Norm Definition and 279 Threads

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. P

    Maximizing Norms of Matrices: A Scientific Approach

    Homework Statement I am trying to show that (1) ||A||_1 = \max_j \sum_i |a_{ij}| (2) ||A||_2 = \sqrt{max\_eigval\_ of\_{ } A^* A} where A* is the conjugate transpose (3) ||A||_\infty = \max_i \sum_i |a_{ij}| Homework Equations In general, ||A||_p = max_{x\neq 0}...
  2. H

    How can I find the norm of X in terms of a and b when X is orthogonal to (-a,b)?

    Homework Statement I'm stuck on this review problem for our final: The projection of X onto (a,b) = (a,b) X is orthogonal to (-a,b) Describe the norm of X in terms of a and b. The Attempt at a Solution I drew everything out on a Cartesian system, with the vector X being perpendicular to...
  3. S

    Linear Algebra - Minimize the Norm

    Homework Statement In R4, let U = span((1, 1, 0, 0), (1, 1, 1, 2)). Find u in U such that ||u - (1, 2, 3, 4)|| is as small as possible. Homework Equations The Attempt at a Solution I came up with a vector u = (-.5, -.5, 0, 0) + (2, 2, 2, 4) = (1.5, 1.5, 2, 4). Then u - (1, 2, 3, 4) = (0.5...
  4. P

    Norm of a Function vs. Length of a Vector

    Suppose f(x)= -2x+1 is a vector in the vector space C[0,1]. Calculating the norm (f,f) results in 1/3. I'm a little confused. So on [0,1] the function is a straight line from (0,1) to (0,-1). So I thought I could simply takes this line segment and turn it into a directed line segment...
  5. G

    Proving Norm of Matrix Inequality for Homework

    Homework Statement Let A = [a_{ij}] be a mxn matrix. Show that max_{ij}|a_{ij}| ≤ ‖A‖ ≤ √(∑_{ij}|a_{ij})|Homework Equations The Attempt at a Solution By the definition ‖A‖=max_{||x||≤1}‖A(x)‖ for all x ∈ Rⁿ.So, ‖A‖≥‖A∘(x₁,..,x_{n})^{T}‖ for x = (0,...,1,...0) with 1 is in the i^{ij} position...
  6. T

    Convergence with L2 norm functions

    Homework Statement (I'm posting this because my proofs seem to be lousy. I want to see if I'm missing anything.) Show that if f_n \in L^2(a,b) and f_n \rightarrow f in norm, then <f_n,g> \rightarrow <f,g> for all g \in L^2(a,b) Homework Equations L^2(a,b) is the space of...
  7. C

    What is the difference between norm and modulus?

    norm is defined to be the length of the vector and we put we denote it by ||a||. However, modulus |a| also means the length of a from the origin? So, what is the difference between the symbol || || and | |?
  8. E

    Proof Norm |x_i| ≤ ||x|| for All x ∈ ℝⁿ

    I have the next problem. I have to proof that \left\vert x_{i}\right\vert\leq\left\vert\left\vert x\right\vert\right\vert \forall x\in\mathbb{R}^{n} with the usual scalar product and norm. It's obvious that x_{i}^{2}=\left\vert x_{i}\right\vert^{2}\leq \max\left\{\left\vert...
  9. F

    Prove that Associates Have the Same Norm

    Homework Statement I need to prove that any two quadratic integers that are associates must also have the same norm. Homework Equations If α = a + b√d, the norm of α is N(α) = a^2 - b^2*d. If two quadratic integers α and β are associates, α divides β, β divides α, and α/β and β/α both equal...
  10. P

    Field trace and norm (Equivalence between definitions)

    I'm sure whoever is familiar with this subject has already seen this several times. I've seen it several times myself, and I even remember proving it in detail a couple of years ago, but now I'm stuck. I'm quoting what my professor did in class. Given some separable extension L/K, say for...
  11. Y

    How Do You Solve Norm and Matrix Approximation Problems in Linear Algebra?

    Q1: how do we find a vector x so that ||A|| = ||Ax||/||x||(using the infinite norm) totally no clue on this question.. Q2: Suppose that A is an n×n invertible matrix, and B is an approximation of A's inverse A^-1 such that AB = I + E for some matrix E. Show that the relative error in...
  12. D

    L^p Norm of a Function on $\mathbb{T}$

    Suppose \mathbb{T}=[-\pi,\pi] and we have a function in L^p(\mathbb{T}) with some measure. If we know the Fourier coefficients of f, what is the L^p norm of f? Is it (\sum f_i^p)^{1/p}? where fi are the coefs.
  13. K

    Proving Matrix Norm Inequality for Frobenius-Norm and Operator Norm

    Homework Statement Let F(AB) be the Frobenius-Norm in respect of the matrix A*B. And let ||A||2 be the operator norm. I have to show that F(AB)<=F(B)*||A||2 2. The attempt at a solution I wrote F(AB) in terms of sums and then tried to go on. But I don't know how I could include the...
  14. M

    Minimizing L_infty Norm: Finding Closest Points to b on x-axis and y=x

    This is a routine minimization problem, find the closest point or points to b = (-1,2)^T that lie on (a) the x-axis and (b) the line y=x. First I am supposed to solve it with the Euclidian norm, which is no problem, but then we are supposed to solve with the L_\infty norm. I am a little...
  15. M

    How do I calculate the distance of the ∞-norm between two vectors in lR^3?

    Alright, so if I want to find the distance of the ∞-norm between two vectors in lR^3, then would I take the max of the vectors first and then subtract, or should I subtract the vectors and then take the max? I think that the vectors are subtracted, and then the norm is taken, but I just want to...
  16. F

    Explaining Non Archimedean Norm Proof

    can someone explain this proof please, I added a star to the inequalities I don't see/understand. if | | is a norm on a field K and if there is a C > 0 so that for all integers n |n.1| is smaller than or equal to C, the norm is non archimedean (ie the strong triangle inequality is true)...
  17. N

    Using a different definition of norm

    Here's a question from Apostol's Calculus Vol1 Suppose that instead of the usual definition of norm of a vector in V_n, we define it the following way, ||A|| = \sum_{k=1}^{n}|a_k|. Using this definition in V_2 describe on a figure the set of all points (x,y) of norm 1. Is...
  18. S

    Maximizing and Minimizing Norm of Vector v - w: A Geometric Explanation

    I don't know if I'm just having a slow day or what is going on but I am being stumped by this: If ||v|| = 2 and ||w|| = 3 what are the largest and smallest values possible for ||v - w||. Give a geometric explanation. Would it be as simple as just adding the two values for the largest, and...
  19. C

    Generalized solutions for the smallest Euclidean norm

    Hi folks, I have to find the generalized solution for the following Ax=y : [1 2 3 4;0 -1 -2 2;0 0 0 1]x=[3;2;1] The rank of A is 3 so there is one nullity so the generalized solution is: X= x+alpha.n (where alpha is a constant , and n represents the nullity) I found the...
  20. D

    How does a norm differ from an absolute value?

    How does a norm differ from an absolute value? For example, is \|\mathbf{x}\| = \sqrt{x_1^2 + \cdots + x_n^2} any different than |\mathbf{x}| = \sqrt{x_1^2 + \cdots + x_n^2} ??
  21. E

    How can we show that the space c_0 is complete with the l^\infty norm?

    Show that the space c_0 of all sequences of real numbers that converge to 0 is a complete space with the l^\infty norm. First I let A^j=\{a_k^j\}_{k=1}^\infty be a sequence of sequences converging to zero and I assume that it is norm summable: \sum \limits_{j=1}^\infty ||A^j||_\infty <...
  22. E

    What is the Definition and Equivalence of the Norm of a Bounded Operator?

    I'm having trouble with this for some reason. If A:\mathcal{H}\to \mathcal{H} is a bounded operator between Hilbert spaces, the norm of A is ||A|| = \inf\limits_{\psi \neq 0} \frac{||A\psi||}{||\psi||}. My trouble is in verifying that ||A|| is in fact a bound for A in the sense that...
  23. R

    Defining Euclidean Norm in Phase Space: A Differential Geometry Analysis

    Hi to everynoe! I have a bit of trouble in understanding the following thing : Suppose we have a phase space, in which a dynamical system evolves: for example a two dimensional vector space: temperature and time. Now, does it make a sense to define the euclidean norm of a vector in such...
  24. Oxymoron

    Smallest Norm in a Hilbert Space

    I have this problem which I want to do before I go back to uni. The context was not covered in class before the break, but I want to get my head around the problem before we resume classes. So any help on this is greatly appreciated. Question Suppose C is a nonempty closed convex set in a...
  25. S

    Hilbert space and infinite norm vectors

    Quickly can we define a hilbert space (H, <,>) where the vectors of this space have infinite norm? (i.e. the union of finite + infinite norm vectors form a complete space). If yes, can you give a link to a paper available on the web? If no, can you briefly describe why? Thanks in advance...
  26. S

    Names of 4-Vector Norms & Physical Quantities

    Are there names for the Lorentz invariant norm of the four-potential and four-current? I assume that they are invariant under the transformations. Also, is it true that any physical quantities which form a four-vector have an invariant quantity associated with them (i.e. the norm of the...
  27. N

    Calculating the Second Norm of a Matrix: Formula and Explanation

    Hi, I have forgotten the formula for calculating the second norm of matrix. Does anyone know the formula? Regards, Niko
  28. C

    Proving Frobenius Norm of Matrix A

    Hi I'm in the process of proving a matrix norm. The Frobenius norm is defined by an nxn matrix A by ||A||_F=sum[(|aij|^2)^(1/2) i=1..n,j=1..n] I'm having trouble showing ||A+B|| <= ||A|| + ||B|| thanks for the help
  29. M

    Help Norm an IQ Test - Allocated 25 mins to Participate

    I am currently trying to norm an iq test. I would appreciate your participation. I am sure that you will find it interesting; go to: http://www.geocities.com/uiowa52405/iq.htm and click on logical iq test-the test is timed, for maximum time of 25 minutes.
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