What is Characteristic: Definition and 306 Discussions
In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive identity the ring is said to have characteristic zero.
That is, char(R) is the smallest positive number n such that
1
+
⋯
+
1
⏟
n
summands
=
0
{\displaystyle \underbrace {1+\cdots +1} _{n{\text{ summands}}}=0}
if such a number n exists, and 0 otherwise.
The special definition of the characteristic zero is motivated by the equivalent definitions given in § Other equivalent characterizations, where the characteristic zero is not required to be considered separately.
The characteristic may also be taken to be the exponent of the ring's additive group, that is, the smallest positive n such that
a
+
⋯
+
a
⏟
n
summands
=
0
{\displaystyle \underbrace {a+\cdots +a} _{n{\text{ summands}}}=0}
for every element a of the ring (again, if n exists; otherwise zero). Some authors do not include the multiplicative identity element in their requirements for a ring (see Multiplicative identity: mandatory vs. optional), and this definition is suitable for that convention; otherwise the two definitions are equivalent due to the distributive law in rings.
Hi I am a little confused as to what length to use as the characteristic length of the Reynolds number for a body in and incompressible flow that has no boundary conditions(ie there all open).
im going to try and draw a few examples to help me out..(i have no idea why..) from what i have read...
I have an x-ray tube. When I see the spectrum, I notice the characteristic lines of the anode. What do I have to do if I want to absorb a characteristic line?
I have thought that I can add a filter. But.. which filter? Should it be an element such that the bound energy for the K-shell electrons...
Hi, this is not a homework. I need your help to understand basic concepts. I am learning by myself.
In textbooks, I usually see the circuit used to make Ic -Vce characteristics.
I see that there are no resistors at base and collector/emitter.
What will happen if VBE is larger than 1V or...
We all know that the Euler characteristic is a topological invariant. But let's suppose that we don't know this or anything else about algebraic topology for that matter. We are given only the Gauss-Bonnet theorem, which expresses the Euler characteristic in geometrical terms. In his string...
Hi
This is just a quick question, it should have a pretty simple answer however I can't find the information anywhere on the internet.
When you are calculating/estimating the heat transfer coefficient for a prismatic fin using the reynolds, prandtl, grashof, and nusselt number, what length do...
Homework Statement
In order to determine the characteristic function of a random variable defined by: Z = max(X,0) where X is any continuous rv, i need to prove that:
F_{l,v}(g(l))=[ \phi_{X}(u+v)\phi_{X}(v) ] / (iv)
where F_{l,v}(g(l)) is the Fourier transform of g(l) and...
Homework Statement
Let V:= ℝ_{2}[t]
V \in f: v \mapsto f(v) \in V, \forall v \in V (f(v))(t) := v(2-t)
a) Check that f \in End(V)
b) Calculate the characteristic polynomial of f.
Homework Equations
The Attempt at a Solution
a) Is it sufficient to check that (f+g)(t)=f(t)+g(t) ...
Why does Quantum Mechanics have objective statistical characteristic,but does not have subjective statistical characteristic?
QM has two axioms:
1-To each dynamical variable there coresponds a linear operator,and the possible values of the dynamical variable are the eigenvalues of the...
I've been having this small problem with modulo arithmetic that's just irking me and this is the perfect example to get clarifivation
find the remainder for 75 congruent to x (mod 21)
here's my issue...so I'd set this up to solve and see if I could find this relationship
7\equiv x...
Homework Statement
Hey guys. I am trying to calculate the Voltage Transfer Characteristic (VTC i.e. Vout vs. Vin) of a simulated circuit in Multisim but for some reason it doesn't appear to be working. I pressed Simulate --> Analyses ---> DC Sweep --> Selected the input pulsed voltage as...
Homework Statement
The characteristic function of a set E is given by χe = 1 if x is in E, and χe = 0 if x is not in E. Let N be a natural number, and {an, bn} from n=1 to N, be any real numbers. Use the definition of the integral (Riemann) to show that \int \sum b_{n} X_{ \left\{ a_{n}...
I am reading Protter's book on stochastic integration and differential equations.
On a couple of occasions the author used some properties of characteristic functions I am not familiar with. I would appreciate if someone could point me in a right direction.
1. p22 T31 - We are working with...
Suppose your characteristic equation for the 2nd order equation has complex roots
r+ and r-
These are conjuagtes of each other so the general solution is:
y = Aer+ + Ber-
My book chooses the constants A and B as conjugates of each other for the reason that this constructs a real...
Hello everyone, first time poster, long time reader here!
I'm an ex-math major and while I'm no longer pursuing a degree anymore in mathematics, I still continue onwards in my spare time trying to learn as much as I can about it because it's always been something I've enjoyed partaking in and...
Homework Statement
1)The pion is an unstable subatomic particle whose characteristic decay time in its own frame is 26ns. A beam of such particles emerges from a source at point X in the laboratory traveling with speed ##v = \sqrt{3}/2\, c## in the laboratory frame. Determine i) the decay...
Homework Statement
Given the initial value problem:
\frac{(u)}{(1-e^-(2x))}u_{x}+ \frac{\sqrt{t}}{u}u_{t}=1, with x, t, u > 0
Subject to condition u(x,1)=e^{-x}
Homework Equations
a) Classify given partial differential equation.
b) Write the characteristic equations. By...
A second order system has the following standart form;
http://controls-design.com/mathtex/mathtex.cgi?H%28s%29%3DK%5Cfrac%7B%5Comega_n%5E2%7D%7Bs%5E2%2B2%5Czeta%5Comega_n%20s%2B%5Comega_n%5E2%7D%20%5Cmbox%7B%20for%20%7D%200%20%5Cle%20%5Czeta%20%5Cle%201
However, sometimes the system I...
Homework Statement
Show that the descriminant of the characteristic polynomial of K is greater than 0.
K=\begin{pmatrix}-k_{01}-k_{21} & k_{12}\\
k_{21} & -k_{12}
\end{pmatrix}
And k_i > 0
Homework Equations
b^2-4ac>0
The Attempt at a Solution
I have tried the following...
What is the load line equation for I-V Characteristic plots?
Why is load line slope negative? According to Ohm's Law: i = V/R; so as voltage increases current increases then slope should be positive, right?
Homework Statement
Let X,W,Y be iid with a common geometric density f_x(x)= p(1-p)^x for x nonnegative integer
and p is in the interval (0,1)
What is the characteristic function of A= X-2W+3Y ?
Determine the family of the conditional distribution of X given X+W?
Homework Equations...
Homework Statement
There are a few problems I have with my HW that involve the characteristic size of an atom in a certain state. For example, the hydrogen atom in the ground state.
Homework Equations
-
The Attempt at a Solution
I don't know where to begin to find the characteristic...
Hello guise.
I am familiar to a method of diagonalizing an nxn-matrix which fulfills the following condition:
the sum of the dimensions of the eigenspaces is equal to n.
As to the algorithm itself, it says:
1. Find the characteristic polynomial.
2. Find the roots of the characteristic...
I am reviewing for the mathematics GRE subject exam, and I have this review book, and in the book when they speak of the characteristic of a ring they give the following def "Let R be a ring. The smallest positive integer n such that na = 0 for every a in R is called the characteristic of the...
Characteristic function and preimage?
Homework Statement
Let S be a nonempty subset of ℝ.
Define χs= { 1 if x is in S and 0 if x is not in S
Determine χs-1(Q) [where Q=set of all rational numbers]
and χs-1((0,∞))
We haven't really dealt much with this function, and I really...
Homework Statement
I must find the characteristic function of the Gaussian distribution f_X(x)=\frac{1}{\sqrt{2\pi}\sigma} e^{-\frac{1}{2} \left [ \frac{(x- \langle x \rangle )^2}{\sigma ^2} \right ]}. If you cannot see well the latex, it's the function P(x) in...
Homework Statement
I must calculate the characteristic function as well as the first moments and cumulants of the continuous random variable f_X (x)=\frac{1}{\pi } \frac{c}{x^2+c^2} which is basically a kind of Lorentzian.Homework Equations
The characteristic function is simply a Fourier...
Homework Statement
Hey guys, I'm self studying some probability theory and I'm stuck with the basics.
I must find the characteristic function (also the moments and the cumulants) of the binomial "variable" with parameters n and p.
I checked out wikipedia's article...
Regarding the term "Spin"...
I've see a number of different descriptions. In the book I'm currently reading it says it is "loosely" the measure of the strength of rotational motion, and "technically" the measure of the quantity called angular momentum.
It also states we consider the spin...
Homework Statement
Show that for any square matrices of the same size, A, B, that AB and BA have the same characteristic polynomial.
Homework Equations
The Attempt at a Solution
I understand how to do this if either A or B is invertible, since they would be similar then. I saw a...
In this thread Antiphon had written that
When I first encountered that, I thought there's something wrong with that, but could not figure it out. In order to create a current the electron must flow from one place to another. But in EM radiation although the electron oscillates or jiggles, it...
I was looking back at a proof I did a while ago. Suppose na=0 with n≠0 and a≠0. Then n is a multiple of the characteristic. I supposed p < n is our characteristic, then I simply used the divison algorithm (I divided n by p) and the distributive property which lead to the remainder being 0. From...
Characteristic energy "units" of primordial fluctuations if gaussian
Correct me if wrong, but I think a purely gaussian distribution of the primordial fluctuations could be characterized by a certain unit of energy (which I'll express as mass). If so, then the observed fluctuations are...
Homework Statement
Transmission line is designed so it has selfinductance, L = 7.0x10^-7 Hm^-1 & signal speed 70% of c
Find characteristic impedance
Homework Equations
Z = √L0/C0 Ω
also
v = 1/ √L0C0
The Attempt at a Solution
So
L = 7.0x10^-7 Hm^-1
v = 0.7c (c = 3x10^8...
Hi Guys,
Attached is a problem from an old exam for a Process Control and Instrumentation unit.
I have tried everything I know (which isn't much, it's not the main assessable portion of the unit).
Other questions similar involve giving us either the characteristic closed loop equation...
Homework Statement
Prove that the characteristic function \chi_A: X\rightarrow R, \chi_A(x)=1,x\in A; \chi_A(x)=0, x\notin A, where A is a measurable set of the measurable space (X,\psi) , is measurable.
Homework Equations
a function f: X->R is measurable if for any usual measurable set...
How to compute \chi(\mathbb{C}\mathrm{P}^2)?
This problem is from a class on differential topology, so we have defined the Euler characteristic as the sum of the indices of isolated zeros on a non-vanishing vector field. Off the top of my head, I cannot think of any theorems which really help...
"Given operators σ,τ on a finite-dimensional space V, show that στ=i, and that σ=p(τ) for some polynomial p in F[x]."
The first part was no problem. As for the second, I have a strong suspicion that p is the characteristic polynomial, mostly because I believe I heard of that fact before (that...
If K is a field of characteristic p, and there exists an element a \in K
which is not a pth power (i.e. the Frobenius endomorphism is not
surjective), then I am told we can show x^p - a is an irreducible polynomial
(and since it is not separable our field is imperfect). I see that
x^p - a...
Hi there,
This is my first post...
and be kind on my english please...:)
So here is a problem i cannot solve...I can't reach to something satifactory
your ideas would be very helpful
Homework Statement
The probability density function f×(x) of the random variable X is zero when x<α...
So I know that the characteristic equation for a 2x2 matrix can be given by
t^2 - traceA + |A|
So how would this be generalised for a 4x4 or higher matrix ?
Let $F$ be a field of characteristic $p \neq 0$. Let $K$ be an extension of $F$. Define $T= \{ a \in K : a^{p^n} \in F \text{ for some } n \}$.
a) Prove that $T$ is a subfield of $K$.
b) Show that any automorphism of $K$ leaving every element of $F$ fixed also leaves every element of $T$ fixed...
So we are told in our class that the characteristic impedance of a coaxial cable is impractical to be at a value like 500Ohms. Why is that?
The assumption is that the transmission line is lossless and that the dielectic constant (Er) is 1
Homework Statement
Let A = \begin{pmatrix}1 & 1 & 0 & 0\\-1 & -1 & 0 & 0\\-2 & -2 & 2 & 1\\ 1 & 1 & -1 & 0 \end{pmatrix}
The characteristic polynomial is f(x)=x^2(x-1)^2. Show that f(x) is also the minimal polynomial of A.
Method 1: Find v having degree 4.
Method 2: Find a vector v of...
Hey, I'm working on a proof for a research-related assignment. I posted it under homework, but it's a little abstract and I was hoping someone on this forum might have some advice:
Homework Statement
Suppose T:V \rightarrow V has characteristic polynomial p_{T}(t) = (-1)^{n}t^n.
(a) Are...
Homework Statement
Suppose T:V \rightarrow V has characteristic polynomial p_{T}(t) = (-1)^{n}t^n.
(a) Are all such operators nilpotent? Prove or give a counterexample.
(b) Does the nature of the ground field \textbf{F} matter in answering this question?
Homework Equations
Nilpotent...
Homework Statement
I'm trying to solve an example in Boas's book (Mathematical methods in the physical science page 423) but by using Goldstein's (Classical mechanics) method.
The mechanical system consists of 3 springs (with constants k) and 2 masses (with mass m) this way: |----m----m----|...
Hello all,
We utilized a differential attenuator to display the i-v characteristic of a 1n4005 diode. However, I was a bit confused as to how I could determine what the y-axis was calibrated as. I know that the vertical axis is actually current, but is there some sort of tactic for determining...
This is inspired by Kardar's Statistical Physics of Particles, page 45, and uses similar notation.
Homework Statement
Find the characteristic function, \widetilde{p}(\overrightarrow{k}) for the joint gaussian distribution:
p(\overrightarrow{x})=\frac{1}{\sqrt{(2\pi)^{N}det...