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Categories: Fluid dynamics, Fluid mechanics, Physics.

# Viscosity

The viscosity of a fluid describes how “sticky” its constituent molecules are; when one part of the fluid moves, it “drags” neighbouring parts by an amount proportional to the viscosity.

Imagine a liquid in a canal, flowing in the $$x$$-direction at a velocity $$v(z)$$ as a function of depth $$z$$. Due to the liquid’s viscosity, its molecules are “stuck” to the bottom of the canal $$z = 0$$, such that it is stationary there $$v(0) = 0$$. However, at the surface $$z = z_s$$, there is a flow at $$v(z_s) = v_s$$.

This difference in $$v$$ means that there is a velocity gradient across $$z$$. Each infinitesimal layer of the liquid is dragging on the layers above and below it, meaning there is a nonzero shear stress $$\sigma_{xz}$$ (see Cauchy stress tensor). Formally, the dynamic viscosity $$\eta$$ is defined as follows:

\begin{aligned} \boxed{ \sigma_{xz} = \eta \dv{v}{z} } \end{aligned}

This is Newton’s law of viscosity, and fluids obeying it are known as Newtonian. In a Newtonian fluid at rest, there are no such shear stresses, and the Cauchy stress tensor $$\hat{\sigma}$$ is diagonal:

\begin{aligned} \sigma_{ij} = - p \delta_{ij} \end{aligned}

Where $$p$$ is the pressure, and $$\delta_{ij}$$ is the Kronecker delta. If the fluid flows according to a velocity field $$\va{v}$$, then a more general definition of $$\eta$$ is as follows, in index notation with $$\nabla_i \!=\! \pdv*{x_i}$$:

\begin{aligned} \boxed{ \sigma_{ij} = - p \delta_{ij} + \eta (\nabla_i v_j + \nabla_j v_i) } \end{aligned}

The double term $$\nabla_i v_j + \nabla_j v_i$$ comes from the fact that the stress tensor of a Newtonian fluid is always symmetric; this definition of $$\sigma_{ij}$$ enforces that.

Another quantity is the kinematic viscosity $$\nu$$, which is simply $$\eta$$ divided by the density $$\rho$$:

\begin{aligned} \boxed{ \nu \equiv \frac{\eta}{\rho} } \end{aligned}

With this, Newton’s law of viscosity is written using the momentum density $$P = \rho v$$:

\begin{aligned} \sigma_{xz} = \nu \dv{P}{z} \end{aligned}

Because momentum is “more fundamental” than velocity, is $$\nu$$ often more useful than $$\eta$$. However, this comes at the cost of our intuition: for example, as you would expect, $$\eta_\mathrm{water} > \eta_\mathrm{air}$$, but you may be surprised that $$\nu_\mathrm{water} < \nu_\mathrm{air}$$. Since air is less dense, it is easier to set in motion, hence we expect it to be less viscous than water, but in fact air’s molecules are stickier than water’s.

## References

1. B. Lautrup, Physics of continuous matter: exotic and everyday phenomena in the macroscopic world, 2nd edition, CRC Press.


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# 6.4 Drag force and terminal speed  (Page 8/12)

 Page 8 / 12

If the order of the barges of the preceding exercise is reversed so that the tugboat pulls the $3.00\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{3}\text{-kg}$ barge with a force of $20.0\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{3}\phantom{\rule{0.2em}{0ex}}\text{N},$ what are the acceleration of the barges and the tension in the coupling cable?

An object with mass m moves along the x -axis. Its position at any time is given by $x\left(t\right)=p{t}^{3}+q{t}^{2}$ where p and q are constants. Find the net force on this object for any time t .

m (6 pt + 2 q )

A helicopter with mass $2.35\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{4}\phantom{\rule{0.2em}{0ex}}\text{kg}$ has a position given by $\stackrel{\to }{r}\left(t\right)=\left(0.020\phantom{\rule{0.2em}{0ex}}{t}^{3}\right)\stackrel{^}{i}+\left(2.2t\right)\stackrel{^}{j}-\left(0.060\phantom{\rule{0.2em}{0ex}}{t}^{2}\right)\stackrel{^}{k}.$ Find the net force on the helicopter at $t=3.0\phantom{\rule{0.2em}{0ex}}\text{s}\text{.}$

Located at the origin, an electric car of mass m is at rest and in equilibrium. A time dependent force of $\stackrel{\to }{F}\left(t\right)$ is applied at time $t=0$ , and its components are ${F}_{x}\left(t\right)=p+nt$ and ${F}_{y}\left(t\right)=qt$ where p , q , and n are constants. Find the position $\stackrel{\to }{r}\left(t\right)$ and velocity $\stackrel{\to }{v}\left(t\right)$ as functions of time t .

$\stackrel{\to }{v}\left(t\right)=\left(\frac{pt}{m}+\frac{n{t}^{2}}{2m}\right)\stackrel{^}{i}+\left(\frac{q{t}^{2}}{2}\right)\stackrel{^}{j}$ and $\stackrel{\to }{r}\left(t\right)=\left(\frac{p{t}^{2}}{2m}+\frac{n{t}^{3}}{6m}\right)\stackrel{^}{i}+\left(\frac{q{t}^{3}}{60m}\right)\stackrel{^}{j}$

A particle of mass m is located at the origin. It is at rest and in equilibrium. A time-dependent force of $\stackrel{\to }{F}\left(t\right)$ is applied at time $t=0$ , and its components are ${F}_{x}\left(t\right)=pt$ and ${F}_{y}\left(t\right)=n+qt$ where p , q , and n are constants. Find the position $\stackrel{\to }{r}\left(t\right)$ and velocity $\stackrel{\to }{v}\left(t\right)$ as functions of time t .

A 2.0-kg object has a velocity of $4.0\stackrel{^}{i}\phantom{\rule{0.2em}{0ex}}\text{m/s}$ at $t=0.$ A constant resultant force of $\left(2.0\stackrel{^}{i}+4.0\stackrel{^}{j}\right)\phantom{\rule{0.2em}{0ex}}\text{N}$ then acts on the object for 3.0 s. What is the magnitude of the object’s velocity at the end of the 3.0-s interval?

9.2 m/s

A 1.5-kg mass has an acceleration of $\left(4.0\stackrel{^}{i}-3.0\stackrel{^}{j}\right)\phantom{\rule{0.2em}{0ex}}{\text{m/s}}^{2}.$ Only two forces act on the mass. If one of the forces is $\left(2.0\stackrel{^}{i}-1.4\stackrel{^}{j}\right)\phantom{\rule{0.2em}{0ex}}\text{N,}$ what is the magnitude of the other force?

A box is dropped onto a conveyor belt moving at 3.4 m/s. If the coefficient of friction between the box and the belt is 0.27, how long will it take before the box moves without slipping?

1.3 s

Shown below is a 10.0-kg block being pushed by a horizontal force $\stackrel{\to }{F}$ of magnitude 200.0 N. The coefficient of kinetic friction between the two surfaces is 0.50. Find the acceleration of the block.

As shown below, the mass of block 1 is ${m}_{1}=4.0\phantom{\rule{0.2em}{0ex}}\text{kg,}$ while the mass of block 2 is ${m}_{2}=8.0\phantom{\rule{0.2em}{0ex}}\text{kg}\text{.}$ The coefficient of friction between ${m}_{1}$ and the inclined surface is ${\mu }_{\text{k}}=0.40.$ What is the acceleration of the system?

$5.4\phantom{\rule{0.2em}{0ex}}{\text{m/s}}^{2}$

A student is attempting to move a 30-kg mini-fridge into her dorm room. During a moment of inattention, the mini-fridge slides down a 35 degree incline at constant speed when she applies a force of 25 N acting up and parallel to the incline. What is the coefficient of kinetic friction between the fridge and the surface of the incline?

A crate of mass 100.0 kg rests on a rough surface inclined at an angle of $37.0\text{°}$ with the horizontal. A massless rope to which a force can be applied parallel to the surface is attached to the crate and leads to the top of the incline. In its present state, the crate is just ready to slip and start to move down the plane. The coefficient of friction is $80%$ of that for the static case. (a) What is the coefficient of static friction? (b) What is the maximum force that can be applied upward along the plane on the rope and not move the block? (c) With a slightly greater applied force, the block will slide up the plane. Once it begins to move, what is its acceleration and what reduced force is necessary to keep it moving upward at constant speed? (d) If the block is given a slight nudge to get it started down the plane, what will be its acceleration in that direction? (e) Once the block begins to slide downward, what upward force on the rope is required to keep the block from accelerating downward?

a. 0.60; b. 1200 N; c. $1.2\phantom{\rule{0.2em}{0ex}}{\text{m/s}}^{2}$ and 1080 N; d. $-1.2\phantom{\rule{0.2em}{0ex}}{\text{m/s}}^{2};$ e. 120 N

Clay Matthews, a linebacker for the Green Bay Packers, can reach a speed of 10.0 m/s. At the start of a play, Matthews runs downfield at 43° with respect to the 50-yard line (the +x-axis) and covers 7.8 m in 1 s. He then runs straight down the field at 90° with respect to the 50-yard line (that is, in the +y-direction) for 17 m, with an elapsed time of 2.9 s. (Express your answers in vector form.) (a) What is Matthews's final displacement (in m) from the start of the play?
A machine at a post office sends packages out a chute and down a ramp to be loaded into delivery vehicles. (a) Calculate the acceleration of a box heading down a 17.4° slope, assuming the coefficient of friction for a parcel on waxed wood is 0.100. (b) Find the angle of the slope down which this box could move at a constant velocity. You can neglect air resistance in both parts.
what principle is applicable in projectile motion
does rocket and satellite follow the same principle??? which principle do both of these follow???
According to d'Broglie's concept of matter waves matter behaves like wave and the wavelength is h/p. but actually there is not only a wave but a wave packet wich is defined by a wave function and that wave function can defines everything about the particle but restricted by the uncertainty principle
what phenomenon describes Matter behave as a wave???
simple definition of wave
hello
can anyone help me with this problem
Carls
A projectile is shot at a hill, the base of which is 300 m away. The projectile is shot at 60°60° above the horizontal with an initial speed of 75 m/s. The hill can be approximated by a plane sloped at 20°20° to the horizontal. Relative to the coordinate system shown in the following figure, the equation of this straight line is y=(tan20°)x−109.y=(tan20°)x−109. Where on the hill does the projectile land?
Carls
what is velocity
hi, Musa,this moment a lateral
what is moment
dimesion is defined as a measureable extent of particular kind such as breadth mass
Dimension is a quantity which give us direction of any vector unit like velocity
what is dimension

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# Keyboard

The table below summarizes the keystroke commands available in PFC that relate to working with a listing or while a Listing pane is active. If an ellipsis (…) appears after the command label, it indicates that a dialog will appear when the command is used. These commands are contextual, meaning they will produce a different result if a pane other than a Listing pane is active.

Table 1: Listing Pane Key Commands
Keystroke Command Description/Comment
Ctrl+Alt+S Save [listing name] As… Save the listing in the active pane to a file.
Ctrl + W Close [listing name] Close the listing in the active pane. Information in the listing is lost unless it has been previously saved.
Ctrl + C Copy Copy selected text in a listing to the clipboard.


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# How to copy a range from one tab to another in Calc while preserving the links to the first tab

I've created a spreadsheet, and while creating it I kept it all in one tab for ease of use.

I now need to make it easier to read so I'm separating out sections into different tabs.

The spreadsheet is a financial model so in this case the original tab had assumptions, income statement, balance sheet and cashflow all on the one big sheet. I'd now like to put the income statement, balance sheet and cashflow statements on separate tabs, but I'd like to preserve the link in the formulas to the assumptions in the first tab.

When I try and just cut and paste, the numbers look like they have copied successfully but the formulas refer to the destination sheet. Any modification causes errors all across the sheet. I can link to the original cells and hide those rows but that doesn't seem very efficient. A cut and paste-link fails with errors.

Any thoughts on how I can do this? Thanks

edit retag close merge delete

Sort by » oldest newest most voted

You need to convert your formulas from relative reference to absolute reference.

Example:
If you have a formula in cell B3 which is =A1, copying this formula to any other place, sheet or spreadsheet will paste the formula as relative, i.e. it will get the value from the cell that is one column to the left and two rows above.
However if you convert the formula to absolute (by editing the formula, placing the cursor near A1 and pressing Shift+F4) it will be converted to =$A$1 (obviously you can also type the dollar sign(s) manually)
This formula always refers to cell A1 if you copy or move it to another cell. Similarly moving it anywhere on another sheet will show the formula as =Sheet1.$A$1

Note that first dollar locks the Column (in this case A) and the second locks the row (in this case Row 1). Pressing consecutively Shift+F4 while editing the formula will cycle through several combinations of locking column and/or row.

more

1

Assuming "Sheet" is meant using the word "Tab" and also assuming that there might be a third sheet some time using the same copied formulae it will be purposive to also give the sheet reference by absolute addressing: $Sheet1.$A$1 , e.g. ( 2014-06-19 13:12:55 +0200 )edit @Lupp, good point! ( 2014-06-19 15:32:13 +0200 )edit "cut-and-paste" and "copy-and-paste" behave differently. "copy-and-paste" is as described by Pedro: links to other cells are interpreted as relative unless you specify absolute by using$. So the cell C3 with "=A3B3" when copied to C4 has "=A4B4" which is what we want when generating new content from old.

However if you "cut-and-paste" the cell from C3 to C4 then it retains the links and so will still be "=A3*B3" even though the formula is using relative references. This is what we want when rearranging content. These behaviours are the same whether you cut/copy and paste within or between sheets.

So though you claimed to have "cut and pasted" I suspect you "copy and pasted" - what I'd do as a backup if I didn't realise it made a difference.

more


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# customizing QGIS Layers Panel to show layer CRS next to layer name

Is there a way to customize the Layers Panel so that the layer CRS is showed next to the layer name?

A little bit more context why this would be handy: I am writing some instructions to use (Q)GIS in particular situations. Some of the people are familiar with projections, but the most of them not.

Normaly you check the CRS by opening the layer properties, but it would be very handy for those 'unfamiliar' people to see directly in which CRS a layer is loaded into QGIS.

You could use something like this once your layers have been loaded into QGIS:

for layer in QgsMapLayerRegistry.instance().mapLayers().values():
crs = layer.crs().authid()
layer.setLayerName(layer.name() + ' (' + crs + ')')

Result:

EDIT:

In response to your comment, you can automatically update a layer's name when you add a new layer by connecting the layersAdded signal to a function. Paste the following code into the Python Console and test it by adding layers:

def updateLayerName(layers):
for layer in layers:
crs = layer.crs().authid()
layer.setLayerName(layer.name() + ' (' + crs + ')')


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# How to show that $\sum_{i=1}^n | \langle f, f_i\rangle |^2 \leq \Vert f \Vert^2$

If the set $\{f_1, ..., f_n\}$ is an orthonormal subset of inner product space $E$ and $f\in E$ then how can I show that:

$$\sum_{i=1}^n | \langle f, f_i\rangle |^2 \leq \Vert f \Vert^2.$$

How can I prove this inequality? What is the key step to notice?

Quoting from the wikipedia page for Bessel's inequality, with $f=x$ and $f_i=e_i$, the key point is:
$$0 \le \left\| x - \sum_{k=1}^n \langle x, e_k \rangle e_k\right\|^2 = \|x\|^2 - 2 \sum_{k=1}^n |\langle x, e_k \rangle |^2 + \sum_{k=1}^n | \langle x, e_k \rangle |^2 = \|x\|^2 - \sum_{k=1}^n | \langle x, e_k \rangle |^2$$


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Some new results on finite algebras whose only known proof uses large cardinals (updated)

The large cardinals above hugeness are a very powerful tool for proving results about finite self-distributive algebras, but mathematicians so far have neglected to exploit the tremendous power of these very large cardinals to prove results about finite objects. Initially around the late 80’s and early 90’s, Laver and other mathematicians have established some very good classical results about the Laver tables. On the other hand, from the later 1990’s to the 2010’s, no mathematician has published any original research that directly relates to what I call Laver-like LD-systems. Furthermore, before my work on the algebras of elementary embeddings, nobody has investigated what will happen in the algebras of elementary embeddings generated by more than one element.
There are two results about the classical Laver tables which presumably require large cardinals to establish. One of these results states that the inverse limit of the classical Laver tables contains free left-distributive algebras on one generator while the other result states that in the classical Laver table $A_{n}$, we have $2*_{n}x=2^{n}\Rightarrow 1*_{n}x=2^{n}$. All of the other results about the classical Laver tables are known to hold in ZFC.

In this post, we shall use large cardinals and forcing to prove the existence of certain classes of finite self-distributive algebras with a compatible linear ordering. The results contained in this note shall be included in my (hopefully soon to be on Arxiv) 100+ page paper Generalizations of Laver tables. In this post, I have made no attempt to optimize the large cardinal hypotheses.

For background information, see this post or see Chapter 11 in the Handbook of Set Theory.

We shall let $\mathcal{E}_{\alpha}$ denote the set of all elementary embeddings $j:V_{\alpha}\rightarrow V_{\alpha}.$
By this answer, I have outlined a proof that the algebra $\mathcal{E}_{\lambda}/\equiv^{\gamma}$ is locally finite. We therefore have established a deep connection between the top of the large cardinal hierarchy and finite algebras.

In this note, we shall use two important ideas to construct finite self-distributive algebras. The main idea is to generalize the square root lemma for elementary embeddings so that one obtains elementary embeddings with the desired properties.

$\textbf{Theorem: (Square Root Lemma)}$ Let $j\in\mathcal{E}_{\lambda+1}$. Then there is some $k\in\mathcal{E}_{\lambda}$ where $k*k=j|_{V_{\lambda}}$.

$\mathbf{Proof}:$ By elementarity
$$V_{\lambda+1}\models\exists k\in\mathcal{E}_{\lambda}:k*k=j|_{V_{\lambda}}$$
if and only if
$$V_{\lambda+1}\models\exists k\in\mathcal{E}_{\lambda}:k*k=j(j|_{V_{\lambda}})$$
which is true. Therefore, there is some $k\in\mathcal{E}_{\lambda}$ with $k*k=j|_{V_{\lambda}}$. $\mathbf{QED}$

The other idea is to work in a model such that there is a cardinal $\lambda$ where there are plenty of rank-into-rank embeddings from $V_{\lambda}$ to $V_{\lambda}$ but where $V_{\lambda}\models\text{V=HOD}$. If $V_{\lambda}\models\text{V=HOD}$, then $V_{\lambda}$ has a definable linear ordering which induces a desirable linear ordering on rank-into-rank embeddings and hence linear orderings on finite algebras. The following result can be found in this paper.

$\mathbf{Theorem}$ Suppose that there exists a non-trivial elementary embedding $j:V_{\lambda+1}\rightarrow V_{\lambda+1}$. Then in some forcing extension $V[G]$ there is some elementary embedding $k:V[G]_{\lambda+1}\rightarrow V[G]_{\lambda+1}$ where
$V[G]_{\lambda}\models\text{V=HOD}$.

Therefore it is consistent relative to large cardinals that there exists a non-trivial elementary embedding $j:V_{\lambda+1}\rightarrow V_{\lambda+1}$ such that $V_{\lambda}\models\text{V=HOD}$.

Now suppose that $V_{\lambda}\models\text{V=HOD}$. Then there exists a linear ordering $\ll$ of $V_{\lambda}$ which is definable in $V_{\lambda}$. If $j\in\mathcal{E}_{\lambda}$ and $\gamma$ is a limit ordinal with $\gamma<\lambda$, then define $j\upharpoonright_{\gamma}:V_{\gamma}\rightarrow V_{\gamma+1}$ by $j\upharpoonright_{\gamma}(x)=x\cap V_{\gamma}$ for each $x\in V_{\gamma}.$ Take note that $j\upharpoonright_{\gamma}=k\upharpoonright_{\gamma}$ if and only if $j\equiv^{\gamma}k$. Define a linear ordering $\trianglelefteq$ on $\mathcal{E}_{\lambda}$ where $j\trianglelefteq k$ if and only if $j=k$ or there is a limit ordinal $\alpha$ where $j\upharpoonright_{\alpha}\ll k\upharpoonright_{\alpha}$ but where $j\upharpoonright_{\beta}=k\upharpoonright_{\beta}$ whenever $\beta<\alpha$. Define a linear ordering $\trianglelefteq$ on $\{j\upharpoonright_{\gamma}\mid j\in\mathcal{E}_{\lambda}\}$ by letting $j\upharpoonright_{\gamma}\triangleleft k\upharpoonright_{\gamma}$ if and only if there is some limit ordinal $\beta\leq\gamma$ where $j\upharpoonright_{\beta}\ll k\upharpoonright_{\beta}$ but where $j\upharpoonright_{\alpha}=k\upharpoonright_{\alpha}$ whenever $\alpha$ is a limit ordinal with $\alpha<\beta$. By elementarity, the linear ordering $\trianglelefteq$ satisfies the following compatibility property: if $k\upharpoonright_{\gamma}\trianglelefteq l\upharpoonright_{\gamma}$, then $(j*k)\upharpoonright_{\gamma}\trianglelefteq(j*l)\upharpoonright_{\gamma}$. We say that a linear ordering $\leq$ on a Laver-like LD-system $(X,*)$ is a compatible linear ordering if $y\leq z\Rightarrow x*y\leq x*z$. If $V_{\lambda}\models\text{V=HOD}$, then $\mathcal{E}_{\lambda}/\equiv^{\gamma}$ has a compatible linear ordering defined by $[j]_{\gamma}\leq[k]_{\gamma}$ if and only if $j\upharpoonright_{V_{\gamma}}\trianglelefteq k\upharpoonright_{V_{\gamma}}$.

Using generalized Laver tables, we know that the set $\{\text{crit}(j):j\in\langle j_{1},…,j_{n}\rangle\}$ has order-type $\omega$. Let $\text{crit}_{r}(j_{1},…,j_{n})$ be the $r$-th element of the set $$\{\text{crit}(j):j\in\langle j_{1},…,j_{n}\rangle\}$$ ($\text{crit}_{0}(j_{1},…,j_{n})$ is the least element of $\{\text{crit}(j):j\in\langle j_{1},…,j_{n}\rangle\}$). Let $T:\bigcup_{n\in\omega}\mathcal{E}_{\lambda}^{n}\rightarrow V_{\omega\cdot 2}$ be a mapping definable in $(V_{\lambda+1},\in)$ where $T(j_{1},…,j_{m})=T(k_{1},…,k_{n})$ if and only if $m=n$ and if $\gamma=\text{crit}_{r} (j_{1},…,j_{m})$ and $\delta=\text{crit}_{r}(k_{1},…,k_{n})$, then there is some isomorphism $\phi:\langle j_{1},…,j_{m}\rangle/\equiv^{\gamma}\rightarrow\langle k_{1},…,k_{n}\rangle/\equiv^{\delta}$ where $\phi([j_{i}]_{\gamma})=[k_{i}]_{\delta}$. We remark that if $T(j_{1},…,j_{m})=T(k_{1},…,k_{n})$, then the subspaces $\overline{\langle j_{1},…,j_{m}\rangle}$ and $\overline{\langle k_{1},…,k_{n}\rangle}$ of $\mathcal{E}_{\lambda}$ are homeomorphic by an isomorphism of algebras preserving $*,\circ$ ($\mathcal{E}_{\lambda}$ can be given a complete metric that induces a canonical uniformity on $\mathcal{E}_{\lambda}$).

The following technical result is a generalization of the Square-Root Lemma, and a simplified special case of the following results can be found in this answer that I gave.

$\mathbf{Theorem:}$ Suppose the following:

1. $\ell_{1},…,\ell_{p},j_{1},…,j_{m}\in\mathcal{E}_{\lambda}$ and $(k_{r,s})_{1\leq r\leq n,1\leq s\leq p}\in(\mathcal{E}_{\lambda})^{n\cdot p}$.
2. $\ell_{1},…,\ell_{p}$ are I1 embeddings.
3. $T(\ell_{i}*j_{1},…,\ell_{i}*j_{m},k_{1,i},…,k_{n,i})=x_{i}$ whenever $1\leq i\leq p$.
4. $v$ is a natural number.
5. there is some $\mu<\lambda$ where $\mu=\text{crit}_{v}(k_{1,1},...,k_{n,1})=\ldots=\text{crit}_{v}(k_{1,p},...,k_{n,p})$.
6. $k_{r,1}\equiv^{\mu}…\equiv^{\mu}k_{r,p}$ for $1\leq r\leq n$.
7. $\ell_{1}\equiv^{\mu+\omega}…\equiv^{\mu+\omega}\ell_{p}$.

Then there are $(w_{r,s})_{1\leq r\leq n,1\leq s\leq p}$ in $\mathcal{E}_{\lambda}$ where

1. $T(j_{1},…,j_{m},w_{1,i},…,w_{n,i})=x_{i}$ for $1\leq i\leq p$,
2. there is some $\alpha<\lambda$ where $\text{crit}_{v}(w_{1,i},...,w_{n,i})=\alpha$ for $1\leq i\leq p$, and
3. $w_{r,1}\equiv^{\alpha}\ldots\equiv^{\alpha}w_{r,p}$ for $1\leq r\leq n$.

$\mathbf{Proof:}$ For $1\leq i\leq p$, let $A_{i}$
$$=\{(w_{1}\upharpoonright_{\text{crit}_{v}(w_{1},…,w_{n})},…,w_{n}\upharpoonright_{\text{crit}_{v}(w_{1},…,w_{n})}): T(j_{1},…,j_{m},w_{1},…,w_{n})=x_{i}\}.$$
Then $\ell_{i}(A_{i})$
$$=\{(w_{1}\upharpoonright_{\text{crit}_{v}(w_{1},…,w_{n})},…,w_{n}\upharpoonright_{\text{crit}_{v}(w_{1},…,w_{n})}): T(\ell_{i}*j_{1},…,\ell_{i}*j_{m},w_{1},…,w_{n})=x_{i}\}.$$
Therefore,
$$(k_{1,i}\upharpoonright_{\mu},…,k_{n,i}\upharpoonright_{\mu})\in\ell_{i}(A_{i})$$ for $1\leq i\leq p$. Since
$k_{r,1}\upharpoonright_{\mu}=…=k_{r,p}\upharpoonright_{\mu}$, we have
$$(k_{1,1}\upharpoonright_{\mu},…,k_{n,1}\upharpoonright_{\mu})=…=(k_{1,p}\upharpoonright_{\mu},…,k_{n,p}\upharpoonright_{\mu}).$$
Therefore, let $$(\mathfrak{k}_{1},…,\mathfrak{k}_{n})=(k_{1,1}\upharpoonright_{\mu},…,k_{n,1}\upharpoonright_{\mu}).$$
Then
$$(\mathfrak{k}_{1},…,\mathfrak{k}_{n})\in\ell_{1}(A_{1})\cap…\ell_{p}(A_{p})\cap V_{\mu+\omega}$$
$$=\ell_{1}(A_{1})\cap…\cap\ell_{1}(A_{p})\cap V_{\mu+\omega}$$
$$\subseteq\ell_{1}(A_{1}\cap…\cap A_{p}).$$
Therefore, $A_{1}\cap…\cap A_{p}\neq\emptyset.$

Let $(\mathfrak{w}_{1},…,\mathfrak{w}_{n})\in A_{1}\cap…\cap A_{p}$. Then there are $(w_{r,s})_{1\leq r\leq n,1\leq s\leq p}$ in $\mathcal{E}_{\lambda}$ where
$$(\mathfrak{w}_{1},…,\mathfrak{w}_{n})=(w_{1,i}\upharpoonright_{\text{crit}_{v}(w_{1,i},…,w_{n,i})},…,w_{n,i}\upharpoonright_{\text{crit}_{v}(w_{1,i},…,w_{n,i})})$$
and
$$T(j_{1},…,j_{m},w_{1,i},…,w_{n,i})=x_{i}$$
for $1\leq i\leq p.$ Therefore, there is some $\alpha<\lambda$ with $\text{crit}_{v}(w_{1,s},...,w_{n,s})=\alpha$ for $1\leq s\leq p$ and where $w_{r,1}\equiv^{\alpha}\ldots\equiv^{\alpha}w_{r,p}$ for $1\leq r\leq n$. $\mathbf{QED}$

$\mathbf{Remark:}$ The above theorem can be generalized further by considering the classes of rank-into-rank embeddings
described in this paper.

$\mathbf{Theorem:}$ Suppose that there exists an I1 cardinal. Suppose furthermore that

1. $U_{1},…,U_{p},V_{1},…,V_{m}$ and $(W_{r,s})_{1\leq r\leq n,1\leq s\leq p}$ are unary terms in the language with function symbols $*,\circ$,
2. $L$ is an $np+1$-ary term in the language with function symbols $*,\circ$,
3. $v$ is a natural number,
4. There is some classical Laver table $A_{N}$ where in $A_{N}$, we have
$\mu=\text{crit}_{v}(W_{1,1}(1),…,W_{n,1}(1))=…=\text{crit}_{v}(W_{1,p}(1),…,W_{n,p}(1))<\text{crit}(2^{n}).$
5. For all $N$, we have $W_{r,1}\equiv^{\mu}\ldots\equiv^{\mu}W_{r,p}(1)$ for $1\leq r\leq n$,
6. $U_{1}(1)\equiv^{\mu^{+}}\ldots\equiv^{\mu^{+}}U_{p}(1)$.

If $Y$ is a finite reduced Laver-like LD-system, then let $\approx$ be the relation on $Y^{<\omega}$ where $(x_{1},...,x_{m})\approx(y_{1},...,y_{n})$ if and only if $m=n$ and whenever $\langle x_{1},...,x_{m}\rangle$ and $\langle y_{1},...,y_{n}\rangle$ both have more than $v+1$ critical points, then there is an isomorphism $\iota:\langle x_{1},...,x_{m}\rangle/\equiv^{\text{crit}_{v}(x_{1},...,x_{m})}\rightarrow \langle y_{1},...,y_{n}\rangle/\equiv^{\text{crit}_{v}(y_{1},...,y_{n})}$ where $\iota([x_{i}])=[y_{i}]$ for $1\leq i\leq n$.
Then there is some finite reduced Laver-like LD-system $X$ along with
$x,(y_{r,s})_{1\leq r\leq n,1\leq s\leq p}\in X$
such that

1. $X$ has a compatible linear ordering,
2. $(U_{s}(x)*V_{1}(x),…,U_{s}(x)*V_{m}(x),W_{1,s}(x),…,W_{n,s}(x))$
$\approx(V_{1}(x),…,V_{m}(x),y_{1,s},…,y_{n,s}),$
3. ind

4. there is some critical point $\alpha$ where $\text{crit}_{v}(y_{1,s},…,y_{n,s})=\alpha$ for all $s$, and
5. $y_{r,1}\equiv^{\alpha}\ldots\equiv^{\alpha}y_{r,p}$.
6. $L(x,(y_{r,s})_{1\leq r\leq n,1\leq s\leq p})\neq 1$.
Challenge

I challenge the readers of this post to remove the large cardinal hypotheses from the above theorem (one may still use the freeness of subalgebras $\varprojlim_{n}A_{n}$ and the fact that $2*_{n}x=2^{n}\Rightarrow 1*_{n}x=2^{n}$ though).

So it turns out that by taking stronger large cardinal axioms, one can induce a linear ordering on the algebras of elementary embeddings without having to resort to working in forcing extensions. We say that a cardinal $\delta$ is an I1-tower cardinal if for all $A\subseteq V_{\delta}$ there is some $\kappa<\delta$ such that whenever $\gamma<\delta$ there is some cardinal $\lambda<\delta$ and non-trivial elementary embedding $j:V_{\lambda+1}\rightarrow V_{\lambda+1}$ such that $\text{crit}(j)=\kappa$ and where $j(\kappa)>\delta$ and where $j(A)=A$. If $A$ is a good enough linear ordering on $V_{\delta}$, then $A\cap V_{\lambda}$ induces a compatible linear ordering the set of all elementary embeddings $j:V_{\lambda}\rightarrow V_{\lambda}$ such that $j(A\cap V_{\gamma})=A\cap V_{j(\gamma)}$ for all $\gamma<\lambda$. It is unclear where the I1-tower cardinals stand on the large cardinal hierarchy or whether they are even consistent.

It turns out that we can directly show that if $j:V_{\lambda+1}\rightarrow V_{\lambda+1}$ is a non-trivial elementary embedding, then there is a linear ordering $B$ of $V_{\lambda}$ where $j(B)=B$. In fact, if $j:V_{\lambda}\rightarrow V_{\lambda}$ is a non-trivial elementary embedding, $\mathrm{crit}(j)=\kappa$, and $A$ is a linear ordering of $V_{\lambda}$, then if we let $B=\bigcup_{n}j^{n}(A)$, then $B$ is a linear ordering of $V_{\lambda}$ and $j(B\cap V_{\gamma})=B\cap V_{j(\gamma)}$ whenever $\gamma<\lambda$. In particular, if $j$ extends to an elementary embedding $j^{+}:V_{\lambda+1}\rightarrow V_{\lambda+1}$, then $j^{+}(B)=B$. One can therefore prove the results about finite permutative LD-systems by working with the linear ordering that comes from $B$ instead of the linear ordering that comes from the fact that $V_{\lambda}[G]\models V=HOD$ in some forcing extension. One thing to be cautious of when one announces results before publication is that perhaps the proofs are not optimal and that one can get away with a simpler construction.

Philosophy and research project proposals

In the above results, we have worked in a model $V$ where there are non-trivial maps $j:V_{\lambda}\rightarrow V_{\lambda}$ and where $V_{\lambda}\models\text{V=HOD}$ in order to obtain compatible linear orderings on finite algebras. However, if we work in different forcing extensions with rank-into-rank embeddings instead, then I predict that one may obtain from large cardinals different results about finite algebras.

I predict that in the near future, mathematicians will produce many results about finite or countable self-distributive algebras using forcing and large cardinals where the large cardinal hypotheses cannot be removed. I also predict that rank-into-rank cardinals will soon prove results about structures that at first glance have little to do with self-distributivity.

The question of consistency

I must admit that I am not 100 percent convinced of the consistency of the large cardinals around the rank-into-rank level. My doubt is mainly due to the existence of finite reduced Laver-like LD-systems which cannot be subalgebras of $\mathcal{E}_{\lambda}/\equiv^{\gamma}$. However, if no inconsistency is found, then the results about finite or countable structures that arise from very large cardinals would convince me not only of the consistency of very large cardinals but also the existence of these very large cardinals. Therefore, people should investigate the finite algebras which arise from very large cardinals in order to quell all doubts about the consistency or the existence of these very large cardinals.

Since it is much more likely that the Reinhardt cardinals are inconsistent than say the I1 cardinals are inconsistent, I also propose that we attempt to use the algebras of elementary embeddings to show that Reinhardt cardinals are inconsistent. I have not seen anyone investigate the self-distributive algebras of elementary embeddings at the Reinhardt level. However, I think that investigating the self-distributive algebras of elementary embeddings would be our best hope in proving that the Reinhardt cardinals are inconsistent.


text:
A basic theorem in communications is the matched filter maximizes the SNR at sampling. I'm a little confused on how this relates to discrete time systems and sampling rate.

Normally if you sample at Nyquist rate it's "good enough" but how does this interact with the discrete time correlator? Does sampling at a higher rate benefit the system? Does up-sampling the signal and passing it through the correlator yield the same benefit?

I understand the idea of Noise being White and the output of white noised passed through the correlator limits its PSD but lets say your analog signal is pre-filtered to remove bands outside of some region and your sample rate meets the Nyquist rate for this bandwidth. Is there any benefit to over-sampling and then discrete time matched filtering?

• Just a snide remark: that theorem says that MF maximizes the SNR for a channel where only additive uncorrelated noise happens to the signal. It's not generally true that matched filtering maximizes SNR. – Marcus Müller Apr 22 '19 at 8:05

Normally if you sample at Nyquist rate it's "good enough" but how does this interact with the discrete time correlator?

It interacts very well! See the following:

Does sampling at a higher rate benefit the system?

You mean oversampling.

If you filter your signal to the original signals bandwidth, which is now smaller than the Nyquist rate. You thus get a component that is correlated between samples (which is your signal) and one that isn't (which is your noise). By averaging / downsampling your signal, you increase the variance of your signal component more than that of your noise component – you get an SNR gain.

Does up-sampling the signal and passing it through the correlator yield the same benefit?

No, because upsampling doesn't "uncorrelate" the noise that you already digitize.

Is there any benefit to over-sampling and then discrete time matched filtering?

Yes, as described above, the oversampling allows you to increase your SNR. Think of it this way:

Assume that if critically sampled, your pulse length is $$N$$; thus, your MF does a correlation over $$N$$ samples.

You oversample your analog signal by a factor of $$f>1$$. You interpolate your MF by the same factor, so that the pulse and the filter are now $$Nf$$ in length.

While the power of the samples stays the same, the amplitude of the signal part in the (digital) correlator output has been scaled by a factor of $$f$$, which means the power has been scaled by $$f^2$$.

For the noise component in the correlator output: White Noise samples are uncorrelated, and we know that thus the variance of their sum (which is what a correlator is, weighted sum) is the sum of their variances. Since variance is power for zero-mean noise, that means we get a noise power increase of $$f$$.

Thus, oversampling by $$f$$ gets us an SNR increase of $$\frac{f^2}f$$ :)

• Marcus, are you saying that if I oversample by a factor of one million, I'll get a one million increase in SNR? – MBaz Apr 22 '19 at 16:07
• Thanks, as a couple questions. 1. If the AWGN has already passed through a bandpass filter, it's no longer white and the noise is then correlated? So it seems the argument doesn't work. 2. I still don't understand why over-sampling would increase SNR, if you sample at the Nyquist rate for your bandpass filter, shouldn't you be able to reconstruct the signal perfectly? It seems you could then apply the matched filter in the frequency domain with maybe a lower sample rate(but it would need to be longer impulse response approximation). – FourierFlux Apr 22 '19 at 18:37
• Also I always wondered, do ADCs take the average value over the interval or the instantaneous value at sampling time? Seems the later would be more prone to noise but I guess that's an ASIC question. – FourierFlux Apr 22 '19 at 18:38
• We typically idealize them to do the latter. You'll find that in reality, they inherently do some integration (typically by charging a sample and hold capacitor), or that to be honest things are quite a bit more complex; if you look at sigma-delta converters, you'll find that these, in the process of taking one measurement, have spectral shaping properties. That's a topic worth reading on, by the way – it's an interesting concept :) – Marcus Müller Apr 23 '19 at 5:51
• Ok thanks, any additional input on the other two questions? – FourierFlux Apr 23 '19 at 15:46


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# 5.02 Fundamental group of a CW complex

## Video

Below the video you will find accompanying notes and some pre-class questions.

## Notes

### Example of the 2-torus revisited

(0.00) In the previous video, we saw how to use Van Kampen's theorem to compute the fundamental group of the 2-torus. The result was $\langle a,b\ |\ b^{-1}a^{-1}ba=1\rangle.$ There is a CW structure on the 2-torus (square with opposite sides identified) with one 0-cell (vertex), two 1-cells (edges) and one 2-cell (the square itself). It is no coincidence that the presentation we obtained has:

• one generator for each 1-cell,
• one relation for each 2-cell: the relation comes from looking at the boundary of the 2-cell.

### Fundamental group of a CW complex

(1.24) If $$X$$ is a CW complex with one 0-cell (for simplicity) then $$\pi_1(X)$$ has a presentation where the generators are the 1-cells and the relations come from the 2-cells. More precisely,

• each 1-cell is a loop (as both ends attach to the unique 0-cell). These loops generate $$\pi_1(X)$$;
• (3.05) the boundary of each 2-cell $$e$$ gives a loop $$\partial e$$ in the 1-skeleton which we can write as a word in the 1-cells; since this loop is nullhomotopic in $$e$$ we need the relation $$\partial e=1$$ in $$\pi_1(X)$$. These relations suffice to give a presentation of $$\pi_1(X)$$.

(5.18) Note that this theorem is about cell complexes of any dimension; in other words, the 3-cells, 4-cells, etc. do not affect the fundamental group.

(5.38) Let $$X^1$$ be the 1-skeleton of $$X$$. This is obtained by attaching 1-cells to a point, so $$X^1$$ is just a wedge of 1-cells $$X^1=\bigvee_{1\mbox{-cells}}S^1$$. By induction, using the computation from last time, $$\pi_1(X^1)=\bigstar_{1\mbox{-cells}}\mathbf{Z}$$ (i.e. the free product of copies of the integers, one for each 1-cell, in other words, the group with one generator for each 1-cell and no relations).

(7.42) What happens when we attach a 2-cell?

Claim: If $$Y$$ is any space and $$Z$$ is obtained from $$Y$$ by attaching a 2-cell $$e$$ with some attaching map $$\varphi$$ then $$\pi_1(Z)=\pi_1(Y)/N(\partial e)$$, where $$N(\partial e)$$ is the normal subgroup generated by the boundary of $$e$$ (that is $$[\varphi]\in\pi_1(Y)$$). This is equivalent to imposing the relation $$\partial e=1$$.

(9.48) Proof of claim: This follows from Van Kampen's theorem. If we decompose $$Z=U\cup V$$ where $$U$$ is the interior of the 2-cell $$int(e)$$ and $$V$$ is the union $$Y\cup (e\setminus\{0\})$$ (that is, $$V$$ is everything except one point in the interior of $$e$$).

• The intersection $$U\cap V$$ is $$e\setminus\{0\}$$, the punctured 2-cell. That is homotopy-equivalent to a circle.
• $$U$$ is contractible.
• $$V$$ is homotopy equivalent to $$Y$$ (retracting the punctured 2-cell down onto its boundary in $$Y$$).

(11.45) Van Kampen's theorem tells us that $$\pi_1(Z)=\pi_1(Y)\star_{\mathbf{Z}}\{1\}$$. In other words, we take $$\pi_1(Y)$$ and we add an amalgamated relation $$\partial e=1$$ ($$\partial e$$ generates $$\pi_1(U\cap V)$$: in $$\pi_1(U)$$ is becomes trivial and in $$\pi_1(V)$$ it becomes $$\partial e$$).

(13.28) What about adding higher dimensional cells? By the same argument, where $$e$$ is an $$n$$-cell with $$n>2$$, we get $\pi_1(Z)=\pi_1(Y)\star_{\{1\}}\{1\},$ because $$U\cap V$$ is now homotopy equivalent to the $$(n-1)$$-sphere, which is simply-connected (so there is no new amalgamated relation).

## Pre-class questions

1. Given that the quotient of the octagon by the identifications indicated in the figure below is a genus 2 surface, use Van Kampen's theorem to give a presentation for the fundamental group of a genus 2 surface.


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## [AAR][Class] 18:00 NNitH @10.06.18

Forum rules
This forum can be viewed publicly.

Member

### [AAR][Class] 18:00 NNitH @10.06.18

Ran my second class in EVE, introducing folks to living at the Wormhole Campus! Based on feedback from the first class, reduced the knowledge portion (to be covered in WH 101 and 201) and focused on actual regular activities in the campus.

Participants:
SPOILER WARNING!
Athena FireflyDropbear MaroonGhad MakanenKiuna McGullygoshKranamor AndedareLiability ManNicholas PilgerPhileas LevesqueDevola Gestalt

Formed in Stacmon, reviewed fits and headed out early. We had two folks who had never been in a wormhole so it was exciting to get them to see this vast part of the EVE Universe! We toured the campus wormhole and headed out to farm content in our static C3. We had some activity in the chain and eventually bombers came into system. We had finished several sites and were already heading out so we headed back to WHC quickly. Feedback form was linked and the students returned to Hisec.

Big thanks to Kabal for assisting directly with the class, and all the campus members that prepared by rage rolling and joined during class in welcoming the participants to the campus and helping run the sites.

Learnings
More action less talk was well received for the NNitH
Would be good to get a little PVP, maybe by having a WHC member fleet out hunting while we run PVE.
Didn't have an easy Gas site to get to this time around - maybe roll bacon for something suitable
Needed more time to rage roll a good connection ahead of time. We started 3 hours early, will plan for 4 next time.

Thank you for joining the class and for giving me the opportunity to further learn about running classes. Feedback very much appreciated!
Last edited by Mhzentul Lafarius on 2019.05.06 00:32, edited 1 time in total.

Member

### Re: [AAR][Class] 18:00 NNitH @10.06.18

Hi,
It seems I was forgotten in the Participant list. The young logibro I am always appreciate some gratitude (and a share of the ISK of the fleet, we all have to pay our ships somehow ).

Thanks for the fleet. It was nice for once to be the big fish ruling the wormhole, and not the solo explo frigate who has to hide .

Member


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# I built a Sudoku Solver!

I built a sudoku solver using Javascript by implementing the Exhaustive Recursion and Backtracking algorithms.

###### Abdush Shakoor ● April 29th, 2017

Before you read about this article, try playing the Sudoku Solver above. You can only insert numbers ranging from 1 to 9 in any empty cell. Press "Random Level" to generate a random sudoku puzzle. Made a mistake? Press "Reset" to clear the board. To test the algorithm, Press "Solve" to complete the entire puzzle.

As for the source code, you can view it in my GitHub repository or can be found near the end of this article.

## What is Sudoku?

Sudoku stems from the two Japanese words: "Su" and "doku", when translated to English, it means "single numbers only". Although, many people feel that Sudoku was invented by the Japanese, it was actually invented by an American named Howard Garns in 1979 under the name "Number Place" but died in 1989 before a Japanese publisher named Nikoli made it popular, renamed it as "Sudoku" and which of course, became mainstream in 1986.

Sudoku is a logic-based, combinatorial, 9x9 number grid puzzle. The puzzles are usually a partially completed grid, thus allows the puzzle solver to come up with a single solution to the puzzle. The objective of this game is to fill the empty spaces in the partially completed 9x9 grid in such a way that each row, column and 3x3 subgrids (that compose the grid) contains all digits from 1 to 9.

## Why did I build it?

Well, one day, I downloaded a Sudoku game on my iPhone and I got hooked to it instantly. I would play the game, usually, during metro rides to my university or any long trips to not make myself feel bored. So one day, I thought "What if I built a sudoku solver and see if it could solve all the puzzles from the game?", which made me even more curious to see how a computer would do it.

## Implementation

Okay, I know this is the part that you must be waiting for: "How to implement this?". As I was doing my research on how to implement it, I came across two algorithmic techniques:

### Recursion

Recursion is an algorithmic technique used for breaking down larger problems into smaller instances of the same problem. This is also known as Divide-and-Conquer method. This is usually used in binary search, reversing a file, generating large Fibonacci numbers and the list could go on.

### Backtracking

Backtracking is an algorithmic technique used for searching every possible combination in order to solve a problem in an optimized way. You could be familiar with this technique in popular problems like Chess, the 8 Queen Puzzle or Crossword solvers.

### Explanation

Here's a pseudocode on how this strategy would work:

Find [row,col] of an unassigned cell
If there is none, then return true

For digits from 1 to 9
If there is no conflict for digit at [row,col]
Assign digit to [row,col] and recursively try to fill in the rest of the grid
If this recursion is successful, return true
Otherwise, remove this digit and try another one
If all digits have been tried for this cell and nothing worked out
return false, trigger backtracking and try again

Javascript implementation:

//Sudoku solver
function solveSudoku(grid, row, col) {
var cell = findUnassignedLocation(grid, row, col);
row = cell[0];
col = cell[1];

// base case: if there's no empty cell
if (row == -1) {
return true;
}

for (var num = 1; num <= 9; num++) {

if (noConflicts(grid, row, col, num)) {
grid[row][col] = num;

if (solveSudoku(grid, row, col)) {
return true;
}

// mark cell as empty (with 0)
grid[row][col] = 0;
}
}

// trigger back tracking
return false;
}

//Find an empty cell
function findUnassignedLocation(grid, row, col) {
var done = false;
var res = [-1, -1];

while (!done) {
if (row == 9) {
done = true;
}
else {
if (grid[row][col] == 0) {
res[0] = row;
res[1] = col;
done = true;
}
else {
if (col < 8) {
col++;
}
else {
row++;
col = 0;
}
}
}
}

return res;
}

//Check if there are any conflicts in row, column or 3x3 subgrid
function noConflicts(grid, row, col, num) {
return isRowOk(grid, row, num) && isColOk(grid, col, num) && isBoxOk(grid, row, col, num);
}

//Check if this number is valid in this row
function isRowOk(grid, row, num) {
for (var col = 0; col < 9; col++)
if (grid[row][col] == num)
return false;

return true;
}

//Check if this number is valid in this column
function isColOk(grid, col, num) {
for (var row = 0; row < 9; row++)
if (grid[row][col] == num)
return false;

return true;
}

//check if this number is valid in 3x3 subgrid
function isBoxOk(grid, row, col, num) {
row = Math.floor(row / 3) * 3;
col = Math.floor(col / 3) * 3;

for (var r = 0; r < 3; r++)
for (var c = 0; c < 3; c++)
if (grid[row + r][col + c] == num)
return false;

return true;
}

## What's next?

So now that you have understood the rough concept of how it works, combining and applying these two techniques in a Sudoku puzzle i.e. trying out every possible number to fill an empty square in the 9x9 grid would allow the computer to solve Sudoku puzzles more efficiently and quickly than using a plain Brute Force technique on puzzle that has around 4 x 1038 possibilities (which means, you'll probably require next-gen hardware to compute one puzzle that could take billions of years!)

Oh, you can play around with the Sudoku Solver above by trying out a Sudoku puzzle or generate a random one and solve it. Or best, if you'd like to challenge yourself, you could try to implement this solution on Project Euler's Problem 96 by solving all 50 sudoku puzzles in a row!


text:
## Wednesday, August 13, 2014

### First female Fields medalist

Off-topic: If you want to learn how – as a conference speaker – you should refer to a very nice blog post in a way that is both accurate and entertaining for the audience, see Suvrat Raju's Strings 2014 talk, 25:40-25:50. Thanks, Suvrat. ;-) Some of the people are laughing because by mentioning "a very nice blog post", Suvrat had offered them too little information – approximately by $$k\ln(6,000)$$ because as of today, there are 6,000 very nice TRF blog posts. ;-)
Using the normal distribution, La Griffe du Lion has predicted a female winner of the Fields medal, the most well-known prize in mathematics, to surface once in 103 years. If you have won a Fields medal yourself, you may calculate that four winners are announced every 4 years which means that it's $$4/4=$$ one winner per year in average.

After 70 years of the award, we have the first female Fields medalist. Maryam Mirzakhani is Persian, was born in Tehran in 1977, was trained at Harvard and received her PhD in 2004, and is currently a Stanford professor.

Looking at the list of papers, I wouldn't be able to recognize them from some above-average papers in mathematics – but I don't belong to that culture. But she has surely published an impressive amount of papers.

No one cares about men but
The other three winners this year were Artur Avila of France, Manjul Bhargava of Princeton University in New Jersey, and Martin Hairer of the University of Warwick in Britain.
Maryam has been my fellow Harvard Junior Fellow for some time – I think that we even overlapped for a whole year, 2003-2004 – but my memory is limited. I remember her first name, however. Let me admit that I only realized that an hour after I posted the original version of this blog post.

Congratulations to the winners and especially Maryam!

#### 33 comments:

1. I'm here to point that Artur Ávila is brazilian and studied in Brazil most of the time. Nowadays he works both at IMPA (his alma mater) and CNRS.

2. Courageous CowardAug 13, 2014, 8:14:00 AM

Maryam Mirzakhani is married to Jan Vondrak who is Czech and an alumni of Charles University (BS in Physics, 1995).

3. Great, I didn't know. ;-)

4. Congratulations on 6000 great posts, Lubos! Looking forward to the next 6K.

5. Since Maryam was at first more interested in reading and writing fiction than doing mathematics, I thought she might enjoy Kurt Vonnegut's short clip "On the Shapes of Stories":

https://www.youtube.com/watch?v=oP3c1h8v2ZQ

I doubt it’s possible for Vonnegut to string 5 words together without getting a chuckle.

6. Lubos,

Maryam Mirzakhani was born in Iran... IRAN!!!!!

How the hell was she able to get on to the Iranian IMO team when women were/are treated as second class citizens compared to men under Islamic law in Iran?

http://www.imo-official.org/participant_r.aspx?id=926

It's really laughable that of all the places on the planet where the first female Field's Medalist comes from it's... ladies and gentlemen...

India? likely
Pakistan? dude, that's an Islamic state
US? almost certainly
Iran? you're on crack, get the hell out of here

Nope, it's.. IRAN! IRAN! IRAN! IRAN! IRAN! IRAN! IRAN! IRAN! IRAN!

(leaves Lubos's blog. screaming and waving hands hysterically... IRAN!...iran!..ir...)

7. Lubos, something that needs also to be mentioned: actually Avila is the first brazilian (professor at IMPA, RJ) to be awarded with a Fields medal: http://www.simonsfoundation.org/quanta/20140812-a-brazilian-wunderkind-who-calms-chaos/ ... maybe you could invite him for a post.

8. Thank you very much, Ann, also for being around for this long time!

Today, I also exceeded 100,000 reputation on Stack Exchange so they want me to pick some "swag".

http://stackexchange.com/catalog

What would you choose? ;-)

9. SteveBrooklineMAAug 13, 2014, 4:37:00 PM

This review of Hairer's work is really something:

http://www.simonsfoundation.org/quanta/20140808-in-mathematical-noise-one-who-heard-music/

I wonder these Quanta pieces are a bit over the top. I can't really judge. Congratulations to all the winners!

10. It appears the chances of a woman winning the medal are about the same as that of a Canadian. 2014 was the first for both categories, although it appears Manjul Bhargava was mearly born in Canada. Maybe I can take solace in the fact that the medal was named/founded by a Canadian. That and that there are no Czech winners :)

Somewhat interesting story on how the Fields Medal rose to prominence:

http://www.nytimes.com/2014/08/10/opinion/sunday/how-math-got-its-nobel-.html

11. The black jacket and gray hoodie are kind of cool, and fall is coming along soon. But, it's surely your preference. Congrats on the big points!

12. Blogger Contest - You Can JoinAug 13, 2014, 4:44:00 PM

Blogs with racing you make your blog a candidate to win the promotional prizes. With the support of our sponsors from blogs are organizing events within the first round of our competition. Add your blog and get involved. You refer to your blog. You can find detail information on our website. Thank you for your interested-in.

Sincerely,
Web: http://www.bloggercontest.info
Mail: contact@bloggercontest.info
Cell: +441269200450

13. Quanta Magazine has a very cute picture of Maryam as a child dressed up as a nurse.

http://www.simonsfoundation.org/quanta/20140812-a-tenacious-explorer-of-abstract-surfaces/

Little girls today don't dress up as a nurse like they used to. They will however dress up as a doctor. I suppose they do this because they have a strong desire to grow up to become a rich and powerful figure in an industry that's awash in cash. But there are a growing number of hospital CEOs that got their start as a nurse. And a growing number of these CEOs are provided with an income that far exceeds that of the highest paid physician at the hospital.

Perhaps if little girls understood this shift in pay from hospital physicians to hospital executives, they would go back to dressing up as a nurse. But they need to be informed that these nurses-turned CEOs aren't real nurses. Very few of them ever took care of a patient at the bedside. And the ones that did were so lousy at it that they had to leave the bedside before they were stripped of their license to practice nursing. They merely obtained a nursing degree in order to increase their odds of landing an overpaid, underworked job in the corporate suite. Health care would be much more affordable and the quality of care would be much better if more healthcare dollars would leave the corporate suite and return to the bedside, where they belong and can do the most good for our healthcare system.

14. Lubos's brain must have been exploded. How can this fit into his world view?

15. +1 for the national organization for development of exceptional talents in iran. the uprising of most talented children must be a priority for any educational system. they are precious. it reminds me a very nice blog post ;) on ayn rand, where i found something that could totally apply here : i don't hate the low achieving students; i just don't believe they're the best and most important things in life and that their convenience is (or should be) the driving powering the people creativity.

16. http://arxiv.org/abs/1108.0174 page 31 forward has Mirzakhani's methods
proving a conjecture of Witten (already proved differently by
Kontsevich).

17. Would this make her the Ed Witten of the female world?

18. Apologies, I've seen this excitement elsewhere but I don't understand this excitement at all. Every person is from *some* country and Brazil is large enough to produce some winners but too weak to produce too many winners, so it's to be expected that winners come from similar countries. Maryam is the first Iranian winner, too, is that an equally big deal? A winner is either from a math superpower or not - comparable amounts. In the latter cases, it's rather likely that he or less likely she will be the first from his or her country.

19. Cynthia: Hospital administrators are like aphids---they clone themselves. Unless you eliminate all of them, the population explodes.
Aphids feed on the sap of phloem vessels of plants. Admin. feed off of the hospital budget, and the results on the hospital organism is the same... :) In 197x, the tertiary hospital where I live had one administrator, one secretary. One could see him to discuss patient care any time. There has been much construction since, but also many bed closures, so the number of active beds is equivalent, and now there are hundreds of administrators. They metastasized and took over the entire nursing school bldg (nursing ed moved to the University). And the people wonder why it takes so long now to get tests, or have surgical procedures....

20. It's terrible, Gordon. Who is really in charge of the hospitals? And if it's some physicians, why can't they fight this metastasis?

21. Thanks for linking to Strings 2014, a chance for laymen to listen to the way real physicists talk to each other. I thought Andy Strominger's overview of the state of string theory was particularly good. Steinhardt's review of BICEP 2 and of inflation in general seemed persuasive to me; were there flaws in his argument?

22. Dear Luke, concerning Steinhardt, do you mean this talk?

https://www.youtube.com/watch?v=_RpwuHOHiB4

Sorry, after 6 minutes or so, I got so upset that I won't be able to complete watching this junk. Witten said that Steinhardt was the bad cop. Oh, really? Is Steinhardt is a cop, who is the felon?

From the first minute, it's a demagogy about his being the only advocate of the scientific method - he only applies these talking points to others' claims and discoveries, not to his own musings. They could equally well invited a Shmoit to the conference, too.

Inflation is an extremely robust, universally applicable theory that explains pretty much all the features of cosmology that are left unexplained by the Big Bang with virtually no assumptions, and BICEP2 is the cleverest experiment done so far to learn something that goes beyond the simple "inflation Yes or No" question. Any amount of general preaching about what is science, if it is directed against inflation and/or against BICEP2, only shows that he a dishonest demagogic asshole, and if you find this šit "persuasive", it only shows that it makes absolutely no sense to promote physics because a layman will always find the exact opposite of physics to be at least equally persuasive on the following day.

23. First of all, you know I am a layman (John Q. Public) who comes here to learn. I trust your opinion or I wouldn't bother to ask. ("The trouble with the global village are all the village idiots." -- who said that?)

Anyway, to his credit Steinhardt did say that if the BICEP 2 results hold up then his own theory (of a big crunch before the big bang) could be ruled out.

Would this leave some form of inflation the only theory standing? What about string gas cosmology? Didn't its author state (on this blog) that it made a prediction about the direction of the polarization counter to that made by theories of inflation? Or is that only true about some theories of inflation?

Anyway, if inflation comes in many different forms then it reminds me a little of Darwin's theory of natural selection at the time that he first put it forward: in some sense his was more of a scenario than a theory because he wasn't sure what the exact mechanism of variation was. He didn't know about genes and even thought Lamarck's ideas about the heritability of acquired characteristics might have some merit. Even so Darwin was basically correct -- and not just about natural selection, but sexual selection too, which, it turns out, may be every bit as important, at least for human beings!

Or do you think this is a bad analogy?

As another possible alternative to inflation, what about String Gas Cosmology? I believe the author of that hypothesis claimed (on your blog) that the polarization observed by BICEP 2 was of a kind (blue or red, I forget which) that was predicted by his theory but not by inflationary theories.

24. Dear Luke, people have various views on particular theories. I don't think that string gas cosmology is a viable replacement for inflation in the very mysteries that inflation has actually demystified. As far as I can say, these general features of inflation are much more important for the appraisal of the theory and its alternatives than some - mostly speculative - focus on some technical features of an emerging experiment such as deviations from the scale invariance etc.

So I keep string gas cosmology in mind but its probability isn't comparable to that of inflation, by far.

If you want to argue that Darwin's theory isn't a theory, just a "scenario", then of course one may say the same thing about inflation. I don't understand what's the point of your trying to remove the status of these theories as "theories". They are clearly theories. They are scientific theories about extremely important, "systemic" questions underlying the world - so important that one could call them "metaquestions".

But words like "metaquestions" are just words. Of course that such metaquestions are questions, too, and it just became possible for science to accumulate a large amount of evidence that helps to determine the truth value of such questions.

I think that the comparison between Darwin's theory and inflationary theory is a good one - but your claims are equally wrong about both of them. Darwin's theory is undoubtedly a theory, an existentially important one in biology, and your claims that Darwin was wrong about some things (like that he didn't know sex and the role it played in selections) are just idiotic creationist fairy-tales.

25. "important one in biology, and your claims that Darwin was wrong about some things (like that he didn't know sex and the role it played in selections) are just idiotic creationist fairy-tales.

You should read more carefully what I actually said, namely, that Darwin did identify the important role of sex and sexual selection in evolution. Anyway evolution is most certainly a theory today because of all the genetic evidence which has come to light in the last hundred plus years that supports it in extraordinary detail. I was merely suggesting a possible analogy between the theory of inflation today to the theory of evolution as it existed in Darwin's day. I did not say that analogy was a good one. Instead, I asked your opinion about it.

26. Dear Luke, as far as I can say, Darwin's theory of evolution as we know it today is the very same theory that was discovered by Darwin.

Technical issues in genetics and DNA - molecular genetics - help to produce a more complete picture but I don't think it's right to count these things to Darwin's theory. Darwin's theory is on par with thermodynamics while molecular genetics is on par with statistical physics. Thermodynamics exists even without statistical physics. Statistical physics is just a more constructive, on-the-first-principles based description of these phenomena but its status doesn't mean that there is anything wrong about thermodynamics or Darwin's theory.

Again, I think that the analogy is a good one but I think you are completely wrong if you think that Darwin's theory of the origin of species was incomplete in some major way during Darwin's life.

27. Physicians lost control of the hospitals long ago. They have a non-voting rep on some administrative committees, but the govt apparatchiks have the power. Illustrative of this is that the physician reps are not paid to attend the meetings, while the admin hacks are...but enough whining.

28. --never did like the Turok-Steinhardt models. How does Lisa feel about the Ekyrotic brane collisions?

29. Dear Gordon, I forgot what she's exactly saying. If it is important, I can ask her.

Unlike inflation, I think she has never done research on cyclic/ekpyrotic stuff.

30. Nah, not important, just a stray transient thought. :)

31. The addition of molecular genetics to Darwin's theory is simply Neo Darwinism- http://en.wikipedia.org/wiki/Neo-Darwinism

Nothing wrong with Darwin's theory---just an "evolution" of it to incorporate new information---the thermodynamics/stats mechanics analogy is a good one.

32. Hi, Paul and Lubos.

Question. The tan\beta < 1 limits are extracted from the signal strengths quoted by the LHC collaborations, but it seems to me that the meaning of 'signal strength' has been misinterpreted. It's the ratio of observation to SM of \sigma(production)xBR(decay). The authors have taken it to be \sigma(H->ff)xBR(decay). Doesn't this void the method by which they bound tan\beta?

33. Most Popular Latest Hot Current Affairs on this Network, News updates, hot updates, hot news

HotCurrentAffairs.com


text:
# Voltage drop

Voltage drop describes how the supplied energy of a voltage source is reduced as electric current moves through the passive elements (elements that do not supply voltage) of an electrical circuit. Voltage drops across internal resistances of the source, across conductors, across contacts, and across connectors are undesired; supplied energy is lost (dissipated). Voltage drops across loads and across other active circuit elements are desired; supplied energy performs useful work. Recall that voltage represents energy per unit charge. For example, an electric space heater may have a resistance of ten ohms, and the wires which supply it may have a resistance of 0.2 ohms, about 2% of the total circuit resistance. This means that approximately 2% of the supplied voltage is lost in the wire itself. Excessive voltage drop may result in unsatisfactory operation of, and damage to, electrical and electronic equipments.

National and local electrical codes may set guidelines for the maximum voltage drop allowed in electrical wiring, to ensure efficiency of distribution and proper operation of electrical equipment. The maximum permitted voltage drop varies from one country to another. [1]. In electronic design and power transmission, various techniques are employed to compensate for the effect of voltage drop on long circuits or where voltage levels must be accurately maintained. The simplest way to reduce voltage drop is to increase the diameter of the conductor between the source and the load, which lowers the overall resistance. More sophisticated techniques use active elements to compensate for the undesired voltage drop.

## Voltage drop in direct-current circuits: resistance

The local voltages along a long line decrease gradually from the source to the load

Consider a direct-current circuit with a nine-volt DC source; three resistors of 67 ohms, 100 ohms, and 470 ohms; and a light bulb—all connected in series. The DC source, the conductors (wires), the resistors, and the light bulb (the load) all have resistance; all use and dissipate supplied energy to some degree. Their physical characteristics determine how much energy. For example, the DC resistance of a conductor depends upon the conductor's length, cross-sectional area, type of material, and temperature.

If you measure the voltage between the DC source and the first resistor (67 ohms), you will notice the voltage potential at the first resistor is slightly less than nine volts. The current passes through the conductor (wire) from the DC source to the first resistor; as this occurs, some of the supplied energy is "lost" (unavailable to the load), due to the resistance of the conductor. Voltage drop exists in both the supply and return wires of a circuit. If you measure the voltage across each resistor, you will measure a significant number. That represents the energy used by the resistor. The larger the resistor, the more energy used by that resistor, and the bigger the voltage drop across that resistor.

You can use Ohm's Law to verify voltage drop. In a DC circuit, voltage equals current multiplied by resistance. $V = I R$. Also, Kirchhoff's circuit laws state that in any circuit, the sum of the voltage drops across each component of the circuit is equal to the supply voltage.

## Voltage drop in alternating-current circuits: impedance

In alternating-current circuits, opposition to current flow does occur because of resistance (just as in direct-current circuits). Alternating current circuits also present a second kind of opposition to current flow: reactance. This "total" opposition (resistance "plus" reactance) is called impedance. The impedance in an alternating-current circuit depends on the spacing and dimensions of the elements and conductors, the frequency of the alternating current, and the magnetic permeability of the elements, the conductors, and their surroundings.

The voltage drop in an AC circuit is the product of the current and the impedance (Z) of the circuit. Electrical impedance, like resistance, is expressed in ohms. Electrical impedance is the vector sum of electrical resistance, capacitive reactance, and inductive reactance. It is expressed by the formula $E = I Z$, analogous to Ohm's law for direct-current circuits.

## Voltage drop in building wiring

Most circuits in a house do not have enough current or length to produce a high voltage drop. In the case of very long circuits, for example, connecting a home to a separate building on the same property, it may be necessary to increase the size of conductors over the minimum requirement for the circuit current rating. Heavily-loaded circuits may also require a cable size increase to meet voltage drop requirements in wiring regulations.

Wiring codes or regulations set an upper limit to the allowable voltage drop in a branch circuit. In the United States, the 2005 National Electrical Code (NEC) recommends no more than a 5% voltage drop at the outlet.[1] The Canadian electrical code requires no more than 5% drop between service entrance and point of use.[2] UK regulations limit voltage drop to 4% of supply voltage. Following changes to the BS7671:2008 on consumers' installation, the following has become in force since 1 July 2008:

Type of Supply Voltage drop lighting Voltage drop other
DNO 3% 5%
Private 6% 8%

Voltage drop of a branch circuit is readily calculated, or less accurately it can be measured by observing the voltage before and after applying a load to the circuit. Excessive voltage drop on a residential branch circuit may be a sign of insufficiently sized wiring or of other faults within the wiring system, such as high resistance connections.

Note: voltage drops for installations are specified from the point of common coupling (i.e. where the utility supply is connected to the premises) to the point at which the electrical load is connected[3]

## How to calculate voltage drop

In situations where the circuit conductors span large distances, the voltage drop is calculated. If the voltage drop is too great, the circuit conductor must be increased to maintain the current between the points. The calculations for a single-phase circuit and a three-phase circuit differ slightly.

Single-phase voltage drop calculation:

Three-phase voltage drop calculation:

VD = Voltage drop (conductor temp of 75°C) in volts

VD% = Percentage of voltage drop (VD ÷ source voltage x 100). It is this value that is commonly called "voltage drop" and is cited in the NEC 215.2(A)(4) and throughout the NEC.

L = One-way length of the circuit's feeder (in feet)

R = Resistance factor per NEC Chapter 9, Table 8, in ohm/kft

I = Load current (in amperes)

Source voltage = The voltage of the branch circuit at the source of power. Typically the source voltage is either 120, 208, 240, 277, or 480 V.

For the calculation use NEC Chapter 9 Table 9 "Alternating-Current Resistance and Reactance for 600-Volt Cables, 3-phase, 60 Hz, 75°C (167°F) - Three Single Conductors in Conduit".

Important Note: According to NEC 215.2(A)(4) informational note No. 2, the voltage drop for feeders should not exceed 3% and the voltage drop for branch circuits should not exceed 5%, for efficient operation.[4]

## Using higher voltages

Over long distances, larger conductors become expensive, and it is preferable to redesign the circuit to operate at a higher voltage. Doubling the voltage halves the current required to deliver the same amount of power, halving the voltage drop, and an additional doubling in efficiency is realized because that drop is a smaller fraction of the total voltage.

This is the motivation for commercial high voltage electrical power distribution, and for the use of the +12V power supply rail for high-power loads in modern personal computers.


text:
# Biber suddenly unstable

Suddenly biber (under bibtex) has become unstable in the sense that it occasionally doesn't compile citekeys into references. In practice this means that in the pdf, instead of the reference, the citekey is printed.

When manually running Tools -> Commands -> Biber it works again ... for some time.

Any suggestions on how to fix that?

The problem occurred again (after several successful compilations). I have added the log file (is there better way to add it?):

'firstinits' option is deprecated, use 'giveninits' instead.
Upgrade package 'polyglossia' to >v1.42.0 recommended.
'babel/polyglossia' detected but 'csquotes' missing.
Conflicting options.
Not defining \perthousand.
Not defining \micro.
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Please (re)run Biber on the file:(biblatex) Inquiry_based_mathematics_education_and_lesson_study(biblatex) and rerun LaTeX afterwards.

• if there is an error in your bib file (or citation reference) biber may fail to generate a bibliography in which case biblatex will act as it acts on the first run, and show the key. There is no generic solution to avoiding errors, if you show the log file or source with an error someone could suggest a fix. – David Carlisle Jul 19 '18 at 12:31
• @DavidCarlisle, thanks for pointing that out. I should of course have pasted the log file. I'll do that when it occurs again. – jbahn Jul 19 '18 at 12:34
• How are you normally compiling? If Tools -> Commands -> Biber this sounds like whatever tool you use for compiling, it might not always run biber – user36296 Jul 19 '18 at 12:51
• You mean under biblatex and not under BibTeX, right? If you apply changes to your .bib file or change bibliography-sensitive bits of your document, LaTeX will prompt you for a Biber-(re-)run in the .log file/warnings. You should then run Biber. If your editor does not automatically run Biber for you on each compilation that means you have to 'manually' initiate the Biber run. If the Biber run finished successfully everything is fine, if you get warnings or errors from Biber (in the .blg file), you must fix these before you can continue. It is not clear to me what exactly you are doing – moewe Jul 19 '18 at 13:10
• ... Can you give more details? Did you instruct your editor to run Biber at all times automatically for you or do you have to call it manually? When or how does the output become unstable? Are all citations affected or only new (updated, changed) ones? – moewe Jul 19 '18 at 13:11

Usually Biber does not become occasionally unstable. Biber might be more sensitive than BibTeX in some regards, but usually that should not result in erratic occasional problems, but in clear cut errors that are always reproducible (I remember one elusive SIGSEGV with weird name formats a while ago, but that is about it).

There are a few possible explanations for your problems. To understand what is going on, we first need to understand what Biber does for the bibliography. Paul Stanley explains that much better than I ever could in Question mark or bold citation key instead of citation number and you should definitely read his answer (completely!, note in particular that what he says about BibTeX is roughly also applicable to Biber, Biber is biblatex's BibTeX replacement) before you continue here. Just to recap:

• LaTeX logs the entries you cite and the order in which you cite them in an external auxiliary file (with biblatex and Biber that is the .bcf, otherwise the .aux file).
• Biber reads the .bcf and gets the requested entries from the .bib file.
• Biber parses the relevant entries into a format that LaTeX or rather biblatex can understand.
• Then Biber sorts the citations and writes the sorted list of citations into an auxiliary file (the .bbl) in a format that biblatex can easily digest.
• LaTeX then reads the entries from the auxiliary .bbl file and uses the data to construct the citations and the bibliography.

That means that you at least need to compile your document (mydoc.tex, say) with LaTeX, Biber, LaTeX to be sure to obtain a complete bibliography, in particular

• LaTeX: pdflatex mydoc (of course you can also use any other flavour of LaTeX, for example xelatex or lualatex)
• Biber: biber mydoc
• LaTeX: pdflatex mydoc
• LaTeX: pdflatex mydoc

often - but not always - the last LaTeX run is required at the end to get page numbers and cross references right, but that need not concern us here.

The bibliography data is only available and usable for LaTeX after a Biber run. Naturally only the data collected at the point of the Biber run is available. If you change the contents of your .bib file after the Biber run, these changes will only propagate to the output of your document after a new Biber and subsequent LaTeX run.

LaTeX will usually prompt you to run Biber or LaTeX again in the .log files if it can detect that a new run is required. That message is usually classified as a warning, so that your editor should show it quite prominently if it parses the log output for you. But LaTeX can not detect all cases in which a Biber re-run is required, it does for example not monitor your .bib file for changes. An automatic tool like latexmk can help you there.

In general you do not need to call Biber every time you call LaTeX. If you just added a bit of text, a new Biber run is often not necessary. But you need to run Biber in the following cases, the reasons marked with an asterisk are normally detected automatically, so that LaTeX reminds you to run Biber if you happen to forget it.

• An entries cited in the document is modified in the .bib file.
• *An entry that was not cited before is cited (the new entry needs to be added).
• *The order of citations is changed (depending on the sorting scheme and other settings).
• *An entry that was cited before is not cited any more (and needs to be removed).

If you notice certain citations missing or misbehaving it can be a good idea to run Biber (and then LaTeX again).

When LaTeX can not find the data for a specific entry in the .bbl file, it just prints the key of the entry in bold and issues a warning. There are several possible causes for such a warning

• There is a typo in the entry key, i.e. sigfridson instead of sigfridsson.
• The entry is not present in the .bib file.
• The entry is 'new' (either in the .bib file or the .tex file) and was not present when the Biber was called the last time.

At the moment what you are describing suggests that your editor does not run Biber for you all the time. That is, as outlined above, not a bad decision, since a Biber run takes a bit of time and is not always required. That means, however, that you need to initiate the Biber run explicitly when it is called for.

• Call Biber whenever you change something bibliography-related in your document or .bib file.
• Call Biber when LaTeX issues a warning to that effect.
• Call Biber when you notice that your citations appear off or broken.

If the full compilation circle LaTeX, Biber, LaTeX, LaTeX does not help to resolve the bad output, you may want to try to delete the temporary files (.aux, .bbl, .bcf, ...) and recompile with LaTeX, Biber, LaTeX, LaTeX from scratch.

If you get any errors on the first (clean) LaTeX run, these must be fixed before you continue. You can ignore warnings about missing citations and references, but all other warnings should ideally also be fixed before you continue. Errors and warnings on the Biber run should also be fixed. On the second LaTeX run you should not get any new errors and warnings. If you get errors even though the first LaTeX run did not cause errors, the cause for your errors lives in your .bib` file. On the last LaTeX run you should obtain a error- and warning-free document with correct citations and complete bibliography.

You can find further hints for troubleshooting Biber in Troubleshooting for biber


text:
Lars Relund Nielsen
Department of Economics and Business Economics
Fuglesangs Allé 4
DK-8210 Aarhus V
Office 2621-114
Phone: +45 871 65145/+45 61 300 299 (mobile)
E-mail: lars@relund.dk
ORCID: 0000-0002-4802-3071

Cand.scient.oecon. (M.Sc.), Ph.D. 2004 in Mathematics and Economics at the University of Aarhus. Professor at Department of Economics and Business Economics, Business and Social Sciences, Aarhus University.

## Using LocalSolver to solve VRP problems

17. May 2017

The last days I have been playing a bit with LocalSolver and tried to model various vehicle routing problems (VRP). LocalSolver is a heuristic solver based on a heuristic search approach combining different optimization techniques. It includes a math modeling language (LSP) and there is a package so that it can be used from R. You need to download and install LocalSolver (I used v7.0) and get an academic license from their webpage. Læs mere »

## Plotting IP and MO-IP models in R

20. February 2017

I recent released a small package gMOIP which can make 2D plots of linear and integer programming models (LP/IP). With the package you can make plots of the feasible region (or solution space) of an LP, visualize the integer points inside the region and the iso profit curve. Moreover, can also make a plot of a bi-objective criterion space and the non-dominated (Pareto) set. Figures are prepared for LaTeX and can automatically be transformed to TikZ using tikzDevice. Læs mere »

## Markov decision theory

Teaching 28. March 2014

During the this semester I am teaching the course “Markov decision theory” at the Department of Mathematics. The course presents the algorithmic aspects of Markov decision theory and illustrates the wide applicability of this theory to a variety of realistic problems. First, the course consider the Poisson process followed by Renewal/Reward Processes. Next we study the theory of Markov chains. Then follows a thorough presentation of the theory of Markov Decision Processes, including some applications. Finally, further applications are discussed based on reports made by the students. A course plan is given here.

## Course on supervision

13. May 2013

During the past months I have been participating in a course about supervision. The course, which runs over several days, covers master, PhD and fellow supervision and use results from research, practice and experiences from the participants. It aims at strengthen the participants’ overall supervisory skills in the form of

• clear and specific criteria-based guidance
• better feedback when commenting on text from the student
• establishment and maintenance of a mutually binding supervision agreement
• initial, ongoing and final project evaluation and feedback
• using meta-communication of supervision content and processes

The course included video-filming ourselves in doing supervision which was a good way of seeing how one actually perform. The course can be recommended to all university staff doing supervision.

## Sets, sums and constraints – A suvival guide for Business School students

Teaching 30. August 2012

Often when I teach students at our Business School they have a hard time understanding compact linear programming (LP) formulations. So here it is,  a short introduction to some of the concepts you need to know for understanding compact LP formulations.

### Sets

A set is a group of elements, e.g. $\{1,2,4\}$ is a set with 3 elements, namely, $1,2$ and $4$ and $A=\{(2,3),(4,5),(6,8),(5,6)\}$ is a set called $A$ with 4 elements (pairs), namely,  $(2,3),(4,5),(6,8)$ and $(5,6)$. Note that in the last case each element is a pair $(i,j)$. Sets containing pairs are often used when we formulate LPs based on network problems where the pair $(i,j)$ denote the arc/edge from node $i$ to node $j$. Læs mere »

## Using logvrp.com when introducing students to VRP

Teaching 22. December 2011

In my recent course on transportation and distribution systems for undergraduate business students I used a week talking about vehicle routing problems (VRP). Since their math skills aren’t that high the focus was not on algorithms instead I tried to talk about the general structure of the problem. Here the site logvrp.com is very good and provide a way of making a realistic problem case.

Logvrp is a web based vehicle route planner software cable of minimize transportation costs and to minimize fleet mileages. It uses Google maps to display routes and trips. Moreover, it supports many different VRP problems. Current two heuristics are implemented and can be run simultaneously on a problem instance. This is a good thing since the students can see that the heuristics behave differently on the same instance and that there is no general winner.

I created an exercise about a small-sized company located in Aarhus which sell organic fruits and vegetables to companies and private persons. The procedure to create the data to be used at logvrp was as follows: first I created sets of customers using the logvrp site. Each customer group was saved as csv files and modified using R. Next, a sets of orders was generated randomly using R and saved as csv files. The students was now able to answer the different questions using the csv files provided.

A typical solution visualized using crow fly distances.

The solution shown using shortest paths.

## Transportation and distribution systems

Teaching 3. December 2011

During the fall 2011 I have been teaching the course “Transportation and distribution systems” an optional course for undergraduate students. We have considered various topics within transportation such as its role in the global economy, modal choice, 3PL, VRP problems, flow problems, airline management, ITS, green logistics etc.

Each week the students was given an exercise where they in most cases had to formulate and solve an mathematical programming model. I also have tried to introduce CPLEX OPL studio to them with minor success. Mostly they like to formulate their models in Excel and with OpenSolver it is possible even for quite large models.

## Teaching large classes

Teaching 23. February 2011

Yesterday I participated in a short course about how to teach large classes. Focus was on how to align teaching for constructive learning, collaborative learning techniques and other active learning tricks. The main point can be summarized in this statement

“If students are to learn desired outcomes in a reasonably effective manner, then the teacher’s fundamental task is to get students to engage in learning activities that are likely to result in their achieving those outcomes… It is helpful to remember that what the student does is actually more important in determining what is learned than what the teacher does.” (Shuell, 1986: 429)

## Teaching Management Science

Teaching 15. December 2010

Just finished my last Management Science lecture for this semester. This is the first time I theach this introductory course to Business Students. Topics are linear and integer programming, goal programming, simulation and network models. Læs mere »

## Seminar in Business Studies – First Experience

Teaching 4. May 2010

The from February to May this year I have been teaching in “Seminar in Business Studies” a bachelor course (HA – 4. semester).

The course represents a teaching and form of learning where a group of 12 students with one or more teachers discusses selected issues and problems in management science. First the student should write a 10 page report on a specific subject. The report should provide the possibility of in-depth studies of theories, models and methods in management science. While writing the report he can use the assigned advisor for at most 1.5 hours. Second the student, 2 weeks after delivering the report, should give a 10 minutes presentation about their report. Afterwards there is time for 30 minutes of discussion and last 5 minutes for giving a grade.

I was assigned 59 students in 6 groups. In 4 of the groups I was the only teacher. In the last 2 groups we were 3 teachers.

Below you will find some experiences collected during the course. Please post a comment if you have suggestions for improvements. Læs mere »