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. B

    Understanding UFD's with Quadratic Integer & Norm Questions

    Hello PhysicsForums! I was reading up on UFD's and I came up with a few quick questions. 1. Why don't the integers of Q[\sqrt{-5}] form a UFD? I was trying to tie in the quadratic integers that divide 6 to help me understand this, but I am stuck. 2. Why is Z a UFD? 3. Assuming Q[\sqrt{d}] is...
  2. A

    Lower bound for the norm of the resolvent

    Hi all! I hope this is the right section to post such a question... I'm studying the theory of resolvent from the QM books by A. Messiah and I read in a footnote (page 713) that the norm of the resolvent satisfies \|R_A(z)\| = \lVert \frac{1}{A-zI} \rVert \ge \text{dist}(z,\sigma(A))^{-1}...
  3. P

    Numerical Analysis - Matrix Norm equalities.

    Homework Statement Determine constants c and C that do not depend on vector x. For this to be true: Use this result and the definition of matrix norms to determine constants k and K so that: Homework Equations The Attempt at a Solution I've done half of this...
  4. Simfish

    Proof that norm of submatrix must be less than norm of matrix it's embedded in

    Homework Statement http://dl.dropbox.com/u/4027565/2010-10-10_194728.png Homework Equations The Attempt at a Solution ||B|| = ||M_1 * A * M_2 || So from an equality following from the norm, we can get... ||B|| <= ||M_1||*||A||*||M_2||. Now, we know that B is a...
  5. Y

    How to Calculate Norms of Field Extensions in Galois Theory

    Hello everyone, I need some help with finding norms of the field extension. I feel pretty comfortable when representing norms as determinants of linear operators but I seem to be stuck with representing norms as product of isomorphims. I have read Lang's GTM Algebra, but I really would...
  6. D

    Proving the Dot Product and Norm Theorems

    Homework Statement The Attempt at a Solution I let D be the center x = DX & a = DA (x-a) * (x+a)=|x|^2-|a|^2 Dunno what to do with the right side of the equation
  7. mnb96

    Squared norm in Clifford Algebras

    Hello, I know that the squared norm of a multivector M in a Clifford Algebra \mathcal{C}\ell_{n,0} is given by: <M \widetilde{M}>_0 that is the 0-grade part of the product of M and its grade-reversal. Is there a more general definition of squared-norm (for multivectors) that works...
  8. I

    Verifying ||.|| is a Norm on $\Re^{2}$

    Homework Statement check whether ||.|| : \Re^{2} -> \Re_{+} given by (x,y) = |x| + |y|^{2} is a norm on R2 Homework Equations For all a in F and all u and v in V, 1. p(a v) = |a| p(v), (positive homogeneity or positive scalability) 2. p(u + v) ≤ p(u) + p(v) (triangle...
  9. S

    A stupid question on norm and trace of fields

    so i came up with a proof that..well.. Let L/K be a field extension and we have defined the norm and trace of an element in L, call it a, to be the determinant (resp. trace) of the linear transformation L -> L given by x->ax. Now it's well known that the determinant and trace are the...
  10. J

    Norm Ordering for a many electron system

    I'm confused with the definition of a norm ordering of operators. The basic definition of norm ordering as understood by me was "Place the annihilation operators to right and creation operators to the left". However I also read another definition "The motivation of norm ordering is to ensure...
  11. D

    Prove that the dual norm is in fact a norm

    Homework Statement Let ||\cdot || denote any norm on \mathbb{C}^m. The corresponding dual norm ||\cdot ||' is defined by the formula ||x||^=sup_{||y||=1}|y^*x|. Prove that ||\cdot ||' is a norm. Homework Equations I think the Hölder inequality is relevant: |x^*y|\leq ||x||_p ||y||_q...
  12. P

    Constructing Norms on Tensor Products of Finite Dimensional Vector Spaces

    I was wondering about useful norms on tensor products of finite dimensional vector spaces. Let V,W be two such vector spaces with bases \{v_1,\ldots,v_{d_1}\} and \{w_1,\ldots,w_{d_2}\}. We further assume that each is equipped with a norm, ||\cdot||_V and ||\cdot||_W. Then the tensor product...
  13. E

    Where can I find a proof of the supremum norm as a norm?

    Could anyone tell me where to find a proof of the fact that the supremumnorm is a norm? The supremum norm is also known as the uniform, Chebychev or the infinity norm.
  14. S

    Numerical Optimization ( norm minim)

    Homework Statement Consider the half space defined by H = {x ∈ IRn | aT x +alpha ≥ 0} where a ∈ IRn and alpha ∈ IR are given. Formulate and solve the optimization problem for finding the point x in H that has the smallest Euclidean norm. Homework Equations The Attempt at a...
  15. K

    Closed subset of R^n has an element of minimal norm

    Homework Statement a) Let V be a normed vector space. Then show that (by the triangle inequality) the function f(x)=||x|| is a Lipschitz function from V into [0,∞). In particular, f is uniformly continuous on V. b) Show that a closed subset F of contains an element of minimal norm, that...
  16. M

    Proving Limit of Square Matrix Norm Exists

    Let T be any square matrix and let \left\| \cdot \right\| denote any induced norm. Prove that lim_{n \rightarrow _{\infty}} \left\| T^{n} \right\| ^{1/n} exists and equals inf _{n=1,2,\cdots } \left\| T^{n} \right\| ^{1/n} I am not sure how I go about proving that the limit exists. For the...
  17. M

    Norm question (Frobenius norm)

    We show that if P and Q are Hermitian positive definite matrices satisfying x^{*}Px \leq x^{*}Qx for all x \in \textbf{C}^{n} then \left\| P \right\|_{F} \leq \left\| Q \right\|_{F} where \left\| \cdot \right\|_{F} denotes the Frobenius norm (or Hilbert-Schmidt norm) If A is a mXn...
  18. U

    The Dependence of Norm on Basis in Vector Spaces

    Hello. My question is: does the norm on a space depend on the choice of basis for that space? Here's my line of reasoning: If the set of vectors V = \left\{ v_1,v_2\right\} is a basis for the 2-dimensional vector space X and x \in X, then let \left(x\right)_V = \left( c_1,c_2\right)...
  19. J

    Functions not satisfying parallelogram identity with supremum norm

    Homework Statement Find two functions f, g \in C[0,1] (i.e. continuous functions on [0,1]) which do not satisfy 2 ||f||^2_{sup} + 2 ||g||^2_{sup} = ||f+g||^2_{sup} + ||f-g||^2_{sup} (where || \cdot ||_{sup} is the supremum or infinity norm) Homework Equations Parallelogram identity...
  20. G

    How do I prove the continuity of the norm in any n.l.s.?

    Homework Statement x[ Prove the continuity of the norm; ie show that in any n.l.s. N if xn \rightarrow x then \left|\left|x_n\left|\left| \rightarrow \left|\left|x\left|\left| The Attempt at a Solution i don't know where to start this from the definition of convergence xn \rightarrow x...
  21. C

    Norm of a linear transformation

    Homework Statement ||T|| = {max|T(x)| : |x|<=1} show this is equivalent to ||T|| = {max|T(x)| : |x| = 1} The Attempt at a Solution {max |T(x)| : x<=1} = {max ||x|| ||T(x/||x||)|| : |x|<=1} <= {max ||T(x)|| : |x| = 1} does that look right? I need to show equality...
  22. M

    Euclidean metric (L2 norm) versus taxicab metric(L1 norm)

    Homework Statement I was just wondering how I would go about proving that the euclidean metric is always smaller than or equal to the taxicab metric for a given vector x in R^n. The result seems obvious but I am not sure how I would show this. Homework Equations The Attempt at a Solution
  23. Somefantastik

    Polynomial bounded w.r.t supremum norm

    Homework Statement E1 = {pn(t) = nt(1-t)n:n in N}; E2 = {pn(t) = t + (1/2)t2 +...+(1/n)tn: n in N}; where N is set of natural numbers is the polynomial bounded w.r.t the supremum norm on P[0,1]? Homework Equations supremum norm = ||*|| = sup{|pn(t)|: t in [0,1]} The Attempt...
  24. A

    Convergence of a sequence of functions to zero in the L1 norm?

    I just want to make sure I'm straight on the definition. Am I correct in assuming that, if I want to show that a sequence \langle f_n \rangle of functions converges to 0 in the L^1 norm, I have to show that, for every \epsilon > 0, there exists N \in \mathbb N such that \int |f_n| < \epsilon...
  25. M

    Integrating Norm in Unit Ball in Rn-2

    \int|x|2 with respect to the vector x in the unit ball in Rn-2 I'm dealing with volumes of unit balls in Rn and after applying a change of variable to the last 2 components and Fubini's Theorem, I get that integral and can't find a way to integrate it. Any help on this?
  26. L

    Not every metric comes from a norm

    Hello! It is said that not every metric comes from a norm. Consider for example a metric defined on all sequences of real numbers with the metric: d(x,y):=\displaystyle\sum_{i=1}^{\infty}\frac{1}{2^i}\frac{|x_i-y_i|}{1+|x_i-y_i|} I can't grasp how can that be. There is a proof...
  27. M

    What is an Extreme Point in a Convex Set?

    Hi, I've been trying to show that the set of matrices that preserve L1 norm (sum of absolute values of each coordinate) are the complex permutation matrices. Complex permutation matrix is defined as permutation of the columns of complex diagonal matrix with magnitude of each diagonal element...
  28. R

    Is There an Inequality Between L1 and L2 Norms?

    Homework Statement \|x\|_2\le\|x\|_1\le\sqrt{n}\|x\|_2 where |x|1 is the l1 norm and |x|2 is the l2 normHomework Equations See aboveThe Attempt at a Solution I have \|\mathbf{x}\|_1 := \sum_{i=1}^{n} |x_i| and \|x\|_2 = \left(\sum_{i\in\mathbb N}|x_i|^2\right)^{\frac12} I have tried to...
  29. K

    Construction of minimum norm solution matrix

    Homework Statement Consider the linear system of equations Ax = b b is in the range of A Given the SVD of a random matrix A; construct a full rank matrix B for which the solution: x = B^-1*b is the minimum norm solution. Also A is rank deficient by a known value and diagonalizable...
  30. A

    Infinity norm of system matrix

    Homework Statement This problem is related to the system theory, using the H-infinity frame work to determine the maximum gain of a multivarible system. The system is described as G(s) = \begin{pmatrix} \frac{s}{s+1} & \frac{s}{s^2+s+1} \\ \frac{s-1}{s+2} & \frac{s-1}{s+1} \end{pmatrix}...
  31. A

    Determine Infinity Norm of a Transfer Matrix

    I'm trying to understand how the infinity norm of a transfer matrix is calculated. For example, assume a simple transfer matrix G(s) = \begin{pmatrix} \frac{s}{s+1} & \frac{s}{s^2+s+1} \\ \frac{s-1}{s+2} & \frac{s-1}{s+1} \end{pmatrix} Now, I'm trying to compute the \mathcal{H}_{\infty}...
  32. S

    Norm of a function ||f|| & the root mean square of a function.

    norm of a function ||f|| & the "root mean square" of a function. How do I explain the connection between the norm of a function ||f|| & the "root mean square" of a function. You may like to consider as an example C[0,\pi], the inner product space of continuous functions on the interval [0,\pi]...
  33. F

    What Is the Derivative of the Shift Operator in L^2(0,∞)?

    Homework Statement Let A:L^2(0,\infty)\to L^2(0,\infty) be given by f(x)\mapsto f(x+1). What is the derivative A', if it exists, of A? That is, we want a function A':L^2(0,\infty)\to L^2(0,\infty) such that \lim_{\|h\|\to0}\frac{\left\|A(f+h)-Af -hA'f\right\|}{\|h\|}=0. Homework...
  34. H

    How can the l1 norm of a linear function be maximized?

    Hi all, Sorry if this is in the wrong section...first time and i couldn't see a convex analysis section. I'm trying to find a good algorithm/theorem that will maximise the l1 norm (sum of absolute values) of a linear function. Namely, given a function z = c + Ax where z is (nx1), A is...
  35. E

    Trouble understanding fft norm axis sampling frequency etc

    Hello, I have a question regarding fft's. My experience with working with Fourier transforms is pretty much limited to transforming contrived functions pen and paper style. But now I need something and I think the fft is the appropriate tool, but I'm having a hard time understanding some...
  36. mnb96

    L2 Norm of +Infinity: Admitted & Defined

    Hello, I have a (infinite dimensional) vector space and defined an inner product on it. The vectors element are infinite sequence of real numbers (x_1, x_2,\ldots). The inner product has the common form: x_iy_i The problem now is that the vectors have an infinite number of elements, so the...
  37. jbunniii

    Norm Satisfying the Parallelogram Law

    Let V be a vector space over the complex field. If V has an inner product <\cdot,\cdot>, and ||\cdot|| is the induced norm, then it's easy to show that the norm must satisfy the parallelogram law, to wit: ||x+y||^2 + ||x-y||^2 = 2||x||^2 + 2||y||^2 Much more interestingly, given an arbitrary...
  38. G

    Norm Inequality: Proving Max Statement

    Homework Statement Show that \frac{\Vert X(u+v) \Vert}{\Vert u+v \Vert} \leq \max \{ \frac{\Vert Xu \Vert}{\Vert u \Vert}, \frac{\Vert Xv \Vert}{\Vert v \Vert} \} Homework Equations The Attempt at a Solution Tried to rewrite the max statement as an inequality...
  39. W

    Child Marriage in Saudi Arabia: A Disheartening Norm

    I thought that Jerry Springer shows was the tip of it all, but I guess this is the norm over there: http://news.yahoo.com/s/ap/20090501/ap_on_re_mi_ea/ml_saudi_child_marriage
  40. P

    Calculating Norm of a Vector with Two Vectors

    Ok, so I have no idea how to take the norm of a vector composed of two vectors. I have \vec{q}=\vec{pi} - \vec{pf}we are given: |\vec{pi}|=|\vec{pf}|=|\vec{p}| so i know that |\vec{q}| \neq 0, that would be too easy, and it doesn't make sense. now, is the following right? it just doesn't seem...
  41. W

    Ess Sup Norm as limit ->oo of L^p norm

    Hi: I am trying to show that the ess sup norm is the limit of the L^p norms as p-->oo . i.e., ess sup =lim_p->oo ( {Int f^p)^1/p Please tell me if this is correct: 1) Def. ess sup f(t)=inf{M:m(t:f(t)>M)=0 } Then, f(t)>M only in the set S , with m(S)=0 , and f(t So Lim_p->oo...
  42. A

    Norm on Dual Space: X' - Showing ||x*||=|x_1|+...+|x_n|

    Homework Statement X is the space of ordered n-tuples of real numbers and ||x||=max|\xij| where x=(\xi1,...,\xin). What is the corresponding norm on the dual space X'? Homework Equations The Attempt at a Solution I think the answer is that ||x*||=|x_1|+...+|x_n| , but I'm not sure...
  43. Nebula

    Linear Transformation Norm Preserving

    Homework Statement From Calculus on Manifolds by Spivak: 1-7 A Linear Transformation T:Rn -> Rn is Norm Preserving if |T(x)|=|x| and Inner Product Preserving if <Tx,Ty>=<x,y>. Prove that T is Norm Preserving iff T is Inner Product Preserving. Homework Equations T is a Linear...
  44. A

    Eventual boundedness of nth derivative of an analytic function in L2 norm

    I'm trying to show that if f(x) is analytic, then for large enough n, || f^{(n)} (x) || \leq c n! || f(x) ||, where || f ||^2=\int_a^b{|f|^2}dx and f^{(n)} denotes the nth derivative. I tried to use the Taylor series, and then manipulated some inequalities, but I wasn't getting...
  45. M

    Defining scalar product from norm

    Euclidean norm is defined usually as|v|2= g(v,v), where g is a nondegenerate, positive definite, symmetric bilinear form. But how can make it backwards? What properties must norm have that g(v,w) = (|v+w|2 - |v|2 - |w|2)/2 be a positive definite, symmetric bilinear form?
  46. A

    Verifying Norm Equality for Ax

    I just want to verify if the following is correct \left\right\|x\|2.\left\right\|A\|2= \left\right\|Ax\|2 Thanks
  47. D

    Hilbert-Schmidt Norm: Calculation & Solution

    Homework Statement http://img523.imageshack.us/img523/4456/56166304yr3.png Homework Equations http://img356.imageshack.us/img356/2793/40249940is8.png The Attempt at a Solution I defined K:[a,b] --> [a,b] with k(s,t) = \frac{(t-s)^{n-1}}{(n-1)!} I found for the norm: \int_a^b \int_a^b...
  48. D

    Find Norm of Matrix: 2x2, 3x3 & Beyond

    Homework Statement Is it possible to find the norm of a matrix? Not a column or row matrix which is a vector, but like on a 2x2 or 3x3 matrix? Homework Equations The Attempt at a Solution
  49. T

    Solve Inhomogeneous Equation: Show \| u \|_{\infty} \leq \frac{1}{4} \| f \|_1

    Homework Statement I have the solution to an inhomogeneous equation: u(x) = \int_{0}^{1} g(x,t)f(t)dt g(x) = x(1-t) , 0<x<t and g(x) = t(1-x), x<t<1 Show that \| u \|_{\infty} \leq \frac{1}{4} \| f \|_1 Homework Equations I already...
  50. 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}...
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