text:
The Faraday paradox (or Faraday's paradox) is an experiment that illustrates Michael Faraday's law of electromagnetic induction. Faraday deduced this law in 1831, after inventing the first electromagnetic generator or dynamo, but was never satisfied with his own explanation of the paradox.

The equipment

The experiment requires a few simple components (see Figure 1): a cylindrical magnet, a conducting disc with a conducting rim, a conducting axle, some wiring, and a galvanometer. The disc and the magnet are fitted a short distance apart on the axle, on which they are free to rotate about their own axes of symmetry. An electrical circuit is formed by connecting sliding contacts: one to the axle of the disc, the other to its rim. A galvanometer can be inserted in the circuit to measure the current.

The procedure

The experiment proceeds in three steps. First, the magnet is held to prevent it from rotating, while the disc is spun on its axis. The result is that the galvanometer registers a direct current. The apparatus therefore acts as a generator, variously called the Faraday generator, the Faraday disc, or the homopolar (or unipolar) generator.

In the second step, the disc is held stationary while the magnet is spun on its axis. The result is that the galvanometer registers no current.

In the third step, the disc and magnet are spun together. The galvanometer registers a current, as it did in step 1.

The experiment is described by some as a "paradox" as it seems, at first sight, to violate Faraday's law of electromagnetic induction, because the flux through the disc appears to be the same no matter what is rotating. Hence, the EMF is predicted to be zero in all three cases of rotation. The discussion below shows this viewpoint stems from an incorrect choice of surface over which to calculate the flux.

The paradox appears a bit different from the lines of flux viewpoint: in Faraday's model of electromagnetic induction, a magnetic field consisted of imaginary lines of magnetic flux, similar to the lines that appear when iron filings are sprinkled on paper and held near a magnet. The EMF is proposed to be proportional to the rate of cutting lines of flux. If the lines of flux are imagined to originate in the magnet, then they would be stationary in the frame of the magnet, and rotating the disc relative to the magnet, whether by rotating the magnet or the disc, should produce an EMF, but rotating both of them together should not.

In Faraday's model of electromagnetic induction, a circuit received an induced current when it cut lines of magnetic flux. According to this model, the Faraday disc should have worked when either the disc or the magnet was rotated, but not both. Faraday attempted to explain the disagreement with observation by assuming that the magnet's field, complete with its lines of flux, remained stationary as the magnet rotated (a completely accurate picture, but maybe not intuitive in the lines-of-flux model). In other words, the lines of flux have their own frame of reference. As we shall see in the next section, modern physics (since the discovery of the electron) does not need the lines-of-flux picture and dispels the paradox.

Modern explanations

Using the Lorentz force

After the discovery of the electron and the forces that affect it, a microscopic resolution of the paradox became possible. See Figure 1. The metal portions of the apparatus are conducting, and confine a current due to electronic motion to within the metal boundaries. All electrons that move in a magnetic field experience a Lorentz force of F = v × B, where "v" is the velocity of the electrons. This force is perpendicular to both the velocity of the electrons, which is in the plane of the disc, and to the magnetic field, which is normal to the disc. An electron at rest in the frame of the disc moves circularly with the disc relative to the "B"-field, and so experiences a radial Lorentz force. In Figure 1 this force (on a "positive" charge, not an electron) is outward toward the rim according to the right-hand rule.

Of course, this radial force, which is the cause of the current, creates a radial component of electron velocity, generating in turn its own Lorentz force component that opposes the circular motion of the electrons. This reaction is a form of Le Chatelier's principle, tending to slow the disc's rotation, but the electrons retain a component of circular motion that continues to drive the current via the radial Lorentz force.

This mechanism agrees with the observations: an EMF is generated whenever the disc moves relative to the magnetic field, regardless of how that field is generated.

Relation to Faraday's law of induction

The flux through the portion of the path from the brush at the rim, through the outside loop and the axle to the center of the disc is always zero because the magnetic field is in the plane of this path (not perpendicular to it), no matter what is rotating, so the integrated emf around this part of the path is always zero. Therefore, attention is focused on the portion of the path from the axle across the disc to the brush at the rim.

Faraday's law of induction can be stated in words as: [See, for example,cite book
title=Elements of Electromagnetics
year= 2007
page =§9.2 pp. 386 ff
publisher=Oxford University Press
edition=Fourth Edition
location=NY/Oxford UK
isbn=0-19-530048-3
] Quotation|The induced electromotive force or EMF in any closed circuit is equal to the time rate of change of the magnetic flux through the circuit. Mathematically, the law is stated:

:

where ΦB is the flux, and "d" A is a vector element of area of a moving surface Σ("t") bounded by the loop around which the EMF is to be found.

How can this law be connected to the Faraday disc generator, where the flux linkage appears to be just the "B"-field multiplied by the area of the disc?

One approach is to define the notion of "rate of change of flux linkage" by drawing a hypothetical line across the disc from the brush to the axle and asking how much flux linkage is swept past this line per unit time. See Figure 2. Assuming a radius "R" for the disc, a sector of disc with central angle θ has an area:

:$A = frac \left\{ heta\right\}\left\{2pi\right\} pi R^2 ,$

so the rate that flux sweeps past the imaginary line is

:$mathcal\left\{E\right\} = - frac \left\{d Phi_B\right\} \left\{dt\right\} = B frac\left\{dA\right\} \left\{dt\right\} = B frac \left\{R^2\right\}\left\{2\right\} frac \left\{d heta\right\}\left\{dt\right\} =B frac \left\{R^2\right\}\left\{2\right\}omega ,$

with ω = "d" θ / "dt" the angular rate of rotation. The sign is chosen based upon Lenz's law: the field generated by the motion must oppose the change in flux caused by the rotation. [ For example, the circuit with the radial segment in Figure 2 according to the right-hand rule "adds" to the applied B-field, tending to increase the flux linkage. That suggests that the flux through this path is decreasing due to the rotation, so "d" θ / "d t " is "negative.]

This flux-cutting result for EMF can be compared to calculating the work done per unit charge making an infinitesimal test charge traverse the hypothetical line using the Lorentz force / unit charge at radius "r", namely |v × B | = "B v = B r" ω:

:$mathcal\left\{E\right\} = int_0^R dr Br omega = frac \left\{R^2\right\}\left\{2\right\} B omega ,$

which is the same result.

The above methodology for finding the flux cut by the circuit is formalized in the flux law by properly treating the time derivative of the bounding surface Σ ( "t" ). Of course, the time derivative of an integral with time dependent limits is "not" simply the time derivative of the integrand alone, a point often forgotten; see Leibniz integral rule and Lorentz force.

In choosing the surface Σ ( "t" ), the restrictions are that (i) it be bounded by a closed curve around which the EMF is to be found, and (ii) it capture the relative motion of all moving parts of the circuit. It is emphatically "not" required that the bounding curve correspond to a physical line of flow of the current. On the other hand, induction is all about relative motion, and the path emphatically "must" capture any relative motion. In a case like Figure 1 where a portion of the current path is distributed over a region in space, the EMF driving the current can be found using a variety of paths. Figure 2 shows two possibilities. All paths include the obvious return loop, but in the disc two paths are shown: one is a geometrically simple path, the other a tortuous one. We are free to choose whatever path we like, but a portion of any acceptable path is "fixed in the disc itself" and turns with the disc. The flux is calculated though the entire path, return loop "plus" disc segment, and its rate-of change found.

In this example, all these paths lead to the same rate of change of flux, and hence the same EMF. To provide some intuition about this path independence, in Figure 3 the Faraday disc is unwrapped onto a strip, making it resemble a sliding rectangle problem. In the sliding rectangle case, it becomes obvious that the pattern of current flow inside the rectangle is time-independent and therefore irrelevant to the rate of change of flux linking the circuit. There is no need to consider exactly how the current traverses the rectangle (or the disc). Any choice of path connecting the top and bottom of the rectangle (axle- to-brush in the disc) and moving with the rectangle (rotating with the disc) sweeps out the same rate-of-change of flux, and predicts the same EMF. For the disc, this rate-of-change of flux estimation is the same as that done above based upon rotation of the disc past a line joining the brush to the axle.

ome observations

Whether the magnet is "moving" is irrelevant in this analysis, as it does not appear in Faraday's law. In fact, rotating the magnet does not alter the "B"-field. Likewise, rotation of the magnet "and" the disc is the same as rotating the disc and keeping the magnet stationary. The crucial relative motion is that of the disk and the return path, not of the disk and the magnet.

This becomes clearer if a modified Faraday disk is used in which the return path is not a wire but another disk. That is, mount two conducting disks just next to each other on the same axle and let them have sliding electrical contact at the center and at the circumference. The current will be proportional to the relative rotation of the two disks and independent of any rotation of the magnet.

Configuration without a return path

A Faraday disk can also be operated with neither a galvanometer nor a return path. When the disk spins, the electrons collect along the rim and leave a deficit near the axis (or the other way around). It is possible in principle to measure the distribution of charge, for example, through the electromotive force generated between the rim and the axle (though not necessarily easy). This charge separation will be proportional to the magnetic field and the rotational velocity of the disk. The magnetic field will be independent of any rotation of the magnet. In this configuration, the polarisation is determined by the absolute rotation of the disk, that is, the rotation relative to an inertial frame. The relative rotation of the disk and the magnet plays no role.

Figure 4 shows a translating rectangle of material with a narrow conducting strip subject to a magnetic field. This strip of material is rendered conducting at a fixed location by, for example, a strong light beam at this location. The magnetic field also is confined to the same strip. The Lorentz force drives a current from the top rail to the bottom rail through this strip, and the circuit is completed through leads attached to the top and bottom conducting rails. In this example, the circuit does not move, and the magnetic flux through the circuit is not changing, so Faraday's law suggests no current flows. However, the Lorentz force law suggests a current does flow. This example is based upon one devised by Richard Feynman to illustrate the inapplicability of Faraday's law of induction to certain situations (that is, the version of Faraday's law of induction which relates EMF to magnetic flux, which he terms the "flux rule"). Referring to his example, Feynman said:cite book
author=Richard Phillips Feynman, Leighton R B & Sands M L
title=The Feynman Lectures on Physics
year= 2006
page =Vol. II, pp. 17-2, 17-3
location=San Francisco
isbn=0805390499
] Quotation |The "flux rule" does not work in this case. It must be applied to circuits in which the "material" of the circuit remains the same. When the material of the circuit is changing, we must return to the basic laws. The "correct" physics is always given by the two basic laws

:::$mathbf\left\{F\right\}=q\left(mathbf\left\{E\right\}+mathbf\left\{v\right\} imesmathbf\left\{B\right\}\right)$

:::|Richard P Feynman  "The Feynman Lectures on Physics"

Accordingly, he explains the phenomenon using the Lorentz force law, as described above. The point is that the flux law applies only to some situations, albeit some very practical ones.

References and notes

* [http://www.brera.unimi.it/old/Atti-Cosenza-2001/Giuliani.pdf "Electromagnetic induction: physics and flashbacks" (PDF)] by Giuseppe Giuliani - details of the Lorentz force in Faraday's disc
* [http://www.spots.ab.ca/~belfroy/Homopolars/homopoltext.html "Homopolar Electric Dynamo"] - contains derivation of equation for EMF of a Faraday disc
* [http://www.tinaja.com/glib/muse121.pdf Don Lancaster's "Tech Musings" column, Feb 1998] - on practical inefficiencies of Faraday disc
* [http://www.marmet.ca/louis/induction_faraday/kelly/KellyFa3.pdf "Faraday's Final Riddle; Does the Field Rotate with a Magnet?" (PDF)] - contrarian theory, but contains useful references to Faraday's experiments
*P. J. Scanlon, R. N. Henriksen, and J. R. Allen, "Approaches to electromagnetic induction," Am. J. Phys. 37, 698–708 (1969). - describes how to apply Faraday's law to Faraday's disc
*Jorge Guala-Valverde, Pedro Mazzoni, Ricardo Achilles "The homopolar motor: A true relativistic engine," Am. J. Phys. 70 (10), 1052–1055 (Oct. 2002). - argues that only the Lorentz force can explain Faraday's disc and describes some experimental evidence for this
*Frank Munley, Challenges to Faraday's flux rule, Am. J. Phys. 72, 1478 (2004). - an updated discussion of concepts in the Scanlon reference above.
*Richard Feynman, Robert Leighton, Matthew Sands, "The Feynman Lectures on Physics Volume II", Chapter 17 - In addition to the Faraday "paradox" (where linked flux does not change but an emf is induced), he describes the "rocking plates" experiment where linked flux changes but no emf is induced. He shows that the correct physics is always given by the combination of the Lorentz force with the Maxwell-Faraday equation (see quotation box) and poses these two "paradoxes" of his own.
* [http://www.geocities.com/terella1/ The rotation of magnetic field] by Vanja Janezic - describes a simple experiment that anyone can do. Because it only involves two bodies, its result is less ambiguous than the three-body Faraday, Kelly and Guala-Valverde experiments.

ee also

*Moving magnet and conductor problem
*Lorentz force

Wikimedia Foundation. 2010.

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text:
# Prove that $A\setminus (B\setminus C) = (A\setminus B) \cup (A\setminus C^c)$ for sets $A,\ B,\ C$ in some Universal Set $U$.

I'm working on this proof for some students I am tutoring and I've gotten a little stuck. I want to show them how to do a proof in complete, extravagant detail and get them familiar with ''element chasing'' in set proofs for an intro to discrete class. I just got stuck and maybe I'm just too tired to see it. Here's where I am so far.

Notation: $A^c$ is the complement of $A$. $A\backslash B$ means $A \cap B^c$.

Prove that $A\setminus (B\setminus C) = (A\setminus B) \cup (A\setminus C^c)$ for sets $A,\ B,\ C$ in some Universal Set $U$.

Proof: ($\Leftarrow$)

Let $x \in (A \backslash B) \cup (A\backslash C^c)$.

This means that $x\in (A \cap B^c) \cup (A \cap C^{c^c})$.

Simplifying, this means that $x \in (A \cap B^c) \cup (A \cap C)$.

So $x\in (A \cap B^c)$ or $x\in (A\cap C)$.

Now we have two cases.

Case 1: $x \in A \cap B^c.$

So $x \in A$ and $x\in B^c$.

Note: Now we must think outside the box a little bit. We want to show that $x$ must be in $C$ but we don't have anything to do with $C$ in what we are able to derive from our assumptions. We only have that $x$ is in $A$ and not in $B$. This is where we must consider our problem. We have the Universe $U$. We know that $A,B,C$ are all in the universe. So any point we chose in those sets must also be in $U$. So think about this for a moment. We don't know if our $x$ is in $C$ or not. We need it to be in order to have the solution we are after. So where is $x$ in relation to $C$? It's either in it or not in it. So $x \in C$ or $x\in C^c$. So we can examine each of these cases and we will find that if $x \in C^c$, we will get a contradiction.

Case 1.a. Suppose $x \in C$.

Then $x \in A$ and $x \in B^c$ and $x \in C$.

So $x \in A$ and $x\in (B^c \cap C)$.

So $x \in A \cap (B^c \cap C)$.

So $x \in A \cap (B \cap C^c)^c$.

So $x \in A \cap (B\backslash C)^c$.

So $x \in A \backslash (B \backslash C)$.

Case 1.b. Suppose $x \in C^c$.

Then $x \in A$ and $x\in B^c$ and $x\in C^c$.

So $x \in A \cap C^c$ and $x \in A\cap B^c$.

Now I'm stuck. I know that I need to develop a contradiction because $x$ cannot be in $C^c$, but I'm just not seeing it. Any suggestions? If I can see this one, I'll be able to see the similar method I need to develop for case 2: $x\in A \cap C$ where I need to examine whether $x \in B$ or $B^c$.

In case 1: $$x\in A\cap B^c\\x\in A,x\in B^c\\x\in A,x\not\in B\\x\in A,x\not\in B\cap C^c\\x\in A,x\not\in B\backslash C\\x\in A,x\in(B\backslash C)^c\\x\in A\cap(B\backslash C)^c\\x\in A\backslash(B\backslash C)$$

You don't need subcases (in case 1) for $x\in C, x\not\in C$.

(Incidentally, case 2 is as simple: there, you don't need subcases for $x\in B,x\not\in B$.)

• I understand that. And there's a much simpler version of this just unpacking definitions. But I'm really trying to get them to see how to use contradictions and to really examine everything. It's for a class of computer science majors. This proof is more about incorporating all the elements of proof writing (except induction) and making sure they know how to use them. – Tyler Murphy Feb 20 '14 at 7:33
• I don't see that the steps in this answer are different from the types of steps you took. Would you prefer them written in order? $x\in A\cap B^c$; $x\in A,x\in B^c$; $x\in A,x\not\in B$; $x\in A,x\not\in B\cap C^c$; $x\in A,x\not\in B\backslash C$; $x\in A,x\in(B\backslash C)^c$; $x\in A\cap(B\backslash C)^c$; $x\in A\backslash(B\backslash C)$. – msh210 Feb 20 '14 at 7:45
• @TylerMurphy, I've edited. – msh210 Feb 20 '14 at 7:51
• Thanks. It makes more sense now that I see the error I made that the other answer pointed out. I should have just slept on it. Thank you very much. – Tyler Murphy Feb 20 '14 at 8:00

There is yet another solution, namely you can rewrite $$A\setminus (B\setminus C) = (A\setminus B) \cup (A\setminus C^c)$$ using $$X - Y = X \cap Y^c$$ into $$A\cap \color{blue}{(B\cap C^c)^c} = (A\cap \color{red}{B^c}) \color{red}{\cup} (A\cap \color{red}{C}).$$ Now it is enough to use De Morgan law for blue part of LHS and the distributive law for red part of RHS: $$A\cap \color{blue}{(B^c\cup C)} = A \cap (\color{red}{B^c \cup C}).$$

I hope this helps $\ddot\smile$

• This solution infinitely better than all the others. – goblin Feb 20 '14 at 10:07
• @user18921 Thank you $\ddot\smile$ – dtldarek Feb 20 '14 at 10:12

In your proof, there exists an error. Note the following fact :

$x \in A \cap (B^c \cup C)$ iff

$x \in A \cap (B \cap C^c)^c$.

• I knew I needed a fresh pair of eyes. That fixed everything. Thank you. – Tyler Murphy Feb 20 '14 at 7:54

it might be enlightening for your students to do this 'algebraically':

If you go to the indicator functions and the set $\mathbf{2}^U=\mathbb{F}_2^U$ then

The symmetric difference of $a$ and $b$ is a+b

The complement of $a$ is $1+a$

The intersection of $a$ and $b$ is $ab$

The union of $a$ and $b$ is $U(a,b)=a+b+ab$

$a$ set-minus $b$ is $a(1+b)$

So the LHS $$a(1+(b(1+c))=a+ab(1+c)=a+ab+abc$$ and the RHS $$U(a(1+b),a(1+1+c))=U(a+ab,ac)=a+ab+ac+abc+aac=a+ab+abc$$ (note that $x^2=x$ and $x+x=0$)

Here is how I would present this proof.

The approach I learned is to start with the most complex side, calculate the elements of that set using the definitions, and then simplify, and finally work towards the other side of the equality.

In other words, for all $\;x\;$, \begin{align} & x \in (A \setminus B) \cup (A \setminus C^c) \\ \equiv & \qquad \text{"definition of $\;\cup\;$"} \\ & x \in A \setminus B \;\lor\; x \in A \setminus C^c \\ \equiv & \qquad \text{"definition of $\;\setminus\;$, twice; definition of $\;^c\;$"} \\ & (x \in A \land \lnot(x \in B)) \;\lor\; (x \in A \land \lnot(\lnot(x \in C))) \\ \equiv & \qquad \text{"logic, simplify: double negation, factor out common conjunct"} \\ & x \in A \land (\lnot(x \in B) \lor x \in C) \\ \equiv & \qquad \text{"logic: DeMorgan -- to work towards $\;x \in A \land \lnot \ldots\;$, suggested by goal"} \\ & x \in A \land \lnot (x \in B \land \lnot(x \in C)) \\ \equiv & \qquad \text{"definition of $\;\setminus\;$, twice"} \\ & x \in A \setminus (B \setminus C) \\ \end{align} This proves the equality by set extensionality.

The only things needed for this type of proof are knowledge of the definitions, and of the laws of (propositional/predicate) logic.

format: plain


text:
How Do You Find an Antiderivative Using the Substitution Method?

In Monday’s class, student found many antiderivatives using the Substitution Method. The basic process is illustrated below for the antiderivative

$latex \displaystyle \int{4{{\left( {{x}^{2}}-3 \right)}^{3}}\cdot 2x,dx}$.

Let’s look at the basic steps.

1. Choose the expression for u. This is generally the inside part of a composition in the integrand. Use the derivative to find an other expression for du.
2. Match the integrand with u and du. All variables in the original integrand must change to u.
3. Change the integrand so that it is written in terms of u.
4. Work out the antiderivative in terms of u.
5. Put in the expression for u so that the antiderivative is written in terms of the original variable.

Now let’s look at the examples carried out in class.

Problem 1

$latex \displaystyle \int{{{\left( 3{{x}^{2}}+4 \right)}^{4}}\cdot 6x,dx}$

Problem 2

$latex \displaystyle \int{{{\left( 3{{x}^{2}}+4 \right)}^{3}}\cdot 4x,dx}$

Problem 3

$latex \displaystyle \int{{{\left( {{x}^{2}}-1 \right)}^{5}}\cdot x,dx}$

Problem 4

$latex \displaystyle \int{4{{\left( 2x+3 \right)}^{2}},dx}$

Problem 5

$latex \displaystyle \int \frac{2}{{{\left( 2m+1 \right)}^{3}}} dm$

Problem 6

$latex \displaystyle \int{\frac{3}{\sqrt{3u-5}},du}$

Problem 7

$latex \displaystyle \int{-4{{e}^{2p}},dp}$

Problem 8

$latex \displaystyle \int{5{{e}^{-0.3g}},dg}$


text:
Sun is on my face ...a beautiful day without you.be apartmore quotes

# pi: fun

UCD Computational and Molecular Biology Symposium, Dublin, Ireland. 1-2 Dec 2016.

# visualization + design

Typography geek? If you like the geometry and mathematics of these posters, you may enjoy something more letter ed. Visions of type: Type Peep Show: The Private Curves of Letters posters.

# The art of Pi ($pi$), Phi ($phi$) and $e$

This section contains various art work based on $\pi$, $\phi$ and $e$ that I created over the years. $pi$ day and $pi$ approximation day artwork is kept separate.

The accidental similarity number (ASN) is a kind of overlap between numbers. I came up with this concept after creating typographical art about the $i$-ness of $\pi$.

The poster shows the accidental similarity number for $\pi$, $\phi$ and $e$.

The accidental similarity number for $\pi$, $\phi$ and $e$ created from the first 1,000,000 digits of each number. (posters, BUY ARTWORK)
VIEW ALL

# Visualizing Clonal Evolution in Cancer

Thu 02-06-2016

Genomic instability is one of the defining characteristics of cancer and within a tumor, which is an ever-evolving population of cells, there are many genomes. Mutations accumulate and propagate to create subpopulations and these groups of cells, called clones, may respond differently to treatment.

It is now possible to sequence individual cells within a tumor to create a profile of genomes. This profile changes with time, both in the kinds of mutation that are found and in their proportion in the overall population.

Ways to present temporal and phylogenetic evolution of clones in cancer. M Krzywinski (2016) Molecular Cell 62:652-656. (read)

Clone evolution diagrams visualize these data. These diagrams can be qualitative, showing only trends, or quantitative, showing temporal and population changes to scale. In this Molecular Cell forum article I provide guidelines for drawing these diagrams, focusing with how to use color and navigational elements, such as grids, to clarify the relationships between clones.

How to draw clone evolution diagrams better. M Krzywinski (2016) Molecular Cell xxx:xxx-xxx. (read)

I'd like to thank Maia Smith and Cydney Nielsen for assistance in preparing some of the figures in the paper.

Krzywinski, M. (2016) Visualizing Clonal Evolution in Cancer. Mol Cell 62:652-656.

# Binning High-Resolution Data

Wed 01-06-2016

Limitations in print resolution and visual acuity impose limits on data density and detail.

Your printer can print at 1,200 or 2,400 dots per inch. At reading distance, your reader can resolve about 200–300 lines per inch. This large gap—how finely we can print and how well we can see—can create problems when we don't take visual acuity into account.

Nature Methods Points of View column: Binning high-resolution data. (read)

The column provides some guidelines—particularly relevant when showing whole-genome data, where the scale of elements of interest such as genes is below the visual acuity limit—for binning data so that they are represented by elements that can be comfortably discerned.

Krzywinski, M. (2016) Points of view: Binning high-resolution data. Nature Methods 13:463.

# Regression diagnostics

Wed 11-05-2016

Residual plots can be used to validate assumptions about the regression model.

Continuing with our series on regression, we look at how you can identify issues in your regression model.

The difference between the observed value and the model's predicted value is the residual, $r = y_i - \hat{y}_i$, a very useful quantity to identify the effects of outliers and trends in the data that might suggest your model is inadequate.

Nature Methods Points of Significance column: Regression diagnostics? (read)

We also discuss normal probability plots (or Q-Q plots) and show how these can be used to check that the residuals are normally distributed, which is one of the assumptions of regression (constant variance being another).

Altman, N. & Krzywinski, M. (2016) Points of Significance: Analyzing outliers: Influential or nuisance? Nature Methods 13:281-282.

Altman, N. & Krzywinski, M. (2015) Points of Significance: Multiple Linear Regression Nature Methods 12:1103-1104.

Altman, N. & Krzywinski, M. (2015) Points of significance: Simple Linear Regression Nature Methods 12:999-1000.

# Analyzing Outliers: Influential or Nuisance?

Fri 08-04-2016

Some outliers influence the regression fit more than others.

This month our column addresses the effect that outliers have on linear regression.

You may be surprised, but not all outliers have the same influence on the fit (e.g. regression slope) or inference (e.g. confidence or prediction intervals). Outliers with large leverage—points that are far from the sample average—can have a very large effect. On the other hand, if the outlier is close to the sample average, it may not influence the regression slope at all.

Nature Methods Points of Significance column: Analyzing Outliers: Influential or Nuisance? (read)

Quantities such as Cook's distance and the so-called hat matrix, which defines leverage, are useful in assessing the effect of outliers.


text:
# Math Help - Vector field help

1. ## Vector field help

Hello!

One of my assignments is to create a vector field resembling a tornado, shape-wise, at least...

I've gotten the basic shape of the tornado, given by the function f(x,y) = 2*ln(x^2+y^2)... How would I go about converting this function to a vector field resembling this shape?

2. I assume the function you've given is a surface $z=2\ln{(x^2+y^2)}$ where the magnitude of the velocity is constant. You need to also say how the speed changes when you move away from the surface.

So you'll need a function of the form:

$\text{speed}=f(x,y,z)$

To get this function, you'll probably want to simplify the equation for your surface, then write an expression that is constant on your surface and increases when $x^2+y^2$ decreases.

The direction at (x,y,z) will be (y,-x,0), which will make the wind go in a circle clockwise around the z-axis. It needs to be a unit vector, so:

$\text{direction}=(\frac{y}{\sqrt{x^2+y^2}},\frac{-x}{\sqrt{x^2+y^2}},0) \text{ or } \frac{y}{\sqrt{x^2+y^2}}i-\frac{x}{\sqrt{x^2+y^2}}j$

and to get your vector field, just multiply speed by direction.

Post again if you're still having trouble.

Let's say if I were to model the speed with something like 1/e^(x^2+y^2) because it's a tornado and tornadoes have 0 outside and inside of the "wall"... (Well, 0 is impossible without piecewise functions, so really, really small magnitudes will suffice)... How would I go about doing the actual modeling and integrating with the direction vector?

Sorry, I've just gotten into vector fields and my teacher threw this assignment at me so I don't really have the hang of it yet... Please bear with me.

4. Ok. You can make the speed $1/e^{x^2+y^2}$. Of course, that will mean that everything is independent of z. So we don't really need to keep z around - we can just use 2 dimensions. Vector fields still work in 2 dimensions, so that's not really a problem.

A vector field is just a vector for every point, and a vector is just magnitude and direction. If the direction is a unit vector $\frac{u}{r}i+\frac{v}{r}j\text{, where }r=\sqrt{u^2+v^2}$, you just multiply the unit vector by the magnitude r to get the actual vector ui + vj.

If the components u and v of the vector are functions of the coordinates x and y, you have a vector field u(x,y)i + v(x,y)j. In three dimensions, it would look like u(x,y,z)i + v(x,y,z)j + w(x,y,z)k.

So you have the speed (which is the magnitude of the velocity vector) and the direction unit vector (which I gave you in my previous post). Multiply them together to get the equations of the vector field:

$v(x,y)\text{ = speed * direction = }1/e^{x^2+y^2} * \left(\frac{y}{\sqrt{x^2+y^2}}i-\frac{x}{\sqrt{x^2+y^2}}j\right)$, and substituting $r=\sqrt{x^2+y^2}$ gives us:

$v(x,y) = \frac{y}{re^{r^2}}i-\frac{x}{re^{r^2}}j\text{, where }r=\sqrt{x^2+y^2}$.

Hope that helps! Post again if there's something you don't understand.

5. Thank you! I think I understand the concept a lot more now...

But the problem still remains.
My goal is to create a vector field that resembles a tornado and the surface z = 2*ln(x^2+y^2), but I'd assume, as you said, it's a constant function... Is it possible to integrate the properties of 1/e^(x^2+y^2) with the surface z = 2*ln(x^2+y^2)?

Sorry if I sound a little lost!

6. Let's set the speed to be 1 on z = 2*ln(x^2+y^2) and behaving as 1/e^(x^2+y^2) when off the curve.

Let q0 be the value e^(z/2) of x^2+y^2 when speed=1. Let q=x^2+y^2. So we want speed=1 when q=q0 and speed goes like 1/e^q. So the equation is speed=1/e^(q-q0). Substituting in gives:

speed = 1/e^(x^2-y^2-e^(z/2))

Is that what you're looking for?


text:
# State variable filter

State variable filter

A state variable filter is a type of active filter. It consists of one or more integrators, connected in some feedback configuration. Any LTI system can be described as a state-space model, with n state variables for an nth-order system. A state variable filter realizes the state-space model directly. The instantaneous output voltage of one of the integrators corresponds to one of the state-space model's state variables.

The example given below can produce simultaneous lowpass, highpass and bandpass outputs from a single input. This is a second-order (biquad) filter. Its derivation comes from rearranging a high-pass filter's transfer function, which is the ratio of two quadratic functions. The rearrangement reveals that one signal is the sum of integrated copies of another. That is, the rearrangement reveals a state variable filter structure. By using different states as outputs, different kinds of filters can be produced. In more general state variable filter examples, additional filter order is possible with more integrators (i.e., more states).

The signal input is marked Vin; the LP, HP and BP outputs give the lowpass, highpass and bandpass filtered signals respectively.

For simplicity, we set:

$R_\left\{f1\right\} = R_\left\{f2\right\}$

$C_1 = C_2$

$R_1=R_2$

Then:

$F_0 = frac\left\{1\right\}\left\{2pi R_\left\{f1\right\}C_1\right\}$

$Q = left\left(1 + frac\left\{R_4\right\}\left\{R_q\right\} ight\right)left\left(frac\left\{1\right\}\left\{2+frac\left\{R_1\right\}\left\{R_g ight\right)$

The pass-band gain for the LP and HP outputs is given by:

$A_\left\{HP\right\} = A_\left\{LP\right\} = frac\left\{R_1\right\}\left\{R_G\right\}$

It can be seen that the frequency of operation and the Q factor can be varied independently. This and the ability to switch between different filter responses make the state-variable filter widely used in analogue synthesizers.

Values for a resonance frequency of 1 kHz are Rf1=Rf2=10k, C1=C2=15nF and R1=R2=10k

ee also

* Linear filter

References

* Texas Instruments' [http://focus.ti.com/lit/ds/symlink/uaf42.pdf UAF42 Universal Active Filter] datasheet

* Analog Devices [http://www.analog.com/Analog_Root/static/techSupport/designTools/interactiveTools/filter/filter.html Interactive Design Tools]

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• State variable topology — refers to filters with two pole filter topology. State variable filters are available in lowpass, highpass, bandpass, and notch versions. The notch version is not recommended, because it requires too many amplifiers. It is one of three topologies …   Wikipedia

• State variable — A state variable is an element of the set of variables that describe the state of a dynamical system.In case of simple mechanical systems, position coordinates and their derivates are typical state variables. Temperature, pressure, internal… …   Wikipedia

• Electronic filter topology — An elementary filter topology introduces a capacitor into the feedback path of an op amp to achieve an unbalanced active implementation of a low pass transfer function Electronic filter topology defines electronic filter circuits without taking… …   Wikipedia

• Biquad filter — For the digital implementation of a biquad filter, check digital biquad filter. A biquad filter is a type of linear filter that implements a transfer function that is the ratio of two quadratic functions. The name biquad is short for biquadratic …   Wikipedia

• Kalman filter — Roles of the variables in the Kalman filter. (Larger image here) In statistics, the Kalman filter is a mathematical method named after Rudolf E. Kálmán. Its purpose is to use measurements observed over time, containing noise (random variations)… …   Wikipedia

• Digital filter — A general finite impulse response filter with n stages, each with an independent delay, di, and amplification gain, ai. In electronics, computer science and mathematics, a digital filter is a system that performs mathematical operations on a… …   Wikipedia

• Electronic filter — Electronic filters are electronic circuits which perform signal processing functions, specifically intended to remove unwanted signal components and/or enhance wanted ones. Electronic filters can be:*passive or active *analog or digital *discrete …   Wikipedia

• Active filter — An active filter is a type of analog electronic filter, distinguished by the use of one or more active components i.e. voltage amplifiers or buffer amplifiers. Typically this will be a vacuum tube, transistor or operational amplifier.Active… …   Wikipedia

• Extended Kalman filter — In estimation theory, the extended Kalman filter (EKF) is the nonlinear version of the Kalman filter which linearizes about the current mean and covariance. The EKF is often considered the de facto standard in the theory of nonlinear state… …   Wikipedia

• Mu-tron — Musitronics, often shortened to Mu tron, was a manufacturer of electronic musical effects active in the 1970s. Their product line focused on filtering and processing effects derived from synthesizer components. Among their most well known… …   Wikipedia


text:
Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, February 24, 2015.

.
events for the
events containing

More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
     January 2015          February 2015            March 2015
Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
1  2  3    1  2  3  4  5  6  7    1  2  3  4  5  6  7
4  5  6  7  8  9 10    8  9 10 11 12 13 14    8  9 10 11 12 13 14
11 12 13 14 15 16 17   15 16 17 18 19 20 21   15 16 17 18 19 20 21
18 19 20 21 22 23 24   22 23 24 25 26 27 28   22 23 24 25 26 27 28
25 26 27 28 29 30 31                          29 30 31

Tuesday, February 24, 2015

11:00 am in 243 Altgeld Hall,Tuesday, February 24, 2015

#### Non-nilpotent elements in motivic homotopy theory

###### Michael Andrews   [email] (MIT)

Abstract: Classically, the nilpotence theorem of Devinatz, Hopkins, and Smith tells us that non-nilpotent self maps on finite p-local spectra induce nonzero homomorphisms on BP-homology. Motivically, over C, this theorem fails to hold: we have a motivic analog of BP and while $\eta:S^{1,1,}\to S^{0,0}$ induces zero on BP-homology, it is non-nilpotent. Work with Haynes Miller has led to a calculation of $\eta^{-1}\pi_{*,*}(S^{0,0})$, proving a conjecture of Guillou and Isaksen. I’ll introduce the motivic homotopy category and the motivic Adams-Novikov spectral sequence before describing this theorem. Then I’ll show that there are more periodicity operators in chromatic motivic homotopy theory than in the classical story. In particular, I will describe a new non-nilpotent self map.

1:00 pm in 347 Altgeld Hall,Tuesday, February 24, 2015

#### Nonlinear Phenomena in Acoustics: Traveling Waves, Bifurcations, and Singular Surfaces

###### Pedro Jordan (Naval Research Laboratory)

1:00 pm in 345 Altgeld Hall,Tuesday, February 24, 2015

#### Whitney's Extension Theorem in o-minimal structures

###### Athipat Thamrongthanyalak (Ohio State)

Abstract: For a $C^m$-function $f : U \to \mathbb{R}$, a jet of order $m$ of $f$ is the collection, $(D^{\alpha}f)_{\alpha\in\mathbb{N}^n,|\alpha|\leq n}$ of derivatives of $f$. In 1934, H. Whitney asked how can we determine whether a collection of continuous functions on a closed subset of $\mathbb{R}^n$ is a jet of order $m$ of a $C^m$-function. Here, we discuss this problem in o-minimal context.

1:00 pm in 243 Altgeld Hall,Tuesday, February 24, 2015

#### Planes, Trains & Automorphisms

###### Mark Bell (University of Warwick)

Abstract: We will discuss a new approach using train tracks to solve the conjugacy problem in the automorphism group of a punctured plane. This solution relies on the action of the mapping class group on the space of measured laminations and has several connections to veering triangulations of fibred 3--manifolds. Unlike techniques in braid groups using Garside structures, not only is this algorithm is effective it also generalises to all higher genus surfaces.

3:00 pm in 241 Altgeld Hall,Tuesday, February 24, 2015

#### New results for efficient hypergraph covering with application to large prime gaps

###### Kevin Ford   [email] (UIUC Math)

Abstract: Very recently, the author together with Ben Green, Sergei Konyagin, James Maynard and Terence Tao improved the bounds on the largest gaps between consecutive prime numbers. An important ingredient in the proof is a new result on efficient covering of hypergraphs, which extends a well-known theorem of Pippenger and Spencer from 1989. The Pippenger-Spencer theorem ensures the existence of a near-perfect packing of a hypergraph $H=(V,E)$ under three basic assumptions on $H$: (a) uniformity - all hyperedges $e\in E$ have the same (bounded) cardinality $k$; (b) regularity - the degree of each vertex $v\in V$ is asymptotically the same; (c) small codegrees - for all distinct $v,w\in V$, there are "few" edges containing both $v$ and $w$. Our new theorem gives essentially the same conclusion (not necessarily a packing, but an efficient near-covering) with a substantial weakening of hypotheses (a) and (b), while retaining (c) and the main hypothesis.

3:00 pm in 243 Altgeld Hall,Tuesday, February 24, 2015

#### t-structures on elliptic fibrations

###### Jason Lo (UIUC Math)

Abstract: A t-structure is a way to organise objects in the derived category. In various problems involving stabilities, t-structures come up naturally. In this talk, I will describe the various t-structures that appear in the derived category of coherent sheaves on an elliptic fibration with a Fourier-Mukai partner, and explain how understanding these t-structures might lead to progress on various problems on elliptic fibrations.


text:
/ / Uncategorized

The YAWL Process Editor is a workflow management system that uses standardized execution patterns and graphical visualizations to help you manage the workflow of your company.
Key features:
• Support for tasks and nets
• Net handling
• Flow pattern handling
• Multi-level flow templates
• XML-based net schemas
• Visual editor
• Validation
• Debugger
• Support for plugins

This download is a simple proof-of-concept using the create a Net engine as well as create a Net wizard. Using YAWL is better than using bash or PowerShell for a number of reasons. One of the biggest is that the Net is an extensible storage format. This means you can define your own data elements inside the Net and you can store collections of Nets together in a structured container. This level of flexibility is difficult to achieve with bash scripts.
This video will give you an introduction to YAWL and how to use it. I will show you how to build a simple Net using the create a Net wizard and also use the build a Net wizard to create a Net from a collection of files.
YAWL is my attempt to build an out-of-the-box workflow and process engine. It will store all of your workflow definitions in XML. Unlike Bash, YAWL deals well with variable amounts of data coming through the workflow. You can access input and output ports just like you can with Maven plugins.
YAWL XML Format
The YAWL Net represents each workflow or process. This is basically a network of connected tasks and containers which represent input and output. The top level of the Net contains the definition of the workflow and then the next level down contains the actions that define the functionality and workflow.
YAWL contains three basic types of containers: Task, Source and Target.
The task has an identity (XML), which represents the goal of the task. The identity is uniquely identified by the name of the task. The identity is used to look up associated resources.
The source container represents a point in a process where data flows into the workflow. The source container holds the data. The source containers defines a type of the data and the type is not necessarily limited to primitive types or built-in types. For example, you may want to represent a file as a container.
The target container represents a point in a process where data flows out of

## YAWL 1.1.0 Crack+ License Key Full

YAWL is a workflow management language that integrates accurate graphical representation and control-flow patterns, which help it distinguish itself from the other available languages. In addition, it uses XML schemas, XPath and XQuery to capture the data perspective, making it compatible with a wide array of standards.
Analyze work practices and improve efficiency
The YAWL Process Editor is a software utility that enables you to bring order to your business, by creating precise workflow management systems to handle the activity in your company. In addition, the systems can be created from within a practical and intuitive interface, thus eliminating the need for workflow experts and other specialized personnel.
To design a workflow, you can use specifications, which describe the task or the process you intend on representing. Inside the application, these specifications are comprised out of nets that can be manually added and customized according to your preferences, by using the mouse to draw a schema and reproduce the workflow in a graphical manner.
Validate the resulting workflows
In addition, you can also perform a validation on the resulting workflow, in order to spot any problems or detect errors in your design. The issues are displayed into the built-in debugger, which offers you in-depth information about every encountered error. After the validation is complete, you can analyze the entire project, although you need to solve any errors before you are able to perform the analysis.
The YAWL Process Editor also includes plugins to improve your experience, which allow you to configure the nets and tasks that are part of the workflow. Hence, you are able to check the net configuration correctness, as well as create multiple task instances and set up the input/output ports.
Meanwhile, the YAWL Control Panel can be used to start the YAWL engine, access the YAWL logon page, view details about the output log, as well as view and uninstall unnecessary components.
A professional workflow system creator
Although the application was not created with beginners in mind, YAWL is a full-featured utility that delivers a wide array of workflow system-related tools. In addition, the user interface is easy to navigate and you are also offered an extensive manual to get you started with the basic concepts of the application.
YAWL is a workflow management language that integrates accurate graphical representation and control-flow patterns, which help it distinguish itself from the other available languages. In addition, it uses XML schemas, XPath and XQuery to capture the data perspective, making it compatible with a wide
b7e8fdf5c8

## YAWL 1.1.0 With License Key [32|64bit] (2022)

The YAWL Process Editor is a workflow management application that enables you to create precise workflow systems to handle the activity in your company. Furthermore, it has been created with a graphical user interface that allows you to eliminate the need for workflow experts and other specialized personnel in the company. In addition, the application was designed in such a way that they can be configured using a common and intuitive tool, which also renders it suitable for beginners.
This Windows product is compatible with Windows 7, Windows Server 2008, Windows Server 2008 R2, Windows Server 2012, Windows 8, Windows 8.1, Windows 10, Windows Server 2012 R2, Windows 8.1, Windows Server 2012, Windows Server 2012 R2, Windows 10, Windows Server 2016 and Windows Server 2019.

3.4

2/17/2013

Product Features

The YAWL Process Editor is a workflow management application that enables you to create precise workflow systems to handle the activity in your company. Furthermore, it has been created with a graphical user interface that allows you to eliminate the need for workflow experts and other specialized personnel in the company. In addition, the application was designed in such a way that they can be configured using a common and intuitive tool, which also renders it suitable for beginners.
This Windows product is compatible with Windows 7, Windows Server 2008, Windows Server 2008 R2, Windows Server 2012, Windows 8, Windows 8.1, Windows 10, Windows Server 2012 R2, Windows 8.1, Windows Server 2012, Windows 10, Windows Server 2016 and Windows Server 2019.Q:

How to Get New Values Using AJAX without refreshing?

How to get updated value without refreshing or use stored javascript script in jquery?
eg:

$.ajax({ url: ‘test.php’, success: function(data) {$(“#result”).html(data);
}
});

test.php

Error:
I need it
success: function(data) {
$(“#result”).html(data); ## What’s New In? Workflows are a series of activities, which are automated by the use of software and computers. Workflows are used to produce systems that automate processes or tasks that can have a wide array of effects and impact. The production of a workflow is implemented by means of a mechanism called workflow management language (YAWL), which is a special language used in graphical applications such as OOML and MOML.Q: Why can we tell that$\phi^i(p)=x$for some$x\in M$? In a proof by contradiction, we suppose that$\phi^n(p)=p$. Since$\phi^n(p)=p$, then$\phi^n(p)=\phi^n(p)=x$for some$x\in M$. However, I don’t know how to proceed from there. Please help. A: Write down$\phi^n(p)=\phi(\phi(\dots \phi(p) \dots)) = \phi(x)$. Note that$\phi(x)$is the image of$x$under$\phi$(by definition of$\phi$), and since$\phi^n(p)=p$then$\phi(x)$is also in$M$, hence$x \in M\$ as well.

The Clinic of the Faculty of Medicine, University of Belgrade, Serbia.

Ethics Statement {#s4a}
—————-

This study was conducted in accordance with the ethical standards of the institutional and national research committees and with the 1964 Helsinki Declaration and its later amendments or comparable ethical standards. This study was approved by the Medical Ethics Committee of University of Medical Sciences in Belgrade, Serbia (Reference No: 2193-11/2007/10). The parents of the participants, adolescents, and any subjects under the age of 18 were well informed about the study and provided written consent. None of the subjects had any history of any psychiatric disorders or any other serious medical diseases. All participants were informed that they were free to participate or refuse to participate in the study, that they could withdraw from the study at any time, and that their data would be kept confidential. All participants indicated that they understood and agreed to participate in the research.

**Competing Interests:**The authors have declared that no competing interests exist.

**Funding:**This study was financed by the Serbian Ministry of Education, Science and

## System Requirements:

Windows 7 / Windows 8 / Windows 10
OS X 10.6.8 / 10.7 / 10.8 / 10.9 / 10.10 / 10.11 / 10.12 / 10.13 / 10.14
2 GB RAM or more
HDD or SSD 2 GB or more
Video Card: Nvidia GTX 560 / AMD R9 280 / Intel HD 4000 Series
Video Card RAM: 1 GB or more
HDD or SSD 300 MB or more
Makes you flash
Model: C7L02


text:
# SEARCH CONTENT

## You are looking at 1 - 10 of 12,263 items :

• General Mathematics
• Applied Mathematics
Clear All

## Abstract

The main goal of this article is to understand the trace properties of nonlocal minimal graphs in $ℝ3$, i.e. nonlocal minimal surfaces with a graphical structure.

We establish that at any boundary points at which the trace from inside happens to coincide with the exterior datum, also the tangent planes of the traces necessarily coincide with those of the exterior datum.

This very rigid geometric constraint is in sharp contrast with the case of the solutions of the linear equations driven by the fractional Laplacian, since we also show that, in this case, the fractional normal derivative can be prescribed arbitrarily, up to a small error.

We remark that, at a formal level, the linearization of the trace of a nonlocal minimal graph is given by the fractional normal derivative of a fractional Laplace problem, therefore the two problems are formally related. Nevertheless, the nonlinear equations of fractional mean curvature type present very specific properties which are strikingly different from those of other problems of fractional type which are apparently similar, but diverse in structure, and the nonlinear case given by the nonlocal minimal graphs turns out to be significantly more rigid than its linear counterpart.

## Abstract

Rickard complexes in the context of categorified quantum groups can be used to construct braid group actions. We define and study certain natural deformations of these complexes which we call curved Rickard complexes. One application is to obtain deformations of link homologies which generalize those of Batson–Seed [] [J. Batson and C. Seed, A link-splitting spectral sequence in Khovanov homology, Duke Math. J. 164 2015, 5, 801–841] and Gorsky–Hogancamp [E. Gorsky and M. Hogancamp, Hilbert schemes and y-ification of Khovanov–Rozansky homology, preprint 2017] to arbitrary representations/partitions. Another is to relate the deformed homology defined algebro-geometrically in [S. Cautis and J. Kamnitzer, Knot homology via derived categories of coherent sheaves IV, colored links, Quantum Topol. 8 2017, 2, 381–411] to categorified quantum groups (this was the original motivation for this paper).

## Abstract

We study deformations of rational curves and their singularities in positive characteristic. We use this to prove that if a smooth and proper surface in positive characteristic p is dominated by a family of rational curves such that one member has all δ-invariants (resp. Jacobian numbers) strictly less than $12(p-1)$ (resp. p), then the surface has negative Kodaira dimension. We also prove similar, but weaker results hold for higher-dimensional varieties. Moreover, we show by example that our result is in some sense optimal. On our way, we obtain a sufficient criterion in terms of Jacobian numbers for the normalization of a curve over an imperfect field to be smooth.

FREE ACCESS

## Abstract

We show that any proper Lie groupoid admits a compatible real analytic structure. Our proof hinges on a Weyl unitary trick of sorts for appropriate local holomorphic groupoids.

## Abstract

We give a restriction formula on jumping numbers which is a reformulation of Demailly–Ein–Lazarsfeld’s important restriction formula for multiplier ideal sheaves and a generalization of Demailly–Kollár’s important restriction formula on complex singularity exponents, and then we establish necessary conditions for the extremal case in the reformulated formula; we pose the subadditivity property on the complex singularity exponents of plurisubharmonic functions which is a generalization of Demailly–Kollár’s fundamental subadditivity property, and then we establish necessary conditions for the extremal case in the generalization. We also obtain two sharp relations on jumping numbers, introduce a new invariant of plurisubharmonic singularities and get its decreasing property for consecutive differences.

OPEN ACCESS

## Abstract

Let X be a dendrite, f : XX be a monotone map. In the papers by I. Naghmouchi (2011, 2012) it is shown that ω-limit set ω(x, f ) of any point xX has the next properties:

1. (1) $ω(x,f)⊆Per(f)¯$ , where Per( f ) is the set of periodic points of f ;
2. (2) ω(x, f ) is either a periodic orbit or a minimal Cantor set.

In the paper by E. Makhrova, K. Vaniukova (2016 ) it is proved that

1. (3) $Ω(f)=Per(f)¯$ , where Ω( f ) is the set of non-wandering points of f.

The aim of this note is to show that the above results – do not hold for monotone maps on dendroids.

## Abstract

In this paper we investigate stability of the integrability property of skew products of interval maps under small C 1-smooth perturbations satisfying some conditions. We obtain here (sufficient) conditions of the partial integrability for maps under considerations. These conditions are formulated in the terms of properties of the unperturbed skew product. We give also the example of the partially integrable map.

## Abstract

An iterative method (AIM) is one of the numerical method, which is easy to apply and very time convenient for solving nonlinear differential equations. However, if we want to work in a large interval, sometimes it may be difficult to apply AIM. Therefore, a multistage AIM named Multistage Modified Iterative Method (MMIM) is introduced in this article to work in a large computational interval. The applicability of MMIM for increasing the solution domain of the given problems is construed in this article. Some problems are solved numerically using MMIM, which provides a better result in the extended interval as compared to AIM. Comparison tables and some graphs are included to demonstrate the results.


text:
# Speed of a Cumulative Compound DC Motor

by mobin.abrar
Tags: compound, cumulative, dc motor, electric machines, torque-speed
 P: 1 I came across an equation for the Speed ω in Radians/seconds of a Cumulative Compound DC motor. ω= K(Vt-IA[RA+RF)/($\Phi$P+$\Phi$S) How would we calculate K in this equation? Shouldn't the Terminal Voltage Vt be replaced by Armature Voltage Ea? I came across a lot of varying torque speed curves. WIll the curve be exponentially decaying or linear?
Mentor
P: 10,519
Maybe you should explain your formula and the used variables, or at least give its source.
My guess: K is some constant given by the geometry of the motor.

 I came across a lot of varying torque speed curves. WIll the curve be exponentially decaying or linear?
I don't think that any of these curves are perfectly exponentially or linear. In addition, it might depend on the motor (e.g. how its internal friction depends on the angular velocity).

 Related Discussions Electrical Engineering 4 Electrical Engineering 7 Special & General Relativity 9 Electrical Engineering 7 Electrical Engineering 0


text:
# Delete rows with no entries in matrix

I have imported data. {{...},{...},{...},...{...}}

Here a screenshot:

As you can see there are empty entires {,,,,,,}

How can I delete those entries ?

If you want to delete rows that only consist of empty strings:

DeleteCases[data, {"" ..}]

"" .. is a pattern for "1 or more empty strings in a row". .. is shorthand for Repeated.

• Very elegant ! Good explenation! Thank you. – henry Aug 9 '16 at 15:35
• DeleteCases[{{1,""}, {2,""}, {}}, {"" ..}] doesn't give me the same result as I have in my answer. Perhaps I am missing something. – Eric Brown Aug 9 '16 at 15:44

Those are probably empty strings, so:

DeleteCases[{1,"",""}, ""]

will probably work for a single list. For lists of lists, you can use Map:

Map[ DeleteCases[#, ""] &, listOfLists]

If you would like to delete remaining empty rows, you can augment this with another DeleteCases:

DeleteCases[
Map[ DeleteCases[#, ""] &, listOfLists]
, {}]

• Hmm ... it is not working. It is working for your problem, but in my case I have rows witch are only empty strings. – henry Aug 9 '16 at 15:20
• Please try my edited response. – Eric Brown Aug 9 '16 at 15:24
• Very close... it delets all the empty entries but leaves the brakets {}. I have found a way to get rid of them. DeleteCases[Map[DeleteCases[#, ""] &, resMat], {}] – henry Aug 9 '16 at 15:27
• please add my solution to your response, so that I can validate your answer. :) – henry Aug 9 '16 at 15:28
• I made the edit. – Eric Brown Aug 9 '16 at 15:34


text:
### New Interpolated Map¶

Interpolated maps are values representing a piecewise function defined by interpolation between a sequence of breakpoints.

They have been initally defined as piecewise linear functions which have been superseded by NIM (an acronym for New Interpolated Map). NIM extends the idea of interpolation between breakpoints to vectors and to a superset of the interpolation types available in the curve construct.

A NIM is an aggregate data structure that records the breakpoints of the piecewise interpolation and the interpolation type (as in a curve). A NIM is considered an extensional function like a map. It can be applied to a numerical value to return the corresponding image.

NIMs can be used as an argument of the curve construct which allows the breakpoints to be dynamically built, "playing" the function later in time. See section Curve Playing a NIM.

#### Continuous and Discontinuous NIM¶

NIMs can represent continuous or discontinuous functions.

Continuous NIM are defined by an expression of the form:

NIM { x₀ y₀, d₁ y₁  "cubic"
, d₂ y₂ "linear"
, d₃ y₃ "bounce"
, ...
, dᵢ yᵢ "tᵢ"
, ...
}

which specifies a piecewise function $f$: between $\mathtt{x}_i$ and $\mathtt{x}_{i+1} = \mathtt{x}_i + \mathtt{d}_{i+1}$, function $f$ is an interpolation of type $\mathtt{t}_{i+1}$ from $y_i$ to $y_{i+1}$. See the following illustration:

The previous definition specifies a continuous function because the value of $f$ at the beginning of $[\mathtt{x}_i, \mathtt{x}_{i+1}]$ is also the value of $f$ at the end of the previous interval.

A second syntax allows one to define a discontinuous function by specifying a different value for the end of an interval and the beginning of the next one:

NIM { x₀, y₀ d₀ Y₀  "cubic"
, y₁ d₁ Y₁  "linear"
, y₂ d₂ Y₂  "bounce"
, ...
}

The definition is similar to the previous form except that on the interval $[\mathtt{x}_i, \mathtt{x}_{i+1}]$ the function is an interpolation between $\mathtt{y}_i$ and $\mathtt{Y}_i$. See illustration:

The NIM keyword is case-unsensitive and introduces an expression. The parameters $\mathtt{x}_i, \mathtt{y}_i, \mathtt{d}_i, \mathtt{Y}_i, \mathtt{t}_i$ are expressions.

#### Definition of the NIM outside the breakpoints¶

The function $f$ is extended outside $[\mathtt{x}_0, \mathtt{x}_n]$ such that

\begin{eqnarray} f(x) &=& \mathtt{y}_0 \qquad \text{for } x \leq \mathtt{x}_0 \\ f(x) &=& \mathtt{y}_n \qquad \text{for } x \geq \mathtt{x}_n = \mathtt{x}_0 +\sum_{i = 0}^n \mathtt{d}_i \end{eqnarray}

However, the functions @min_key and @max_key return $\mathtt{x}_0$ and $\mathtt{x}_n$ respectively (these functions also perform similarly on maps: maps are functions on a discrete range while NIMs are functions on a continuous range).

#### Interpolation type¶

The type of the interpolation is either a constant string or an expression. In the latter case, however, the expression must be enclosed in parentheses. The names of the allowed interpolation types are the same as those of a curve (see curve interpolation methods), and the interpolation types are illustrated on these figures. Other experimental interpolation methods are currently under development.

The specification of an interpolation type is optional. When not defined, the interpolation type is assumed to be linear.

#### Vectorial NIM¶

The NIM construct accepts tabs as arguments: in this case, the result is a vectorial function. For example:

NIM{ [-1, 0] [0, 10], [2, 3] [1, 20] ["cubic", "linear"] }

defines a vectorial of two variables: $$\vec{f} \binom{x_1}{x_2} = \binom{\,f_1(x_1)\,}{\,f_2(x_2)\,}$$ where $f_1$ is a cubic interpolation between $0$ and $1$ for $x_1$ going from $-1$ to $-1 + 2 = 1$ and $f_2$ is a linear interpolation between $10$ and $20$ for $x_2$ going from $0$ to $0 + 3 = 3$.

The NIM construction is ‟smart”: the parameters of a vectorized NIM may mix scalar $s$ and tabs. In this case, the scalar is extended implicitly into a vector of the correct size whose elements are all equal to $s$. This is the case even for the specification of the interpolation type. For example:

NIM{ 0, 0, [1, 2] 10 }

defines the function $\vec{f} = \binom{f_1}{f_2}$ where

\begin{eqnarray} f_1(x) &=& 0 \text{ if } x < 0 \\ &=& 10 \text{ if } x > 1 \\ &=& 10 x \text{ elsewhere} \end{eqnarray}

and

\begin{eqnarray} f_2(x) &=& 0 \text{ if } x < 0 \\ &=& 10 \text{ if } x > 2 \\ &=& 5 x \text{ elsewhere} \end{eqnarray}

A vectorized NIM is listable: it can be applied to a scalar argument. In this case, the scalar argument $x$ is implicitly extended to a vector of the correct dimension: $$\newcommand{\MYatop}[2]{\genfrac{}{}{0pt}{}{#1}{#2}} \vec{f}(x) = \vec{f}\binom{x}{\MYatop{\displaystyle{x}}{\displaystyle\dots}}$$

The function @dim returns the dimension of the image of a NIM, that is, $1$ for a scalar NIM and $n \not= 1$ for a vectorized NIM, where $n$ is the number of elements of the tab returned by the application of the NIM.

The function @size returns the number of breakpoints of the NIM (which is not equal to the dimension of the NIM). Note that a nim with zero breakpoints is the result of a bad definition.

##### Heterogeneous breakpoints in vectorial NIM¶

A vectorial NIM can be seen as the aggregation of several scalar NIMs.

The functions @projection and @aggregate can be used to respectively extract a scalar nim from a vectorial nim and to aggregate several (scalar or vectorial) nims of dimension $d_1, \dots, d_p$ into a nim of dimension $d_1 + \dots + d_p$.

let $nim1 := NIM{ 0 0, 1 10, 1 0 "sine_in_out" } let$nim2 := NIM{ -1 10, 4 0 "sine_in_out" }
let $nim3 := @aggregate($nim1, $nim2) This process can be used to build vectorial NIMs whose breakpoints are different, as in the specification of vectorial curves. There is no syntax to directly specify such NIMs and the function @aggregate must be used to build such NIM values that are said to have heterogeneous breakpoints. When printing such a NIM, the function @aggregate is used to list its many components: @aggregate( NIM{0 0, 1 10 "linear", 1 0 "sine_in_out"}, NIM{-1 10, 3 0 "sine_in_out"} ) The function @projection can be used to extract a component from a vectorial NIM: @projection($nim3, 1) ==  nim2 true #### Extending a NIM¶ The function @push_back can be used to add a new breakpoints to an existing NIM. They are several variations: @push_back(nim, d, y1) @push_back(nim, d, y1, type) @push_back(nim, y0, d, y1) @push_back(nim, y0, d, y1, type) @push_back(nim₁, nim₂) The four first forms modify the NIM ‟in-place”. The last form builds a new NIM: • The first two forms extend a NIM in a continuous fashion (the value at the start of the added breakpoint is the value of the last breakpoint of the NIM). • The next two forms explicitly specify the starting value of the added breakpoint, enabling the specification of discontinuous function. • The last form extends the nim in the first argument by the breakpoint of the nim in second argument. It effectively builds the function resulting in the “concatenation” of the breakpoints. Note that @push_back is an overloaded function: it can also be used to add (in place) an element at the end of a tab. #### NIM Transformation and Smoothing¶ NIMs can be managed through Nim Related Functions @aggregate, @align_breakpoints, @dim, @filter_max_t, @filter_median_t, @filter_min_t, @integrate, @linearize, @max_key, @max_val, @min, @min_key, @min_val, @projection, @push_back, @push_front, @sample, @simplify_lang_v, @simplify_radial_distance_t, @simplify_radial_distance_v, @window_filter_t See their descriptions in the library. Here, we describe the notions that appear when a NIM is restructured. ##### Sampling, homogeneization and linearization¶ It is possible to convert a NIM with an arbitrary interpolation type to a NIM with a linear interpolation type. The function @sample takes a NIM $nim$ and a number $n$ and returns a NIM with $n$ breakpoints and linear interpolation type that approximates $nim$. The process is illustrated in the following diagram: the NIMnim4 in input

@sample(nim3, 5) @aggregate( nim{0 0, 0.333333 3.33333, 0.333333 6.66667, 0.333333 10, 0.333333 7.5, 0.333333 2.5, 0.333333 0}, nim{-1 10, 0.5 9.33013, 0.5 7.5, 0.5 5, 0.5 2.5, 0.5 0.669873, 0.5 0} ) As can be seen, in the case of a vectorial NIM with heterogeneous components, the approximation is done component by component and the result has heterogeneous breakpoints. The function @align_breakpoints can be used on a NIM with linear interpolation type to obtain an equivalent NIM with homogeneous breakpoints: letnim4 :=  @sample(nim3, 5) will compute a NIM with a heterogeneous breakpoint printed as @align_breakpoints(nim4)
NIM { [-1, -1],
[0, 10] [0.5, 0.5] [0, 10]
[0, 9.33013] [0.5, 0.5] [0, 9.33013]
; ...
[0, 0] [0.333333, 0.333333] [0, 0] }

The function @sample samples its NIM argument homogeneously (component by component). This is not always satisfactory because the variation of the NIM can differ greatly between two intervals. The function @linearize(nim, tol) uses an adaptive sampling step to linearize nim, to align the breakpoints and to achieve an approximation within a given tolerance tol.

@linearize(nim3) NIM { [-1, -1], [0, 10] [0.609946, 0.609946] [0, 9.01426], [0, 9.01426] [0.270743, 0.270743] [0, 8.02012], ; ... [0.563462, 0.0636838] [0.152573, 0.152573] [0, 0] } The application of the @linearize function can be time consuming and care must be taken so as not to perturb the real-time computations, e.g., by precomputing a linearization (see @eval_when_load clause and function @loadvalue). #### NIM simplification¶ NIMs can be used to record a temporal series of data arriving (through time) from the environment, using setvar or OSC messages. It is then often useful to simplify the NIM, in order to have a more compact representation. The underlying idea is that the NIM approximates a curve known by a series of points, the breakpoints of the NIM, and that this curve can be approximated by fewer points. The simplified NIM consists of a subset of the breakpoints that defined the original. Several functions can be used to achieve a reduction of the number of breakpoints of a NIM with linear interpolation. All the simplification functions apply to scalar as well as vectorial NIMs, but consider only the ”$x$-coordinate” given by the breakpoints of the first component2. In case of a vectorial NIM with heterogeneous breakpoints, this can be a drawback. In this case, an equivalent NIM with homogeneous breakpoints can be first built using the @align_breakpoints function. The three following functions are inspired from polyline simplification functions developed in computer graphics1: • @simplify_radial_distance_t(nim, d) simplies each component of the NIM independently by reducing successive breakpoints that are clustered too closely to a single breakpoint. Because the simplification works on each component independantly, a breakpoint in this context is simply a point $(x, y)$ in the plane. The simplification process starts from the first breakpoint and is iterated until it reaches the final breakpoints. The first and the last breakpoints are always part of the simplification. All consecutive breakpoints that fall within a distance from a kept breakpoints are removed. The first encountered breakpoints that lies further away than is kept. • @simplify_radial_distance_v(nim, d): this simplification function is similar to the previous one but instead of working on each component independently, the vectorial nature of the nim is taken into account and the “$x$-coordinate” is ignored. The NIM is seen as a sequence of points, the image of the NIM at coordinates given by the $x$ part of the breakpoint of the first component. It is this series of points that is simplified to build the new NIM. The distance is thus taken in a $n$-dimensional space, where $n$ is the dimension of the NIM. • @simplify_lang_v(nim, tol, d): this simplification function follow the same approach as the previous one and consider a sequence of points in a $n$-dimensional space, where $n$ is the dimension of the nim and the points are given by the image of the breakpoints of the first component. Then, a Lang simplification algorithm is applied to reduce the number of points. The remaining points are used to build the simplified nim. The Lang simplification algorithm defines a fixed size search-interval. The first and last points of that interval form a segment. This segment is used to calculate the perpendicular distance to each intermediate point. If any calculated distance is larger than the specified tolerance , the interval will be shrunk by excluding its last point. This process will continue until all calculated distances fall below the specified tolerance, or when there are no more intermediate points. All intermediate points are removed and a new search interval is defined starting at the last point from the old interval. The effect of these simplification functions on a nim can be observed using the function @size which returns, for a nim, its number of breakpoints. #### Smoothing and Transformation¶ The functions described here work independently on each component of a nim. They see each component as a sequence of points $y$ (the image of the nim) that are smoothed in various way. The resulting points are used to build a new nim with linear interpolation type. The number of breakpoints is preserved, as well as their $x$-coordinate. Only the $y$-part of the breakpoint (the image) is affected. • @filter_median_t(nim, n) smoother the $y$ by replacing every image by the median in its range-$n$ neighborhood. Notice that the median is taken in a sequence of $2 n + 1$ values. The first $n$ points and the last $n$ points are left untouched. • @filter_min_t(nim, n) filters the $y$ by replacing every image by the minimum value in a sequence of length $2 n + 1$ centered on $y$. The first $n$ points and the last $n$ points are left untouched. • @filter_max_t(nim, n) filters the $y$ by replacing every image by the maximum value in a sequence of length $2 n + 1$ centered on $y$. The first $n$ points and the last $n$ points are left untouched. • @window_filter_t(nim, coef, pos) replaces every $y$ by the scalar product of coef (a tab) with a sequence of $n$ values, where $n$ is the size of coef and pos is the position in this window of the current $y$ (numbering starts with 0). The first pos values and the lastn - \$ pos values are left untouched.

For example, @window_filter_t(nim, [2], 0) builds a NIM by scaling the image of by 2. @window_filter_t(nim,[0.1, 0.2, 0.5, 0.2, 0.1], 2) is a moving weighted average with symmetric weights around $y$.

1. The following references gives technical details on polyline simplification. Description and evaluation of several algorithms in a cartographic context are given in this paper. The Ramer–Douglas–Peucker algorithm follows an idea similar to the Lang algorithm used by @simplify_lang_v but the shrinking of the current interval is dichotomous. This project provides an implementation of the main algorithms is in C++.

2. If the recorded value is a time series, then the $x$-coordinate corresponds to time and the NIM will be homogeneous as it is incrementally built using the function @push_back. The only way to build a non-homogeneous nim is to use the function @aggregate


text:
# what  actually  is  reaction  quotient?

The reaction quotient usually denoted as - Q measures the relative amount of products and reactants taking part in a reaction at a particular point in time. The reaction quotient helps in finding out in which direction a reaction is more likely to proceed, if the  pressures or the concentrations of the reactants and the products are known. The reaction quotient - Q value can be compared to K, which determines the direction of the reaction that is happening.

• 1

The reaction quotient (Q) measures the relative amount of products and reactants present during a reaction at a particular point in time. The reaction quotient aids in figuring out which direction a reaction is likely to proceed, given either the pressures or the concentrations of the reactants and the products.

• 0

Definition: Reaction quotient is the ratio of the concentrations of the products of a reaction to the concentrations of the reactants.

Each concentration is raised to the power of the stoichiometric coefficient in the chemical formula.

In general, for the reaction:

aA + bB → cC + dD
The reaction quotient, Q is

Q = [C]c[D]d/[A]a[B]b

• 1

Reaction quotient is the ratio of the initial concentrations of the products of a reaction to the initial concentrations of the reactants.

Each concentration is raised to the power of the stoichiometric coefficient in the chemical formula.

In general, for the reaction:

aA + bB → cC + dD
The reaction quotient, Q is

Q = [C]c[D]d/[A]a[B]b

• 0
What are you looking for?


text:
Schyper

# COMPUTE THE DISTANCE BETWEEN THE TWO POINTS

BIRTHDAY

Born:

Place:

This program computes the distance between the two points (x_{1},y_{1}) and (x_{2},y_{2}), given by $\sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}$
INPUT

Compute the distance between (0,0) and (0,0)
RESULT
Quod Erat Inveniendum
FUNCTIONS


text:
## Physics: Principles with Applications (7th Edition)

The pressure difference is due to the depth of water. Use equation 10–3b, and round the answer to 2 significant figures. $$\Delta P=\rho g \Delta h$$ $$\Delta h= \frac{\Delta P}{\rho g}$$ $$=\frac{(85 mm-Hg)((133 N/m^2)/(1mm-Hg))}{(1.00 \times 10^3 kg/m^3)(9.80 m/s^2)} \approx 1.2m$$ His maximum depth would be about 1.2 m.


text:
# What is the effect of altitude on true airspeed?

I know that wind is taken into account when we compute the TAS(true airspeed) and GS(ground speed). However what is the relationship between the TAS and altitude?

For example: What is the true airspeed on the level flight if the headwind is 20 mph and ground speed is 100 mph at 30,000 feet altitude?

• Learn to make the mental calculation of adding 2% to indicated airspeed for every 1000 ft above SL, and you can get a TAS value that is accurate enough for most situations. Apr 18 '19 at 14:13

If you know GS and local wind speed, the TAS always the same, regardless of altitude. The true airspeed is called "true" because it is exactly how fast you are moving relative to the air. Picture it as a person in a weather balloon (which does not move relative to the surrounding air, i.e., TAS=0), pointing a radar gun at your aircraft to measure its speed. The results would be the same regardless of altitude, temperature, pressure, etc. This is what sets TAS apart from pretty much any other speed like IAS, CAS, EAS and Mach.

Only at truly ridiculous altitudes you would have some problems with calculating TAS, due to the increased orbital radius from the center of the Earth. However, at this point, you're well into the near vacuum of space which means that the whole concept of 'air' speed is quite silly.

• Thanks for valuable answer.. Apr 18 '19 at 12:01
• Not to mention, in re your last paragraph, that you are dealing as much in plasma dynamics as fluid dynamics at those altitudes. Apr 18 '19 at 12:40

The true airspeed (TAS) can be calculated from the indicated airspeed (IAS), which is derived from the pitot tubes and static ports, as follows:

$$\mathrm{TAS} = \mathrm{IAS} \sqrt{\frac{\rho_0}{\rho(a)}} ,$$

where $$\rho_0$$ is the air density at sea level and $$\rho(a)$$ the air density at altitude $$a$$, which depends on pressure $$P$$ and temperature $$T$$:

$$\rho(a) = \frac{M \cdot P(a)}{R \cdot T(a)} ,$$

where $$M$$ is the molar mass of air and $$R$$ is the universal gas constant.

Using the international standard atmosphere for $$P(a)$$ and $$T(a)$$, one can plot the TAS as a function of altitude:

The ground speed (GS) is then given by the vector addition of the TAS and the wind speed: $$\mathrm{GS} = \mathrm{TAS} + v_\mathrm{wind} \cos(\alpha) ,$$ where $$\alpha$$ is the angle between the wind direction and the track of the aircraft ($$\alpha = 0^\circ$$ for tailwind, $$\alpha = 180^\circ$$ for headwind). This step is independent of pressure or temperature and as such independent of altitude. This means for a given TAS and headwind component, the ground speed is the same at all altitudes (for your example: 100 mph GS with 20 mph headwind implies 120 mph TAS at all altitudes).


text:
# How to measure RF power with an oscilloscope?

I'd like to measure the RF power coming from an RF amplifier using an oscilloscope.

The signals are sinusoids with a frequency of 50-120 MHz. The maximal power coming from the amplifier is 20 W, which is attenuated by a -3 dB RF attenuator. This power is delivered directly to the load (50 ohms), which is an ultrasonic transducer. The cables used are SMA coaxial cables.

I'm planning to put a T-splitter between the load and the attenuator, and use that to send the signal to a high frequeny digital oscilloscope set to 1 megaohms.

However, the oscilloscope can only handle 9 V RMS on its inputs at such a high frequency. So I thought that I could use a compensated 1:10 passive probe to attenuate the signal. But now I see that these probes aren't designed for frequencies like 100 MHz, and to cut the coaxial cable arrangement (for the probe) may not be the best idea, either. What is your opinion?

Alternatively, we've got some 50 ohms -3 dB RF attenuators, so I could wire them in series and put them between the T-splitter and the scope set to 50 ohms. Is this going to work? Won't they present a different load to the amplifier? Do I need to set the scope to 50 ohms in this case, or can I leave it on 1M (so it can handle a greater voltage)?

• Directional coupler would be easiest. You can pick them up from Minicircuits. – Peter Oct 31 '15 at 21:52

## 2 Answers

As your signals may go up to 120MHz, it's best to leave the scope on 50$\Omega$ input impedance.

Ideally you'll split the power with either a directional coupler, a 6dB power splitter, or a Wilkinson divider. To preserve accuracy you want something that is matched on all ports.

Then stack your 3dB attenuators in series until you are down to a power your scope input can handle.

• This seems like a good plan. The OP is planning to use a 3dB attenuator anyway. Maybe he can use a splitter instead of an attenuator at that location. The splitter insertion loss will probably be slightly higher than 3dB. If that is acceptable, that should work. He would then stack up as many attenuators as needed, as you suggest, on the path to the oscilloscope. – mkeith Oct 31 '15 at 20:53

Here is a diagram of my idea. It is a bit idealized. I think the splitter will attenuate the signal by a bit more than 3 dB. If the oscilloscope is not connected to out1, it would be a good idea to terminate it with a 50 Ohm load. Otherwise the insertion loss may be reduced.


text:
# Find the additive and multiplicative inverse of $-36$

Find the additive and multiplicative inverse of $-36$.

I don't remember how to do these problems. If you can help that would be great. thanks!

Hint: To be an inverse, you apply the operation to that and your number and need to get the identity. What is the additive identity? What can you add to $-36$ to get that? Same for multiplication.
For additive inverse you want a number $x$ such that $-36+x=0$ For multiplicative inverse you want a number $y$ such thast $-36*y=1$(For example in this one the multiplicative inverse for $2$ is $\frac{1}{2}$ since $\frac{1}{2}*2=1=2*\frac{1}{2})$


text:
# Some issues with the latest update of MikTeX

Baris Erkus bariserkus at hotmail.com
Sat Nov 9 13:40:59 UTC 2019

```On 09-Nov-19 1:33 PM, Kees Zeelenberg (kzStats) wrote:
> -----Oorspronkelijk bericht-----
> Van: lyx-users <lyx-users-bounces at lists.lyx.org> Namens Baris Erkus
> Verzonden: donderdag 7 november 2019 09:10
> Aan: lyx-users at lists.lyx.org
> Onderwerp: Some issues with the latest update of MikTeX
>
>> ....
>> LyX reported that iftex.sty was missing. I checked MikTeX and indeed iftex package was somehow not loaded in the package list, > which is awkward since I have installed "all" MikTeX packages and everything is expected to be there. It does not make sense
>>   MikTeX remove some standard packages during a MikTeX update.
>> ...
>> Please do not install the latest MikTeX update till you make sure it is safe to do so.
> On my machine (Windows 10, x64, LyX 2.3.3, MikTeX installed as administrator in C:\Program Files) everything still works fine after updating the MikTeX packages. The location of the iftex package has been changed: it is now in tex\generic instead of tex\latex; the latex\iftex folder is now indeed empty, but the generic\iftex folder is full. I did not have to reconfigure LyX, iftex.sty is still found when included in the preamble.
> Perhaps somehow the MikTeX file name database was not refreshed?
>
That explains it. As you have sad, probably something happened to MikTex
and could not properly address the correct location of the iftex pacakge.

--

↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓


text:
## Tuesday, November 18, 2014

### Codeforces Round #277.5 Div. 2

Problem A:
A. SwapSort

Solution:
You simply need to implement a Selection Sort since it will require at most N number of swaps. Why? Imagine placing N items to N correct positions in increasing order, then for each i in N there is only 2 possibility: item on i is already at its correct position, or we can find the item that should actually be there. So worst case scenario is you have to do the swapping N times.

Problem B:
B. BerSU Ball

Solution:
Greedy approach: Sort each stacks, and going from the top of each stack we pair what we can pair. If we find that the two elements on the top cannot be paired, eliminate the one which is bigger.

Problem C:
C. Given Length and Sum of Digits...

Solution:
To find the biggest possible:
From left to right: Greedily try to fill in 9. If can't be done, then fill in the biggest possible, and fill the rest with 0.
To find the smallest:
From left to right: Try to fill in i = 0, 1, 2, ..., 9. If remaining sum - i <= 9*(remaining length), then we can fill i. Otherwise increase i. Caution: the leftmost digit cannot be 0.

Problem D:
D. Unbearable Controversy of Being

Solution:
Very interesting problem. Notice that to find the number of damn rhombus between vertex u and v, we need to find M[u][v], the number of vertices i such that edge u-i and edge i-v exists, and then the number of damn rhombus between u and v is just the number of ways to choose those vertices i. Now the problem reduces to finding an efficient way to get that information for all pairs u-v. This can be done by going through all edges i-j that is provided, and for each k such that j-k is connected, then we increase M[i][k] by 1. Hence we have an $$O(VE + V^2)$$ running time overall.

Problem E:
Discussion on Problem E

Problem F:
Discussion on Problem F


text:
## Project Part 08 (G. Leobacher): Adapting QMC Algorithms to the Simulation Problem

### SFB funding period 2 (2018-2022)

In this project, we want to make problems from applications, which are not accessible by exact calculation, and for which Monte Carlo methods are the only means of getting an approximate solution, (more) suited for simulation using low-discrepancy point sets and sequences. Latter have been developed to achieve faster convergence rates than the usual order $1/2$ for Monte Carlo. This works well for certain spaces of functions on the unit cube.

Practical problems, especially from finance, can in general not be transformed into ones on the unit cube in a way such that established error bounds hold. Nevertheless, some of the concepts from certain spaces of functions on the unit cube carry over to certain spaces of functions on the ${\mathbb R}^d$, the so-called Hermite-Spaces. In particular, the concepts of weighted spaces and related notions of tractability can be studied and answered in these spaces as well. But still most problems can be transformed into ones which have smaller error bounds in those spaces.

We develop several transforms, linear (Brownian-bridge) and non-linear (smoothing techniques), to make integrands more suitable for the treatment with QMC. We study theoretical foundations of existing techniques and develop new ones, using theoretical findings.

The classes of models which will be studied will include stochastic differential equations (in particular) with irregular coefficients,  piecewise deterministic Markov processes and backward stochastic differential equations. All of these processes have wide areas of application, in particular in mathematical finance.

### SFB funding period 1 (2014-2017)

The project is located at the interface between quasi-Monte Carlo methods (QMC) and applications in finance and natural sciences. Hereby the main questions are how to reformulate a given high-dimensional integration problem to make it more suitable for QMC, which used to be recommended as a method for problems of low to moderate dimensions.

One of the most fruitful approaches known is to express the problem as an expectation of a function depending on independent standard normal variables and concatenate the function with a carefully chosen orthogonal transform. For very high-dimensional problems another important requirement is that the transform can be computed sufficiently fast. Examples of such transforms are the Brownian bridge construction and the principal component analysis construction.

In the present project we aim to further strengthen the theoretical basis for the use of orthogonal transforms with QMC and to find algorithms for constructing fast transforms for different types of problems from mathematical finance.


text:
Posted on June 16, 2015 on haskell

When you need something to go faster, hit it with the Yoneda lemma. Still isn’t fast enough? Use it again. —— Edward Kmett

Yoneda lemma may be the first understanding challenge in category theory. For an arbitrary functor $$F$$ from a category $$C$$ to the category $$Set$$, Yoneda lemma says: for any object $$A$$ of $$C$$, the natural transformations from the hom-functor $$h_A$$ to $$F$$ are in one-to-one correspondence with the elements in $$F(A)$$. That’s is,

$\mathrm{Nat}(h_A,F) \cong F(A).$

Doesn’t it hurt your brain? The proof is even more tedious. In another way, however, it just states that every object can be determined by how other objects map into it. The interesting thing is, once translated into Haskell, the Yoneda lemma becomes more readable and understandable.

In Haskell the category $$C$$ and $$Set$$ are both $$Hask$$, the category of all types in Haskell. And a natural transformation is just a polymorphic function which, in this case, from functor (->) a to functor f. So in Haskell we can write Yoneda lemma as

(forall b. (a -> b) -> f b) ~ f a

which means an isomorphism between a polymorphic function and a type. This can be interpreted in the following way: for a fixed type a and a functor f, if there is a polymorphic function g that knows how to convert any a -> b into f b, then this g is just something holding a f a (and remembering to fmap any a -> b on it), and vice versa.

To prove this isomorphism, we need to show that for the following two conversion functions:

fw :: (Functor f) => (forall b . (a -> b) -> f b) -> f a
fw f = f id

bw :: (Functor f) => f a -> (forall b . (a -> b) -> f b)
bw x f = fmap f x

the following two statements are both true:

fw . bw = id
bw . fw = id

The first one is easy:

(fw . bw) fa
= fw (bw fa)
= (bw fa) id
= bw fa id
= fmap id fa
= fa

and there are two steps notable in the second proof:

bw (fw g) a2b
= fmap a2b (g id)
= g (fmap a2b id) (*)
= g (a2b . id)    (**)
= g a2b

In (*) we swap g and fmap a2b when apply them to id in sequence. This is because the forall in the polymorphic function g makes it a natural transformation from functor (-> a) to functor f and ensure that the following diagram commute:

                  g
(a -> b) --------> f b
^                ^
|                |
fmap a2b |                | fmap a2b
|                |
|                |
(a -> a) --------> f a
g

Commutation means the both ways from bottom left to top right are the same. So we can swap g and fmap a2b. Also notice that the bottom left is just id. The (**) step is an application of the fmap definition of functor (-> a), which is just function composition (.). QED.


text:
### Overview

Instructor
Jeffrey Wong ( main site)
Email
jtwong at math dot duke dot edu
Office
Physics 029B
Office Hours
TBA or by appt.
Lectures
MTWThF 12:30-1:45 PM; through Zoom
Syllabus
Textbook(s)
(None required)

##### Course Website
During the semester, we will use Piazza for the course website. Resources will periodically be updated here.

##### Course Description
This is an introductory programming course in python that will provide a foundational background for programming in a mathematical setting. Students will learn the basics of object oriented programming: memory storage and variable scoping, recursion, objects and classes, and basic data structures. A variety of numerical methods will be introduced, with a focus on their practical implementation, through a series of modules covering a variety of subjects such as physical modeling, genetics, and optimization. The course is separated into two parts: In the first, we cover programming fundamentals in python. The second part is comprised of a series of modules on applications in computational mathematics, which provide a practical setting in which to develop these programming skills. In addition, towards the end of the course, there will be a final project in which the student chooses a topic to explore in depth.

##### Prerequisites
Students should have taken at least linearr algebra (Math 216, 218 or 221), while no programming background is required. Not open to students who have taken CS 201.

##### Technology
Access to the following (free) resources is required: Zoom (for lectures), github (to share and turn in work), Python 3.x and an editor (e.g. a text editor or an IDE like PyCharm), LaTeX, (for some homework and project writing). Information for getting started will be provided.

##### Homework and Exams
• Homework will be assigned approximately twice each week; typically due on the date of the next assignment.
• Homework should be turned in by the deadline to ensure full credit. If late, the homework should still be turned in for feedback (and partial credit).
• Working and studying in groups is strongly encouraged, as well as discussing code (writing code to be read by others is an excellent way to improve computational skills!). However, the final product - solutions and code - should be your own. Further guidelines for code and computational results will be provided throughout the course.
• Solutions should be complete arguments and code should be functional out of the box' (i.e. it runs and produces the required outputs). W When explanations are asked for, they may reference code but should be complete and easy to follow (it should not be necessary to dig through' the code to get to solutions).
• Typing solutions in LaTeX is suggested; if handwritten, make sure solutions are readable when scanned (a high quality phone camera with good lighting should be sufficient).

##### Exams
There is one midterm exam that covers programming fundamentals and no final exam. The midterm will be `take-home' and open notes (with some restrictions), to be taken during a subset of a continuous three hour period of the student's choice.

Course grades will be based on the following components:
• Midterm exam (20%)
• Homework (lowest score dropped; 45%)
• Final Project (30%): Students select a topic involving some application of computational mathematics. The project will involve learning and implementing algorithms to solve that problem and a written report.
• Participation (5%); this includes active participation in out-of-class activities such as posting questions/answers on Piazza, attending office hours and final project discussions with the instructor.
Note that participation in synchronous class meetings is strongly encouraged but, if missed, is not penalized in the grade.

##### Ethics
Students are expected to follow the Duke Community Standard. If a student is found responsible for academic dishonesty through the Office of Student Conduct, the student will receive a core of zero for that assignment. If a student’s admitted academic shonesty is resolved directly through a faculty-student resolution agreement approved by the Office of Student Conduct, the terms of that ent will dictate the grading response to the assignment at issue.