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# 11.3: Ligand Field Stabilisation Energy (LFSE)

There is a variation on how to think about d orbital splitting diagrams that can be useful in deciding how the d electrons are configured in transition metal complexes. We can use the relative energy levels of the d orbitals in a given complex to calculate whether the overall energy would be higher or lower in a high-spin vs. a low-spin case, for example. The calculation provides us with a value that is called the ligand field stabilisation energy (LFSE). Although we have been thinking of bonding in transition metal complexes in terms of molecular orbital ideas, ligand field stabilisation energy actually has its roots in a separate approach called crystal field theory.

## Origin Story: LFSE originally comes from CFT

Crystal field theory (CFT) was independently developed around 1930 by German physicist Hans Bethe and American physicist John Hasbrouck van Vleck; the two later became professors of physics at Cornell and Harvard, respectively. CFT actually pre-dates the molecular orbital approach that we have been using so far, but it reaches similar conclusions about transition metal electron configurations.

Both scientists were interested in the magnetic properties of metals and metal salts. They knew these properties were related to unpaired electrons. Compounds with unpaired electrons (paramagnetic) are attracted by magnetic fields, whereas compounds having only paired electrons (diamagnetic) are not. They were interested in the factors that influenced the d electron configuration of transition metal salts, because how the d electrons filled could result in different numbers of unpaired electrons. Those differences influenced how strongly the compounds interacted with magnetic fields.

Crystal field theory does not consider any bonding interactions in transition metal compounds. It focuses only on the repulsion between the electrons on an anion and the electrons on a metal cation. Of course, all of these electrons have negative charges. As an anion approaches a metal, the physicists reasoned, both sets of electrons would experience repulsive forces that would increase their energy level.

If the metal were surrounded by a spherical field of electrons (physicists like to assume things are spherical because it makes the math easier), then all of the electrons on the metal would be raised equally high in energy by these repulsive forces.

But in a metal salt, electrons are not approaching the metal from all directions. The geometry comes in because of the fact that the cations and anions pack together in specific arrays. For example, if a metal cation is sitting in an octahedral hole, electrons would only approach from six directions: at each end of the x, y, and z axes.

In that case, only the electrons in d orbitals that were aligned along the x, y and z axes would be raised significantly by repulsion with the approaching electrons on the anions. The electrons in the off-axis orbitals would be lower in energy than they would have been in a perfectly spherical field.

Scientists later adapted this idea to octahedral coordination complexes, in which the metal sits amidst six ligands rather than getting packed among six anions.

That result leads to a picture that is pretty similar to what we get from molecular orbital theory, at least as far as the d orbitals are concerned. In fact, the true mathematical approach to molecular orbital theory does take these electron-electron repulsions into account, but it also factors in attractions between the electrons of the ligands and the nucleus of the metal (and vice versa). The biggest difference is that molecular orbital theory includes what happens to the kinetic energy of the ligand lone pairs as they become shared with the metal (it goes down; that's a big part of why the bonds form).

## The calculation of LFSE

You don't really have to appreciate where this approach came from to be able to see how it is commonly used. If we take the five d orbitals after they have been placed in an octahedral environment, we see that they are at two different energy levels. The average energy level is not, as it might first appear, halfway between these levels. That's because there are three lower energy level orbitals and two higher energy level orbitals. The average is two-fifths of the way up from the bottom three, or three-fifths of the way down from the top two.

That average energy level of the five d orbitals is called the "barycenter"; it is assigned a relative energy of zero. The difference between the top level and the bottom level is the field splitting; for an octahedral complex, this field splitting is given the symbol Δo; here, Δ stands for the energy difference and o stands for octahedral.

If the barycenter is at an energy level of zero, then the lower orbitals are below zero. How far? They are two-fifths of the total Δo below the barycenter. Any electrons in those orbitals are -0.4Δo below 0 in energy. Electrons in the upper orbitals are +0.6Δo above zero in energy.

### Example $$d^4$$:

Suppose there are four electrons. The first three electrons sit in each of the three lower orbitals, the ones we sometimes label the t2g orbitals. The fourth will either go into the upper level, which we sometimes call the eg level, or else pair up with another electron at the lower level.

We can now calculate the energy difference between these two possible cases. We can calculate what is called the ligand field stabilisation energy, LFSE (sometimes called crystal field stabilisation energy, or CFSE). It's just the sum of the energies of each of the electrons.

$LFSE = [(0.6 \times number \: of \: e_{g} \: electrons) - (0.4 \times number \: of \: t_{2g} \: electrons)] \Delta_{o}$

or if that's too much jargon, $$LFSE = [(0.6 \times \: \# upper \: e^{-}) - (0.4 \times \: \# lower \: e^{-})] \Delta_{o}$$

For the $$d^4$$ high-spin case:

$LFSE = [(0.6 \times 1) - (0.4 \times 3)] \Delta_{o} = [0.6-1.2] \Delta_{o} = -0.6 \Delta_{o}$

For the $$d^4 low-spin case: $LFSE = [(0.6 \times 0) - (0.4 \times 4)] \Delta_{o} = -1.6 \Delta_{o}$ So far, it certainly seems like the low-spin case is at lower energy and thus more stable. Remember, though, that low-spin configuration requires that we place two electrons together into the same orbital. Pairing electrons in the same orbital is going to cost some energy. But how much? This repulsion between a pair of electrons in one orbital is called the pairing energy (PE). For each pair of electrons that occupy the same orbital, that energy must be added to take that repulsion into account. As a result, a calculation of the overall stabilization energy (SE) includes both the LFSE and the PE. $SE = LFSE + PE$ If there were two sets of paired electrons, we would add 2PE; if there were three sets of paired electrons, we would add 3PE, and so on. Overall, deciding quantitatively whether a complex will be high spin or low spin is the most useful application of an LFSE calculation. However, you do have to know the values of the parameters (the field splitting and the pairing energy) for a definitive decision. ##### Exercise \(\PageIndex{1}$$

Use SE calculations to determine stabilisation in both high spin and low spin cases. Just leave your answer expressed in terms of Δo and PE.

a. Fe+2

b. Co+2

c. Co+3

d. Mn+2

e. Ti+3

Fe+2

d6 low spin
SE = [-0.4(6)+ 0.6(0)] Δo + 3PE
= [-2.4] Δo + 3PE

d6 high spin
SE = [-0.4(4)+ 0.6(2)] Δo + 1PE
= [-.4] Δo + 1PE

Co+2

d7 low spin
SE = [-0.4(6)+ 0.6(1)] Δo + 3PE
= [-1.8] Δo + 3PE

d7 high spin
SE = [-0.4(5)+ 0.6(2)] Δo + 2PE
= [-.8] Δo + 2PE

Co+3
d6 so this looks the same as Fe+2

Mn+2

d5 low spin
SE = [-0.4(5)+ 0.6(0)] Δo + 2PE
= [-2.0] Δo + 2PE

d5 high spin
SE = [-0.4(3)+ 0.6(2)] Δo + 0PE
= [0] Δo

Ti+3
d1 so there is no possibility of low spin or high spin

## SE depends on the Magnitude of $$\Delta_o$$ and total PE

Right now, let's think about Δo. How much energy is it? How big is the gap between the d orbitals? And how do we know? Well, we can easily measure this gap using a simple spectrophotometer. A spectrophotometer measures how much light is absorbed by a sample. Furthermore, it measures what specific colors of light are absorbed by the sample. As it happens, the absorption of ultraviolet and visible light by a material is associated with electrons in that material becoming promoted to a higher energy level. Shine a light on a transition metal complex, and an electron may jump the gap.

So all we have to do is shine different colors of monochromatic light on the complex and see whether it absorbs one of the colors.

This experiment is quite reproducible. A specific compound will always absorb the same colors of light. That's why specific materials have specific colors; the color we see represents the light that is not absorbed by the material.

Different colors of light have different amounts of energy. Blue light has higher energy than orange light, which has higher energy than red light. Max Planck and Albert Einstein worked out that the energy of a photon (a "particle" or "packet" of light) is directly proportional to the frequency at which the photon oscillates or vibrates. Blue light has a higher frequency than red light, so it has higher energy.

Planck and Einstein expressed this idea with an equation,

E = hν

in which the Greek letter ν, pronounced "noo," stands for the frequency (some people just use f for frequency) and h stands for a sort of conversion factor called Planck's constant.

Blue light has too much energy to promote an electron to the next energy level in this case. It would send the electron way past the next level, and quantum mechanics does not allow that sort of thing. Red light does not have enough energy for the electron to get there. Orange light, in this case, is just right. It has exactly the right amount of energy to jump the gap.

So, if we know the frequency of orange light, we know how much energy there is in an orange photon, and we know how big the field splitting is between the d orbitals.

Historically, it turns out that people have most often described visible light in terms of wavelength rather than frequencies. The wavelength is just the distance from one "peak" to the next as the wave of the photon rolls along. The higher the frequency, the closer these peaks are together, and the shorter the wavelength. Blue light has a shorter wavelength, around 400 nm, than red light, around 700 nm. A nanometer (nm) is 10-9 meters (m); a meter is around a yard.

Orange light has a wavelength of around 600 nm, or 600 x 10-9 m, or 6 x 10-7 m. For reasons we won't get into, spectroscopists in the past (people who measure the interaction of light and matter) sometimes preferred to work in centimeters, cm; there are 100 cm in 1 m, so orange light has a wavelength of 6 x 10-5 cm. Now, because they knew there was an inverse relationship between wavelength and energy (the longer the wavelength, the lower the energy; the shorter the wavelength, the higher the energy), they simply took the reciprocal of the wavelength in centimeters to get a number in cm-1, which they called wavenumbers. They used this as a unit of energy. An orange photon has an energy of 1/0.00006 cm = 16,000 cm-1.

So the gap, the field splitting, Δo in this complex is 16,000 cm-1.

Let's go back to our comparison between the high-spin and low-spin case. For high spin:

$LFSE = [(0.6 \times 1) - (0.4 \times 3)] \Delta_{o} = [0.6 -1.2] \Delta_{o} = -0.6 \Delta_{o} = -0.6 \times 16000 cm^{-1} = -9600 cm^{-1}$

For the low-spin case:

$LFSE = [(0.6 \times 0) -(0.4 \times 4)] \Delta_{o} = -1.6 \Delta_{o} = -1.6 \times 16000 cm^{-1} = -25600 cm^{-1}$

These LFSE calculations show that the low-spin case is lower in energy, by 14,000 cm-1. However, we still need to include the pairing energy. Like the field splitting, the pairing energy varies from one complex to another. 20,000 cm-1 is a ballpark estimate of a typical pairing energy.

If we use this average value for PE in the example we were discussing above for the high-spin case:

$SE = LFSE + PE = -9600 + 0 cm^{-1} = -9600 cm^{-1}$

For the low-spin case:

$SE = LFSE + PE = -25600 + 20000 cm^{-1} = -5600 cm^{-1}$

Taking the pairing energy into account in this case suggests that the high-spin case is favored. It cost less energy to jump the gap and put an electron in a high-lying eg orbital than it did to pair electrons in a low-lying t2g orbital.

### Reminder: Factors that influence $$\Delta_o$$

Let's take a look at some real examples of field splitting values to get an idea of how large they are, and what factors they depend on.

#### The metal influences $$\Delta_o$$

For example, we can look at the charge on the metal ion:

 Table of Field Splitting Values, Δo, in Hexaquo Complexes of Differing Chargesa Metal Complex [Mn(OH2)6]2+ [Mn(OH2)6]3+ [Fe(OH2)6]2+ [Fe(OH2)6]3+ [Co(OH2)6]2+ [Co(OH2)6]3+ Δo (cm-1) 7,800 21,100 10,400 13,800 9,300 18,300 a) Holleman, A. F.; Wiberg, E.; Wiberg, N. Inorganic Chemistry, 34th Ed. Academic Press: Berlin, 2001.

In this case, it looks like increased charge on the metal ion results in an increased field splitting. The same thing happens in all three examples: manganese, iron, and cobalt. As the metal ion becomes more charged, attraction of the d electrons toward the nucleus increases. The lower-lying t2g level is more strongly attracted to the nucleus because it is a little closer; consequently, it drops a little further than the eg level, and the gap gets bigger.

Charge on the metal ion is one of two key factors that influences the size of the field splitting. The other factor is the period or row in the periodic table.

 Table of Field Splitting Values, Δo, in Hexammine Complexes of Group 9a,b Metal Complex [Co(NH3)6]3+ [Rh(NH3)6]3+ [Ir(NH3)6]3+ Δo (cm-1) 21,500 33,100 41,100 a) Holleman, A. F.; Wiberg, E.; Wiberg, N. Inorganic Chemistry, 34th Ed. Academic Press: Berlin, 2001. b) Miessler, G. L.; Tarr, D. A. Inorganic Chemistry, 4th Ed. Pearson, 2010.

Moving from one row to the next in group 9 of the periodic table, we see that the field splitting increases by about 10,000 cm-1 each row.

In general:

• Δo increases with charge on the metal
• Δo increases with the period in the periodic table

#### The ligand identity influences $$\Delta_o$$

The metal ion is not the only factor that affects the field splitting. The ligands also play an important role, as seen in the table below.

 Table of Field Splitting Values, Δo, in Assorted Complexesa Metal Complex [CrCl6]3- [CrF6]3- [Fe(OH2)6]2+ [Fe(CN)6]4- [Co(OH2)6]3+ [Co(NH3)6]3+ [Co(CN)6]3- Δo (cm-1) 13,300 15,300 10,400 33,200 18,300 23,000 33,700 a) Holleman, A. F.; Wiberg, E.; Wiberg, N. Inorganic Chemistry, 34th Ed. Academic Press: Berlin, 2001.

If we sort these ligands into three broad categories, we can form a trend for these effects. It may be easiest to start with the three entries on the right, involving cobalt. The oxygen donor in water has two lone pairs, whereas the nitrogen donor in ammonia has only one. The extra lone pair allows water to act as a π (pi) donor; the lone pair can donate to an orbital on the metal to form a pi bond. In this case, it looks like the π donor (water) results in a smaller Δo than the normal σ (sigma) donor (ammonia).

The cyanide ligand has a donor atom that is also participating in a π bond within the ligand; there is a triple bond between the C and N of the cyanide. That means there is also an associated antibonding orbital, π*. That antibonding orbital raises the possibility of back-bonding from the metal. The metal can donate into the π* orbital to make a pi bond. Cyanide is a π-acceptor. Its field splitting is much larger than the sigma donor.

Those conclusions can be confirmed by looking at the entries for iron in the middle of the table. Cyanide is a π-acceptor whereas water is a π-donor. The field splitting in the aquo complex should be much smaller than the field splitting in the cyano complex, and it is.

The two entries for chromium on the left both show halides, which are π-donors. They illustrate a factor that can be used to predict field strength between two ligands from the same group, such as two π-donors or two π-acceptors. In general, the more basic the ligand, the greater the field splitting. Fluoride ion is more basic than chloride ion (because chloride is a more stable anion than fluoride) so it results in a slightly greater value of Δo.

### The Pairing Energy

In order to thoroughly estimate the stabilisation energy, we also need reliable values for the pairing energy. Pairing energies can be calculated for free metal ions; correction factors can be applied to arrive at a corresponding value in a complex. In general, the values in coordination complexes are somewhat lower than the values in free metal ions. In the table below, we have made a general estimate that the value in the complex is about 20% lower than in the free metal ion.

 Table of Pairing Energies, PE, in Free Ions, with Estimated PE in Complexes Metal Ion Mn2+ Mn3+ Fe2+ Fe3+ Co3+ Ru3+ PE, free ion (cm-1) 28,000 25,500 17,700 30,100 21,100 ~15,000b PE, complex (est., cm-1) 20,400 22,900 14,200 24,100 16,900 ~12,000b a) Holleman, A. F.; Wiberg, E.; Wiberg, N. Inorganic Chemistry, 34th Ed. Academic Press: Berlin, 2001. b) Estimate.

If we compare the pairing energies (PE) of the 2+ ions to those of the 3+ ions, we see that PE is always lower in the more highly charged ion. That's the opposite of the trend in Δo. Pairing energy is largely dependent on the size of the ion. The smaller the ion, the smaller the orbital, and the more repulsion between two electrons in the same orbital. Because a Mn3+ ion is smaller than a Mn2+ ion (Coulomb's Law says there is greater attraction for the electrons in the former case, so the ion shrinks), a pair of electrons on Mn3+ is closer together than a pair of electrons on Mn2+.

In general, pairing energies also get smaller upon moving down a column in the periodic table. Second row transition metals are larger than first-row transition metals, so the pairing energy is smaller in the second row than in the first. Third row transition metals are about the same size as first row metals, so their pairing energies are similar to those of the second row. That's because of a phenomenon called "the lanthanide contraction": third row transition metals contain an extra set of protons in their nuclei because of the f block elements before them; consequently they are a little smaller than might be expected.

##### Exercise $$\PageIndex{2}$$

Compare pairing energies to field splitting values to determine whether the following complexes would be high-spin or low spin.

a) Mn(OH2)62+ b) Fe(CN)64- c) Co(NH3)63+

Δo < PE, so high-spin.

Δo > PE, so low-spin.


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# Infinite Series Formula

Infinite series is one of the important concepts in mathematics. It tells about the sum of a series of numbers that do not have limits. If the series contains infinite terms, it is called an infinite series, and the sum of the first n terms, Sn, is called a partial sum of the given infinite series. If the partial sum, i.e. the sum of the first n terms, Sn, given a limit as n tends to infinity, the limit is called the sum to infinity of the series, and the result is called the sum of infinite of series.

## Sum of Infinite Series Formula

The sum of infinite for an arithmetic series is undefined since the sum of terms leads to ±∞.

The sum to infinity for a geometric series is also undefined when |r| > 1.

If |r| < 1, the sum to infinity of a geometric series can be calculated.

Thus, the sum of infinite series is given by the formula:

$$\begin{array}{l}\large \mathbf{S_{\infty}=\frac{a}{1-r}}\end{array}$$

Or

$$\begin{array}{l}\large \lim_{n\rightarrow \infty}S_n = S_{\infty}=\frac{a}{1-r}\end{array}$$

### Solved Examples

Question 1: Evaluate:

$$\begin{array}{l}\sum_{0}^{\infty }(\frac{1}{2})^{n}\end{array}$$

Solution:

The sum of given series is,

$$\begin{array}{l}\sum_{0}^{\infty }(\frac{1}{2})^{n}\end{array}$$

So the series can be written as,

$$\begin{array}{l}\sum_{0}^{\infty }(\frac{1}{2})^{n} = (\frac{1}{2})^{0}+(\frac{1}{2})^{1}+(\frac{1}{2})^{2}+(\frac{1}{2})^{3}+…………..\end{array}$$

Here,

First term, a = 1

Common ratio, r = 1/2.
Infinite series formula is,

$$\begin{array}{l}S_{\infty} = \frac{a}{1-r}\end{array}$$

So,

$$\begin{array}{l}S_{\infty} = \frac{1}{1-\frac{1}{2}} = \frac{2}{2-1}=2\end{array}$$

Question 2:

Find the sum to infinity for the series 24 + 12 + 6 + …. (if exist).

Solution:

Given,

64 + 16 + 4 + ….

Here,

First term, a = 64

Common ratio = 16/64 = 1/4

r = 1/4 < 1, so we can find the sum to infinity.

$$\begin{array}{l}S_{\infty} = \frac{a}{1-r}\end{array}$$

So,

$$\begin{array}{l}S_{\infty} = \frac{64}{1-\frac{1}{4}} = \frac{64\times 4}{4-1}=\frac{256}{3}\end{array}$$

Therefore, the sum of infinite terms of the given series is 256/3.

format: plain


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# How do you write the trigonometric form of 1+6i?

Oct 26, 2017

$\sqrt{37} \left(\cos {80.5}^{o} + i \cdot \sin {80.5}^{o}\right) \mathmr{and} \sqrt{37} \cdot c i s {80.5}^{o}$

#### Explanation:

Converting complex numbers into trigonometric form is very similar to converting rectangular coordinates (aka Cartesian Coordinates) into polar coordinates. There are 2 formulas that will assist (assuming the complex number is in $a + b i$ format):

$r = \sqrt{{a}^{2} + {b}^{2}}$

$\tan {\theta}_{r e f} = | \frac{b}{a} |$

We begin by finding the value $r$:

$r = \sqrt{{1}^{2} + {6}^{2}} = \sqrt{1 + 36} = \sqrt{37}$

Next, we can determine the reference angle. For this problem, I am using degrees, though the same work could be done with radians as the measure instead.

$\tan {\theta}_{r e f} = | \frac{6}{1} |$

${\theta}_{r e f} = {80.5}^{o}$

This is only the reference angle, which will always be in the first quadrant when done this way. The complex number $1 + 6 i$ lies in the first quadrant in the complex plane, and so this angle will be sufficient for our needs.

The trigonometric form of a complex number $a + b i$ is written one of two ways:

$\left\{\begin{matrix}r \left(\cos \theta + i \cdot \sin \theta\right) & \text{Expanded Form" \\ r*cis theta & "Condensed Form}\end{matrix}\right.$

$\sqrt{37} \left(\cos {80.5}^{o} + i \cdot \sin {80.5}^{o}\right) \mathmr{and} \sqrt{37} \cdot c i s {80.5}^{o}$


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Ellipse problem : Find the slope of a common tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and a concentric circle of radius r.

Problem :

Find the slope of a common tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and a concentric circle of radius r.

Few concepts about Ellipse :

Equation of Tangent to ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at point $(x_1,y_1)$ is given by $T = \frac{xx_1}{a^2} +\frac{yy_1}{b^2}-1$

Also point of contact where line $y =mx +c$ touched the ellipse .

The line is y =mx +c touches the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ when $c =\pm \sqrt{a^2m^2+b^2}$

$(x_1,y_1) =(\pm \frac{a^2m}{\sqrt{a^2m^2 +b^2}}, \pm \frac{b^2}{\sqrt{a^2m^2+b^2}})$

Now, we know that equation of any tangent to the given ellipse is $y =mx \pm \sqrt{a^2m^2+b^2}$

Now if it touches $x^2+y^2=r^2$

Then

$\sqrt{a^2m^2+b^2} = r\sqrt{1+m^2}$ I am unable to understand this condition ..please guide for this particular condition only... thanks.

If we use parametric forms of the ellipse $(a\cos\phi,b\sin\phi)$ and of the circle $(r\cos\psi,r\sin\psi)$

we get the tangents to be $\displaystyle x\frac{\cos\phi}a+y\frac{\sin\phi}b=1\iff x b\cos\phi+y a\sin\phi=ab$

and for the circle $x\cos\psi+y\sin\psi=r$ with slope $\displaystyle-\frac{\cos\psi}{\sin\psi}=-\cot\psi\ \ \ \ (1)$

These two equations have to be identical implies $$\frac{b\cos\phi}{\cos\psi}=\frac{a\sin\phi}{\sin\psi}=\frac{ab}r$$

So, $\displaystyle\cos\phi=\frac{a\cos\psi}r,\sin\phi=\frac{b\sin\psi}r$

Squaring & adding we get, $\displaystyle(a\cos\psi)^2+(b\sin\psi)^2=r^2$

Divide either sides by $\sin^2\psi,$

$\displaystyle(a\cot\psi)^2+b^2=r^2(1+\cot^2\psi)\implies \cot^2\psi=\frac{r^2-b^2}{a^2-r^2}$ which must be $\ge0$

$\displaystyle\implies(r^2-b^2)(a^2-r^2)\ge0\iff (r^2-a^2)(r^2-b^2)\le0$

$\displaystyle\implies$ min$(a,b)\le r\le$max$(a,b)$

The value of $r$ will dictate the value of the slope in $(1)$

You have done sufficient hard work.

As $m$ is the slope of the common tangent, we have $\displaystyle \pm \sqrt{a^2m^2+b^2} = \pm r\sqrt{1+m^2}$

Squaring we get $\displaystyle a^2m^2+b^2=r^2(1+m^2)$

What is $m$, compare with my other answer?


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# Finding the end

In our examples so far, we've known how much data there is in the worksheet. But in RL, the amount of data changes from day to day. For example, you have 280 sales records on Monday, 318 on Tuesday, 278 on Wednesday, and so on. You have to write programs that can deal with that.

# What ends the data?

Here's a typical column of data:

And another:

And another:

After a while, the data just stops. There's an empty cell after the data. Hmm… how about we read the data until we get to an empty cell?

We can't use a For-Next for that. Remember the general form:

For [index variable] = [start] To [end]

We don't know what [end] is. But we can use a regular Do loop.

# Counting rows

Let's have a variable rows, that we initialize to zero. Then ask: is the next cell (rows + 1 = 1) empty?

No, so add 1 to rows (it's now 1) and keep going. This is look ahead. We decide what to do, based on what the next value is.

rows is 1. Is the next cell (rows + 1 = 2) empty?

No, so add 1 to rows (it's now 2) and keep going.

rows is 2. Is the next cell (rows + 1 = 3) empty?

No, so add 1 to rows (it's now 3) and keep going.

rows is 3. Is the next cell (rows + 1 = 4) empty?

Yes, the cell is empty, so stop the loop. rows is 3. That's how much data we have.

Here's code for it.

1. Private Sub cmdRun_Click()
2.     Dim rows As Integer
3.     rows = 0
4.     Do While Cells(rows + 1, 1) <> ""
5.         rows = rows + 1
6.     Loop
7.     Cells(4, 3) = rows
8. End Sub
Line 4 shows the look ahead. Instead of Cells(rows, 1), it tests Cells(rows + 1, 1), looking ahead one cell.

Pattern

Find the end of a data set

Situation:
The number of items in a data set varies. Maybe 502 one day, and 534 the next. You want to find the number of items.
Actions:
Use a Do loop and an index variable. Check cells one by one, until there's an empty cell.

There's something else different about the Do. Previously, we've seen loops like this:

Do
[code]
Loop Until [logical expression]
Do
[code]
Loop While [logical expression]

These are post-tested loops, that is, the test is at the end of the loop. The [code] always executes at least once.

There are also pretested loops, where the test is at the start of the loop.

Do Until [logical expression]
[code]
Loop
Do While [logical expression]
[code]
Loop

It's possible that [code] won't execute at all. For example:

1. x = 7
2. Do While x < 6
3.     x = x + 1
4.     MsgBox "x is " & x
5. Loop
The loop body will never execute.

That might happen in our find-the-end-of-the-data program. Here's the code again:

1. Private Sub cmdRun_Click()
2.     Dim rows As Integer
3.     rows = 0
4.     Do While Cells(rows + 1, 1) <> ""
5.         rows = rows + 1
6.     Loop
7.     Cells(4, 3) = rows
8. End Sub
Suppose there's no data at all, that is, Cells(1, 1) is empty. What's the code going to do? Line 3 initializes rows to zero. The While loop's test fails, because Cells(rows + 1, 1) is "". The loop doesn't run even once. When line 7 outputs rows, it outputs 0, the correct value.

Which loop form should you use? Pretested, post-tested, While, Until… Use whichever makes the most sense to you, for the task you're working on. If the program is easier to understand when you use a post-tested loop, use that one.

# Processing while counting

You can do other computations while finding the last element. For example:

1. Private Sub cmdRun_Click()
2.     Dim rows As Integer
3.     Dim total As Single
4.     'Initialize
5.     total = 0
6.     rows = 0
7.     'Process
8.     Do While Cells(rows + 1, 1) <> ""
9.         rows = rows + 1
10.         total = total + Cells(rows, 1)
11.     Loop
12.     'Output
13.     Cells(4, 3) = rows
14.     Cells(5, 3) = total
15. End Sub
This works for totaling, but highest and lowest are a bit messier. So is validation. So is filtering, that we'll get to later. In fact, the more processing you mix in with the counting code, the more confusing things get. More bugs. More screaming.

# Thunking

To keep from screaming, split processing up. Remember IPO? That's the input-processing-output pattern.

Pattern

Input-processing-output

Situation:
Data needs to be transformed into other data, without user interaction after input.

Actions:
Use abstraction to define each of the three parts, and the way they communicate. Abstraction means that you:
• Name each subroutine.
• Specify parameters and, for functions, return values.

Replace input with counting, and it works quite well.

Let's write a program that reads an unknown amount of data and computes the total, highest, and lowest values. Let's add some validation: it won't run if there's fewer than 2 pieces of data.

Here's the main program:

1. Private Sub cmdRun_Click()
2.     Dim rows As Integer
3.     Dim countOK As Boolean
4.     Dim total As Single
5.     Dim highest As Single
6.     Dim lowest As Single
7.     'Count data cells.
8.     countDataCells rows, countOK
9.     If Not countOK Then
10.         End
11.     End If
12.     'Processing.
13.     computeStats rows, total, highest, lowest
14.     'Output
15.     outputStats rows, total, highest, lowest
16. End Sub
All countDataCells does is, er, count the data cells. It returns the number of cells, and a flag showing whether everything went OK. If it didn't, the program stops.

Here's the code for the sub:

1. 'Count the number of data cells.
2. '  rows: number of cells.
3. '  countOK: flag showing whether there's a problem with the data.
4. Sub countDataCells(rows As Integer, countOK As Boolean)
5.     'Initialize.
6.     rows = 0
7.     'Count.
8.     Do While Cells(rows + 1, 1) <> ""
9.         rows = rows + 1
10.     Loop
11.     'Validate.
12.     countOK = True
13.     If rows < 2 Then
14.         MsgBox "Sorry, you must have at least two data items."
15.         countOK = False
16.     End If
17. End Sub
Lines 8 to 10 are the look ahead loop we saw before. The countOK flag is set to True (line 12). It's switched to False only if there's a problem with too little data.

Once we have a count, what's next? Here's part of the main program again:

1. 'Count data cells.
2. countDataCells rows, countOK
3. If Not countOK Then
4.     End
5. End If
6. 'Processing.
7. computeStats rows, total, highest, lowest
We send rows into computeStats, and it sends back total, highest, and lowest:

1. 'Compute statistics.
2. '  rows: number of data items.
3. '  total: sum of all items.
4. '  highest: highest value.
5. '  lowest: lowest value.
6. Sub computeStats(rows As Integer, total As Single, _
7.         highest As Single, lowest As Single)
8.     Dim row As Integer
9.     'Initialize.
10.     total = Cells(1, 1)
11.     highest = Cells(1, 1)
12.     lowest = Cells(1, 1)
13.     For row = 1 To rows
14.         total = total + Cells(row, 1)
15.         If Cells(row, 1) > highest Then
16.             highest = Cells(row, 1)
17.         End If
18.         If Cells(row, 1) < lowest Then
19.             lowest = Cells(row, 1)
20.         End If
21.     Next row
22. End Sub
Hey! Now that we know how many data elements there are, we can use a For-Next loop. Woohoo! Just copy the code from before.

The main program again:

1. 'Count data cells.
2. countDataCells rows, countOK
3. If Not countOK Then
4.     End
5. End If
6. 'Processing.
7. computeStats rows, total, highest, lowest
8. 'Output
9. outputStats rows, total, highest, lowest
We send rows, total, highest, and lowest into outputStats:

1. 'Output statistics.
2. '  rows: number of data items.
3. '  total: sum of all items.
4. '  highest: highest value.
5. '  lowest: lowest value.
6. Sub outputStats(rows As Integer, total As Single, _
7.         highest As Single, lowest As Single)
8.     Cells(4, 3) = rows
9.     Cells(5, 3) = total
10.     Cells(6, 3) = highest
11.     Cells(7, 3) = lowest
12. End Sub
Thunking the program makes it easier to write, test, and read. For example, when we write countDataCells, we only have to think about counting cells. Toss the stats and output stuff out of our heads, for now. Simpler code. Fewer bugs. Less screaming.

We can also have different people write the subs, then put the whole thing together.

We can replace one of the subs, without touching the others. For example, suppose The Boss wants the output to show in message boxes. Easy peasy.

1. 'Output statistics.
2. '  rows: number of data items.
3. '  total: sum of all items.
4. '  highest: highest value.
5. '  lowest: lowest value.
6. Sub outputStats(rows As Integer, total As Single, _
7.         highest As Single, lowest As Single)
8.     MsgBox "Elements: " & rows
9.     MsgBox "Total: " &  total
10.     MsgBox "Highest: " & highest
11.     MsgBox "Lowest: " & lowest
12. End Sub
Nothing else changes. We don't have to think about, test, debug, or scream about the rest of the program.

# Let's thunk

From now on, let's organize our code in chunks like this. Our programs are going to get more and more complex. Thunking will keep the complexity under control. Less screaming.

That's a Good Thing.

# Exercises

Exercise: Count the data values
Download this workbook. It looks like this:

Add code that counts the number of data values in column 2, and shows the count in row 5, column 1. The usual coding standards apply.

Upload your workbook.

Attachments

count-data.xlsm

(If you were logged in as a student, you could submit an exercise solution, and get some feedback.)

Exercise: Comparing regional sales
Download a workbook with sales for two regions:

The first column has sales for the north region. The second has sales for the west region. The number of sales in each column might be different.

Write a program that will total the first column and the second. If the first column is higher, output "North" in row 5, column 3. If the second is higher, output "West". If they are the same, output "Equal".

Some of the code has been given for you. Use the code, without change.

The usual coding standards apply.

Upload your workbook.

Attachments

compare-regions.xlsm

(If you were logged in as a student, you could submit an exercise solution, and get some feedback.)

# Summary

So far, we've known how much data there is in the worksheet. But in RL, the amount of data changes from day to day. We can't use a For-Next loop, because we don't know where to end.

We can use a regular Do loop, with look ahead. We decide what to do, based on what the next value is.

The more processing you mix in with the counting code, the more confusing things get. A way around this is to split processing up. Thunking the program makes it easier to write, test, and read. We can also have different people write the subs, then put the whole thing together.


text:
# Cofactorial automorphism

This article defines an automorphism property, viz a property of group automorphisms. Hence, it also defines a function property (property of functions from a group to itself)
View other automorphism properties OR View other function properties

## Definition

### For a finite group

An automorphism $\sigma$ of a finite group $G$ is termed a cofactorial automorphism for $G$ if every prime dividing the order of $\sigma$ (viewed as an element of the automorphism group of $G$) also divides the order of $G$.

### For a periodic group

An automorphism $\sigma$ of a periodic group $G$ is termed a cofactorial automorphism for $G$ if every prime dividing the order of $\sigma$ equals the order of some non-identity element of $G$.

### For a group with elements of infinite order

If the group has any element of infinite order, then every automorphism of the group is considered to be a cofactorial automorphism.


text:
## Trigonometry (11th Edition) Clone

$\cos\theta=3$ is wrong. The domain of $\cos\theta$ only ranges from $-1$ to $1$.
The problem here is with the statement $\cos\theta=3$. As we know, the domain of $\cos\theta$ only ranges from $-1$ to $1$. In other words, the value of $\cos\theta$ cannot exceed $1$. Therefore, $\cos\theta=3$ is impossible.


text:
# Weighted hypergeometric distribution

The probablility of $M$ successes out of $N$ draws without replacement from a population of $S$ success and $F$ failure states, assuming that each remaining state is equally likely, is given by

$\dfrac{\binom{S}{M} \binom{F}{N-M}}{\binom{S+F}{N}}$.

We can think of it of drawing $N$ marbles from an urn containing $S$ white and $B$ black marbles, and winding up with $M$ out of those $M$ marbles being white. This is known as hypergeometric distribution.

The assumption that each remaining state is equally likely is to say that for a single draw from a population of $S$ success and $F$ failure states, the probability of success is $\frac{S}{S+F}$.

Now my problem: assume that the success states have a weight $W$, which can be either larger of smaller than $1$. In a population of $S$ success and $F$ failure states, the probability of drawing a success state is $\frac{WS}{WS + F}$. In the urn model, this can mean that one color of marbles is easier to get a grip on than the other.

I can certainly manage to find a probability mass function, but my expression gets a rather complex sum in the denominator.

Does this distribution have a name that can help me find known properties for it? My primary interest is finding an efficient algorithm for computing the probability mass function and the cumulative distribution. So, for instance, I would be interested in a mass function expression that is as simple as possible. Approximations for different parameter settings are also of interest.

-

The non-central hypergeometric distributions fit the bill exactly.

However, the subtlety is that with the weighted drawing is that the distribution of white balls in the sample depends on the sampling policy, unlike with the hypergeometric distribution. If $k$ balls are sampled one after another one in a competitive manner, this is known as the Wallenius' non-central hypergeometric distribution, whereas if $k$ balls as sampled independently, i.e. at once, the distribution is known as the Fisher's non-central hypergeometric distribution.

Of course, both distributions agree for $k=1$. Say, using Mathematica:

In[236]:= With[{s = 3, f = 5},
PDF[FisherHypergeometricDistribution[1, s, s + f, w], 1]]

Out[236]= (3 w)/(5 + 3 w)

In[238]:=
With[{s = 3, f = 5},
PDF[WalleniusHypergeometricDistribution[1, s, s + f, w],
1]] // Simplify

Out[238]= (3 w)/(5 + 3 w)

-
Thanks a lot! I am actually not sure whether Wallenius or Fisher applies in my case. Have to give this some thought. –  Jesper Larsson May 8 '12 at 9:19


text:
WIKISKY.ORG
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# NGC 525

Contents

### Images

DSS Images   Other Images

### Related articles

 Properties of isolated disk galaxiesWe present a new sample of northern isolated galaxies, which are definedby the physical criterion that they were not affected by other galaxiesin their evolution during the last few Gyr. To find them we used thelogarithmic ratio, f, between inner and tidal forces acting upon thecandidate galaxy by a possible perturber. The analysis of thedistribution of the f-values for the galaxies in the Coma cluster leadus to adopt the criterion f ≤ -4.5 for isolated galaxies. Thecandidates were chosen from the CfA catalog of galaxies within thevolume defined by cz ≤5000 km s-1, galactic latitudehigher than 40o and declination ≥-2.5o. Theselection of the sample, based on redshift values (when available),magnitudes and sizes of the candidate galaxies and possible perturberspresent in the same field is discussed. The final list of selectedisolated galaxies includes 203 objects from the initial 1706. The listcontains only truly isolated galaxies in the sense defined, but it is byno means complete, since all the galaxies with possible companions underthe f-criterion but with unknown redshift were discarded. We alsoselected a sample of perturbed galaxies comprised of all the diskgalaxies from the initial list with companions (with known redshift)satisfying f ≥ -2 and \Delta(cz) ≤500 km s-1; a totalof 130 objects. The statistical comparison of both samples showssignificant differences in morphology, sizes, masses, luminosities andcolor indices. Confirming previous results, we found that late spiral,Sc-type galaxies are, in particular, more frequent among isolatedgalaxies, whereas Lenticular galaxies are more abundant among perturbedgalaxies. Isolated systems appear to be smaller, less luminous and bluerthan interacting objects. We also found that bars are twice as frequentamong perturbed galaxies compared to isolated galaxies, in particularfor early Spirals and Lenticulars. The perturbed galaxies have higherLFIR/LB and Mmol/LB ratios,but the atomic gas content is similar for the two samples. The analysisof the luminosity-size and mass-luminosity relations shows similartrends for both families, the main difference being the almost totalabsence of big, bright and massive galaxies among the family of isolatedsystems, together with the almost total absence of small, faint and lowmass galaxies among the perturbed systems. All these aspects indicatethat the evolution induced by interactions with neighbors would proceedfrom late, small, faint and low mass Spirals to earlier, bigger, moreluminous and more massive spiral and lenticular galaxies, producing atthe same time a larger fraction of barred galaxies but preserving thesame relations between global parameters. The properties we found forour sample of isolated galaxies appear similar to those of high redshiftgalaxies, suggesting that the present-day isolated galaxies could bequietly evolved, unused building blocks surviving in low densityenvironments.Tables \ref{t1} and \ref{t2} are only available in electronic form athttp://www.edpsciences.org Redshift-Distance Survey of Early-Type Galaxies: Spectroscopic DataWe present central velocity dispersions and Mg2 line indicesfor an all-sky sample of ~1178 elliptical and S0 galaxies, of which 984had no previous measures. This sample contains the largest set ofhomogeneous spectroscopic data for a uniform sample of ellipticalgalaxies in the nearby universe. These galaxies were observed as part ofthe ENEAR project, designed to study the peculiar motions and internalproperties of the local early-type galaxies. Using 523 repeatedobservations of 317 galaxies obtained during different runs, the dataare brought to a common zero point. These multiple observations, takenduring the many runs and different instrumental setups employed for thisproject, are used to derive statistical corrections to the data and arefound to be relatively small, typically <~5% of the velocitydispersion and 0.01 mag in the Mg2 line strength. Typicalerrors are about 8% in velocity dispersion and 0.01 mag inMg2, in good agreement with values published elsewhere. Star Formation Histories of Early-Type Galaxies. I. Higher Order Balmer Lines as Age IndicatorsWe have obtained blue integrated spectra of 175 nearby early-typegalaxies, covering a wide range in galaxy velocity dispersion andemphasizing those with σ<100 km s-1. Galaxies havebeen observed both in the Virgo Cluster and in lower densityenvironments. The main goals are the evaluation of higher order Balmerlines as age indicators and differences in stellar populations as afunction of mass, environment, and morphology. In this first paper, ouremphasis is on presenting the methods used to characterize the behaviorof the Balmer lines through evolutionary population synthesis models.Lower σ galaxies exhibit a substantially greater intrinsicscatter, in a variety of line-strength indicators, than do higherσ galaxies, with the large intrinsic scatter setting in below aσ of 100 km s-1. Moreover, a greater contrast inscatter is present in the Balmer lines than in the lines of metalfeatures. Evolutionary synthesis modeling of the observed spectralindexes indicates that the strong Balmer lines found primarily among thelow-σ galaxies are caused by young age, rather than by lowmetallicity. Thus we find a trend between the population age and thecentral velocity dispersion, such that low-σ galaxies have youngerluminosity-weighted mean ages. We have repeated this analysis usingseveral different Balmer lines and find consistent results from onespectral indicator to another. A new catalogue of ISM content of normal galaxiesWe have compiled a catalogue of the gas content for a sample of 1916galaxies, considered to be a fair representation of normality''. Thedefinition of a normal'' galaxy adopted in this work implies that wehave purposely excluded from the catalogue galaxies having distortedmorphology (such as interaction bridges, tails or lopsidedness) and/orany signature of peculiar kinematics (such as polar rings,counterrotating disks or other decoupled components). In contrast, wehave included systems hosting active galactic nuclei (AGN) in thecatalogue. This catalogue revises previous compendia on the ISM contentof galaxies published by \citet{bregman} and \citet{casoli}, andcompiles data available in the literature from several small samples ofgalaxies. Masses for warm dust, atomic and molecular gas, as well asX-ray luminosities have been converted to a uniform distance scale takenfrom the Catalogue of Principal Galaxies (PGC). We have used twodifferent normalization factors to explore the variation of the gascontent along the Hubble sequence: the blue luminosity (LB)and the square of linear diameter (D225). Ourcatalogue significantly improves the statistics of previous referencecatalogues and can be used in future studies to define a template ISMcontent for normal'' galaxies along the Hubble sequence. The cataloguecan be accessed on-line and is also available at the Centre desDonnées Stellaires (CDS).The catalogue is available in electronic form athttp://dipastro.pd.astro.it/galletta/ismcat and at the CDS via anonymousftp to\ cdsarc.u-strasbg.fr (130.79.128.5) or via\http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/405/5 Supernovae studied by the M-1 group in 2000.Not Available Arcsecond Positions of UGC GalaxiesWe present accurate B1950 and J2000 positions for all confirmed galaxiesin the Uppsala General Catalog (UGC). The positions were measuredvisually from Digitized Sky Survey images with rms uncertaintiesσ<=[(1.2")2+(θ/100)2]1/2,where θ is the major-axis diameter. We compared each galaxymeasured with the original UGC description to ensure high reliability.The full position list is available in the electronic version only. Groups of galaxies. III. Some empirical characteristics.Not Available An Einstein X-Ray Survey of Optically Selected Galaxies. I. DataWe present the results of a complete Einstein imaging proportionalcounter X-ray survey of optically selected galaxies from theShapley-Ames Catalog, the Uppsala General Catalogue, and the EuropeanSouthern Observatory Catalog. Well-defined optical criteria are used toselect the galaxies, and X-ray fluxes are measured at the opticallydefined positions. The result is a comprehensive list of X-ray detectionand upper limit measurements for 1018 galaxies. Of these, 827 haveeither independent distance estimates or radial velocities. Associatedoptical, redshift, and distance data have been assembled for thesegalaxies, and their distances come from a combination of directlypredicted distances and those predicted from the Faber-Burstein GreatAttractor/Virgocentric infall model. The accuracy of the X-ray fluxeshas been checked in three different ways; all are consistent with thederived X-ray fluxes being of <=0.1 dex accuracy. In particular,there is agreement with previously published X-ray fluxes for galaxiesin common with a 1991 study by Roberts et al. and a 1992 study byFabbiano et al. The data presented here will be used in further studiesto characterize the X-ray output of galaxies of various morphologicaltypes and thus to enable the determination of the major sourcescontributing to the X-ray emission from galaxies. An image database. II. Catalogue between δ=-30deg and δ=70deg.A preliminary list of 68.040 galaxies was built from extraction of35.841 digitized images of the Palomar Sky Survey (Paper I). For eachgalaxy, the basic parameters are obtained: coordinates, diameter, axisratio, total magnitude, position angle. On this preliminary list, weapply severe selection rules to get a catalog of 28.000 galaxies, wellidentified and well documented. For each parameter, a comparison is madewith standard measurements. The accuracy of the raw photometricparameters is quite good despite of the simplicity of the method.Without any local correction, the standard error on the total magnitudeis about 0.5 magnitude up to a total magnitude of B_T_=17. Significantsecondary effects are detected concerning the magnitudes: distance toplate center effect and air-mass effect. A revised catalog of CfA1 galaxy groups in the Virgo/Great Attractor flow fieldA new identification of groups and clusters in the CfA1 Catalog ofHuchra et al. is presented, using a percolation algorithm to identifydensity enhancements. It is shown that in the resulting catalog,contamination by interlopers is significantly reduced. The Schechterluminosity function is redetermined, including the Malmquist bias. Nearby Groups of Galaxies in the Pisces / Cetus Area - Part One - the NGC524 GroupNot Available On the relationship between radio emission and optical properties in early-type galaxiesTo study the origin of radio activity in early-type galaxies, thepossible dependence of their radio emission on basic optical parameters,such as the absolute magnitude, the central velocity dispersion sigma,and the mean surface brightness mu is explored. A sample of 743 E and SOgalaxies is used which is based on three independent radio surveys ofoptically selected galaxies with virtually complete information onmagnitudes, morphological types, redshift distances, diameters, andradio fluxes. For both E and SO galaxies, only the absolute magnitudeappears to be directly related to the radio activity, while sigma and mudo not. Also, a significant dependence of the apparent flattening onradio power is confirmed for E galaxies. Some relevant implications ofthese results are discussed. A search for environmental effects on the optical properties of galaxies in groupsEnvironmental density-related modifications of basic optical properties(luminosities, sizes, axial ratios, and colors) of galaxies belonging toGeller and Huchra's (1983) groups have been investigated. Remarkably, itis found that the broad maxima of the distributions of luminosities anddiameters of spirals and the whole corresponding distributions oflenticulars tend to move to lower values as one goes to groups of highcompactness, whereas the luminosity-diameter relationship of spiralstends to become flatter. No color and axial ratio differences betweengalaxies of high- and low-compactness groups have been detected. A survey of galaxy redshifts. IV - The dataThe complete list of the best available radial velocities for the 2401galaxies in the merged Zwicky-Nilson catalog brighter than 14.5mz and with b (II) above +40 deg or below -30 deg ispresented. Almost 60 percent of the redshifts are from the CfA surveyand are accurate to typically 35 km/s. Groups of galaxies. III - The CfA surveyA statistically homogeneous group catalog (CfA) is based on the CfAredshift survey (Huchra et al.) is presented. Groups in the catalog areall density enhancements in redshift space of a factor greater than 20.Group members are identified according to the procedure described in aprevious study (Huchra and Geller) of a shallower whole-sky sample. Allgroups contain at least three members. Of the 176 groups in the CfAcatalog, 102 have been identified in one or more previous studies.Because the utilized algorithm searches for volume rather than surfacedensity enhancements, the groups in a given region generally change onlythrough the addition of fainter members when the magnitude limit of thegalaxy catalog increases. In the region of overlap, agreement betweenthe shallow catalog and the CfA catalog is excellent.
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text:
Physics Help Forum Help me understand correct interpretation of Gauss' Magnetic Law

 Aug 16th 2014, 12:06 AM #1 Senior Member   Join Date: Nov 2013 Location: New Zealand Posts: 550 Help me understand correct interpretation of Gauss' Magnetic Law I have been looking at http://www.maxwells-equations.com and they had a different interpretation to what I expected to Gauss' magnetic law. $\triangledown&space;\cdot&space;\mathbf{B}&space;=&space;0$ I know this is the divergence operator but for the case of a bar magnet inside a sphere (completely) I imagined that it was the number of magnetic force lines leaving one pole and re-entering the other are always equal. However, their explanation is simply that there are no magnetic monopoles. What is the correct interpretation of this law and is that all there is to Gauss' magnetic law, that there exist no monopoles?
 Aug 16th 2014, 06:00 AM #2 Physics Team     Join Date: Jun 2010 Location: Morristown, NJ USA Posts: 2,344 The origination of Gauss's law for magnetic fields is that there are no magnetic monopoles. The form you cited: $\triangledown&space;\cdot&space;\mathbf{B}&space;=&space;0$ expresses that. The divergence operator tells us the magnitude of "stuff" flowing away from, or into, a point - I like to think of it as the magnitude of flow away from or into a source or sink. For example water flowing out of a tap has positive divergence, while water flowing into a drain has negative divergence. If the divergence of the magnetic field is zero, that means that the flow of magnetic field away from every point must equal the flow of magnetic field back into the point. Thus there can be no source of sink of magnetic flux in isolation. This is what is meant by saying that there are no magnetic monopoles. This is different than the case of electric field lines - if you have an electric charge by itself then it can be the source (or sink) or electric field lines, and the divergence of the electric field is equal to the charge density. Regarding the bar magnet example - the sphere that you described does not need to enclose the entire magnet. If you envision a smaller sphere around just the north pole, it would experience magnetic field line flowing from the north pole out through the surface of the sphere to the south pole looping external to the bar magnet. But there is also an equal number of field lines flowing from the south pole back through the bar magnet itself to the north pole. Thus the total flux through the surface of the sphere is always 0.
 Aug 16th 2014, 10:17 PM #3 Senior Member   Join Date: Nov 2013 Location: New Zealand Posts: 550 Ok, I thought the purpose of the sphere was that it completely enclosed the bar magnet. Are you thinking that the magnetic field lines continue in a closed loop through the centre of the bar magnet? I am not quite getting why the divergence should be zero in such a case if this is not so. Last edited by kiwiheretic; Aug 16th 2014 at 11:16 PM. Reason: I mean "field lines", not "force lines"
Aug 17th 2014, 06:32 AM   #4
Physics Team

Join Date: Jun 2010
Location: Morristown, NJ USA
Posts: 2,344
 Originally Posted by kiwiheretic Are you thinking that the magnetic field lines continue in a closed loop through the centre of the bar magnet? I am not quite getting why the divergence should be zero in such a case if this is not so.
Yes, that's right - the field lines form a complete, continuous loop by returning to the north pole through the interior of the magnet itself. The fact that divergence = 0 means the sphere can be infinitesimally small and would still not be a net source or sink of magnetic field lines.

Aug 18th 2014, 08:21 AM   #5
Senior Member

Join Date: May 2014
Location: Poole, UK
Posts: 132
 Originally Posted by kiwiheretic ...What is the correct interpretation of this law and is that all there is to Gauss' magnetic law, that there exist no monopoles?
Chip has given the "official" answer, and IMHO expressed it well.

But when you dig around, there are some problems with it. There is no actual source or sink. Nothing is actually flowing out of the magnet's North pole and back into the South pole. In John Jackson’s authoritative textbook Classical Electrodynamics you have to wait until section 11.10 before he says "one should properly speak of the electromagnetic field Fμv rather than E or B separately". The thing is this: an electron doesn't have an electric field, it doesn't have a magnetic field, it has an electromagnetic field. If you are at rest with respect to an electron, it seems to have an electric field. But when you move, it seems to have a magnetic field too. You didn't create the latter just by moving, you just saw a different aspect of the electron's electromagnetic field. And whilst that electron is a charged particle and we talk of electric charge, it's actually electromagnetic charge. So there is no such thing as "magnetic charge".

 Aug 18th 2014, 08:46 AM #6 Physics Team     Join Date: Jun 2010 Location: Morristown, NJ USA Posts: 2,344 Indeed there is nothing "flowing" in a magnetic (or electromagnetic) field - I use that analogy as it makes the discussion about divergence of the field a bit easier to visualize and understand. Similarly it is common to draw lines closer together to represent a stronger field, which vaguely implies that a stronger field has more field lines, but that's not strictly correct either. Yet it's a useful depiction because it's a bit more intuitive - you can see that the number of field lines entering the sphere in question is equal to the number of field lines exiting, and hence Div B is zero. But I agree that it should be pointed out that these are just convenient short cuts.
 Aug 18th 2014, 09:30 AM #7 Senior Member   Join Date: May 2014 Location: Poole, UK Posts: 132 Sorry Chip, I didn't mean to be critical of you, I thought you gave the stock explanation very well. I guess I just don't like the way electromagnetism is taught.
 Aug 18th 2014, 09:36 PM #8 Senior Member   Join Date: Nov 2013 Location: New Zealand Posts: 550 Those are some interesting points. So are we saying that the magnetic force is a "fictional" force? I understand that a static electric charge has field lines but no magnetic field. However a uniformly moving charge has both but to which observer is the charge moving? Even more confusing if one has two parallel wires with electric current flowing in opposite direction in the wires. The wires are drawn together by the interaction of the magnetic fields between them. However if we only considers the elctric fields the charges should repel as like charges repel each other. Also when only considering the electric field, say from a single electron, we can place a point charge in the vicinity and note the force on the second electron (presumably by its direction and acceleration away from the first electron). However, what kind of "magnetic charge" can we use to test the strength of a magnetic field? As far as I know there is no magnetic point charge. If there was you could place a magnetic point charge at one pole of a bar magnet and watch it "flow" from one pole to the other along a magnetic field line. Thoughts?
Aug 19th 2014, 05:03 AM   #9
Physics Team

Join Date: Jun 2010
Location: Morristown, NJ USA
Posts: 2,344
 Originally Posted by kiwiheretic Those are some interesting points. So are we saying that the magnetic force is a "fictional" force?
No, not at all - magnetic forces are real. But the concept of field lines is a mathematical construct - simply a model that helps us do calculations, but not necessarily real. In the quantum mechanical view of electromagnetism there are no fields at all, just particles interacting.

 Originally Posted by kiwiheretic I understand that a static electric charge has field lines but no magnetic field. However a uniformly moving charge has both but to which observer is the charge moving?
Interesting point. To an observer traveling with the charge it is stationary, so no magnetic field. But to an observer in motion relative to the electric charge there IS a magnetic field.

 Originally Posted by kiwiheretic Even more confusing if one has two parallel wires with electric current flowing in opposite direction in the wires. The wires are drawn together by the interaction of the magnetic fields between them. However if we only considers the elctric fields the charges should repel as like charges repel each other.
Wires don't normally have a charge - they have moving electrons, but the charge on the wire at all points is zero, so there is no electrostatic force at play between the wires.

 Originally Posted by kiwiheretic However, what kind of "magnetic charge" can we use to test the strength of a magnetic field? As far as I know there is no magnetic point charge.
The strength of a magnetic field can be tested either in several ways. A couple of possibilities: (a) seeing how much torque is created on a standard bar magnet, (b) passing a current through a wire and seeing how much force is created in that wire, or (c) shooting an electron through the field and seeing how much it is deflected.

Last edited by ChipB; Aug 19th 2014 at 05:43 AM.

 Aug 19th 2014, 05:11 AM #10 Senior Member     Join Date: Apr 2008 Location: Bedford, England Posts: 668 There is some debate over the existance, or otherwise, of magnetic monopoles (the magnetic equivalent of a point charge). Reading the other posts, I get the impression that magnetism is actually as much product of the electron as electricity. If the electron is moving (relative to the measuring equipment) then a magnetic effect will be observed. This seems to me to indicate that the electron is as much the magnetic point charge as it is the electrical point charge. In my imagination, the magnetic monopole would just be an electron, seen from an alternative perspective. In you question about the parallel wires, the effect of all the protons in the wire will cancel out the electric repulsion effect leaving just the magnetic effect. However, the experiment could be re-stated to use two parallel electron beams fired in oposite directions (in a vacuum) I don't know what the result would be, perhaps someone else does... Last edited by MBW; Aug 19th 2014 at 05:13 AM.

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• # question_answer Saloni took a piece of burning charcoal and collected the gas evolved in a test tube.                                  (i) How will she find the nature of the gas? (ii) Write down word equations of all reactions taking place in this process.

 (i) When charcoal is burnt, carbon dioxide gas is produced. This gas turns lime water milky. The nature of the gas can be tested by using moist red and blue litmus paper. No effect on red litmus but the gas turns blue litmus to red, so it is acidic in nature. (ii) Equations of all the reactions are: \begin{align} & \,\,\,\,\,\,\,C\,\,\,\,\,\,+\,\,\,\,\,\,\,\,\,{{O}_{2}}\,\,\,\,\,\,\to \,\,\,C{{O}_{2}} \\ & \operatorname{Carbon}\,\,\,\,\,Oxygen\,\,\,\,\,\,\,Carbon\,dioxide \\ \end{align} $\operatorname{Carbon}\,dixide\,\,+\,\,Lime\,water\,\,\to \,\,Milky$ \begin{align} & \operatorname{Carbon}\,dixide\,\,+\,\,\operatorname{Blue}\,Litmus\,\,\to \,\,\operatorname{Red} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(moist) \\ \end{align}


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linalg(deprecated)/ihermite - Help

linalg(deprecated)

 ihermite
 integer-only Hermite Normal Form

 Calling Sequence ihermite(A) ihermite(A, U)

Parameters

 A - rectangular matrix of integers U - name

Description

 • Important: The linalg package has been deprecated. Use the superseding packages LinearAlgebra[HermiteForm], instead.
 - For information on migrating linalg code to the new packages, see examples/LinearAlgebraMigration.
 • The function ihermite computes the Hermite Normal Form (reduced row echelon form) of a rectangular matrix of integers.
 • The Hermite normal form of A is an upper triangular matrix H with rank(A) = the number of nonzero rows of H.  If A is an n by n matrix of full rank then $\prod _{i],i=1\mathrm{..}n}H[i$.
 • This is not an efficient method for computing the rank or determinant except that this may yield a partial factorization of $\mathrm{det}\left(A\right)$ without doing any explicit factorizations.
 • The Hermite normal form is obtained by doing elementary row operations. This includes interchanging rows, multiplying through a row by -1, and adding an integral multiple of one row to another.
 • One can use transposes to obtain column form of the Hermite Normal Form.
 • In the case of two arguments, the second argument U will be assigned the transformation matrix on output, such that the following holds: ihermite(A) = U A.
 • The command with(linalg,ihermite) allows the use of the abbreviated form of this command.

Examples

Important: The linalg package has been deprecated. Use the superseding packages, LinearAlgebra and VectorCalculus, instead.

 > $\mathrm{with}\left(\mathrm{linalg}\right):$
 > $H≔\mathrm{array}\left(\left[\left[9,-36,30\right],\left[-36,192,-180\right],\left[30,-180,180\right]\right]\right)$
 ${H}{≔}\left[\begin{array}{ccc}{9}& {-36}& {30}\\ {-36}& {192}& {-180}\\ {30}& {-180}& {180}\end{array}\right]$ (1)
 > $\mathrm{ihermite}\left(H\right)$
 $\left[\begin{array}{ccc}{3}& {0}& {30}\\ {0}& {12}& {0}\\ {0}& {0}& {60}\end{array}\right]$ (2)

To obtain column form of Hermite Normal Form

 > $\mathrm{transpose}\left(\mathrm{ihermite}\left(\mathrm{transpose}\left(H\right)\right)\right)$
 $\left[\begin{array}{ccc}{3}& {0}& {0}\\ {0}& {12}& {0}\\ {30}& {0}& {60}\end{array}\right]$ (3)
 > $A≔\mathrm{array}\left(\left[\left[3,7,11\right],\left[13,17,19\right]\right]\right):$
 > $B≔\mathrm{ihermite}\left(A,U\right)$
 ${B}{≔}\left[\begin{array}{ccc}{1}& {29}& {61}\\ {0}& {40}& {86}\end{array}\right]$ (4)
 > $\mathrm{eval}\left(U\right)$
 $\left[\begin{array}{cc}{9}& {-2}\\ {13}& {-3}\end{array}\right]$ (5)
 > $\mathrm{evalm}\left(U\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&*\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}A-B\right)$
 $\left[\begin{array}{ccc}{0}& {0}& {0}\\ {0}& {0}& {0}\end{array}\right]$ (6)


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# (a) Let A and B be square matrices that are unitarily equivalent. Prove that ||A||= ||B||.(b) Let T be a linear operator on a finite-dimensional inner product space V. Define Prove that ||T||= || [T]β||, where β is any orthonormal basis for V.(c) Let V be an infinite-dimensional inner product space with an orthonormal basis {v1, v2, . . .}. Let T be the linear operator on V such that T(vk) = kvk. Prove that ||T|| (defined in (b)) does not exist.

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(a) Let A and B be square matrices that are unitarily equivalent. Prove that ||A||= ||B||.

(b) Let T be a linear operator on a finite-dimensional inner product space V. Define Prove that ||T||= || [T]β||, where β is any orthonormal basis for V.

(c) Let V be an infinite-dimensional inner product space with an orthonormal basis {v1, v2, . . .}. Let T be the linear operator on V such that T(vk) = kvk. Prove that ||T|| (defined in (b)) does not exist.

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# Tezos : Recent Questions and Answers

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## Appendix: Frequencies Representable by a Geometric Sequence

Consider , with . Then we can write in polar form as

where , , and are real numbers.

Forming a geometric sequence based on yields the sequence

where . Thus, successive integer powers of produce a sampled complex sinusoid with unit amplitude, and zero phase. Defining the sampling interval as in seconds provides that is the radian frequency in radians per second.

A natural question to investigate is what frequencies are possible. The angular step of the point along the unit circle in the complex plane is . Since , an angular step is indistinguishable from the angular step . Therefore, we must restrict the angular step to a length range in order to avoid ambiguity.

Recall from §4.3.3 that we need support for both positive and negative frequencies since equal magnitudes of each must be summed to produce real sinusoids, as indicated by the identities

The length range which is symmetric about zero is

which, since , corresponds to the frequency range

However, there is a problem with the point at : Both and correspond to the same point in the -plane. Technically, this can be viewed as aliasing of the frequency on top of , or vice versa. The practical impact is that it is not possible in general to reconstruct a sinusoid from its samples at this frequency. For an obvious example, consider the sinusoid at half the sampling-rate sampled on its zero-crossings: . We cannot be expected to reconstruct a nonzero signal from a sequence of zeros! For the signal , on the other hand, we sample the positive and negative peaks, and everything looks fine. In general, we either do not know the amplitude, or we do not know phase of a sinusoid sampled at exactly twice its frequency, and if we hit the zero crossings, we lose it altogether.

In view of the foregoing, we may define the valid range of digital frequencies'' to be

While one might have expected the open interval , we are free to give the point either the largest positive or largest negative representable frequency. Here, we chose the largest negative frequency since it corresponds to the assignment of numbers in two's complement arithmetic, which is often used to implement phase registers in sinusoidal oscillators. Since there is no corresponding positive-frequency component, samples at must be interpreted as samples of clockwise circular motion around the unit circle at two points per revolution. Such signals appear as an alternating sequence of the form , where can be complex. The amplitude at is then defined as , and the phase is .

We have seen that uniformly spaced samples can represent frequencies only in the range , where denotes the sampling rate. Frequencies outside this range yield sampled sinusoids indistinguishable from frequencies inside the range.

Suppose we henceforth agree to sample at higher than twice the highest frequency in our continuous-time signal. This is normally ensured in practice by lowpass-filtering the input signal to remove all signal energy at and above. Such a filter is called an anti-aliasing filter, and it is a standard first stage in an Analog-to-Digital (A/D) Converter (ADC). Nowadays, ADCs are normally implemented within a single integrated circuit chip, such as a CODEC (for coder/decoder'') or multimedia chip''.

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Informal Derivation of Taylor Series
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Sampling Theorem


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 Mat. Zametki, 1977, Volume 22, Issue 1, Pages 13–21 (Mi mz8020)

Generalized theorems of Liénard and Shepherd

G. F. Korsakov

Abstract: The paper considers a real polynomial $p(x)=a_0+a_1x+…+a_nx^n$ ($a_0>0$) for which there hold inequalities $\Delta_1>0, \Delta_3>0,…$ or $\Delta_2>0, \Delta_4>0$, where $\Delta_1,\Delta_2,…,\Delta_n$ are the Hurwitz determinants for polynomial $p(x)$. It is proven that polynomial $p(x)$ can have, in the right half-plane, only real roots, where the quantity of positive roots of polynomial $p(x)$ equals the quantity of changes of sign in the system of coefficients $a_0,a_2,…,a_n$, when $n$ is even, and $a_0,a_2,…,a_{n-1},a_n$, when $n$ is odd. From the proven theorem, in particular, there follows the Liénard and Shepherd criterion of stability.

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English version:
Mathematical Notes, 1977, 22:1, 498–503

Bibliographic databases:

UDC: 512

Citation: G. F. Korsakov, “Generalized theorems of Liénard and Shepherd”, Mat. Zametki, 22:1 (1977), 13–21; Math. Notes, 22:1 (1977), 498–503

Citation in format AMSBIB
\Bibitem{Kor77} \by G.~F.~Korsakov \paper Generalized theorems of Li\'enard and Shepherd \jour Mat. Zametki \yr 1977 \vol 22 \issue 1 \pages 13--21 \mathnet{http://mi.mathnet.ru/mz8020} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=476098} \zmath{https://zbmath.org/?q=an:0442.26009} \transl \jour Math. Notes \yr 1977 \vol 22 \issue 1 \pages 498--503 \crossref{https://doi.org/10.1007/BF01147688}