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. U

    Kronecker product decomposition

    hi everybody Today I have a question about Kronecker products, If you have a direct answer it is perfect but if not, any kind of paper reference might work as well. now say I have to matrices A and B in general there is nothing special about them. They are not hermitian or triangular or what...
  2. O

    Octave partial fraction decomposition help

    So I'm trying to figure out how to decompose the following using octave: 85000/[(s^2+250^2)(0.2s^2+40s+10000)] I tried using the residue command but I think that only works if the polynomials have real roots, which these don't. When I do use residue I get the following: b =...
  3. askhetan

    Decomposition of total Wavefunction for N electron problem

    A. Given the exact locations of N electrons - can we find out the total wave function and the individual wave functions by some law of nature? B. Given the exact total wave function - can we find out the coordinates of each electron and the indiviual wave function of each electron by some law...
  4. D

    Examine Structurally Distinct Abelian Groups with Primary Decomposition Thm

    I am just into reviewing abstract algebra and came across a theorem I'd forgotten: http://en.wikipedia.org/wiki/Finitely-generated_abelian_group#Primary_decomposition (I linked to the theorem instead of writing it here just because I'm not sure how to write all those symbols here) Anyway...
  5. S

    Spectral decomposition of a diagonal matrix

    Homework Statement I have J=\begin{bmatrix} \frac{\pi}{2}&0&0\\ 1&\frac{\pi}{2}&0\\ 0&1&\frac{\pi}{2}\\ \end{bmatrix} I need to find \sin(J) \text{ and } \cos(J) \text{ and show that } \sin^{2}(J)+\cos^{2}(J)=I Homework Equations The Attempt at a Solution I have the...
  6. T

    Proof of Singular Value Decomposition

    Homework Statement A is arbitrary linear map of the complex vector space s.t. A: V-->V. 1) Show that there exist unitary matrices s.t A=V*DU where D is diagonal and its entires are non-negative. 2) Show that part 1 holds if and only if there exist orthonormal bases {u_i} and {v_i} s.t...
  7. H

    Temperature of the products of decomposition of hydrazine

    Dear all, Homework Statement For practice, I'm trying to solve an exercise I've found on the Internet. A screen shot of it is attached. I would like to be sure about my approach, especially when it comes to question 2. Let's say, for the sake of simplification, that the reaction is as...
  8. D

    Half-life decomposition, find final concentration

    Homework Statement In a first order decomposition in which the rate constant is 0.03 sec-1, how much of the compound (in mol/L) is left after 39 sec, if there was 2.00 mol/L at the start? I'm using a few equations and trying to plug it in but I don't know whether they are appropriate or not...
  9. H

    Investigating the Hydrothermal Decomposition of Hemicellulose

    I'm working on the hydrothermal decomposition of hemicellulose and one of the reaction products is acetic acid. The incomplete reaction model I'm using says 20% of my original hemicellulose is converted into acetic acid. In comparison to other papers this seems to be too high. So were does my...
  10. caffeinemachine

    MHB Graph Theory. Decomposition of K_{2n+1} into hamiltonian cycles.

    Theorem: Prove that there exist $n$ edge disjoint Hamiltonian cycles in the complete graph $K_{2n+1}$. ---------------------------------------------------------------------------------- I have found two constructive proofs of this over the internet. But I would like to prove it...
  11. T

    Partial fraction decomposition

    I have been having trouble of late with partial fraction decomposition. Not so much the maths, but the intuition behind it. What I mean by this, but a question in front of me, I now what procedure to follow to get the answer, but I don't get why you follow the said produced. I will give an...
  12. G

    QR Decomposition of Matrix: Solve for Answers

    Homework Statement Decompose the following matrix using QR decomposition \begin{bmatrix} 4 & 1 \\ 3 & -1 \end{bmatrix} the answer is \begin{bmatrix} .8 & .6 \\ .6 & .8 \end{bmatrix} The following matrix is supposed to be next to the previous but I can't figure out how to do that. Any...
  13. Q

    Decomposition of a Rotation Matrix

    Hey guys! I'm new here, so forgive me if I'm posting in the wrong section. I recently picked up a book on robotics and it had a section about rotation matrices. I'm having a difficult time with the decomposition of rotation matrices. Everywhere I look, I can find the the equations for the roll...
  14. M

    Decomposition of Vector Addition to lead to unique solution

    I have a puzzling question that a gentleman discussed and have differnt view on the problem, the probelm goes as follows: What is know is the Resultant of a Vector Addition, the goal is to find the two vectors from which the resultant came from. The other piece of information that is know is...
  15. N

    Helmholtz Decomposition: Magnitude of Irrot. & Solenoidal Comp.

    Hi, Consider P = \boldsymbol{\nabla} f +\boldsymbol{\nabla}\times \bold{A} where f and A are scalar and vector potentials, respectively, and P is strictly positive and well behaved, and only nonzero in a domain \mathcal{D}. I want to find how the magnitude of \int \boldsymbol{\nabla} f dV...
  16. B

    Decomposing Functions: Finding the Pattern

    Homework Statement If g(x) = 2x + 1 and h(x) = 4x^2 +4x + 7 , find a function f such that f o g = h Homework Equations The Attempt at a Solution Well, I know I have to decompose h into two parts: the give one being g, and the other f. But I can't seem to perceive any relation between g and h...
  17. A

    Complex and Real Representations, their differences by decomposition

    Homework Statement Decompose \mathbb{C}^{5}, the 5 dimensional complex Euclidean space) into invariant subspaces irreducible with respect to the group C_{5} \cong \mathbb{Z}_{5} of cyclic permutations of the basis vectors e_{1} through e_{5}. Hint: The group is Abelian, so all the irreps...
  18. M

    Reif Ch7, Decomposition of partition function

    Homework Statement For a system A consists of two parts A' and A'' which interact only weakly with each other, if the states of A' and A'' are labeled respectively by r and s, then a state of A can be specified by the pair of numbers r,s and its corresponding energy E_{rs} is simply...
  19. T

    Partial fraction decomposition with complex function

    As part of a project I have been working on I fin it necessary to manipulate the following expression. e^(icx)/(x^2 + a^2)^2 for a,c > 0 I would like to decomp it into the form A/(x^2 + a^2) + B/(x^2 + a^2) = e^(icx)/(x^2 + a^2)^2 but I am having trouble getting a usable outcome.
  20. S

    PA=LU decomposition (w/ Partial Pivoting)

    I'm have a little trouble understanding PA=LU, I have no problems with A=LU but can't seem to figure out the Permutation matrix. So I have summarised the process I am using let me know where it can be improved. Step 1: Using Gaussian Elimination with partial pivoting reduce A to form a...
  21. J

    Prove dual space has the direct sum decomposition

    I apologize for not having any attempted work, but I have no idea how to even begin tackling this proof. Any direction would be greatly appreciated! Mike Homework Statement Let V be a vector space, Let W1, ..., Wk be subspaces of V, and, Let Vj = W1 + ... + Wj-1 + Wj+1 + ...
  22. H

    LDU decomposition - sparse matrix

    I have a large sparse symmetric matrix and I'd like to know the number of its negative eigenvalues. To this end, I should perform an LDLT decomposition of the matrix and count the number of negative diagonal entries of the D matrix. This would be equal to the number of negative eigenvalues...
  23. Shackleford

    Spectral Decomposition

    \begin{bmatrix} 1 & 2 \\ 2 & 1 \\ \end{bmatrix} = A The eigenvalues of A are 3 and -1. The eigenvectors are (1,1) and (-1,1), respectively. I'm not sure how to proceed.
  24. R

    Partial fraction decomposition

    Homework Statement \frac{2e^3}{((s^2)-6s+9)*s^3} you can factorize the denominator into s,s,s,(s-3),(s-3) that gives you 5 residuals. the first 3 should all be the same value but that's apparently not correct, so where am I going wrong?
  25. T

    Help With Partial Fraction Decomposition

    Homework Statement I'm supposed to decompose 1 / x(x2 + 1)2 Also, we haven't learned matrices yet so I can't use that technique to solve it. Homework Equations None. The Attempt at a Solution 1 / x(x2 + 1)2 = A/x + (Bx + C) / (x2 + 1) + (Dx + E) / (x2 + 1)2 I multiplied...
  26. E

    How can QR decomposition be used to solve least squares problems?

    Homework Statement Okay so I'm supposed to find the least squares solution of a set of equations, which I can do, but it adds that I must use QR decomposition. I don't really know how to apply QR decomposition to this problem. Problem: Find the least squares solution of x_1 + x_2 = 4...
  27. T

    Proof about the decomposition of the reals into two sets.

    Homework Statement Let S and T be nonempty sets of real numbers such that every real number is in S or T and if s \in S and t \in T, then s < t. Prove that there is a unique real number β such that every real number less than β is in S and every real number greater than β is in T. The Attempt...
  28. R

    Calculating Heat Energy Released in Ozone Decomposition Reaction

    I have been doing some research on ozone recently, and I have a question that none of my resources have answered... The following reaction is an exothermic reaction, but how would I calculate the amount of heat energy is released? O_{3} \underbrace{\rightarrow}_{uv\ light} O_{2} + O...
  29. M

    LU decomposition of matrix

    Actually I am new to this topic. I read few tutorials about LU decomposition of matrix in the net. A = LU ; A - actual matrix, L - Lower triangular matrix, U - Upper triangular matrix.Few people say that, principal diagonal elements of L should be unity. Some others say that, principal...
  30. B

    Eigenvalue VS Cholesky Decomposition

    Assuming the matrix is positive definite (necessary for cholesky decomposition). Which is faster? Which is more accurate? Is there a reliable source that has all the most common decompositions listed in order of accuracy and speed?
  31. B

    Fortran eigenvalue decomposition

    I've been trying to invert a real symmetric matrix and the inverse that I compute via eigenvalue decomposition is not the inverse (using QV^-1Q^T), the stranger thing is that QVQ^T gets back my orginal matrix matrix. Even more unusual is that the matrix starts off at approximately identity (in...
  32. R

    Decomposition of SU(3) and particles

    As we know the algebra of SU(3) consist of two Cartan generators and 6 raising and lowering operators. We define the eigenstates of the Cartan operators as u,d,s, correspoding to the three lightest quarks. Now when we study the 3\otimes 3 tensor product we can show that the Hilbert space of...
  33. B

    Can Eigenvectors Diagonalize a Matrix with Only One Eigenvalue?

    Let F= R or C, and A = [1 2 3] is considered as linear operator in F3 [0 1 2] [0 0 1] then the minimal polynomial of A = (x-1)^3, can we say that the primary decomposition thm doesn't give any decomposition, can we find an invertible P s.t P^-1*A*p is a block diagonal matrix?
  34. D

    Help Factorial Partial Fraction Decomposition

    Homework Statement Show that n/(n+1)!=(1/n)-(1/(n+1)!) I am totally lost on the algebraic steps taken to come to this conclusion. It is for an Infinite series. Thanks
  35. S

    First Passage Decomposition

    Homework Statement I have a Stochastic Processes test coming up soon and we are taught about the first passage decomposition. I understand this but it then says that it changes it to a generating function Homework Equations Generating function Pij(s) = Ʃ pij(n)sn...
  36. F

    What is the Principle of Cluster Decomposition in Quantum Field Theory?

    Recently a student brought the following to my attention from Weinberg’s Quantum Theory of Fields, Volume I, from page 177, which I must admit that it stumped me. Here Weinberg introduces the concept of Cluster Decomposition: “It is one of the fundamental principles of physics (indeed, of all...
  37. H

    Solving Ax=b; when to use LU decomposition?

    I want to solve the linear equation below: Ax = b For this purpose, I'm writing a C++ code. I have written both routines for decomposing A matrix to L and U matrices, and for calculating inverse of A matrix. I may multiply both sides with A-1: Ax = b A^{-1}Ax = A^{-1}b x =...
  38. X

    Online tools for Gaussian elimination / LU decomposition ?

    I can not find any useful online tool that solves you a system of equations using Gaussian elimination and LU decomposition. So just a system like: -3X+4X+9X+4x=-2 9X+2X+1X-5X=2 etc. Just about 4 lines. several lines. So basically an online tool where you can just plug in the...
  39. Fredrik

    Proof of Hahn decomposition theorem

    I'm trying to read this proof, and I'm stuck on the inequality on page 27 following the statement "It follows that every measurable subset..." Why does it hold? The theorem is about signed measures, i.e. functions that are like measures, but can assign both positive and negative "sizes" to...
  40. B

    Partial fraction decomposition

    Yes, another of these. How do you decompose (z3+1)/(z(1-z)2) ? I've tried A/z + B/(1-z) + (Cz+D)/(1-z)2 and a slew of others that don't work. Thanks a bunch!
  41. N

    What is the condition number of a rectangular matrix and how can it be finite?

    Homework Statement Hi If we have a matrix M, we can always make a singular value decomposition. If the matrix has full column rank (= is invertible), then the singular values are all nonzero, otherwise they are not all nonzero. Now, we can also associate a condition number to a matrix...
  42. C

    Jordan Matrix Decomposition

    A = PD[P][/-1]; A: square matrix; D: is a matrix of Jordan canonical form P:is EigenVectors..(p1,p2,p3,p4...pr) Is it possible to permute the sequence p1,p2,p3,p4...pr into other form? I know it should be possible to permute... How should permute it and what significance it has? Why is...
  43. K

    The plane wave decomposition is mathematically universal?

    The plane wave decomposition is mathematically universal? 1. My questions is: Can "expi[wt-(w/c)*r]/r with r=sqrt(x**2+y**2+z**2), the frequency w is given, monochromatic wave" be represented as a sum of uniform plane waves? Note: r=0 is a singularity. This is actually the potential...
  44. K

    Decomposition of numbered permutations.

    Say I have three elements: A, B, C. I can list all the permutations by going alphabetical in the first element, then the second, then the third, and so on, like so: ABC ACB BAC BCA CAB CBA What I'm wondering, is given a number N, how do I decompose this into knowing what permutation it...
  45. K

    Cyclic Decomposition Theorem

    I took an Intermediate Linear Algebra course all last year (two semesters worth) and we covered the CDT. My professor didn't teach it well, and I got my first B- in university because of it (didn't affect my GPA but still irritating). I didn't understand a lot of the canonical form stuff...
  46. jfy4

    Decomposition of vector potential

    Hi, The vector potential for elctrodynamics, A_{\mu}, can be decomposed A_{\mu}\in\mathbf{0}\oplus\mathbf{1} but the \mathbf{0} part we ignore. My question is: do we ignore the scalar part because of experiment, or is it ignored for mathematical reasons I am naive about? Thanks,
  47. S

    Spectral Decomposition of a Matrix

    Homework Statement Find the projectors of matrix A (I am not sure how to write it in matrix form here so I have written out each row of the matrix) Row 1 : [ 2 1 1] Row 2: [ 2 3 2] Row 3: [ 1 1 2] Homework Equations A theorem in my textbook says the Lagrange...
  48. D

    Proving diagonal entries of R in QR decomposition nonzero

    Homework Statement Prove that the diagonal entries of R in the QR decomposition are non-zero. Homework Equations ui = qi + <ui, q1>q1 + <ui, q2>q2 + ... + <ui, qi-1>qi-1 => From the general equation for the Gram-Schmidt process. <ui, qi> => From the diagonal entries of QR. The...
  49. M

    Eigen decomposition of a large spasre matrix

    I have 12000 x 12000 R(data class single) sparse matrix, and I want to find its eigen values and vectors, but I can't do it in matlab, it gives out of memory error by doing [V D] = eigs(R) and I am using 4GB RAM , online help suggested using ARPACK or ARPACK++ with C/C++. please any...
  50. D

    Singular Value Decomposition for a Given Matrix

    Homework Statement I want to do a singular value decomposition for the following matrix: M = \left\lceil 1 \ \ -1 \right\rceil \ \ \ \ \ \ \ \:\:\left\lfloor 1 \ \ -1 \right\rfloor Homework Equations M=U\Sigma V^{\ T} M^{\ T}M MM^{\ T} The Attempt at a Solution To...
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