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.
Homework Statement
Which of the symptoms is characteristic for food allergy
bumps (pimple) on the face
difficulty in breathing
flue
none of these
Homework EquationsThe Attempt at a Solution
I'm between the first and second.
Hi all
I have the following question that I require help with.
1. Homework Statement
A supply fan must deliver 2.5m3s of air against a system pressure of 500 Pa.The selected fir this has a performance characteristic as listed below when running at a speed of 1800 revs/min.
i) Plot the system...
Homework Statement
I am trying to figure out following problem.
Let A ⊂ R. Then we can define the characteristic function:
\begin{align}
\chi_A : R → \{0, 1\}, x = \begin{cases}
1 & \text{if } x \in A \\
0 & \text{else }
\end{cases}
\end{align}
Let a be bigger than 0. I am...
In an experiment one observes the characteristic Ka X-rays of two elements at energies of 6.930 and 7.478 eV. The higher energy line is due to Ni. What element is responsible for the lower energy line?
Solve the following Cauchy problem using the Method of characteristic curves:
$x_1u_{x_1}+x_2u_{x_2}=\alpha u$ in $\mathbb{R}^{+}\times \mathbb{R}^{+}$
$u(x_1,1)=g(x)$ for all $x_1\in \mathbb{R}^{+}.$
Find the local solution for the problem.Hello. I get as solution $g(s)=s$, I want to know if...
Homework Statement
Hello. This should be rather easy. I am trying to figure out what the Characteristic length I should use in calculating the grashof number for a greenhouse. The dimensions given for the greenhouse are 1.5 m wide x 2 m long and 2 m high. What characteristic length should I...
Hello all,
Homework Statement
$$x{u_{xy}} - y{u_{yy}} = 0$$ Assume $$x,y \in {\rm{Reals}}$$
Homework Equations
I have been able to solve this using different methods, but my classmates and I are trying to figure out if there is a way to do this using the methods from the course's text. The...
I have come across this inequality:$$ g≤ log\ n <g + 1$$
We assume that the base of the log is b >1 and n is all the natural numbers. I would like to know if anyone could provide a proof regarding this and mention for what g ? Is it for all the g which are integers ?
Homework Statement
If I have the characteristic equation:
-λ3 + 3λ2 + 9λ + 5
And I'm told that one of its eigenvalues is -1.
How do I find the rest of the eigenvalues?
Homework Equations
-λ3 + 3λ2 + 9λ + 5
The Attempt at a Solution
The furthest I can get is:
-λ3 + 3λ2 + 9λ + 5 = (λ + 1) x...
Hi everybody!
The problem which I am dealing with is surprisingly very well depicted in open source lecture notes made in October 29th of 2012, titled: " MAE 5310: COMBUSTION FUNDAMENTALS", available on the web...
THIS IS NOT A HW/COURSEWORK QUESTION!
I am simply revising for an exam and am trying to better understand the behavior of the stripline as shown in the file attached. What would be the effect on the characteristic impedance, as W/b/epsilon_r/t increases (each separately)?
Homework Statement
Using the diagram for the I-V characteristic of a semi-conductor (attached). Answer the following questions:
1. Why is the resistance of an I-V characteristic not determined by the gradient of the graph
2. Explain why the temperature of semiconductor increases
3. Explain why...
Homework Statement
Given an NxN symetric tri-diagonal matrix, derive the recursion relation for the characteristic polynomial Pn(λ)
Homework Equations
Pn(λ) = |A -λI |
Pn(λ) = (An,n - λ)Pn-1(λ) - A2n,n-1Pn-2(λ)
The Attempt at a Solution
This was easy to do by induction, but I am always...
Hey! :o
We have the equation $$2u_{xx}-u_{tt}+u_{xt}=f(x, t)$$
This is equal to $$\left (\frac{2\partial^2}{\partial{x^2}}-\frac{\partial ^2}{\partial{t^2}}+\frac{\partial ^2}{\partial{x}\partial{t}}\right )u=f$$
To find the characteristics do we solve the homogeneous equation...
Homework Statement
I'm looking to find the characteristic function of
p(x)=\frac{1}{\pi}\frac{1}{1+x^2}[/B]
Homework Equations
The characteristic function is defined as
\int_{-\infty}^{\infty} e^{ikx}p(x)dx
3. The Attempt at a Solution
I attempted to solve this using integration by...
Homework Statement
What are the energies of the characteristic Kα line and Kβ lines?
Homework Equations
K-shell ( n =1):-69.5 keV
L-shell ( n =2 ): -11.3 keV
M-shell ( n =3 ): -2.30 keV
Z = 74
E = (energy level) (Z-1)2/n2
The Attempt at a Solution
for alpha i have
(-11.3keV) (74-1)2/22 for n...
Firstly, apologies if this is in the wrong thread.
I'm currently writing a presentation on the physics of getting a spacecraft from Earth to Mars in the near future. In my research I've come up against Porkchop plots which seem to plot contours of equal characteristic energy so you can find...
Dear all!
I have a question about the impedance characteristics in the distance relay used for transmission line protection. I understand that in the first quadrant we have a inductive load and in the fourth quadrant we have a capacitive load. I, however, do not understand what we measure in...
We can find differential of alloy and composite. MMC is came from 2 differential of metal, one from it aren't melting same as producing alloy. but there are might produce new reaction. ex: C with Fe, the name is carbon steel. the nano-composite just too, but the particle aren't big, and the size...
I am looking for a local field of positive characteristic, like Q22 was used in this article:
http://8pic.ir/images/s9oiiuqqkq989w3posu9.png
in fact, i need an another Example of a local field of positive characteristic like Q22 .
Homework Statement
Hi guys,
I am not sure what the characteristic time of something is. I've been looking online but am getting confused with the definitions. Say, for example, the characteristic time for some particles suspended it water to descend to the bottom.
Homework EquationsThe...
Hello! (Wave)
In my notes, there is the following theorem:
Theorem:
Let $Q(X_1, X_2, \dots, X_n)=\sum_{i,j=1}^n a_{ij} X_i X_j$
$$a_{ij}=a_{ji}, \ \ \ \ ch K \neq 2 ( \text{ Characteristic of polynomial })$$
symmetric quadratic form of $n$ variables.
There is a linear transformation of the...
In physics one often invokes arguments about a characteristic scale. For example at distances much larger than the Compton wavelength (a characteristic length scale) one might ignore quantum phenomena and calculate classically.
I would like to read up some more on the thinking behind such...
I have just finished a junior-level lab assignment in which we used X-ray diffraction to determine the lattice constants of unknown materials. In the theory section of the lab write-up, it briefly explains the K-alpha doublet of the characteristic spectrum. I understand that it is the result...
Find the characteristic function for the PMF \(p_X[k] = \frac{1}{5}\) for \(k = -2, -1,\ldots, 2\).
The characteristic function can be found with
\begin{align*}
\phi_X(\omega) &= E[\exp(i\omega X)]\\...
I am integrating the characteristic equation in order to recover the PMF, but I am going to get the answer to be zero so something went wrong.
\begin{align*}
p_X[k]...
Hello, All Outstanding electrical engineering student, I have a some question.
As posted,
1. At low frequency, the inversion electrons will be able to respond to the ac voltage
=> CG (ω= 0) = Cox (Why?).
2. At high frequency, inversion electrons are not able to respond to ac voltage...
Hello Everyone,
I am currently reading about when the Quantum Mechanical Theory becomes relevant. A term that continually emerges is "characteristic length." Could someone provide me with a thorough definition of this concept?
Thank you!
I am working on impedance measurements of close-wound electromagnetic coils. I am using an Agilent LCR meter to measure the impedance of these coils over a frequency range of 20Hz-2MHz. When I perform the measurements, I get impedance magnitude and phase angle in degrees. I would like to...
Hello everyone!
In the following speed-torque characteristic of a DC shunt motor, we see that if the speed exceeds a certain value, then the machine becomes a generator.
I don't understand something. How is it possible for the machine to have a negative torque and a positive speed...
Homework Statement
Find the output signal for an amplifier with transfer characteristic:
vO = 10-10-11e40*vi
output bias of 5V
input bias of 0.673 V
and an input signal of 1 mV superimposed on the bias
Homework Equations
vO = bias out + signal out
vI = bias in + signal in
The...
Hello guys, I posted this question in the classical mechanics forum (thinking stochastic mechanics falls into the classical mechanics category) but I haven't gotten an answer there. I was told I'd be better off posting the question here. I don't know how to move a thread, so I'll just copy and...
Homework Statement
suppose u(x,y) satisfies the partial differential equation:
-4y\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=0
Find the characteristic curves for this equation and name the shape they form
The Attempt at a Solution
\frac{dy}{dt}=1 \Longrightarrow y=t+y_0...
Now w/ Attachment: Telephone Wire w/ Load and Characteristic Impedance
Homework Statement
Please see attachment.
Homework Equations
I(z)=I+(exp(-γz)) + I-(exp(γz))
V(z)=V+(exp(-γz)) + V-(exp(γz))
The Attempt at a Solution
I solved easily for the characteristic impedance and gamma...
Homework Statement
A:B→B a linear operator
Show r is multiple root for minimal polynomial u(x) iff
>$$\{0\}\subset \ker(A - rI) \subset \ker(A - rI)^2$$
note: it is proper subsetHomework Equations
The Attempt at a Solution
Homework Statement
My thought:
I know ker(A−rI) is basically {{0}...
Dear all,
Greetings! I was given a problem from Reichl's Statistical Physics book. Thank you very much for taking time to read my post.
Homework Statement
The stochastic variables X and Y are independent and Gaussian distributed with
first moment <x> = <y> = 0 and standard deviation...
Homework Statement
The stochastic variables X and Y are independent and Gaussian distributed with
first moment <x> = <y> = 0 and standard deviation σx = σy = 1. Find the characteristic function
for the random variable Z = X2+Y2, and compute the moments <z>, <z2> and <z3>. Find the first 3...
Find characteristic functions of
1. The random variable X uniformly distributed on[-1..1]
2. The random variable Y distributed exponentially (with exponent λ)
3. The random variable Z=X+Y
Homework Statement
See attached image
Homework Equations
The Attempt at a Solution
For the first half of the question, ordered pairs would be (1, [1]), since 1 and [1] are the multiplicative identities in these rings. but no matter how many times we add (1, [1]) to itself...
This was something I noticed as I was trying to practice solving PDEs using the method of characteristics.
The text has the following example: $$\frac{\partial u}{\partial x} + x \frac{\partial u}{\partial y} = 0$$
This should be easy enough. I let p(x,y) = x and solve for \frac{\partial...
Similar matrices share certain properties, such as the determinant, trace, eigenvalues, and characteristic polynomial. In fact, all of these properties can be determined from the character polynomial alone.
However, similar matrices also share the same rank. I was wondering if the rank is...
Hi there,
I'm an instructor teaching a Physics course for Radiography students designed for students with little to no Physics background. In prepping the course, I've come up with a couple of questions that are bothering me and hope someone here can be of assistance.
My questions pertain to...
I'm just starting to read up on RF circuits, and I'm a little confused about what the characteristic impedance of a system is (not the characteristic impedance of a transmission line, I understand that pretty well). For example, in Pozar's book he talks about a 3 dB attenuator with a 50 Ω...
I've been looking for proof of the fact that the characteristic polynomial of an n by n matrix has degree n with leading coefficient ## (-1)^{n} ##.
I first tried proving it myself but my method is a bit strange (it does use induction though) and I am doubting the rigor, so could perhaps...
Homework Statement
The pump characteristic curve of a centrifugal pump can be plotted from the following details:
(See attachment)
The pump delivers 90 L/s when pumping against a static head of 12 m. Plot the pump characteristic curve from the details given and calculate:
1. The...
Homework Statement
The given quantities are the shunt resistances across each of the 3 diode junctions,assume them to be Rsh1,Rsh2,Rsh3; Tunnel diode resistances as Rt1 and Rt2, Photocurrent for each of the junction be Ip1,Ip2,Ip3 and bandgap of each subcell be Eg1,Eg2,Eg3 are given.Let V & I...