What is Bound: Definition and 501 Discussions

Outward Bound (OB) is an international network of outdoor education organizations that was founded in the United Kingdom by Lawrence Holt and Kurt Hahn in 1941. Today there are organizations, called schools, in over 35 countries which are attended by more than 150,000 people each year. Outward Bound International is a non-profit membership and licensing organisation for the international network of Outward Bound schools. The Outward Bound Trust is an educational charity established in 1946 to operate the schools in the United Kingdom. Separate organizations operate the schools in each of the other countries in which Outward Bound operates.Outward Bound helped to shape the U.S. Peace Corps and numerous other outdoor adventure programs. Its aim is to foster the personal growth and social skills of participants by using challenging expeditions in the outdoors.

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  1. Mathelogician

    MHB A question on "Change of bound variables" Theorem (predicate logic)

    Hi all; I need some clarification in red part; in how it is deduced from the theorem 2.5.6! I know how the blue is deduced from the theorem but don't even know how to get blue form red in practice!(No algorithm is suggested...) Anyway, any explanation is thanked... Regards.
  2. W

    What are upper and lower bounds and why are they important in mathematics?

    At: http://en.wikipedia.org/wiki/Upper_and_lower_bounds in example it says that "2 and 5 are both lower bounds for the set { 5, 10, 34, 13934 }, but 8 is not" Why "2"? as 2 is not in that set. Also, at: http://en.wikipedia.org/wiki/Supremum in example it says that "The...
  3. C

    Can the 'mass' of bound states show up full propagator?

    The result of the Kallen-Lehmann spectral representation is that the two point correlation (and thus also the full propagator) has a pole in the physical mass of the particle. In Peskin and Schroeder it is also argued that multiparticle states show up as a cut, but bound states can also show up...
  4. S

    Q* (the set of rational cuts) has least upper bound property or not?

    I am struggling to draw this point home: To prove that R has LUB property, we used the following reasoning: First we defined R to be set of cuts (having certain properties) where each cut corresponds to a real number and then we proved any subset A of R has LUB (least upper bound) property...
  5. nomadreid

    Non-locality and the Bekenstein bound?

    On one side, the amount of information is bounded above for any fixed volume of space: this would seem (?) to indicate that information content is local. Yet physical states are not necessarily local, as non-local entanglement shows. So how do you have local information content of a non-local...
  6. S

    Optimizing Simpson's Rule for Error Bound: Finding the Minimum Value of n

    Homework Statement Calculate the value of n so that the approximation is within 0.0001. b = 2, a = 1. f(x) = 1/x. Homework Equations f4(x) = 24/x^5 (Think this is correct) Error <= (b-a)^5/180n^4(MAXx [a,b](f4(x)) The Attempt at a Solution Well, 24/x^5 obtains it's max at x =1...
  7. C

    Are black holes bound to galactic revolution?

    A couple of quick questions after watching a video on the helectic model that the solar system follows on it's course around the galactic center. Please bare with me these maybe idiotic questions. A) Are black holes bound to the spin of the galaxy or do they sit in place on the galactic plane...
  8. J

    MHB Proving Lower Bound of $\int_{0}^{ \pi}\sin^{7}(x) \, dx$ is $(\pi/2)^7$

    Hi guys I have a doubt. How can I prove that (∫ (from 0 to pi) sin^7 xdx)(∫ (from 0 to pi) sin^(7/6) xdx)^6 is at most 128 But how can I prove that the lower bound of this expression is (pi/2)^7I think is a very interesting and not an easy question so any ideas? A guidance or something...
  9. B

    MHB An approximated lower bound of an expression.

    Hii All, $ \sum_{i=1}^{x}i^{N}:N>2 $. Is there any approximated lower bound for the above summation? Is it > $ \frac{1}{N+1}x^{(N+1)}$ ? If yes, how to prove that?regards, Bincy
  10. M

    How to Estimate the Operator Norm ||A||_2 for a Difference Operator?

    Greetings everyone! I have a set of tasks I need to solve using using operator norms, inner product... and have some problems with the task in the attachment. I would really appreciate your help and advice. This is what I have been thinking about so far: I have to calculate a non trivial upper...
  11. H

    Excitons: Opposite Movements of Bound Electron-Hole Pairs

    How a bound electron-hole pair (exciton) can move together while the velocity of the free electron in the conduction band is opposite to that of the corresponding hole in the valence band?
  12. Saitama

    Variable Magnetic field bound in a cylindrical region

    Homework Statement There is a uniform but variable magnetic field ##\vec{B}=(B_0 t)(-\hat{k})##, in a cylindrical region, whose boundary is described by ##x^2+y^2=a^2##. ##\displaystyle \int_P^{Q} \vec{E} \cdot \vec{dy}## is (see attachment 1) A)0 B)##\frac{\pi}{4}(B_0 a^2)##...
  13. 7

    Energies and numbers of bound states in finite potential well

    Hello I understand how to approach finite potential well. However i am disturbed by equation which describes number of states ##N## for a finite potential well (##d## is a width of a well and ##W_p## is potential): $$ N \approx \dfrac{\sqrt{2m W_p}d}{\hbar \pi} $$ I am sure it has something to...
  14. A

    Brownian Particle bound by a Spring / internal Energy

    Hi, i regard a Brownian Particle connectet to a Spring and there is a heat-reservoir. The distribution of the x-coordinate of the particle follows the Diffusion-Equation (Fokker-Planck-Equation): \partial_{t}P(x,t)=\frac{D}{2} \partial_{x}^{2}P(x,t)- \Gamma\partial_{x}[f(x)P(x,t)] A...
  15. M

    Lower and Upper bound proof in R

    I am getting lost in the proof in the 5th line when it says there are 10 numbers that have the same kth digit as x. Why 10? I don't understand where this number is coming from and it doesn't seem arbitrary. the rest of the proof...
  16. michael879

    What is the maximum entropy a system can have based on its energy and size?

    ok so my main question here is about the normal bekenstein bound, but I will go into why I'm asking too in case anyone has any comments on that. The way I understand the derivation of the bekenstein bound is: If you have some closed system with energy E bounded by radius R, you can derive the...
  17. P

    Vacuum stability bound on higgs mass

    Hi I have been working through and want to plot the graph (fig1.2) on page 10 also found here http://www.amazon.com/dp/0198509545/?tag=pfamazon01-20 or here http://arxiv.org/pdf/hep-ph/0003170v1.pdf I have worked through and got the formula for the triviality bound and that is fine and got...
  18. H

    Help with Problem on Bekenstein Bound

    Hi Guys, I've been struggling over a problem with the Bekenstein Bound, and I wonder if someone can help, please. The Bekenstein Bound is derived from the entropy of black holes, and says that the maximum information content of a region of space is proportional the area of that region, not...
  19. O

    MHB Least upper bound - greatest lower bound duality

    Hello everyone! There's a point I didn't get in Rudin's theorem 1.11 that says: Suppose S is an ordered set with the LUB property, and B $\subset$ S, B is not empty and B is bounded below. Let L be the set of lower bounds of B. Then a = sup L exists in S, and a - inf B. In particular inf B...
  20. M

    Production of bound states of slow fermions- Peskin 5.3

    Hi all, I've been reading section 5.3 of Peskin and Schroeder, in which the authors discuss the production of a bound state of a muon-antimuon pair close to threshold in electron-positron collisions. Here \xi,\xi' are the Weyl spinors used to construct the Dirac spinors for the muon and...
  21. N

    Doubt regarding derivation of bound charges in dielectric

    In Griffiths, for deriving the bound charges for a given polarization P , the formula used is the general formula for dipoles .i.e ( equation 4.9) {Here the potential at r is calculated due to the dipole at r' ) V(r) = ∫\frac{x.P(r')}{X^2}d\tau' Here X = r - r' , and x = unit vector in...
  22. H

    Positive lower bound in the punctured rectangle

    Homework Statement Let p(x,y) be a positive polynomial of degree n ,p(x,y)=0 only at the origin.Is it possible that the quotient p(x,y)/[absolute value(x)+absval(y)]^n will have a positive lower bound in the punctured rectangle [-1,1]x[-1,1]-{(0,0)}? Homework Equations The...
  23. P

    Bound State Wavefunctions vs Non-Bound State Wavefunctions

    Bound vs "not"bound states Homework Statement Hi, I do not understand how two bound state wavefunctions differ from not bound state wavefunctions. To be more precise I m thinking about the graphical representation. [b]ons[/b2. Relevant equati The Attempt at a Solution I speculate that bound...
  24. A

    Analytic continuation to find scattering bound states

    Hello, I am trying to understand the idea of using analytic continuation to find bound states in a scattering problem. What do the poles of the reflection coefficent have to do with bound states? In a problem that my quantum professor did in class (from a previous final), we looked at the 1D...
  25. P

    Li atom and its valence e bound energy.

    Homework Statement Find bond energy of valence electron in principal state in Li atom 2S. If first line of the sharp series is 0.813microm and short wave boundary is 0.349 microm. Homework Equations i think I have to use \tilde{v}=R[( 1/(x-(Δ)) )^2-( 1/(n-(Δ)) )^2] Δ-quantum...
  26. S

    What is the Lower Bound of this Sequence?

    Homework Statement Homework Equations The Attempt at a Solution This is what I have so far: x_{n+1}=\frac{x^5_n + 1}{5x_n}=1 x_{n+2}=\frac{x^5_{n+1} + 1}{5x_{n+1}}=\frac{1^5+1}{5} I think you have to do some sort of repeated substitution but I don't quite see it. Any help? Thanks.
  27. phosgene

    Minimum energy of electron bound in nucleus

    Homework Statement An electron is confined to a nucleus of radius 4 femtometres. Estimate its minimum energy. Homework Equations ΔxΔp_x=h/4\pi E^2=p^2c^2 + m^2c^4 As the electron's rest energy will be much less than it's kinetic energy, E=pc The Attempt at a Solution So I...
  28. 8

    IS it possible to have total bound current NOT equal to 0?

    I have a permanent magnetization \vec{M}=(ks)\hat{z}, k is just a const, s is the cylindrical coor. Then it turns out that the total bound current not equal to 0. i wonder is it possible? the magnetization is stored inside the shell of a cylinder of inner radius a and outer radius b. thanks in...
  29. M

    How to Prove an Upper Bound for a Set of Real Numbers?

    Homework Statement Let A be a set of real numbers. If b is the supremum (least upper bound) of the set A then whenever c<b there exist an a in A such that a>c. Homework Equations The Attempt at a Solution I considered two cases. The first one when the supremum b is attained by...
  30. T

    Understanding Free & Bound Charges: H & D Fields Explained

    My teacher's notes don't explain this. What are free and bound charges, and why are the H and D field defined like they are?
  31. R

    Index out of bound because numel(w)=11

    Hello,, I try to convert a fortran program to matlab. I want to make an absorbing boundary model. But when I run it, I keep getting an error says: ? Attempted to access w(12); index out of bounds because numel(w)=11. Error in ==> absorb_bound_coba at 45...
  32. G

    Bound states in propagator

    Why must it be true that a system that has a bound state must have its scattering amplitude have a pole in the upper half of the complex wave-number plane? For example, if the scattering amplitude as a function of the initial wave number magnitude |k| is: A=\frac{1}{|k|-iB} with B>0, then...
  33. tom.stoer

    Number of bound states and index theorems in quantum mechanics?

    Just an idea: is there an index theorem for an n-dimensional Hamiltonian H = -\triangle^{(n)} + V(x) which "counts" the bound states (H - E) \,u_E(x) = 0 i.e. eigenfunctions and eigenvalues in the discrete spectrum of H?
  34. E

    Greatest lower bound problem - Rudin POMA Ch1 Exercise 5

    Homework Statement 5. Let ##A## be a nonempty set of real numbers which is bounded below. Let ##-A## be the set of all numbers ##-x##, where ##x\in A##.Prove that $$\inf A=-\sup(-A)\text{.}$$ Homework Equations The Attempt at a Solution Does the proof below look OK? I am a bit uneasy...
  35. R

    Difference Between Bound and Free charge

    Homework Statement So I'm having a bit of trouble getting my head around this concept and was hoping someone would be able to shed some light on it. I know the definition. i.e free charge isn't bound to a nucleus whereas bound is. But physically what difference does this make. i.e are free...
  36. B

    Least upper bound of open interval.

    I am having trouble understanding how there could be a least upper bound for an open interval. If I have (a,b) and i am looking for the least upper bound X which is the number that is less than or equal to the set of Y such that Y> all the numbers in the interval (a,b) when I think about it I...
  37. D

    Clarifications on the least upper bound property and the irrational numbers

    Hello everyone. I desperately need clarifications on the least upper bound property (as the title suggests). Here's the main question: Why doesn't the set of rational numbers ℚ satisfy the least upper bound property? Every textbook/website answer I have found uses this example: Let...
  38. A

    Bound charge in linear dielectric

    How can we show that the bound charge in a homogeneous linear dielectric is proportional to the density of the free charge. I have a handful of equations but still I can't work this out.
  39. B

    Least Upper Bound and Supremum

    Are the least upper bound and supremum of a ordered field same thing? If so, then why do we have two different terms and why do textbooks do not use them interchangeably. That also means that greatest lower bound and infimum are also the same thing.
  40. H

    MHB How Accurate is Lagrange Interpolation for Approximating Cos(0.75)?

    Problem: Use the Lagrange interpolating polynomial of degree three or less and four digit chopping arithmetic to approximate cos(.750) using the following values. Find an error bound for the approximation. cos(.6980) = 0.7661 cos(.7330) = 0.7432 cos(.7680) = 0.7193 cos(.8030) = 0.6946 The...
  41. STEMucator

    Upper Bound Proof of Sup(SUT)=max{sup(S), sup(T)}

    Homework Statement Prove or disapprove, for non-empty, bounded sets S and T in ℝ : sup(SUT) = max{sup(S), sup(T)} Homework Equations The least upper bound axiom of course. The Attempt at a Solution Since we know S and T are non-empty and bounded in the reals, each of them...
  42. C

    Defining an upper/lower bound in lexicographically ordered C.

    If I have a lexicographic ordering on ℂ, and I define a subset, A = \{z \in ℂ: z = a+bi; a,b \in ℝ, a<0\}. I have an upper bound, say α = 0+di. My question is does only the real part, Re(α) = 0 define the upper bound? Or does the Im(α) = d have nothing to do with bounds in general? Since it...
  43. U

    Real Analysis Least Upper Bound Question

    Homework Statement If S1, S2 are nonempty subsets of ℝ that are bounded from above, prove that l.u.b. {x+y : x \in S1, y \in S2 } = l.u.b. S1 + l.u.b. S2 Homework Equations Least Upper Bound Property The Attempt at a Solution Using the least upper bound property, let us suppose...
  44. bcrowell

    Effect of cosmological expansion on bound systems in realistic cosmologies

    This paper dates to 1998: Cooperstock, Faraoni, and Vollick, "The influence of the cosmological expansion on local systems," http://arxiv.org/abs/astro-ph/9803097v1 They show that systems such as the solar system, galaxies, and clusters of galaxies experience nonzero effects from cosmological...
  45. V

    Exploring Bound Electrons: Discrete Energy Levels and Wave Functions Explained

    I have a question, and I'm positive it has a really simple answer, but I can't think of it right now. In the infinite square well (the simplest bound problem), the wave functions have discrete energy values. We can have a wave function that's a linear superposition of any number of these so...
  46. M

    Definite Integration with Upper bound as another integral

    i have a similar one. f(x) = \int\frac{dt}{\sqrt{1+t^3}} on (0, g(x)) g(x) = \int(1+sin(t^2))dt on (0, cos(x)) that is, these are definite integrals on the interval from zero up to the given function. the question is to solve f'(pi/2). the correct answer is -1 but i don't understand...
  47. iVenky

    Lower Bound on Q(x) for X ~ Gaussian RV

    I think everyone knows that Q(x)= P(X>x) where X is a Gaussian Random variable. Now I was reading about it and it says that Q(x) is bounded as follows Q(x)≤ (1/2)(e-x2/2) for x≥0 and Q(x)< [1/(√(2∏)x)](e-x2/2) for x≥0 and the lower bound is Q(x)> [1/(√(2∏)x)](1-1/x2) e-x2/2 for x≥0 Can...
  48. R

    The least upper bound property and the irrationals.

    Hi Does anybody know if the irrational numbers have the least upper bound property?
  49. QuestForInsight

    MHB What is the definition of greatest/least upper bound in a partially ordered set?

    Let $a$, $b$ and $c$ be elements of a partially ordered set $P$. My book defines $c$ as the greatest upper bound of $a$ and $b$ if, for each $x \in L$, we have $x \le c$ if and only if $x \le a$ and $x \le b$. Similarly, it defines $c$ as the least upper bound of $a$ and $b$ if, for each $x \in...
  50. B

    DE: Lower Bound for radius of convergence

    Prb:(x^4+4*x^2+16)y"+4(x-1)y'+6xy=0 P=(x^4+4*x^2+16) Q=4(x-1) R=6x P=0 for - 1 - 3^(1/2)*i 1 - 3^(1/2)*i - 1 + 3^(1/2)*i 1 + 3^(1/2)*i Q=0 for 1 R=0 for 0 Do we ignore Q & R, plotting P, then find shortest distance which would equal 2?
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