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.
I woud like to find the characteristic curves for ##u_t + (1-2u)u_x = -1/4, u(x,0) = f(x)## where ##f(x) = \begin{cases} \frac{1}{4} & x > 0 \\ \frac{3}{4} & x < 0 \end{cases}##.
By using the method of chacteristics, I obtain the Charpit-Lagrange system of ODEs: ##dt/ds = 1##, ##dx/ds = 1 -...
I was reading about characteristic x rays. I have a question I did not find an answer for it in the book (Concepts of Modern Physics-Sixth Edition-Arthur Beiser) or in the internet search. My question is:
How do the atom in the target (in the x ray production experiment) go to the normal state...
The characteristic equation ## m^3 -6m^2 + 12m -8 = 0## has just one single, I mean all three are equal, root ##m=2##. So, one of the particular solution is ##y_1 = e^{2x}##. How can we find the other two? The technique ##y_2 = u(x) e^{2x}## doesn't seem to work, and even if it were to work how...
Before starting, I will leave the link to the article I am talking about here: http://www.msc.univ-paris-diderot.fr/~phyexp/uploads/LaimantParesseux/aimant2.pdf
I am conducting a similar experiment to the one discussed in the paper above. Basically, I am rolling a neodium supermagnet down a...
Can someone provide more information about this method to measure chracteristic impedance using resistance paper?. Kraus' book claims that the characteristic impedance can be measured by simple dc measurement. It even shows a case to mesure the impedance of a coaxial cable with square outer...
intermolecular distance means distance between particles. So, I imagine a sphere.
$$\frac{4}{3} \pi d^3 = \frac{V}{N}$$
However, Griffitfhs pictures a box instead, where
$$d^3 = \frac{V}{N}$$
And the difference between both models is a factor of ##(4\pi/3)^{2/5} \approx 1.8##, which is...
good evening everyone!
Decided to solve the problems from last year's exams. I came across this example. Honestly, I didn't understand it. Who can help a young student? :)
Find characteristic equation of the matrix A in the form of the polynomial of degree of 3 (you do not need to find...
dx/dt =1, x(0,s)=0, dy/dt=x, y(0,s) = s, du/dt=(y-1/2x^2)^2, u(0,s)=e^s
I did well at the beginning to get x(t,s) =t and y(t,s)=1/2t^2 + s, but got stuck with the du/dt part.
You can sub in x=t and y=1/2t^2 +s for x and y to get du/dt = s^2. But that's still three variables, and I can't see...
I found my energies for Potassium. I have 3.3 KeV for the k alpha nd 254.6 ev for L alpha, using z=19 and n=3. Are these values correct?
Edit: I found the ratio to be .077
I have the characteristic function of the Cauchy distribution ##C(t)= e^{-(\mid t \mid)}##. Now, how would I show that the Cauchy distribution has no moments using this? I think you have to show it has no Taylor expansion around the origin. I am not sure how to do this.
hello everyone!
i have some problem in solution of this problem should i transfert the vertical stree to the horizontal stress and solve it ?
problem :
The equivalent static stiffness of the soil as an infinite elastic half-space under an infinite strip foundation is given in the vertical...
Image 1: Image 2:
I am attempting to learn about transmission lines and am having problems with this homework problem.
For part a, I have derived an equation for the maximum electric field within the dielectric. I came up with: with r being the radius and the electric field decreasing with...
Hello,
I've carried out an experiment to plot the characteristic curves (Ic vs Vce) for a BC108 transistor and then attempted to find where the load-line intersects those curves. Below are my results:
...as you can see, the load-line doesn't intersect the characteristic curves at all...
Could I get a conceptual answer, supported by math, explaining why coax has less characteristic impedance than open wire feeds?I’m new to EE. Thanks for you patience.
I'm trying to get saturation characteristic of current transformer in simulink...
In simulink, there was an example of a current transformer : https://www.mathworks.com/help/physmod/sps/examples/current-transformer-saturation.html
And in this document, it said : the CT is assumed to saturate...
What are the critical Reynolds number for fluid and characteristic length when a cylinder immerses and then rotates in the fluid (The fluid is initially static)?
Please suggest the critical number for the transition from concentric flow to laminar and from laminar to turbulent flow.
I would...
Homework Statement
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I am trying to determien the characteristic function of the function:
$$ f(x)= ae^{-ax}$$
$$\therefore E(e^{itx}) =\int_0^\infty e^{itx}ae^{-ax} dx = a \cdot \frac{e}{it-a} |_0 ^ \infty $$
But I am not sure how to evaluate the integral.
Wolfram alpha suggests this...
Do we have reflection when the intrinsic impedance η=E/H between two media are matched but not necessarily the characteristic impedance (assuming a transmission line)?
Basically, I have a case here shown below
I have two parts with different geometries (this may not be a transmission-line, if...
Hello,
A few years ago I heard about a list named the "10 characteristic experiences of physics", may I know if a such list exist?
A few years ago, at the physics workshops at the Palais de la découverte, I saw the so-called "triple point of water" experience.
The experimenter told us that...
Consider a scattering between two particles a and b that produces two particles c and d: d is stable, while c decays in two other different particles e and f.
The first interaction is by strong force (time of interaction ##t_1\sim 10^{-23}s##, which is also the time of generation of c and d)...
Homework Statement
I am trying to understand the very last equality for (let me replace the tilda with a hat ) ##\hat{P_{X}(K)}=\hat{P(k_1=k_2=...=k_{N}=k)}##(1)
Homework Equations
I also thought that the following imaginary exponential delta identity may be useful, due to the equality of...
This thread will be a collection of multiple questions I asked before over different forums. I will start from the beginning, and I hope someone will follow the steps with me, because I did it before alone, and I ended with a numerical integration that is not finite, which doesn't make sense...
Is there a way to find the CDF of a random variable from its characteristic function directly, without first finding the PDF through inverse Fourier transform, and then integrate the PDF to get the CDFÉ
Homework Statement
The stability of a spinning body may be explored by using equation (3.40), with no
torque components present. It will be assumed here that the spin is about the z -axis and
has a rate ωZ = S.
Homework Equations
$$I_{xx}\dot{ω} - (I_{yy}-I_{zz})Sω_y = 0$$
$$I_{yy}\dot{y} -...
Suppose that ##Y=\sum_{k=1}^KX_{(k)}##, where ##X_{(1)}\leq X_{(2)}\leq\cdots X_{(N)}## and (##N\geq K##). I want to find the characteristic function of ##Y## as
\phi(jvY)=E\left[e^{jvY}\right]=E\left[e^{jv\sum_{k=1}^KX_{(k)}}\right]
In the case where ##\{X\}## are i.i.d random variables, the...
Homework Statement
This problem is from Gray and Searle's Electronic Principles:
P1.1 - The MOS transistor characteristics of Fig. 1.8 are a graphical presentation of the functional relationship
## i_{D} = i_{D}(v_{GS},v_{DS} )##
in which ##v_{GS}## is taken as a...
Homework Statement
x2 d2y/dx2 + 3x dy/dx + 5y = g(x)Homework Equations
How do we find Characteristic equation for it.
The Attempt at a Solution
x2λ2 + 3xλ + 5 = 0
λ1 = 1/2 [-x2 + √ (x4 + 20 ) ]
λ2 = 1/2[ -x2 - √(x4 + 20) ]
I used 1/3 -/+ a √(a2 + 4b)
where
a = x2
b = 5
Will, I have weird question , when the glass is broken for example, its take some shap like
So Why this happen , I mean if i throw a ball on glass does the weakest place are broken ,does this right?
These are the links to the specific product I am looking at and its noise characteristic graph.
http://www.emelectronics.co.uk/a20.html
http://www.emelectronics.co.uk/graphs/a20graph.htm
I understand the graph for the most part, but I am confused about which one of the four filters that are...
The Euler Characterist of the projective plane and sphere is given by V - E + F. V is vertices, E is edges, F is faces.
A presentation of the projective plane is {a | aa} and a presentation of the sphere is {b | bb^1}
Yet the Euler characteristic is 2 for the sphere and 2-n for the connected...
How can I choose the characteristic linear dimension? For example in pipe it is its diameter, but on a surface is the length, on a flat plane it can be measured as 4A/P. I was having problems determining the characteristic linear dimension for a diffusion problem in a "rectangular" pipe. I don't...
By definition, the characteristic of a field is the smallest number of times one must use the ring's multiplicative identity element (1) in a sum to get the additive identity element (0). Can we use the same rule for the set of natural numbers?
If yes, I found a problem, that has something to...
As we know $$Re= \frac{ρ.v.l}{μ}$$ The characteristic length (l) is said to be representative of the problem but according to what should we choose this length? I mean if we want to calculate Reynolds number of a wing of plane, should we choose the length or the width?
Hi,
The x-ray consists of the bremsstrahlung spectrum and the characteristic spectrum. We can get x-rays by using Coolidge tube where there are an applied voltage on the tube between the filament and the target. The bremsstrahlung is depeding on the PD between the filament and the target, but it...
Homework Statement
Hi,
I have the probabilty density: ##p_{n}=(1-p)^{n}p , n=0,1,2... ##
and I am asked to find the characteristic function: ##p(k)= <e^{ikn}> ## and then use this to determine the mean and variance of the distribution.
Homework Equations
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I have the general expression...
Hello, guys. I am trying to solve for characteristic function of normal distribution and I've got to the point where some manipulation has been made with the term in integrands exponent. And a new term of t2σ2/2 has appeared. Could you be so kind and explain that to me, please...
Hi,
Does anyone know if there are such characteristics that are tell tale signs of diode degradation? I'm not sure if diodes fail short or open, I'd have thought open, but I'm not sure.
I was wondering if as a diode gets old the forward bias voltage might increase? Or the diode power loss per...
How do I go about proving that for two matrices of same size, $M$ and $N$, that the characteristic polynomials of $MN$ and $NM$ are the same? If $M$ and $N$ are inversible, then the proof are very straightforward, for example, I can have
$|MN - \lambda I| = |MN - \lambda MM^{-1}| = |M(N -...
The question is : Is it true that two matrices with the same characteristic polynomials have the same trace?
I know that similar matrices have the same trace because they share the same eigenvalues, and I know that if two matrices have the same eigenvalues, they have the same trace. But I am...
Homework Statement
I have a LED
Given: u-i-characteristic curve -> thus the wavelengh
It's Part of a curcuit.
Also (After the LED) There is a resistance of R (known)
An a capacitor, (known)
Homework Equations
If i have a Voltage of U, how can i find the current Running through the LED and...
Hi,
Just curious if someone knows of any Characteristic class used to determine if a manifold allows
a Complex structure? It seems strange that Complex Space C^n is topologically Identical to R^{2n}
yet I believe not all R^{2n}s ( if any) allow Complex structures. Thanks for any comments, refs...
Homework Statement
A lossless transmission line has an inductance of 9.0 nH / m and a capacitance of 3.6 pF / m.
a) What is the line's characteristic impedance?
b) Calculate the phase constant of a frequency of 1 GHz.
Homework Equations
Equation of characteristic impedance of a transmission...
Hello all,
I'm trying to find the characteristic function of the random variable ##X## whose PDF is ##f_X(x)=1/(x+1)^2## where ##X\in[0,\,\infty)##. I started like this:
\phi_X(j\nu)=E\left[e^{j\nu X}\right]=\int_0^{\infty}\frac{e^{j\nu X}}{(x+1)^2}\,dx
where ##j=\sqrt{-1}##. I searched the...
Hi all,
In an x-ray spectrum, the curve part represents the bremsstrahlung part, and the spikes are the characteristics x-rays. Characteristic x-rays represent a discrete energy. However, in many textbooks, I noticed that the characteristic x-rays are often represented as a peak, which implies...
Hi everyone.
I'm trying to derive the formula for the characteristic age of a pulsar.
What I'm starting with is the following differential equation.
dP/dt=K*P2-n
What i think is odd, is several places they say solving this differential equation gives the following solution...
There are characteristic lengths for Reynold's, Grashof, Nusselt, and Biot number but the method of obtaining them is not given in my notes. I would like to know how to do so for a plane wall, cylinder and sphere. Thank you.
Hello
I have to prove mathematically all the characteristics of bravais lattices.
I googled it but they just explain the cubic lattices. I want a source where i can find all the 7 crystal systems.
What i need to know are:
Volume, conventional cell
Lattice points per unit cell
Volume of...
An ODE of second order with constants coefficients, linear and homogeneous: Af''(x) + Bf'(x) +Cf(x) = 0 has how caractherisc equation this equation here: Ax^2 + Bx +C = 0 and has how solution this equation here: f(x) = a \exp(u x) + b \exp(v x) where u and v are the solutions (roots) of the...