What is Decomposition: Definition and 411 Discussions

Decomposition is the process by which dead organic substances are broken down into simpler organic or inorganic matter such as carbon dioxide, water, simple sugars and mineral salts. The process is a part of the nutrient cycle and is essential for recycling the finite matter that occupies physical space in the biosphere. Bodies of living organisms begin to decompose shortly after death. Animals, such as worms, also help decompose the organic materials. Organisms that do this are known as decomposers. Although no two organisms decompose in the same way, they all undergo the same sequential stages of decomposition. The science which studies decomposition is generally referred to as taphonomy from the Greek word taphos, meaning tomb. Decomposition can also be a gradual process for organisms that have extended periods of dormancy.One can differentiate abiotic from biotic substance (biodegradation). The former means "degradation of a substance by chemical or physical processes, e.g., hydrolysis. The latter means "the metabolic breakdown of materials into simpler components by living organisms", typically by microorganisms.

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  1. P

    Partial Fractional Decomposition to obtain an integral.

    Homework Statement Find the exact values of A, B, and C in the following partial fraction decomposition.Then obtain the integral using those values. 1/(x^3-3x^2) = A/x + B/x^2 + C/(x-3)Homework Equations I'm not sure at this point.The Attempt at a Solution To make the denominator on the right...
  2. Fredrik

    Schmidt decomposition and bipartite system

    I asked a question about Schmidt decomposition in one of the math forums, but haven't gotten any replies yet. Link. The question is about why the state of a bipartite system (a system that consist of two component systems) can be expressed in the form \sum_{n=1}^\infty...
  3. Fredrik

    Schmidt decomposition theorem proof

    I would be interested in seeing a correct statement and proof of the Schmidt decomposition theorem that (I think) says that if x\otimes y\in H_1\otimes H_2, there exists a choice of bases for these Hilbert spaces, such that x\otimes y=\sum_{n=1}^\infty \sqrt{p_n}\ a_n\otimes b_n with...
  4. K

    Ozone Decomposition: Half-Life & Pressure Effects

    I have recently found that the half-life of ozone at a temperature of 250°C is 1.5 seconds. However, this source (http://www.lenntech.com/library/ozone/decomposition/ozone-decomposition.htm) failed to give any statistics on the effect of pressure on ozone decomposition. My question is; if the...
  5. B

    Partial Fractions Complex Decomposition

    Homework Statement Sorry I don't have equation editor working 1/(z+1)(z2 + 2z + 2) Homework Equations The Attempt at a Solution (z2 + 2z + 2) z2 + 1) can be factor as (z - i)(z + i) However, I'm having trouble seeing the pattern on what (z2 + 2z + 2) would become, I...
  6. J

    Can a matrix A be decomposed if it doesn't have n distinct eigenvalues?

    If a n x n matrix A has an eigenvalue decomposition, so if it has n different eigenvalues, by the way, is it correct that a n x n matrix that doesn't have n different eigenvalues can't be decomposed? Are the more situations in which it can't be decomposed? Why can't I just put the same...
  7. A

    Cluster decomposition of the S-matrix

    Hi, I am reading through Weinberg's "Quantum Theory of Fields" (vol. 1) and I am somewhat confused about the signs in the cluster decomposition of the S-matrix. Specifically, referring to eq. 4.3.2, let's say the term coming from the partition \alpha \to \alpha_1\alpha_2,\beta \to...
  8. S

    Show that this orthogonal diagonalization is a singular value decomposition.

    Homework Statement Prove that if A is an nxn positive definite symmetric matrix, then an orthogonal diagonalization A = PDP' is a singular value decomposition. (where P' = transpose(P))2. The attempt at a solution. I really don't know how to start this problem off. I know that the singular...
  9. S

    Singular Value Decomposition of an nxn matrix?

    I was just wondering if it was possible to find the singular value decomposition of an nxn matrix such as 1 1 -1 1 I tried this but then when finding the eigenvectors of A^T*A I found there were none (non-trivial anyhow). So, is this not possible? EDIT: How embarrassing I made an...
  10. D

    Monitoring First-Order Decomposition of Species X with Spectrophotometer

    The first-order decomposition of a colored chemical species, X, into colorless products is monitored with a spectrophotometer by measuring changes in absorbance over time. Species X has a molar absorptivity constant of 5.00 x 10^3 cm^ -1 M^ -1 and the path length of the cuvette containing the...
  11. K

    Fourier Transform Decomposition

    Hello, If I've a real signal, and I do a forward Fourier transformation I receive two parts: Real and Imaginary, what's the difference between them? i need to represent the transform in a software program, which part do i represent ?
  12. D

    LU Decomposition: Calculating Determinant of Square Matrix (Example Included)

    How would I calculate the determinant of a square matrix using LU Decomposition. Please be plain, I am not good with technical terms. An example would be nice. Thank you!
  13. C

    Even function has a Laurent decomposition of even functions and even powers of z

    Hi, if let's say that there's an even function f(z) then how do we know if its Laurent decomposition (i.e. f(z) = f0(z) + f1(z) ) will be even functions and have even powers of z? Any help will be greatly appreciated.
  14. R

    Partial Fraction Decomposition

    Homework Statement 1/[s*(s^2 + 4)] Homework Equations The Attempt at a Solution 1/[s*(s^2 + 4)] = A/(s) + (Bs + C)/(s^2 + 4) => 1 = A(s^2 + 4) + (Bs + C)s s = 0 1 = A(0 + 4) + (B*0 + C)*0 A = 1/4 s = i 1 = A(i^2 + 4) + (Bi + C)i 1 = A(i^2 + 4) + Bi^2 + Ci 1 =...
  15. R

    Partial Fraction Decomposition

    Homework Statement I just can't understand it I've read plenty of guides online, I just can't figure it out. How do you do partial fraction decomposition the farthest i can get is below 1/[(s^2 + 1)(s+1)] Homework Equations The Attempt at a Solution 1/[(s^2 + 1)(s+1)] = A/(s+1)...
  16. M

    Proving Unique Decomposition of a Square Matrix

    Homework Statement An nxn matrix C is skew symmetric if C^t = -C. Prove that every square matrix A can be written uniquely as A = B + C where B is symmetric and C is skew symmetric. Homework Equations The Attempt at a Solution No clue.
  17. B

    Schmidt Decomposition: Is It Enough to Find States?

    Is entangle enough to find that given states have Schmidt decomposition?
  18. P

    Spectral density and wold decomposition

    Hello, I would greatly appreciate any comment on the following problem: Suppose that I estimate the spectral density of a weakly stationary series, say, nonparametrically by smoothing the periodogram. Is there a simple way to recover the coefficients of the wold decomposition of the process...
  19. S

    Find the decomposition of the standard two-dimensional rotation

    Homework Statement Find the decomposition of the standard two-dimensional rotation representation of the cyclic group Cn by rotations into irreducible representations The Attempt at a Solution Ok i did this directly, finding complementary 1-dimensional G-invariant subspaces. but...
  20. J

    Partial Fraction Decomposition

    I am just coming back to math after a, oh 30 year or so, vacation. In the class I'm taking, we are studying Partial Fraction Decomposition ( Px/Qx:Qx). It doesn't entirely make sense to me, tho like a monkey typing the great American novel, I can solve them given enough time. I am just having...
  21. S

    Help with Newtonian Potentials for Helmholtz Decomposition

    I'm trying to find a divergenceless vector field based on its curl, and discovered that I could use a http://en.wikipedia.org/wiki/Helmholtz_decomposition" , and the article I found on this didn't make much sense to me. First, can someone confirm that the dimension referred to in the...
  22. S

    Newtonian potential in Helmholtz decomposition

    I'm trying to find a divergenceless vector field based on its curl, and discovered that I could use a http://en.wikipedia.org/wiki/Helmholtz_decomposition" , and the article I found on this didn't make much sense to me. First, can someone confirm that the dimension referred to in the...
  23. J

    What is the difference between the two definitions of Schur decomposition?

    Wikipedia defines the Shur decomposition of matrix A as A = Q U Q^{-1} where Q is unitary and U is upper triangular. http://en.wikipedia.org/wiki/Schur_decomposition Mathworld defines the Shur decomposition of matrix A as Q^H A Q = T, where Q is unitary and T is upper...
  24. B

    Space Junk: Aging & Decomposition Rates

    Has there been any real scientific testing to determine the ratio of aging or decomposition of space junk above our atmosphere compared to atmospheric aging? I ask this because if space junk is going to be "out there" for millions of years as is, why would we even consider adding more in the...
  25. R

    Using QR decomposition to find a nontrivial solution to Ax=0

    Homework Statement Supposedly the process to solve Ax=0 is to solve Transpose(R).Ry=z (where z is a random vector) and then x=y/(norm-2 of y). Homework Equations The Attempt at a Solution Ay=b for some random vector b Transpose(A).Ay= Transpose(A).b...
  26. A

    Polynomial of Jordan Decomposition

    Homework Statement Let A be a real or complex nxn matrix with Jordan decomposition A = X \Lambda X^{-1} where \Lambda is a diagonal matrix with diagonal elements \lambda_1,..., \lambda_n. Show that for any polynomial p(x): p(A)=Xp(\Lambda)X^{-1} p(\Lambda) should really be the matrix with...
  27. Z

    Help: Mode Decomposition and Retarded Feilds and Green's functions

    Hello, I studying General Relativity at University of Chicago and I am looking for information regarding "Fourier-Harmonic Mode Decomposition" or mode decomposition in general and "Retarded Greens Functions" and retarded fields in general. Does anyone have advice about where is a good...
  28. A

    Jordan Decomposition to Schur Decomposition

    Homework Statement Let A be a complex or real square matrix. Suppose we have a Jordan decomposition A = XJX-1, where X is non-singular and J is upper bidiagonal. Show how you can obtain a Schur Decomposition from a Jordan Decomposition. Homework Equations Schur Decomposition: A = QTQ*...
  29. N

    Partial fraction decomposition

    Homework Statement Find the partial fraction decomposition of : \frac{x^2}{(1-x^4)^2} The Attempt at a Solution \frac{x^2}{(1-x^4)^2}=\frac{A}{(1-x^4)}+\frac {B}{(1-x^4)^2} =A(1-x^4)+B when x=1 1=A(1-1^4)+B Hence B=1 and A=0 \frac{x^2}{(1-x^4)^2}=\frac{0}{(1-x^4)}+ \frac{1}{(1-x^4)^2}...
  30. S

    Thermal Decomposition of Carbonates

    Hi, I have a practical assessment tomorrow which deals with the Thermal Decomposition of a metal Carbonate. Is it possible to write Ionic Equations for the reaction? Ionic solids and solutions are involved, but the reactions seem to be Double Decomposition, especially the test for CO2 Also...
  31. D

    Proving the Existence of Cholesky Decomposition: Lemma on Positive Matrices

    I'm supposed to prove this step as part of my proof for existence of Cholesky Decomposition. I can see how to use it in my proof, but I can't seem to be able to prove this lemma: For any positive (nxn) matrix A and any non-singular (nxn) matrix X, prove that B=X^{\dagger}A X is...
  32. A

    What are the singular values of a matrix multiplied by the identity matrix?

    Homework Statement Let A be a real mxn matrix, m>=n, with singular values \sigmaj.Show that the singular values of (\stackrel{I_{n}}{A}) are equal to \sqrt{1+\sigma_j^2}. Homework Equations The Attempt at a Solution I know that an SVD for A is A = U(\stackrel{\Sigma}{0})v^T and...
  33. L

    First Order Decomposition -> % of compound. Please help

    First Order Decomposition -> % of compound. Please help! Homework Statement In a first order decomposition, the constant is 0.00313 sec-1. What percentage of the compound has decomposed after 7.17 minutes? Homework Equations rate = ln[A]o - ln[A]t = kt The Attempt at a Solution I...
  34. T

    Why Is Checkerboard Decomposition Used in Quantum Monte Carlo Simulations?

    Checkerboard decomposition? I need help from guys working with Quantum Monte Carlo. I'm reading about worldline method applied to Heisenberg chain (or nearest-neighbor Hubbard model) and after calculation of nonzero matrix elements I'm immediately thrown to the checkerboard representation of...
  35. K

    HELP on Partial Fraction Decomposition Problem - Heavy Side Technique

    HELP! on Partial Fraction Decomposition Problem! - Heavy Side Technique I am doing a partial fraction decomposition problem for my calc 2 class. We use the Heavy Side Technique, but I will take help either way! \frac{2x-1}{x(x^2+1)^2} Thank you!
  36. quasar987

    What are examples of cellular decomposition?

    Say X is a CW-complex. Then for any n, the n-skeleton X^n of X is obtained from the (n-1)-skeleton X^(n-1) by gluing some n-cells on X^(n-1) along their boundary. From what I read, it seems that the way to obtain X^n from X^(n-1) in this way is not unique. Is this non-uniqueness superfluous...
  37. M

    Decomposition of a R.V. into dependent and independent parts

    Given random variables X and Y, which are not independent, is it always possible to find a random variable W which is independent from X, such that Y = f(X,W), for some function f? Example: let the joint distribution of X and Y be Y 0 1 X+------- 0|1/3 1/6 1|1/6 1/3 Then if we...
  38. S

    Jordan Decomposition: Solve 4x4 Matrix Eqn

    Homework Statement I need to find a Jordan decomposition for: \[ \left( \begin{array}{cccc} 2 & 0 & 1 & 2 \\ -1 & 3 & 0 & -1 \\ 2 & -2 & 4 & 6 \\ -1 & 1 & -1 & -1 \end{array} \right)\] Homework Equations The Attempt at a Solution I found the eigenvalues: 2 (m=4). I also found the eigenvectors...
  39. A

    Pseudo Inverse & QR Decomposition

    Homework Statement Let A\inRmxn be factorized as A=QR where Q\inRmxn has orthonormal columns. Prove that A+=R+QT Homework Equations R is an upper triangular matrix The Attempt at a Solution I tried to apply the definition A+=(ATA)+A+ I ended up here: R+(RR+)TQT I'm not sure if...
  40. M

    Spectral Decomposition of Linear Operator T

    [b]1. Let T be the linear operator on R^n that has the given matrix A relative to the standard basis. Find the spectral decomposition of T. A= 7, 3, 3, 2 0, 1, 2,-4 -8,-4,-5,0 2, 1, 2, 3 [b]3. eigen values are 1 (mulitplicity 1), -1 (mult. 1), 3 (mult. 2). And associated eigen...
  41. S

    Symmetric group (direct product and decomposition)

    I am looking for a mechanism to find a decomposition of symmetric groups. For finitely generated abelian group G, there is a mechanism to decompose G such that G is isomorphic to a direct sum of cyclic groups. For symmetric groups, it seems a bit complex for me to find it. For example...
  42. W

    Decomposition of a complex vector space into 2 T-invariant subspaces

    Homework Statement Suppose V is a complex vector space and T \in L(V). Prove that there does not exist a direct sum decomposition of V into two proper subspaces invariant under T if and only if the minimal polynomial of T is of the form (z - \lambda)^{dim V} for some \lambda \in C. Homework...
  43. T

    Cycle decomposition of n-cycle's power

    Homework Statement let a=(b_1,...,b_n) n-cycle in the permutation group S_n . prove that the cycle decomposition of a^k consist of gcd(n,k) cycles of n/gcd(n,k) size. The Attempt at a Solution I know that a^k(b_i)=b_{i+k (mod n)} how can it help me ?
  44. P

    Partial Fraction Decomposition

    Homework Statement \frac{4x^{4}-8x^{3}+5x^{2}-2x-1}{2x^{2}-3x-2} Homework Equations The Attempt at a Solution I started of by breaking the bottom part down into (2x+1)(x-2) which then allowed me to set... \frac{A}{(2x+1)}+\frac{B}{(x-2)} The problem is from here I tried...
  45. D

    LU Decomposition: Solving for A

    Homework Statement http://img296.imageshack.us/img296/6584/93872041bz2.png Homework Equations http://img208.imageshack.us/img208/2062/55585340nj2.png The Attempt at a Solution I think the key to the solution is the matrix A being non-singular but I can't see how.
  46. M

    Linear operator decomposition

    In finite dimensions, a matrix can be decomposed into the sum of rank-1 matrices. This got me thinking - in what situations can a bounded linear operator mapping between infinite dimensional spaces be written as an (infinite) sum of rank-1 operators? eg, let A be a bounded linear operator...
  47. D

    RQ decomposition from QR decomposition

    What I'm wondering is: Q and R in the QR decomposition of A are the same Q and R in the RQ decomposition of which matrix? I found some MATLAB code which will get RQ from QR, but I don't understand how you would do those operations FIRST, then find the QR decomposition. ReverseRows = [0...
  48. R

    Proving Matrix X rank Decomposition

    How can you prove that matrix X with rank n can be written as the sum of matrices Y and Z where Y has rank n-1 and Z has rank of 1. Thanks!
  49. P

    Partial fractions decomposition?

    I'm trying to solve this integral but I'm not sure if I'm on the right track. My question is: can this integral be solved by partial fractions decomposition? I solved the problem that way but I'm not sure if it is the right answer. thanks! ∫1-x+2x^2-x^3 ÷ x(x^2+1)^2
  50. E

    Fast Polar Decomposition Of A sTrAnGe Square Matrix - Help

    Fast Polar Decomposition Of A sTrAnGe Square Matrix --!Help! Homework Statement | 2 0 4 | | 0 1 0 | = A | 3 0 2 | Calculate the Polar Decomposition of A: A = LP L =^{t}L > 0 L is symmetric and positive defined P^{t}P = I...
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