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# L1 vs L2 Regularization: The intuitive difference

A lot of people usually get confused which regularization technique is better to avoid overfitting while training a machine learning model.

Source ? http://laid.delanover.com/difference-between-l1-and-l2-regularization-implementation-and-visualization-in-tensorflow/

I won?t go about much in detail about the maths side. Instead will try to explain the intuitive difference between them.

The main intuitive difference between the L1 and L2 regularization is that L1 regularization tries to estimate the median of the data while the L2 regularization tries to estimate the mean of the data to avoid overfitting.

As we can see from the formula of L1 and L2 regularization, L1 regularization adds the penalty term in cost function by adding the absolute value of weight(Wj) parameters, while L2 regularization adds the squared value of weights(Wj) in the cost function.

While taking derivative of the cost function, in L1 regularization it will estimate around the median of the data. Let me explain it in this way ? Suppose you take an arbitrary value from the data (assume data is spread along a horizontal line). If you then move in one direction to some distance d, suppose in the backward direction, then while calculating loss, the values to the one side (let say left side) of the chosen point will have a lesser loss value while on another side will contribute more in the loss function calculation.

Therefore, to minimize the loss function, we should try to estimate a value that should lie at the mid of the data distribution. That value will also be the median of the data distribution mathematically.

While in L2 regularization, while calculating the loss function in the gradient calculation step, the loss function tries to minimize the loss by subtracting it from the average of the data distribution.

That?s the main intuitive difference between the L1 (Lasso) and L2 (Ridge) regularization technique.

Another difference between them is that L1 regularization helps in feature selection by eliminating the features that are not important. This is helpful when the number of feature points are large in number.

That?s all from my side this time. Hope I am able to explain the intuitive difference between both the regularization techniques.


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• What does the Schmidt number describe?
• How is the Schmidt number defined?
• What is the analogy between the Schmidt number and Prandtl number?

Not only the transport of momentum and the heat transfer influence each other as already explained in detail in the article on the Prandtl number. Momentum transport and mass transfer also interact with each other. Among other things, the momentum transport is due to Brownian motion. In this way, fluid particles diffuse across the fluid layers and have to adapt to the new velocities, so to speak (transfer of momentum). If there is also a concentration gradient, the fluid particles diffuse into other fluid layers not only due to the random Brownian motion, but also due to the concentration gradient.

The ratio of momentum and mass transfer is expressed by the so-called Schmidt number $$Sc$$. It is, so to speak, a measure for the ratio of the boundary layer thicknesses between hydrodynamic boundary layer and concentration boundary layer. The Schmidt number puts the viscosity $$\nu$$ of a fluid in relation to its diffusion coefficient $$D$$:

\begin{align}
&\text{Schmidt number} = \frac{\text{transport of momentum}}{\text{transport of mass}} \\[5px]
\label{sc}
&\boxed{Sc = \frac{\nu}{D}} ~~~~~\text{Schmidt number}\\[5px]
\end{align}

The Schmidt number can be considered analogous to the Prandtl number. While the Prandtl number describes the diffusion of heat, the Schmidt number characterizes the diffusion of matter. In the same way that the Prandtl number is used to determine the Nusselt number as a dimensionless heat transfer coefficient, the Schmidt number is used to determine the Sherwood number, which has its significance as a dimensionless mass transfer coefficient. Both dimensionless transfer numbers (Nusselt number and Sherwood number) also have in common that they are influenced by the Reynolds number $$Re$$.

Just like the Prandtl number, the Schmidt number is also a dimensionless similarity parameter. This means that mass and momentum transport in two differently sized systems are only physically similar if the Schmidt numbers are identical. Only then, for example, is it possible to transfer the knowledge gained in a model experiment to real applications.

The Schmidt number is a dimensionless similarity parameter to describe mass and momentum transfer. Only with identical Schmidt numbers one obtains physically similar mass and momentum transfers, independent of the size of the system.

format: plain


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F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF08JVF (ZSTEDC)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F08JVF (ZSTEDC) computes all the eigenvalues and, optionally, all the eigenvectors of a real $n$ by $n$ symmetric tridiagonal matrix, or of a complex full or banded Hermitian matrix which has been reduced to tridiagonal form.

## 2  Specification

 SUBROUTINE F08JVF ( COMPZ, N, D, E, Z, LDZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO)
 INTEGER N, LDZ, LWORK, LRWORK, IWORK(max(1,LIWORK)), LIWORK, INFO REAL (KIND=nag_wp) D(*), E(*), RWORK(max(1,LRWORK)) COMPLEX (KIND=nag_wp) Z(LDZ,*), WORK(max(1,LWORK)) CHARACTER(1) COMPZ
The routine may be called by its LAPACK name zstedc.

## 3  Description

F08JVF (ZSTEDC) computes all the eigenvalues and, optionally, the eigenvectors of a real symmetric tridiagonal matrix $T$. That is, the routine computes the spectral factorization of $T$ given by
 $T = Z Λ ZT ,$
where $\Lambda$ is a diagonal matrix whose diagonal elements are the eigenvalues, ${\lambda }_{i}$, of $T$ and $Z$ is an orthogonal matrix whose columns are the eigenvectors, ${z}_{i}$, of $T$. Thus
 $Tzi = λi zi , i = 1,2,…,n .$
The routine may also be used to compute all the eigenvalues and eigenvectors of a complex full, or banded, Hermitian matrix $A$ which has been reduced to real tridiagonal form $T$ as
 $A = QTQH ,$
where $Q$ is unitary. The spectral factorization of $A$ is then given by
 $A = QZ Λ QZH .$
In this case $Q$ must be formed explicitly and passed to F08JVF (ZSTEDC) in the array Z, and the routine called with ${\mathbf{COMPZ}}=\text{'V'}$. Routines which may be called to form $T$ and $Q$ are
 full matrix F08FSF (ZHETRD) and F08FTF (ZUNGTR) full matrix, packed storage F08GSF (ZHPTRD) and F08GTF (ZUPGTR) band matrix F08HSF (ZHBTRD), with ${\mathbf{VECT}}=\text{'V'}$
When only eigenvalues are required then this routine calls F08JFF (DSTERF) to compute the eigenvalues of the tridiagonal matrix $T$, but when eigenvectors of $T$ are also required and the matrix is not too small, then a divide and conquer method is used, which can be much faster than F08JSF (ZSTEQR), although more storage is required.

## 4  References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5  Parameters

1:     COMPZ – CHARACTER(1)Input
On entry: indicates whether the eigenvectors are to be computed.
${\mathbf{COMPZ}}=\text{'N'}$
Only the eigenvalues are computed (and the array Z is not referenced).
${\mathbf{COMPZ}}=\text{'I'}$
The eigenvalues and eigenvectors of $T$ are computed (and the array Z is initialized by the routine).
${\mathbf{COMPZ}}=\text{'V'}$
The eigenvalues and eigenvectors of $A$ are computed (and the array Z must contain the matrix $Q$ on entry).
Constraint: ${\mathbf{COMPZ}}=\text{'N'}$, $\text{'V'}$ or $\text{'I'}$.
2:     N – INTEGERInput
On entry: $n$, the order of the symmetric tridiagonal matrix $T$.
Constraint: ${\mathbf{N}}\ge 0$.
3:     D($*$) – REAL (KIND=nag_wp) arrayInput/Output
Note: the dimension of the array D must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: the diagonal elements of the tridiagonal matrix.
On exit: if ${\mathbf{INFO}}={\mathbf{0}}$, the eigenvalues in ascending order.
4:     E($*$) – REAL (KIND=nag_wp) arrayInput/Output
Note: the dimension of the array E must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}-1\right)$.
On entry: the subdiagonal elements of the tridiagonal matrix.
On exit: E is overwritten.
5:     Z(LDZ,$*$) – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array Z must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$ if ${\mathbf{COMPZ}}=\text{'V'}$ or $\text{'I'}$, and at least $1$ otherwise.
On entry: if ${\mathbf{COMPZ}}=\text{'V'}$, Z must contain the unitary matrix $Q$ used in the reduction to tridiagonal form.
On exit: if ${\mathbf{COMPZ}}=\text{'V'}$, Z contains the orthonormal eigenvectors of the original Hermitian matrix $A$, and if ${\mathbf{COMPZ}}=\text{'I'}$, Z contains the orthonormal eigenvectors of the symmetric tridiagonal matrix $T$.
If ${\mathbf{COMPZ}}=\text{'N'}$, Z is not referenced.
6:     LDZ – INTEGERInput
On entry: the first dimension of the array Z as declared in the (sub)program from which F08JVF (ZSTEDC) is called.
Constraints:
• if ${\mathbf{COMPZ}}=\text{'V'}$ or $\text{'I'}$, ${\mathbf{LDZ}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$;
• otherwise ${\mathbf{LDZ}}\ge 1$.
7:     WORK($\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{LWORK}}\right)$) – COMPLEX (KIND=nag_wp) arrayWorkspace
On exit: if ${\mathbf{INFO}}={\mathbf{0}}$, the real part of ${\mathbf{WORK}}\left(1\right)$ contains the minimum value of LWORK required for optimal performance.
8:     LWORK – INTEGERInput
On entry: the dimension of the array WORK as declared in the (sub)program from which F08JVF (ZSTEDC) is called.
If ${\mathbf{LWORK}}=-1$, a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK, LRWORK or LIWORK is issued.
Constraints:
if ${\mathbf{LWORK}}\ne -1$,
• if ${\mathbf{COMPZ}}=\text{'N'}$ or $\text{'I'}$ or ${\mathbf{N}}\le 1$, LWORK must be at least $1$;
• if ${\mathbf{COMPZ}}=\text{'V'}$ and ${\mathbf{N}}>1$, LWORK must be at least ${{\mathbf{N}}}^{2}$.
Note: that for ${\mathbf{COMPZ}}=\text{'V'}$, then if N is less than or equal to the minimum divide size, usually $25$, then LWORK need only be $1$.
9:     RWORK($\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{LRWORK}}\right)$) – REAL (KIND=nag_wp) arrayWorkspace
On exit: if ${\mathbf{INFO}}={\mathbf{0}}$, ${\mathbf{RWORK}}\left(1\right)$ returns the optimal LRWORK.
10:   LRWORK – INTEGERInput
On entry: the dimension of the array RWORK as declared in the (sub)program from which F08JVF (ZSTEDC) is called.
If ${\mathbf{LRWORK}}=-1$, a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK, LRWORK or LIWORK is issued.
Constraints:
if ${\mathbf{LRWORK}}\ne -1$,
• if ${\mathbf{COMPZ}}=\text{'N'}$ or ${\mathbf{N}}\le 1$, LRWORK must be at least $1$;
• if ${\mathbf{COMPZ}}=\text{'V'}$ and ${\mathbf{N}}>1$, LRWORK must be at least $1+3×{\mathbf{N}}+2×{\mathbf{N}}×lg\left({\mathbf{N}}\right)+4×{{\mathbf{N}}}^{2}$, where $lg\left({\mathbf{N}}\right)=\text{}$smallest integer $k$ such that ${2}^{k}\ge {\mathbf{N}}$;
• if ${\mathbf{COMPZ}}=\text{'I'}$ and ${\mathbf{N}}>1$, LRWORK must be at least $1+4×{\mathbf{N}}+2×{{\mathbf{N}}}^{2}$.
Note: that for ${\mathbf{COMPZ}}=\text{'I'}$ or $\text{'V'}$ then if N is less than or equal to the minimum divide size, usually $25$, then LRWORK need only be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2×\left({\mathbf{N}}-1\right)\right)$.
11:   IWORK($\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{LIWORK}}\right)$) – INTEGER arrayWorkspace
On exit: if ${\mathbf{INFO}}={\mathbf{0}}$, ${\mathbf{IWORK}}\left(1\right)$ returns the optimal LIWORK.
12:   LIWORK – INTEGERInput
On entry: the dimension of the array IWORK as declared in the (sub)program from which F08JVF (ZSTEDC) is called.
If ${\mathbf{LIWORK}}=-1$, a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK, LRWORK or LIWORK is issued.
Constraints:
if ${\mathbf{LIWORK}}\ne -1$,
• if ${\mathbf{COMPZ}}=\text{'N'}$ or ${\mathbf{N}}\le 1$, LIWORK must be at least $1$;
• if ${\mathbf{COMPZ}}=\text{'V'}$ or ${\mathbf{N}}>1$, LIWORK must be at least $6+6×{\mathbf{N}}+5×{\mathbf{N}}×lg\left({\mathbf{N}}\right)$;
• if ${\mathbf{COMPZ}}=\text{'I'}$ or ${\mathbf{N}}>1$, LIWORK must be at least $3+5×{\mathbf{N}}$.
13:   INFO – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error Indicators and Warnings

Errors or warnings detected by the routine:
${\mathbf{INFO}}<0$
If ${\mathbf{INFO}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{INFO}}>0$
The algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and columns ${\mathbf{INFO}}/\left({\mathbf{N}}+1\right)$ through .

## 7  Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrix $\left(T+E\right)$, where
 $E2 = Oε T2 ,$
and $\epsilon$ is the machine precision.
If ${\lambda }_{i}$ is an exact eigenvalue and ${\stackrel{~}{\lambda }}_{i}$ is the corresponding computed value, then
 $λ~i - λi ≤ c n ε T2 ,$
where $c\left(n\right)$ is a modestly increasing function of $n$.
If ${z}_{i}$ is the corresponding exact eigenvector, and ${\stackrel{~}{z}}_{i}$ is the corresponding computed eigenvector, then the angle $\theta \left({\stackrel{~}{z}}_{i},{z}_{i}\right)$ between them is bounded as follows:
 $θ z~i,zi ≤ cnεT2 mini≠jλi-λj .$
Thus the accuracy of a computed eigenvector depends on the gap between its eigenvalue and all the other eigenvalues.
See Section 4.7 of Anderson et al. (1999) for further details. See also F08FLF (DDISNA).

If only eigenvalues are required, the total number of floating point operations is approximately proportional to ${n}^{2}$. When eigenvectors are required the number of operations is bounded above by approximately the same number of operations as F08JSF (ZSTEQR), but for large matrices F08JVF (ZSTEDC) is usually much faster.
The real analogue of this routine is F08JHF (DSTEDC).

## 9  Example

This example finds all the eigenvalues and eigenvectors of the Hermitian band matrix
 $A = -3.13i+0.00 1.94-2.10i -3.40+0.25i 0.00i+0.00 1.94+2.10i -1.91i+0.00 -0.82-0.89i -0.67+0.34i -3.40-0.25i -0.82+0.89i -2.87i+0.00 -2.10-0.16i 0.00i+0.00 -0.67-0.34i -2.10+0.16i 0.50i+0.00 .$
$A$ is first reduced to tridiagonal form by a call to F08HSF (ZHBTRD).

### 9.1  Program Text

Program Text (f08jvfe.f90)

### 9.2  Program Data

Program Data (f08jvfe.d)

### 9.3  Program Results

Program Results (f08jvfe.r)


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# Uniform Linear Array (ULA) beamwidth and angular resolution using FFT

I have a four element uniform linear array receiver and I want to know what will be the angular resolution if I perform perform Fast Fourier Transform (FFT) over antenna elements. As I think that it will be same as the beam width of the array.

The antenna element spacing is lambda/2

Question1: What is the beamwidth of the array structure

Question2: If I perform FFT over the element array will I get the same angular resolution as beamwidth

Question3: How each bin in antenna FFT will represent my region of space (e.g Frequency bin 1 = theta 0 to 30 degrees)

# Theoretical Derivation

The antenna electric field pattern of array antenna consists of isotropic radiator can be given by

$\begin{vmatrix}&space;E(\Theta&space;)&space;\end{vmatrix}&space;=&space;\begin{vmatrix}&space;\frac{&space;sin\begin{bmatrix}&space;N\left&space;(&space;\pi&space;d&space;/\lambda\right&space;)sin\Theta&space;)&space;\end{bmatrix}}{sin\left&space;[&space;(\pi&space;d/\lambda&space;)sin\Theta&space;\right&space;]}&space;\end{vmatrix}$

where N is the number of antenna elements

d is the spacing between antenna elements

In order to find the beamwidth (3 dB), the above equation should be equated to $\frac{1}{\sqrt{2}}$ and solve for $\Theta$

The solution will come to be as $0.89\frac{\lambda&space;}{D}$

where D is the total aperture distance and can be approximated as $Nd$

For spacing of $d&space;=&space;\lambda&space;/2$ the equation simplifies and can be approximated as $2/N$

Thus beam width of planar antenna can be represented as

$\Delta&space;\theta_{3dB}=&space;2/N$

# Angular Resolution Using FFT Over Antenna Elements

In order the prove the lemma that the angular resolution obtained by performing the FFT across antenna dimension equals to the beamwidth of the antenna array, we have to obtain the angular resolution using FFT.

The below diagram shows the frequency obtained due to path difference between the antenna elements which occurred due to the angle of arrival other than the broadside angle

The frequency resolution using FFT can be represented as $\frac{2\pi}{N}$ where N is the number of samples but in our case it is equal to number of antenna elements.

The angular resolution $\Delta&space;\theta$ can be found by equating the difference in frequency of different angles $w_1&space;=&space;\frac{2&space;\pi&space;d&space;sin\Theta&space;}{\lambda}$ and $w_2&space;=&space;\frac{2&space;\pi&space;d&space;sin\left(\Theta&space;+&space;\Delta&space;\Theta&space;\right&space;)&space;}{\lambda}$

The frequency resolution becomes: $w_2&space;-&space;w_1&space;=&space;\frac{2\pi}{N}&space;=&space;\frac{2&space;\pi&space;d&space;sin\left(\Theta&space;+&space;\Delta&space;\Theta&space;\right&space;)&space;}{\lambda}&space;-&space;\frac{2&space;\pi&space;d&space;sin\Theta&space;}{\lambda}$

Solving we get $\Delta&space;\Theta&space;=&space;\frac{\lambda&space;}{Ndcos\Theta&space;}$ and if we put $\theta&space;=&space;0$ and $d&space;=&space;\lambda&space;/2$ we get the same resolution as the beamwidth i.e
$\Delta&space;\Theta&space;=&space;2/N&space;=&space;\Delta&space;\Theta_{3dB}$

# Summary

Thus for summarizing the above discussion using FFT we can only achieve the max angular resolution equals to the beamwidth of the antenna array which is equal to angular resolution = antenna array beamwidth = 2/N

Edit: *The derivation holds only for isotropic radiating element, in case of non isotropic element "array factor" should compensate the specific antenna's field pattern.

*Also the beam width is approximated to 2/N i.e. N should be large enough to get the equality

Reference: Fundamentals of Radar Signal Processing by Mark A. Richards, 2005

• I like your answer very much. I hope the answer 2/N is in radians and we have to convert it into degrees. I would also like to know whether you can provide some reference (Journal paper or textbook) for the above derivation, so that I can cite it in my reference list. Thanks. – Ashish Aug 13 '17 at 16:56
• Thanks for the appreciation. Yes, answer is in radians. I will share the reference shortly. Feel free to up vote the answer, if it helps you in any case. – Zeeshan Aug 13 '17 at 21:04
• You can see "Fundamentals of Radar Signal Processing by Mark A. Richards, 2005". – Zeeshan Aug 16 '17 at 5:30
• This derivation is for an isotropic radiator, this "array factor" will be modified by the electric field pattern of the actual array antenna elements used. – Captainj2001 Sep 20 '17 at 13:33
• sure, array factor incorporated into the note – Zeeshan Sep 20 '17 at 13:41

I think estimating the HPBW as 2/N is not that accurate. The more accurate answer would be 0.89*2/N. I have checked this by actually plotting the array response in MATLAB. For four antenna elements the answer is off by about 2.5% but the answer is very accurate as the number of elements is increased to ten. I guess the answer would get progressively more accurate as the number of elements is increased.

• For sure, its not exactly 2/N its approximated to 2/N for N large enough to estimate. This is also being quoted in my answer too, "For spacing of [d = \lambda /2] the equation simplifies and can be ~approximated~ as [2/N]" I think that is what you are also observing using the MATLAB plots for array length of less than 10. – Zeeshan Jan 25 '18 at 13:49
• In my own view, it should be a comment under the answer not an answer itself. – Zeeshan Jan 25 '18 at 13:55
• Often the approximation $\lambda/D$ is used for beamwidth, where D = length of array = $(N-1)*d$. If $d = \lambda/2$, then this simplifies to $2/(N-1)$. – Gillespie Feb 12 at 15:43


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# NEET Physics Ray Optics and Optical Instruments Questions Solved

NEET - 2015

In an astronomical telescope in normal adjustment a straight black line of length L is drawn on inside part of objective lens. The eye-piece forms a real image of this line. The length of this image is l. The magnification of the telescope is

(a)L/l+1

(b)L/l-1

(c)L+1/L-1

(d)L/l

We know,magnification of telescope,we have

M=$\frac{{f}_{0}}{fe}$

Here, =>fe/fe+u=-l/L

=> fo/fe-(fo+fo)=-l/L

=>fe/fo=l/L

i.e. M=L/l

Difficulty Level:

• 19%
• 25%
• 23%
• 36%
Crack NEET with Online Course - Free Trial (Offer Valid Till September 17, 2019)


text:
# Plus sign

+
Mathematical signs
arithmetic
Plus sign +
Minus sign - , ./.
Mark , ×
Divided sign : , ÷ , /
Plus minus sign ± ,
Comparison sign < , , = , , >
Root sign
Percent sign %
Analysis
Sum symbol Σ
Product mark Π
Difference sign , Nabla ,
Prime
Partial differential
Integral sign
Concatenation characters
Infinity symbol
geometry
Angle sign , , ,
Vertical , parallel ,
Triangle , square ,
Diameter sign
Set theory
Union , cut ,
Difference , complement ,
Element character
Subset , superset , , ,
Empty set
logic
Universal quantifier
Existential quantifier
Conjunction , disjunction ,
Negation sign ¬

The plus sign (+) is used as a mathematical operator for addition or as a sign for positive numbers or quantities .

## History and creation

The cross has been known as a symbol since the Stone Age ( e.g. in stone carvings ), but it is usually not understandable. As a mathematical symbol, it is of more recent date. In the 15th century it was still common to spell out the words “plus” and “minus”.

The first plus sign in print appeared in the work Mercantile Arithmetic or Behende und hüpsche Rechenung auff alle Kauffmanschektiven by Johannes Widmann , published in Leipzig in 1489 .

## use

The plus sign is used in the following roles:

## Typography and appearance

The typographically correct plus sign consists of two bars of the same size, which cross each other at right angles at their center and one of which runs parallel to the base line. Thus, it resembles a Greek cross , but the bars are usually narrower in relation to the overall size of the sign.

Arithmetic symbols in a print from 1834

In 19th century pamphlets, its width usually resembled that of an em dash . The height of the vertical bar varied between capital height and square height . Thus (in contrast to today's shapes) the vertical bar was often slightly shorter than the horizontal one.

In today's fonts , the width of the plus sign is usually matched to the appearance of the decimal digits. Thus, its width is mostly similar to that of the digit zero. The vertical bar is always the same length as the horizontal one, so the height of the character is always less than the capital height. Vertically it is mostly positioned slightly (about a bar width) above the baseline, so it does not take the x-line into account .

Like all common arithmetic symbols, it is always sans serif regardless of the style of the font in which it is contained . In italics , the vertical bar appears in many fonts without an incline (i.e. the font is not always taken into account).

The plus sign in the fonts Arial , Times New Roman , Cambria , Linux Libertine , Adobe Garamond , Andron Mega Corpus, Consolas and Breitkopf-Fraktur , in the second line in italics in the same fonts, together with “x” / zero to mark the x - or capital height

### Basis for the form of other arithmetic symbols

The design of the plus sign in a font usually determines the design of other arithmetic symbols.

• The (typographical) minus sign (-) is identical to the horizontal bar of the plus sign in terms of its shape (length, line width, if necessary shape of the ends).
• The equal sign (=) consists of two parallel horizontal bars with the same shape as the horizontal bar of the plus sign. The bars are offset equally far up or down compared to the center of the plus sign.
• In many fonts the center of the cross (×) is exactly as high as the center of the plus sign.

### Hebrew plus sign

A different form of the plus sign was designed for the Hebrew script , which lacks the lower part of the vertical bar, so that its shape does not resemble a (Christian) cross . There has been a (not universally used) tradition in Judaism since the 19th century, as it was later in individual Israeli schools, to symbolize addition in this way. It is also used on El Al aircraft for seat adjustment control. However, most adults use the international plus sign. The Israeli religious scholar Adin Steinsaltz described the Hebrew plus sign in a 1994 interview with Spiegel as "absurd externality".

The character is contained in Unicode as U + FB29 ﬩ hebrew letter alternative plus sign .

## Representation in computer systems

### Coding

The plus sign is defined and coded as follows:

International character  coding standard Unicode
and coding in the Internet document format HTML
character Unicode
position
Unicode
designation
designation HTML
HTML
decimal
HTML
named
URL
encoded
Unicode block
+ U + 002B plus sign Plus sign %2 B Basic Latin

The block mathematical operators , in which the typographical minus is coded, does not contain a special typographical plus.

In TeX / LaTeX it is simply encoded by its ASCII characters.

### Variants of the plus sign

character Unicode
position
Unicode
designation
designation block Latex
+ U + 002B plus sign Plus sign Basic Latin +
˖ U + 02D6 modifier letter plus sign Modifying plus sign Spacing Modifier Letters
◌̟ U + 031F combining plus sign below COMBINING SUB-PLUS SIGN COMBINING DIACRITICS \textsubplus
U + 207A superscript plus sign Superscript plus sign Superscripts and subscripts
U + 208A subscript plus sign Subscript plus sign Superscripts and subscripts
U + 2795 heavy plus sign Bold plus sign Dingbats
U + 29FA double plus Double plus sign Various math symbols-B
U + 29FB triple plus Triple plus sign Various math symbols-B
U + 29FE tiny Tiny ( combinatorial game theory ) Various math symbols-B
U + FB29 hebrew letter alternative plus sign hebrew alternative plus sign Alphabetical forms of presentation
U + FE62 small plus sign Small plus sign Small variations in shape
U + FF0B fullwidth plus sign Full-width plus sign Half-width and full-width shapes

### Similar characters

character Unicode
position
Unicode
designation
designation block Latex
U + 1429 canadian syllabics final plus (a character from the Cree script ) Unified indigenous Canadian syllables
U + 205C dotted cross dotted cross General punctuation
U + 22B9 hermitian conjugate matrix Hermitian conjugated matrix Mathematical operators
U + 2629 cross of jerusalem Crutch cross (1) Different symbols \kreuz
U + 2719 outlined greek cross Outlined Greek cross Dingbats \PlusOutline
U + 271A heavy greek cross Fat Greek cross Dingbats \Plus
U + 271B open center cross Open center cross Dingbats
U + 271C heavy open center cross Bold cross with an open center Dingbats \PlusThinCenterOpen
U + 2722 four teardrop-spoked asterisk Four-armed teardrop star (teardrop cross) Dingbats \FourAsterisk
U + 2723 four balloon-spoked asterisk Four-armed ball star (ball cross) Dingbats \JackStar
U + 2724 heavy four balloon-spoked asterisk Fat four-armed ball star Dingbats \JackStarBold
U + 2725 four club-spoked asterisk Shamrock cross Dingbats \CrossClowerTips
U + 2E2D five dot mark Five point mark Additional punctuation \fivedots
U + 3038 hangzhou numeral Suzhou number sign ten CJK symbols and punctuation
U + 5341 (a CJK sign) Unified CJK ideograms
? U + 10932 lydian letter q Lydian letter Q Lydian
? U + 11055 brahmi number four Brahmi number four Brahmi
? U + 16869 bamum letter phase-b shet njaq (Historical letter of the Bamum script) Bamum, addition
? U + 169B7 bamum letter phase-e ket (Historical letter of the Bamum script) Bamum, addition
? U + 1F542 cross pommee Apple cross Various pictographic symbols
? U + 1F70A alchemical symbol for vinegar Alchemical symbol for vinegar Alchemical symbols
(1)In German, " Jerusalem Cross " denotes another symbol.

### Sign with plus sign as a component

character Unicode
position
Unicode
designation
designation block Latex
± U + 00B1 plus-minus sign Plus minus sign Latin-1, supplement \pm
U + 2213 minus-or-plus sign Minus plus sign Mathematical operators \mp
U + 2214 dot plus Plus sign with dot Mathematical operators \dotplus
U + 2295 circled plus sign Circled plus sign Mathematical operators \oplus, \varoplus
U + 229E squared plus sign Framed plus sign Mathematical operators \boxplus
U + 27F4 right arrow with circled plus Right arrow with plus sign circled Additional arrows-A
U + 293D top arc anticlockwise arrow with plus Left high arch arrow over plus sign Additional arrows-B
U + 2945 rightwards arrow with plus below Right arrow over plus sign Additional arrows-B
U + 2946 leftwards arrow with plus below Left arrow over plus sign Additional arrows-B
U + 2A01 n-ary circled plus operator N-digit circled plus sign Additional math operators \bigoplus
U + 2A04 n-ary union operator with plus N-digit union sign with plus sign Additional math operators \bigcupplus
U + 2A22 plus sign with small circle above Plus sign with ring above Additional math operators \circplus
U + 2A23 plus sign with circumflex accent above Plus sign with circumflex above Additional math operators
U + 2A24 plus sign with tilde above Plus sign with tilde above Additional math operators
U + 2A25 plus sign with dot below Plus sign with a dot below Additional math operators
U + 2A26 plus sign with tilde below Plus sign with tilde below Additional math operators
U + 2A27 plus sign with subscript two Plus sign with subscript two Additional math operators
U + 2A28 plus sign with black triangle Plus sign with black triangle Additional math operators
U + 2A71 equals sign above plus sign Equal sign over plus sign Additional math operators
U + 2A72 plus sign above equals sign Plus sign over equal sign Additional math operators
⪿ U + 2ABF subset with plus sign below Subset over plus sign Additional math operators
U + 2AC0 superset with plus sign below Superset over plus signs Additional math operators
U + 2B32 left arrow with circled plus Left arrow with plus sign circled Different symbols and arrows

A plus sign-like cross as a component also contains alchemical-astronomical symbols such as  " Earth (planet), church ( map sign ), antimony " (U + 2641 earth ),  "female, Venus , copper " (U + 2640 female sign ),  Mercury , Mercury (U + 263F mercury ) - see astronomical symbols , planetary metals - and many others.

## Individual evidence

1. ^ PT Meissner, Chemische Aequivalenten- oder Atomlehre , bookshop of the widow of the blessed knight JG von Mösle, Vienna 1834, part 1, p. 227.
2. ^ The Holocaust in Three Generations: Families of Victims and Perpetrators of the Nazi Regime, Gabriele Rosenthal, A&C Black, August 13, 1998, p. 107
3. ^ A b Christian-Jewish Dialogue: Theological Foundations By Peter von der Osten-Sacken (1986 - Fortress Press - ISBN 0-8006-0771-6 ) p. 96 ( preview in Google Book Search)
4. Stefan Simons, Dieter Wild: "A Pain for the Church". The Orthodox Israeli Rabbi Adin Steinsaltz on Jews and Christians . In: Der Spiegel . No. 15 , 1994, pp. 214-216 ( online - 11 April 1994 ).
5. a b c Scott Pakin: The Comprehensive LaTeX Symbol List. (PDF, 8.7 MB) January 19, 2017, archived from the original on September 28, 2017 ; Retrieved on September 28, 2017 (English, linking the original results in a mirror of CTAN , the archive link compare file: Comprehensive LaTeX Symbol list.pdf ).


text:
## Finite-temperature photoemission in the extended Falicov-Kimball model: a case study for Ta$_2$NiSe$_5$

Satoshi Ejima, Florian Lange, Holger Fehske

SciPost Phys. 10, 077 (2021) · published 26 March 2021

### Abstract

Utilizing the unbiased time-dependent density-matrix renormalization group technique, we examine the photoemission spectra in the extended Falicov-Kimball model at zero and finite temperatures, particularly with regard to the excitonic insulator state most likely observed in the quasi-one-dimensional material Ta$_2$NiSe$_5$. Working with infinite boundary conditions, we are able to simulate all dynamical correlation functions directly in the thermodynamic limit. For model parameters best suited for Ta$_2$NiSe$_5$ the photoemission spectra show a weak but clearly visible two-peak structure, around the Fermi momenta $k\simeq\pm k_{\rm F}$, which suggests that Ta$_2$NiSe$_5$ develops an excitonic insulator of BCS-like type. At higher temperatures, the leakage of the conduction-electron band beyond the Fermi energy becomes distinct, which provides a possible explanation for the bare non-interacting band structure seen in time- and angle-resolved photoemission spectroscopy experiments.

### Authors / Affiliation: mappings to Contributors and Organizations

See all Organizations.
Funder for the research work leading to this publication


text:
# How to typeset the whole article efficiently? [duplicate]

Possible Duplicate:
Undefined citation warnings
Question mark instead of citation number

I am a new user of LaTeX. I am writing a paper using TeXworks, using Jabref for the bibliography. I had to typeset so many times to get my article to have correct citation format, otherwise it's showing me a question (?) mark! Sometimes, I had to close all the files and re-open them to get correct output. Sometimes it's not working.

Can anyone please give me any idea to resolve the issue? It's really time consuming having to typeset again and again but not getting the output. In fact, I even don't know in which conditions this happens.

-
Duplicate of Undefined citation warnings –  Alan Munn Jul 18 '12 at 18:12
If you're looking for a way to automate the compilation, take a look at this question: Configure latexmk in TeXworks. –  Alan Munn Jul 18 '12 at 18:15
thank you alan munn. but my prob;em is not yet resolved following the way you described here. any other advice? –  Nusrat Jul 18 '12 at 18:43
@Nusrat If none of the advice here is working, then you need to give us much more specific information about what you are doing. Does this happen on all documents or just one particular one? Given the first question I linked to, you will minimally need to run `latex bibtex latex latex` to get correct references on any document. If you're doing that and still having problems, then there's more you're not telling us. If you want to add more information, you can do so by editing your question. –  Alan Munn Jul 18 '12 at 18:58


text:
# How important to you is parking when job hunting?

Serious question.

Hi Everyone,

Hopefully there are going to be some interesting points raised here that can help feed into a discussion we are having in regards to recruiting currently.

We have a few satellite offices which have minimal parking availability. One of these offices was a bit of a rush to get it up and running and we are now in the process of looking to relocate the office.
Our H/O has a large free car park, our main IT office has a deal with a local organization that allows us to use their car park cheaply.
The IT Director for our IT Satelitte office (that we are looking to redirect) has made it clear that parking is not to be a main decision maker in the new office location. The team leader in this office has raised a fair concern that parking is fairly important for our job.

TL;DR: When you look for a new job, how important is the parking availability to you and what cost would you accept to pay per day to park if there wasn't onsite parking?

Many Thanks

How important to you is parking when job hunting? Closed | 141 votes by 123 users
Important but negotiable 36%
Very Important (Deal Breaker) 18%
Has never been a factor in a job decision 13%
Unconcerned 13%
Bonus if there is but not important 9%
Pay = Offered $- Transportation Cost (driving distance/time, parking, etc. all included) 8% 11 Votes Don't have a car 4% 5 Votes Popular Topics in IT Jobs / Careers Spiceworks Help Desk The help desk software for IT. Free. Track users' IT needs, easily, and with only the features you need. ## 41 Replies · · · Datil OP At my last position, we were downtown. They did not provide parking and at first I was paying$150/month for parking.  I was able to find a park and ride a little farther out that dropped my parking to $85/month, but it took longer for me to get to my car if I had to go onsite for a client. I just dealt with it and when I had to go onsite - I told them that it would just take longer for me to get there. 1 · · · Mace OP It's right up there with free massages on lunch and free, catered food 5 days a week. I work in the DFW suburbs, and every job i've ever had has had free parking right in front of the building, or provided free access to the parking garage. 2 · · · Mace OP Parking needs to be relatively close (for me that's up to half a mile but it will differ for each person). For areas with good mass transit that will suffice as long as everyone acknowledges and takes the vagaries of bus and train schedules into account. 1 · · · Thai Pepper OP I work in central London, I just pretend parking spaces don't exist. 4 · · · Jalapeno OP This is one of those where the office is. I'm currently working out in the countryside, with nothing approaching public transport to help me get here. Parking's a bit important to me right now. The job I was going for recently was up in London - a short walk from Waterloo station, and I can get to the station in my home town easily - parking isn't an issue at all. 1 · · · Thai Pepper OP I will not work at a place where i have to pay for parking everyday. With the cold winters and hot and humid summers, i would rather have free parking near my workplace. I worked downtown for 8 months and it was terrible. I could not find parking sometimes and i had to walk a few blocks to work in the cold. Whether the employer reimburses for parking or not, its not worth my effort. 3 · · · Datil OP Ross42. wrote: It's right up there with free massages on lunch and free, catered food 5 days a week. I work in the DFW suburbs, and every job i've ever had has had free parking right in front of the building, or provided free access to the parking garage. I'm in the same area, and I've come to recognize the extreme benefits of covered parking. Our last office had 5 covered parking spots and it was always a race to get there first during the summers. Current office has a garage, but it's too small, and I sometimes end up on the roof with no cover. Just brutal in August.... 1 · · · Ghost Chili OP This totally depends on your location. Unless you are in a really big city, parking is most likely of little concern. London, New York, Chicago, LA, yeah parking can be an important factor in choosing an office location. Where I am (town of 3000) my decision is to park 20 yards from the door, or 50 yards... Even when I worked in a larger city in the area (250,000 people) parking was still not much of an issue. 2 · · · Datil OP It all depends for me. If the pay is right, I'll take a job where I have to pay for parking. But, that paying for parking just entered my analysis of the appropriate pay. Meaning, hypothetically, if I get offered the exact same job in a suburb where I can park for free and downtown where I have to pay to park, I expect to get more pay to compensate at the downtown job. 2 · · · Jalapeno OP Outside of the high costs that some people need to pay for parking, safety is also a concern. Someone I know used to work for a large company (way up on the Fortune 500) and she had to pay to park about 1/4 mile away in one of the company lots and walk through a very bad neighborhood. Regardless if people were alone\in groups, day\night, there were frequent muggings and the company did absolutely nothing about it. You'd think they would at least offer a shuttle, but no. In situations like this, parking is a concern. 1 · · · Mace OP Personally it's massive if you live and work somewhere where public transport isn't an option. I don't know how much car parking costs in the US and if parking on the street is common. 1 · · · Mace OP Who cares! As long as I have a job the inconvenience of walking some minimal distance isn't even worth worrying about. I know people that have come unglued because they don't have an assigned parking spot....get a freaking life you self absorbed a-hole!! 1 · · · Datil OP Parking is pretty important to me. I'm one of those people who actually refuses to get out of my car until it has stopped completely. I know, if I were really a faithful employee I'd just jump out while on the interstate and let physics and chance park the car, but I just can't bring myself to do that. I park my car myself. Oh, that wasn't quite the point? Let me rethink ... If a company has guaranteed free parking within ½ mile, or guaranteed non-free parking within ½ mile which they pay for, then I will consider working there the same as anywhere else. If the parking is non-free and I have to pay for it, that will impact my expected salary and benefits. If parking is non-guaranteed, that will impact my expected working hours. If the parking is more than ½ mile away, there will need to be transportation between locations, and that will impact my expected working hours too. If the transportation is not either free or paid for by the company, that will impact my expected salary and benefits again. If the parking is more than ½ mile away (free or paid for or not doesn't matter), and there's no transportation between locations, it's pretty much a deal breaker unless I'm getting a substantial pay rise from whatever I'm currently making. How substantial? Start at 50% of my current salary and we'll negotiate up. I am not walking over ½ mile each way between parking and desk for an insignificant raise; my time and comfort is worth more, at least to me, than that. That being said, if a company does want me badly enough to give me a 50% or greater raise in exchange for walking a couple miles a day, I'll do the legwork. 3 · · · Ghost Chili OP Since I live in Kansas, everyone has parking. Even downtown. Everyone drives here so if you did not have parking, you would not have employees. Fortunately land is cheap in KS, so everyone has big lots with plenty of places to park. For some numbers on why, Great Britain has 22 times the population, and is about the same size as my state. So we have plenty of space to park. Edited Dec 30, 2016 at 14:41 UTC 0 · · · Thai Pepper OP I work in Downtown St. Louis, MO. Right next to Busch Stadium where the Cardinals play. My office reimburses me 75.00 a month for parking. They setup a deal with the garage that has a walkway that connects to our building and we pay 94 a month. So I pay out of my own pocket only 19.00 dollars a month to park. Plus if I go to a Cards game I can park free, which is usually 20 to 30 dollars otherwise. I usually go to about 4 games a year. That saves me enough money that it actually comes out to 12.67 a month. Talking about less than a dollar a day to park. I can deal with that. Edit: there is a nice feeling when everyone around you is paying$20 to 30 dollars for a game and you show them your badge and they wave you through, kinda makes you feel important. 3 · · · Datil OP Very important. Living in Toronto, in one of the much maligned 'burbs where transit is almost nonexistent and an office which isn't in the city core, means parking is a requirement. Paid parking pass, or a free lot - whatever way it is provided. Again though, depends on the location - if it was on one of the train lines that would be fine as the train stations have free parking. Toronto Transit has been getting rid of the vehicle lots and selling them to put up more condos - so TTC not only sucks, but now they are getting rid of the last incentive for drivers to park and commute on the TTC. 0 · · · Chipotle OP I live and work in a small town, where we have lots at our building. I get to park in the top lot, rather than the back lot due to where my office is, which is super nice. However it is amazing that when we have meetings being held here, we have to park in the back lot so guests can park close to the building, and I have become so spoiled I get upset about walking about 100 more feet. Or the fact that this morning, since most people aren't working this week, there are 4 cars (including mine) in our lot (with assigned parking). Someone is in my spot and I was actually MAD about this. (Seriously? There are far more things to be angry about!) Having never lived in a big city, I don't know what it is like, so I'd probably be the one that is all "I NEED A CLOSE PARKING SPOT, or I'm taking my talents elsewhere!" 2 · · · Thai Pepper OP It's not something I really need to worry about. I live in the Dayton, OH area, so there are very few places that don't have their own parking lots. My current employer's corporate office is on the second floor of our distribution center, so we have a massive parking lot since the DC is kind of in the middle of nowhere. 0 · · · Pure Capsaicin OP I agree that parking shouldn't be the primary concern when shopping for office space. It is a factor, however, in the total usability of the location. An a prospective employee, my expectation is that you provide a place for me to park if you want me to work there. Either a space or compensation toward one. 1 · · · Thai Pepper OP Rockn wrote: Who cares! As long as I have a job the inconvenience of walking some minimal distance isn't even worth worrying about. I know people that have come unglued because they don't have an assigned parking spot....get a freaking life you self absorbed a-hole!! Wow, that escalated quickly. 3 · · · Datil OP PaplooThePirate wrote: I live and work in a small town, where we have lots at our building. I get to park in the top lot, rather than the back lot due to where my office is, which is super nice. However it is amazing that when we have meetings being held here, we have to park in the back lot so guests can park close to the building, and I have become so spoiled I get upset about walking about 100 more feet. Or the fact that this morning, since most people aren't working this week, there are 4 cars (including mine) in our lot (with assigned parking). Someone is in my spot and I was actually MAD about this. (Seriously? There are far more things to be angry about!) Having never lived in a big city, I don't know what it is like, so I'd probably be the one that is all "I NEED A CLOSE PARKING SPOT, or I'm taking my talents elsewhere!" At least you have enough perspective to notice this ... even comment on it! 0 · · · Ghost Chili OP It depends. Some places, parking is a real issue. (Even worse if you have to pay for it, or for the Congestion Charge). In many of those areas, public transport is usually accessible, but not always. It's unlikely that I would turn down a job for not having parking, but if it caused too many issues, I might well on a bit sooner. 0 · · · Habanero OP We have to contend with the cab stand next door that masquerades as a restaurant for parking. I do get a little tweaked when someone takes "my" spot, but I do drive a company truck and I do quite often have to load stuff into it. I'm not lugging servers and UPSes across the lot because you can't be bothered to walk an additional 15 feet for convenience. Also, with several thousands of worth of equipment in my van, I would prefer not sitting it in the open air in this less than upscale neighborhood.

0
· · ·
Datil
OP

I live in the business suburbs of a driving city. We don't have parking problems. Even if I worked downtown, I would take the light rail in and wouldn't have to deal with that nonsense.

0

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# Confused about wheel's tractive force...

This topic is 1112 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

## Recommended Posts

Hello,

I been developing a simple car physics system (for learning purposes) in Unity flowing the famous Marco Monster tutorial about Car Physics

http://www.asawicki.info/Mirror/Car%20Physics%20for%20Games/Car%20Physics%20for%20Games.html however I am getting really confused about how compute the wheel's traction force, I simple cant get (understand) enough from the tutorial to compute it, so I really appreciate if you could give me some advice or point to some tutorials about it.

Here is part of my code to compute up to drive torque:

// My gear ratios (unity AnimationCurve)

var gearRatio = _gearRatiosCurve.Evaluate(_currentGear);

// wheel rpm (for now I am just using the easy way) to get my project running...

var wheelRpm = (speed / rearLeftWheel.radius) * 60.0f / (2.0f * Mathf.PI);

var engineRpm = MinEngineRpm + wheelRpm * gearRatio * _differentialRatio;

var engineTorque = _torqueCurve.Evaluate(engineRpm) * _throttle;

var driveTorque = engineTorque * gearRatio * _differentialRatio * _transmissionEfficiency;

// At this point, I need compute the traction torques from both wheels so I can get the total torque on wheel...

Worth not that I am not trying to do a full simulation, so simpler (arcade like) formulas should work for my project.

Kromen

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The maximum tangential force that a tire can produce can be simulated as mu * normal-force where the tire touches the ground. mu is the coefficient of friction between the tire and the surface. normal-force is the force normal to the tire's surface where it meets the "ground."

Depending on wet-to-dry conditions, and the type of tire, mu is commonly modeled as 0.3 (wet, slippery) to 0.75 (dry). I don't know how you're modeling the suspension, but the downward force under static conditions (normal-force straight "down") on one of four tires is approx. chassis-mass * gravity / 4, assuming equal weight distribution front-to-back and side-to-side.

Given drive torque, assuming that's the torque to the drive axle, the maximum available force for a tire at the surface, assuming 2 drive wheels and torque applied (divided) equally to each wheel, is drive-torque / ( 2 * radius ).

The tangential force on a drive tire is, then, the lesser of mu*normal-force and drive-torque / (2*radius).

Edited by Buckeye

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Hello Buckeye, thanks for your answer, it really helped me as now I can simulate different terrains and in overall its working, however I cant get the correct wheel angular velocity, as I commented above, I am using the formulas from the Marco Monster tutorial.

Here is my current implementation (for one of the two powered wheels) without suspension force:

// The wheel's drive force (driveTorque is the torque delivered via gearbox)

var driveForce = driveTorque / _radius;

// Normal force in this wheel

var normalForce = _rigidbody.mass * 9.81f * 0.25f * _mu;

// Traction force, this is the force I apply to my ridigbody

var tractionForce = Mathf.Min(driveForce, normalForce);

// Now I try to compute the wheel angular velocity (not working... :( )

var tractionTorque = tractionForce * _radius;

// The drive torque minus the friction torques that counteract it (braking torque if you're braking and traction torque from the contact with the road surface)

totalTorque = driveTorque - tractionTorque - brakeTorque;

// totalTorque / cylinder inertia.

var angularAcceleration = totalTorque / (_mass * _radius * _radius * 0.5f);

_angularVelocity += angularAcceleration * Time.deltaTime;

The above in theory should work (i guess ), but unfortunately the wheel angular velocity seems not right at all, so I appreciate any help about this.

Regards,

Kromen

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About what, exactly? As you're the only one that knows what "seems not right at all" means, the only suggestion I can make is to change your code to get what you want, or change your expectations.

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About what, exactly? As you're the only one that knows what "seems not right at all" means, the only suggestion I can make is to change your code to get what you want, or change your expectations.

Ha you right, that was a late response :), what I mean about this, is if I plug my computed angular velocity back to my wheelRpm (used to get the engine rpm), the car moves much more slow and wheels slip, much like if the car were on wet or ice surface even with high friction coefficient (mu = 1).

So I ended with a simpler way to compute the wheel's angular velocity, using its relative forward (longitudinal) velocity over its radius.

Anyway I really appreciate you kind help.

Best Regards,

Kromen

##### Share on other sites

Yeah, handling feedback RPM is tricky. You have to decide if the wheel is slipping to determine what mu to use (sliding friction is less than rolling friction), so it gets circular. You have to decide how complicated you want your calcs to be. Simplification as you've done works pretty well. That's why it's called simulation!


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Stage: 2 Short Challenge Level:

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

Product Sudoku

Stage: 3, 4 and 5 Challenge Level:

The clues for this Sudoku are the product of the numbers in adjacent squares.

Cayley

Stage: 3 Challenge Level:

The letters in the following addition sum represent the digits 1 ... 9. If A=3 and D=2, what number is represented by "CAYLEY"?

Ratio Sudoku 1

Stage: 3 and 4 Challenge Level:

A Sudoku with clues as ratios.

Product Sudoku 2

Stage: 3 and 4 Challenge Level:

Given the products of diagonally opposite cells - can you complete this Sudoku?

Ratio Sudoku 3

Stage: 3 and 4 Challenge Level:

A Sudoku with clues as ratios or fractions.

Seasonal Twin Sudokus

Stage: 3 and 4 Challenge Level:

This pair of linked Sudokus matches letters with numbers and hides a seasonal greeting. Can you find it?

Corresponding Sudokus

Stage: 3, 4 and 5

This second Sudoku article discusses "Corresponding Sudokus" which are pairs of Sudokus with terms that can be matched using a substitution rule.

My New Patio

Stage: 2 Challenge Level:

What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?

Sums and Differences 1

Stage: 2 Challenge Level:

This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

Consecutive Numbers

Stage: 2 Challenge Level:

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

Stage: 3 and 4 Challenge Level:

Four numbers on an intersection that need to be placed in the surrounding cells. That is all you need to know to solve this sudoku.

Problem Solving, Using and Applying and Functional Mathematics

Stage: 1, 2, 3, 4 and 5 Challenge Level:

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

Fault-free Rectangles

Stage: 2 Challenge Level:

Find out what a "fault-free" rectangle is and try to make some of your own.

Sums and Differences 2

Stage: 2 Challenge Level:

Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?


text: Generated by the Physics Derivation Graph. Eq. \ref{eq:8882098} is an initial equation. $$ax^2 + bx + c = 0 \label{eq:8882098}$$ Divide both sides of Eq. \ref{eq:8882098} by $$a$$; yields Eq. \ref{eq:7777621}. $$x^2 + (b/a)x+(c/a) = 0 \label{eq:7777621}$$ Subtract $$c/a$$ from both sides of Eq. \ref{eq:7777621}; yields Eq. \ref{eq:1212129}. $$x^2 + (b/a)x = -c/a \label{eq:1212129}$$ Add $$(b/(2 a))^2$$ to both sides of Eq. \ref{eq:1212129}; yields Eq. \ref{eq:1120000}. $$x^2 + (b/a)x + (b/(2 a))^2 = -c/a + (b/(2 a))^2 \label{eq:1120000}$$ Change variable $$h$$ to $$b/(2 a)$$ in Eq. \ref{eq:9091270}; yields Eq. \ref{eq:1239010}. $$x^2 + 2 x (b/(2 a)) + (b/(2 a))^2 = (x + (b/(2 a)))^2 \label{eq:1239010}$$ Simplify Eq. \ref{eq:1239010}; yields Eq. \ref{eq:1734000}. $$x^2 + x(b/a) + (b/(2 a))^2 = (x+(b/(2 a)))^2 \label{eq:1734000}$$ Eq. \ref{eq:9091270} is an initial equation. $$x^2 + 2 x h + h^2 = (x + h)^2 \label{eq:9091270}$$ LHS of Eq. \ref{eq:1120000} is equal to LHS of Eq. \ref{eq:1734000}; yields Eq. \ref{eq:2985412}. $$(x+(b/(2 a)))^2 = -(c/a) + (b/(2 a))^2 \label{eq:2985412}$$ Take the square root of both sides of Eq. \ref{eq:2985412}; yields Eq. \ref{eq:6608123} and Eq. \ref{eq:6608102}. $$x+(b/(2 a)) = -\sqrt{(b/(2 a))^2 - (c/a)} \label{eq:6608123}$$ $$x+(b/(2 a)) = \sqrt{(b/(2 a))^2 - (c/a)} \label{eq:6608102}$$ Subtract $$b/(2 a)$$ from both sides of Eq. \ref{eq:6608123}; yields Eq. \ref{eq:4433112}. $$x = \sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a)) \label{eq:4433112}$$ Subtract $$b/(2 a)$$ from both sides of Eq. \ref{eq:6608102}; yields Eq. \ref{eq:2657355}. $$x = -\sqrt{(b/(2 a))^2 - (c/a)}-(b/(2 a)) \label{eq:2657355}$$ Simplify Eq. \ref{eq:2657355}; yields Eq. \ref{eq:8811221}. $$x = \frac{-b-\sqrt{b^2-4ac}}{2 a} \label{eq:8811221}$$ Simplify Eq. \ref{eq:4433112}; yields Eq. \ref{eq:8761200}. $$x = \frac{-b+\sqrt{b^2-4ac}}{2 a} \label{eq:8761200}$$ Eq. \ref{eq:8811221} is one of the final equations. Eq. \ref{eq:8761200} is one of the final equations.


text:
# zbMATH — the first resource for mathematics

## Lu, Jianfeng

Compute Distance To:
 Author ID: lu.jianfeng Published as: Lu, J.; Lu, J. F.; Lu, J.-F.; Lu, Jian-feng; Lu, Jianfeng; Lü, J.-F.; Lü, Jian-Feng; Lü, Jianfeng External Links: ORCID
 Documents Indexed: 111 Publications since 1986, including 1 Book
all top 5

#### Co-Authors

 1 single-authored 20 E, Weinan 11 Ying, Lexing 9 Lin, Lin 7 Yang, Xu 5 Li, Qin 4 Liu, Jianguo 4 Sun, Weiran 4 Yang, Haizhao 4 Yang, Jingyu 3 Cao, Yu 3 Ding, Jing 3 Lai, Rongjie 3 Ming, Pingbing 2 Cai, Zhenning 2 Daubechies, Ingrid C. 2 Delgadillo, Ricardo 2 Gao, Yuan 2 García-Cervera, Carlos J. 2 Han, Jianmin 2 Li, Xingjie Helen 2 Liu, Jingchen 2 Lu, Yulong 2 Marzuola, Jeremy L. 2 Nolen, James 2 Peng, Hao 2 Thicke, Kyle 2 Vanden-Eijnden, Eric 2 Weinstein, Michael I. 2 Yang, Chao 2 Zhou, Xiang 1 Caelli, Terry M. 1 Cai, Ying 1 Car, Roberto 1 Chen, Huajie 1 Chen, Jingrun 1 Chen, Xiaoyun 1 Cook, Alexis B. 1 Cornelis, Bruno 1 Dai, Jinwen 1 Dai, Shibin 1 Du, Haishun 1 Du, Qiang 1 Fan, Hong-Yi 1 Fang, Di 1 Gauckler, Ludwig J. 1 Goodfriend, Alex 1 Gu, Xingjian 1 Huang, Hui 1 Huang, Yufang 1 Jackson, Trachette L. 1 Jin, Lei 1 Kohn, Robert Vita 1 Li, Bo 1 Li, Cuiling 1 Li, Lei 1 Li, Lian 1 Li, Tiejun 1 Li, Xiantao 1 Li, Xiaoou 1 Li, Yingzhou 1 Lin, Ling 1 Lu, Yanting 1 Lu, Zhiqi 1 Ma, Zhixin 1 Mendl, Christian B. 1 Meza, Juan C. 1 Moroz, Vitaly 1 Muratov, Cyrill B. 1 Ocon, Noelle 1 Ortner, Christoph 1 Osher, Stanley Joel 1 Otto, Felix 1 Pan, Zhigeng 1 Ren, Weiqing 1 Rousset, Frédéric 1 Schratz, Katharina 1 Schweizer, Ben 1 Shen, Chunhua 1 Spiliopoulos, Konstantinos V. 1 Steinerberger, Stefan 1 Tian, Xiaochuan 1 Wang, Liantao 1 Wang, Sheng 1 Watson, Alexander B. 1 Wu, Hau-Tieng 1 Xu, Dewu 1 Xu, Manwu 1 Yan, Wei 1 Yan, Xuan 1 Yang, Jian 1 Yang, Jianping 1 Yao, Yuan 1 Zeng, Fancong 1 Zhang, David D. 1 Zhang, Hao 1 Zhang, Hao 1 Zhang, Minming 1 Zhao, Rongyong 1 Zhou, Zhennan 1 Ziazadeh, Daniel R.
all top 5

#### Serials

 12 Journal of Computational Physics 7 Multiscale Modeling & Simulation 7 Communications in Mathematical Sciences 5 Communications on Pure and Applied Mathematics 5 SIAM Journal on Scientific Computing 4 Journal of Mathematical Physics 4 Mathematics of Computation 3 Archive for Rational Mechanics and Analysis 3 International Journal of Heat and Mass Transfer 3 Journal of Statistical Physics 3 SIAM Journal on Numerical Analysis 3 Pattern Recognition 2 Chinese Annals of Mathematics. Series B 2 SIAM Journal on Mathematical Analysis 2 Journal of Nonlinear Science 2 Communications in Nonlinear Science and Numerical Simulation 2 Annals of Mathematical Sciences and Applications 1 Communications in Mathematical Physics 1 Information Processing Letters 1 Physics Letters. A 1 ACM Transactions on Mathematical Software 1 Memoirs of the American Mathematical Society 1 Acta Mathematicae Applicatae Sinica. English Series 1 Probability Theory and Related Fields 1 Journal of Computer Science and Technology 1 Asymptotic Analysis 1 Journal of Scientific Computing 1 The Annals of Applied Probability 1 M$^3$AS. Mathematical Models & Methods in Applied Sciences 1 Proceedings of the National Academy of Sciences of the United States of America 1 Applied and Computational Harmonic Analysis 1 Journal of Huazhong University of Science and Technology 1 Methods and Applications of Analysis 1 Discrete and Continuous Dynamical Systems 1 Journal of Nanjing Normal University. Natural Science Edition 1 Acta Mathematica Sinica. English Series 1 IEEE Transactions on Image Processing 1 Control and Decision 1 Mathematical Biosciences and Engineering 1 European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis 1 Journal of Guangxi Normal University. Natural Science Edition 1 Communications in Computational Physics 1 Communications in Theoretical Physics 1 Science China. Mathematics 1 Proceedings of the Royal Society of London. A. Mathematical, Physical and Engineering Sciences 1 Transactions of Mathematics and its Applications
all top 5

#### Fields

 42 Numerical analysis (65-XX) 38 Partial differential equations (35-XX) 26 Statistical mechanics, structure of matter (82-XX) 22 Quantum Theory (81-XX) 13 Mechanics of deformable solids (74-XX) 11 Probability theory and stochastic processes (60-XX) 11 Computer science (68-XX) 6 Dynamical systems and ergodic theory (37-XX) 6 Fluid mechanics (76-XX) 5 Calculus of variations and optimal control; optimization (49-XX) 4 Ordinary differential equations (34-XX) 4 Biology and other natural sciences (92-XX) 3 Functional analysis (46-XX) 3 Operator theory (47-XX) 3 Statistics (62-XX) 3 Optics, electromagnetic theory (78-XX) 3 Systems theory; control (93-XX) 2 Linear and multilinear algebra; matrix theory (15-XX) 2 Approximation and expansions (41-XX) 2 Integral equations (45-XX) 2 Classical thermodynamics, heat transfer (80-XX) 2 Operations research, mathematical programming (90-XX) 2 Information and communication, circuits (94-XX) 1 General mathematics (00-XX) 1 Real functions (26-XX) 1 Harmonic analysis on Euclidean spaces (42-XX) 1 Algebraic topology (55-XX) 1 Mechanics of particles and systems (70-XX)


text:
# StrucDiv

## StrucDiv: Spatial structural diversity quantification in raster data

Spatial structural diversity refers to the spatial, i.e. horizontal, configuration of landscape elements. The StrucDiv package provides methods to quantify spatial structural diversity, hereafter structural diversity, in raster data, i.e. data in gridded field format. This means that landscape elements are pixels. The methods are based on the spatial arrangement of pixels as pairs, based on Haralick, Shanmugam, and Dinstein (1973). Pixels are considered as pairs at user-specified distances and angles. Distance refers to the distance between the two pixels that are considered a pair. Angle refers to the angle at which two pixels are considered a pair. Angles can be horizontal or vertical direction, or the diagonals at 45° or 135°. The direction-invariant version considers all angles.

The frequencies of pixel pair occurrences are normalized by the total number of pixel pairs, which returns the gray level co-occurrence matrix (GLCM). The GLCM contains the empirical probabilities $$P_{i,j}$$ that pixel values v$$_{i}$$ and v$$_{j}$$ occur as a pair at the specified distance and angle(s), in the area within which pixel pairs are counted. The total number of pixel pairs depends on the extent of the area within which pixel pairs are counted, i.e. on the spatial scale. The spatial scale is defined by the window side length (WSL) of a moving window.

Pixel values v$$_{i}$$ and v$$_{j}$$ that occur as a pair enter the GLCM as row and column names, hence $$i$$ and $$j$$ index the position of a pixel value in the GLCM. The values at positions $$i$$ and $$j$$ can differ between GLCMs. Instead of pixel values v$$_{i}$$ and v$$_{j}$$, ranks (r, s) can be considered. Ranks refer to the position of the pixel value in the GLCM, hence the value associated with a rank can differ between GLCMs. The total (GLCM-specific) number of values is denoted with $$m$$, and the total (GLCM-specific) number of ranks is denoted with $$l$$.

Diversity metrics are calculated for every element of the GLCM, and their sum is assigned to the center pixel of the moving window and represents structural diversity of the window. The final map is called a ‘structural diversity map.’

Diversity metrics include second-order texture metrics: contrast, dissimilarity, homogeneity and entropy (i.e. Shannon entropy). Additionally, structural diversity entropy includes a difference weight $$\delta \in \{1,2\}$$, which weighs the difference between pixel values v$$_{i}$$ and v$$_{j}$$, either by absolute, or by squared differences. When $$\delta$$ = 0, structural diversity entropy corresponds to Shannon entropy. Dissimilarity naturally employs $$\delta = 1$$, contrast employs $$\delta = 2$$, and homogeneity employs $$\delta = 2$$ in the denominator. Furthermore, normalized entropy is available. Normalized entropy is Shannon entropy normalized over maximum entropy. Maximum entropy depends on the extent of the area within which pixel pairs are counted, i.e. on the spatial scale. These methods can be applied to any continuous raster data, or continuous raster data that is binned to reduce the number of gray levels (GL). The methods can also be applied to categorical data if categories are numbered in a meaningful way, or if entropy or normalized entropy are used. This is because otherwise $$\delta$$ cannot be computed or is meaningless.

For entropy, normalized entropy, and homogeneity, the possible number of GL is restricted, i.e. high numbers of GL lead to structureless diversity maps (Schuh et al. 2020).

With these methods, structural diversity features can be detected. Structural diversity features are also called latent landscape features. Typical structural diversity features include narrow borders, line features, patches and hotspots Schuh et al. (2021).

### Structural diversity metrics using pixel values

\begin{align} \text{Structural diversity entropy} & = \ - \sum_{i,j=1}^{m} P_{ij} \ln P_{ij} \ |v_{i}-v_{j}| ^ \delta, \delta \in \{0,1,2\} \label{DivEnt} \\ \text{Contrast} & = \sum_{i,j=1}^{m} P_{ij} \ |v_{i}-v_{j}| ^ \delta, \delta = 2 \label{ValCon} \\ \text{Dissimilarity} & = \sum_{i,j=1}^{m} P_{ij} \ |v_{i}-v_{j}| ^\delta, \delta = 1 \label{ValDis} \\ \text{Homogeneity} & = \sum_{i,j=1}^{m} P_{ij} / (1 + |i-j|^\delta), \delta = 2 \label{ValHom} \end{align}

### Structural diversity metrics using ranks

\begin{align} \text{Structural diversity entropy} & = \ - \sum_{r,s=1}^{l} P_{rs} \ln P_{rs} \ |r-s| ^ \delta, \delta \in \{0,1,2\} \label{RankDivEnt}\\ \text{Contrast} & = \sum_{r,s=1}^{l} P_{rs} \ |r-s| ^ \delta, \delta = 2 \label{RankCon} \\ \text{Dissimilarity} & = \sum_{r,s=1}^{l} P_{rs} \ |r-s| ^ \delta, \delta = 1 \label{ConDis} \\ \text{Homogeneity} & = \sum_{r,s=1}^{l} P_{rs} / (1 + |r-s|^\delta), \delta = 2 \label{HomRank} \end{align}

## Installation

You can install the released version of StrucDiv from CRAN with:

install.packages("StrucDiv")

## Example

Calculate normalized entropy on Normalized Difference Vegetation Index (NDVI) data, which was binned to 15 GL (see data documentation). We define the size of the moving window with WSL three, and we consider distance one between pixels (i.e. direct neighbors), and a direction-invariant approach.

entNorm <- StrucDiv(ndvi.15gl, wsl = 5, dist = 1, angle = "all", fun = entropyNorm,
na.handling = na.pass)

## References

Haralick, R. M., K. Shanmugam, and I. Dinstein. 1973. “Textural Features for Image Classification.” IEEE Transactions on Systems, Man, and Cybernetics SMC-3 (6): 610–21. https://doi.org/10.1109/TSMC.1973.4309314.
Schuh, L., M. J. Santos, R. de Jong, M. Schaepman, and R. Furrer. 2021. Structural Diversity Entropy: A Unified Diversity Measure to Detect Latent Landscape Features. Under review.
Schuh, L., M. Schaepman, M. J. Santos, R. de Jong, and R. Furrer. 2020. “Advancing Texture Metrics to Model Landscape Heterogeneity.” In IGARSS 2020 - 2020 IEEE International Geoscience and Remote Sensing Symposium, 2735–38. https://doi.org/10.1109/IGARSS39084.2020.9323930.


text:
Definite intergration area under curve bounded with line

1. May 25, 2008

thomas49th

1. The problem statement, all variables and given/known data

A cruve has the equation $$y = x{3} - 8x^{2} + 20x$$. The curve has stationary points A and B. There is a line through B parallel to y axis and meets the x axis at the point N. The region R is bounded by the curve , the x-axis and the line from A to N. Find the exact area under the curve

2. Relevant equations

3. The attempt at a solution

Well I found the x co-ords of A and B, which is $$\frac{10}{3}$$ or 2. I intergrated the curve and got

$$\frac{4x^{3}}{4} - \frac{8x^{3}}{3}+10x^{2}$$

no +C as we'll be having limits i presume

But i dont know how to get the area of region R... as there is a stupid line in the way!!!

Can somebody show/help me to do it.

Thanks :)

2. May 25, 2008

eys_physics

Hey

My first advice is to a picture of your problem. After that you notice that the exercise is to calculate the integral from x=A to x=B of f, i.e. integration of a polynomial. I expect you know how to do that.

3. May 25, 2008

thomas49th

i can intergrate a polynominal easily and in the question paper there is a picture of the question. But because of this AN line, it's thrown me. How would you go about doing it.

Thanks

4. May 25, 2008

tiny-tim

Hi thomas!

If I've understood the question right, all you have to do is add a triangle (whose area is obvious), and you get the standard integral.

5. May 26, 2008

thomas49th

ahhh i see cheerz :)


text:
# A Magic System Based on Einstein's e=mc² Equation

I have been thinking about a magic system that is about "e=mc²". Main concept is it transforms certain kind of ink to energy it does this via runes and tattoos.(They could extract this ink from a special type of Cristal as an essance.) I know how e=mc² works but I don't have any idea about its details.

This 'magic' could used in different purposes:

• Lightning and fire for armors, swords, shields, etc.

• Heating for smithing tools, swords and armors etc.

• Energy shields for armors and castles

• Pushing objects and making winds

• Power ups like strength,speed and stamina etc.

• Tattoos

• Teleportation

My main problem is about energy. What would be the total energy of a fantasy lightning etc. How many times could you use it for a energy that equals a gram of matter. How much would teleportation cost.

Explanation for tattoos: They are very similar to others. they basically puts tattoos that represents certain actions like fire, lightning, stamina, strength etc.

• The basic conversion in MKS units is that for every kilogram of matter, you get (299792458^2) = 8.98755 * 10^16 joules of energy. So if you convert 1 milligram of matter to energy, that's one one-thousandth of a gram or one one-millionth of a kilogram, so you'd get one one-millionth of that energy, or 8.98755 * 10^10 joules (by comparison a bolt of lightning is typically about 10^9 joules, so you may want less, but you could convert less mass) – Hypnosifl Sep 26 '19 at 19:27
• Could you explicitly state what the question is, because at the moment we're faced with a great deal of possibilities to explore, but no determinate direction. 7 Bullet points of possible directions in-fact. Care to narrow it down? – Tantalus' touch. Sep 26 '19 at 19:30

Good idea, but that's an equation that applies at the atomic level.

Einstein's Mass-Energy Equivalence comes into play at reaaaally small intervals or super-high energy situations. This equation is derived from the theory of Special Relativity after all. For that reason, I wouldn't want to be using it on my skin. If you want one gram of mass in terms of energy:

$$E = mc^2 = 0.001*(299,792,458)^2 = 8.99*10^{13}\,Joules$$

As humans, we know that we can turn mass into energy through fission and fusion (read into mass defects). Atomic mass is lost during fusion and fission as it's converted into energy. Lots of this is just speeding the particles up, but some energy is released in the form of gamma radiation. In the form of a diagram:

There's a lot going on, but I hope you get the idea. It's an explosive, near-uncontrollable mess that would likely disintegrate you if you ever tried it.

However, in the interests of being helpful and not (totally) blowing holes in your ideas, there's been a fair few questions on here about using energy from 'the quantum realm' (oooo). If your ink can somehow stabilise particles in superposition (these blip in and out of our reality), you could harness some serious energy through particle-antiparticle annihilation. Look into it- it sounds cool, doesn't need to be explained to death, and with a little magic it might just work.

Hope it helps!

When constructing a magic system how would I measure the energy needed for certain spells to create something of an "equal energy exchange". Correct?

You wouldn't. For my current project I created an extensive system with the prime principle of equal energy exchange.

You cannot do it in any scientifically exact way and I explain why below.

I can recommend the following approach (That's how I have done it):

Rank you spells by how you FEEL would be an accurate energy in RELATION to the other spells. Example:

Heating a bottle of water is 1. Fireball is 1250. And so on.

Now, that you got that, think about what you want to happen.

All magic systems are narrative tools, since magic in essence is narration. How powerful do you want your mages to be? Think about all implications with regard to food, travel, warfare and so on.

Now you can scale your (energy) cost to the level of the spellcaster. Tune your numbers until it fits your desired scenarios.

NOTE: Why you cannot do it scientifically:

While the numbers for heating spells would be relativly straightforward, the cost of something like Power-ups is not quantifiably (You could calculate the energy cost of the increased metabolism, but not of the effect itself).

You can look some numbers up (for example heat spells) and use them as sound bones for the system. That's generally a good idea. But to try and quantify energy needs for a complete made-up-system (a story basically) might be fun, but will not be possible to the degree you want to do and has no added narrative benefit for going too far into science.


text:
What are Bitcoins?

One trending topic recently has been Bitcoin, and when I say recently I mean more than a year. I’ve had many friends coming at me to ask “Hey… mark, since you are into that stuff, do you know what Bitcoins are?” (The only thing that confuses me is what that stuff is. :S)

What are Bitcoins?

Well then, this is an easy answer: Bitcoins are a form of cryptocurrency, a virtual currency achieved through cryptography. Confusing, huh? Well it’s simpler than you might imagine. The real mechanism behind Bitcoins is pretty complicated and explaining it here is out of question; let’s put it this way: Bitcoins are “Bits in your computer”. How these bits are stored and secured is a complicated matter too, but using them is much simpler than how they work, proving that is anyone can use Bitcoins with near-to-zero knowledge of how they work. As a matter of fact, to start using Bitcoins you just need three things:

• A computer
• A Bitcoin client
• And a bit of patience. : )

Bitcoin is decentralized

What does this mean? It means there is no huge-server controlling all the traffic without which everything wouldn’t work. If you decide to run Bitcoin on your home computer you’ll become officially a node of the Bitcoin network. As a member of this network you will be able to send and receive Bitcoins, and until your machine stays up the Bitcoin network will continue to exist. This means you are one of many and that we’re all the same for the Bitcoin network. That’s what decentralized means: so long we run Bitcoin clients on our machines the network will exist, and so it would result difficult for any big company in the world to control all the nodes in the network.

Bitcoin has low fees

This is probably one of the biggest reasons why Bitcoin is seen as evil by governments/banks etc. We all know that if we use Banks, Paypal or Credit Cards to send/receive money we’ll have fees added to our orders. With Bitcoin this is not the case. Unless you perform a micro-transaction (usually < 0.01 BTC, but there are other cases) you will have no transaction fee added to your transaction! You may, however, pay an arbitrary fee to speed up the transaction so that it is confirmed in even less than a minute.

Bitcoin is anonymous

When you use Bitcoin you are just a node in the network. Transactions are performed through Bitcoin Addresses, and these are unique and belong to you! Once you use a address it becomes part of the network and everyone can see you used it, but this doesn’t mean they will be able to trace the address back to you. You can generate endless addresses and each time you generate one, no-one in earth can know you’re the same owner as the precedent, it’s like switching your identity. For this reason Bitcoin may have a bad impact on money laundering and other evil activities.

Bitcoin is secure

Well, secure is an abused word nowadays but when secure is alongside cryptography, it gains a whole new meaning. Bitcoin uses a “military-level” security policy to assure everything is safe. This security is associated with one tiny file called wallet.dat, this file contains the keys you use to perform transactions and that are the base to generate addresses. Losing your wallet.dat file means losing control over Bitcoin, much like giving a stranger your house’s master key. Fortunately you can use a password/passphrase to further secure your wallet.dat (and I suggest you to).

Bitcoin is highly fluctuating

This is kind of a double-edged blade. When I started using Bitcoins in 2012-2013, one BTC was worth 20$and they were not that used or known. Soon after in April 2013 one BTC reached 100$, the price was almost the same until the 25th November 2013 when BTC exchange rate suddenly rose up to 1:900, yeah 1 BTC = 900$, almost 900%! Nearly two weeks after (16th December) the price was 600$. On the 16th January 2014 it was 800$. Today the price is near 300$. Meaning investing in Bitcoin can be a great risk, but you know, no pain no gain, so you might get rich or lose everything you invested in a matter of days. But from your point of view it shouldn’t be that different from Forex. Keep in mind investing in Bitcoin is without doubt a great risk.

Bitcoins are auto-generated

This one is particularly confusing. Actually did you even question “If there is no one printing Bitcoins, how are they created?” or “How many Bitcoins are in circulation?”. The answers you seek are in the Bitcoin itself. To produce Bitcoins there is a specified action: mining. Mining is the term Bitcoin users use to describe “producing Bitcoins” or “confirming transactions”. The process itself is kind of simple: you use your computer’s ability to solve mathematical problems to solve very difficult problems. These problems are gradually more difficult as time passes, and also the size of Bitcoin network matters (the bigger the more difficult). In the early days of Bitcoin, a CPU was enough to mine Bitcoins (one could’ve earned 50 BTC within a few weeks). Then the Bitcoin network became bigger and CPUs weren’t producing enough power to be still earning Bitcoins, so everyone switched to GPUs. Nowadays even GPUs aren’t powerful enough, so that we came to use specific hardware for mining, known as Bitcoin ASIC. This method was thought to auto-regulate the network (freeing it from a central authority). At the time I’m writing this article there are a total of 13,620,350 BTC in circulation (you can check the exact number in a given time here). But there’s a upper limit to this Bitcoin fantasy: there can’t be more than 21,000,000 BTC in circulation. Even though there are many rumors about the cancellation of this rule.

That’s all for it today, next time I’ll show you how you can start using Bitcoins in a few steps!

Image courtesy of Jason Benjamin

Image courtesy of Jason Benjamin

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The IT guy with a slight look of boredom in his eyes. Freelancer. Current interests: Kubernetes, Tensorflow, shiny new things.

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Math Insight

Picture of divergence as expansion

We have shown that the divergence measures expansion or compression of a vector field. We immersed a sphere into a vector field that had positive divergence everywhere, as shown below. No matter where one moves the sphere (using the sliders), more fluid flows out of the sphere than into the sphere, indicating the fluid is expanding.

Movable sphere embedded in a divergent vector field. This sphere can be moved by dragging the blue sliders with the mouse to change its $x$, $y$, and $z$ coordinates. If the vector field represents the velocity of fluid flow, then no matter the sphere's position, more fluid is flowing out of the sphere than into the sphere. This fact suggests that the vector field has positive divergence everywhere.

The vector field pictured is \begin{align*} \dlvf(x,y,z) = (x,y,z). \label{vecfield1} \end{align*} Its divergence is \begin{align*} \div \dlvf(x,y,z) = \pdiff{}{x}x + \pdiff{}{y}y + \pdiff{}{z}z = 1+1+1=3, \end{align*} which is a positive constant independent of the point $(x,y,z)$. The picture of the vector field looks like fluid exploding outward, so it makes sense that the fluid is expanding.

Can a picture be misleading?

As one becomes more sophisticated in mathematical thinking, one discovers that pictures can sometimes be misleading. (One reason mathematicians demand mathematical proof is to ensure they aren't fooling themselves into jumping to conclusions based on incomplete information, such as the information gained solely by exploring pictures.) With regard to divergence, one might wonder if an outward flow, such as pictured above, always means that the divergence of the vector field is positive?

Here's a picture of a different vector field showing fluid flowing outward from the origin. However, it differs from the above vector field in that the arrows get shorter the further they are from the origin. Is the divergence of this vector field positive? In other words, is the fluid expanding as it may look like from the picture?

Outward flowing vector field with zero divergence. This sphere can be moved by dragging the blue sliders with the mouse to change its $x$, $y$, and $z$ coordinates. Although the vector field is radiating outward, the fluid whose velocity is represented by the vector field is not expanding. As long the the sphere is away from the origin (where the vector field is not defined), the same amount of fluid is flowing into as out of the sphere. The divergence of the vector field is zero.

To answer this question, we have to compute the divergence. This vector field is \begin{align*} \dlvf(x,y,z) = \frac{(x,y,z)}{(x^2+y^2+z^2)^{3/2}}, \label{vecfield2} \end{align*} for $(x,y,z) \ne (0,0,0)$. (It is not defined at the origin.) This new vector field is the same as the first vector field above except that we have divided it by its magnitude raised to the third power. (We could write this vector field as $\dlvf(\vc{x}) = \frac{\vc{x}}{\norm{\vc{x}}^3}$, where $\vc{x}=(x,y,z)$.) In this way, the vector field gets smaller as one moves away from the origin.

We calculate the divergence of $\dlvf$: \begin{align*} \div \dlvf(x,y,z) &= \pdiff{}{x}\frac{x}{(x^2+y^2+z^2)^{3/2}} + \pdiff{}{y}\frac{y}{(x^2+y^2+z^2)^{3/2}} + \pdiff{}{z}\frac{z}{(x^2+y^2+z^2)^{3/2}} \\ &= \frac{(x^2+y^2+z^2) - 3x^2} {(x^2+y^2+z^2)^{5/2}} + \frac{(x^2+y^2+z^2) - 3y^2} {(x^2+y^2+z^2)^{5/2}} + \frac{(x^2+y^2+z^2) - 3z^2} {(x^2+y^2+z^2)^{5/2}}\\ &= \frac{3(x^2+y^2+z^2) - 3(x^2+y^2+z^2)} {(x^2+y^2+z^2)^{5/2}} =0 \end{align*}

Hence, as long as we are not at the origin, the divergence is zero and the flow is neither expanding nor contracting.

How can we reconcile this fact with the picture? If the sphere is at the origin, clearly the flow is out of the sphere. But the divergence is not defined at the origin, so we have to ignore that point. If you move the sphere away from the origin, it is not clear if there is more fluid flowing into the sphere or more fluid flowing out. On one hand, the flow out of the sphere is slower than the flow into the sphere, as the arrows are getting shorter. On the other hand, because the flow is radiating outward, the fluid is flowing out of the sphere across more than half of its surface. For this particular vector field, I balanced those two effects (by carefully choosing how quickly the vector field shrinks as one moves away from the origin) so that the net flow into the sphere is exactly equal to the net flow out of the sphere. If we stay away from the origin, the fluid is neither expanding nor compressing and the divergence is zero.

Dependence on dimension

Here's one more subtlety just for fun. To make the divergence zero in the above example, I balanced the outward flow of the vector field by shrinking the vector field as one moves away from the origin. The flow out of the sphere was equal to the flow into the sphere and there was no expansion or compression.

What happens if I take the two-dimensional version of the previous vector field? The 2D vector field is \begin{align*} \dlvf(x,y) = \frac{(x,y)}{(x^2+y^2)^{3/2}}, \end{align*} for $(x,y) \ne (0,0)$. (It is not defined at the origin.) This vector field is shown below along with a circle that you can move by dragging its top red point with your mouse. Move the circle so that it is away from the origin. In this case, is the divergence positive, negative, or zero?

Outward flowing 2D vector field with negative divergence. This circle can be moved by dragging the red point. Although the vector field is radiating outward, it is actually compressing. As long the the circle is away from the origin (where the vector field is not defined), there is a net flux into the circle. The divergence of the vector field is negative.

We calculate the divergence of $\dlvf$: \begin{align*} \div \dlvf(x,y) &= \pdiff{}{x}\frac{x}{(x^2+y^2)^{3/2}} + \pdiff{}{y}\frac{y}{(x^2+y^2)^{3/2}} \\ &= \frac{(x^2+y^2) - 3x^2} {(x^2+y^2)^{5/2}} + \frac{(x^2+y^2) - 3y^2} {(x^2+y^2)^{5/2}} \\ &= \frac{2(x^2+y^2) - 3(x^2+y^2)} {(x^2+y^2)^{5/2}} = \frac{-1} {(x^2+y^2)^{3/2}} <0 \end{align*}

In this case, away from the origin, the divergence is negative. The fluid is compressing even though it is flowing outward.

Why did the dimension make a difference? One can see the difference from the calculations, but what is the difference in the geometric picture? As in the three-dimensional case, the fluid flows into the circle faster than it flows out of the circle, as the arrows are getting shorter. And, as in the three-dimensional case, because the flow is radiating outward, the fluid is flowing out of the circle over more than half the boundary of the circle. But, because we are only in two dimensions, the effect from the boundary is smaller. I chose the vector field to balance the two effects and make the divergence zero in three dimensions. But, this makes the divergence of the two-dimensional analogue be negative.

You can check that the divergence of the vector field \begin{align*} \dlvf(x,y) = \frac{(x,y)}{x^2+y^2} \end{align*} is zero but that the divergence of the three-dimensional analogue \begin{align*} \dlvf(x,y,z) = \frac{(x,y,z)}{x^2+y^2+z^2} \end{align*} is positive. In general, for a number $p$, the divergence of the vector field \begin{align*} \dlvf(\vc{x}) = \frac{\vc{x}}{\norm{\vc{x}}^p} \end{align*} is $\div \dlvf(\vc{x}) = (3-p)/\norm{\vc{x}}^p$ in three dimensions and is $\div \dlvf(\vc{x}) = (2-p)/\norm{\vc{x}}^p$ in two dimensions. So you need $p=3$ to have zero divergence in three dimensions and $p=2$ to have zero divergence in two dimensions.


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# Beer Lambert Law Calculator

Top
This online calculator helps to determine the intensity of a monochromatic light beam after passing through an absorbing medium. According to the law, the intensity of the monochromatic light decreases exponentially with the concentration and thickness of the medium. The formula is given by,
I = $I_{0}e^{-\mu x}$
where,
I = intensity after absorption,
$I_{0}$ = initial intensity,
$\mu$ = absorption coefficient,
x = thickness of the medium.

## Steps

Step 1 : From the given question, note down the values of the parameter.

Step 2 : Substitute these values in the given formula and find out the required value,

I = $I_{0}e^{-\mu x}$.

## Problems

Let us discuss the problems related to Beer Lambert Law.

### Solved Examples

Question 1: Calculate the intensity of a monochromatic light which is passed through a medium of thickness 0.1m. The initial intensity is given as 25mW and the absorption coefficient is 0.3m-1.

Solution:

Given that,
$I_{0}$ = 25mW,
$\mu$ = 0.3m-1,
x = 0.1m.

Intensity of a monochromatic light is,
I = $I_{0}e^{-\mu x}$
I = 25$\times e^{-0.3\times0.1}$
I = 25$\times$0.970445 = 24.26

Question 2: Calculate the intensity of a monochromatic light which is passed through a medium of thickness 0.05m. The initial intensity is given as 15mW and the absorption coefficient is 0.6m-1.

Solution:

Given that,
$I_{0}$ = 15mW,
$\mu$ = 0.6m-1,
x = 0.05m.

Intensity of a monochromatic light is,
I = $I_{0}e^{-\mu x}$
I = 15$\times e^{-0.6\times0.05}$
I = 15$\times$0.970445 = 14.556


text:
Publication
Title
Measurement of the inclusive **ep** scattering cross section at low $Q^{2}$ and **x** at HERA
Author
Institution/Organisation
H1 Collaboration
Abstract
 A measurement of the inclusive ep scattering cross section is presented in the region of low momentum transfers, 0.2 GeV(2) a parts per thousand currency signQ (2)a parts per thousand currency sign12 GeV(2), and low Bjorken x, 5a <...10(-6)a parts per thousand currency signxa parts per thousand currency sign0.02. The result is based on two data sets collected in dedicated runs by the H1 Collaboration at HERA at beam energies of 27.6 GeV and 920 GeV for positrons and protons, respectively. A combination with data previously published by H1 leads to a cross section measurement of a few percent accuracy. A kinematic reconstruction method exploiting radiative ep events extends the measurement to lower Q (2) and larger x. The data are compared with theoretical models which apply to the transition region from photoproduction to deep inelastic scattering.
Language
English
Source (journal)
European physical journal : C : particles and fields. - Berlin
Publication
Berlin : 2009
ISSN
1434-6044
Volume/pages
63:4(2009), p. 625-678
ISI
000270978300005
Full text (Publisher's DOI)
Full text (open access)
UAntwerpen
 Faculty/Department Research group Publication type Subject Affiliation Publications with a UAntwerp address