Characteristic Definition and 306 Threads

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.

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  1. BloonAinte

    I Characteristic curves for ##u_t + (1-2u)u_x = -1/4, u(x,0) = f(x)##

    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 -...
  2. Viona

    I Characteristic X Rays: Understanding Atom Reversion

    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...
  3. H

    Repeated roots of a characteristic equation of third order ODE

    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...
  4. emus

    I Defining Characteristic Time and Distance in an Acceleration Function

    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...
  5. iochoa2016

    I Measuring characteristic impedance as suggested by KRAUS' book

    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...
  6. yucheng

    Griffiths Quantum Mechanics Problem 1.18: Characteristic Size of System

    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...
  7. W

    MHB Solving Matrix A: Characteristic Equation and Eigenvectors

    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...
  8. T

    Solving a Partial Differential Equation with the Characteristic Method

    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...
  9. C

    Find the ratio of the energy of the La characteristic x-ray to the Ka

    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
  10. Neothilic

    I How to find the moments using the characteristic function?

    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.
  11. Zouatine

    Finding Characteristic Depth of Strip Foundation

    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...
  12. E

    Characteristic Impedance of a coaxial cable

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  13. N

    Transistor Load line doesn't intersect characteristic curves....

    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...
  14. Q

    Characteristic Impedance differences

    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.
  15. M

    Simulink: Saturation characteristic of saturable transformer

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  16. ntdiemai

    Critical Reynolds number and characteristic length

    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...
  17. CivilSigma

    Characteristic Function Integrand Evaluation

    Homework Statement [/B] 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...
  18. iVenky

    Reflection (intrinsic and characteristic impedance effects)

    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...
  19. C

    A list named the"10 characteristic experiences of physics" does it exist?

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  20. C

    I Measuring characteristic time of strong and weak interaction

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  21. binbagsss

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  22. E

    I The CDF from the characteristic function

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  23. E

    I The CDF from the Characteristic Function

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  24. T

    Characteristic equation in terms of the Laplace operator

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  25. E

    I The characteristic function of order statistics

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  26. E

    B Can Characteristic Functions Determine CDF Without Defined Moments?

    Suppose I have a random variable whose moments are not defined, can I still use the characteristic function to find the CDF of that random variable?
  27. T

    Forward Transfer Characteristic of Triode - Gray Searle P1.1

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  28. K

    Characteristic equation with x^2 coefficient

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  29. Moayd Shagaf

    I Glass Breaking -- What causes this characteristic fracture pattern?

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  30. A

    Help reading Noise Characteristic graph for sub-nanovolt amp

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  31. PsychonautQQ

    I Euler Characteristic of the Projective plane and sphere?

    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...
  32. D

    Characteristic linear dimension (Reynolds' number)

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  33. Cathr

    A What if the (semi) field characteristic of N is not zero?

    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...
  34. mastermechanic

    Characteristic lengh in Reynolds number

    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?
  35. Samar A

    B What is the minimum voltage required to get characteristic spectrum

    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...
  36. binbagsss

    Moments from characteristic function geometric distribution

    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 [/B] I have the general expression...
  37. senobim

    A Gaussian distribution characteristic function

    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...
  38. T

    What are the Signs of Diode Degradation and Aging?

    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...
  39. A

    MHB Characteristic Equations and Commutativity

    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 -...
  40. MrsM

    Linear Algebra: characteristic polynomials and trace

    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...
  41. T

    LED Characteristic Homework: Find Current & Voltage in Circuit

    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...
  42. W

    A Existence of Complex Structures and Characteristic Classes

    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...
  43. B

    Lossless transmission line characteristic phase constant

    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...
  44. E

    B How can I find this characteristic function

    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...
  45. A

    I X-ray spectra- bremsstrahlung and characteristic x-rays

    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...
  46. R

    B How is the characteristic age of pulsars typically calculated?

    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...
  47. G

    How to determine characteristic length?

    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.
  48. P

    Characteristic of all Bravais lattices

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  49. B

    Characteristic equation for recurrence equation

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