text:
prefix:Centi

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'centi' is a decimal prefix for expressing a value with a scaling of $$10^{-2}$$.

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1.0E-2
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'centi' is a decimal prefix for expressing a value with a scaling of $$10^{-2}$$.
Centi(en)

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# Calculating Torque

## Main Question or Discussion Point

I am trying to calculate how much torque will be exerted on a half shaft on the rear axel of a car. I am a little confused as the power of the engine is put through a chain-drive and then turned by 90 degrees through the differential before it reaches the half shafts.

Do you just assume the power is inline (parallel) with the axel and if so what information do you need to know? I will assume that the half shafts are only taking the torque of the engine and are not supporting any weight.

I am a bit lost about where to start on this and need some help!

Thanks

Cheerfulbloke

Last edited:

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Mech_Engineer
Gold Member

The torque at the flywheel of the engine will go through the transmission (which depending on the gear, will be multiplied by the gear's ratio) and then the differential (which also has an associated gear ratio.) Also keep in mind the differential divides the torque by two, equal parts for each wheel.

By the way, you mention a chain-drive in your driveline, is this "car" (truck) 4WD? Are you given operating efficiencies for each part of the driveline, or the driveline as a whole, or are you supposed to assume no losses?

So, for an example, not taking into account efficiencies:

Assume 350lb-ft at the flywheel of the engine (475 N-m).

1st gear's ratio is 3.643:1, so after the transmission, the torque is multiplied to 1275 lb-ft (1730 N-m).

The differential's gear ratio is 4.300:1, so the differential multiplies the total torque to 5482 lb-ft (7440 N-m), but this is then divided by 2 since the torque is split equally (assumption) between the two tires. So, each tire (assuming it's 2WD) can get a maximum of about 2741 lb-ft (3720 N-m) of torque.

From this number you can calculate the maximum instantaneous force applied to the road as well, given the tire's diameter (assuming no-slip).

Last edited:
it is actually a 2WD car-it is using a motorbike engine to drive the rear wheels using the chain.

Thank you for your help-it all seems so simple now...I wasn't sure if there were other conversions to do because of the change in direction on the force, but what you've said seems to make sense.

Thanks again for the help. I am one happy chappie!

I have looked at several publications regarding calculating the final drive wheel torque and none of them mention dividing the diff torque by two to get the final drive wheel toruqe?

The formula they give
drive wheel torque = flywheel torque x first gear ratio x final drive ratio x 0.85

So I'm a little confussed?

brewnog


text: Preview; Assign Practice; Preview. $2{\text{NH}}_{3}\left(g\right)\rightleftharpoons{\text{N}}_{2}\left(g\right)+3{\text{H}}_{2}\left(g\right)\Delta H=92\text{kJ}$, ${\text{N}}_{2}\left(g\right)+{\text{O}}_{2}\left(g\right)\rightleftharpoons2\text{NO}\left(g\right)\Delta H=181\text{kJ}$, $2{\text{O}}_{3}\left(g\right)\rightleftharpoons3{\text{O}}_{2}\left(g\right)\Delta H=-285\text{kJ}$, $\text{CaO}\left(s\right)+{\text{CO}}_{2}\left(g\right)\rightleftharpoons{\text{CaCO}}_{3}\left(s\right)\Delta H=-176\text{kJ}$, $2{\text{H}}_{2}\text{O}\left(g\right)\rightleftharpoons2{\text{H}}_{2}\left(g\right)+{\text{O}}_{2}\left(g\right)\Delta H=484\text{kJ}$, ${\text{N}}_{2}\left(g\right)+3{\text{H}}_{2}\left(g\right)\rightleftharpoons2{\text{NH}}_{3}\left(g\right)\Delta H=-92.2\text{kJ}$, $2\text{Br}\left(g\right)\rightleftharpoons{\text{Br}}_{2}\left(g\right)\Delta H=-224\text{kJ}$, ${\text{H}}_{2}\left(g\right)+{\text{I}}_{2}\left(s\right)\rightleftharpoons2\text{HI}\left(g\right)\Delta H=53\text{kJ}$. In (b), the addition of HCl causes a reaction with NH3 to form more ${\text{NH}}_{4}{}^{+}$ by removing OH− as it reacts with the acid to form water. If we add additional product to a system, the equilibrium will shift to the left, in order to produce more reactants. Ammonia is a weak base that reacts with water according to this equation: ${\text{NH}}_{3}\left(aq\right)+{\text{H}}_{2}\text{O}\left(l\right)\rightleftharpoons{\text{NH}}_{4}{}^{+}\left(aq\right)+{\text{OH}}^{-}\left(aq\right)$, Acetic acid is a weak acid that reacts with water according to this equation: ${\text{CH}}_{3}{\text{CO}}_{2}\text{H}\left(aq\right)+{\text{H}}_{2}\text{O}\left(aq\right)\rightleftharpoons{\text{H}}_{3}{\text{O}}^{+}\left(aq\right)+{\text{CH}}_{3}{\text{CO}}_{2}{}^{-}\left(aq\right)$, Suggest two ways in which the equilibrium concentration of Ag, How can the pressure of water vapor be increased in the following equilibrium? Or, if we remove reactants from the system, equilibrium will … 1. Le Chatelier's principle. The principle is named after French chemist Henry Louis Le Chatelier, and sometimes also credited to Karl Ferdinand Braun, who discovered it independently. Thus, increasing the temperature to increase the rate lowers the yield. The concentration of colorless N2O4 increases, and the concentration of brown NO2 decreases, causing the brown color to fade. What will happen to the concentration of each reactant and product at equilibrium if more C is added? In order to get as much ammonia as possible in the equilibrium mixture, you need as low a temperature as possible. Does the equilibrium constant for the reaction increase, decrease, or remain about the same as the temperature increases? Le Chatelier’s principle states that if a dynamic equilibrium is disturbed by changing the conditions (such as concentration, temperature and pressure changes) , the position of equilibrium shifts to counteract the change to reestablish an equilibrium. It is helpful in predicting the effect of a change in conditions on the chemical equilibrium. If a chemical reaction is at equilibrium and experiences a change in pressure, temperature, or concentration of products or reactants, the equilibrium shifts in the opposite direction to offset the change. Commercial production of ammonia requires heavy equipment to handle the high temperatures and pressures required. However, if we have a mixture of reactants and products that have not yet reached equilibrium, the changes necessary to reach equilibrium may not be so obvious. 2489 views As described in the previous paragraph, the disturbance causes a change in $$Q$$; the reaction will shift to re-establish $$Q = K$$. By the same logic, reducing the concentration of any product will also shift equilibrium to the right. (credit: modification of work by Mark Ott). It states that “If an external stress is applied to a reacting system at equilibrium, the system will adjust itself in such a way that the effect of the stress is nullified”. Thus, addition of a gas not involved in the equilibrium will not perturb the equilibrium. Le Chatelier’s principle states that, if a system already in equilibrium is disturbed by change in conditions, such as temperature, pressure, or concentrations, the equilibrium will … During World War I, he played a major role in the development of poisonous gases used for trench warfare. However, if we have a mixture of reactants and products that have not yet reached equilibrium, the changes necessary to … Le Chatelier was born on 8 October 1850 in Paris and was the son of French materials engineer Louis Le Chatelier and Louise Durand. However, if we have a mixture of reactants and products that have not yet reached equilibrium, the changes necessary to reach equilibrium may not be so obvious. Since this stress affects the concentrations of the reactants and the products, the value of Q will no longer equal the value of K. To re-establish equilibrium, the system will either shift toward the products (if Q < K) or the reactants (if Q > K) until Q returns to the same value as K. This process is described by Le Châtelier’s principle: When a chemical system at equilibrium is disturbed, it returns to equilibrium by counteracting the disturbance. What will happen to the concentrations of N, Write the expression for the equilibrium constant for the reversible reaction. However, if we have a mixture of reactants and products that have not yet reached equilibrium, the changes necessary to reach equilibrium may not be so obvious. Temperature affects the equilibrium between NO2 and N2O4 in this reaction, ${\text{N}}_{2}{\text{O}}_{4}\left(g\right)\rightleftharpoons2{\text{NO}}_{2}\left(g\right)\Delta H=57.20\text{kJ}$, The positive ΔH value tells us that the reaction is endothermic and could be written, $\text{heat}+{\text{N}}_{2}{\text{O}}_{4}\left(g\right)\rightleftharpoons2{\text{NO}}_{2}\left(g\right)$. What will happen to the concentration of each reactant and product at equilibrium if the temperature of the system is increased? The Le Chatelier’s principle is of great importance in chemical industry because it can help to The formation of additional amounts of NO2 decreases the total number of molecules in the system, because each time two molecules of NO2 form, a total of three molecules of NO and O2 are consumed. And we could use Le Chatelier's principle to exploit these properties. This principle was named after Henry Louis Le Chatelier and Karl Ferdinand Braun. The stress on the system in Figure 1 is the reduction of the equilibrium concentration of SCN– (lowering the concentration of one of the reactants would cause Q to be larger than K). The principle is named after the French chemist Henry Louis Le Chatelier. How might Le Chateliers principle be useful in the chemical industry For from CHEM MISC at Ateneo de Davao University The reverse reaction would be favored by a decrease in pressure. We next address what happens when a system at equilibrium is disturbed so that Q is no longer equal to K. If a system at equilibrium is subjected to a perturbance or stress (such as a change in concentration) the position of equilibrium changes. How was Le Chatelier's principle used to determine the extinction coefficient? around the world. Each year, ammonia is among the top 10 chemicals, by mass, manufactured in the world. A chemical system at equilibrium can be temporarily shifted out of equilibrium by adding or removing one or more of the reactants or products. The reaction shifts to the left to relieve the stress, and there is an increase in the concentration of H2 and I2 and a reduction in the concentration of HI. At room temperature, for example, the reaction is so slow that if we prepared a mixture of N2 and H2, no detectable amount of ammonia would form during our lifetime. Note the double arrows. How will an increase in temperature affect each of the following equilibria? Early life. Define the optimum conditions for the chemical processes employed in industry; Reduce undesirable reversibility; Predict the effect of an altered factor on the equilibrium position of an untried reaction. ${\text{H}}_{2}\left(g\right)+{\text{I}}_{2}\left(g\right)\rightleftharpoons2\text{HI}\left(g\right)+\text{heat}$. In such a case, we can compare the values of Q and K for the system to predict the changes. Regarding his role in these developments, Haber said, “During peace time a scientist belongs to the World, but during war time he belongs to his country.”[1] Haber defended the use of gas warfare against accusations that it was inhumane, saying that death was death, by whatever means it was inflicted. To be practical, an industrial process must give a large yield of product relatively quickly. Le-Chatelier’s principle of equilibrium is used in the industrial applications as the reaction scheme involves parameters like temperature, pressure, concentration of reaction species a change in even single parameter results in the change of equilibrium leads to undesired product formation. As we learned during our study of kinetics, a catalyst can speed up the rate of a reaction. Le Chatelier’s principles, also known as the equilibrium law, are used to predict the effect of some changes on a system in chemical equilibrium (such as the change in temperature or pressure). of pressure. It can be stated as: When any system at equilibrium for a long period of time is … Suggest four ways in which the concentration of hydrazine, N, Suggest four ways in which the concentration of PH. ${\text{H}}_{2}\text{O}\left(l\right)\rightleftharpoons{\text{H}}_{2}\text{O}\left(g\right)\Delta H=41\text{kJ}$, Additional solid silver sulfate, a slightly soluble solid, is added to a solution of silver ion and sulfate ion at equilibrium with solid silver sulfate: $2{\text{Ag}}^{+}\left(aq\right)+{\text{SO}}_{4}{}^{2-}\left(aq\right)\rightleftharpoons{\text{Ag}}_{2}{\text{SO}}_{4}\left(s\right)$, Additional silver sulfate will form and precipitate from solution as Ag, ${K}_{c}=\frac{\left[{\text{CH}}_{3}\text{OH}\right]}{{\left[{\text{H}}_{2}\right]}^{2}\left[\text{CO}\right]}$, ${K}_{c}=\frac{\left[\text{CO}\right]\left[{\text{H}}_{2}\right]}{\left[{\text{H}}_{2}\text{O}\right]}$, Herrlich, P. “The Responsibility of the Scientist: What Can History Teach Us About How Scientists Should Handle Research That Has the Potential to Create Harm?”. He stands as an example of the ethical dilemmas that face scientists in times of war and the double-edged nature of the sword of science. Secondly, in order to introduce a catalyst, the temperature is increased to over 400 degrees Celsius. Le Châtelier’s principle can be used to predict changes in equilibrium concentrations when a system that is at equilibrium is subjected to a stress. How is Le Chatelier's Principle used in the Haber process? Ag+(aq) + SCN−(aq) ⇌ AgSCN(s). Industrial Applications. Industrial Applications. It is helpful in predicting the effect of a change in conditions on the chemical equilibrium. The exothermic nature of the reaction is to cause equilibrium to the equilibrium constant to... The fathers of chemical reactions do not result in a disturbance of the reaction quotient ( Q ) equal. Perturb the equilibrium constant ( K ) the volume of the increased.! 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text:
Domain of One Variable Function – A function with sum of ln’s – Exercise 2443

Exercise

Determine the domain of the function:

$$f(x)=\ln (x+2)+\ln (x-3)$$

$$x>3$$

Solution

Let’s find the domain of the function:

$$f(x)=\ln (x+2)+\ln (x-3)$$

Because there are ln’s, we need the expressions inside the ln’s to be greater than zero:

$$x+2>0\text{ and }x-3>0$$

We got two inequalities. We’ll arrange them:

$$x+2>0 \Longrightarrow x>-2$$

$$x-3>0 \Longrightarrow x>3$$

Intersect both inequalities:

$$x>-2 \text{ and } x>3 \Longrightarrow x>3$$

Hence, the domain of the function is

$$x>3$$

Have a question? Found a mistake? – Write a comment below!
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text:
# What is the domain and range of x = y^2 -9?

Nov 8, 2017

"D:" {x inRR | x >= ~9}.
$\text{R:} \left\{y \in \mathbb{R} | y \ge 0\right\}$.

#### Explanation:

Rather than just say the domain and range, I'll show you how I got the answer, step by step.

First off, let's isolate $y$.

$x = {y}^{2} - 9$

$x + 9 = {y}^{2}$

$\sqrt{x + 9} = y$

Now, we can identify the type of function.

Let's describe the transformations of the function before we go on to the domain and range.

$y = \sqrt{x + 9}$

• There is only a horizontal translation of $9$ units to the left.

Now that that is done with, let's graph the function, so it is easier to determine the domain and range. Graphing is not necessary, but it makes it much easier.

The easiest way to graph this function is to sub in values for $x$ and solve for $y$. Graph the variables you subbed and solved for.

graph{y=sqrt(x+9) [-10, 10, -5, 5]}

We can see that the domain can only be values that are equal to or greater than ~9, thus, the domain is {x inRR | x >= ~9}.

As for the range, it can only be values that are equal to or greater than $0$, thus, the range is $\left\{y \in \mathbb{R} | y \ge 0\right\}$.

Hope this helps :)


text:
I am reading the section about moving average models in Hyndman & Athanasopoulos Forecasting: principles and practice. I am trying to understand the MA(q) model in words.

What is white noise? Is this a differenced series which is normally distributed with mean zero? Is it the difference between an observation and the mean of all observations? I do not know what the book is talking about when it says "white noise".

I can understand what a differenced series is. I can understand what sum of square error means. But what is this "white noise" and where did it come from? What is an error term? What does it mean? Who made this up? Can I see an actual example that I can work out in Excel?

When forecasting with an MA(q) model, do you add the moving average series to the mean to get a forecast? How does it actually work? An Excel document or an example involving actual numbers would really help.

I am having a lot of difficulty understanding what is actually going on in the formula (reproduced below). Some examples with actual numbers would be great. $$y_t=c+e_t+θ_1e_{t−1}+θ_2e_{t−2}+⋯+θ_qe_{t−q}$$

## closed as too broad by gung♦, Nick Stauner, Scortchi♦, Peter Flom♦Jun 6 '14 at 10:43

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• Colors of noise: en.wikipedia.org/wiki/Colors_of_noise – Alexis Jun 5 '14 at 22:50
• There is an awful lot here, can you focus your question somewhat? What is "the formula" you are referring to? Can you list it? What is "the book" you are referring to? We'd be happy to help you, but I'm not sure this is answerable at the moment. – gung Jun 5 '14 at 22:57
• The text you cite defines white noise earlier: otexts.org/fpp/2/2 You just need to search in the same document. – Nick Cox Jun 5 '14 at 23:12
• There is an excellent reference book called Applied Time Series Analysis for the Social Sciences by McCleary and Hay or Forecasting with Univariate Box - Jenkins Models: Concepts and Cases by Pankratz that you can consult to learn more about what MA means. – forecaster Jun 6 '14 at 0:47
• I'd suggest if you're looking for intuition, first begin by trying to understand an MA(1). – Glen_b Jun 6 '14 at 2:38

It sounds like you are reading about statistical models. Such models include:

1. A deterministic part (i.e. something that looks like an algebraic relationship; e.g. a line like $y = a + bx$ is a deterministic relationship where $y$ is determined by a linear function of $x$); and

2. A random part (i.e. something, like noise, that is more or less unknowable... or only knowable in an aggregate sense, like a normal distribution, or some other distribution.). The random part may be called 'noise' or 'error' or something else, depending on the conventions of talking about statistics in a particular discipline. The difference between an observation and the mean of all observations (e.g. $X_{i} - \bar{X}$) is often termed error.

In a moving average($q$) model—e.g. $y_{t} = \mu + \varepsilon_{t} + \theta_{1}\varepsilon_{t-1} + \theta_{2}\varepsilon_{t-2} + \dots + \theta_{q}\varepsilon_{t-q}$—you are explaining $y$ as determined by some mean $\mu$ plus some amount of noise (i.e. a random quantity), plus some amount ($\theta_{1}$) of noise ($\varepsilon_{t-1}$) from last time ($t-1$), plus some (possibly different) amounts of noises to $t-q$ times ago.

I do not know the history of who made the MA(q) model up. Some jerk? Some awesome person? No idea.

I am not gonna post an excel spreadsheet, but it's not too hard to apply. Suppose the contribution of noise at time $t$ is inversely proportional to how long ago the noise happened. Then $\theta_{1} = 1$, $\theta_{2} = 1/2$, $\dots$, and $\theta_{q} = 1/q$, and the MA(q=3) is:

$y_{t} = \mu + \varepsilon_{t} + \varepsilon_{t-1} + \frac{1}{2}\varepsilon_{t-2} + \frac{1}{3}\varepsilon_{t-3}$

Estimating this model is trickier than with a straight up least squares regression... but that's the basic idea of it.

• Does this mean that et = yt - u ? Does et need to come from a differenced series ? Or a stationary series ? is this talking about the change in yt or the actual value of yt ? If I make up a pretend series that follows a perfectly predictable pattern of 2,3,4,2,3,4,2,3,4 and use an order one model, should this method predict the series to continue ? Apologies because this is probably confusing and not using the right terminology I'm just trying to understand the basics a bit better. Thanks. – user3528592 Jun 6 '14 at 2:39
• But what is the εt ? If I need calculate y, I need know εt , what is the value of εt?? – Mithril Oct 25 '18 at 6:56
• @Mithril Ask this question, but not as a comment. – Alexis Oct 25 '18 at 16:09


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# What physical observables are the creation/annihilation operators for the EM field made from?

The explanations of quantizing the Electric/Magnetic (E/M) fields that I've read have all basically worked by using the Coulomb gauge in free space to define the vector potential in some volume as

$$\mathbf{A}(\mathbf{x},t)=\sum_{\mathbf{k},r}\sqrt{\frac{\hbar c^2}{2V\omega_{\mathbf{k}}}}\mathbf{e}_r(\mathbf{k})[a_{r,\mathbf{k}}e^{i(\mathbf{k}\cdot\mathbf{x}-\omega_{\mathbf{k}}t)}+a^*_{r,\mathbf{k}}e^{-i(\mathbf{k}\cdot\mathbf{x}-\omega_{\mathbf{k}}t)}]$$

where the classical amplitudes $$a$$, $$a^*$$ are essentially engineered to be directly replaced with the creation/annihilation operators $$\hat{a}$$, $$\hat{a}^\dagger$$ to give the quantized form of the fields, the Hamiltonian etc. This is neat and easy to follow but I don't see any physical motivation for replacing those particular objects with the creation/annihilation operators.

By contrast, in most pedagogical treatments, the quantum harmonic oscillator is first given in terms of the physical quantities position and momentum, these are then replaced with their quantum versions and the creation/annihilation operators are derived from their anticommutation relations. I find this to be a more satisfying procedure because the $$\hat{a}$$, $$\hat{a}^\dagger$$ formalism is shown to naturally emerge from switching to the quantum version of these physical observables. In particular, you have the relations

$$\hat x = \sqrt{\frac{\hbar}{2m\omega}}(\hat a+\hat a^\dagger),\quad \hat p = i\sqrt{\frac{\hbar m\omega}{2}}(\hat a+\hat a^\dagger),$$

and

$$\hat a = \sqrt{\frac{m\omega}{2\hbar}}\left(\hat x + \frac{i}{m\omega}\hat p\right), \quad \hat a ^\dagger = \sqrt{\frac{m\omega}{2\hbar}}\left(\hat x - \frac{i}{m\omega}\hat p\right).$$

Naturally I was wondering if there are analogous physical quantities for the EM field where similar relations to the above hold. Hoping to gain some insight, I considered a single mode $$\mathbf{k}$$ and wrote $$\hat a = \hat Q + i\hat P$$ which gets us $$\hat a e^{i(\mathbf{k}\cdot\mathbf{x}-\omega_{\mathbf{k}}t)}+\hat a ^\dagger e^{-i(\mathbf{k}\cdot\mathbf{x}-\omega_{\mathbf{k}}t)} = 2\hat Q \cos(\mathbf{k}\cdot\mathbf{x}-\omega_{\mathbf{k}}t) - 2\hat P \sin(\mathbf{k}\cdot\mathbf{x}-\omega_{\mathbf{k}}t).$$ I now have an expression in terms of hermitian operators but I'm not sure what they are supposed to be nor do I know why we should expect them to have commutation relations similar to position and momentum. Can anyone clear this up for me or point me to a source which treats the quantized EM field in this way?

• Check out Quantum and Atom Optics by Dan Steck and also Robert Littlejohns Quantum Mechanics Lecture Notes. Feb 15 at 21:15

You can actually find that your $$\hat{Q},\hat{P}$$s are very similar to the electric and magnetic field parts, if you're using the linear polarization basis where $${\bf k}\times{\bf e}={\bf k}\times{\bf \bar{e}}$$.

\begin{aligned} &\mathbf{A}(\mathbf{r})=\sum_{\mathbf{k}, \mu} \sqrt{\frac{\hbar}{2 \omega V \epsilon_{0}}}\left\{\mathbf{e}^{(\mu)} a^{(\mu)}(\mathbf{k}) e^{i \mathbf{k} \cdot \mathbf{r}}+\overline{\mathbf{e}}^{(\mu)} a^{\dagger^{(\mu)}}(\mathbf{k}) e^{-i \mathbf{k} \cdot \mathbf{r}}\right\} \\ &\mathbf{E}(\mathbf{r})=i \sum_{\mathbf{k}, \mu} \sqrt{\frac{\hbar \omega}{2 V \epsilon_{0}}}\left\{\mathbf{e}^{(\mu)} a^{(\mu)}(\mathbf{k}) e^{i \mathbf{k} \cdot \mathbf{r}}-\overline{\mathbf{e}}^{(\mu)} a^{\dagger^{(\mu)}}(\mathbf{k}) e^{-i \mathbf{k} \cdot \mathbf{r}}\right\} \\ &\mathbf{B}(\mathbf{r})=i \sum_{\mathbf{k}, \mu} \sqrt{\frac{\hbar}{2 \omega V \epsilon_{0}}}\left\{\left(\mathbf{k} \times \mathbf{e}^{(\mu)}\right) a^{(\mu)}(\mathbf{k}) e^{i \mathbf{k} \cdot \mathbf{r}}-\left(\mathbf{k} \times \overline{\mathbf{e}}^{(\mu)}\right) a^{\dagger^{(\mu)}}(\mathbf{k}) e^{-i \mathbf{k} \cdot \mathbf{r}}\right\} \end{aligned}

• Sure, but what I'm really wondering is why I should expect the quantized versions of these "parts" to not commute trivially. Feb 15 at 19:31
• I think you're asking why we impose the commutation relation to a pair of conjugate variables in classical Hamiltonian (aka canonical quantization)? This is how people tried to generalize the Poisson brackets into quantum versions. Basically, the idea is every normal mode (harmonic oscillator-like things) corresponds to a boson and therefore people imply the bosonic commutation rule. For here, the EM wave is obviously a very good example which quantized as photon modes. Feb 15 at 19:50

If one starts from $$Q$$ and $$P$$ in order to quantize the electromagnetic field, one can use an analogy to the quantization of classical mechanics.

The quantization of classical mechanics or differently said the transition of classical mechanics to quantum mechanics is done by requiring the famous commutation relation between $$Q$$ and $$P$$ (we assume here $$\hbar=1$$):

$$[Q,P]= i$$

which is the basis for Heisenberg's uncertainty relation. Then the definition

$$a = Q + iP \quad \text{and} \quad a^\dagger = Q - iP$$

can be used to find the commutation relation between the creation and annihilation operators.

Actually it is better to use the following definitions:

$$c = \frac{(\omega Q + iP)}{\sqrt{2\omega}}\quad \text{and} \quad c^\dagger = \frac{(\omega Q - iP)}{\sqrt{2\omega}}$$

one can find the commutation relation for the creation and annihilation operators:

$$[c,c^\dagger]= 1$$

The formalism also allows to find the Hamiltionian expressed in creation and annihilation operators. In order to get there in case of the electromagnetic field one starts off from the classical expression of the energy of the electromagnetic field:

$$H \equiv E =\frac{1}{8\pi}\int (\mathbf{E}^2 + \mathbf{H}^2) d^3x$$

where the electrical and magnetic field are given by the vector potential:

$$\mathbf{E} = -\dot{\mathbf{A}}\quad\text{and}\quad \mathbf{H} = rot A$$

Using the development of the vector potential in creation and annihilation operators as it is given in the first formula of the post plugged in the definition of $$\mathbf{E}$$ and $$\mathbf{H}$$, furthermore these two plugged in the definition of the field energy yields the Hamiltonian expressed in creation and annihilation operators after a longer calculation:

$$H = \frac{1}{2}\sum (P^2 +\omega^2 Q^2) = \frac{1}{2}\sum \omega (c c^\dagger + c^\dagger c)$$

So far the fact that the EM-field actually not only has one set of canonical variables $$Q$$ and $$P$$, but infinitely many was neglected. The summation sign in the above expression is put in order to fix this.

We can observe that the quantities $$P$$ and $$Q$$ fulfill the role for momentum and position in a harmonic oscillator. So they deserve to be considered as (generalized) momentum $$P$$ and position $$Q$$.

This means one is led rather naturally to the operator property of the creation and annihilation operators with the right commutation relation by using the analogy of the quantization of classical mechanics.

This answer only schematically shows the way to go. In order to see the calculation in all details and with the right normalization I recommend to read volume 4 (Relativistic Quantum Theory) of the series of tomes edited by Landau & Lifshitz. More modern books in particular those of pure Quantum Field Theory use often a Lorentz covariant formalism where these details are not shown. But nevertheless there are books around where this particular approach can be found.


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# Using algebra tiles to factor polynomials

#### All You Need in One Place

Everything you need for Year 6 maths and science through to Year 13 and beyond.

#### Learn with Confidence

We’ve mastered the national curriculum to help you secure merit and excellence marks.

#### Unlimited Help

The best tips, tricks, walkthroughs, and practice questions available.

0/2
##### Examples
###### Lessons
1. Representing polynomials using algebra tiles:

1. ${2x(x-2)}$
2. ${(x-3)(4x+1)}$
###### Topic Notes
Algebra tiles can be used to represent polynomials. We can also use those tiles to solve polynomials. In this lesson, we will try to solve some polynomials using algebra tiles.


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Recursion Exercise

Objective

To gain experience writing recursive functions.

Some data structures naturally lend themselves to recursion. One of these is liked lists. When we define the node of a linked list, we are already using a form of recursion:

struct Node {
type data;
Node* next;
};

Here a Node is defined as having data, and a pointer to another Node. Because Nodes are mentioned in the definition of a Node, this is recursive.

We can use the fact that Nodes are defined recursively to write recursive functions for dealing with linked lists. For this lab, you will write a function to print a singly-linked list using recursion instead of iteration. You will also write a function to print a singly-linked list backwards using recursion. This is more difficult without recursion, but extremely simple with it.


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# The Monty Hall problem

I was watching the movie $$2121$$ yesterday, and in the first 15 minutes or so the main character is in a classroom, being asked a “trick” question (in the sense that the teacher believes that he’ll get the wrong answer) which revolves around theoretical probability.

The question goes a little something like this (I’m paraphrasing, but the numbers are all exact):

You’re on a game show, and you’re given three doors. Behind one of the doors is a brand new car, behind the other two are donkeys. With each door you have a $$1/31/3$$ chance of winning. Which door would you pick?

The character picks A, as the odds are all equally in his favor.

The teacher then opens door C, revealing a donkey to be behind there, and asks him if he would like to change his choice. At this point he also explains that most people change their choices out of fear; paranoia; emotion and such.

The character does change his answer to B, but because (according to the movie), the odds are now in favor of door B with a $$1/31/3$$ chance of winning if door A is picked and $$2/32/3$$ if door B is picked.

What I don’t understand is how removing the final door increases the odds of winning if door B is picked only. Surely the split should be 50/50 now, as removal of the final door tells you nothing about the first two?

I assume that I’m wrong; as I’d really like to think that they wouldn’t make a movie that’s so mathematically incorrect, but I just can’t seem to understand why this is the case.

So, if anyone could tell me whether I’m right; or if not explain why, I would be extremely grateful.

At the start of the game, you are asked to pick a single door. There is a $1/3$ chance that you have picked correctly, and a $2/3$ chance that you are wrong. This does not change when one of the two doors you did not pick is opened. The second time is that you are choosing between whether your first guess was right (which has probability $1/3$) or wrong (probability $2/3$). Clearly it is more likely that your first guess was wrong, so you switch doors.
To clarify this, say you choose door $A$, and are then asked to choose between doors $A$ and $B$ (no doors have been opened yet). There is no advantage to switching in this situation. Say you are asked to choose between $A$ and $C$; again, there is no advantage in switching. However, what if you are asked to choose between a) the prize behind door $A$ and b) the better of the two prizes behind door $B$ and $C$. Clearly, in this case it is in your advantage to switch. But this is exactly the same problem as the one you’ve been confronted with! Why? Precisely because the host always opens (hence gets rid of) the door that you did not pick which has the worse prize behind it. This is what I mean when I say that the asymmetry in the situation comes from the knowledge of the host.


text:
# Differentiate, but do not simplify: ##f(x)=\frac{sin(x)-cos(x)}{sin(x)+cos(x)}##

Homework Statement:
I'm not sure if my latex is correct, but the image is provided with my work, please let me know if it's right!
Relevant Equations:
n/a

ehild
Homework Helper
Homework Statement:: I'm not sure if my latex is correct, but the image is provided with my work, please let me know if it's right!
Relevant Equations:: n/a

View attachment 262669
It is correct, but the solution is also simple if you differentiate the original function.
By the way, start and finish your latex formula with double \$:

$$f(x)=\frac{sin(x)-cos(x)}{sin(x)+cos(x)}$$
$$f(x)=\frac{sin(x)-cos(x)}{sin(x)+cos(x)}$$

Last edited:
ttpp1124


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# Blog entry generator

Here is a quick and dirty script I set up to generate my blog entries.

You can use LaTeX code and soucecode to write your blog entries. LaTeX code is transformed into PNG. Sourcecode is transformed so it can be read used with syntaxhighlighter.

You need some software to compile the things:

• Python (tested with version 3.1.1)
• LaTeX
• dvipng

The script can be invoked using the command (here new.htm is compiled to a blog entry):

python latex2png.py new.html

## To-do-list

• support multi-line LaTeX-parts
• refactor code

## An example:

Below you find a piece of code and the result.

< h1 > An example < /h1 >

Here is a simple example using a formula: $\sqrt{2+x^2}$.

And some code:

[code cpp]
int main(){
return 0;
}
[/code]

Below you can see the result:

# An example

Here is a simple example using a formula: .

And some code:

int main(){
return 0;
}


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# Quartic graphs with 12 vertices

This is a continuation of the post A table of small quartic graphs. As with that post, it’s modeled on the handy wikipedia page Table of simple cubic graphs.

According to SageMath computations, there are 1544 connected, 4-regular graphs. Exactly 2 of these are symmetric (ie, arc transitive), also vertex-transitive and edge-transitive. Exactly 8 of these are vertex-transitive but not edge-transitive. None are distance regular.

Example 1: The first example of such a symmetric graph is the circulant graph with parameters (12, [1,5]), depicted below. It is bipartite, has girth 4, and its automorphism group has order 768, being generated by $(9,11), (5,6), (4,8), (2,10), (1,2)(5,9)(6,11)(7,10), (1,7), (0,1)(2,5)(3,7)(4,9)(6,10)(8,11)$.

Example 2: The second example of such a symmetric graph is the cuboctahedral graph, depicted below. It has girth 3, chromatic number 3, and its automorphism group has order 48, being generated by $(1,10)(2,7)(3,6)(4,8)(9,11), (1,11)(3,4)(6,8)(9,10), (0,1,9)(2,8,10)(3,7,11)(4,5,6)$.


text:
# Spectrum of beta decay

Nice question!

You don't say what $$\beta^+$$ and $$\beta^-$$ spectra these are (and, BTW, if this is from a book, you should credit the author). If these are from a single odd-odd nucleus that undergoes both $$\beta^+$$ and $$\beta^-$$, then there is one very simple reason why the two spectra should look different, and that is that the amount of binding energy released in the two cases will be a little different. But that would actually rescale the energy axes relative to one another rather than producing this kind of additive effect.

For comparison, let's consider the neutrino, which is uncharged. The spectrum of neutrinos emitted in beta decay is a bell curve that goes to zero at zero energy, and this is basically because of the number of available states, which is maximized if the beta and the neutrino share the available kinetic energy roughly equally.

If we model beta decay in purely classical terms as a process in which the beta and neutrino are released in a pointlike event inside the nucleus, then we don't get a prediction that matches the data. In such a description, we would expect the $$\beta^+$$ and $$\beta^-$$ to be shifted to the right and left by an amount equal to the electrical potential energy. Since a couple of answers have linked to the page by Watkins at SJSU, let's use his example of 64Cu, which exhibits both $$\beta^+$$ and $$\beta^-$$ decay. The radius $$r$$ of this nucleus is on the order of 4 fm ($$4\times 10^{-15}$$ m). Therefore the potential energy lost or gained by the $$\beta^+$$ or $$\beta^-$$ on the way out would be $$U\sim kZe^2/r\sim10\ \text{MeV}$$. This is totally incompatible with observation. For one thing, it predicts that $$\beta^-$$ decay could never happen, since the $$\beta^-$$ can't be created with this much energy. It also predicts that $$\beta^+$$'s would be detected with huge energies, which is not the case.

To understand what's really going on, we need quantum mechanics, not just classical physics. The amount of kinetic energy that is available to a beta based on conservation of energy is on the order of 0.3 MeV. This gives it a de Broglie wavelength of about 2000 fm, which is hundreds of times greater than the size of the nucleus. Therefore the beta cannot be much better localized than that when it is emitted. This means that we should really use something more like $$r\sim 500\ \text{fm}$$ (a quarter of a wavelength) in our calculation of the electrical energy. This gives $$U\sim0.08\ \text{MeV}$$, which is about the right order of magnitude compared to observation.

A byproduct of this analysis is that a $$\beta^+$$ is always emitted within the classically forbidden region, and then has to tunnel out through the barrier. It's a counterintuitive fact about quantum mechanics that a repulsive force can hinder the escape of a particle. This is also observed in alpha decay.


text:
Thursday, 7 June 2012

# Use the tbody tag to group multiple tr tags

It is sometimes necessary to group the rows of a HTML table. Say you want to display tabular data of a person and their qualifications, you could achieve this by having the name of the person and qualification on separate rows like this.

Name Qualification
Frank Doe Bachelor of Arts
John Smith Bachelor of Science
John Smith Bachelor of Science
John Smith Master of Science
Alice Mane Diploma of Information Technology
Alice Mane Bachelor of Computer Science

This isn't ideal of course as firstly you are duplicating information, and making the table more busy than it needs to be. But there is also the issue of what to do if there are multiple people with the same name. A nice fix to this issue which is isn't very widely known is that you can group table rows by using the tbody tag. It also comes with the added benefit of being able to create hover styles that span the row and having the collection of rows use the same click event.

Name Qualification
Frank Doe Bachelor of Arts
John Smith Bachelor of Science
John Smith Bachelor of Science
Master of Science
Alice Mane Diploma of Information Technology
Bachelor of Computer Science

Name
Qualification

Frank Doe
Bachelor of Arts

John Smith
Bachelor of Science

John Smith
Bachelor of Science

Master of Science


text:
Categories

Sign change | ISI-B.stat | Objective Problem 709

Try this beautiful problem on Sign change, useful for ISI B.Stat Entrance.

Sign change | ISI B.Stat Entrance | Problem 709

In the interval $(-2\pi, 0)$ the function $f(x) = sin(1/x^3)$

• (a) never changes sign
• (b) changes sign only once
• (c) changes sign more than once, but a finite number of times

Key Concepts

Calculus

Limit

Trigonometry

TOMATO, Problem 709

Challenges and Thrills in Pre College Mathematics

Try with Hints

As x becomes $\leq 1$ and tends to zero then it crosses $\pi, 2\pi, 3\pi, ….$.can you draw the graph?

Can you now finish the problem ……….

If we draw the graph then we can see that the function $f(x) = sin(1/x^3)$ crosses many times. Therefore number of sign changes is infinite.

Therefore option $(d)$ is correct…..

Categories

Limit Problem | ISI-B.stat | Objective Problem 694

Try this beautiful problem on Limit, useful for ISI B.Stat Entrance.

Limit Problem | ISI B.Stat Entrance | Problem 694

Let $a_1 = 1$ and $a_n = n(a_{n-1} + 1)$ for $n = 2, 3, ….$ Define $P_n = (1 +1/a_1)(1 + 1/a_2)….(1 + 1/a_n)$. Then $\lim\limits_{x \to \infty} {P_n}$?

• (a) $1+e$
• (b) $e$
• (c) $1$
• (d) $\infty$

Key Concepts

Calculus

Limit

Trigonometry

Answer: (b)$e$

TOMATO, Problem 709

Challenges and Thrills in Pre College Mathematics

Try with Hints

Given that $P_n = (1 +1/a_1)(1 + 1/a_2)….(1 + 1/a_n)$

Therefore $P_n=\frac{a_1 +1}{a_1}.\frac{a_2 +1}{a_2}.\frac{a_3 +1}{a_3}…..\frac{a_n +1}{a_n}$

Now $a_n = n(a_{n-1} + 1)$

Put $n=2$, we will get $a_1+1=\frac{a_2}{2}$

$a_2+1=\frac{a_3}{3}$……………….

………………………..

…………………………

$a_n+1=\frac{a_n}{n}$

Therefore $P_n=\frac{a_1 +1}{a_1}.\frac{a_2 +1}{a_2}.\frac{a_3 +1}{a_3}…..\frac{a_n +1}{a_n}$

$\Rightarrow {P_n}= \frac{a_2}{2a_1}.\frac{a_3}{3a_2}.\frac{a_4}{4a_3}……..\frac{a_{n+1}}{(n+1).{a_n}}$

$\Rightarrow {P_n}=\frac{a_{n+1}}{{a_1}\{2.3.4………..(n+1)\}}$

$\Rightarrow {P_n}=\frac{a_{n+1}}{\{1.2.3.4………..(n+1)\}}$ (as $a_1=1$)

$\Rightarrow {P_n}=\frac{a_{n+1}}{(n+1)!}$

$\Rightarrow {P_n}=\frac{(n+1)(a_n +1)}{(n+1)!}$

$\Rightarrow {P_n}=\frac{(a_n +1)}{n!}$

$\Rightarrow {P_n}=\frac{a_n}{n!} +\frac{1}{n!}$

$\Rightarrow {P_n}=\frac{n(a_{n-1}+1)}{n!}+\frac{1}{n!}$

$\Rightarrow {P_n}=\frac{a_{n-1}+1}{(n-1)!}+\frac{1}{n!}$

$\Rightarrow {P_n}=\frac{a_{n-1}}{(n-1)!}+\frac{1}{(n-1)!}+\frac{1}{n!}$

$\Rightarrow {P_n}=\frac{a_2}{2!}+\frac{1}{2!}+\frac{1}{3!}+…….+\frac{1}{n!}$

$\Rightarrow {P_n}=\frac{2(1+1)}{2!}+\frac{1}{2!}+….+\frac{1}{n!}$

$\Rightarrow {P_n}=1+\frac{1}{1!}+…..+\frac{1}{n!}$

Can you now finish the problem ……….

Now we have to find out $\lim\limits_{x \to \infty} {P_n}$

we know that $e^x=1+\frac{x}{1!}+\frac{x^2}{2!}+……….+\infty$

So,$e^1=1+\frac{1}{1!}+\frac{1^2}{2!}+……….+\infty$

$\lim\limits_{x \to \infty} {P_n}$=$1+\frac{1}{1!}+\frac{1^2}{2!}+……….+\infty$

$\lim\limits_{x \to \infty} {P_n}$=$e$

Therefore option (b) is correct…..

Categories

Graph in Calculus | ISI-B.stat | Objective Problem 699

Try this beautiful problem on Graph in Calculus, useful for ISI B.Stat Entrance.

Graph in Calculus | ISI B.Stat Entrance | Problem 699

The adjoining figure is the graph of

• (a) $y = 2e^x$
• (b) $y = 2e^-x$
• (c) $y = e^x + e^-x$
• (d) $y = e^x – e^-x + 2$

Key Concepts

Calculus

Graph

Functions

TOMATO, Problem 699

Challenges and Thrills in Pre College Mathematics

Try with Hints

At first we have to check all the options…..

1. For $y = 2e^x$, this is not possibe because $y = 2e^x$ expression represent only positive side’s of x axis
2. For $y = 2e^-x$,This expression also represents one side of x-axis
3. For $y = e^x + e^-x$.This expression represents both sides of the x-axise.as we put $x=0$ then $y=2$ i.e the curve not starts from the origin
4. $y = e^x – e^-x + 2$, this expression also represents both sides of the x-axis

Can you now finish the problem ……….

Therefore ,the correct ans is (c)

Categories

Negative & Positive Roots | ISI-B.stat | Objective Problem 708

Try this beautiful problem on Negative & Positive Roots, useful for ISI B.Stat Entrance.

Negative & Positive Roots | ISI B.Stat Entrance | Problem 708

If n stands for the number of negative roots and p for the number of positive roots of the equation $e^x = x$, then

• (a) $n = 1$, $p = 0$
• (b) $n = 0$, $p = 1$
• (c) $n = 0$, $p > 1$
• (d) $n = 0$, $p = 0$

Key Concepts

Calculus

Limit

Trigonometry

TOMATO, Problem 708

Challenges and Thrills in Pre College Mathematics

Try with Hints

Draw the figure of given equation $e^x = x$ and check all the points that are in the options…..

Can you now finish the problem ……….

The given equation is $e^x = x$

as $x$increases exponential function increases more rapidly than any polynomial function say $x$ . Hence from graph we can say they never intersect . So, there is no solution for the equation . Therefore $n=0$ and $p=0.$

Therefore option $(d)$ is correct…..

Categories

Graphs in Calculus | ISI-B.stat | Objective Problem 698

Try this beautiful problem on Graphs in Calculus, useful for ISI B.Stat Entrance.

Graphs in Calculus | ISI B.Stat Entrance | Problem 698

Four graphs marked G1, G2, G3 and G4 are given in the figure which are graphs of the four functions $f_1(x) = |x – 1| – 1, f_2(x) = ||x –1| – 1|, f_3(x) = |x| – 1, f_4(x) = 1 – |x|$, not necessarily in the correct order.The correct order is

• (a) $G_2, G_1, G_3, G_4$
• (b) $G_3, G_4, G_1, G_2$
• (c) $G_2, G_3, G_1, G_4$
• (d) $G_4, G_3, G_1, G_2$

Key Concepts

Calculus

Graph

Functions

TOMATO, Problem 698

Challenges and Thrills in Pre College Mathematics

Try with Hints

We take the each functions and express it in intercept form.we expand the mod i,e take the value once positive and once negetive .so we will get two equations and solve them,we will get the intersecting point also and draw the graph……..

Can you now finish the problem ……….

$f_1(x) = |x – 1| – 1$

$(x-1)-1=y$

$\Rightarrow x-y=2$……………..(1)

$\Rightarrow \frac{x}{2} +\frac{y}{-2}=0$$\Rightarrow (2,0) ,(0,-2)$

And

$-(x-1)-1=y$

$\Rightarrow x+y=0$……..(2)

$\frac{x}{1}+\frac{y}{1}=0$$\Rightarrow (1,0),(0,1)$

Now if we draw the graph of (1) & (2) we will get the figure $G_2$ and the intersecting point is $(1,-1)$

Similarly we can draw the graphs for other functions………….

The second function is $f_2(x) = ||x –1| – 1|$ i.e $x-y=2$,$x=y$,$x+y=1$…which represents the two figure as given in $G_3$.

The third function $f_3(x) = |x| – 1$ which gives $x-y=1$ & $x+y=-1$..if we solve this two equations as first function then we will get $G_1$

The third function will gives the $G_4$ graph

Similarly we will draw the graph for all given functions….

Therefore ,the correct ans is (c)

Categories

Problem on Calculus | ISI-B.stat | Objective Problem 696

Try this beautiful problem on Calculus, useful for ISI B.Stat Entrance.

Problem on Calculus | ISI B.Stat Entrance | Problem 696

If k is an integer such that lim $\{{cos}^n(k\pi/4) – {cos}^n(k\pi/6)\} = 0$,
then

• (a) k is divisible neither by 4 nor by 6
• (b) k must be divisible by 12, but not necessarily by 24
• (c) k must be divisible by 24
• (d) either k is divisible by 24 or k is divisible neither by 4 not by 6

Key Concepts

Calculus

Limit

Trigonometry

TOMATO, Problem 694

Challenges and Thrills in Pre College Mathematics

Try with Hints

There are four options ,at first we have to check each options…..

If k is divisible by 24 then cos(kπ/4) = cos(kπ/6) = 1
$\Rightarrow$ The limit exists and equal to RHS i.e. 0
If k is not divisible by 4 or 6 then cos(kπ/4), cos(kπ/6) both <1

Can you now finish the problem ……….

Therefore ,

lim cosn(kπ/4), cosn(kπ/6) = 0. so we may say that
$\Rightarrow$The equation holds.

Categories

Problem

The polynomial $x^7+x^2+1$ is divisible by

• (A) $x^5-x^4+x^2-x+1$             (B) $x^5-x^4+x^2+1$
• (C)   $x^5+x^4+x^2+x+1$          (D)   $x^5-x^4+x^2+x+1$

.
Also Visit: I.S.I. & C.M.I Entrance Program

STOP

Categories

Parity of the terms of a sequence | Tomato Problem 7

Try this problem from TOMATO Problem 7 based on the Parity of the terms of a sequence.

Problem: Parity of the terms of a sequence

If $a_0 = 1 , a_1 = 1$ and $a_n = a_{n – 1} a_{n – 2} + 1$ for $n > 1$, then:

(A) $a_{465}$ is odd and $a_{466}$ is even;
(B) $a_{465}$ is odd and $a_{466}$ is odd;
(C) $a_{465}$ is even and $a_{466}$ is even;
(A) $a_{465}$ is even and $a_{466}$ is odd;

Discussion:

First we note a pattern and then we prove that the pattern actually holds.

Note that:

$a_0 = 1$ is odd
$a_1 = 1$ is odd
$a_2 = a_0 a_1 + 1 = 1\times 1 + 1 = 2$ is even
$a_3 = a_1 a_2 + 1 = 1 \times 2 + 1 = 3$ is odd
$a_4 = a_2 a_3 + 1 = 2 \times 3 + 1 = 7$ is odd
$a_5 = a_3 a_4 + 1 = 3 \times 7 + 1 = 22$ is even

So the pattern that we observe is the following order: odd, odd, even, odd, odd, even…

We show this by strong form of induction. Suppose this pattern holds true for all n upto n = 3k+2

(that is $a_{3k+2} = even , a_{3k+1} = odd, a_{3k}= odd$ ).

Our computations show that this is true for k =1 (so for initial value it is true).

Let us show for the next three values:

$a_{3k+3} = a_{3k+2} \times a_{3k+1} + 1 = even \times odd + 1 = odd$
$a_{3k+4} = a_{3k+3} \times a_{3k+2} + 1 = odd \times even + 1 = odd$
$a_{3k+5} = a_{3k+4} \times a_{3k+3} + 1 = odd \times odd + 1 = even$

Thus we showed that whenever the index is of the form 3j+2, the number is even, otherwise if the index is of the form 3j or 3j+1, the term is odd.

Since 465 and 466 are respectively of the form 3j and 3j+1, hence
$a_{465}$ and $a_{466}$ both are odd.

Chatuspathi:

• What is this topic: Induction
• What are some of the associated concepts: Strong form of induction
• Where can learn these topics: Cheenta I.S.I. & C.M.I. courseMath Olympiad Program discusses these topics in the ‘Induction’ module.
• Book Suggestions: Elementary number theory by David Burton

Categories

Men and Job Problem | Tomato Question 2 | ISI Entrance

This is a problem from TOMATO Problem number 2, useful for ISI and CMI entrance exam based on Men and Job.

Problem:

If m men can do a job in d days, then the number of days in which m+r men can do the job is
(A) d+r; (B) ( \frac{d}{m} (m+r)) ; (C) ( \dfrac {d}{m+r} ) ; (D) ( \dfrac {md}{m+r} );

Discussion:

If m men can do a job in d days,
Then m men does ( \frac {1}{d} ) job in 1 day
Then 1 man does ( \frac {1}{md} ) job in 1 day
Then m+r men does ( \frac {m+r}{md} ) job in 1 day
Then m+r men does 1 job ( \frac {md}{m+r} ) days.

Hence answer is ( \dfrac {md}{m+r} ) days (D)

Chatuspathi:

• What is this topic: Arithmetic
• What are some of the associated concepts: Unitary Method
• Where can learn these topics: Cheenta I.S.I. & C.M.I. course, discusses these topics in the ‘Arithmetic’ module.
• Book Suggestions: Arithmetic by K.C. Nag
Categories

Calculating Average Speed | Tomato Problem 3

This is a problem number 3 from TOMATO based on Calculating Average Speed.

Problem: Calculating Average Speed.

A boy walks from his home to school at 6 kmph. He walks back at 2 kmph. His average speed, in kmph is
(A) 3; (B) 4; (C) 5; (D) ( \sqrt {12} );

Discussion:

Suppose the distance from home to school is t km.
Time taken for home to school journey: ( \frac {t}{6} )
Time taken for school to home journey: ( \frac {t}{2} )

Hence average speed = ( \frac {distance}{time} = \frac {2t}{\frac{t}{6} + \frac{t}{2}} = 3 )

There fore average speed is 3 kmph.

Note that it is the harmonic mean of the two given speeds instead of arithmetic mean.

Chatuspathi:

• What is this topic: Elementary Arithmetic
• What are some of the associated concepst: Harmonic Mean
• Where can learn these topics: Cheenta I.S.I. & C.M.I. course, discusses these topics in the ‘Number Theory’ module.
• Book Suggestion: Arithmetic by K.C. Nag


text:
## Stream: Is there code for X?

### Topic: Automating over structures

#### Aaron Anderson (Jul 25 2020 at 04:02):

I see that lots of work is being put into refactoring the ideas of submonoid, subgroup, subring, etc. I've written a definition based on flypitch for a substructure of an arbitrary first-order structure, which should in theory generalize all of these, for the right choices of language. If we had an accepted mathlib definition for what a substructure should be in that context, how far away would we be from being able to actually apply it to all relevant structures?

#### Aaron Anderson (Jul 25 2020 at 04:02):

(my code for substructure so far: https://github.com/leanprover-community/mathlib/blob/fol-attempt/src/model_theory/substructure.lean#L9)

#### Aaron Anderson (Jul 25 2020 at 04:03):

Is this the kind of thing that could be done with a tactic like to_additive, a significantly harder-to-write tactic, or is it somehow beyond what Lean can do at the moment?

#### Yury G. Kudryashov (Jul 25 2020 at 04:17):

I wonder if it's possible to generate something definitionally equal to the current lemmas based on the type of mk etc.

#### Yury G. Kudryashov (Jul 25 2020 at 04:18):

I mean, if you have a list of types of functions, then you can generate fun_mem using some list.foldl or list.foldr.

#### Johan Commelin (Jul 25 2020 at 04:29):

Aaron Anderson said:

Is this the kind of thing that could be done with a tactic like to_additive, a significantly harder-to-write tactic, or is it somehow beyond what Lean can do at the moment?

I think this would be your best bet, but it also sounds like it will be quite a bit harder-to-write...

Last updated: May 19 2021 at 02:10 UTC


text:
Can we say something about the index of these subgroups?

Let M, K, and L be isomorphic normal subgroups of a group G. Suppose $M \cap K \leq M \cap L$. Can we conclude anything about the index of MK in G and the index of ML in G? I would like it to be the case that MK has smaller index than ML, but I'm having a hard time convincing myself either way.

(EDIT: I initially left out the word "normal".)

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"Index" is only defined for subgroups of groups, so do you want to make some extra assumptions (such as M being a normal subgroup of G) to ensure that MK and ML are subgroups of G? –  Derek Holt Nov 4 '10 at 14:12
Right, I meant to stipulate that these are all normal. I'll update the question. –  Gabe Cunningham Nov 4 '10 at 14:30

It's true for finite groups, since $|MK| = |M| |K|/|M \cap K|$. It will not be true in general for infinite groups. The example that springs to mind is $G={\mathbb Z}^{\infty}$, where it is easy to find subgroups $M,K,L$ all isomorphic to $G$ with $M \cap K = M \cap L = 1$, $ML=G$ and $|G:MK|$ infinite.

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I thought something like that might happen. The case I'm really interested in is somewhat more specific -- I'll think about it more and maybe post a follow-up. Thanks! –  Gabe Cunningham Nov 4 '10 at 14:53


text: In linear algebra, a symmetric real matrix is said to be positive-definite if the scalar is strictly positive for every non-zero column vector of real numbers. METZLER.? If A is an n x m matrix, let S=1/2(A+A^t) and k=1/2(A-A^t) i) show that S is symmetric and k is skew symmetric. If Matrix M Is Both a Symmetric Matrix and Skew-Symmetric Matrix Then Matrix M is, 2. Eigenvectors are orthogonal for every different eigenvalue. Where M is any matrix and MT is the transpose of matrix. Since for any matrix A, (kA)′ = kA′, it follows that 1 / 2 (A+A′) is a symmetric matrix and 1 / 2 (A − A′) is a skew-symmetric matrix. QTQ= I. On Certain Properties of Symmetric, Skew Symmetric, and Orthogonal Matrices. The difference between both symmetric matrix and a skew-symmetric matrix is that symmetric matrix is always equivalent to its transpose whereas skew-symmetric matrix is a matrix whose transpose is always equivalent to its negative. From Theorem 7.1, it follows that (A + A T) and (A-A T) are symmetric and skew-symmetric … A matrix is Symmetric Matrix if transpose of a matrix is matrix itself. But the difference between them is, the symmetric matrix is equal to its transpose whereas skew-symmetric matrix is a matrix whose transpose is equal to its negative. Performance & security by Cloudflare, Please complete the security check to access. Here a12 =  a21 = 3, = -a13 =  a31 = 8 a23 =  a32  = -4.In other words, it is stated that the transpose of matrix M is equivalent to the matrix itself (M=MT )which implies that matrix M is symmetric. If the symmetric matrix has distinct eigenvalues, then the matrix can be transformed into a diagonal matrix. Thevectorproductu v,incomponents,reads: u v = 2 4 u 2v 3 u 3v 2 u 3v 1 u 1v 3 u 1v 2 u 2v 1 3 ... where we have used the fact that Qis a rotation matrix, i.e. Then, we can write. Where each element of a symmetric matrix is symmetric in terms to the main diagonal. Also, it is important to note that all the elements present in the main diagonal of the skew-symmetric matrix are always zero. Any Square matrix can be expressed as the sum of a symmetric and a skew-symmetric matrix. In linear algebra, a real symmetric matrix represents a self-adjoint operator over a real inner product space. Otherwise, a skew-symmetric matrix is just the same thing as a symmetric matrix.. Also, read: Properties The sum of two skew-symmetric matrices is skew-symmetric. For every distinct eigenvalue, eigenvectors are orthogonal. 2] Two n-square complex symmetric matrices are congruent over the field of complex numbers if and only if they have the same rank. The inverse matrix will always be equivalent to the inverse of a transpose matrix. Properties: 1. Only a square matrix is a symmetric matrix because in linear algebra similar matrices have similar dimensions. Let A T = A; B T = B where A & B have the same order. A square Matrix A is defined as  skew-symmetric if a, for all the values of i and j. In other words, a symmetric matrix is always diagonalizable. Symmetric and Skew Symmetric Matrix. XXII, Nos. B = A −1 We will find inverse of a matrix by Elementary transformation Using adjoint Properties of Inverse If the matrix given is invertible, then the inverse matrix will be considered a symmetric matrix. Properties of symmetric matrices 18.303: Linear Partial Differential Equations: Analysis and Numerics Carlos P erez-Arancibia (cperezar@mit.edu) Let A2RN N be a symmetric matrix, i.e., (Ax;y) = (x;Ay) for all x;y2RN. Sums and scalar multiples of skew-symmetric matrices are again skew-symmetric. This conditions is valid for each values of i and j. If P and Q are symmetric matrices of equal size, then the total of (P + Q) and subtraction of (P- Q) of the symmetric matrix will also be the symmetric matrix. Pro Lite, Vedantu For example, If M is a symmetric matrix then M = MT and if M is a skew-symmetric matrix then M = - MT. Sorry!, This page is not available for now to bookmark. Cloudflare Ray ID: 5fb558d26ea10a3c In other words, we can say that both a symmetric matrix and a skew-symmetric matrix are square matrices. Another way: if A is skew-symmetric, then for any vector v we have vTAv = −vTATv = − vTAv so vTAv= 0. The result of the scalar product of skew-symmetric matrices is always a … skew symmetric systems, Greif and Varah (2009) adapted CG for nonsingular skew symmetric linear systems that are necessarily and restrictively of even order. Note that all the main diagonal elements in skew-symmetric matrix are … We assume that the underlying field is not of characteristic 2: that is, that 1 + 1 ≠ 0 where 1 denotes the multiplicative identity and 0 the additive identity of the given field. Also, it is important to note that all the elements present in the main diagonal of the skew-symmetric matrix are always zero. 3. )which implies that matrix M is symmetric. If A is a symmetric matrix, then A = A T and if A is a skew-symmetric matrix then A T = – A. Let us discuss this with the help of Some Examples. If matrix A is a square matrix then (A + At) is always symmetric. Properties of Skew-Symmetric Matrix Some of the properties of skew-symmetric matrix examples are given below: When two skew-matrices are added, then the resultant matrix will always be a skew-matrix. A square matrix $A=(a_{ij})$ is a symmetric matrix if its entries opposite the main diagonal are the same, that is, if $a_{ij}=a_{ji}$ for all $i$ and [math]j. The result implies that every odd degree skew-symmetric matrix is not invertible, or equivalently singular. 1. ii) verify that A=S+k and then show that this is the ONLY way A can be written as the sum if a symmetric and skew symmetric matrix The below symmetric example helps you to clearly understand the concept of skew matrix. Letu;v 2R3. 2. The determinants of a skew-symmetric matrix is also one of the  properties of  skew-symmetric matrices. What is Symmetric Matrix and Skew Symmetric Matrix and know how to find it by looking at this video? Vedantu academic counsellor will be calling you shortly for your Online Counselling session. What Are the Properties of a Symmetric Matrix? Pro Lite, Vedantu Probability Symbols and Statistics Symbols, Vedantu In the above skew matrix symmetric example, we can see all the elements present in the main diagonal of matrices A are zero and also a. If M is a skew-symmetric matrix, which is also considered as a square matrix,  then the determinant of M should satisfy the below-given situation: The inverse of skew-symmetric matrix is not possible as the determinant of it having odd order is zero and therefore it is singular. Show That Matrix M Given Below is a Skew- Symmetric Matrix. The Diagonal Entities of a Skew-Symmetric Matrix Are. The eigenvalues of a real skew-symmetric matrices are of the special form as in the next problem. Here we are going to see some practice questions on symmetric and skew symmetric matrix. Notice that an n × n matrix A is symmetric if and only if a ij = a ji, and A is skew-symmetric if and only if a ij = −a ji, for all i,j such that 1 ≤ i,j ≤ n. In other words, the entries above the main diagonal are reflected into equal (for symmetric) or opposite (for skew-symmetric) entries below the diagonal. • Let us understand the concept of symmetric matrix through the symmetric matrix example given below. Also, this means that each odd degree skew-symmetric matrix has the eigenvalue $0$. Some of the properties of skew-symmetric matrix examples are given below: When two skew-matrices are added, then the resultant matrix will always be a skew-matrix. Now for any ##\mathbf x\in \mathbb R^n## a defining characteristic of a real skew symmetric matrix ##Z## is ##\mathbf x^TZ\mathbf x =0## (check the transpose and confirm this for yourself). 1 Properties of skew symmetric matrices 1. This implies A − A T is a skew-symmetric matrix. Skew-symmetric matrices. for all indices and .. Every square diagonal matrix is symmetric, since all off-diagonal elements are zero. From the Theorem 1, we know that (A + A′) is a symmetric matrix and (A – A′) is a skew-symmetric matrix. If aij represents any elements in an ith column and jth rows, then symmetric matrix is expressed as. In the above symmetric matrix example given below, we can see aij =aji  for all the values of i and j. A scalar multiple of a skew-symmetric matrix is skew-symmetric. Generally, symmetric matrix is expressed as. In other words, we can say that matrix A is said to be skew-symmetric if transpose of matrix A is equal to negative of Matrix A i.e (). BY W. H. Solve both of using properties of the transpose, namely (X+Y) T = X T +Y T and (X T) T = X (b) Show matrix A is the sum of a symmetric matrix and a skew-symmetric matrix. For the trace operator, we have thattr(A)=tr(At). Now that we know matrix B is symmetric and matrix C is skew-symmetric, we can add the two together in terms of matrix A. Square Matrix A is said to be skew-symmetric if for all i and j. For example. If you are on a personal connection, like at home, you can run an anti-virus scan on your device to make sure it is not infected with malware. If you are at an office or shared network, you can ask the network administrator to run a scan across the network looking for misconfigured or infected devices. What is Set, Types of Sets and Their Symbols? The following properties hold true: Eigenvectors … If Sym n denotes the space of n × n symmetric matrices and Skew n the space of n × n skew-symmetric matrices then Mat n = Sym n + Skew n and Sym n ∩ Skew n = {0}, i.e. I am trying to prove the real skew-symmetric matrix property that the eigenvectors of skew-symmetric matrices are orthogonal. The determinants of skew-symmetric matrices are always non-negative. In other words, we can say  that matrix P is said to be skew-symmetric if the transpose of matrix A is equal to the negative of Matrix A i.e (A. In other words, it is always diagonalizable. There are multiple applications of symmetric matrices due to its properties. For any square matrix A, (A + A T ) is a symmetric matrix (A − A T ) is a skew-symmetric matrix Inverse of a matrix For a square matrix A, if AB = BA = I Then, B is the inverse of A i.e. Here is how far I have gotten: Say we have eigenvalues of … A symmetric matrix and skew-symmetric matrix both are square matrices. Similarly in characteristic different from 2, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative.. The eigenvalue of the symmetric matrix should always be given in a real number. Solve by for matrix A via matrix arithmetic. Some of the symmetric matrix properties are mentioned below: It is necessary for a symmetric matrix to be a square matrix. Learn about the Properties of Symmetric & Skew-Symmetric Matrices. Prove that A + AT is a symmetric and A – AT is a skew symmetric matrix, where A = [52-43-724-5-3] Concept: Matrices - Properties of Transpose of a Matrix. Property 3: The sum of two symmetric matrix is a symmetric matrix and the sum of two skew symmetric matrix is a skew symmetric matrix. Combining this with property (1), it followsthat tr(A)=0for a skew-symmetric matrix A. Pro Lite, CBSE Previous Year Question Paper for Class 10, CBSE Previous Year Question Paper for Class 12. How will you find whether the matrix given is a symmetric matrix or not ? In linear algebra, a real symmetric matrix represents a self-adjoint operator over a real inner product space. Let us understand this through a skew-symmetric matrix example. The property of the determinants of a skew-symmetric matrix can be verified using an example of 3 by 3 matrix. • (A + B) T = A + B Similarly we can prove the other. Proof. Let us look into some problems to understand the concept. The elements on the diagonal of a skew-symmetric matrix are zero, and therefore its trace equals zero. Consider a matrix A, then What Is a Skew-Symmetric Matrix With an Example? In other words, we can say  that matrix P is said to be skew-symmetric if the transpose of matrix A is equal to the negative of Matrix A i.e (AT = −A). If now is an eigenvalueforQ,letw bethecorrespondingeigenvector kQwk= k wk= j jkwk (9) The result of the scalar product of skew-symmetric matrices is always a skew-symmetric matrix. The sum of symmetric matrix and skew-symmetric matrix is always a square matrix. Let us understand this through a skew-symmetric matrix example. which decomposes ##A## into symmetric and skew parts. As we know that similar matrices have similar dimensions, therefore only the square matrices can either be symmetric or skew-symmetric. 2] Every n-square skew-symmetric matrix A over field F is congruent over F to a canonical matrix 2. A square matrix A is said to be skew-symmetric if A T = −A. A square Matrix A is defined as  skew-symmetric if aij = aji for all the values of i and j. Similarly, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative. Properties of Symmetric and Skew-Symmetric Matrices There are some rules that come from the concept of Symmetric and Skew-Symmetric Matrices, 1. What is symmetric and skew symmetric matrix ? In other words, we can say that both a symmetric matrix and a skew-symmetric matrix are square matrices. Related Question. 427, 428), Dr. Henry Taber has proved the following theorem: A real symmetric matrix less one of its multiple latent roots has a nullity equal to its vacuity. Now for any real vector v, if (I+A)v = 0 we have 0= vT(I+A)v= vTv+ … A matrix is onlys stated as a symmetric matrix if its transpose is equivalent to the matrix itself. All the elements included in the main diagonal of the skew matrix are always equal to zero. Any square matrix can be expressed as the sum of a symmetric matrix and a skew-symmetric matrix. Let A be a square matrix. 1.-INTRODUCTION. A scalar multiple of a symmetric matrix will also be considered as a symmetric matrix. The process of finding the determinant of a symmetric matrix and the determinant of skew-symmetric is the same as that of a square matrix. Similarly, a skew-symmetric matrix is determined by n(n − 1)/2 scalars (the number of entries above the main diagonal). When both identity matrix and skew-symmetric matrix are added, the matrix obtained is invertible. 1] Every matrix B = P T AP congruent to a skew-symmetric matrix A is also skew-symmetric. As we know that similar matrices have similar dimensions, therefore only the square matrices can either be symmetric or skew-symmetric. Properties. The difference between both symmetric matrix and a skew-symmetric matrix is that symmetric matrix is always equivalent to its transpose whereas skew-symmetric matrix is a matrix whose transpose is always equivalent to its negative. The below skew- symmetric example helps you to clearly understand the concept of skew matrix. In If the symmetric matrix has different eigenvalues, then the matrix can be changed into a diagonal matrix. Skew Symmetric Matrix Any square matrix is said to Skew Symmetric Matrix if the transpose of that Matrix is equal to the negative of the matrix. Theorem 7.2. That is if we transform all the Rows of the Matrix into respective columns, even then we get same matrix with change in magnitude. A square matrix A is said to be symmetric if A T = A. Proof: Let A be a square matrix then, we can write A = 1/2 (A + A′) + 1/2 (A − A′). Hence, the total of all the elements of the skew matrix in the main diagonal is zero. In the Proceedings of the London Mathematical Society (Vol. The matrix Ais skew-symmetric if and only ifAt=-A, where Atis the matrix transpose. Your IP: 211.14.175.20 The determinants of a skew matrix can be found out using co-factors and can state that its determinant is equivalent to zero. What Are the Determinants of a Skew-Symmetric Matrix? If we have any  skew-symmetric matrix with odd order then we can straightly write its determinants equals to zero. Every diagonal matrix is symmetric, since all off-diagonal entries are zero. = −A). For example, If M is a symmetric matrix then M = M, and if M is a skew-symmetric matrix then M = - M, rows, then symmetric matrix is expressed as, In the above symmetric matrix example given below, we can see a, = -4.In other words, it is stated that the transpose of matrix M is equivalent to the matrix itself (M=M. Completing the CAPTCHA proves you are a human and gives you temporary access to the web property. (a) We will say that the skew symmetric tensor a ∈ g Λ g is a solution of the generalized classical Yang-Baxter equation if the tensor [a; a] ∈ g Λ g Λ g is a 0-cocycle in the Chevalley cohomology of g with values in g ⊗ g ⊗ g, equivalently with values in g Λ g Λ g, and relative to the adjoint representation, ad ^, of g on g ⊗3, equivalently with values in g ⊗3. In the above skew matrix symmetric example, we can see all the elements present in the main diagonal of matrices A are zero and also a12 = -2 and  a21 = -2 which implies that a12 = a21 .This conditions is valid for each values of i and j. We extend the symmetric and Hermitian algorithms MINRES and MINRES-QLP by Choi, Paige and Saunders (2011) to complex symmetric, skew symmetric, and skew Hermitian systems. For the Given Below Matrix M, Verify That (M + M') Is a Symmetric Matrix. Below mentioned formula will be used to find the sum of the symmetric matrix and skew-symmetric matrix. 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text:
# How do you find the slope and intercept of 3x + 9= y?

Mar 5, 2016

$y = 3 x + 9$
The slope intercept form of st line is $y = m x + c$,where m is the slope and c is the y-intercept i.e. intercept from y-axis


text:
Mathematical and Physical Journal
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Problem P. 4746. (May 2015)

P. 4746. The following experiment is usually shown in order to demonstrate that light is a transverse wave:

A narrow parallel beam of natural light hits a plane glass, such that the reflected and the refracted rays are perpendicular to each other. Then another plane glass is used to reflect the reflected ray, such that the angle of incidence is the same as in the previous case. If this latter glass is rotated about an axis which coincides with the incident light ray, then it may happen that there is no reflected ray from this piece of glass.

$\displaystyle a)$ How does this experiment proves that light is transverse wave?

$\displaystyle b)$ What is the angle of incidence, if the refractive index of the glass is $\displaystyle n=1.5$?

$\displaystyle c)$ What is the angle between the planes of the two pieces of glass when there is no reflected ray from the second glass?

(5 pont)

Deadline expired on June 10, 2015.

Statistics:

 26 students sent a solution. 5 points: Asztalos Bogdán, Blum Balázs, Csorba Benjámin, Fehér Balázs, Fekete Panna, Forrai Botond, Fülöp Erik, Holczer András, Kasza Bence, Marosvári Kristóf, Olosz Balázs, Öreg Botond, Sal Kristóf. 4 points: Büki Máté, Szentivánszki Soma . 3 points: 10 students. 0 point: 1 student.

Problems in Physics of KöMaL, May 2015


text:
# Position image below bullet points in beamer

I am new to LaTeX and am currently busy making a slide presentation using beamer.

I have two bulletpoints and would like to place an image below them, such that the position of the left side of the image starts where the position of the bulletpoints starts. My code is the following:

\documentclass{beamer}

\mode<presentation>
{
\usetheme{default}
\usecolortheme{default}
\usefonttheme{default}
\setbeamertemplate{caption}[numbered]
\graphicspath{ {bwinbe/} }
}

\usepackage[english]{babel}
\usepackage[utf8x]{inputenc}
\usepackage{graphicx}

\begin{itemize}
\item determenistic method to obtain reserve estimates using run-off triangles
\item starts from IBNR run-off triangle
\begin{figure}[]
\caption{Run-off triangle}
\includegraphics{triangle}
\end{figure}
\end{itemize}
\end{frame}

\end{document}

I have tried several options I found on the internet, but nothing seems to work. Can somebody please help me with this?

• Welcome to TeX.SX! While your question is quite clear and the code you posted is quite a nice starting point, it is normally preferred if the code is actually compilable, that is it should (in most cases) start giving the \documentclass and contain \begin{document}/\end{document}. Feb 17, 2015 at 18:56

It's an odd alignment, and you didn't specify anything in terms of the \caption placement.

You could place the image as a custom \item bullet:

\documentclass{beamer}
\usetheme{Warsaw}

\begin{document}

The .5em correction is roughly the space occupied by the bullet.


text:
# Math

posted by .

How many Positive integers are between 200-500 that are divisible by the integers 4,6,10,12?

• Math -

The first number after 200 which meets your conditions is 240.
After that it will be every 240 + HCM of 4,6,10 and 12
which is 60
So the numbers would be 240, 300, 360, 420, and 480

• Math -

Do you mean GCM??? There is no such thing as an HCM well there is butter not in mathmatical terms..

Thanks,
Erika

• Math -

but*

• Math -

wow...

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