What is Norm: Definition and 278 Discussions

Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the environment, such as uranium, thorium and potassium and any of their decay products, such as radium and radon. Produced water discharges and spills are a good example of entering NORMs into the surrounding environment. Natural radioactive elements are present in very low concentrations in Earth's crust, and are brought to the surface through human activities such as oil and gas exploration or mining, and through natural processes like leakage of radon gas to the atmosphere or through dissolution in ground water. Another example of TENORM is coal ash produced from coal burning in power plants. If radioactivity is much higher than background level, handling TENORM may cause problems in many industries and transportation.

View More On Wikipedia.org
Recent contents Top users
  1. S

    Help with this notation -- some sort of norm?

    I need help understanding this notation, what does this mean? Squared of 2-norm? 1. Homework Statement Thanks
  2. E

    A Understanding Sobolev Norms: A Beginner's Guide

    Can someone give me in very abstract terms what a sobolev norm is or means?
  3. lep11

    Norm inequality, find coefficients

    Homework Statement Find coefficients a,b>0 such that a||x||∞≤||x||≤b||x||∞.Homework EquationsThe Attempt at a Solution No idea how to get started. Help will be appreciated.
  4. G

    I Newtonian 4-Momentum Norm Analogue

    Hi. I read that the Lorentz invariance Minkowski norm of the four-momentum $$E^2-c^2\cdot \mathbf{p}^2=m^2\cdot c^4$$ has no analogue in Newtonian physics. But what about $$E-\frac{\mathbf{p}^2}{2m}=0\quad ?$$ It might look trivial by the definition of kinetic energy, but it's still a relation...
  5. P

    B Does the L2 norm of a vector destroy all directional info?

    Sorry I'm a little rusty with my math and proof logic, and this feels like a dumb question, but oh well! The Euclidian norm of a vector in ℝ3 is \|{v}\| = \sqrt{x^2 + y^2 + z^2} where \|{v}\| \geq 0. I'm trying to show that there is always an infinite number of solutions for arbitrary...
  6. JulienB

    Contraction, integral, maximum norm

    Homework Statement Hi everybody! Here is another problem about contraction and Banach fixed-point theorem that I don't get: The function ƒ: C([0,½]) → C([0,½]) is defined by: [f(x)](t) := 1 + \int_{0}^{t} x(s) ds ∀ t∈[0,\frac{1}{2}]. Is ƒ a contraction with respect to the norm || ⋅ ||∞? If...
  7. JulienB

    Maximum norm and Banach fixed-point theorem

    Homework Statement Hi everybody! I have a math problem to solve, I'd like to check if I understand well the Banach fixed-point theorem in the case of Euclidean norm and how to deal with maximum norm. Check if the following functions ƒ: ℝ2 → ℝ2 are strictly contractive in relation to the given...
  8. Telemachus

    Proving the Inequality between Lp Norms: A Demonstration for Homework

    Homework Statement Hi there. I have to prove this inequality: ##||x||_2 \leq ||x||_1 \leq \sqrt{n} ||x||_2## Where ##||x||_2## is the ##l_p## norm with p=2, so that: ##||x||_2=(|x_1|^2+|x_2|^2+...+|x_n|^2)^{\frac{1}{2}}## And similarly ##||x||_1=|x_1|+|x_2|+...+|x_n|## is the ##l_1##...
  9. P

    Is the Induced Weighted Matrix Norm Equal to WAW^-1?

    Homework Statement The weighted vector norm is defined as ##||x||_W = ||Wx||##. W is an invertible matrix. The induced weighted matrix norm is induced by the above vector norm and is written as: ##||A||_W = sup_{x\neq 0} \frac{||Ax||_W}{||x||_W}## A is a matrix. Need to show ##||A||_W =...
  10. P

    Proving Unitary Matrix Norm: $$||UA||_2 = ||AU||_2$$

    Homework Statement Prove $$||UA||_2 = ||AU||_2$$ where ##U## is a n-by-n unitary matrix and A is a n-by-m unitary matrix. Homework Equations For any matrix A, ##||A||_2 = \rho(A^*A)^.5##, ##\rho## is the spectral radius (maximum eigenvalue) where ##A^*## presents the complex conjugate of A. U...
  11. P

    Is this proof of an ##\infty## norm valid?

    I am trying to prove ##||A||_{\infty} = max_i \sum_{j} |a_{ij}|## which reads as the ##\infty## norm is the max row sum of matrix A. ##i## is the row index and ##j## is the column index. Here is what I thought of: ##||A||_{\infty} = sup_{x\neq 0} \frac{||Ax||_{\infty}}{||x||_{\infty}}## The...
  12. C

    Question about induced matrix norm

    The induced matrix norm for a square matrix ##A## is defined as: ##\lVert A \rVert= sup\frac{\lVert Ax \rVert}{\lVert x \rVert}## where ##\lVert x \rVert## is a vector norm. sup = supremum My question is: is the numerator ##\lVert Ax \rVert## a vector norm?
  13. T

    Derivation of Minkowski norm of the four-momentum

    I have attached a derivation of the Minkowski norm of the four-momentum but just don't quite see how the writer arrived at ## -m^2 c^2 ## from what was given. How exactly does this quantity follow from ## -\frac {E^2}{c^2} + p^2##? I feel like it might be very obvious, so any explanation would...
  14. C

    Is the Topology of the Schwartz Space Normable?

    Hello : CONTEXT : let be E the space of rapidly decreasing functions on ##\mathbb{R}^{n}## in ##\mathbb{R}##. I define $$(||.||_{i})_{i \in \mathbb{R}^{n+1}}$$ with forall ##i = (k, m_{1},\ldots, m_{n}) = (k, m)## we define for ##f## in ##E## ##||f||_{i} = \sup_{x \in \mathbb{R}^{n}} \Big|(1...
  15. A

    What's the meaning of the norm of Poynting 4-vector?

    Assuming a four velocity ##u_{\nu}=(1,0,0,0)## we can use the Maxwell Energy-Momentum Tensor to build a 4-vector in the following way ##P^{\mu}=u_{\nu}T^{\mu\nu}=\left(\frac{E^{2}+B^{2}}{2},\mathbf{E}\times\mathbf{B}\right)## So, we have a vector whose time component is the energy density of...
  16. RicardoMP

    Which vectorial norm should I use?

    I am to study how fast an iterative method for nonlinear system of equations converges to a certain root and found out that I can evaluate my rate of convergence by using the following formula: ##r^{(k)}=\frac{||x^{(k+1)}-x^{(k)}||_V}{||x^{(k)}-x^{(k-1)}||_V}##. My question is which vectorial...
  17. C

    Sequence is norm convergent implies it's strongly convergent

    If a sequence of operators \{T_n\} converges in the norm operator topology then: $$\forall \epsilon>0$$ $$\exists N_1 : \forall n>N_1$$ $$\implies \parallel T - T_n \parallel \le \epsilon$$ If the sequence converges in the strong operator topology then: $$\forall \psi \in H$$...
  18. RJLiberator

    Unitary Matrix preserves the norm Proof

    Homework Statement Let |v> ∈ ℂ^2 and |w> = A|v> where A is an nxn unitary matrix. Show that <v|v> = <w|w>. Homework Equations * = complex conjugate † = hermitian conjugate The Attempt at a Solution Start: <v|v> = <w|w> Use definition of w <v|v>=<A|v>A|v>> Here's the interesting part Using...
  19. S

    MHB How Can I Calculate the Norm of the Operator \(I-L^{-1}K\)?

    I have a linear integral operator $K\psi=\int_{a}^{b} \,k(x,s) \psi(s) ds$ $L\psi=\int_{a}^{b} \,l(x,s) \psi(s) ds$ both are continuous I know how to obtain the eigenvalues of each alone. But how can I calculate the eigenvalues of the operator $I-{L}^{-1} K$ or at least the norm...
  20. I

    Understanding convergence in norm, uniform convergence

    Homework Statement Find an example of a sequence ##\{ f_n \}## in ##L^2(0,\infty)## such that ##f_n\to 0 ## uniformly but ##f_n \nrightarrow 0## in norm. Homework Equations As I understand it we have norm convergence if ##||f_n-f|| \to 0## as ##n\to \infty## and uniform convergence if there...
  21. Diffie Heltrix

    Norm indueced by a matrix with eigenvalues bigger than 1

    Suppose we pick a matrix M\in M_n(ℝ) s.t. all its eigenvalues are strictly bigger than 1. In the question here the user said it induces some norm (|||⋅|||) which "expands" vector in sense that exists constant c∈ℝ s.t. ∀x∈ℝ^n |||Ax||| ≥ |||x||| . I still cannot understand why it's correct. How...
  22. FOIWATER

    Can the Induced Matrix Norm be Proven with Triangle Inequality?

    Hi, I found a statement without a proof. It seems simple enough, but I am having trouble proving it because I am not positive about induced matrix norms. The statement is that $$||A^k|| \leq||A||^{k}$$ for some matrix A and positive integer k. I have found that the norm of a matrix is the...
  23. T

    Why is one significant figure the norm in uncertainties?

    Why is it that only one significant figure for any uncertainty is taught? It seems like a nice general rule, but isn't it an unnecessary constraint and ultimately a poor rule in many cases? For example ## 5 \pm 1 ## could mean an error of either 0.5 or 1.5, which is a very large range. Wouldn't...
  24. Z

    How do I prove the existence of this norm?

    I am reading an article[1] that states: Let k be a fixed local field. Then there is an integer q=pr, where p is a fixed prime element of k and r is a positive integer, and a norm |.| on k such that for all x∈k we have |x|≥0 and for each x∈k\{0} we get |x|=qm for some integer m. This norm is...
  25. S

    MHB Solving Linear Integral Equation: Norm of Resolvent

    I have a linear integral operator (related to integral equations) $(Ky)(x)=\int_{a}^{b} \,k(x,s)y(s)ds$ If $|b|. ||K||<1$ (b is a scalar) can I say $||(I-bK)-1||< 1 / (1-|b|.||K||)$ I think it's correct is it? thanks
  26. evinda

    MHB Why do we deduce that the functional is continuous in respect to the other norm?

    Hello! (Wave) Let $V=C^1([a,b])$. Show that if $J$ is a continuous functional in respect to the norm $||y||_1:=||y||_{\infty}+||y'||_{\infty}, y \in V$ then it is also continuous in respect to the norm $||y||:=||y||_{\infty}$. Also, show that the inverse of the above claim does not hold. Let...
  27. J

    Norm of a vector related to coherent states

    Hi, For the past couple of days I've been attempting to derive the equality (for any normalised ##\varphi##) ||(a+za^\dagger+ \lambda)\varphi ||^2 = ||(a^\dagger+\bar{z}a+ \bar{\lambda} )\varphi ||^2 + (|z|^2-1)||\varphi ||^2 First of the summation seemed like a typo. So I first tried to prove...
  28. S

    MHB Is the Triangle Inequality Applicable to Norms of Integral Operators?

    Can I always say without reservation that for any two integral operators $K$ and $L$ defined as follows say $(Ky)(x)=\int_{a}^{b} \,k(x,s)y(s)ds$ that $||L||+||K-L||\ge||K||$ thanks Sarrah
  29. S

    MHB When Does the Norm of the Sum Equal the Sum of Norms for Integral Operators?

    Hi I have 2 linear integral operators $(Ku)(x)=\int_{a}^{b} \,k(x,t)u(t)dt$ $(Mu)(x)=\int_{a}^{b} \,m(x,t)u(t)dt$ I am defining $||K||=max([x\in[a,b]\int_{a}^{b} \,|k(x,t| dt$ same for $L$ when does $||K+M||=||K||+||M||$ thanks sarrah
  30. S

    MHB Norm of a linear integral operator

    i have a problem concerning the norm of linear integral operator. ii found the answer in a book called unbounded linear operators theory and applications by Dover books author seymour Goldberg. the proof runs as follows ||T|| is less than max over x in [a,b] of integral (|k(x,y)|dy) Then he...
  31. M

    P-adic norm, valuation, and expansion

    In this book I'm working from (P-adic numbers: An introduction by Fernando Gouvea), he gives an explanation of the p-adic expansion of a rational number which I'm pretty sure I understand this, but a bit later he talks about the p-adic absolute value, which from what I understand is the same as...
  32. Th3HoopMan

    [College Level] Norm of Matrices

    So I'm studying norm of matrices in my calc class, and most resources I've looked at seem like it's just the square root of all the entry values over the sum of all the entries, but when given the matrix [1 3] [0 1] Sqrt((11+sqrt(117))/2) The 11 is the sqrt of 12 + 32 + 12 the 2 is the root...
  33. C

    When is the norm of the sum of 2 vectors=sum of norms

    Homework Statement When is |x+y|=|x|+|y| for arbitrary non-zero vectors x,y∈Rn ie, when does equality hold for the well known inequality |x+y|≤|x|+|y| Homework Equations |x|2=<x,x>=Σixi2 |x+y|≤|x|+|y| The Attempt at a Solution squaring both sides of the inequality we have...
  34. P

    Norm of Vector Formed by Two Vectors

    If we have a vector $$\partial_v$$ and we want o find its norm, we easily say (According to the given metric) that the norm of that vector is:$$ g^{vv}\partial_v\partial_v$$. My question what if we have a vector that is combination of 2 vectors like: $$\phi =\partial_v + a\partial_x$$ where $a$...
  35. P

    How to compute the norm of a root from the Cartan matrix?

    Hello! As far as I understand, the Cartan matrix is associated with a unique semi simple algebra. How can we compute the norm of a root α from it since its components are invariant under rescaling (if all the simple roots are multiplied by the same constant, the Cartan matrix remains unchanged)...
  36. L

    MHB Proving ||u||_d = 0 and u=0 in Metric Spaces

    Let (X, d) be a metric space, AE_0(X) = \{ u : X \rightarrow \mathbb{R} \ : \ u^{-1} (\mathbb{R} \setminus \{0 \} ) \ \ \text{is finite}, \ \sum_{x \in X} u(x)=0 \}, for x,y \in X, \ x \neq y, \ m_{xy} \in AE_0(X), \ \ m_{xy} (x)=1, \ m_{xy}(y)=-1, \ m_{xy}(z)=0 for z \neq x, y and m_{xx}...
  37. M

    MHB What Does Norm Convergence Mean in $L^p$ Spaces?

    Hey! :o If $f_n, f \in L^p, 1\leq p < +\infty$ and $f_n \rightarrow f$ almost everywhere, and $||f_n||_p \rightarrow ||f||_p$, then $f_n\rightarrow f$ as for the norm. Could you give me some hints how to show it?? (Wondering) What does convergence as for the norm mean?? (Wondering)
  38. M

    MHB Norm in R^d with Lebesgue measure

    Hey! :o In $\mathbb{R}^d$ with the Lebesgue measure if $f \in L^p, 1 \leq p < +\infty$, and if for each $y$ we set $f_y(x)=f(x+y)$ then: $f_y \in L^p$ and $||f||_p=||f_y||_p$ $\lim_{y \rightarrow 0} ||f-f_y||_p=0$ Could you give me some hints how to show that?? (Wondering)
  39. Schnurmann

    Least Squares Estimation - Problem with Symbols

    Hi folks, 1. Homework Statement I don't fully understand the question statement, how is it supposed to be read? Question: Give a formula for the minimizer x* (to be read as x-star) of the function ƒ:ℝn → ℝ, x → ƒ(x) = ||Ax-b||22, where A∈ℝm×n and b∈ℝm are given. You can assume that A has rank...
  40. Julio1

    MHB Show $\|.\|$ is an Norm: Prove Triangle Inequality

    If $\|(x,y)\|=\sqrt{x^2+4y^2}.$ Show that $\|.\|$ is an norm. Hello MHB! :) The cases $\|(x,y)\|\ge 0$ and $\|\lambda (x,y)\|=|\lambda|\|(x,y)\|$ are obvious, but the case $\|(x,y)+(x',y')\|\le \|(x,y)\|+\|(x',y')\|$ I don't know how it do? Help me!
  41. Barioth

    MHB How Do You Differentiate a Matrix Norm Function?

    Hi Everyone I'm trying to evaluate: \frac{d}{dW}\sum_{i=1}^{n}||t_i-W^Tx_i||^2+\sum_{i=1}^{n}||w_i||^2 Where t_i,x_i avec vector and w_i is a column of W. Would the solution be: \frac{d}{dW}\sum_{i=1}^{n}(t_i-W^Tx_i)x_i+\sum_{i=1}^{n}2*w_{i,j} does that make sense? edit: Please don't...
  42. S

    How to prove that the L2 norm is a non-increasing function of time?

    Homework Statement Homework Equations How can I start the proof? Shall I use the Poincare inequality? The Attempt at a Solution Well, I know that this norm is defined by , but still I don't know how to start constructing the proof?
  43. S

    MHB Solving Problem w/ Norm Space Proof: Advice & Resources

    Hi guys, I've attached a problem that I've been struggling with for a while now. I was wondering if anyone had some advice on how to approach it (in particular part a) or some resources they could recommend to me?Thanks in advance, Sam
  44. gfd43tg

    Linear program with multiple norm vectors

    Homework Statement In the previous few modules you studied the problem of minimizing ##\| Ax -b \|_{2}## by choice of ##x##. So far you've done this in Matlab using either the backslash operator or the command pinv. Now that you've been exposed to linear programming, you have the tools to solve...
  45. S

    MHB Norm of Integrals: Bounding the Matrix Product

    Hi I have an integral over [0,1] of product of two matrices say A(t). B(t) and I wish to bound its norm. Can you say that ||integral (AB)||<||B(t)||.||integral (A)|. is there some conditions on that to occur thanks sarrah
  46. A

    Yarman Geometric Norm or Impedence Normalization

    I'm working with some old software to optimize antenna networks, and I've come across some stuff that I don't understand. For the software to run, all impedence values entered must be normalized to 1 ohm. What does it mean to normalize an impedence and how do I do it? Also, the manual for the...
  47. K

    MHB How to Calculate Matrix Norm in a Banach Space?

    How can I calculate the following matrix norm in a Banach Space: $$ A=\begin{pmatrix} 5 & -2 \\ 1 & -1 \\ \end{pmatrix} ?$$ I have tried $$\|A\|=\sup\limits_{\|x\|=1}\|Az\|$$ and then did $$Az=\begin{pmatrix} 5 & -2 \\ 1 & -1 \\...
  48. D

    Unitary matrix and preservation of vector norm in arbitrary basis

    Hi PF people! I am not sure my question can elegantly fit in the template, but I 'll try. Homework Statement I am self-studying the 8th chapter of "Mathematical Methods for Physics and Engineering", 3rd edition by Riley, Hobson, Bence. In the section about unitary matrices, it is stated that...
  49. J

    Kantorovich Norm Explained: Pushforward Measure & Integration

    I've seen a few references to it, but can't find it defined anywhere. What is it? Something with integrating distances between measures and something. Pushforward measure or something else?
  50. B

    Is every norm preserved under a unitary map?

    I am a bit confused, so this question may not make much sense. A unitary operator from one vector space to another is one whose inverse and Hermitian transpose are identical. It can be proved that unitary operators are norm preserving and inner product preserving. Which raises the question...
Back
Top