in random directions. This process sets a lower limit—called the Doppler limit—on the temperature achievable using simple laser cooling. The Heisenberg energy-time uncertainty principle states that longer timescales between these random, spontaneous photon-emission events are needed to achieve lower energies, and therefore, temperatures. Cooling methods that go below the Doppler limit therefore rely on engineering coherent quantum states that last longer than the short-lived excited electronic state. Magnetic fields can scramble the wave functions of the molecules and destroy this coherence, meaning magneto-optical traps (MOTs), which employ magnetic fields to confine and compress molecules, are used at the expense of higher temperatures.

The Colorado group’s scheme is less sensitive to these magnetic fields because the strong electron-nucleus interaction in YO makes the electron and nucleus “pair up” such that their magnetic moments cancel out. This lets the researchers utilize MOTs without compromising the cooling effects of sub-Doppler methods, and it allows them to combine low temperatures with high densities (Fig. 1). With this technique, the researchers approximately match the 5 $𝜇\text{K}$ temperature obtained using sub-Doppler methods to cool CaF, the previous low-temperature record holder [6, 7]. But they achieve a density about 4 times higher. In terms of phase-space density—a measure of how “quantum” a gas is, based on the degree to which the molecules’ de Broglie waves overlap—Ye’s team breaks the record for molecules in free space. This puts their ultracold YO gas closer to the quantum regime than researchers have managed with other molecular species.

Such trapped polar molecules with high phase-space density are a frontier in the study of strongly interacting quantum systems, as strong dipole-dipole interactions are more robust in gases of molecules than of atoms. To realize many of the exciting applications that this property makes possible, researchers need to hold ultracold gases in optical dipole or tweezer traps. Unlike MOTs, these “conservative” traps can control ultracold gases for long periods of time without changing the internal states of the molecules. But loading these traps requires sub-Doppler temperatures and high densities. The team’s new method will therefore be extremely useful in this respect.

Achieving a record high density is noteworthy in itself; doing so with YO offers unique opportunities in molecular physics. For example, using SrF and CaF, researchers have had access only to molecules with single fluorine bonds. As the first ultracold oxide, YO has a double bond to a different species, making it a compelling molecule for the study of ultracold chemical reactions. Additionally, while the structure of YO that enables these new methods is different compared to existing laser-cooled molecules, it is not unique among all molecules. Importantly, many other polar species that could be cooled using this new method also include a heavy metal with a nuclear spin, like YO. As heavier species contain more charges and therefore stronger internal electromagnetic fields, other ultracold molecules could be even more sensitive to nuclear interactions, such as the weak force, and possibly, interactions that aren’t described by the standard model.

This research is published in Physical Review X.

## References

1. S. Ding et al., “Sub-Doppler cooling and compressed trapping of YO molecules at $𝜇$K temperatures,” Phys. Rev. X 10, 021049 (2020).
2. J. F. Barry et al., “Magneto-optical trapping of a diatomic molecule,” Nature 512, 286 (2014).
3. L. Anderegg et al., “Radio frequency magneto-optical trapping of CaF with high density,” Phys. Rev. Lett. 119, 103201 (2017).
4. S. Truppe et al., “Molecules cooled below the Doppler limit,” Nat. Phys. 13, 1173 (2017).
5. A. L. Collopy et al., “3D magneto-optical trap of yttrium monoxide,” Phys. Rev. Lett. 121, 213201 (2018).
6. L. Caldwell et al., “Deep laser cooling and efficient magnetic compression of molecules,” Phys. Rev. Lett. 123, 033202 (2019).
7. L. W. Cheuk et al., “$\Lambda$-enhanced imaging of molecules in an optical trap,” Phys. Rev. Lett. 121, 083201 (2018).

Nick Hutzler received his bachelor’s degree in mathematics from the California Institute of Technology (Caltech), his Ph.D. in physics at Harvard, and performed postdoctoral research at the MIT-Harvard Center for Ultracold Atoms (CUA). He is currently an Assistant Professor at Caltech, developing techniques to control cold molecules for a variety of applications, especially precision measurement searches for physics beyond the standard model.

## Subject Areas

Atomic and Molecular Physics

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text:
None is zero

Logic Level 2

$\large \begin{array} { l l l l l } & &D & O & G\\ +& &C & A & T \\ \hline & 1 & 0 & 0 & 0 \\ \end{array}$

The above shows a long addition, where each of the symbols represents a distinct single-digit positive integer.

What is the sum $D + O + G + C + A + T ?$

×


text:
# Can A Utility Function Take On Negative Values?

Can someone provide a rigorous definition of a utility function? I had thought that a utility function only needs to the preserve the order of preferences. Thus a utility function can take on negative values as long as it preserves the order of preferences. Others have told me that a utility function cannot take on negative values. Is this a condition of a rigorous definition of a utility function?

A utility function can certainly be negative. The utility function is nothing more than a way to represent a preference relationship. This is an important conceptual point. In several theorems that typically show up in introductory texts, we show that sets of preferences that satisfy certain regularity conditions can be represented as utility function.

Also, there are different decision theory frameworks that allow the utility function to be transformed. You alluded to something like this in your question. In the traditional framework without uncertainty, the utility function is defined up to a monotonic transformation. Under certain kinds of uncertainty, we get Von Neumann–Morgenstern utility functions that are unique up to affine transformations. You can read more about this elsewhere. For now, the consider the following definition of a utility function. It is taken from Advanced Microeconomic Theory by Jehle and Reny (3rd edition):

A preference relation $\succeq$ is defined as follows:

where the axioms references are these:

Axiom 1: Completeness. For all $x^1$ and $x^2$ in $X$, either $x^1 \succeq x^2$ or $x^2 \succeq x^1$.

Axiom 2: Transitivity. For any three elements $x^1$, $x^2$, and $x^3$ in $X$, if $x^1 \succeq x^2$ and $x^2 \succeq x^3$, then $x^1 \succeq x^3$.

• Nice answer! Solved my question. May 19 '19 at 15:20

Here is one possible rigorous definition of a utility function:

Let $X$ be a set of alternatives. Let $\succeq$ be a preference relation over those alternatives. $U: X \to \mathbb{R}$ is a utility function means that $U(x) \geq U(y) \iff x \succeq y$.

Then if for example $X$ is 'amounts of money you might be given', and $x \succeq y$ only if $x \geq y$, then some possible utility functions are $U(x) = x, U(x) = e^x, U(x) = \log(x)$...

Some of these are negative.

One could of course require that $U(x) > 0$. Maybe this makes it easier to swallow as an interpretation of the 'well-being' of an individual. But that would rule out many commonly used utility functions, such as $U(x) = x$ or $U(x) = \log(x)$.

• "Maybe this makes it easier to swallow as an interpretation of the 'well-being' of an individual." But of course it's just best to think of a utility function as a representation of the preference relation $\succeq$ and nothing more. Some utility functions are always non-positive (e.g., the CRRA type $C^{1-\gamma}/(1-\gamma)$ where $\gamma > 1$). Mar 13 '15 at 17:19
• I wouldn't say it's 'best'. For welfare analysis or for thinking about things like attitudes toward risk we sometimes give more meaning to the utility function than as merely representing a preference relation. One could debate the philosophy of it but there's a long history of interpreting the utility function as representing something real. Apr 17 '15 at 14:15
• "there's a long history of interpeting the utility function as representing something real" Yes. But to be specific, this is only the case with "cardinal utliity" as opposed to "ordinal utility." But even VnM utliity functions (which are cardinal and are the utility functions that are most commonly used) are defined up to affine transformations. Thus, the sign of the utility is in some sense still arbitrary. Apr 22 '15 at 22:09

Like jmbejara says generally in economics utility is not measured in anything but preference relations, so it's called ordinal utility (which contrasts cardinal utility). So a bundle giving utility of -1 is preferred to any bundle giving less than -1. The number -1 doesn't tell us anything else.

"Ordinal utility theory states that while the utility of a particular good or service cannot be measured using a numerical scale bearing economic meaning in and of itself, pairs of alternative bundles (combinations) of goods can be ordered such that one is considered by an individual to be worse than, equal to, or better than the other. This contrasts with cardinal utility theory, which generally treats utility as something whose numerical value is meaningful in its own right." (source: http://en.wikipedia.org/wiki/Ordinal_utility)


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# 17.3. Cross-Validation¶

We can use the train-test paradigm to help use choose a model. The idea is to further divide the train set into separate parts where we fit the model on one part and evaluate it on another. This approach is called cross-validation. We describe one version of it, called $$k$$-fold cross-validation*. Figure 17.3 shows the idea behind this division of the data.

Fig. 17.3 Five-fold Cross-Validation. The train set is divide into 5 parts, which are used in turn to validate models built on the remainder of the data.

Cross-validation can help select the general form of a model. By this we mean, the degree of the polynomial, the number of features in the model, or a cut-off for a regularization penalty (see the next section). The basic steps behind $$k$$-fold cross-validation go as follows.

• Divide the train set into $$k$$ parts of roughly the same size, each of these parts are called folds. Use the same technique that was used to create the train and test sets to make the folds. Typically, we divide the data at random.

• Set one fold aside to act as a test set.

• Fit all models on the remainder of the train data (the training data less the particular fold).

• Use the fold you set aside to evaluate all of these models.

• Repeat this process $$k-1$$ more times, where each time you set aside one fold, use the rest of the train set to fit models, and evaluate them on the fold that was set aside.

• Combine the error in fitting each model across the folds, and choose the model with the smallest error.

These fitted models will not have identical coefficients across folds. As an example, when we fit a polynomial of, say, degree 3, we average the MSE across the $$k$$ folds to get a value of polynomials of degree 3. Then, we do the same for polynomials of degree 4, and so on. We then compare the MSEs and choose the degree of the polynomial with the lowest MSE. The actual coefficients for the $$x$$, $$x^2$$ and $$x^3$$ terms in the cubic polynomial are not be the same in each fit. Once the degree is selected, we refit the model using all of the training data and evaluate it on the test set. (We haven’t used the test set in any of the earlier steps to select the model).

Typically, we use between 5 or 10 folds. Another popular choice puts one observation in each fold. This special case is called leave-one-out cross-validation. It’s popularity stems from the simplicity in adjusting a least squares fit to have one fewer observation.

Generally, $$k$$-fold cross-validation takes some computation time since we typically have to refit each model from scratch for each fold. The scikit-learn library provides a convenient sklearn.model_selection.KFold class to implement $$k$$-fold cross-validation. We provide an example of $$k$$-fold cross-validation to give an idea of how it works.

## 17.3.1. Example: Fitting a Bent Line Model with Cross-validation¶

We demonstrate $$k$$-fold cross-validation on the gas consumption example. However, this time we fit a different set of models than polynomials. The original scatter plot of the data looks like the points fall along two connected line segments. In cold temperatures the relationship between gas consumption and temperature looks roughly linear with a negative slope of about $$-4$$cubic ft/degree, and in warmer months, the relationship appears nearly flat. So, rather than fit a polynomial, we can fit a bent line to the data.

Let’s start by fitting a line with a bend at 65 degrees. To do this, we create a second feature that enables the points with temperatures above 65 to have a different slope. The model is

$y = \theta_0 + \theta_1x + \theta_2(x-65)^+$

Here the $$(~)^+$$ stands for “positive part” so when $$x$$ is less than 65 it evaluates to 0, and when $$x$$ is 65 or greater, it is just $$x-65$$.

y = heat_df[['ccf']]
X = heat_df[['temp']]

X["temp65p"] = (X['temp'] - 65) * (X['temp'] >= 65 )
bend = LinearRegression().fit(X, y)

Let’s overlay this fitted “curve” on the scatter plot to see how well it captures the shape of the data.

fig = go.Figure()

go.Scatter(x=heat_df['temp'], y=heat_df["ccf"], mode='markers'))

go.Scatter(x=Xs['temp'], y=y_hats))

fig.update_layout(showlegend=False, width=350, height=250,
xaxis_title='Temperature (Farenheit)',
yaxis_title='Gas (cubic ft)')

fig.show()

This model appears to fit the data much better than the earlier polynomials. There are many bent line models possible. The line might bend at 55 degrees or 60 degrees, and so on. We can use $$k$$-fold cross-validation to choose the temperature value at which the line bends. We can consider models with bends at $$40, 41, 42, \ldots, 69$$ degrees. For each of these, we need to create the additional feature to enable the line to bend there.

for i in np.arange(40, 70, 1):
heat_df[ 'temp'+ i.astype('str') + 'p'] = (heat_df['temp'] - i) * (heat_df['temp'] >= i)

heat_df

temp ccf temp40p temp41p ... temp66p temp67p temp68p temp69p
0 29 166 0 0 ... 0 0 0 0
1 31 179 0 0 ... 0 0 0 0
2 15 224 0 0 ... 0 0 0 0
... ... ... ... ... ... ... ... ... ...
96 76 11 36 35 ... 10 9 8 7
97 55 32 15 14 ... 0 0 0 0
98 39 91 0 0 ... 0 0 0 0

97 rows × 32 columns

The first step in cross-validation is to create our train and test sets. We choose 22 observations at random to be placed in the test set. That leaves 75 for the train set.

y = heat_df[['ccf']]
X = heat_df.drop(['ccf'], axis=1)

test_size = 22

X_train, X_test, y_train, y_test = train_test_split(
X, y, test_size=test_size, random_state=0)

print(f'  Training set size: {len(X_train)}')
print(f'      Test set size: {len(X_test)}')

  Training set size: 75
Test set size: 22

Now we can divide the train set into folds. We use three folds so that we have 25 observations in each fold. For each fold, we fit 30 models, one for each bend in the line.

from sklearn.model_selection import KFold

kf = KFold(n_splits=3, shuffle=True, random_state=66)

validation_errors = np.zeros((3, 30))
i = 0

for train_idx, valid_idx in kf.split(X_train):

split_X_train, split_X_valid = X_train.iloc[train_idx, :], X_train.iloc[valid_idx, :]
split_Y_train, split_Y_valid = y_train.iloc[train_idx, :], y_train.iloc[valid_idx, :]

bent_models = [LinearRegression().fit(split_X_train.iloc[ :, [0,j]], split_Y_train)
for j in range(1,31)]

bent_preds = [bent_models[j].predict(split_X_valid.iloc[ :, [0,(j+1)]])
for j in range(30)]

error_bend = [mean_squared_error(split_Y_valid, bent_preds[j].flatten())
for j in range(30)]

validation_errors[i][ : ] = error_bend
i = i+1

Then, we combine the validation errors across the 3-folds and plot them against the location of the bend.

totals = [ sum(x)/3 for x in validation_errors.transpose() ]

The MSE looks quite flat for 57, 58, 59, and 60 degrees. The minimizing value occurs at 58 degrees so we choose that particular model. To assess this model with the test set, we first we fit it using the entire train set.

bend_final = LinearRegression().fit(X_train.loc[:, ["temp", "temp58p"]], y_train)

82.68661029815688

Then we use the fitted model to predict gas consumption for the test set.

y_pred_test = bend_final.predict(X_test.loc[:, ["temp", "temp60p"]])

mean_squared_error(y_test, y_pred_test.flatten())

82.68661029815688

Let’s overlay the bent-line fit on the scatter plot and examine the residuals to get a better idea as to the quality of the fit.

The fitted curve looks reasonable, and the size of the residuals is much smaller than those from the polynomial fit.

Note that the scikit-learn library has a cross_val_predict method to automatically perform cross-validation, so we don’t have to break the data into training and validation sets ourselves.

Using cross-validation to manage model complexity has a couple critical limitations: typically it requires the complexity to vary discretely; and there may not be a natural way to order the models. Rather than changing the dimensions of a sequence of models, we can instead fit a large model and apply constraints on the size of the coefficients. This notion is called regularization and is the topic of the next section.


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Introduction Slides

Reference Slides

Adding the corrected pipeline diagram for the question discussed in the video:

posted Jan 27, 2019
edited Jan 28, 2019 | 1,884 views
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video link is not working Sir.
Fixed now

@Arjun Sir, you said in video that in True dependency consider the previous write only. I want to confirm that in Anti dependency and Output dependency can we consider out of order dependency?

R4 <- R1 + R4

R4 <- R2 + R4

R4 <- R4 + R5

there are following dependencies-

1) RAW(True) - (I1-I2) , (I2-I3) we'll not count (I1-I3) so 2.

2) WAR(Anti)- (I1-I2),(I1-I3) and(I2-I3) so 3.

3) WAW(Output) - (I1-I2),(I2-I3) and (I1-I3) so 3.

there is dependency but it is not causing hazards
@Shubh+Gupta Yes. For WAW and WAR the dependency come to play only when instruction reordering happens.

For RAW dependency as per Bernstein's condition I1-I3 is also a dependency. But in none of the standard references nor dependency outputting software that is output and so you should not count it in GATE also.

@Arjun Sir

in the video you told that WAR and WAW hazards cannot happen in an inorder pipeline..But in this link https://people.engr.ncsu.edu/efg/521/s06/common/lectures/notes/lec18.pdf

It is shown that WAR and WAW hazards can happen in an inorder pipeline also..where am I missing out?Can you pls explain Sir?

In the slide it is mentioned rt -- either instruction reordering or newer instruction jumping ahead to a stage before older one -- this won't happen in classical RISC pipeline.
Got it Sir..thanks! :)

@Arjun Sir, if there is no data forwarding, why is the decode stage in I2 before WB in I1? (Same with decode in I4 before WB in I3). Won't the register files get old data in the decode stage? And if so, then how EX stage is getting the new if no forwarding is there?

that is because of split phasing WB writes in first half cycle & ID reads operands in 2nd half of a clock cycle.

@Shubhgupta Okay so we can consider split phase at all times even if its not mentioned right? Thank you!! :)

Can you clear me one more thing.. why there is a RAW hazard b/w I2 and I4? I4 anyway have to wait for I3. So I2 isn't causing any extra stall.

yes we need to always consider the best case. according to all previous gate questions they are using split phase in question if mentioned or not mentioned.

@Shubhgupta

Please check my comment again. I have edited it.

at first I also thought this question but think if I3 is not present in pipeline then I4 needs to wait till I2 writes back so this will cause stall in pipeline in finding of hazard we consider all those dependency which cause stall in pipeline not those which consider extra stall. So at 7th clock both I2 and I3 causing stall in pipeline.

@Arjun sir

Sir why we are not considering $IF$ stage of $I3$ from $3^{rd}$ clock cycle, which might lead to stalls if considered in such a way.

@Arjun sir i have the same doubt

@Mk Utkarsh Because $I_2$ hasn't started it's $ID$ phase yet. Observe that only when $I_2$ has started it's $ID$ phase in $5_{th}$ clock cycle, $I_3$ was able to start it's $IF$ phase during the same cycle.

 CC1 CC2 CC3 CC4 CC5 CC6 CC7 CC8 CC9 CC10 CC11 CC12 I1 IF ID EX M WB I2 IF ID ID ID EX M WB I3 IF IF IF ID EX M WB I4 IF ID ID ID EX MA WB

@pranay562

If we consider table in such a way what is the problem we are facing. Please guide me where I am going wrong. Thanks.

As in CC3 as I1 start in the EX stage surely I2 will jump on ID stage as well as I3 will jump on IF stage. So where I am going wrong.

@Arjun  sir

https://gateoverflow.in/753/gate2001-12

In this question which you have answered, you have taken $I3-I1$ as a RAW hazard. Why ? I think there is a RAW dependency b/w $I3-I1$ but that is not leading to hazard but because of $I2$ stalling it is effecting $I3$ also. And hence there is hazard between $I3-I1$.

But then why we are not considering $I3-I1$ hazard in this question which is similar case as of gate question ?


text:
# Running tests¶

AiiDA-VASP utilizes pytest which makes it easy to run and construct new tests. We also utilize the fixture managers that are present, see the AiiDA documentation.

In order to run all tests execute, in the root folder:

%/\$ pytest

This is a useful test to see if the plugin works and that everything is installed correctly.

For developers, one principle rule: always try to construct tests when submitting a PR.


text:
# How do I rewrite the expression $r+(3/2)s-(1/3)t$

Can someone please explain how to rewrite the expression $r+(3/2)s-(1/3)t$ in terms of $a, b,$ and $c$, where $r= \log (a), s=\log(b) \text{ and } t=2\log(c)$. The rewritten expression should be in the form of $\log (g(a,b,c))$

-
Use $\log(x)+\log(y)=\log(xy)$, $\log(x)-\log(y)=\log(x/y)$, $k\log(x)=\log(x^k)$. –  lhf Aug 11 '11 at 11:46

## 1 Answer

I'm assuming that by "loge" you mean the natural logarithm with base $\mathrm e$, which I will write as "$\log$".

The key here are the relations $\log(xy)=\log x + \log y$ and $\log x^y=y\log x$. They allow you to rewrite as follows:

$$r+\frac32s-\frac13t=\log a+\frac32\log b-\frac132\log c=\log a+\log b^{3/2}+\log c^{-2/3}=\log(ab^{3/2}c^{-2/3})\;.$$

-
Thanks joriki - appreciate your help –  Michelle Aug 11 '11 at 12:00


text:
Test of Mathematics Solution Subjective 77 - Rational Function Inequality

This is a Test of Mathematics Solution Subjective 77 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.

Problem

For ${x > 0}$, show that ${\displaystyle{\frac{x^n - 1}{x - 1}}{\ge}{n{x^{\frac{n - 1}{2}}}}}$, where ${n}$ is a positive integer.

solution

${\displaystyle{\frac{x^n - 1}{x - 1}}{\ge}{n{x^{\frac{n - 1}{2}}}}}$

${\Leftrightarrow}$ ${\displaystyle{\frac{(x - 1)(x^{n - 1} + x^{n - 2} + ......... + x + 1)}{x - 1}}}$ ${> n x^{\frac{n - 1}{2}}}$

${\Leftrightarrow}$ ${\displaystyle{\frac{x^{n - 1} + x^{n - 2} + ......... + x^1 + x^0}{n}}}$ ${> x^{\frac{n - 1}{2}}} (\dagger)$

Now to prove $( \dagger)$ we observe:

But  $\displaystyle{\frac{x^{n - 1} + x^{n - 2} + ......... + x^1 + x^0}{n} \\ > \{x^{n-1}\cdot x^{n-2} \cdots x^0 \}^{\frac{1}{n}} \\ = \{x^{(n-1) + \cdots 0} \}^{\frac{1}{n}} \\ =\{ x^{\frac{n(n-1)}{2}}\}^\frac{1}{n}} \\ = x^{\frac{(n-1)}{2}}$

Now this follows directly from AM-GM inequality.

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
CAREERTEAM
support@cheenta.com


text:
<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
You are reading an older version of this FlexBook® textbook: CK-12 Texas Instruments Algebra I Teacher's Edition Go to the latest version.

# 2.1: Back in Time?

Difficulty Level: At Grade Created by: CK-12

This activity is intended to supplement Algebra I, Chapter 1, Lesson 5.

ID: 11572

Time Required: 15 minutes

## Activity Overview

Students will explore the definition of a function graphically, with a set of ordered pairs, and by using an input-output model with the graphing calculator. This model dynamically allows students to discover the function by experimenting with input values that produce the desired output. Function notation is also reinforced.

Topic: Functions & Relations

• Function definition & notation
• Using ordered pairs, function machines and graphs.

Teacher Preparation and Notes

Associated Materials

## Problem 1 – Graphical

The students are asked to consider a graph of position vs. time. They are asked if this is a function. Since there is not one unique output of $d$ for each value of $t$, it is not a function.

Students then discuss with their neighbor why the answer is ‘no’ based on the definition, and under what circumstance could the graph occur. It could only occur if Marty went back in time.

Students then redraw the graph so that it is a function. An example of a solution is to the right.

Use this opportunity to discuss the concept of the vertical line test.

## Problem 2 – Set of ordered pairs

To review the notation of a set of ordered pairs this section begins with a multiple choice question with multiple solutions. Students should realize that as long as there is only one unique output for each input then it describes a function, otherwise it can only be referred to as a relation.

Another question that can be asked is “What are some examples of relations that are not functions?” Answers include inequalities, like $y > 3x, x$ as a function of $y$ when $y = x^2$, and $x$ as a function of $y$ when $y = |x + 2|$.

Students will calculate the output values for given input values using the formula $d=\frac{1}{2} \cdot a \cdot t^2$ for when $a = 12 \ ft/s^2$. The formula can be simplified to $d=6 \cdot t^2$.

To compute $d$, students can substitute the $t-$values directly into the formula as shown in the first screenshot.

To compute $t$, students need to either solve for $t \left( t=\pm \sqrt{\frac{d}{6}} \right)$ and then substitute the $d-$values into the formula as shown in the second screenshot.

Or they can use the solve command.

The solve command has parameters

solve(expression, variable, guess, {low bound, high bound}).

Note that the low bound and high bound are option, but if used must be placed in braces.

The expression needs to be equal to zero, but low and high bounds do not have to entered.

Press $2^{nd}$ [CATALOG][LN] and scroll down to select solve(.

4. Functions are $A, B$ and $D$ because for every input value there is only one output value.

5. $(0, 0), (1, 6), (2, 24), (6, 216)$

6. $(0, 0), \left( \frac{2}{3}, \frac{1}{3}\right), \left(\frac{2}{3}, - \frac{1}{3}\right), (6, 1), (6, -1)$

7. The first set is a function because it has only one output for every input.

8. A. $d$ is a function of $t$

## Problems 3 – Function notation

In this problem, students will explore using function notation on the calculator entering input values to find the output value.

Students will enter the function next to $Y1$ on the $Y=$ screen. Then to find $f(4)$, they will enter $Y1(4)$ on the Home screen. For Question 10, students will need to update the function next to $Y1$.

9. $f(4) = 4^2 - 2(4) + 3 = 11$

10. $f(2) = 3(2)^2 + 5(2) + 3 = 25$

## Problem 4 – Function Machine

For this problem, students will need the program MACHINE. This program has 3 options, each option is a different function. When students enter an input value, it will return the output value.

Students need to determine which input will give the goal output and then determine the correct function that determines the given output.

11. $17$

12. $f(x) = 0.5x$

13. $-4$

14. $f(x) = x + 10$

15. $20$

16. $f(x) = 4x + 3$

Feb 22, 2012

Oct 31, 2014

# Reviews

Image Detail
Sizes: Medium | Original

TI.MAT.ENG.TE.1.Algebra-I.2.1


text:
User muon - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T22:33:41Z http://mathoverflow.net/feeds/user/21640 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/125196/stability-of-t-mathbbp2-and-omega-mathbbp2 Stability of $T_{\mathbb{P}^2}$ and $\Omega_{\mathbb{P}^2}$? Muon 2013-03-21T19:48:06Z 2013-05-02T22:22:00Z
How can one prove that the tangent bundle $T_{\mathbb{P}^2}$ and its dual $\Omega_{\mathbb{P}^2}$ are stable vector bundles with respect to $\mathcal{O}_{\mathbb{P}^2}(1)$? Similarly, is it true that $T_{\mathbb{P}^n}$ and its dual $\Omega_{\mathbb{P}^n}$ are stable vector bundle for any $n \in \mathbb{N}$? By stable I mean polynomial stability.
 http://mathoverflow.net/questions/128213/is-pi-1-widetildex-g-always-finite-if-pi-1x-is-finite Is $\pi_1(\widetilde{X/G})$ always finite if $\pi_1(X)$ is finite? Muon 2013-04-21T07:46:38Z 2013-04-21T12:55:36Z
Let $X$ be a smooth complex manifold with finite fundamental group. Suppose that a finite group $G$ acts on $X$ and let $\widetilde{X/G}$ be a resolution of singularities. Is $\pi_1(\widetilde{X/G})$ always finite?

I think this is true but don't know a way to prove or a reference.
 http://mathoverflow.net/questions/119139/are-rational-sections-of-a-vector-bundle-useful Are rational sections of a vector bundle useful? Muon 2013-01-17T06:28:39Z 2013-01-17T15:40:12Z
Let $X$ be a complex manifold or variety and $L$ a line bundle on it. Given a rational section $s$ of $L$, we get a divisor $D=Div(s)$ and may recover $L$ as $\mathcal{O}(D)$. What about vector bundles? I think it makes sense to speak about rational section $t$ of a vector bundle $E$, as a section is locally a collection of functions. My question is, is this $t$ useful? For example, can we say anything about Chern classes of $E$ from $t$?
 http://mathoverflow.net/questions/117317/what-do-correlation-functions-compute-in-cft What do correlation functions compute in CFT? Muon 2012-12-27T10:46:16Z 2012-12-28T04:40:47Z
I would like to understand what correlation functions compute in Conformal Field Theory in mathematics. Let me begin with basic definitions. We define a free boson field $\phi(z)$ as a formal power series $$\phi(z)=q+a_0\log(z)-\sum_{n\ne0}\frac{a_n}{n}z^{-n},$$ where $q,a_n$ for $n\in \mathbb{Z}$ are operators satisfying relations $$[a_n,q]=\delta_{n,0}, \ \ \ [a_m,a_n]=m\delta_{m+n,0}.$$ Here these operators acts on the $space$ generated by $vacuum$ vector $|0\rangle$ with properties $$a_n |0\rangle=0\ (n\ge0), \ \ \langle0|q=\langle0|a_n=0\ (n<0),$$ where $\langle0|$ is the dual of the vacuum, i.e. $\langle0|0\rangle=1$. We then define the current $J(z)$ of $\phi(z)$ as a formal derivative $$J(z)=\sum_{n\in \mathbb{Z}}a_nz^{-n-1}.$$

Here is my question. In CFT one is interested in $correlation$ $functions$ such as $$\langle0|J(z_1)J(z_2)|0\rangle, \ \ \langle0|J(z_1)J(z_2)J(z_3)J(z_4)|0\rangle$$ What do they compute in this context?

In physics, correlation function like $\langle0|\phi(x)\phi(y)|0\rangle$ computes the possibility of the field $\phi(x)$ to become $\phi(y)$ etc where $x,y$ are coordinate in for example Minkowski space. How should one interprete the correlation function in our case?
 http://mathoverflow.net/questions/117287/motivation-of-virasoro-algebra Motivation of Virasoro algebra Muon 2012-12-26T23:37:33Z 2012-12-27T16:19:18Z
I have a question on definition/motivation of Virasoro algebra. Recall that Virasoro algebra is an infinite Lie algebra generated by elements $L_n$ $(n\in \mathbb{Z})$ and $c$ over $\mathbb{C}$ with relations $$[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}(m^3-m)\delta_{m+n,0}.$$ A typical explanation of this definition is the following.

Define vector fields $l_n=-z^n\frac{\partial}{\partial z}$ on $\mathbb{C}\setminus {0}$. They form a Lie algebra of infinitesimal conformal transformation $$[l_m,l_n]=(m-n)l_{m+n}.$$ So the Virasoro algebra is a central extension of this algebra by $c$. $c$ is called the central charge.

My questions are

How can one see that the Lie algebra above is associate to infinitesimal conformal transformation?

What is the central charge $c$ intuitively? Why are we interested in such a central extension?

As to second question, I don't have enough physics background to check what the central charge $c$ means in physics literature.

At this point, I don't have any intuition and have trouble in digesting the concept. I would really appreciate your help.
 http://mathoverflow.net/questions/103651/global-dimensions-of-non-commutative-rings Global dimensions of non-commutative rings Muon 2012-08-01T02:33:00Z 2012-11-28T19:59:26Z
This is related to my previous question: When is a quantum affine space $\mathbb{A}^{n}$ Calabi-Yau? I now would like to know the global dimension of the ring $R=\mathbb{C}\langle x_1,\dots,x_n\rangle/I$, where I is the two-sided ideal generated by $x_ix_j=a_{ij}x_jx_i$ for some $a_{ij}=a_{ji}^{-1}\in \mathbb{C}$ for $1≤i,j≤n$. Recall that the global dimension $gl\dim(S)$ of a ring $S$ is defined to be the supremum of the set of projective dimensions of all left $S$-modules.

In case your ring $R$ is defined by a quiver with relations, there seem some techniques. Are there another approach to compute $gl\dim(R)$? Moreover, are there any standard way to compute $gl\dim(S)$ when $S$ is non-commutative?

There are many characterization of $gl\dim(R)$ when $R$ is commutative and I am aware of similar questions, such as Commutative Ring of Finite Global Dimension.
 http://mathoverflow.net/questions/112558/boundedness-of-modules-on-as-regular-algebras Boundedness of modules on AS regular algebras Muon 2012-11-16T07:10:47Z 2012-11-16T07:10:47Z
Let $k$ be an algebraically closed field and $A$ be an Artin-Shelter regular $k$-algebra. Fix a numerical polynomial $H(t)$. I would like to know whether or not semi-stable f.g. graded $A$-modules with Hilbert polynomial $H(t)$ is bounded. More precisely we regard g.f. graded modules $M$ and $N$ as isomorphic is they are isomorphic up to torsion modules. Semi-stability and boundedness are defined in the same manner as commutative case.

The boundedness may not be hold for general AS regular algebras, but then I would like to know for which class of algebra this kind of boundedness result hold. I would appreciate any reference, comments and suggestion.
 http://mathoverflow.net/questions/108183/strongly-noetherian-propoerty-when-is-the-tensor-a-otimes-kb-noetherian-for Strongly Noetherian propoerty. When is the tensor $A\otimes_{k}B$ Noetherian for Noetherian rings $A$ and $B$? Muon 2012-09-26T17:10:04Z 2012-11-12T19:00:32Z
Let $k$ be a field. It is well-known that $A\otimes_{k}B$ is not necessarily Noetherian even if $k$-algebras $A$ and $B$ are Noetherian. For example $\mathbb{R}\otimes_{\mathbb{Q}}\mathbb{R}$.

When is the tensor $A\otimes_{k}B$ Noetherian for Noetherian "commutative" $k$-algebras $A$ and $B$?

What if $A$ is noncommutative? Are there any good criteria for $A\otimes_{k}B$ to be Noetherian? If $B$ is a finitely generated $k$-algebra, Hilbert's basis theorem implies that $A\otimes_{k}B$ is again Noetherian. So we need to check this with quite nasty $B$.

My primary motivation to ask these questions is the second question. Such ring $A$ is called a "strongly Noethrian ring" and has a lot of good properties, but I don't know many examples. Moreover I realized that things are not very clear even in commutative case and I need to understand commutative case first. I would appreciate it if experts on MO could let me know good criteria for this property and provide me with examples.

Rings I have in my mind are weakly noncommutative in the sense that they are commutative up to scalar multiplication such as quantum planes and their $good$ hypersurfaces.
 http://mathoverflow.net/questions/111545/does-castelnuovo-mumford-regularity-hold-for-this-mathbbc-algebra Does Castelnuovo-Mumford regularity hold for this $\mathbb{C}$-algebra$? Muon 2012-11-05T10:48:43Z 2012-11-05T10:48:43Z
Let$R$be a noncommutative finitely generated$\mathbb{C}$-algebra such that its center$S$is smooth (in commutative sense) and$R$is finite over$S$. Is there Castelnuovo-Mumford regularity theorem for this kind of$\mathbb{C}$-algebra$R$? I have $$R=\mathbb{C}\langle x,y,z \rangle/(xy=\zeta_1yx,yz=\zeta_2zy,zx=\zeta_3xz,x^3+y^3+z^3),$$ where$\zeta_i$is a 3rd root of unity. I am aware of CM regularity theorem for AS regular algebra but not sure if it works for this example. Thank you in advance.
 http://mathoverflow.net/questions/111543/balanced-dualizing-complex-vs-rigid-dualizing-complex Balanced dualizing complex vs rigid dualizing complex? Muon 2012-11-05T10:27:57Z 2012-11-05T10:27:57Z
In noncommutative projective geometry, there is a counterpart of dualizing complex in commutative world. It seems to me that they are called either a balanced dualizing complex or rigid dualizing complex. I am aware that they are different objects but cannot really understand how they are different. I would appreciate it if someone kindly explain to me the difference.

P.S. I am reading this paper by M. van den Bergh
 http://mathoverflow.net/questions/103581/diophantine-theory-of-homogeneous-cubic-polynomials Diophantine theory of homogeneous cubic polynomials Muon 2012-07-31T04:33:51Z 2012-10-13T03:29:22Z
Arithmetic of quadratic forms over$\mathbb{Z}$(or lattices theory) has received much attention and there are many applications in broad area of mathematics (such as intersection forms on fourfolds). I now wonder whether or not a similar theory for cubic forms can be developed. I recently found a beautiful theorem about binary cubic forms:

Theorem (B. N. Delone and D. K. Faddeev,W.-T. Gan, B. H. Gross, and G. Savin) There is a canonical bijection between isomorphism classes of cubic rings and the set of$GL_{2}(\mathbb{Z})$-equivalence classes of integral binary cubic forms. Under this bijection, the discriminant of a cubic ring is equal to the discriminant of the corresponding binary cubic form.

I don't know how useful this theorem is because I don't know how difficult to classify cubic rings. My question is, are there any classification theory when the number of variable is small? I would appreciate it if anyone could give me a reference for recent development of cubic forms.
 http://mathoverflow.net/questions/109347/point-modules-of-quantum-projective-space-mathbbpn Point modules of quantum projective space$\mathbb{P}^n$Muon 2012-10-11T01:42:57Z 2012-10-11T14:57:03Z
Let$A$be a quantum$\mathbb{P}^n$defined by $$A=\mathbb{C}\langle x_1,x_2,\dots,x_{n+1}\rangle/(x_ix_j-r_{ij}x_jx_i)_{1\le i < j\le n+1}.$$ I would like to know the set$X$of isomorphism classes of point modules for$A$. Here a point module is a cyclic graded right$A$-module$M$such that each graded piece of$M$is one-dimensional.

The set$Y$of isomorphism classes of of point modules for the quantum$\mathbb{P}^2$$$\mathbb{C}\langle x_1,x_2,x_3\rangle/(x_ix_j-r_{ij}x_jx_i)_{1\le i < j\le 3}$$ is known to be$\mathbb{P}^2$or a union of three lines in$\mathbb{P}^2$.

It seems known that$X$is projective for$n=4$, but does anyone know explicit description of$X$? I tried to compute$X$in a similar manner as the quantum$\mathbb{P}^2$case, but it seems quite complicated. I would also appreciate it if someone give me a good description for higher dimensional$n$case (especially$n=3,4$).

Thank you very much.
 http://mathoverflow.net/questions/109362/complex-manifold-x-with-dim-h0x-omega-dim-mathbbc-x2 Complex manifold$X$with$\dim H^0(X,\Omega^{\dim_{\mathbb{C}} X})>2$Muon 2012-10-11T08:30:50Z 2012-10-11T08:53:41Z
Are there any complex surface or threefold$X$with $$\dim H^0(X,\Omega^{\dim_{\mathbb{C}} X})>2?$$ I am asking this because I don't know any example while there are complex curves of genus greater than one. I guess that there are no such example. If so, could someone kindly explain why? Any counter example is also welcome.

Edit My question turns out to be a silly question. Please ignore this.
 http://mathoverflow.net/questions/108456/how-to-understand-ext-mathcalo-y-mathcalo-z-for-subvarieties-y-z-su How to understand$Ext(\mathcal{O}_{Y},\mathcal{O}_{Z})$for subvarieties$Y,Z\subset X$? Muon 2012-09-30T09:26:51Z 2012-10-01T10:13:27Z
By standard homological algebra we know that$Ext(A,B)$of$R$-modules classifies certain equivalence classes of short exact sequences$0\rightarrow B\rightarrow C \rightarrow A \rightarrow 0$of$R$-modules, where$R$is a commutative ring. I now would like to understand this fact in geometry.

Let$X$be a variety (or a scheme if you want), how should I understand$Ext(O_{Y},O_{Z})$for subvarieties$Y,Z\subset X$? Of course it classifies extensions of$O_{Z}$by$O_{Y}$, but are there any geometric or intuitive way to understand$Ext(O_{Y},O_{Z})$?

More generally are there any geometric way to understand$Ext(\mathcal{E},\mathcal{F})$for coherent$O_X$-modules$\mathcal{E},\mathcal{F}$?

I would appreciate any idea about "seeing" these extensions.
 http://mathoverflow.net/questions/107133/moduli-space-of-modules-over-non-commutative-rings Moduli space of modules over non-commutative rings Muon 2012-09-13T23:39:11Z 2012-09-14T09:10:57Z
Let$X=Proj(A)$be a projective scheme, one can the moduli space of coherent sheaves on$X$with fixed Hilbert polynomial and stability. Since coherent sheaves on$X$are all obtained as the sheafification$\widetilde{M}$of a graded$A$-module$M$, it is reasonable to ask whether we can construct the moduli space of sheaves on non-commutative space, i.e. the moduli space of graded right$B$-modules for some$good$non-commutative graded ring$B$with some fixed data. I am pretty sure that some works in this direction have been done (what condition on$B$and what data of modules should be fixed, etc).

Could anyone tell me a reference or paper which discuss the construction? Thank you very much.

Edit I have for example a quantum plane in my mind.
 http://mathoverflow.net/questions/106798/deformation-of-modules-over-noncommutaitve-rings Deformation of modules over noncommutaitve rings Muon 2012-09-10T09:33:35Z 2012-09-11T22:58:21Z
Let$M$be a finitely generated module over a commutative ring$R$. The first order deformation of module$M$is parametrized by$Ext^{1}(M,M)$and the obstruction is parametrized by$Ext^{2}(M,M)$. Is there a similar story for noncommutative$R$? I don't expect this to be true for any noncommutative rings but wonder if this is still true for some$good$ones. I would appreciate any reference suggestion, comments, and ideas.

Edit I would like naively to compute the tangent space of the moduli space$X$of module with some data if it exists. At$M\in X$, the tangent space can be understood as the set of extensions of the$R$-module$M$to some$R\otimes_{k} k[\epsilon]/(\epsilon^2)$-module. The obstruction is defined in the same manner.
 http://mathoverflow.net/questions/89228/online-number-theory-video Online Number Theory Video? Muon 2012-02-22T21:30:05Z 2012-02-22T21:30:05Z
Are there any graduate level number theory course available on line ? The only video series I am aware of are some MSRI videos, and Ted Chinburg's courses http://www.math.upenn.edu/~ted/noframes.html, which are hard to see sometimes. I am also looking for good video lectures on algebraic k-theory. Thank you for your sharing information :)
 http://mathoverflow.net/questions/128213/is-pi-1-widetildex-g-always-finite-if-pi-1x-is-finite/128220#128220 Comment by Muon Muon 2013-04-22T04:42:06Z 2013-04-22T04:42:06Z Thank you for the nice answer, Francesco. I was a bit surprised to know that my question was not as easy as I had initially expected. http://mathoverflow.net/questions/125196/stability-of-t-mathbbp2-and-omega-mathbbp2/125201#125201 Comment by Muon Muon 2013-03-21T23:20:43Z 2013-03-21T23:20:43Z Thank you for the answer, but could you explain a bit more? I don't know why it is enough to check$H^0(T_{\mathbb{P}^2}(-2))=0$. I am not familiar with this field. http://mathoverflow.net/questions/86583/geometric-treatment-of-the-ward-takahashi-identity Comment by Muon Muon 2013-02-27T01:25:05Z 2013-02-27T01:25:05Z Could you explain what you mean by Ward-Takahashi identity in your question, please? http://mathoverflow.net/questions/121242/lifting-a-birational-map-of-x-g-to-a-birational-map-of-x/121250#121250 Comment by Muon Muon 2013-02-11T00:40:12Z 2013-02-11T00:40:12Z Doesn't your example have fixed locus$f=0$? http://mathoverflow.net/questions/119139/are-rational-sections-of-a-vector-bundle-useful/119177#119177 Comment by Muon Muon 2013-01-19T10:13:06Z 2013-01-19T10:13:06Z Thank ou for the answer. http://mathoverflow.net/questions/117317/what-do-correlation-functions-compute-in-cft Comment by Muon Muon 2012-12-28T17:50:11Z 2012-12-28T17:50:11Z @Kelly What is "moment" of the partition function? http://mathoverflow.net/questions/117317/what-do-correlation-functions-compute-in-cft/117386#117386 Comment by Muon Muon 2012-12-28T07:29:31Z 2012-12-28T07:29:31Z Thank you for the detailed answer providing physic background. I don't fully understand your answer, but now get a feeling about what CFT try to understand. http://mathoverflow.net/questions/117317/what-do-correlation-functions-compute-in-cft/117373#117373 Comment by Muon Muon 2012-12-28T02:54:03Z 2012-12-28T02:54:03Z Thank you for the answer, Igor. I am now happy to know that holomorphicity may be interpreted as a field equation. May I ask what the fields$\phi(z)$and$J(z)$stands for in this context? Is$\langle0| J(z)J(w)|0\rangle$the amplitude to observe a particle at$z$after a particle is created at$w$or something? http://mathoverflow.net/questions/117317/what-do-correlation-functions-compute-in-cft Comment by Muon Muon 2012-12-28T02:48:17Z 2012-12-28T02:48:17Z @Jeff I would rather want to know the physical interpretation. It would be nice if you could provide an answer. By the way, I don't know if there is any "purely mathematical interpretation" of correlation function. http://mathoverflow.net/questions/117317/what-do-correlation-functions-compute-in-cft Comment by Muon Muon 2012-12-28T02:45:40Z 2012-12-28T02:45:40Z @user1540 My question is a bit vague. I would like to know both the complex coordinates and the physical meaning of$J(z)$, and they must be related. My problem is that I don't have any intuition behind what I am computing. http://mathoverflow.net/questions/117287/motivation-of-virasoro-algebra/117336#117336 Comment by Muon Muon 2012-12-27T22:53:46Z 2012-12-27T22:53:46Z Thanks you for letting us know the projective representation point of view. That makes sense. http://mathoverflow.net/questions/117287/motivation-of-virasoro-algebra Comment by Muon Muon 2012-12-27T06:30:24Z 2012-12-27T06:30:24Z I don't quite understand what you try to mean in the second comment. I would appreciate it if you could post a bit more detail as an answer. http://mathoverflow.net/questions/117287/motivation-of-virasoro-algebra/117290#117290 Comment by Muon Muon 2012-12-27T06:29:05Z 2012-12-27T06:29:05Z Thanks for your writing "why$c=D\$" above. http://mathoverflow.net/questions/117287/motivation-of-virasoro-algebra/117290#117290 Comment by Muon Muon 2012-12-27T00:48:59Z 2012-12-27T00:48:59Z Thank you for the answer, Chris. May I ask why multiplication by the unit vector is related to the space dimension? http://mathoverflow.net/questions/117287/motivation-of-virasoro-algebra Comment by Muon Muon 2012-12-27T00:42:42Z 2012-12-27T00:42:42Z @Abdelmalek Thanks for pointing out the typo.


text:
#### (A new question of the week)

Real life questions of probability often require information that we don’t have – they become a job for statistics instead. But sometimes just trying some plausible numbers, as in a Fermi problem, can yield interesting results. Here we consider the probability of an injury when kids play near a baseball field. Then we’ll jump from that to an older question about masks and COVID, using actual data. In both cases, we end up looking at long-term risks.

## How likely is it that a child will be hit?

The first question I want to look at came in late June:

Hi,

We have a play park right behind our baseball field.

The play park is about 16 000 sq ft.

The city wants to close the baseball field because they are afraid a child will get hurt.

What are the possibilities of a softball hitting a child in this area with a home run?

I’d say a child occupy 4 sq ft and a softball might be .10 sq feet?

To visualize this, suppose that the baseball field extends about 300 feet in each direction from home plate at the lower left corner (bases being 90 feet apart), for a total of 90,000 square feet, and the play area measures 300 by a little more than 50 feet. Here is a possible layout:

We don’t know the actual shape or placement of the play area, so this is just conceptual. I didn’t make such model before answering, and in fact envisioned the play area being closer to the size of the ballfield, with only part being near the outfield. What I’ve drawn here is perhaps the worst case.

I answered, skeptical of the whole question:

Hi, Michael.

A question like this really can’t be answered without statistical data, such as how many children are in the park during a typical game, the distribution of locations where home runs will land, and more.

These missing details are very important in reality. If the play area were packed solid with children, and if batters habitually hit home runs over the fence, the situation would be far worse than if there were one kid sitting at the far corner, and children batting who could barely hit into the outfield!

I started out just trying to demonstrate these ideas by making up some totally arbitrary facts that would be somewhat plausible:

For example, suppose that during a game you can expect 10 home runs that would land in the play area, which would be uniformly distributed over half of that area (the other half being too far away). And suppose that there are 20 children uniformly distributed over the entire play area during the game.

Then we could estimate (roughly) that children take up about 20 4-ft2 patches out of 16 000/2 = 8000 ft2, consisting of 8000/4=2000 such patches, on which a ball might land. So the probability that any one ball would land on a child would be 20/2000 = 0.01 = 1%.

The assumption of half the play area being too far away to be hit is not reasonable based on my drawing, but seemed so at the time; and I mostly wanted to illustrate the idea of a (non-uniform) distribution. The suggested size of a child is probably too large based on a top view, but perhaps too small considering the ball would be coming in at an angle. My calculations suggest that children might take up about 1/2000 = 0.05% of the area the ball might hit, so that each ball that reaches the play area has that much chance of hitting one. With each child on a different spot, the probability of the ball hitting someone is $$20\times0.05\%=1\%$$. (It was entirely by chance that my randomly chosen numbers worked out so nicely, but that motivated me not to adjust them!)

Out of 10 such hits, the probability that none would hit a child would be 0.99^10 ≈ 0.90 = 90%. So the probability of a child being hit during a given game would be 10%. If there were 100 games during a season, the probability of no child being hit becomes 0.90^100 = 0.000043, and the probability of at least one being hit would be 0.999957 – almost certain.

That 1% sounds like a small probability; but for something you really, really don’t want to happen, you need to take into account how many opportunities there are. My proposals of 10 long hits per game, and 100 games per season, are probably too high, but … it made a point. (With only 5 hits into the play area per game and 50 games per season, the probability becomes $$1-0.99^5=0.049=4.9%$$ per game, and $$1-0.951^{50}=0.919=91.9%$$ for the season, which is still significant.)

I didn’t do this as a serious answer to your question, because my assumptions are intended only as wild guesses, and probably contrary to fact; but it interestingly demonstrates that the probability can increase considerably as the number of opportunities increases!

I was expecting a small chance, but instead I think I’ve shown why they would want to be cautious. Not allowing games when children are present, or not allowing children during games, might be very reasonable, IF there are many hits into a large enough part of the play area. Those are the sort of statistics you’d need to gather. Has a hit ever gone into that area in the past? How often?

Questions of safety can be tricky. The real probabilities may well be too low to be a major worry, but we’ve seen enough to want to be cautious, and gather the additional data – or just make small changes to the rules.

## How risky is a loose mask?

That question about risk reminds me of a question from last December, from Laura in Germany:

I am wondering about the probability of getting infected with COVID when you wear a mask. A recent study said that if you wear a good mask but it’s not tight around your nose, the probability of getting infected if you are with an infected person for 20 minutes is 4%. (There are more details but that would make this too long). That seems like a small risk to me. But my husband said I have to add the risk for every time we meet people. So if we go shopping once a week it would be 4% times 52 weeks after a year. A very big risk. That seems crazy to me. But he’s a scientist so he’s probably right. Can you clarify?

This gives us another chance to do the same kind of calculation we did above, and to dig in a little deeper: How do you find the long-term risk when you know the risk for each exposure? How much is it affected by changes in parameters?

Hi, Laura.

The details you are ignoring may change the ultimate conclusions considerably; but I’ll just look at your specific question.

We will assume that each week you spend enough time close enough to one infected person that the probability that you will catch the infection on that occasion is 4%.

Your husband is wrong about the specific calculation to determine the probability of being infected in one of 52 such incidents; you can’t just add probabilities. (In particular 52*4% = 208%, which is impossible as a probability! The highest a probability can be is 100%, which is 1.) But the probabilities do “add up”, figuratively.

The calculation he suggested was a rough first estimate, which we can fix:

Here’s the correct calculation (again, assuming the data are correct):

The probability that you will get infected at least once is 1 minus the probability that you will not be infected on any of the occasions.

The probability that you will not be infected on any one occasion is 1 – 0.04 = 0.96 (that is, 96%). The probability that this will happen 52 times in a row is (0.96)52 = 0.1197.

So the probability that you will be infected at least once in the year is 1 – 0.1197 = 0.88 (that is, 88%).

Let’s turn this into a general formula. Suppose that the probability of “success” (which in our case means getting sick!) at any one “trial” is $$p$$. Then the probability of “failure” is $$1-p$$; and if the trial is repeated $$n$$ times, the probability of failure every time is $$(1-p)^n$$. The probability of at least one success is the complement of no success, namely $$1-(1-p)^n$$. Here is a graph of this in our case, with $$p=0.04$$, as $$n$$ increases up to 104 (two years of weekly events):

The blue line shows what would happen if we just multiplied by $$n$$, using the naïve guess about the effect of repeated exposure. Reality looks quite different, but still rises inexorably.

So, yes, the probability increases with each exposure; and it does eventually get quite large. If you had provided the article, I might be able to check out other mathematical issues in it, but I’m sure we both know there are a lot of complexities to be kept in mind. In particular, I doubt that you spend 20 minutes that close to any one person at the store. But that’s not my area of expertise.

We’ll see the article in a moment.

### Reality

Laura replied:

Thanks so much. I can send a link to the study if you are interested but it’s in German (at least the link I have). I agree that in reality the risk would be much smaller. In the study itself, they say that under real conditions the risk would be 10 to 100 times smaller!

I was interested in the risk calculation in theory. I didn’t realize one needed to add them at all. That applies to all areas of life, then. Driving, for example. Kind of scary.

So the study’s numbers are very conservative, even given the assumption of spending 20 minutes with a contagious person. We should be concerned, but not terrified (as long as we are masked!). And, yes, that’s true for all of life. Everything we do has some risk, and if we do anything often enough, the risk adds up. We have to make decisions based on how serious the consequences are (for ourselves and others), and avoid doing risky things too often.

Yes, risks “add up” when you do anything enough times. Though it isn’t literal addition, it does grow.

On the other hand, the same can be said of benefits!

Just to take this one step further, if the risk was really 0.4% per occasion, 1/10 as much, then after 52 times, the total risk would be 1 – (1 – 0.004)52 = 0.188 = 18.8%.

It could be interesting to see the study, though there are enough of them around. If you feel like it, send me the link and I’ll at least let Google translate it for me (though I did take German in college many years ago, and so did my daughter).

Here is the graph for this 0.4% risk for a poorly fitting mask:

At such a small probability, the increase is much closer to linear (that is, to literally adding up); after two years, it is still less than 35%. This illustrates how modifying the assumptions can affect our results.

Now think about what I said about the benefits also adding up. Since wearing a good mask reduces the probability of getting sick on one occasion considerably, it is even more valuable over multiple exposures, and more so when more people wear them. So is vaccination. Every measure you take helps. In particular, I’m reminded of the Swiss Cheese Model.

Laura wrote back:

Hi Doctor Peterson. I figure I can just summarize the study in a few sentences. (By the way: I live in Germany which is why I am reading studies in German). This one is from the Max Planck Institute (Eberhard Bodenschatz). They say that if you wear a FFP2 or KN95 mask (I don’t know if you call them that in the US), and you have an infected person and a healthy but unvaccinated person in close proximity indoors, the risk of the healthy person getting infected after 20 minutes is 0,1% — if the mask is worn properly — ie if the metal thing is pushed tightly against the nose. If the mask isn’t worn properly, the risk rises to 4%. If you wear the normal surgical mask (the blue ones — at least here they’re blue), the risk is 10%. And if you wear no mask at all, the risk is 90% after only a few minutes, even at 3 meters distance. The researchers stress, however, that the estimates are very conservative. In normal life situations, the risk is “surely” 10 to 100 times less.

It’s worth noting that while our original question was about why a 4% risk is not as low as it seems, the point of the article is that even a poorly worn mask is better than none, and a well-fitted mask is far more effective than the researchers expected.

Thinking about your answer about adding up risks (I know: not literally adding up), it makes perfect sense. If I drive drunk once, I might be OK but if I drive drunk often — bad idea. I just hadn’t really thought about it. We talk a lot about the fact that normal people are so unaware of statistics (I don’t think it was taught in school at my time) and how important it is. I will keep trying to inform myself!

Apart from the mathematical details, what we’ve been discussing about repeated risks should be common sense; but it is easy to overlook!

I replied:

Thanks. A quick search turned up this article that appears to be an English language version of the same report:

I work in a tutoring center at my school, which requires everyone to be masked, and we sit (mostly) on opposite sides of wide tables. But many people wear surgical type masks very loosely, and I often wonder what good it is to require masks at all. But the article does say, “Surgical masks already reduce the amount significantly, even if they fit poorly.”

### The actual article

Here are some quotes from the article:

The Göttingen study confirms that FFP2 or KN95 masks are particularly effective in filtering infectious particles from the air breathed – especially if they are as tightly sealed as possible at the face. If both the infected and the non-infected person wear well-fitting FFP2 masks, the maximum risk of infection after 20 minutes is hardly more than one per thousand, even at the shortest distance. If their masks fit poorly, the probability of infection increases to about four percent. If both wear well-fitting medical masks, the virus is likely to be transmitted within 20 minutes with a maximum probability of ten percent.

Here is our graph for all four cases (using their conservative numbers):

As before, the horizontal axis is the number of 20-minute interactions, and the vertical is the probability of at least one infection. A good fit makes a big difference.

The infection probabilities determined by the Max Planck team indicate the upper limit of the risk in each case. “In daily life, the actual probability of infection is certainly 10 to 100 times smaller,” says Eberhard Bodenschatz. This is because the air that flows out of the mask at the edges is diluted, so you don’t get all the unfiltered breathing air. But we assumed this because we can’t measure for all situations how much breathing air from one mask wearer reaches another person, and because we wanted to calculate the risk as conservatively as possible,” Bodenschatz explains. “Under these conditions, if even the largest theoretical risk is small, then you’re on the very safe side under real conditions.”

Here is our graph for all four cases assuming the probabilities are 1/10 of their worst case:

Even here, without a mask, a year of weekly interactions lead to almost certain infection; a well-fitting KN95 gives impressive protection.

Although the detailed analysis by the Max Planck researchers in Göttingen shows that tight-fitting FFP2 masks provide 75 times better protection compared to well-fitting surgical masks and that the way a mask is worn makes a huge difference;  even medical masks significantly reduce the risk of infection compared to a situation without any mouth-nose protection at all.

Laura responded:

Cool. And thanks for the English link. I can think of a few people who might be interested in it. I know plenty of teachers, for example. (Although: maybe they would be annoyed — hmmm… have to think about that one)

The fact that masks worn tightly is so efficient is good news to me. That means that in highly uncertain situations, I can wear one and feel quite safe. And the fact that loosely worn masks help somewhat is good news too. It means I can feel fairly safe then too. Like just running into the drugstore quickly.

The COVID era is teaching lots of us about math and statistics. Exponential growth is a good one. For most of us that was just a theoretical idea until now. Suddenly it is really important to understand.

Anyway — I’m glad I discovered this site. I’m pretty sure my husband knew you don’t add up things literally but we were in an argument so he probably fudged a bit to try to convince me. Ah, life!

Until my next statistical emergency!

Laura

There is certainly far more that could be said in practical terms, given more data; but our goal here is just to think about how short-term risks become long-term risks, not to decide policies or judge the accuracy of a study. As I said two years ago, our goal is to illustrate math, not to analyze reality.

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text:
# How to move the handles of a sequence via scripting?

## There are the following commands:

### create sequence editor

bpy.context.scene.sequence_editor_create()

### add strip (name, filepath, channel, frame_start)

bpy.context.scene.sequence_editor.sequences.new_image("coolName", "path/db.png", 4, 1)

Until now I have created a sequence on channel 4, starting at frame 1 like in the following picture:

I want to control the starting frame of the slider and the ending frame of this strip. How can I achieve this using blender script?

seq = bpy.context.scene.sequence_editor.sequences.new_image("coolName", "somePath/img.PNG", 4, 1)

Then you can edit the frame_final_start and frame_final_end to edit the sequence start and finish, same as the handles:
seq.frame_final_start = 10


text:
# Formula for finding number of colors to color a map such that no two adjacent counties have the same color

Is there a formula for determining the minimum number of colors to color a map with n counties such that no adjacent county have the same color?

• I suppose instead of "formula" you mean an "algorithm"? – dkaeae Nov 30 at 8:56

No.

• The famous four-colour theorem says that four colours are always enough (assuming that counties are contiguous areas of land).

• It's easy to tell if one colour is enough: this is the case only if every county is an island, so has no border with any other county.

• It's easy to tell if two colours is enough: this corresponds to the adjacency graph being bipartite. You can determine this by colouring an arbitrary county red, then colouring all its neighbours blue, then colouring all their neighbours red, and so on. If this process produces a legal colouring, you have your colouring; if it fails, it turns out that there is no colouring with two colours.

• If two colours isn't enough, then it's NP-complete to tell whether three is enough or you have to use four. This means that, as far as we know, there's no efficent algorithm to determine whether an arbitrary map can be coloured with three colours. This is a result due to Garey, Johnson and Stockmeyer (Some simplified NP-complete graph problems, Theoretical Computer Science, 1(3):237--267, 1976; Science Direct).

As I recall, it's also hard to find a four-colouring of a given map (unless there's a one- or two-colouring).

• In your last paragraph, did you mean "a three-colouring of a given map"? It's indeed true as well, i.e., planar 3-coloring is NPC. – Juho Nov 30 at 16:35

In this case the problem is NP-complete. By the way we know that:
$$\text{Given a graph }G\text{, let }X_G \text{ be the number of colors of G and } D_G\text{ the maximum number of adjacents in G,}$$we have that $$X_G\le D_G+1$$