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. M

    MHB LR decomposition with column pivoting

    Hey! :o We have the matrix $$A=\begin{pmatrix}1 & -2 & 1 \\ 3 & -1 & 2 \\ -2 & -2 & 1\end{pmatrix}$$ I want to apply the LR decomposition with column pivoting. First we permutate the first two rows and we get $$A=\begin{pmatrix}3 & -1 & 2 \\ 1 & -2 & 1 \\ -2 & -2 & 1\end{pmatrix}$$ Then we...
  2. Telemachus

    MPI: domain decomposition strategy

    Hi. I'm trying to parallelize my code. I am new at MPI, I'm learning the basics. I want to use a domain decomposition strategy in my code. I've been reading a bit, and wanted to use this subroutine to exchange points between neighbors. I've started by trying to modify a code presented by Gropp...
  3. N

    Clebsch-Gordan Decomposition for 6 x 3

    Homework Statement [/B] I am trying to get the C-G Decomposition for 6 ⊗ 3. 2. Homework Equations Neglecting coefficients a tensor can be decomposed into a symmetric part and an antisymmetric part. For the 6 ⊗ 3 = (2,0) ⊗ (1,0) this is: Tij ⊗ Tk = Qijk = (Q{ij}k + Q{ji}k) + (Q[ij]k +...
  4. Y

    I Schmidt decomposition and entropy of the W state

    Hello, The state | W \rangle = \frac { 1 } { \sqrt { 3 } } ( | 001 \rangle + | 010 \rangle + | 100 \rangle ) is entangled. The Schmidt decomposition is : What would the Schmidt decomposition be for | W \rangle ? I am also intersted in writing the reduced density matrix but I need the basis...
  5. D

    Integration of Spherical Harmonics with a Gaussian (QM)

    Homework Statement I wish to solve this integral $$b_{lm}(k) = \frac{1}{2(\hbar)^{9/4}(2\pi)^{5/2}\sqrt{\sigma_{px} \sigma_{py} \sigma_{pz}}} \int_{\theta_k = 0}^{\pi}\int_{\varphi_k = 0}^{2\pi} i^l \text{exp}\left[ - \frac{1}{(2\hbar)^2}\left(\frac{(k_z - k_{z0})^2}{\sigma_{pz}^2} + \frac{(k_y...
  6. opus

    Partial Fraction Decomposition

    Homework Statement Find the partial fraction decomposition for: ##\frac{1}{\left(x^2-1\right)^2}## Homework EquationsThe Attempt at a Solution Please see my attached images. I think the image shows my thought process better and it would take me well over an hour to type all that out! Im...
  7. opus

    B Partial Fraction Decomposition - "Telescoping sum"

    There is a problem in a PreCalculus book that I'm going over that states: Express the sum ##\frac{1}{2⋅3}+\frac{1}{3⋅4}+\frac{1}{4⋅5}+...+\frac{1}{2019⋅2020}## as a fraction of whole numbers in lowest terms. It goes on to state that each term in the sum is of the form...
  8. D

    Mde decomposition of quantum field in a box

    Homework Statement I am considering the Klein Gordon Equation in a box with Dirichlet conditions (i.e., ##\hat{\phi}(x,t)|_{boundary} = 0 ##). 1-D functions that obey the Dirichlet condition on interval ##[0,L]## are of the form below (using the discrete Fourier sine transform) $$f(x) =...
  9. J

    Decomposition of linearly polarized field MRI

    Homework Statement Hi, I am having trouble understanding how the B1 field as described by (3.48) in the image attached in MRI which is described as a linearly polarized field is decomposed into it's final two circularly polarized field as described by (3.49) in the image attached. Homework...
  10. L

    MHB Proving Matrix Equality Using Singular Value Decomposition

    Hi, I have another question, if A and B are mxn matrices, how do I prove that $AA^T = BB^T$ iff $A = BO$ where $O$ is some orthogonal matrix? I think I need to use a singular value decomposition but I am not sure. Thanks!
  11. Q

    Stuck on this integral (using partial fraction decomposition)

    Homework Statement \int\frac{x^2}{\sqrt{x^2+4}}dx Homework Equations n/a The Attempt at a Solution Letting x=2tan\theta and dx=2sec^2\theta d\theta \int\frac{x^2}{\sqrt{x^2+4}}dx=\int\frac{4tan^2\theta}{\sqrt{4+4tan^2\theta}}2sec^2\theta d\theta=\int\frac{8tan^2\theta...
  12. P

    Potential flows and Helmholtz decomposition

    Hi. I'm studying fluid dynamics and in particular potential flows. I know that for an irrotational flow the velocity field is a conservative field and it can be rapresented by the gradient of a scalar field v=-∇Φ. In this case the explicit form of Φ is something like a line integral between a...
  13. H

    Full thermal decomposition of metal oxides?

    I haven't been able to find much information on the thermal decomposition of metal oxides into their corresponding metals and oxygen. What temperature would Fe3O4 decompose mostly(80%) into its base elements? Additionally, how can this information be determined based upon bond...
  14. Kaneki123

    Relation between Decomposition and Reversible reactions....

    Okay...I read that the decomposition of water is a reversible reaction (because the constituents can react to form water and water can decompose to form constituents)...This lead me to another thought that almost all compounds can be decomposed (although it is true that their conditions for...
  15. Mr Davis 97

    Partial fraction decomposition with cos() in the numerator

    Homework Statement See below Homework EquationsThe Attempt at a Solution I am looking at a particular integral, and to get started, my text gives the indication that one should use partial fraction decomposition with ##\displaystyle \frac{\cos (ax)}{b^2 - x^2}##. Specifically, it says "then...
  16. Mr Davis 97

    Partial fraction decomposition

    Homework Statement Find the partial fraction decomposition of ##\displaystyle \frac{1}{x^4 + 2x^2 \cosh (2 \alpha) + 1}## Homework EquationsThe Attempt at a Solution Using the identity ##\displaystyle \cosh (2 \alpha) = \frac{e^{2 \alpha} + e^{- 2\alpha}}{2}##, we can get the fraction to the...
  17. Adgorn

    I Proving a lemma on decomposition of V to T-cyclic subspace

    I am reading Schaum's outlines linear algebra, and have reached an explanation of the following lemma: Let ##T:V→V## be a linear operator whose minimal polynomial is ##f(t)^n## where ##f(t)## is a monic irreducible polynomial. Then V is the direct sum ##V=Z(v_1,T)⊕...⊕Z(v_r,T)## of T-cyclic...
  18. B

    MHB How can I factorize this polynomial?

    Decompose $$6a^2-3ab-11ac+12ad-18b^2+36bc-45bd-10c^2+27cd-18d^2$$ I noticed that the factorized form would be $$(Aa+Bb+Cc+Dd)(Wa + Xb + Yc + Zd)$$ Which is similar to the factorized form $$(Aa+Bb+Cc)(Wa+Xb+Yc)$$ $$Yc(Aa+Bb)+Cc(Wa+Xb) = c(CX+BY)$$ Is there a way that I can somehow use...
  19. BH1988

    Fe2O3 Thermal Decomposition Question

    Hi, I've been asked to heat a sample that contains Fe2O3, amongst some other things, to a temperature where Fe2O3 will decompose. I am unable to find the temperature at which I would need to heat the sample and I'm not entirely convinced this is actually possible with the equipment I have...
  20. karush

    MHB 206.08.05.44 partial fraction decomposition

    $\tiny{206.08.05.44}$ $\textsf{Use the method of partial fraction decomposition}\\$ \begin{align*} \displaystyle I_{44} &=\int \frac{4x^3+6x^2+128x}{x^5+32x^3+256x}dx\\ &=4\int \frac{1}{x^2+16} \, dx +6\int \frac{x}{(x^2+16)^2} \, dx +64\int \frac{1}{(x^2+16)^2} \, dx \end{align*}$\textsf{so far...
  21. karush

    MHB Partial fraction decomposition

    $\tiny{206.8.5,42}\\$ $\textsf{partial fraction decomostion}\\$ \begin{align} \displaystyle && I_{42}&=\int\frac{3x^2+x-18}{x^3+9x}\, dx& &(1)&\\ && \frac{3x^2+x-18}{x^3+9x} &=\frac{Ax+B}{x^2+9} +\frac{C}{x} & &(2)& \end{align} $\textit{just seeing if this is set up ok before finding values} $
  22. iCloud

    A Regression analysis and Time Series decomposition

    If we can use Regression analysis to forecast, why do we use “Time Series Decomposition”? What's the difference between the 2? Thanks
  23. A

    Calculators How to perform singular value decomposition using MS Excel?

    I have a matrix of data and I want to do SVD using excel, is it possible?
  24. A

    Car collision: decomposing momentum in x- and y-direction

    Homework Statement Two cars collide at an intersection. Car A, with mass 2000 kg, is going from west to east, while car B, with mass 1500 kg, is going from north to south at 15 m/s. As a result of this collision, the two cars become enmeshed and move as one afterward. In your role as an expert...
  25. Cosmophile

    I Partial Fraction Decomposition With Quadratic Term

    Hey, all! I'm learning partial fraction decomposition from Serge Lang's "A First Course in Calculus." In it, he gives the following example: \int\frac{x+1}{(x-1)^2(x-2)}dx He then decomposes this into the following sum: \frac{x+1}{(x-1)^2(x-2)} =...
  26. Turbodog66

    Solve Vector Decomposition Homework: F=-11j, v=-i-5j

    Homework Statement The force on an object is F = -11j. For the vector v =-i-5j, find: 1. The component of F parallel to v 2. The component of F perpendicular to v 3. The work, W, done by force F through displacement v Homework Equations ProjvF = v dot F/ |v|2 OrthvF = F - ProjvF W = D...
  27. Ken Gallock

    I What is the name of the matrix decomposition with specific properties?

    Hi everyone. There is the ##2\times 2## matrix ##B## $$B= \left[ \begin{array}{cc} B_{11} &B_{12} \\ B_{21}&B_{22} \end{array} \right],~B_{ij}\in \mathbb{C} $$ with property $$\vert B_{11}\vert^2 + \vert B_{12}\vert^2=1,$$ $$\vert B_{21}\vert^2 + \vert B_{22}\vert^2=1,$$...
  28. Anchovy

    A SU(5), 'Standard Model decomposition', direct sum etc.

    This has turned out to be a long question to type out so I apologise, but I don't think it's too hard to follow or read through quickly and I believe the actual question itself may not be too complicated once I get round to asking it. You can possibly skip to the last few paragraphs and still be...
  29. D

    I Decomposing Vectors Using Row Reduction: A Practical Approach

    Hello, I am trying to figure out how to best decompose a vector into a best fit linear superposition of other, given vectors. For instance is there a way of finding the best linear sum of: (3,5,7,0,1) (0,0,4,5,7) (8,9,2,0,4) That most closely gives you (1,2,3,4,5) My problem contains...
  30. mnb96

    I Question about decomposition of matrices in SL(2,R)

    Hello, we are given a 2×2 matrix S such that det(S)=1. I would like to find a 2x2 invertible matrix A such that: A S A^{-1} = R, where R is an orthogonal matrix. Note that the problem can be alternatively reformulated as: Is it possible to decompose a matrix S∈SL(2,ℝ) in the following way...
  31. T

    MHB Partial Fraction Decomposition when denominator can't be further factored

    I have this fraction $$x^2 / (x^2 + 9)$$ I'm not sure how to approach this problem since the denominator can't be further factored. What is the right approach for this type of problem?
  32. S

    I If pair of polynomials have Greatest Common Factor as 1 ....

    NOTE: presume real coefficients If a pair of polynomials have the Greatest Common Factor (GCF) as 1, it would seem that any root of one of the pair cannot possibly be a root of the other, and vice-versa, since as per the Fundamental Theorem of Algebra, any polynomial can be decomposed into a...
  33. D

    Partial fraction decomposition exercise 2

    Homework Statement Hello! Here is my second post on the subject partial fraction decomposition. The subject looks pretty easy to learn, but when I try exercises, I do not get to the correct answer. Please, take a look at the exercise below and help me to see my mistakes. Homework Equations...
  34. D

    Partial fraction decomposition using matrix

    Homework Statement Hello! I am doing a chapter on partial fraction decomposition, and it seems I do not understand it correctly. Here is the exercise doing which I get wrong answers. Please, take a look at the way I proceed and, please, let me know what is wrong in my understanding. Homework...
  35. Augbrah

    Representation decomposition

    Homework Statement Show that vector representation 5 and adjoint representation 10 in SO(5) decompose respectively into representations of SO(4) as: 5 →4⊕1 10→6⊕4 Homework EquationsThe Attempt at a Solution [/B] I understand that 5 is rep of SO(5) corresponding to Dynkin labels (1, 0). 1 is...
  36. Y

    Linear Algebra - Singular Value Decomposition Problem

    Homework Statement Find the SVD of Homework EquationsThe Attempt at a Solution I'm stuck My question is why in the solution it originally finds u_2=[1/5,-2/5]' but then says u_2=[1/sqrt(5),-2/sqrt(5)]'. I don't see what math was done in the solution to change the denominator from 5...
  37. T

    A Bessel decomposition for arbitrary function

    Orthogonality condition for the 1st-kind Bessel function J_m $$\int_0^R J_m(\alpha_{mp})J_m(\alpha_{mq})rdr=\delta_{pq}\frac{R^2}{2}J_{m \pm 1}^2(\alpha_{mn}),$$ where α_{mn} is the n^{th} positive root of J_m(r), suggests that an original function f(r) could be decomposed into a series of 1-st...
  38. Theengr7

    The decomposition of the numerator

    (a) Find a power series representation for the function. I'm struggling on the decomposition of the numerator. This exercise is from chapter 8, section 6 of Th Stewart Calculus book.
  39. P

    MHB Jamal's question via email about solving a system with a PLU decomposition.

    To start with, since $\displaystyle \begin{align*} A = P\,L\,U \end{align*}$, that means in our system we have $\displaystyle \begin{align*} P\,L\,U\,\mathbf{x} = \mathbf{b} \end{align*}$. Normally to solve for $\displaystyle \begin{align*} \mathbf{x} \end{align*}$ we would use inverses, so we...
  40. BobTheLawyer

    Derivation of Cholesky Decomposition

    Homework Statement Derive Cholesky Decomposition for a 3x3 matrix Homework Equations IN: S is Real matrix with dimensions 3x3 and is Symmetric and semi-definite Out: L is a Real matrix with dimensions 3x3 such that S=L*L^t L is lower-triangular The Attempt at a Solution We learned this in...
  41. Geek007

    Decomposition of apriodic and periodic signals

    Hi there, why the decomposition of periodic Composite signal give discrete frequencies and decomposition of aperiodic signal give continuous(in decimal) frequencies. please kindly do explain the concept behind in as simple words possible. Thanks
  42. P

    Helmotz decomposition definitions.

    I' m studing the hodge helmotz decomposition of a flow Field, and i have Found different definitions. I'm Not sure to have assigned the rigth meaning to the terms of the decomposition. Look At The picture( i don't write here cose there are several equations).
  43. R

    Sodium Bicarbonate Thermal Decomposition

    I have a question about the thermal decomposition of sodium bicarbonate (baking soda). I need to use baking soda for an experiment but will not be using a burner to heat it. We are heating it to about 90-100C and I was wondering if we put vessel the baking soda is in under vacuum -0.5bar or...
  44. Sace Ver

    Decomposition Rxn: (NH4)2CO3 → 2NH3 + H2O + CO2

    1. Homework Statement (NH4)2CO3 = 2NH3 + H2O + CO2 Homework Equations Decomposition Rxn The Attempt at a Solution I'm not quite sure why this reaction occurred can someone please explain it to me.
  45. L

    MHB Partial fraction decomposition

    Hello everybody! I have to decompose to simple fractions the following function: V(z)=\frac{z^2-4z+4}{(z-3)(z-1)^2}. I know I can see the function as: V(z)=\frac{A}{z-3}+\frac{B}{(z-1)^2}+\frac{C}{z-1}, and that the terms A, B, C can be calculated respectively as the residues in 3 (single pole)...
  46. Mark44

    Insights Partial Fractions Decomposition - Comments

    Mark44 submitted a new PF Insights post Partial Fractions Decomposition Continue reading the Original PF Insights Post.
  47. P

    Is it possible to decompose a vector into non-perpendicular components?

    Homework Statement [/B]Homework Equations [/B]The Attempt at a Solution When I have to describe a motion I'm supposed to decompose a vector in two directions, for example in an inclined plane is decompose the weight in these directions: the normal to the plane and the parallel to the plane...
  48. U

    Spinodal Decomposition: Exploring Crossover Points & Metastable States

    Referring to first figure attached, if we put a system of a certain composition at a T and P corresponding to a point inside the spinodal, will it (with very small fluctuations) just jump to the corresponding binodal compositions at that temperature and pressure? As per this figure, the system...
  49. ognik

    MHB Prove hermitian decomposition

    If C NOT Hermitian, show we can decompose C into $\frac{1}{2}\left( C + {C}^{\dagger} \right) +\frac{1}{2i}i\left( C- {C}^{\dagger} \right) $ I've managed to prove C = C a couple of times, EG taking Hermitian or conjugate of both sides, probably there is a bit of info I am not thinking of or...
  50. V

    Estimating singular values from QR decomposition

    I have a matrix for which I know its QR decomposition: A = QR. I want to estimate the largest and smallest singular values of A (\sigma_1 and \sigma_n) however in my application it is too expensive to compute the full SVD of A. Is it possible to estimate the largest/smallest singular values...
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