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. icesalmon

    Partial Fraction Decomposition

    Homework Statement use partial fraction decomposition to re-write 1/(s2(s2+4) The Attempt at a Solution I thought it would break down into (A/s) + (B/s2) + ((cx+d)/(s2+4) but it doesn't.
  2. maajdl

    Singular value decomposition and eigenvalue problem:

    Could you explain me: what the difference is between singular value decomposition and eigenvalue problem, when square matrices are involved. Thanks
  3. S

    MHB Partial Fraction Decomposition

    When I'm evaluating a problem like \int \frac{2x^2 + 8x + 9}{(x^2 + 2x + 5)(x + 2)} = \frac{Ax + B}{x^2 + 2x + 5} + \frac{C}{x+2} I understand how to get the C part, that's simple. But what is a Good trick to know that I need to have Ax + B over the x^2 + 2x + 5 denominator? Is there a way I...
  4. P

    MHB New to Linear Algebra - LU Decomposition

    Just came across LU decomposition and I am not sure how to work on this problem: Let L and L1 be invertible lower triangular matrices, and let U and U1 be invertible upper triangular matrices. Show that LU=L1U1 if and only if there exists an invertible diagonal matrix D such that L1=LD and...
  5. F

    MHB Spectral decomposition of compact operators

    Let T be a compact operator on an infinite dimensional hilbert space H. I am proving the theorem which says that $Tx=\sum_{n=1}^{\infty}{\lambda}_{n}\langle x,x_{n}\rangle y_{n}$ where ($x_{n}$) is an orthonormal sequence consisting of the eigenvectors of $|T|=(T^*T)^{0.5}$, (${\lambda}_{n}$)...
  6. G

    Different solutions to x/(1-x) by substitution and decomposition

    Homework Statement Derive x/(1-x) Homework Equations By substitution: u = (1-x) du = -dx ∫x/(1-x)dx = -∫(1-u)/u du = ∫1-(1/u)du = u-log(u)+c = (1-x) - log(1-x) + c By decomposition: x/(1-x) = 1/(1-x)-1 ∫1/(1-x)-1dx = -log(1-x)-x+c The Attempt at a Solution Which solutions...
  7. jdawg

    Partial Fraction Decomposition with Integration

    Homework Statement ∫(2x3-4x-8)/(x2-x)(x2+4) dx Homework Equations The Attempt at a Solution ∫(2x3-4x-8)/x(x-1)(x2+4) dx Next I left off the integral sign so I could do the partial fractions: 2x3-4x-8=(A/x)+(B/(x-1))+((Cx+D)/(x2+4))...
  8. S

    MHB Partial Fraction Decomposition Evaluation

    Ok I'm stuck I have \int \frac{x^2 - 5x + 16}{(2x + 1)(x - 2)^2} \, dx and I got to this part: x^2 - 5x + 16 = A(x - 2)^2 + B(x - 2)(x + \frac{1}{2}) + c(x + \frac{1}{2})So do i need to distribute all of these and factor out or is there a simpler way? I found a solution where they are just...
  9. S

    MHB Partial Fraction Decomposition

    Quick question... I know that if the numerator is greater than the denominator I need to divide out by long division BUT If the numerator is equal to the denominator (the exponent is what I'm talking about to be specific) then, do I need to do anything? Because I'm stuck on this problem \int...
  10. jdawg

    Partial Fraction Decomposition

    Homework Statement (t4+9)/(t4+9t2) Homework Equations The Attempt at a Solution I'm not completely sure if I'm using the correct method to solve this. Since the degrees of the numerator and denominator are the same, wouldn't you divide the denominator into the numerator? Here is...
  11. W

    Thermal decomposition of calcium carbonate - problem

    Hello, here goes the problem: The mass of calcium carbonate upon thermal decomposition decreased by 1/5. (a) How many molecules of CaCO3 per 100 molecules were decomposed to CaO and CO2. (b) The content of CaO in the final sample express in molar fraction. My solution: I set the starting...
  12. M

    Thermal decomposition of calcium carbonate ?

    In general, we heat CaCO3 to temperature of approximately 825°C it decomposes into calcium oxide and liberates carbon dioxide gas: CaCO3 →825°C→ CaO + CO2 Is it possible to heat the calcium carbonate at temperatures below 825 degrees Celsius, so the decomposition into calcium oxide.
  13. J

    MHB Partial fraction decomposition

    Q3.) Express as partial fractions. a) \frac{3x+4}{x^2+3x+2} b) \frac{5x^2+5x+8}{(x+2)\left(x^2+2 \right)} c) \frac{x^2+15x+21}{(x+2)^2(x-3)}
  14. C

    Chemical transformation (Decomposition)

    Hi, I was wondering when you need to write the decomposition of a substance, how do you know if the number is going to be a coefficient or subscript ? Example: 2H2O ===>2H2 + 02 would be the answer But why not 2H2O ==> 2H2 + 2O Knowing that we have 2 moles of O in the beginning...
  15. MarkFL

    MHB Integration by Partial Fraction Decomposition - Yahoo Answers

    Here is the question: I have posted a link there so the OP can view my work.
  16. J

    Can Ferromagnetic Objects Be Decomposed Using Magnetism?

    I have came up with a theory, it could be completely wrong or not but i had the idea of ferromagnetic decomposition and what i mean by this is breaking a ferromagnetic object apart (such as iron) using magnetism. Is this possible or not?
  17. J

    Number of jordan blocks in Jordan decomposition

    Given a matrix $$A$$. Is it possible to have a Jordan block form like: $$\begin{pmatrix} \lambda & 1 & 0 & 0\\ 0 & \lambda & 0 & 0 \\ 0 & 0 & \lambda & 1\\ 0 & 0 & 0 & \lambda\\ \end{pmatrix}$$ ?
  18. Hercuflea

    How to use QR decomposition to find eigenvalues?

    Homework Statement I need to understand how I would go about using QR decomposition of a matrix to find the matrix's eigenvalues. I know how to find the factorization, just stuck on how I would use that factorization to find the eigenvalues. Homework Equations A=QR where Q is an...
  19. J

    Is the Matrix R Necessarily Lower Triangular in Matrix Decomposition?

    I would like to learn bit more about matrices and their decomposition. Let ##\mathbf C## be symmetric real-valued square matrix. Let ##\mathbf R## be such that $$ \mathbf R\mathbf R^T = \mathbf C. $$ Is the matrix ##\mathbf R## necessarily lower triangular (I suspect not)? Cholesky...
  20. H

    A Question on Lebesgue Decomposition

    Homework Statement Let μ be the counting measure and m be the Lebesgue measure. Then show that on the interval [0,1] m has no Lebesgue decomposition with respect to μ. Homework Equations If such a decomposition exists, then the following holds true where X is the whole space, E is a subset...
  21. twoski

    Operation Count for LU Decomposition

    Homework Statement Consider the n x n matrix A = diag[1,3,1] and vector x: (1,2,3) Determine the number of operations needed to compute the LU decomposition of this n x n matrix. The Attempt at a Solution So for a general n x n matrix, my prof's notes say that LU decomposition...
  22. NATURE.M

    Matrix Diagonalization & Eigen Decomposition

    Do these terms practically refer to the same thing? Like a matrix is diagonalizable iff it can be expressed in the form A=PDP^{-1}, where A is n×n matrix, P is an invertible n×n matrix, and D is a diagonal matrix Now, this relationship between the eigenvalues/eigenvectors is sometimes...
  23. K

    MHB Decomposition formulas for rotational symmetries of a cube

    I have a problem that I would like to check my work on. I am also stuck on the verifications for $E$ and $F$. Any help would be greatly appreciated. Thanks in advance. **Problem statement:** Let $G$ be the group of rotational symmetries of a cube, let $G_v, G_e, G_f$ be the stabilizers of a...
  24. G

    MHB QR Decomposition and Full Column Rank of A

    Hey guys, I have a problem where I am supposed to prove that R is nonsingular iff A is of full column rank in a QR decomposition. I feel like I fully understand the two major processes for obtaining a QR decomposition (Gram-Schimdt and Householder Transformations), however, I am not entirely...
  25. L

    I interpreting my results for a thermal decomposition lab

    Homework Statement I performed an experiment recently, which required me to heat a flask and an unknown compound to constant weight. The unknown was supposed to be a mixture of calcium sulfate dihydrate and NaCl. My goal for this lab was to determine the % NaCl in the mixture. My percent error...
  26. D

    Finding the Jordan decomposition

    Homework Statement I have a linear transformation T defined by $$ T(v_{1})=v_{1}+iv_{2}\\ T(v_{2})=-iv_{1}+v_{2}\\ $$ and I want to find a triangular matrix B of T and an invertible matrix S such that SB=AS where A is the matrix of T with respect to the basis ##\{v_{1},v_{2}\}##.The Attempt at...
  27. J

    Partial Fraction Decomposition

    My professor asks us to solve the integral of: [x/(x^4 + 1)]dx This expression is not factorable; what should I do? She is asking us to solve specifically using PFD, not u-substitution.
  28. twoski

    Checking Correctness of LU Decomposition

    Homework Statement compute the LU decomposition of the 3x3 matrix: A= 2, 1, 1/2 1/2, 2, 1 1, 1/2, 2 Let f be the vector (5,3,6) Solve Lg=f and then Ux=g. You can check your answer with A_{3}x=f The Attempt at a Solution I finished the calculations: U = 1, 1/2, 1/4...
  29. twoski

    Operation Count w/ LU Decomposition

    Homework Statement Fidn the total multiplications and divisions needed for the LU Decomp. of a general n x n matrix A, whose entries satisfy a_{ij} = 0 if j ≤ i - 2 Assume n=5. Also f ind the total multiplications and divisions needed for solving the lower triangular system Lg=f and for...
  30. I

    I don't understand partial fraction decomposition

    if there is something like (x^2+3x+6) in the denominator for one of the terms in a partial fraction problem, why do we put Ax+B instead of just A? and if the denominator is (x^2+3x+6)^2, why do we do {(Ax+B)/(x^2+3x+6)}+{(Cx+D)/(x^2+3x+6)^2}? i was always just told to memorize it, but why do we...
  31. S

    Partial Fraction Decomposition

    Hello I am stuck on an ODE involving substitution. I have done the correct substitutions, but have become stuck on decomposing the fraction. i have the following ∫(1/x)dx + ∫(u+1)/(u^2+1)du = 0 Im stuck on breaking the u down into a partial decomposition. Could anyone offer some advice on...
  32. I

    MHB Singular value decomposition

    Hi Suppose that A \in \mathbb{R}^{3 \times 3} who maps the unit sphere in \mathbb{R}^3 to an ellips with the following semi-axes; x = \left(-\frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}},0\right)^{T} \mapsto Ax = (2,0,0)^{T} x=\left(-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}},0\right)^{T} \mapsto Ax =...
  33. N

    Fraction decomposition for inverse Laplace

    Homework Statement Find the solution of the givien initial value problem and draw its graph y''+2y'+2y = δ(t-π) y(0) = 1, y'(0) = 0 Homework Equations A Laplace transform chart would be very useful The Attempt at a Solution I chose to solve the equation with Laplace...
  34. chwala

    Adomian Decomposition method research Ph.D ADVISE NEEDED

    Hi, I hope i have posted this in the right forum I am requesting to get a supervisor who is willing to supervise me in the areas of differential equations hence get an admission for Ph.D overseas I did my M.sc project in ADM METHOD IN SOLVING THE STURM LIUVILLE EIGENVALUE PROBLEM(see...
  35. P

    Deriving decomposition of transverse acceleration

    Homework Statement The question asks one to derive the acceleration vector, $$\vec{a} = [\frac{d^2r}{dt^2} - r(\frac{d\Theta}{dt})^2]\vec{u}_r + [\frac{1}{r}\frac{d}{dt}(r^2\frac{d\Theta }{dt})]\vec{u}_\Theta$$ from the velocity vector. $$\vec{v} = \vec{u}_r\frac{dr}{dt} +...
  36. B

    Decomposition of the acceleration vector

    A basis for any 3-dimensional vector space must have 3 vectors in it. So the acceleration of any object in ℝ^{3} can be decomposed into the standard basis vectors for ℝ^{3}. However, I have seen another decomposition, namely, into the tangential and normal (centripetal) acceleration vectors...
  37. N

    Partial fraction decomposition: One quick question

    Homework Statement Give the partial fraction decomposition of 1/z4+z2 Homework Equations The Attempt at a Solution My question is about the final answer. The book gives the answer to be 1/z2+ 1/2i(z+i)- 1/2i(z-i). For my answer I keep getting a negative for both of the 1/2i...
  38. MarkFL

    MHB DW123's question at Yahoo Answers regarding partial fraction decomposition

    Here is the question: Here is a link to the question: Decompose the equation into two simpler fractions? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  39. PhizKid

    What is the Mass of LiHCO_3 in the Original Mixture?

    Homework Statement Given the reaction: 2LiHCO_3 + SiO_2 \rightarrow Li_{2}CO_3 + SiO_2 + CO_2 + H_{2}O where SiO_2 is unaffected, the mass of the 2LiHCO_3 + SiO_2 is 9.62 g. and Li_{2}CO_3 + SiO_2 is 6.85 g., find: a) mass loss due to CO_2 + H_{2}O b) mass of LiHCO_3 in the original...
  40. Jameson

    MHB Best method for LU decomposition

    Hi all, I realize there might not be a "best method" but I want to ask if anyone has any improvements to the method taught in my class. I've looked at the Wikipedia page on this already. Let's use the matrix \left( \begin{array}{ccc} 2 & -1 & 2 \\ -6 & 0 & -2 \\ 8 & -1 & 5 \end{array} \right)...
  41. R

    2D FIR Filter & Finite Decomposition

    Hi All, I would like to know how can I call or express the following process! I use a (3x3) 2D FIR Filter for imaging processing with DC = 0, like this: 0 1 1 2 O 2 /8 1 1 0 My filter is such that I can decompose it into finite sates, as my image (medical) can take 9...
  42. M

    The Schmidt Decomposition: Looking for some intuition

    Hi, I finished reading about the Schmidt decomposition from Preskill's notes today. I understand and follow his derivation but it still seems completely non intuitive to me. We have \mid\psi\rangle_{AB}=\sum_{i,u}a_{iu}\mid i\rangle_{A}\mid u\rangle_{B}=\sum_{i}\mid...
  43. MarkFL

    MHB Partial Fraction Decomposition Help - Calculus BC

    Here is the question: Here is a link to the question: Help with Calculus BC: partial fractions!? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  44. B

    QM I - Decomposition of countable basic states into coherent states

    Homework Statement Consider a quantum system with a countable number of basic states \left|n\right\rangle. Calculate the decomposition into a basis of coherent states \left|λ \right\rangle all obeying \hat{a} \left|λ \right\rangle = λ \left|λ \right\rangle Homework Equations \hat{a}...
  45. L

    Enthelpy of the decomposition reaction of N2H4 gas

    I am trying to determine the change of heat for: N2H4(g) -> N2(g) + 2H2(g) Here is what I did and what I got the correct answer is -86.0 which I am clearly not getting [(163)+ (2*436)] - [(4*391) + (163)] = -692 thanks for any help!
  46. I

    Help with LU Decomposition in C

    Hey, I have this code for lu decomposition but It doesn't quite work. If anyone could help me with the problem I'd be very appreciative. for(j=0; j<N; j++) for(i=j+1; i<N; i++) U[i][j]=0; for(j=0; j<N; j++) for(i=j+1; i<N; i++)...
  47. matqkks

    MHB Exploring the Importance and Applications of Singular Value Decomposition (SVD)

    What is the best way of introducing singular value decomposition (SVD) on a linear algebra course? Why is it so important? Are there any applications which have a real impact?
  48. matqkks

    How can we use the Singular Value Decomposition (SVD) for image compression?

    What is the best way of introducing singular value decomposition (SVD) on a linear algebra course? Why is it so important? Are there any applications which have a real impact?
  49. H

    Nightclub physics: Fourier decomposition with cocktail glasses

    This is purely conceptual and I'm just looking for opinions on whether its misguided or, indeed, plausible. From what I understand about Fourier decomposition we can break down an analog signal into component sinusoidal waves. My thinking is that the sound system at a nightclub can be...
  50. Einj

    Isospin decomposition of K->ππ decay

    I'm studying the decay K->ππ and I have some doubts on the isospin decomposition. We know that the state (\pi\pi) can have total isospin 0 or 2. Now, if we remember that in the isospin representation we have |\pi^+\langle=|1,1\langle, |\pi^0\rangle=|1,0\rangle and |\pi^-\rangle=|1,-1\rangle...
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