What is Weak: Definition and 503 Discussions

In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a topological vector space (such as a normed vector space) with respect to its continuous dual. The remainder of this article will deal with this case, which is one of the concepts of functional analysis.
One may call subsets of a topological vector space weakly closed (respectively, weakly compact, etc.) if they are closed (respectively, compact, etc.) with respect to the weak topology. Likewise, functions are sometimes called weakly continuous (respectively, weakly differentiable, weakly analytic, etc.) if they are continuous (respectively, differentiable, analytic, etc.) with respect to the weak topology.

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  1. binbagsss

    Weak Field Approx, algebra geodesic equation

    My book says in the slow motion approx, so ## v << c ##, ##v=\frac{dx^{i}}{dt}=O(\epsilon) ## It then states: i) ##\frac{dx^{i}}{ds}=\frac{dt}{ds}\frac{dx^{i}}{dt}=O(\epsilon) ## ii) ## \frac{dx^{0}}{ds}=\frac{dt}{ds}=1+O(\epsilon) ## The geodesic equation reduces from...
  2. D

    Gauge Invariance of Weak Gravity Approximation

    Hey guys, So I have a question about the gauge invariance of the weak field approximation. So if I write the approximation as \Box h^{\mu\nu} -\partial_{\alpha}(\partial^{\mu}h^{\nu\alpha}+\partial^{\nu}h^{\mu\alpha})+\partial^{\mu}\partial^{\nu}h=0 then this is invariant under the gauge...
  3. C

    FEM: How the weak form is related to an inner product

    Hi all, I am a final year maths student and am doing my dissertation in the finite element method. I have gotten a little stuck with some parts though. I have the weak form as a(u,v)=l(v) where: $$a(u,v)=\int_{\Omega}(\bigtriangledown u \cdot\bigtriangledown v)$$ and $$l(v)=\int_\Omega...
  4. E

    How To Fix My Weak Physics Foundations?

    Hello PhysicsForums! I am a Physics/Math double major undergraduate junior. I've always loved math, and to study the universe; so I decided to combine both and become a physicist! I did my Gen Phys 1-2 (with Calc) and most of my math at a state college and then I transferred to a university...
  5. D

    Weak Nuclear Force: Electron Ejection Explained

    Is the weak nuclear force really a force? How can an electron be ejected from a nucleus which has many protons; wouldn't positive charges hold it back?
  6. C

    MHB Proving coercivity for weak form

    I have the weak form of the poisson equation as $a(u,v)=l(v)$ where $$a(u,v)=\int_\Omega \bigtriangledown u\cdot\bigtriangledown v$$ I have been proving existence and uniqueness of the solution using the Lax-Milgram Thm. I am stuck on proving that the bilinear form $$a(u,v)$$ is coercive. I...
  7. X

    I think I have a weak formal education in mathematics

    For example: $$ y^{2} = 25- x^{2} $$ $$ y = \sqrt{25- x^{2}} , -5\leq x\leq 5 $$ This part: $$ , -5\leq x\leq 5 $$ What is the name of this?If it were a function, it would be the domain. And for equations, and resolutions of equations, what is the name? Other example, the equation: xy = 1...
  8. Larry Pendarvis

    Can a maximal charged black hole have "flavor" or weak hair?

    Consider a Maximal (negatively) charged black hole. Can it have "flavor" or weak charge? Suppose a muon collided with that hole. Would an electron be emitted? What if electron neutrinos collided with that black hole; would electron neutrinos more likely be emitted as Hawking Radiation...
  9. F

    Compare & contrast properties of strong & weak interactions

    Hi all - in the question below I could do with some help on the "..explanation for the relative strengths..." bit, as I can't think of anything to say. 1. Homework Statement Compare and contrast the properties of the strong and weak interactions. Include in your account an explanation for the...
  10. ChrisVer

    Weak Interactions violate Lorentz?

    I was wondering. For fermions someone can find that the Lorentz group is isomorphic to SU_L(2) \times SU_R(2) . However in the Standard Model there is only left-handed neutrinos interacting with the rest of matter. If Lorentz was not partially violated, wouldn't someone expect for the left and...
  11. S

    How do I know what if an acid is strong or weak?

    Homework Statement The question asks me to use the modified Arrhenius theory of acids to classify the following as strong or weak acids, HNO3(aq), HI(aq) and HF(aq). Homework Equations But the book only says that the modified Arrhenius theory for acids is, "an Arrhenius acid reacts with...
  12. Dilawar Ali

    Strong and weak nuclear forces?

    I can't understand completely the concept of strong nuclear force and weak nuclear force,,,
  13. P

    Maxwell Lagrangian at weak fields

    In http://arxiv.org/abs/hep-th/9506035 the author said after writing this equation: $$\frac{1}{4}\eta^{\mu\nu\lambda\rho} F_{\mu\nu}F_{\lambda\rho} = \eta_{\sigma\tau\alpha\beta}\frac{\partial L}{\partial F_{\sigma\tau}} \frac{\partial L}{\partial F_{\alpha\beta} } + 2C$$ where C was arbitrary...
  14. R

    How do the W and Z Gauge Bosons work in the weak nuclear force?

    I've seen explanations that when a neutrino with a W+ Boson comes near a neutron, it affects one of the bottom quarks and changes it to a up quark which effectively turns the neutron into a proton. The neutrino then turns into an electron. Source: (2:20 onwards) I've seen other explanations...
  15. C

    The No Cloning theorem vs weak measurement

    Question: hypothesis Would it be possible to determine/ capture the total wave function of an ensemble using quantum tomography & weak measurement ? As seen in the following article.. Its been done on a photon.. My question is, could this same technique be done on an ensemble of particles...
  16. M

    Range of Weak Values: Explaining Eigenvalues of A

    The weak value of an observable A with pre-selected state ##\psi## and post-selected state ##\phi## is defined as: A_w =\langle \phi | A | \psi \rangle / \langle \phi | \psi \rangle References like Wikipedia then include a comment that the weak value is not bounded by the range of eigenvalues...
  17. mokeejoe5

    Understanding the Weak Interaction: What Causes It & What is Its Range?

    Recently it struck me that I'm not sure I understand the weak interaction at all. What causes it to happen? I know that its mediated by the W and Z bosons and has a short range as a result of the large mass these bosons posses, but what does that range refer to? Range from what?!
  18. TheFerruccio

    Weak Form with Weighted Residuals

    Homework Statement Given a strong form boundary value problem, derive the weak form using weighted residuals. Homework Equations ##(2-x)u''(x)-u'(x)+u(x)=f(x)## for ##x\in(0,1)## with $$u(0)=u(1)=0$$ The Attempt at a Solution I must multiply both sides of this equation by an arbitrary test...
  19. W

    Schrodinger in a weak periodic potential

    Hi, I have problems figuring out what both c-coefficients means. Can someone explain this in a conceptual way? Also, Vg and V-g are the coefficients of the Fourier series if you expand V, but what does it mean physically? Thanks in advance.
  20. cosmik debris

    Weak measurement with post selection

    I'm reading stuff about weak measurements with post selection and its likely ability to improve computational performance. What I'm curious about is whether weak measurement is a real thing or not. There are many papers showing how to do the measurements and the theory behind them, and then...
  21. D

    Can light produce weak gravity waves?

    I have read online that light can produce a weak gravitational field (for example antiparallel beams should, in principle, attract weakly). This made me wonder if light can produce minute gravitational waves. Even if the waves were extremely weak (no disregarding of those high order terms in the...
  22. M

    I'm new this -- super force splitting into weak and strong forces?

    Forgive me if this is a foolish question but, what caused the super force in the early universe to split into the weak and strong forces and obviously without them the universe as we know it wouldn't exsist.
  23. Demystifier

    What is the true meaning of weak measurements in quantum mechanics?

    Here I criticize the whole concept of weak measurements and weak values in quantum mechanics, first proposed by Aharonov, Albert and Vaidman: http://prola.aps.org/abstract/PRL/v60/i14/p1351_1 1. By a suitable choice of the post-selected state, the weak value can be made ARBITRARILY large. In...
  24. J

    What happens to particles in synchrotron accel. if weak magentic field

    Homework Statement "Describe what might happen to the particles in the accelerator if the magnetic field was too weak." Homework Equations N/A The Attempt at a Solution I am merely guessing here. I am looking on the internet for info regarding this but I can't find anything...
  25. wolfspirit

    Hadron interaction via strong yet k mesons decay through weak

    Hi, I have been getting a little confused; the textbook (and from what i can see on-line) are saying that Hadrons only interact through the strong interaction (along with the electromagnetic force if charged Q) and yet K Mesons decay through the weak interaction. If my understanding is correct...
  26. M

    Weak Form of the Effective Mass Schrodinger Equation

    Hi, I am numerically solving the 2D effective-mass Schrodinger equation \nabla \cdot (\frac{-\hbar^2}{2} c \nabla \psi) + (U - \epsilon) \psi = 0 where c is the effective mass matrix \left( \begin{array}{cc} 1/m^*_x & 1/m^*_{xy} \\ 1/m^*_{yx} & 1/m^*_y \\ \end{array} \right) I know that...
  27. M

    Weak gravity at quantum level?

    Why is that gravity is weak at the quantum level?
  28. K

    Why EM and weak interaction are unified

    In SM, people say the ##SU(2)\otimes U(1)## theory unify the EM and weak interaction. I understand that the two interactions both originate from the electroweak theory in a mixed way. But does this mean they are unified? Since when we talk about unification like GUT, we require the couplings run...
  29. Primroses

    How to Determine the Range of Strong and Weak Interaction

    How can we determine the ranges of strong interactions and weak interactions? Can calculations of relevant scattering cross sections do this? Thanks.
  30. M

    Retro causation in weak measurements?

    Hello, I have been working for a while in weak measurements and I have noticed that some physicists such as Aharonov, Vaidmenn or Popescu have suggested that weak measurements reveal some sort of retro causation phenomena. One of the things that confuse me is that they state that when we weakly...
  31. M

    Weak Form of the Poisson Problem

    Hi, I know the weak form of the Poisson problem \nabla^2 \phi = -f looks like \int \nabla \phi \cdot \nabla v = \int f v for all suitable v. Is there a similarly well-known form for the slightly more complicated poisson problem? \nabla (\psi \nabla \phi ) = -f I am writing some finite...
  32. M

    Universe With No Weak Interaction

    Hello, I have a homework problem posing the following hypothetical situation Imagine a hypothetical universe in which weak interactions do not exist and only first-generation quarks and leptons are present (i.e. there are no charm, strange, top or bottom quarks, and no muons, muon...
  33. B

    The weak force, first step of nuclear fusion in the Sun

    Hi. I need help please with a textbook illustration that has confused me. Is the caption a typo? Should be "neutrons into protons" instead? Thanks.
  34. C

    Understanding the weak base NH3 in water - .

    Hi i have a general question concerning the behavior of a weak base NH3 in water. If we have a reaction between a weak base and water NH3 + H2O = NH4(+) + OH(-) NH3 is a weak base with respect to the base ,OH(-), produced from water. This means that the lone pair of OH(-) will Attack a...
  35. C

    The weak acid conjugate base - something i can't understand.

    I seem to have a problem understanding this: If 1% of a weak acid dissociates in pure water. I would assume that 99% of it’s conjugate base would dissociate to form HA in pure water, but this is not the case: I tried to set up a situation below: Please help me understand what I’m...
  36. W

    Weak fundamentals, but easily understand hard concepts

    Hello everyone. I have a rather unusual issue in which I do not seem to some know/understand basic concepts in prealgebra, but do rather well in something like calculus. By basic concepts, I don't mean something like solving equations, but I don't understand why certain things work the way they...
  37. M

    Diprotic acids and weak acids/bases

    Hi, I was reviewing some of my notes and was wondering what the correct steps for this would be (assuming in aqueous solution). If you have ## H_2CO_3 ⇔ H^+ + HCO_3^- ## finding the final concentration of HCO3- is simple enough. Now, for ## HCO_3^- ⇔ H^+ + CO_3^{2-} ## this would change the...
  38. T

    Weak Field Dynamics Homework: Equations of Motion

    Homework Statement The metric for a given particle traveling in the presence of a gravitational field is g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}, where \eta_{\mu\nu} is the Minkowski metric, h_{00}=-2\phi (\phi the Newtonian gravitational potential); h_{i0}=0; and h_{ij}=-2\phi\delta_{ij}. Units...
  39. T

    Why is the strength of weak nuclear force important ?

    I am just a student. I read that if the strength of weak nuclear force were stronger than current value, this would cause the rarity of neutrons. And, if the strength of weak nuclear force were weaker than current value, this would cause most of hydrogen to convert to helium. I can't understand...
  40. P

    Weak in Circuit Analysis what to do?

    Hey, I've found circuit analysis super difficult all my life. Just not one of my things I was born to do. However, now I am doing electronics engineering so that topic often in lectures. In class we have to do analysis of complex transistor cirucuit analysis using 2 port network and norton...
  41. A

    Weak Localization: Explaining the Argument

    A very handwaving argument for weak localization is the following: In a conductor the electron can take many paths leading back to its origin. Consider two of these, being the time reversed of each other and denote them +,-. The classical return probability is: P_classical = A+2 + A_2 While...
  42. A

    Titration of Weak Acid HCN: Calculating pH on Zumdahl's Chemistry (9th edition)

    Hello, on Zumdahl's Chemistry (9th edition), p.736: HCN, weak acid (Ka=6.2x10^-10), is dissolved in water. 50ml sample of 0.100M HCN is titrated with 0.100M NaOH. Calculate pH after 8.00 mL of 0.100M NaOH has been added. The steps are given. Then final answer is 8.49. That makes no sense to...
  43. J

    Weak Localization, Anderson Localization and disorder

    Hey, I was wondering if anybody could give me a brief overview (if possible) of how weak localization differs from strong localization, and any rigorous definitions of "disorder" and "localization" (if they exist) and perhaps recommend some good papers to read on the subject. Thanks
  44. A

    PH for this Weak Acid-Strong Base titration.

    \:Homework Statement What is the pH when 10 ml of 0.1 M NaOH is added to 25 ml of 0.1 M H_{}C_{2}H_{3}O_{2}? Homework Equations pH = -log[H_{3}O] The Attempt at a Solution The main thing confusing me about this type of problem is that my teacher taught me a way which is different from the...
  45. A

    Acid-Base low concentration of weak acid

    My textbook assumes that whenever K-constant of a weak acid is bigger than K-constant of water, the water should be a negligible source of H+ ions. However, how do we know that the concentration of weak acid could be too low and the H+ ions from water can actually have a significant effect?
  46. R

    Hydrolysis of salts of weak acid-weak base

    I just came across the hydrolysis of the salts and the formula pH=7+1/2(pKa-pKb) What's bugging me about this is how can this be independent of Concentration? I mean if we dilute it why won't it affect the pH? That means that even a negligible amount of such a salt can affect the pH of a...
  47. PsychonautQQ

    Strong vs Weak Decay: Lifetimes Explained

    I read in my textbook that a decay process involving the strong interaction has a very short lifetime (10^-23s) whereas a decay process involving the weak interaction has a longer lifetime (10^-10s). Why is this?
  48. A

    What is the reason for weak force?

    what property of elementary particles is reason for weak interaction? what I mean is as color charge is reason for strong force ,as electric charge is reason for EM force and as mass is reason for gravitational force what is reason for weak force?
  49. K

    Is my background too weak to study superconductivity?

    Hi everyone, I tried to study BCS theory but got stuck, here are my notes: I have a civil engineering degree but now I am doing a master course in physics. I know most of the undergraduate level physics including modern physics (base on Young and Freedman's book), although I cannot say...
  50. darida

    How do I use bivectors to find the electric field in a weak magnetic field?

    Geodesic equation: m_{0}\frac{du^{\alpha}}{d\tau}+\Gamma^{\alpha}_{\mu\nu}u^{\mu}u^{\nu}= qF^{\alpha\beta}u_{\beta} Weak-field: ds^{2}= - (1+2\phi)dt^{2}+(1-2\phi)(dx^{2}+dy^{2}+dz^{2}) Magnetic field, B is set to be zero. I want to find electric field, E, but don't know where to start, so...
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