text:
## Function

 In this task we calculate the function

a) What is the natural definition of the amount of functional expression?

b) Find the asymptotes of the function.

c) Calculate f `(x) and explain why f (x) is increasing.

d) What is the equation of the tangent at the point (3/5, f(3/5))?

e) Find the area bounded by the x-axis, y-axis and the graph.


text:
# A number theory problem.

$\large \frac {1978 \cdot 1979 + 1980 \cdot 21 + 1958}{1980 \cdot 1979 - 1978 \cdot 1979 } = ?$

Solve the above without using a calculator.

×

format: plain


text:
# Combinatorial interpretation of explicit formula for Stirling numbers of the second kind

I see that the below formula is the explicit formula of the Stirling numbers of the second kind. I know that the Stirling number of the second kind is the number of ways to partition set of $n$ objects into $k$ non-empty subsets. But, I don't at all see from where the below formula comes from. Clearly, there is some kind of inclusion-exclusion going on, but I cannot figure it out. If someone can give me some kind of combinatorial interpretation of the formula, I would be glad.

$$\begin{Bmatrix} n\\ k \end{Bmatrix} =\frac{1}{k!}\sum_{j=0}^{k}(-1)^{k-j}\binom{k}{j}j^n$$

$\begin{Bmatrix} n\\ k \end{Bmatrix}$ is the number of ways to distribute $n$ distinct objects into $k$ identical boxes with no empty boxes allowed/each box containing at least $1$ object. We will first consider distributions into $k$ distinct boxes.

The total number of ways to distribute $n$ distinct objects into $k$ distinct boxes with empty boxes allowed is: $$k^n$$

Let $A_i$ be the number of ways to perform the distribution with the condition that box $i$ must be an empty box. There are no restrictions on the other boxes. (This is equivalent to distributing the $n$ objects into the remaining $k-1$ boxes, empty boxes allowed.)

For example $A_1$ is the number of ways to distribute the objects into the $k$ boxes with box 1 being empty and is equivalent to distributing the objects into the remaining $k-1$ boxes, empty boxes allowed.

$$\implies |A_1| = (k-1)^n$$

Now we notice that $|A_1|=|A_2|=|A_3|=\dots$ since it doesn't matter which box is empty. (We are always distributing into the remaining $k-1$ boxes).

We know there are $\binom{k}{1}$ possible $A_i$

$$\sum_{i=1}^k|A_i| = \binom{k}{1}(k-1)^n$$

Using a similar argument, $|A_i\cap A_j|=(k-2)^n$ because it involves distribution into $k-2$ remaining boxes. Also, using the fact that $|A_1\cap A_2|=|A_1\cap A_3|=|A_1\cap A_4|...$ and that there are $\binom{k}{2}$ such terms,

$$\sum_{1\leq i,j\leq k}|A_i\cap A_j| = \binom{k}{2}(k-2)^n$$

On a similar note, we try extending this to when $l$ boxes must be empty:

$$\sum_{1\leq i,j\dots l\leq k}|A_i\cap A_j\cap \dots\cap A_l| = \binom{k}{l}(k-l)^n$$

$|A_1\cup A_2\cup \dots \cup A_k|$ is the number of ways to distribute $n$ distinct objects into $k$ distinct boxes with at least one empty box.

We are trying to find the distribution with no empty boxes: $$|\overline{A_1\cup A_2\cup \dots \cup A_k}|=k^n -|A_1\cup A_2\cup \dots \cup A_k|$$

By the principle of inclusion and exclusion,

$$=k^n-(|A_1|+|A_2|+|A_3|+...)+(|A_1\cap A_2|+|A_1\cap A_3|+|A_1\cap A_4|+...)-...$$

$$=k^n-\binom{k}{1}(k-1)^n+\binom{k}{2}(k-2)^n-\binom{k}{3}(k-3)^n+\dots +(-1)^l\binom{k}{j}(k-j)^n$$

$$=\sum_{l=0}^{k}(-1)^l\binom{k}{l}(k-l)^n$$

Let $j=k-l$, when $l=0$, $j=k$. When $l=k$, $j=0$.

$$=\sum_{j=k}^{0}(-1)^{k-j}\binom{k}{k-(k-j)}j^n (\text{this is technically wrong notation})$$

This is just the sum backwards. So we have:

$$=\sum_{j=0}^{k}(-1)^{k-j}\binom{k}{j}j^n$$

Now we divide by $k!$ since we are dealing with identical boxes:

$$\begin{Bmatrix} n\\ k \end{Bmatrix} =\frac{1}{k!}\sum_{j=0}^{k}(-1)^{k-j}\binom{k}{j}j^n$$

• I didn't exactly understood the sums at the top, what do they correspond to, how did we find them, and why do we need to check all the intersections? In the below sum $\sum_{l=0}^{k}(-1)^l\binom{k}{l}(k-l)^n$, shouldn't be go up to $j$ and not $k$, cause in the inclusion exclusion we have $(-1)^l\binom{k}{j}(k-j)^n$, and lastly how do we get that these two sums are equal $\sum_{l=0}^{k}(-1)^l\binom{k}{l}(k-l)^n=\sum_{j=0}^{k}(-1)^{k-j}\binom{k}{j}j^n$? Please don't jump over any algebraic step. – user72151 Nov 16 '15 at 16:00
• What do you mean with distinct and identical boxes? Does identical mean that all the boxes have same size, and distinct they all are of different size? – user72151 Nov 16 '15 at 18:09
• @terett Let's say we have objects $a,b,c$. If we were to distribute them into two identical boxes, then $(a)(b,c)$ would be the same as $(b,c)(a)$. Now, if we were distribute them into two distinct boxes, say, box $A$ and $B$, $A:(a)$ $B:(b,c)$ would not be the same as $A:(b,c)$ $B:(a)$ – Nicholas Nov 17 '15 at 4:24
• @Nicholas So, all you did was actually label the two boxes so you can distinguish them, you see that they are separate. – user72151 Nov 17 '15 at 8:15


text:
# Revision history [back]

### Cmake Project to Catkin

I have the following folder structure of a pure cmake project:

 projectfolder/
-  code/
...
-  build/
...
-  cmake/
CompilerOptions.cmake
...
-  other/
...

which can be compiled by the following commands without problem:

$cd build$ cmake ..
$make  I would like to catkinize this package without interfering with its complex dependency structure. Basically build the package normally from the same root "/build" and than link the created libraries to the catkin libraries. Unfortunately when I use catkin_make the root of the cmake is changing. This can be seen as I get an cmake error for not finding CompilerOptions.cmake (which is in build). I tried to Set(CMAKE_SOURCE_DIR${CMAKE_SOURCE_DIR}/src/projectfolder), but without the desired behavior. How can I change the catkin_make path accordingly, if I have the structure:

catkinws/
-    build
-    devel
-    src/
projectfolder/
-  code/
...
-  build/
...
-  cmake/
CompilerOptions.cmake
...
-  other/
...

Thank you very much for your help.

### Cmake Project to Catkin

Hey every one, I have the following folder structure of a pure cmake project:

 projectfolder/
-  code/
...
-  build/
...
-  cmake/
CompilerOptions.cmake
...
-  other/
...

which can be compiled by the following commands without problem:

$cd build$ cmake ..
$make  I would like to catkinize this package without interfering with its complex dependency structure. Basically build the package normally from the same root "/build" and than link the created libraries to the catkin libraries. Unfortunately when I use catkin_make the root of the cmake is changing. This can be seen as I get an cmake error for not finding CompilerOptions.cmake (which is in build). I tried to Set(CMAKE_SOURCE_DIR${CMAKE_SOURCE_DIR}/src/projectfolder), but without the desired behavior. How can I change the catkin_make path accordingly, if I have the structure:

catkinws/
-    build
-    devel
-    src/
projectfolder/
-  code/
...
-  build/
...
-  cmake/
CompilerOptions.cmake
...
-  other/
...

Thank you very much for your help.


text:
# Electro-osmosis

Electroosmotic flow (or electro-osmotic flow, often abbreviated EOF; synonymous with electroosmosis or electroendosmosis) is the motion of liquid induced by an applied potential across a porous material, capillary tube, membrane, microchannel, or any other fluid conduit. Because electroosmotic velocities are independent of conduit size, as long as the electrical double layer is much smaller than the characteristic length scale of the channel, electroosmotic flow is most significant when in small channels. Electroosmotic flow is an essential component in chemical separation techniques, notably capillary electrophoresis. Electroosmotic flow can occur in natural unfiltered water, as well as buffered solutions.

Electroosmotic flow schematic

## History

Electroosmotic flow was first reported in 1809 by F. F. Reuss in the Proceedings of the Imperial Society of Naturalists of Moscow. He showed that water could be made to flow through a plug of clay by applying an electric voltage. Clay is composed of closely packed particles of silica and other minerals, and water flows through the narrow spaces between these particles just as it would through a narrow glass tube. Any combination of an electrolyte (a fluid containing dissolved ions) and an insulating solid would generate electro-osmotic flow, though for water/silica the effect is particularly large. Even so, flow speeds are typically only a few millimeters per second.

## Cause

Electroosmotic flow is caused by the Coulomb force induced by an electric field on net mobile electric charge in a solution. Because the chemical equilibrium between a solid surface and an electrolyte solution typically leads to the interface acquiring a net fixed electrical charge, a layer of mobile ions, known as an electrical double layer or Debye layer, forms in the region near the interface. When an electric field is applied to the fluid (usually via electrodes placed at inlets and outlets), the net charge in the electrical double layer is induced to move by the resulting Coulomb force. The resulting flow is termed electroosmotic flow.

## Description

The resulting flow from applying a voltage is a plug flow. Unlike a parabolic profile flow generated from a pressure differential, a plug flow’s velocity profile is approximately planar, with slight variation near the electric double layer. This offers significantly less deleterious dispersive effects and can be controlled without valves, offering a high performance method for fluid separation, although many complex factors prove this control to be difficult. Because of difficulties measuring and monitoring flow in micro fluidic channels, primarily disrupting the flow pattern, most analysis is done through numerical methods and simulation.[1] Electroosmotic flow through micro channels can be modeled after the Navier-Stokes equation with the driving force deriving from the electric field and not the pressure differential. Thus it is governed by the continuity equation

${\displaystyle \nabla \cdot \mathbf {U} =0}$

and momentum

${\displaystyle \rho {\frac {D\mathbf {U} }{Dt}}=-\nabla p+\mu \nabla ^{2}\mathbf {U} +\rho _{e}\nabla \left(\psi +\phi \right),}$

where U is the velocity vector, ρ is the density of the fluid, ${\displaystyle D/Dt}$ is the material derivative, μ is the viscosity of the fluid, ρe is the electric charge density, Φ is the applied electric field, and ψ is the electric field due to the zeta potential at the walls.

Laplace’s equation can describe the external electric field

${\displaystyle \nabla ^{2}\phi =0,}$

while the potential within the electric double layer is governed by

${\displaystyle \nabla ^{2}\psi ={\frac {-\rho _{e}}{\epsilon \epsilon _{0}}},}$

where ε is the dielectric constant of the electrolyte solution and ε0 is the vacuum permittivity. This equation can be further simplified using the Debye-Hückel approximation

${\displaystyle \nabla ^{2}\psi =k^{2}\psi ,}$

where 1 / k is the Debye Length, used to describe the characteristic thickness of the electric double layer. The equations for potential field within the double layer can be combined as

${\displaystyle \rho _{e}=-\epsilon \epsilon _{0}k^{2}\psi .}$

## Applications

Electroosmotic flow is commonly used in microfluidic devices,[2][3] soil analysis and processing,[4] and chemical analysis,[5] all of which routinely involve systems with highly charged surfaces, often of oxides. One example is capillary electrophoresis,[3][5] in which electric fields are used to separate chemicals according to their electrophoretic mobility by applying an electric field to a narrow capillary, usually made of silica. In electrophoretic separations, the electroosmotic flow affects the elution time of the analytes.

It is projected that micro fluidic devices utilizing electroosmotic flow will have great application with medical research. Once controlling this flow is better understood and implemented being able to separate fluids on the atomic level will be a vital component for drug dischargers.[6] Mixing fluids at the micro scale is currently troublesome. It is believed that electrically controlling fluids will be the method in which small fluids are mixed.[6]

A controversially use of electro-osmotic systems is being done to control rising damp in structure of a building.[7] Whilst there is little evidence to suggest that these systems can be useful in moving salts in walls, such systems are claimed to be specially efficient on structures with very thick walls. However some claim that there is no scientific base for those systems, and cite several examples for their failure.[8]

### Physics

In fuel cells, electro-osmosis causes protons moving through a proton exchange membrane (PEM) to drag water molecules from one side (anode) to the other (cathode).

### Vascular plant biology

In vascular plant biology, electro-osmosis is also used as an alternative or supplemental explanation for the movement of polar liquids via the phloem that differs from the cohesion-tension theory supplied in the mass flow hypothesis and others, such as cytoplasmic streaming.[9] Companion cells are involved in the "cyclic" withdrawal of ions (K+) from sieve tubes, and their secretion parallel to their position of withdrawal between sieve plates, resulting in polarisation of sieve plate elements alongside potential difference in pressure, and results in polar water molecules and other solutes present moved upward through the phloem.[9]

In 2003, St Petersburg University graduates applied direct electric current to 10 mm segments of mesocotyls of maize seedlings alongside one-year linden shoots; electrolyte solutions present in the tissues moved toward the cathode that was in place, suggesting that electro-osmosis might play a role in solution transport through conductive plant tissues.[10]

Maintaining an electric field in an electrolyte requires Faradaic reactions to occur at the anode and cathode. This is typically electrolysis of water, which generates hydrogen peroxide, hydrogen ions (acid) and hydroxide (base) as well as oxygen and hydrogen gas bubbles. The hydrogen peroxide and/or pH changes generated can adversely affect biological cells and biomolecules such as proteins, while gas bubbles tend to "clog" microfluidic systems. These problems can be alleviated by using alternative electrode materials such as conjugated polymers which can undergo the Faradaic reactions themselves, dramatically reducing electrolysis.[11]

## References

1. ^ G. F. Yao, H. A Computational Model for Simulation of Electroosmotic Flow in Microsystems.
2. ^ Bruus, H. (2007). Theoretical Microfluidics. ISBN 0-19-923509-0.
3. ^ a b Kirby, B. J. (2010). Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices: Chapter 6: Electroosmosis. Cambridge University Press.
4. ^ Wise, D. L. and Trantolo, D. J., eds. Remediation of Hazardous Waste Contaminated Soils.
5. ^ a b Skoog. Principles of Instrumental Analysis. ISBN 0-495-12570-9.
6. ^ a b Ducree, Jen. myFluidix.com.
7. ^ Ottosen, Lisbeth; Anne J. Pedersen; Inge Rorig-Dalgaard. Salt related problems in brick masonry and electrokinetic removal of salts. Journal of Building Appraisal.
8. ^
9. ^ a b Clegg, C. J., Mackean, D. G. (2006) "Advanced Biology – principles & applications" Hodder Stoughton Publhishers, pp. 340–343.
10. ^ Polevoi, V. V. (2003). "Electroosmotic Phenomena in Plant Tissues". Biology Bulletin 30 (2): 133. doi:10.1023/A:1023285121361.
11. ^ Erlandsson, P. G.; Robinson, N. D. (2011). "Electrolysis-reducing electrodes for electrokinetic devices". Electrophoresis 32 (6–7): 784–790. doi:10.1002/elps.201000617. PMID 21425174.


text:
# Python and Numpy

This guide is obsolete, it was written during the early stages when I was learning python. A more up to date guide is available here:

Other python guides including installation of python are available in:

## NumPy: The Numerical Python Library

For science, mathematics and engineering the numeric Python library is fundamental. This library makes it far easier to work with arrays of numbers and has a large number of numerical functions for working with numerical arrays.

### Importing NumPy

To use the numerical python library numpy we need to import it. As it is commonly used and referenced, it is usually imported as a two letter abbreviation at the top of a Python script:

Python

It can be imported at the top of a script or it can be imported by typing directly into the console.

If we try to use numpy without running the script.

We will get the following error:

If we launch the script with the command to import numpy or run the command in the console.

We can access the commands in this case the command np.pi gives the constant π

Which is the value 3.14… as expected.

If we however go to Console and then Restart Kernal, or close Spyder or restart the computer.

Then once again if we try and access any of the commands within the library it won't be able to.

### Always Enabling NumPy

As NumPy and MatPlotLib are so commonly used for plotting, it is also possible to load the libraries by default. To do this select Tools and then Preferences:

To the left hand side select iPython Console:

To the top right select Graphics, then Automatically Load PyLab and NumPy Modules. Also it is recommended to change the Backend to Automatic. Select Apply:

Then OK. Changes will not be made until either the Kernal is restarted, Spyder is closed and reopened or the computer is restarted.

## Looking Up All Commands Using Dot and Tab

Next it is worth looking through the library, this can be done by typing either in the script or console, np followed by a dot and then tab:

Python

This will give a large list of entries by alphabetical order; items stating with a capital are listed in alphabetical order first and then all the lower case functions are likewise listed in alphabetical order.

Take your time and scroll through the list by either using the scroll bar or the [↓] and [↑] keys. Scrolling through the list, in the console, we can see the function pi, which will return pi:

We can select it:

And then execute it:

Python
```
3.141592653589793

```

pi returns the value of pi, π.

## Functions Without Input Arguments

Python
```
Type:        float
String form: 3.141592653589793
Docstring:   Convert a string or number to a floating point number, if possible.

```

This prints the help in the console and explains briefly how to use the command.

Likewise we can have a look at

Python
```
Type:        float
String form: 2.718281828459045
Docstring:   Convert a string or number to a floating point number, if possible.

```

We can also highlight the e and press [Ctrl] + [ i ] to get the information about the command (i for information).

This shows only the doc string for these basic strings

Since this is a constant, we can just use the command to generate it.

Python
```
2.718281828459045

```

## Functions with Input Arguments

There are a number of functions that can use a scalar input, for instance the square root function. It is worthwhile reading the help of these:

Python

Doing so prints out all the help in the console and you can take your time to scroll through it.

```
Call signature:   np.sqrt(*args, **kwargs)
Type:            ufunc
String form:     <ufunc 'sqrt'>
File:            c:\users\pyip\appdata\local\continuum\anaconda3\lib\site-packages\numpy\__init__.py
Docstring:
sqrt(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

Return the non-negative square-root of an array, element-wise.

Parameters
----------
x : array_like
The values whose square-roots are required.
out : ndarray, None, or tuple of ndarray and None, optional
A location into which the result is stored. If provided, it must have
a shape that the inputs broadcast to. If not provided or `None`,
a freshly-allocated array is returned. A tuple (possible only as a
keyword argument) must have length equal to the number of outputs.
where : array_like, optional
Values of True indicate to calculate the ufunc at that position, values
of False indicate to leave the value in the output alone.
**kwargs
For other keyword-only arguments, see the
:ref:`ufunc docs <ufuncs.kwargs>`.

Returns
-------
y : ndarray
An array of the same shape as `x`, containing the positive
square-root of each element in `x`.  If any element in `x` is
complex, a complex array is returned (and the square-roots of
negative reals are calculated).  If all of the elements in `x`
are real, so is `y`, with negative elements returning ``nan``.
If `out` was provided, `y` is a reference to it.
This is a scalar if `x` is a scalar.

--------
lib.scimath.sqrt
A version which returns complex numbers when given negative reals.

Notes
-----
*sqrt* has--consistent with common convention--as its branch cut the
real "interval" [`-inf`, 0), and is continuous from above on it.
A branch cut is a curve in the complex plane across which a given
complex function fails to be continuous.

Examples
--------
np.sqrt([1,4,9])
array([ 1.,  2.,  3.])

np.sqrt([4, -1, -3+4J])
array([ 2.+0.j,  0.+1.j,  1.+2.j])

np.sqrt([4, -1, numpy.inf])
array([  2.,  NaN,  Inf])
Class docstring:
Functions that operate element by element on whole arrays.

To see the documentation for a specific ufunc, use `info`.  For
example, ``np.info(np.sin)``.  Because ufuncs are written in C
(for speed) and linked into Python with NumPy's ufunc facility,
on a ufunc.

A detailed explanation of ufuncs can be found in the docs for :ref:`ufuncs`.

Calling ufuncs:
===============

op(*x[, out], where=True, **kwargs)
Apply `op` to the arguments `*x` elementwise, broadcasting the arguments.

* Dimensions of length 1 may be prepended to either array.
* Arrays may be repeated along dimensions of length 1.

Parameters
----------
*x : array_like
Input arrays.
out : ndarray, None, or tuple of ndarray and None, optional
Alternate array object(s) in which to put the result; if provided, it
must have a shape that the inputs broadcast to. A tuple of arrays
(possible only as a keyword argument) must have length equal to the
number of outputs; use `None` for uninitialized outputs to be
allocated by the ufunc.
where : array_like, optional
Values of True indicate to calculate the ufunc at that position, values
of False indicate to leave the value in the output alone.  Note that if
an uninitialized return array is created via the default ``out=None``,
then the elements where the values are False will remain uninitialized.
**kwargs
For other keyword-only arguments, see the :ref:`ufunc docs <ufuncs.kwargs>`.

Returns
-------
r : ndarray or tuple of ndarray
`r` will have the shape that the arrays in `x` broadcast to; if `out` is
provided, it will be returned. If not, `r` will be allocated and
may contain uninitialized values. If the function has more than one
output, then the result will be a tuple of arrays.

```

However you may not want the help for every command you want to look up filling up your console. Instead it is recommended to type in the command and leave the parenthesis for the input argument(s) open ended.

Python

Here you will see the form of the input arguments.

In our case, we are only interested in the argument (args) which go first and do not care about the optional keyword argument (kwargs).

You can now highlight the command and the open left bracket in this case sqrt(

Then press [Ctrl] + [ i ] to get the information, you can expand the help to the top right in order to read it better. This gives you the on demand information without clogging up your console. So we see the input argument we are interested in is x.

The first example the help gives is a row vector, however we can also use a simple positive scalar input for simplicity. The help also gives an example with a complex number and an infinite number. In our case we can use the function with a scalar input argument of 2:

Python
```
1.4142135623730951

```

The answer has a sequence of non-converging numbers following the decimal point. It may be useful to round this number, we can look at the round function. Once again we can type in:

Python

This will give details on the input arguments:

Here we see we have the arg a, which is the number to be rounded, and optional input arguments which have argument names decimal and out. The default values of these latter arguments if not specified are 0 and None.

Once again we can type in the following to print out the help in the console:

Python
```
Signature:  np.round(a, decimals=0, out=None)
Docstring:
Round an array to the given number of decimals.

--------
around : equivalent function; see for details.
File:      c:\users\pyip\appdata\local\continuum\anaconda3\lib\site-packages\numpy\core\fromnumeric.py
Type:      function

```

However in order to avoid clogging up the console, we can highlight the function and press [Ctrl] + [ i ] to get the information

Here we are told the first input argument is our array a, in our case our scalar input argument can directly be input:

Python
```
1.0

```

Note in the expression above, the outside brackets ( ) surround the input arguments of round, with the function round being called from the library np (abbreviated from the full library name numpy). The inside brackets ( ) surround the input arguments of the function sqrt and sqrt is once again a function being called from the library np.

In np.sqrt there is a single input argument, the scalar 2. In np.round there is a single input argument np.sqrt(2) and as only a single input argument was specified for np.round, the default setting decimal=0 and out=None were used.

A second input argument to round can be specified as 1 to round to 1 decimal place. In this case the comma , is used to separate out the two input arguments.

Take the time to understand calling functions from the np library using the dot . aswell as the use of ( ) to surround input arguments and the use of , to separate out the input arguments.

Python
```
1.4

```

If instead we wanted 2 decimal places:

Python
```
1.41

```

Note that we do not need to use the argument name decimal, merely placing it in the correct order works. We can also look at the function power. Once again looking up help for this function we can see a description:

Python

This unnecessarily clogs up the console.

```
Call signature:  np.power(*args, **kwargs)
Type:            ufunc
String form:     <ufunc 'power'>
File:            c:\users\pyip\appdata\local\continuum\anaconda3\lib\site-packages\numpy\__init__.py
Docstring:
power(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

First array elements raised to powers from second array, element-wise.

Raise each base in `x1` to the positionally-corresponding power in
`x2`.  `x1` and `x2` must be broadcastable to the same shape. Note that an
integer type raised to a negative integer power will raise a ValueError.

Parameters
----------
x1 : array_like
The bases.
x2 : array_like
The exponents.
out : ndarray, None, or tuple of ndarray and None, optional
A location into which the result is stored. If provided, it must have
a shape that the inputs broadcast to. If not provided or `None`,
a freshly-allocated array is returned. A tuple (possible only as a
keyword argument) must have length equal to the number of outputs.
where : array_like, optional
Values of True indicate to calculate the ufunc at that position, values
of False indicate to leave the value in the output alone.
**kwargs
For other keyword-only arguments, see the
:ref:`ufunc docs <ufuncs.kwargs>`.

Returns
-------
y : ndarray
The bases in `x1` raised to the exponents in `x2`.
This is a scalar if both `x1` and `x2` are scalars.

--------
float_power : power function that promotes integers to float

Examples
--------
Cube each element in a list.

x1 = range(6)
x1
[0, 1, 2, 3, 4, 5]
np.power(x1, 3)
array([  0,   1,   8,  27,  64, 125])

Raise the bases to different exponents.

x2 = [1.0, 2.0, 3.0, 3.0, 2.0, 1.0]
np.power(x1, x2)
array([  0.,   1.,   8.,  27.,  16.,   5.])

x2 = np.array([[1, 2, 3, 3, 2, 1], [1, 2, 3, 3, 2, 1]])
x2
array([[1, 2, 3, 3, 2, 1],
[1, 2, 3, 3, 2, 1]])
np.power(x1, x2)
array([[ 0,  1,  8, 27, 16,  5],
[ 0,  1,  8, 27, 16,  5]])
Class docstring:
Functions that operate element by element on whole arrays.

To see the documentation for a specific ufunc, use `info`.  For
example, ``np.info(np.sin)``.  Because ufuncs are written in C
(for speed) and linked into Python with NumPy's ufunc facility,
on a ufunc.

A detailed explanation of ufuncs can be found in the docs for :ref:`ufuncs`.

Calling ufuncs:
===============

op(*x[, out], where=True, **kwargs)
Apply `op` to the arguments `*x` elementwise, broadcasting the arguments.

* Dimensions of length 1 may be prepended to either array.
* Arrays may be repeated along dimensions of length 1.

Parameters
----------
*x : array_like
Input arrays.
out : ndarray, None, or tuple of ndarray and None, optional
Alternate array object(s) in which to put the result; if provided, it
must have a shape that the inputs broadcast to. A tuple of arrays
(possible only as a keyword argument) must have length equal to the
number of outputs; use `None` for uninitialized outputs to be
allocated by the ufunc.
where : array_like, optional
Values of True indicate to calculate the ufunc at that position, values
of False indicate to leave the value in the output alone.  Note that if
an uninitialized return array is created via the default ``out=None``,
then the elements where the values are False will remain uninitialized.
**kwargs
For other keyword-only arguments, see the :ref:`ufunc docs <ufuncs.kwargs>`.

Returns
-------
r : ndarray or tuple of ndarray
`r` will have the shape that the arrays in `x` broadcast to; if `out` is
provided, it will be returned. If not, `r` will be allocated and
may contain uninitialized values. If the function has more than one
output, then the result will be a tuple of arrays.

```

We can instead just type in the command with an open parenthesis

Python

This once gives us the form of the input arguments. We can also have a look at the information by highlighting the function and then pressing [Ctrl] + [ i ]

This will give the information. Here we see we have two input arguments x1 and x2.

In this example we are only looking at scalars, so the following will calculate:

$\displaystyle {{3}^{3}}$

Python
```
27

```

This can be used also in place of the sqrt function above. Once again we can see that the outside brackets ( ) surround the input arguments of round, with the function round being called from the library np (abbreviated from the full library name numpy). The inside brackets ( ) surround the input arguments of the function power and power is once again a function being called from the library np. There are two input arguments for the function power 2 and 1/2, these are once again separated by a comma , and the inside function and its input arguments i.e. np.power(2,1/2) is an input argument alongside 2 for the outside function np.round, once again separated by a comma.

Python
```
1.41

```

The power function can also be used to get the cube root or nth root:

Python
```
3

```

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# Tuning domain wall dynamics by shaping nanowires cross-sections

## Abstract

The understanding of the domain wall (DW) dynamics along magnetic nanowires is crucial for spintronic applications. In this work, we perform a detailed analysis of the transverse DW motion along nanowires with polygonal cross-sections. If the DW displaces under a magnetic field above the Walker limit, the oscillatory motion of the DW is observed. The amplitude, the frequency of oscillations, and the DW velocity depend on the number of sides of the nanowire cross-section, being the DW velocity in a wire with a triangular cross-section one order of magnitude larger than that in a circular nanowire. The decrease in the nanowire cross-section area yields a DW behavior similar to the one presented in a cylindrical nanowire, which is explained using an analytical model based on the general kinetic momentum theorem. Micromagnetic simulations reveal that the oscillatory behavior of the DW comes from energy changes due to deformations of the DW shape during the rotation around the nanowire.

## Introduction

Progress in nanofabrication enables the development of nanomagnets with different and well-controlled shapes and sizes1,2,3,4,5. Some examples are self-organized nanowires with a triangular cross-section grown on uniaxially grooved W(110) surfaces by Borca et al.6,7. Also FeCo nanowires with a three-dimensional structure customizable by design were grown along arbitrary crystal directions by Tao et al.8. In addition, 3D focused electron beam induced deposition9,10,11,12 has opened new perspectives in science and technology, allowing the fabrication in one single direct-write step of nanostructures with small feature sizes, complex and diverse architectures and materials. These studies confirm the possibility of manufacturing nanowires with different cross sections, and make it necessary to analyze theoretically geometry-induced effects in the magnetization properties of nanomagnets9,13,14. Due to an increase in the surface/volume ratio, the shape of an element at the nanoscale is a crucial factor to determine its magnetization dynamic and static properties. Therefore, much effort has been done to understand how the shape of a nanoparticle changes its magnetic properties15,16,17,18,19,20,21,22,23,24,25,26.

Among several issues regarding magnetism at the nanoscale, the propagation of domain walls (DW) is at the heart of technological proposals that promote the use of their dynamical properties in devices such as “race-track” memory27, random access memory28, nano-oscillators29,30, interconnections between spin logic gates31, logic devices32,33, and magnetic field generation34. From the theoretical point of view, it was shown that the geometry of the nanowire cross-section is crucial for describing the DW dynamics. Rectangular nanowires exhibit the so-called Walker limit35,36, which is the loss of the linearity of the DW velocity as a funciton of magnetic field when the latter exceeds a critical value, known as Walker field ($$H_W$$). At this threshold field, there is the transition from a DW rigid-body purely translational motion to an oscillatory translational motion with a DW rotation. Indeed, Porter and Donahue37 showed that when a DW is propagating along a rectangular nanowire under the Walker regime, two movements are superimposed. The first one consists of an oscillation along the wire axis, while the second is a DW rotation around the nanostripe. When the DW rotates in 180$${}^{\circ }$$, it exhibits a complete oscillation of the DW position along the stripe. Therefore, the rotation frequency is twice the oscillation one. The theoretical model derived by Mougin et al.36 also predicts that the $$H_W=0$$ when the DW is displacing along nanostripes with a square cross-section. Nanowires with a circular cross-section also present $$H_W=0$$, and the velocity of a transverse DW varies linearly with the magnetic field38. Nevertheless, the analysis of the DW displacement along bent nanowires with cylindrical cross-section showed that the $$H_W\propto \kappa$$39,40,41, where $$\kappa$$ is the curvature of the nanowire. In this case, the oscillating and rotational frequencies are the same. We can then conjecture that the difference between the DW dynamics along nanostructures with rectangular and circular cross-sections can be explained because of the existence of two equivalent preferred directions (DW phase) for the DW center to point while displacing along a rectangular wire. At the same time, there exists just one preferred direction in a circular bent wire42, and no preferential direction in a cylindrical straight wire.

Planar magnetic nanowires are highly attractive for spintronic and magnetic-based applications. However, and in addition to race-track memories27, it is possible to use magnetic nanowires as different components of nanoscale devices. Such applications demand a full control of the domain wall velocity under different external stimuli and magnetic parameters. Based on the above, we investigate intrinsically 3D nanostructures where DWs may oscillate around and along the nanowire. This double oscillation allows other applications of the DW dynamics, as sources of radio-frequency electromagnetic emission39,41,43 or nano-oscillators29,30,44. Therefore, because the number of preferential directions of the DW phase plays a role in the oscillation frequency, the engineering of future 3D applications demands to understand how the shape of the nanowire cross-section may influence this regime as well as the transition between it and the regime of the linear with field velocity. In this line, aiming to get a better understanding of the oscillatory phenomenon of a DW as a function of the energy minima, we explore the dynamics of a transverse DW propagating along nanowires with different polygonal cross-sections, driven by a constant external magnetic field. The performed micromagnetic simulations reveal that depending on the wire geometry, two regimes of the DW dynamics are observed: (i) a regime in which the DW displaces linearly as a function of the external magnetic field; and (ii) a regime in which the DW presents the oscillatory behavior characterized by the Walker breakdown. In this last case, the DW oscillation frequency is strongly dependent on the shape of the 3D nanowire cross-section. We have explained the first regime with a theoretical model considering the propagation of a rigid DW. The second regime is explained by considering structural changes in the DW when it is rotating around the nanowire. Therefore, our work shows clearly that the oscillatory dynamics of a DW displacing in a straight nanowire with a polygonal cross-section depends on the number of times that the DW changes its shape during one rotation around the nanowire. That is, the frequency of the backward motion of the DW is related with the number of sides of the polygon describing the nanowire cross-section. This fact implies that the assumption of a rigid DW is valid just for nanowires with cross-sections with small area and its validity eliminates the DW oscillatory motion characterizing the Walker regime in the analyzed structures. Thus, our work provides a deeper understanding of the Walker breakdown phenomena. Here we investigated the transition between DW dynamics in triangular to circular nanowires cross-sections, showing that the Walker breakdown is specifically related to the change of the domain wall shape during the DW rotation.

## Model and results

In this work, we have calculated the domain wall dynamics in nanowires with different regular cross-sections by solving the Landau–Lifshitz–Gilbert equation45,46 using the public micromagnetic code NMAG47. This open software package is based on a finite element discretization method, which results in general more suitable to describe the different shapes of cross-sections of nanostructures considered in this work. The demagnetizing field is calculated by using the hybrid finite element method/boundary element method, allowing to discretize the space only in the regions occupied by the magnetic material. The systems we study consist of nanowires of length $$\ell = 1000$$ nm, with regular cross-sections such as triangle (TNW), square (SNW), pentagon (PNW), and hexagon (HNW). We start considering a set (S1) of dimensions such that all nanowires have a similar cross-section areas of 700 nm$$^2$$ (see Fig 1b). *** The magnetic parametrization corresponds to a soft material such as permalloy to avoid screening of the effect of shaping different cross-sections into the DW dynamics by the magnetocrystalline anisotropy. Indeed, currently, the growth conditions or chemical synthesis of magnetic nanowires produce mostly polycrystalline materials48,49. In these circumstances it is natural first to center our study on the case with no crystalline anisotropy. We use a mesh size of 2 nm, $$\mu _0 M_s=1$$ T, where $$M_s$$ is the saturation magnetization, the exchange stiffness $$A =1.3\times 10^{-11}$$ J m$$^{-1}$$, the damping and gyromagnetic factor are $$\alpha =0.01$$ and $$\gamma = 1.76 \times 10^{11}$$ rad T$$^{-1}$$ s$$^{-1}$$ respectively, and no magnetocrystalline anisotropy is taken into account. For all the geometries, except for the square, a tail to tail transverse DW of 30 nm width is set up in the system, pointing along the positive y-direction (with the coordinate system as shown in Fig. 1a).*** In the square, the initial DW points along one of the lateral faces of the nanowire (in particular, we choose the positive xy-plane direction, as shown in the coordinate system of Fig. 1a).*** After relaxing these initial configurations for a few nanoseconds, DWs remain traverse, both placed and pointing in the same direction as before, but with different widths, corresponding to each equilibrium state.

Starting from this state, we apply a constant magnetic field of 10 mT along the $$z+$$ direction during 10 ns, to determine the influence of the cross-section shape on the DW dynamics. The time dependence of the magnetization for the S1 wires, as obtained by micromagnetic modelling, is displayed in Fig. 2, where the label n represents the number of sides of the polygon and, as it will be shown after, it is also related to the number of easy directions of the nanowire cross-section. The component $$m_z$$ is related to the DW position, and the components $$m_x$$ and $$m_y$$ describe the DW phase. During its motion, we observe a DW rotation around the nanowire axis, while displacing along the z-axis direction with an oscillatory behavior. An interesting result is that the frequency of the DW rotation around the z-axis is shape-independent, being the same as that in a cylindrical nanowire under equivalent conditions38. This result agrees with those obtained for straight and bent nanowires with cylindrical cross-section34,36,38,39,40,41, which revealed that once the DW is displacing in the Walker regime, the rotation frequency is a function of the external magnetic field and current density. On the other side, the frequency and amplitude of the DW oscillatory motion along the z-axis, are strongly shape dependent. Indeed, Fig. 3 reveals that a DW propagating along a nanowire with a cross-section given by a polygon of n sides oscillates n times during one rotation. Additionally, one can notice that the oscillation amplitude is inversely proportional to n, being very small in the HNW. These aspects of the DW dynamics in multifaceted nanowires can be understood from the fact that the DW rotation is affected just by the external magnetic field, which is pointing along the z-axis direction. That is, the magnetic field produces a torque in the xy-plane, which makes the DW to rotate, producing changes in the exchange and dipolar interactions. Therefore, the oscillatory motion is governed by changes in the dipolar and exchange effective fields, which are strongly dependent on n. Thus, while the external magnetic field plays a role on the rotation motion, changes in the dipolar and exchange effective fields when the DW rotates around the z-axis direction affect the DW motion along the wire axis, according to the Landau–Lifshitz equation45. In conclusion, the rotational motion does not depend on n, but the oscillatory motion in the z-axis direction depends on it.

To be more precise, Fig. 3 illustrates the DW position as a function of time for each of the considered cross-sections. To determine the DW position, we use the following procedure: first, we extract from our results the normalized magnetization, $$\mathbf{m} (z)=\mathcal {M}/M_S$$, along the nanowire axis for each time step ($$\sim$$ ps). After that, we fit the obtained magnetization configuration to the well-known transverse DW profile $$m(z)=\tanh \left[ (z-z_0)\pi /\Delta _{DW}\right]$$, where $$z_0$$ and $$\Delta _{DW}$$ are respectively the DW position and width for each time step. Thus, $$z_0$$ is calculated for each time step. Effectively, previous conclusions become more clear, i.e., both frequency and amplitude of the oscillatory motion strongly depend on the geometry. Furthermore, the domain wall velocity also depends on n, and the higher is the n value, the smaller is the velocity, recovering analytical predictions for cylindrical nanowires when increasing n. Indeed, for the HNW case the oscillation amplitude is small, and its velocity is matching with a cylindrical one. In this way, in cylindrical wires DW propagates slower than in wires of any other geometry. From the DW position, we can determine the DW average velocity $$\langle v\rangle =d/t_1$$ for each considered nanowire. Here, d is the distance that the DW has displaced after $$t_1=3.6$$ ns (a complete rotation of the DW around z-axis). Figure 4 shows the obtained values to $$\langle v\rangle$$ as a function of n. One can notice that this figure depicts the behavior observed in Fig. 3, in which $$\langle v\rangle _3\approx 20\, \langle v\rangle _4$$, where $$\langle v\rangle _n$$ is the average velocity for a DW displacing in a nanowire with cross-section of n sides. The inset in Fig. 4 presents a zoom in the region between $$n=5$$ and $$n=6$$. In this case, we have that $$\langle v\rangle _5\approx 1.04\, \langle v\rangle _6$$, evidencing that the larger the n, the lower the difference between $$\langle v\rangle _{n}$$ and $$\langle v\rangle _{n+1}$$.

To better understand the influence of nanowire faceting on the DW dynamics, we consider a second set of sizes, S2, (see Fig. 1b), with polygons with a smaller cross section area of approximately 360 nm$$^2$$. The main results regarding the magnetization dynamics are presented in Fig. 5. It can be observed that for the rotation motion, we get the same results as for wires from set S1. However, the difference is observed in the oscillatory behavior along the axis. The amplitude of the oscillations is smaller in all cases. Indeed, in contrast with S1 simulations, oscillations have disappeared both for the PNW and HNW. Furthermore, micromagnetic simulations for smaller structures revealed that this tendency continues, i.e., the smaller is the cross-section, the smaller is the oscillation amplitude, disappearing for all the structures at a specific value of side size. In our calculations, different fields were considered, showing the same qualitative results, with a slower propagations of the DW under lower magnetic fields, as well as lower rotation frequencies.

Aiming at understanding why the oscillations disappear for very thin polygons, independently of the cross-section shape, we performed analytical calculations using the general kinetic momentum theorem, in which it is possible to relate the components of the torque produced on the magnetization, $$\varvec{\Gamma }_\theta$$ and $$\varvec{\Gamma }_\phi$$, and their associated rotation velocities $$\partial {\theta }/\partial t$$ and $$\partial {\phi }/\partial t$$, that is

\begin{aligned} \frac{\partial \theta }{\partial t}=-\frac{\gamma }{M_s}\varvec{\Gamma }_\theta \, \text {and}\frac{\partial \phi }{\partial t}=-\frac{\gamma }{M_s}\varvec{\Gamma }_\phi . \end{aligned}
(1)

The torques can be written in a spherical basis36 by using the transfer matrix

\begin{aligned} \varvec{\Gamma }_{r,\theta ,\phi }=\left( \begin{array}{ccc} \sin \theta \cos \phi &{} \sin \theta \sin \phi &{}\cos \theta \\ \cos \theta \cos \phi &{} \cos \theta \sin \phi &{} -\sin \theta \\ -\sin \phi &{} \cos \phi &{} 0 \end{array}\right) \varvec{\Gamma }_{x,y,z} , \end{aligned}
(2)

where $$\varvec{\Gamma }_{x,y,z}$$ are the torques expressed in Cartesian coordinates. Because the considered nanowires are very thin, we adopt a rigid domain wall model, in which the DW does not change its width or shape during is displacement, we do not consider the torque coming from the exchange energy effective field36. Therefore, in our calculations we include only the torques produced by the Zeeman and dipolar fields, and the effective damping term. The DW velocity is determined from the fact that the velocity v of any spin embedded in a Bloch wall is related to its characteristic width $$\Delta$$ by $$v = -{\Delta \,{\partial _t\theta }}/{\sin \theta }$$.

In our analysis the magnetic body is a wire with length $$\ell =2L$$, with a polygonal cross-section consisting of n sides, each of size 2a. By considering that the wire cross-section is circunscribed by a cylinder of radius r, we have that $$a=r\,\cos [\pi (n-2)/2n]$$ and consequently, a decreases when the number of sides increases in such a way that $$a\rightarrow 0$$ for $$n\rightarrow \infty$$ (cylindrical cross-section). Furthermore, from this assumption, we can determine the distance between the surface S1 and the geometrical center of the wire, $$h=r\,\sin [\pi (n-2)/2n]$$ and then, $$h\rightarrow r$$ for $$n\rightarrow \infty$$.

The torques produced by Zeeman and damping terms have already been obtained in Ref.36. On the other hand, in this work we are not considering a rectangular-shaped cross-section, and therefore the torque produced by the demagnetizing field will be calculated from the definition of the total demagnetizing tensor $$\mathbf {N}$$ of a prism with n sides. Generally, the point-function demagnetization tensor (PFDT) can be given by50

\begin{aligned} \mathbf {N}_p(\mathbf {r})=\frac{1}{4\pi }\nabla \oint _{s'} \frac{1}{|\mathbf {r}-\mathbf {r}'|}d\mathbf {S}', \end{aligned}
(3)

where $$\mathbf {r}$$ and $$\mathbf {r}'$$ are respectively the field and source point locations, and $$\mathbf {S}'$$ is the surface enclosing the body. The PFDT has two important properties50: (i) the trace of the tensor is equal to 1 inside the body and 0 outside; (ii) the tensor satisfies $$N_p^{\mu \nu }=N_p^{\nu \mu }$$, being symmetric for a closed surface. In this work we are considering regular prisms with two uniaxial rotational periodicities of order n along the z-axis and of order two around the y-axis (see Fig. 1c). The use of the first property of the point-function demagnetization tensor makes it possible to define the matrix representation of the magnetometric demagnetization tensor as50

\begin{aligned} \mathbf {N}_M=\frac{1}{V}\int _{V}\mathbf {N}_{p}(\mathbf {r}) dV=\left( \begin{array}{ccc} N_{1}&{}0&{}0\\ 0&{}N_1&{}0\\ 0&{}0&{}N_2\end{array}\right) , \end{aligned}
(4)

where $$N_2=1-2\,N_1$$ inside the magnetic body, and $$N_2=-2\,N_1$$ outside the magnetic body.

The demagnetization tensor of a prism can also be given by the sum of the $$n+2$$ surfaces’ demagnetizing tensors associated with each one of the n sides of the considered polygon and the two surfaces on the prism borders51. That is, if $$\mathbf {N}_{Sj}$$ is the demagnetizing tensor of the rectangular surface of the side j of the prism, we have that $$\mathbf {N}_p(\mathbf {r})=\sum _{j=1}^{n}\mathbf {N}_{Sj}+2\mathbf {N}_{b}$$, where $$\mathbf {N}_b$$ is the demagnetizing tensor of the borders of the prism. By considering a nanowire in which $$L\gg a$$, we can assume $$\mathbf {N}_b=0$$. Under this framework, the point-function demagnetizing field will be then defined as $$\mathbf {H}_d(\mathbf {r})=-4\pi \mathbf {N}_p(\mathbf {r})\cdot \mathbf {M}$$.

Without lost of generality, we will assume that the basis of the wire is in the xz-plane, in such a way that S1 is the surface whose normal vector points along $$-\hat{y}$$ (see Fig. 1c). In this context, we obtain

\begin{aligned} \mathbf {N}_{S1}=\left( \begin{array}{ccc} 0\,{}N_{_{S1}}^{xy}\,{}0\\ 0\,{}N_{_{S1}}^{yy}\,{}0\\ 0\,{}N_{_{S1}}^{zy}\,{}0 \end{array}\right) , \end{aligned}
(5)

where the elements $$N_{_{S1}}^{\mu y}$$ are given by52

\begin{aligned} N_{_{S1}}^{\mu y}=\frac{1}{4\pi }\int _{-L}^{L}dz'\int _{-a}^{a}dx'\, \frac{\mu -\mu '}{\rho ^{3/2}}\,, \end{aligned}
(6)

with $$\mu =(x,y,z)$$, and $$\rho =(x-x')^2+(y-y')^2+(z-z')^2$$. Because the prism has rotational periodicity about the y-axis, it has zero shape anisotropy in the directions transverse to this axis50, and we can state that the average of $$N^{xy}_{S1}$$ and $$N^{zy}_{S1}$$ over the whole volume of the prism vanishes50,51.

The demagnetizing tensor $$\mathbf {N}_{Sj}$$ of an arbitrary side j can be obtained from $$\mathbf {N}_{Sj}=\mathbf {R}^{-1}_j,\mathbf {N}_{S1},\mathbf {R}_j$$, where $$\mathbf {R}_{j}$$ is the transformation matrix defining a rotation along z-axis,

\begin{aligned} \mathbf {R}_{j}=\left( \begin{array}{ccc} \cos \varphi _j&{}\sin \varphi _j&{}0\\ -\sin \varphi _j&{}\cos \varphi _j&{}0\\ 0&{}0&{}1 \end{array}\right) , \end{aligned}
(7)

where $$\varphi _j=2\pi (j-1)/n$$. These assumptions allow us to determine the point-function demagnetizing tensor, evaluated as

\begin{aligned} \mathbf {N}_p(\mathbf {r})=\sum _{j=1}^{n}\mathbf {N}_{Sj} =\left( \begin{array}{ccc} \frac{n }{2}\,N_y&{}0&{}0\\ 0&{}\frac{n}{2}\,N_y&{}0\\ 0&{}0&{}1-n\,N_y \end{array}\right) , \end{aligned}
(8)

where $$N_y=V^{-1}\int N_{_{S1}}^{yy}\,dV$$. The calculation of this integral for an arbitrary geometry is very hard. Nevertheless, from the properties of the demagnetizing tensor, we can determine $$N_y$$. Indeed, since the nanowire is considered infinite because $$L\gg a$$, we have that $$1-n\,N_y=0\Rightarrow N_y=1/n$$. This result is in agreement with the theorem that states that if a sample has uniaxial rotational periodicity $$n \ge 3$$, then it has zero-shape anisotropy along the directions transverse to that axis50. Due to this zero-shape anisotropy, one can notice that in the approximation of infinitely thin and long nanowire, the demagnetizing tensor components along the x and y directions of any regular polygon are equal. Then, from the dynamical equations obtained by Mougin et al.36, we can state that the dynamics of a rigid DW propagating along a nanowire with cross-section given by a thin regular polygon follows the same behavior observed in a cylindrical nanowire38. That is, the dynamics of a DW displacing in thin nanowires is independent of the cross-section geometry, and no Walker regime is observed. Therefore, the velocity of the DW center ($$\theta =\pi /2$$) is

\begin{aligned} v=\frac{H\,\alpha \,\Delta }{1+\alpha ^2}. \end{aligned}
(9)

The above results allow us to conjecture that the oscillations of DW motion, detected by micromagnetic simulations and dependent on the polygon shape, appear as a consequence of the violation of the theoretical assumption of rigid domain wall. Indeed, when the dimensions of the nanowire cross-section diminish, the DW does not present changes in its structure and shape during the rotation around the nanowire cross-section, and then no oscillations appear in its motion. From the set of DW thicknesses shown in Fig. 1b, we conclude that the narrower is the nanowire, the thinner is $$\Delta _{\text {DW}}$$, approaching the analytical value $$\Delta _{\text {DW}}=\pi \sqrt{4A/(\mu _0M_s^2)}\approx 26$$ nm, corresponding to an infinite long and narrow nanowire when decreasing the cross section as it does in cylindrical nanowires53. The impact of this fact into the DW dynamics is crucial as the DW velocity is proportional to $$\Delta _{\text {DW}}$$. Effectively, the DW is slower for the S2 ($$v=2$$ m/s) than for the S1 ($$v=6.5$$ m/s).

To understand the oscillations observed during the DW motion, we analyze the energy of the DW during its rotation around the nanowire. Our results are depicted in Fig. 6, evidencing that during one rotation the DW energy varies n times for wires in set S1. The amplitude of the energy variation diminishes with n in such a way that it is practically constant for the HNW. The amplitude of energy changes also decreases with the size of each side, as evidenced in Fig. 6b, where we present the energy of the DW for set S2. Because the magnetic field necessary to ensure that the DW will overtake the energy barrier, starting to rotate around the nanowire cross-section, increases with the energy amplitude, these results allow us to state that the Walker field for which the DW starts its forth and back motion depends on both n and the side size of the regular polygon, in such a way that the lower n, the higher the Walker limit. Additionally, the greater the side size, the higher the magnetic field generating the oscillatory behavior for the DW motion. The analysis of this figure reveals also that the DW energy in the PNW and HNW is almost constant along a rotation. The changes of the DW energy comes also from changes in the DW width and shape during the rotation around a nanowire, as it is shown in Figs. 1b and 7. Indeed, Fig. 1b evidences the existence of variations in the DW width, which are more pronounced when the DW rotates around the geometries characterized by the S1 system. The amplitudes of the DW width variation ($$\Delta w$$) are intrinsically related with the DW energy variations and consequently, with the amplitude of the oscillations of the DW position. For instance, one can notice that for the pentagon, $$\Delta w_p\approx 1$$ nm for S1 and $$\Delta w_p\approx 0.2$$ nm for S2. Therefore, in the second case, the DW moves almost without changes its width. Additionally, Fig. 7 depicts the snapshots of $$m_y$$ component of the DW center when it is pointing along different directions during its rotation for the TNW and PNW cases in S1 (a) and S2 (b) systems. Again, it can be observed that for TNW the shape of the domain wall changes significantly during one rotation, which traduces in changes of the energy. Nevertheless, the DW rotating around a PNW practically does not change its structure (width and shape), and therefore presents a very small variation in its energy. Furthermore, the reduction in the cross-section dimensions lead to smaller changes in the DW structure in such a way that in the PNW of S2, the changes are almost imperceptible in such a way that the DW can be considered as a rigid structure and then there is no Walker regime in this case, as observed in Fig. 5c. We can then conclude that periodic changes in the energy associated with changes in the DW width and shape during its rotation are responsible for the oscillatory motion observed when the DW displaces in nanowires whose side size is large enough to allow structural changes in the DW. Indeed, the DW deformations during its dynamics have previously been suggested as the reason of the Walker breakdown54. All the analysis performed from the presented micromagnetic simulations are done for a particular set of magnetic parameters. Nevertheless, since the demagnetizing field is crucial to determine the DW oscillatory motion, we can state that changes in the relation $$A/\mu _0M_s^2$$ (exchange to dipolar energies) should bring important effects in the DW dynamics. Despite a complete analysis of these effects is out of the scope of this work, we can state that the increase in the $$M_s$$ value will be associated to a decrease of $$\Delta _{\text {DW}}$$, with larger amplitudes of the oscillatory motion because the torque exerted by the dipolar effective field would increase. Indeed, it was previously shown that the amplitude of the oscillatory motion of a DW displacing in the Walker regime strongly depends on the energy change during the DW rotation motion39,40,41, i.e., the bigger the changes in the DW energy during a rotation, the bigger the amplitude of the DW motion oscillations. Therefore, because changes in the DW dipolar energy when the DW rotates will be smaller for higher values of $$M_s$$, the amplitude of the oscillations must be affected. Nevertheless, because the rotation frequency is a function just of the external magnetic field, the oscillation frequency will remain dependent on n.

## Conclusions

In conclusion, we analyzed the domain wall dynamics in magnetic nanowires with different cross-sections described by regular polygons. Two regimes for the DW can be observed. The first one is characterized by oscillations along the nanowire, whose frequency depends on the number of sides of the nanowire cross-section. In the second regime no oscillations are observed and DW displaces with a velocity that varies linearly with the magnetic field, independent of the number of sides of the wire polygonal cross-section. In both regimes DW also rotates around the NW. The analysis of the energetics of the DW reveals that changes in the DW shape and structure when it rotates around the nanowire are responsible for changes in the DW energy. The changes in the energy are periodic and depend on the number of sides of the cross-section. It is important to highlight that the propagation velocity of a domain wall in a wire with a triangular cross section is an order of magnitude greater than that of a circular wire. We have also developed an analytical model that describes the rigid DW motion along the considered nanowires, and it was shown that if the DW does not change its structure and shape, its energy is constant and so, no oscillatory motion occurs in this case.

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## Acknowledgements

We thank the financial support from Financiamiento Basal AFB 180001 para Centros Científicos y Tecnológicos de Excelencia and proyecto FONDECYT 1200867. In Brazil, we thank CNPq (Grant numbers 401132/2016-1 and 302084/2019-3) and Fapemig for finantial support. YPI acknowledges the support from the state task of the Ministry of Science and Higher Education of the Russian Federation No. 0657-2020-0005. O.C.-F. thanks the support from the Spanish MINECO under Grant MAT2016-76824-C3-1-R. R.C. acknowledge University of Santiago de Chile through Grant Dicyt 041831AD.

## Author information

Authors

### Contributions

D.A. and V.L.C.-S. contributed the main ideas and wrote the main manuscript text. R.M.C., R.M., and Y.P.I. performed micromagnetic simulations and prepared the figures. J.M.F. and O.C.-F. analyzed the results. All authors reviewed the manuscript.

### Corresponding author

Correspondence to Vagson L. Carvalho-Santos.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

### Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Altbir, D., Fonseca, J.M., Chubykalo-Fesenko, O. et al. Tuning domain wall dynamics by shaping nanowires cross-sections. Sci Rep 10, 21911 (2020). https://doi.org/10.1038/s41598-020-78761-w

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text:
# PIERRE DE FERMAT MATHEMATICIAN

## Biography

 Pierre de Fermat (1601-1665)

Another Frenchman of the 17th Century, Pierre de Fermat, effectively invented modern number theory virtually single-handedly, despite being a small-town amateur mathematician. Stimulated and inspired by the “Arithmetica” of the Hellenistic mathematician Diophantus, he went on to discover several new patterns in numbers which had defeated mathematicians for centuries, and throughout his life he devised a wide range of conjectures and theorems. He is also given credit for early developments that led to modern calculus, and for early progress in probability theory.

Although he showed an early interest in mathematics, he went on study law at Orléans and received the title of councillor at the High Court of Judicature in Toulouse in 1631, which he held for the rest of his life. He was fluent in Latin, Greek, Italian and Spanish, and was praised for his written verse in several languages, and eagerly sought for advice on the emendation of Greek texts.

Fermat’s mathematical work was communicated mainly in letters to friends, often with little or no proof of his theorems. Although he himself claimed to have proved all his arithmetic theorems, few records of his proofs have survived, and many mathematicians have doubted some of his claims, especially given the difficulty of some of the problems and the limited mathematical tools available to Fermat.

## The Two Square Theorem

 Fermat’s Theorem on Sums of Two Squares

One example of his many theorems is the Two Square Theorem, which shows that any prime number which, when divided by 4, leaves a remainder of 1 (i.e. can be written in the form 4n + 1), can always be re-written as the sum of two square numbers (see image at right for examples).

His so-called Little Theorem is often used in the testing of large prime numbers, and is the basis of the codes which protect our credit cards in Internet transactions today. In simple (sic) terms, it says that if we have two numbers a and p, where p is a prime number and not a factor of a, then a multiplied by itself p-1 times and then divided by p, will always leave a remainder of 1. In mathematical terms, this is written: ap-1 = 1(mod p). For example, if a = 7 and p = 3, then 72 ÷ 3 should leave a remainder of 1, and 49 ÷ 3 does in fact leave a remainder of 1.

## Fermat numbers

Fermat identified a subset of numbers, now known as Fermat numbers, which are of the form of one less than 2 to the power of a power of 2, or, written mathematically, 22n + 1. The first five such numbers are: 21 + 1 = 3; 22 + 1 = 5; 24 + 1 = 17; 28 + 1 = 257; and 216 + 1 = 65,537. Interestingly, these are all prime numbers (and are known as Fermat primes), but all the higher Fermat numbers which have been painstakingly identified over the years are NOT prime numbers, which just goes to to show the value of inductive proof in mathematics.

## Last Theorem

 Fermat’s Last Theorem

Fermat’s pièce de résistance, though, was his famous Last Theorem, a conjecture left unproven at his death, and which puzzled mathematicians for over 350 years. The theorem, originally described in a scribbled note in the margin of his copy of Diophantus‘ “Arithmetica”, states that no three positive integers a, b and c can satisfy the equation an + bn = cn for any integer value of n greater than two (i.e. squared). This seemingly simple conjecture has proved to be one of the world’s hardest mathematical problems to prove.

There are clearly many solutions – indeed, an infinite number – when n = 2 (namely, all the Pythagorean triples), but no solution could be found for cubes or higher powers. Tantalizingly, Fermat himself claimed to have a proof, but wrote that “this margin is too small to contain it”. As far as we know from the papers which have come down to us, however, Fermat only managed to partially prove the theorem for the special case of n = 4, as did several other mathematicians who applied themselves to it (and indeed as had earlier mathematicians dating back to Fibonacci, albeit not with the same intent).

Over the centuries, several mathematical and scientific academies offered substantial prizes for a proof of the theorem, and to some extent it single-handedly stimulated the development of algebraic number theory in the 19th and 20th Centuries. It was finally proved for ALL numbers only in 1995 (a proof usually attributed to British mathematician Andrew Wiles, although in reality it was a joint effort of several steps involving many mathematicians over several years). The final proof made use of complex modern mathematics, such as the modularity theorem for semi-stable elliptic curves, Galois representations and Ribet’s epsilon theorem, all of which were unavailable in Fermat’s time, so it seems clear that Fermat’s claim to have solved his last theorem was almost certainly an exaggeration (or at least a misunderstanding).

In addition to his work in number theory, Fermat anticipated the development of calculus to some extent, and his work in this field was invaluable later to Newton and Leibniz. While investigating a technique for finding the centres of gravity of various plane and solid figures, he developed a method for determining maxima, minima and tangents to various curves that was essentially equivalent to differentiation. Also, using an ingenious trick, he was able to reduce the integral of general power functions to the sums of geometric series.

Fermat’s correspondence with his friend Pascal also helped mathematicians grasp a very important concept in basic probability which, although perhaps intuitive to us now, was revolutionary in 1654, namely the idea of equally probable outcomes and expected values.


text:
# How are these expressions equal?

I came across this step in an inductive proof, but my algebra skills seem a bit rusty..

$7(7^{k+2}) + 64(8^{2k+1}) = 7(7^{k+2}+8^{2k+1})+57(8^{2k+1})$

How did they do this?

Note: The point was to show that the expressions are divisible by 57.

-

$64=7+57$, so $64\cdot 8^{2k+1}=(7+57)\cdot 8^{2k+1}=7\cdot 8^{2k+1}+57 \cdot 8^{2k+1}$.
Hint $\$ With $\rm\: A = 7^{\,k+2}\:$ and $\rm\: B = 8^{\,2k+1}\:$ it is more easy to comprehend
$$\rm 7\, A + 64\ B\ =\ 7\,(A + B) + 57\ B$$


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text:
# C3 Basic Trig

Watch
Announcements
Thread starter 3 years ago
#1
tan2θcosec2θ
Simplify
I did tan2θ * 1/sin2θ = tan2θ/sin2θ
However, it is either wrong or i have not simplified far enough?
0
3 years ago
#2
is that tan2feta, or tan squared feta?
0
Thread starter 3 years ago
#3
(Original post by DeGoldenGamer)
is that tan2feta, or tan squared feta?
tan 2 feta
0
3 years ago
#4
(Original post by DeGoldenGamer)
is that tan2feta, or tan squared feta?
Big man what's feta? Lmao, You making me hungry.
2
3 years ago
#5
(Original post by Desultory)
tan2θcosec2θ
Simplify
I did tan2θ * 1/sin2θ = tan2θ/sin2θ
However, it is either wrong or i have not simplified far enough?
You didn't really simplify it, you just changed the cosec to a sin

Given these are both functions of , you may be expected to use the double angle formulae to change these into expressions of just and

Give that a go and let us know what you come up with
0
Thread starter 3 years ago
#6
(Original post by Ed5)
You didn't really simplify it, you just changed the cosec to a sin

Given these are both functions of , you may be expected to use the double angle formulae to change these into expressions of just and

Give that a go and let us know what you come up with
This is c3 year 13 maths? Sorry i dont know the double angle formulae
0
3 years ago
#7
(Original post by Desultory)
This is c3 year 13 maths? Sorry i dont know the double angle formulae
If you're on edexcel, aye it's C3 and these formulae are a key part of it. They let you break down trignometric functions into more basic ones, e.g.

You should definitely cover it in class at some point so it's likely you just haven't reached the topic yet
0
3 years ago
#8
(Original post by Desultory)
tan2θcosec2θ
Simplify
I did tan2θ * 1/sin2θ = tan2θ/sin2θ
However, it is either wrong or i have not simplified far enough?
Write and see what you notice.

You don't need to use double angle formulae for this question.
0
Thread starter 3 years ago
#9
(Original post by Ed5)
If you're on edexcel, aye it's C3 and these formulae are a key part of it. They let you break down trignometric functions into more basic ones, e.g.

You should definitely cover it in class at some point so it's likely you just haven't reached the topic yet
That is true but this is a section 6 question which is basic trig not section 7 which is further trig so i am not expexted to use that so i need another way?
0
Thread starter 3 years ago
#10
(Original post by Notnek)
Write and see what you notice.

You don't need to use double angle formulae for this question.
i know tan theta = sin theta / cos theta but if its tan 2 theta, are you allowed to put a 2 infront of the sin and cos theta?
I didnt know that?
0
3 years ago
#11
(Original post by Desultory)
i know tan theta = sin theta / cos theta but if its tan 2 theta, are you allowed to put a 2 infront of the sin and cos theta?
I didnt know that?
Yes, the identity is and since it's an identity, it's true for all so you can replace with anything.

E.g.

or
0
3 years ago
#12
(Original post by Desultory)
That is true but this is a section 6 question which is basic trig not section 7 which is further trig so i am not expexted to use that so i need another way?
(Original post by Notnek)
Write and see what you notice.

You don't need to use double angle formulae for this question.
Must just be like this then. I wasn't really sure given you didn't mention the module, and 'simplify' can be quite open ended
0
Thread starter 3 years ago
#13
(Original post by Notnek)
Yes, the identity is () and since it's an identity, it's true for all so you can replace with anything.

E.g.

or
Okay that makes sense thanks i worked it out now
However, if there is a question where its tan^2 theta, can that equal sin^2 theta / cos ^2 theta or not?
0
Thread starter 3 years ago
#14
(Original post by Ed5)
Must just be like this then. I wasn't really sure given you didn't mention the module, and 'simplify' can be quite open ended
I appreciated the advice!
0
3 years ago
#15
(Original post by Desultory)
Okay that makes sense thanks i worked it out now
However, if there is a question where its tan^2 theta, can that equal sin^2 theta / cos ^2 theta or not?
Yes that's also true but for a different reason:

0
Thread starter 3 years ago
#16
(Original post by Notnek)
Yes that's also true but for a different reason:

Okay thanks now these questions seem easy!
1
3 years ago
#17
(Original post by Desultory)
Okay thanks now these questions seem easy!
Getting good at C3/4 trigonometry is all about practice. Some students think they can get by if they just learn the identities but this doesn't work. You have to try as many different questions as you can to see how/where identities can be used and to pick up all the little tricks.

When a student tells me that they hate C3 trig (common), I know that they probably haven't practiced enough
0
Thread starter 3 years ago
#18
(Original post by Notnek)
Getting good at C3/4 trigonometry is all about practice. Some students think they can get by if they just learn the identities but this doesn't work. You have to try as many different questions as you can to see how/where identities can be used and to pick up all the little tricks.

When a student tells me that they hate C3 trig (common), I know that they probably haven't practiced enough
haha i completely understand
are you a teacher of maths?
reccommend any good websites for tutors or even yourself?
0
3 years ago
#19
(Original post by Desultory)
haha i completely understand
are you a teacher of maths?
reccommend any good websites for tutors or even yourself?
Yes I teach/tutor. If you google "maths tutor" then you'll find websites where you can find tutors near you.
0
Thread starter 3 years ago
#20
(Original post by Notnek)
Yes I teach/tutor. If you google "maths tutor" then you'll find websites where you can find tutors near you.
ok thanks

sec A - sec A * sin^2 A

i did 1/cos A - 1/cos A * 1-cos^2A) but it doesnt really simplify
have i done this wrong?
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text:
A reduction in atomic size with increase in atomic number is a characterstic of elements of

$\begin{array}{1 1}(a)\;d-block&(b)\;f-block\\(c)\;radio\;active\;series&(d)\;high\;atomic\;masses\end{array}$

A reduction in atomic size with increase in atomic number is a characterstic of elements of f-block.
Hence (b) is the correct answer.