Elements Definition and 1000 Threads

This is a basic question about the behavior of $a^k$ for $a\in\mathbb{Z}/n\mathbb{Z}$ and $k=1,2,\dots$. I know that Fermat's little theorem says $a^{p-1}=1$ in $\mathbb{Z}/p\mathbb{Z}$ for prime $p$; thus, the sequence $1,a,a^2,a^3,\dots$ is periodic with period at most $p-1$. More generally...

Consider the following extract taken from page 60 of Matthew Schwartz's 'Introduction to Quantum Field Theory':We usually calculate ##S##-matrix elements perturbatively. In a free theory, where there are no interactions, the ##S##-matrix is simply the identity matrix ##\mathbb{1}##. We can...

So I've been studying advanced linear algebra and have started learning about modules. However, I am having a hard time understanding the difference between a zero divisor and a torsion elements. The definitions seem extremely similar. Can someone offer a good definition of each and an...

Homework Statement Let x and y be conjugate elements of a Group G. Prove that x^n = e if and only if y^n = e, hence x and y have the same order. Homework Equations Conjugate elements : http://mathworld.wolfram.com/ConjugateElement.html The Attempt at a Solution Since y is a conjugate of x...

Hey! :o I am looking at the following exercise: An element $r\in R$ is called nilpotent if $r^n=0$ for some integer $n=1,2, \dots $. Show that if $r$ is nilpotent then $1-r$ is invertible in $R$. Show that if $R$ is commutative then the set $N(R)$ of nilpotent elements is an ideal of...

Can someone please explain the deconstruction and elements of this set. I understand it to be.. F3[X] = {f(x)=a(0)+a(1)X+...+a(n)X^n : a(i) in F3] <x^2+1> = {g(x)(x^2+1): g(x) in F3[x]] So an element in the quotient should be something like f(x)+<x^2+1> Yet, research shows there are nine...

Homework Statement These questions are related to an experiment where we had to record the current for varying values of potential difference (10 readings), then repeat the process with the applied potential difference was reversed a) Why is the resistance of a light bulb not constant? b) Does...

In discussing proportions (a topic to which I have not been properly exposed) Legendre states that, adding the antecedent of a proportion to the consequent, and comparing the sum to the antecedent, one obtains a proportion equal to the original plus unity. Legendre's book is apparently...

I've been looking all over the internet, can calcium and iron form an ionic bond, Gold and Radium, Sodium and Tin. If those elements happened to run into each other. Also, another thing I don't understand is why don't the atoms in my hand bond with atoms on the wall when they come into contact.

I don't understand how it able to skip one element to the other in order to multiply the numbers by 2. It be helpful if anyone can explain.Double any element's value that is less than minVal. Ex: If minVal = 10, then dataPoints = {2, 12, 9, 20} becomes {4, 12, 18, 20}. #include <iostream>...

I am stuck on now to decrement the numbers and would like some help please. Write a loop that subtracts 1 from each element in lowerScores. If the element was already 0 or negative, assign 0 to the element. Ex: lowerScores = {5, 0, 2, -3} becomes {4, 0, 1, 0}. #include <iostream> #include...

I know about quantum numbers and how we do it as 1s,2s,2p,3s,3p,3d etc...but how do I find the number of valence electrons by doing this ? I watched so many videos about this topic and I still don't get it.Possibly because I am trying to learn the portions before even it is taught at school.Can...

Do more unstable elements decay faster than more stable ones? Thanks!

I am having problems with printing out each elements of the array using a for loop It's suppose to look like this: Enter non-negative numbers (ints) terminated by -99 1 2 3 4 5 6 -99 Original list (6 values): 1, 2, 3, 4, 5, 6. the code's output looks like: Enter non-negative numbers (ints)...

I'm designing an oven and want to ensure that the insulation I specify has a low enough Thermal Conductivity (k) to resist excessive heat loss. I determine heat loss (Hout)with the following equation: Hout=A*U*(T1-T0). U is dependent on k (U=k/L). I omitted the heat transfer coefficient in...

Homework Statement I did poorly on my exam, which I thought was very fair, and am now trying to understand certain aspects of perturbation theory. There are a total of three, semi related problems which i have questions about. They are mainly qualitative in nature and involve an intuitive...

I am looking at one idea to passive vibration supression that involves "a resistive shunt to a piezoelectric material" but I can not find a lot of info outside of papers that I would have to pay for. does anyone know and can explain some detail about this method?

In glass, the elements that make it up, like silicon, and sodium. Does the electron eV levels change as these elements become molecules [PLAIN]http://www.chemicalforums.com/Smileys/classic/grin.gif, and become solid glass. To put it another way, if silicon, sodium, and calcium atoms which have...

Ok, so I read that a tuple is an ordered list of n elements, where n is a non-negative integer. Sometimes I get confused with tuples and partially ordered sets. So if I organized a list of all the airplanes in a particular airport, would that be a tuple, or would that be a set? I'm leaning...

Hey! :o Let $G$ be a finite group. I want to show that the probability that two elements of $G$ commute is $\frac{m}{|G|}$, where $m$ is the number of conjugacy classes of $G$. A conjugacy class is $O_x=\{g*x\mid g\in G\}=\{g^{-1}xg\mid g\in G\}$, right? (Wondering) Do we maybe take $x$...

There is no longer a history section in PDF, but this is not suitable for any speciality forum. Just yesterday*, as usual a step ahead and with uncanny prescience, I was recommending a student to familiarise himself with Moseley's law as reinforcement and more in the learning of atomic physics...

Let $S = \left\{x_1, \ldots, x_k\right\}$ be independent set consisting of k elements in a linear space V and let $l(S)$ be the subspace spanned by $S$. Then every set of k+1 elements in l(S) is dependent. I've gotten nowhere with this - but I'm posting my attempt nonetheless if only for the...

I know that Arsenic sublimes directly when heated. So I want to know which other elements like this sublimates on heating throughout the periodic table. I also know dry ice. thanks

Is there a way to create a grid here on PF that has square elements? I ask because I'm yearning to make a post that will require a Cayley table.

Hey! :o To show that in the additive group $\mathbb{Z}_2\times\mathbb{Z}$ there are non-zero elements $A,B$ of infinite order such that $A+B$ has finite order, we have to find such $A$ and $B$, right? (Wondering) How do the elements of $\mathbb{Z}_2\times\mathbb{Z}$ look like? (Wondering)

Homework Statement A flat part with constant thickness b is loaded in tension as shown. The height changes from 50 to 87 mm with a radius r = 4.0 mm. Find how much lower a load can be transmitted through the bar if the height increases from 50 to 100 mm and the radius increases from 4.0 to 10...

Homework Statement If elements with principal quantum number n > 4 were not allowed in nature, the number of possible elements would be (A) 60 (B) 32 (C) 4 (D) 64 Homework EquationsThe Attempt at a Solution I got 36 from the periodic table (2+8+8+18) However, that isn't even an option. The...

Good morning, I was making some investigation regarding the calculation of mean orbital elements - not osculating. Is this done simply by fitting (with the best equation) several osculating elements across time, or is there a mathematical approach already settled? Cheers,

Homework Statement Homework Equations δ = Pl/AE σ = P/A The Attempt at a Solution I had calculated the deformation for both the steel and aluminum, and added them together to give me δ = 0.481 mm, which is incorrect. I also thought I could simply plug in the force P = 200kN and the...

Hello, Suppose we have a semigroup S with a subgroup G≤S. Assume there is an element s∈S that commutes with all the elements in G. Does this statement implies (or is equivalent to) another statement? If hypothetically the element s would have been in G, then we could have said that s was an...

Sorry to disturb you again, but I thought of what you said in my previous questions. Dichroic mirrors, polarizers are used in the above entanglement experiment but according to their encyclopedic definition, they only filter light by color and polarization. Half-wave plates and quarter-wave...

Today's APOD http://apod.nasa.gov/apod/ap160125.html Shows this fascinating table. What a pity that is doesn't show isotopes. Strange are Ba, La, Ce which are shown as large star but not supernova. The article says that the origin of Cu is not well known. Fascinating. I'm curious about the...

Hello all, I have been doing a number of analysis that require the use of Workbench and Spaclaim and Classic (APDL). SpaceClaim is used to bring in the geometry and defeature it and to create the proper components with elimination of contacts thru the use of the Share Topology feature. This is...

What do you call metals under metalloids on the periodic table? Since they arent metalloids nor transition metals. Do they have a specific name? Thanks!

Greetings all, In Hartle's Gravitation, it seems like a given the way he manipulates the "differentials" when he gives a line element. The thing is that it works, and I know it's not mathematically rigorous, but first of all, I'm wondering when treating the differentials in the line elements in...

Although strictly quantum mechanics is defined in ##L_2## (square integrable function space)， non normalizable states exists in literature. In this case, textbooks adopt an alternative normalization condition. for example, for ##\psi_p(x)=\frac{1}{2\pi\hbar}e^{ipx/\hbar}## ##...

Elements of Lie algebra are generators. So for example Pauli matrices are generators of rotation and the elements of Lie algebra. And multiplication in Lie algebra is commutator. Right? What about if there is only one generator. As in case in rotation in plane. What is Lie algebra product in...

So I know that all the elements above 92 on the periodic table do not occur naturally, does that mean that we've discovered all the naturally occurring elements? Is it possible that there can be elements with a higher atomic number than 118 that we just haven't discovered yet in nature?

Most books discribe NMR as a diagnotic technique for Hydrogen and its isotopes. Could Nuclear Magnetic Resonance Spectroscopy be applied for other elements with odd atomic number besides H? If so, what is the major challenge for a multi-element scanning?

Right now I think there's 118 elements. If somewhere in space it was a certain temperature, with certain pressure, and other certain factors, could there be more elements. Let's say these conditions are too extreme to be made on Earth like a Trillion degrees Celsius. So if there were there...

One of the classes of the group C5v is written C=(2*C5). So why do we mean by 2*C5 and how we can know the elements of this class from this writing ? Thanks

If S={a,b,c}, what does it mean that S=T when T={a,a,a,a,a,a,b,c}? The mapping that confirms the definition of equality assumes that the duplicate symbols in a set are representative of the same entity or idea. If S={1/2} and T={ .5 , 2/4 , ,25/,5 , 4/8 } are these sets equal? At what point...

Homework Statement A and B are both spans. A = span(a1,a2,a3) B = span(b1,b2) Then how many elements would AUB have? Homework Equations N/A The Attempt at a Solution [/B] I'm almost certain the answer is 2, with the two elements being span(a1,a2,a3) and span(b1,b2) but I'm unsure as to...

Homework Statement Give an element-wise proof for the following: If A⊆B and B⊆C', then A ∩ C = ∅ Homework Equations A is a subset of B (written A ⊆ B) if every element in the set A is also an element in the set B. Formally, this means that fore every x, if x ∈ A, then x ∈ B. A ∩ B = { x ∈ U ...

I'm trying to understand the maths of QM from Shankar's book - Principles of Quantum Mechanics: On page 21 of that book, there is a general derivation that if we have a relation: |v'> = Ω|v> Where Ω is a operator on |v> transfroming it into |v'>, then the matrix entries of the operator can be...

Good afternoon, I wonder if someone can help me in a small doubt. I'm trying to calculate the orbital parameters of some solar system bodies. It's quite easy knowing their positions and velocities. But my question is this: these two components, position and velocity, must they be heliocentric...

p here stands to power (v in power equation is magnitude), in addition to this : is it also true that in the first one : V = RI (V upper - V lower) and in the second one : V = - RI (V lower - V upper) ? (current is magnitude in all previous equations)

The periodic table is globally known! But, it relates EACH element separately! Exist some table, some lists, some spreadsheet, etc, that relates the combinations between the elements and give us the information about each combination? PS: for "combined elements" I want to say simples substance...

Please I need your help in such problems.. in terms of ladder operators to simplify the calculation of matrix elements... calculate those i) <u+2|P2|u> ii) <u+1| X3|u> If u is different in both sides, then the value is 0? is it right it is 0 fir both i and ii? when exactly equals 0, please...

Homework Statement Determine the voltage across all elements Homework Equations i=v/r v1+v2+v3+..=0 The Attempt at a Solution I'm not sure which of the nodes I should use to take my KVL from. The a and b are labeled, but should I combine the set of 2-6-6 resistors and label that as an...