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.

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  1. L

    What is the formula for the norm of a vector cross product?

    Hi everyone, I'm having problems with task c In the task, the norm has already been defined, i.e. ##||\vec{c}||=\sqrt{\langle \vec{c}, \vec{c} \rangle }## I therefore first wanted to calculate the scalar product of the cross product, i.e. ##\langle \vec{a} \times \vec{b} , \vec{a} \times...
  2. P

    I On a bound on the norm of a matrix with a simple pole

    Let ##A(z)## be a matrix function with a simple pole at the origin; in other words, we can expand it into a Laurent series of the form ##\frac1{z}A_{-1}+A_0+zA_1+\ldots##, where ##A_i## are constant matrices and ##A_{-1}\neq 0##. Fix ##\theta_0\in[0,2\pi)## and ##c\in(0,1)## (here ##1## could...
  3. C

    Does each norm on vector space become discontinuous when restricted to S^1?

    Dear Everybody, I am having trouble with last part of this question. I believe the answer is no. But I have to proof the general case. Here is my work for the problem: Suppose that we have two distinct norms on the same vector space ##X## over complex numbers. Then there exists no ##K## in...
  4. K

    Complex Matrix in vector norm

    TL;DR Summary: For every Complex matrix proove that: (Y^*) * X = complex conjugate of {(X^*) * Y} Here (Y^*) and (X^*) is equal to complex conjugate of (Y^T) and complex conjugate of (X^T) where T presents transponse of matrix I think we need to use (A*B)^T= (B^T) * (A^T) and Can you help...
  5. P

    A Norm 2, f Integrable function, show: ##||f-g||_2<\epsilon##

    Let ##F:[0,2\pi] --> Complex## ##F## is integrable riemman. show for all ##\epsilon>0## you can find a ##g##, continuous and periodic ##2\pi## s,t: ##||f-g||_2<\epsilon## What I tried ( in short ), which is nothing almost, but all I know: because g in continuous and periodic, according to...
  6. P

    V Space With Norm $||*||$ - Fourier Series

    Hi, a question regarding something I could not really understand The question is: Let V be a space with Norm $||*||$ Prove if $v_n$ converges to vector $v$. and if $v_n$ converges to vector $w$ so $v=w$ and show it by defintion. The question is simple, the thing I dont understand, what...
  7. M

    Optimization: Dual for L1 norm minimization with equality constraint

    Hi, I was reading through some notes on standard problems and their corresponding dual problems. I came across the L2 norm minimization for an equality constraint, and then I thought how one might formulate the dual problem if we had an L1-norm instead. Question: Consider the following...
  8. H

    Prove that ##\langle x, y \rangle = 0 \iff ||x + cy|| \geq ||x||##.

    (We are working in a real Euclidean space) So, we have to show two things: (1)the arrow goes from left to right, (2) the arrow comes from right to left. (1) if we're given ##\langle x, y \rangle = 0 ## $$ || x+ cy||^2 = \langle x,x \rangle + 2c\langle x,y\rangle +c^2 \langle y,y \rangle $$ $$...
  9. redtree

    B The norm of the derivative of a vector

    Is the following true? ##\left| \frac{d\vec{u}}{d t} \right| \overset{?}{=} \frac{d |\vec{u}|}{d |t|}##
  10. nomadreid

    I Shouldn't this definition of a metric include a square root?

    In https://mathworld.wolfram.com/InnerProduct.html, it states "Every inner product space is a metric space. The metric is given by g(v,w)= <v-w,v-w>." In https://en.wikipedia.org/wiki/Inner_product_space , on the other hand, "As for every normed vector space, an inner product space is a metric...
  11. LCSphysicist

    I Unit Norm Axis Rotation in R3: Exploring Representation & Algebra

    Now, i am extremelly confused about all this thing. More preciselly, i can't understand how 1.29 was obtained. It was used the A representation? How do he uses it? There is something to do with the canonical basis?
  12. A

    I How can the Lp Norm be used to prove inequalities?

  13. B

    I Why is (N dot N) different for magnitude than for X, Y, Z components?

    I have to perform a calculation on my data. Here is an example of data from just one time step (data from other time steps would appear as additional rows). X Y Z Total 2 2 1 3 Total = SQRT(X2 + Y2 + Z2). The calculation I have to do is: (N • N), where "N" is an average. I tried...
  14. F

    Determine the norm of an operator Tf(t)

    I don't know how to start to find the bounded condition nor the norm. I thought about finding a maximal norm to show that it is bounded but I don't know how to continue.
  15. arcTomato

    A Why the Poisson noise level is set to 2 after applying Leahy norm

    I am studying about power spectrum analysis in high energy astrophysics. I cannot understand why the Poisson noise level is set to 2 after applying Leahy normalization. $$P_{j}=2 /_{N \mathrm{ph}}\left|a_{j}\right|^{2}$$ The above is the equation for leahy norm, Can I expand the equation from...
  16. SchroedingersLion

    A Derivative of norm of function w.r.t real-part of function

    Greetings, suppose we have ##h(u)=\frac{1}{2} \left\|Au-b \right\|_{2}^2## with ##A## a complex matrix and ##b,u## complex vectors of suitable dimensions. Write ##u=u_1 + iu_2## with ##u_1## and ##u_2## as the real and imaginary part of ##u##, respectively. Show that ##\frac {\partial h}...
  17. SchroedingersLion

    A Is the Chain Rule Applicable to the Euclidean Norm in Calculating Derivatives?

    Greetings, suppose we have 3d vectors ##\mathbf{x}_k, \mathbf{y}_k, \mathbf{b}## for ##k=1,...,N## and a 3x3 matrix ##\mathbf{W}## with real elements ##w_{i,j}##. Are the following two results correct? $$ \frac{\partial}{\partial \mathbf{b}} \sum_k ||\mathbf{Wx}_k+\mathbf{b}-\mathbf{y}_k||² =...
  18. P

    I Prove that the norm squared of a superposition of two states is +ve

    This is what I have so far: $$ |\alpha\Psi_1 + \beta\Psi_2|^2 = |\alpha|^2|\Psi_1|^2 + |\beta|^2|\Psi_2|^2 + \alpha^*\beta\Psi_1^*\Psi_2 + \alpha\beta^*\Psi_1\Psi_2^* $$ $$=> |\alpha\Psi_1 + \beta\Psi_2|^2 = |\alpha|^2|\Psi_1|^2 + |\beta|^2|\Psi_2|^2 + 2Re(\alpha^*\beta\Psi_1^*\Psi_2) $$ I am...
  19. CCMarie

    Closest point to a set with l1 norm

    I tried to find the element of best approximation ||t_0||≤||t||, ∀ y ∈ π Then |x_0|+|y_0|+|z_0| ≤|x|+|y|+|z| and we have x_0+2y_0+z=1 and x+2y+z=1. But I don't know hoe to continue...
  20. Math Amateur

    MHB Understanding Andrew Browder's Prop 8.7: Operator Norm and Sequences

    I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ... I need yet further help in fully understanding the proof of Proposition 8.7 ...Proposition...
  21. Math Amateur

    MHB Understanding Proposition 8.7: Operator Norm and Sequences

    I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ... I need some further help in fully understanding the proof of Proposition 8.7 ...Proposition...
  22. Math Amateur

    MHB Operator Norm and Cauchy Sequence .... Browder, Proposition 8.7 ....

    I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ... I need some help in fully understanding the proof of Proposition 8.7 ...Proposition 8.7 and...
  23. Math Amateur

    MHB Operator Norm and Distance Function .... Browder, Proposition 8.6 ....

    I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ... I need some help in fully understanding the concepts in Proposition 8.6 ...Proposition 8.6...
  24. Math Amateur

    MHB Operator norm .... Field, Theorem 9.2.9 ....

    I am reading Michael Field's book: "Essential Real Analysis" ... ... I am currently reading Chapter 9: Differential Calculus in \mathbb{R}^m and am specifically focused on Section 9.2.1 Normed Vector Spaces of Linear Maps ... I need some help in fully understanding Theorem 9.2.9 (3) ...
  25. Math Amateur

    MHB Operator norm --- Remarks by Browder After Lemma 8.4 ....

    I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ... I need some help in fully understanding some remarks by Browder after Lemma 8.4 pertaining to...
  26. Math Amateur

    MHB Operator Norm .... differences between Browder and Field ....

    I am reader Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra ... I need some help in fully understanding the differences between Andrew Browder and Michael...
  27. Math Amateur

    MHB Uniform norm .... Garling Section 11.2 Normed Saces .... also Example 11.5.7

    I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume II: Metric and Topological Spaces, Functions of a Vector Variable" ... ... I am focused on Chapter 11: Metric Spaces and Normed Spaces ... ... I need some help in order to understand some...
  28. Math Amateur

    MHB Norm bounded Sets .... remarks by Garling in Section 11.2 Normed Spaces ....

    I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume II: Metric and Topological Spaces, Functions of a Vector Variable" ... ... I am focused on Chapter 11: Metric Spaces and Normed Spaces ... ... I need some help in order to understand some...
  29. Math Amateur

    MHB Bounded in Norm .... Garling, Section 11.2: Normed Spaces ....

    I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume II: Metric and Topological Spaces, Functions of a Vector Variable" ... ... I am focused on Chapter 11: Metric Spaces and Normed Spaces ... ... I need some help with some remarks by Garling concerning a...
  30. Math Amateur

    I Bounded in Norm .... Garling, Section 11.2: Normed Spaces ....

    I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume II: Metric and Topological Spaces, Functions of a Vector Variable" ... ... I am focused on Chapter 11: Metric Spaces and Normed Spaces ... ... I need some help with some remarks by Garling concerning a subset...
  31. Ventrella

    A All complex integers of the same norm = associates?

    Are all complex integers that have the same norm associates of each other? I have seen definitions saying that an associate of a complex number is a multiple of that number with a unit. And I understand that the conjugate of a complex number is also an associate. But I am looking for a...
  32. R

    Constructing a cube with a Norm

    Homework Statement Let X = ##\mathbb{R^m}## and ||.|| be a Norm on X. The dual norm is defined as ##||y||_*:=sup({\langle\,x,y\rangle :||x|| \leq 1})## a) Show that ##||.||_*## is also a norm b) Construct two norms ##||.||^O## and ##||.||^C## so that: {##x:||x||^O=1##} is a regular octahedron...
  33. mertcan

    A Matrix norm and Matrix measure

    Hi, initially I would like to share this link: https://books.google.com.tr/books?id=gWeVPoBmBZ8C&pg=PA25&lpg=PA25&dq=matrix+measure+properties&source=bl&ots=N1unizFvG6&sig=kxijoOVlPAacZDEdyyCwam4RQnQ&hl=en&sa=X&ved=2ahUKEwjd7o-Ap53dAhWJGuwKHdRbAO04ChDoATABegQICBAB#v=onepage&q=matrix measure...
  34. E

    A Vec norm in polar coordinates differs from norm in Cartesian coordinates

    I am really confused about coordinate transformations right now, specifically, from cartesian to polar coordinates. A vector in cartesian coordinates is given by ##x=x^i \partial_i## with ##\partial_x, \partial_y \in T_p \mathcal{M}## of some manifold ##\mathcal{M}## and and ##x^i## being some...
  35. S

    MHB Norm of an integral operator

    Hello A simple question. I have a linear integral operator (self-adjoint) $$(Kx)(t)=\int_{a}^{b} \, k(t,s)\,x(s)\,ds$$ where $k$ is the kernel. Can I say that its norm (I believe in $L^2$) equals the spectral radius of $K?$ Thanks! Sarah
  36. Ventrella

    A Differences between Gaussian integers with norm 25

    I am exploring Gaussian integers in terms of roots, powers, primes, and composites. I understand that multiplying two integers with norm 5 result in an integer with norm 25. I get the impression that there are twelve unique integers with norm 25, and they come in two flavors: (1) Four of them...
  37. M

    MHB Convergence of iteration method - Relation between norm and eigenvalue

    Hey! :o Let $G$ be the iteration matrix of an iteration method. So that the iteration method converges is the only condition that the spectral radius id less than $1$, $\rho (G)<1$, no matter what holds for the norms of $G$ ? I mean if it holds that $\|G\|_{\infty}=3$ and $\rho (G)=0.3<1$ or...
  38. Math Amateur

    MHB Norm of a Linear Transformation: Proving Homogeneity From Definition - Peter

    I am reading Hugo D. Junghenn's book: "A Course in Real Analysis" ... I am currently focused on Chapter 9: "Differentiation on \mathbb{R}^n" I need some help with the proof of Proposition 9.2.3 ... Proposition 9.2.3 and the preceding relevant Definition 9.2.2 read as follows...
  39. Math Amateur

    MHB Help with Proof of Junghenn Proposition 9.2.3 - A Course in Real Analysis

    I am reading Hugo D. Junghenn's book: "A Course in Real Analysis" ... I am currently focused on Chapter 9: "Differentiation on \mathbb{R}^n" I need some help with the proof of Proposition 9.2.3 ... Proposition 9.2.3 and the preceding relevant Definition 9.2.2 read as follows: In the above...
  40. Math Amateur

    I Norm of a Linear Transformation .... Another question ....

    I am reading Hugo D. Junghenn's book: "A Course in Real Analysis" ... I am currently focused on Chapter 9: "Differentiation on ##\mathbb{R}^n##" I need some help with the proof of Proposition 9.2.3 ... Proposition 9.2.3 and the preceding relevant Definition 9.2.2 read as follows: In the...
  41. Math Amateur

    MHB Norm of a Linear Transformation .... Junnheng Proposition 9.2.3 .... ....

    I am reading Hugo D. Junghenn's book: "A Course in Real Analysis" ... I am currently focused on Chapter 9: "Differentiation on \mathbb{R}^n" I need some help with the proof of Proposition 9.2.3 ... Proposition 9.2.3 and the preceding relevant Definition 9.2.2 read as follows...
  42. Math Amateur

    I Norm of a Linear Transformation .... Junghenn Propn 9.2.3 ....

    I am reading Hugo D. Junghenn's book: "A Course in Real Analysis" ... I am currently focused on Chapter 9: "Differentiation on ##\mathbb{R}^n##" I need some help with the proof of Proposition 9.2.3 ... Proposition 9.2.3 and the preceding relevant Definition 9.2.2 read as follows: In the...
  43. fresh_42

    Insights Hilbert Spaces and Their Relatives - Comments

    Greg Bernhardt submitted a new PF Insights post Hilbert Spaces and Their Relatives Continue reading the Original PF Insights Post.
  44. Math Amateur

    MHB The Euclidean Norm is Lipschitz Continuous .... D&K Example 1.3.5 .... ....

    I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of Example 1.3.5 ... ... The start of Duistermaat and Kolk's Example 1.3.5 reads as...
  45. S

    B How to interpret the integral of the absolute value?

    This is rather basic, and may be a misconception of the notation, however, I can't make the following sum up: The following is given: x_n(t) = 1 -nt , (if 0 <= t <= 1/n) and 0, (if 1/n < t <= 1) However, this part I can't grasp this part in the book: \begin{equation} ||x_n||^2 = \int_0^1...
  46. S

    I Is the Frobenius Norm a Reliable Indicator of Matrix Conditioning?

    I have calculated that a matrix has a Frobenius norm of 1.45, however I cannot find any text on the web that states whether this is an ill-posed or well-posed indication. Is there a rule for Frobenius norms that directly relates to well- and ill-posed matrices? Thanks
  47. S

    I Norm of a Functional and wavefunction analysis

    Hi, I am working on a home-task to analyse the properties of a ODE and its solution in a Hilbert space, and in this context I have: 1. Generated a matrix form of the ODE, and analysed its phase-portrait, eigenvalues and eigenvectors, the limits of the solution and the condition number of the...
  48. Math Amateur

    MHB The "Operator Norm" for Linear Transformations .... Browder, Lemma 8.4, Section 8.1, Ch. 8 .... ....

    The "Operator Norm" for Linear Transformations ... Browder, Lemma 8.4, Section 8.1, Ch. 8 ... ... I am reader Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra...
  49. Math Amateur

    MHB Operator Norm for Linear Transformations: Browder Ch. 8, Section 8.1, Page 179

    The "Operator Norm" for Linear Transfomations ... Browder, page 179, Section 8.1, Ch. 8 ... ... I am reader Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.1 Linear Algebra...
  50. user2010

    I Norm of Laplacian Let: Formula for | ∇X|² in Coordinates

    Let ##(M,g)## a manifold with a Levi-Civita connection ## \nabla ## and ##X## is a vector field. What is the formula of ## | \nabla X|^2 ## in coordinates-form? I know that ##|X|^2= g(X,X)## is equivalent to ## X^2= g_{ij} X^iX^j## and ##\nabla X## to ##\nabla_i X^j = \partial_i X^j +...
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