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. Mastermind01

    A Hodge decomposition of a 1-form on a torus

    I was reading Dunne's review paper on Chern-Simons theory (Les-Houches School 1998) and I don't get how he decomposes the gauge potential on the torus. My own knowledge of differential geometry is sketchy. I do know that the Hodge decomposition theorem states that a differential form can be...
  2. A

    I Proving SL_2(C) Homeomorphic to SU(2)xT & Simple Connectedness

    Using the QR decomposition (the complex version) I want to prove that ##SL_2(C)## is homeomorphic to the product ##SU(2) × T## where ##T## is the set of upper-triangular 2×2-complex matrices with real positive entries at the diagonal. Deduce that ##SL(2, C)## is simply-connect. So, I can define...
  3. C

    Finding ##A^{-1}## of a matrix given three submatrices

    For this problem, Find ##A^{-1}## given, The solution is, However, in the first image, why are we allowed to put together the submatrices in random order? In general does someone please know why we are allowed to decompose matrices like this? Many thanks!
  4. K

    I Tensor decomposition, Sym representations and irreps.

    New to group theory. I have 3 questions: 1. Tensor decomposition into Tab = T[ab] +T(traceless){ab} + Tr(T{ab}) leads to invariant subspaces. Is this enough to imply these subreps are irreducible? 2. The Symn representations of a group are irreps. Why? 3. What is the connection between...
  5. C

    Spectral decomposition of 4x4 matrix

    ## A = \pmatrix{ -4 & -3 & 3 & 3 \\ -3 & -4 & 3 & 3 \\ -6 & -3 & 5 & 3 \\ -3 & -6 & 3 & 5 } ## over ## \mathbb{R}##. Let ## T_A: \mathbb{R}^4 \to \mathbb{R}^4 ## be defined as ## T_A v = Av ##. Thus, ## T_A ## represents ## A ## in the standard basis, meaning ## [ T_A]_{e} = A ##. I've...
  6. H

    I Plane wave decomposition method in scalar optics

    Suppose an optical scalar wave traveling in Z direction. Using the diffraction theory of Fourier Optics, we can predict its new distribution after a distance Z. The core idea of Fourier Optics is to decompose a scalar wave into plane waves traveling in different directions. But this...
  7. Lynch101

    I Spectral Decomposition of the State Space

    I'm looking to check my understanding of the information below and ultimately get a better understanding of it. Is spectral decomposition a mathematical procedure? Does "the state space of the measured system" refer to the possible values that the system could take, when measured?
  8. N

    How to Calculate the Pseudoinverse Using the SVD?

    My Answer: I am still beginner in this area so it s quite hard for me to understand this one. I am not sure what the output that this question asked me. I thought it might be asked about the value of x1, x2, x3, and x4
  9. T

    I Proof of Wick Decomposition: Expectation Values of Density Matrices

    Hey all, I am currently looking at a proof on the Wick Decomposition from this paper: https://www.sciencedirect.com/science/article/pii/0003491684900927 Specifically, the part that proves if a state satisfies the Wick Decomposition, then it has a density matrix of a specific exponential form...
  10. Physics Slayer

    A doubt in Partial fraction decomposition

    Say you want to find the following Integrals $$\int \frac{1}{(x-1)(x+2)} (dx)$$ $$\int \frac{1}{(x-1)(x^2 + 2)} (dx)$$ The easiest way to solve them will be by using partial fraction decomposition on both the given functions. Decomposing the first function, $$\frac{1}{(x-1)(x+2)} =...
  11. Arquimedes

    A Schmidt decomposition - How do I find the matrix related to the state?

    Hello, I am currently studying the Schmidt decomposition and how to use it to determine if a state is entangled or not and I can't understand how to write the state as a matrix so I can apply the Singular Value Decomposition and find the Schmidt coefficients. The exercise I am trying to complete...
  12. A

    Partial fraction decomposition with Laplace transformation in ODE

    Hello! Im having some trouble with solving ODE's using Laplace transformation,specifically ODE's that require partial fraction decomposition.Now I know how to do partial fraction decomposition,and have done it many times on standard polynoms but here some things just are not clear to me.For...
  13. H

    Engineering Decomposition minimal phase & all pass

    Hey everybody! I have put G1 = (1-s)/(2-10s) & G2 = (2-10s)/ (2 +10s) but than I read that all poles and zeroes should be inside the unit circle, and I don't know how to move the Zero S_01 = 1 to the unit circle
  14. PeterDonis

    I Kinematic Decomposition for "Rod and Hole" Relativity Paradox

    In a recent thread, I said that if there was interest, I would post in a separate thread the calculations for the kinematic decomposition of the congruence of worldlines describing the rod in the "rod and hole" relativity paradox discussed in that thread. Since there was interest, I am posting...
  15. mattlfang

    Stick leaning on the wall, find the acceleration from the initial position

    So the acceleration of point A was given by a force F exerted on cylinder that's along the direction of the stick, decomposed into the horizontal direction. so aA = F cos Θ The same force along the opposite direction is exerted on stick, and if we decompose that in vertical and horizontal...
  16. Poetria

    Vector decomposition - gravity

    It's a puzzle. I have decomposed vector v by using formulas known from physics: m*g*sin(theta) and m*g*cos(theta). I got: ##\vec v = (5, 5*\sqrt{3})## But it has been marked as wrong. Consequently, the rest of my calculations is not correct. Could you tell me, why?
  17. M

    MHB QR decomposition with permutation matrix

    Hey! :giggle: At the QR-decomposition with permutation matrix is the matrix $R$ equal to $R=G_3^{-1}P_1G_2^{-1}P_0G_1^{-1}A$ or $G_3P_1G_2P_0G_1A=R$? Which is the correct one? Or are these two equivalent? In general, it holds that $QR=PA$, right? :unsure:
  18. I

    Chemistry Decomposition of C5H6O3 equilibrium

    My solution: partial pressure of C5H6O3 = mRT/MV = (5.63 g)(0.08206 L*atm/mol*K)(473 K) / (114.098 g/mol)(2.50 L) = 0.766 atm equilibrium partial pressure of C5H6O3 = 0.766 - x equilibrium partial pressure of C2H6 = x equilibrium partial pressure of CO = 3x total pressure = 0.766 atm - x + x +...
  19. Rabindranath

    I Meaning of terms in a direct sum decomposition of an algebra

    Let's say I want to study subalgebras of the indefinite orthogonal algebra ##\mathfrak{o}(m,n)## (corresponding to the group ##O(m,n)##, with ##m## and ##n## being some positive integers), and am told that it can be decomposed into the direct sum $$\mathfrak{o}(m,n) = \mathfrak{o}(m-x,n-x)...
  20. I

    Chemistry Decomposition of Sucrose: Understanding the Rate and Order of Reaction

    I think this is a first order reaction because ln[C12H22O11] vs. time is linear. The k value is the negative of the slope. Therefore, my answer is rate = 0.45 hr-1 [C12H22O11]. The correct solution is rate = -0.45 hr-1 [C12H22O11]0. I don't understand why this is a zero order reaction, or why...
  21. I

    Chemistry Equilibrium of Methanol Vapor Decomposition

    The solution says that when the effusion rate ratio is multiplied by the equilibrium mole ratio of H2 to CH3OH, the effused mixture will have 33.0 times as much H2 as CH3OH. I don't understand why. I just set the equilibrium mole ratio of H2 to CH3OH as equal to 33.0 times, Why is this...
  22. D

    Chemistry Volume of a Gas from a thermal decomposition

    Ca(HCO3)2 -> CaCO3 + H2O + CO2 First I evaluate the moles of calcium carbonate (don't mind the units, just to save time) ##\frac {80.0}{40,00+12.01+3*16,00}= 0,799 mol## From the equation, correct me if I am wrong , one mole of CaCO3 is proportional to one mole of CO2, so from this I can...
  23. D

    I Decompose SL(2C) Matrix: Real Parameters from Complex

    Hi, suppose I am given an SL(2C) matrix of the form ##\exp(i\alpha/2 \vec{t}\cdot\vec{\sigma})## where ##\alpha## is the complex rotation angle, ##\vec{t}## the complex rotation axis and ##\vec{\sigma}## the vector of the three Pauli matrices. I would like to decompose this vector into...
  24. V

    I Blinder–Oaxaca decomposition confusion

    The following three equations illustrate this decomposition. Estimate separate linear wage regressions for individuals i in groups A and B: {\displaystyle {\begin{aligned}(1)\qquad \ln({\text{wages}}_{A_{i}})&=X_{A_{i}}\beta _{A}+\mu _{A_{i}}\\(2)\qquad...
  25. M

    MHB B=Q^TAQ has a LU decomposition

    Hey! 😊 I saw the below sentence in some notes: Let $A\in \mathbb{R}^{n\times n}$ be a not necessarily symmetric, strictly positive definite matrix, $x^TAx>0$, $x\neq 0$ und $Q\in \mathbb{R}^{n\times n}$ an orthogonal matrix, then $B=Q^TAQ$ has a LU decomposition. I want to understand...
  26. M

    MHB LU decomposition: With or without pivoting?

    Hey! 😊 We consider the matrix $$A=\begin{pmatrix}1 & -2 & 5 & 0 & 5\\ 1 & 0 & 2 & 0 & 3\\ 1 & 2 & 5 & 4 & 6 \\ -2 & 2 & -4 & 1 & -6 \\ 3 & 4 & 9 & 5 & 11\end{pmatrix}$$ I want to find the LU decomposition. How do we know if we have to do the decomposition with pivoting or without? :unsure:
  27. J

    I Decomposition per the Fundamental Theorem of Finite Abelian Groups

    According to the book I am using, one can decompose a finite abelian group uniquely as a direct sum of cyclic groups with prime power orders. Uniquely meaning that the structures in the group somehow force you to one particular decomposition for any given group. Unfortunately, the book gives no...
  28. ilper

    B Understanding Photon in a Box: Eigenmodes, Reflection, and Energy Measurement

    The eigen wavelengths λn(WL) of EM radiation in box are 2d/n where d is the size of the box. If I put a photon in a box with WL>2d via an optic cable trough a hole it must reflect on the perfect mirror walls and be a running wave. Maybe it is possible to decompose it as a set of eigenmodes of...
  29. M

    MHB Number of multiplications and divisions for LU decomposition

    Hey! 😊 Let $A$ a $n\times n$ matrix with known LU decomposition, let $u\in \mathbb{R}^n, v\in \mathbb{R}^{n+1}$. Show that the number of multiplications and divisions that are needed to get a LU decomposition of the $(n+1)\times (n+1)$ matrix $$\begin{pmatrix}A & u \\ v^T\end{pmatrix}$$ is at...
  30. M

    MHB Such a decomposition exists iff A is positive definite

    Hey! 😊 Let $A=L^TDL$ be the Cholesky decomposition of a symmetric matrix, at which the left upper triangular $L$ hat only $1$ on the diagonal and $D$ is a diagonal matrix with positiv elements on the diagonal. I want to show that such a decomposition exists if and only if $A$ is positive...
  31. M

    MHB Determine the cycle decomposition of the permutations

    Hey! :o We have the following permutations in $\text{Sym}(14)$ : - $\pi_1=(1 \ 2\ 4 \ 9)\circ(1 \ 3)\circ (6 \ 8\ 12)$ - $\pi_2=(2 \ 4\ 5 \ 8\ 7)\circ (1 \ 12 \ 6)\circ \ (13 \ 14)$ - $\pi_3=(1 \ 4 \ 5\ 8 \ 11)\circ (2 \ 4\ 6 \ 5 \ 1)$ 1. Determine the cycle decomposition of $\pi_1...
  32. B

    Pivoted Cholesky decomposition algorithm

    Hi at all! I need to implement the Pivoted Cholesky Decomposition in C++ and I know that is possible implement it without rows permutations. Where can I find the algorithm described clearly and/or codes example in other language to replicate in C++? Thanks!
  33. PainterGuy

    B Decomposition of a function into even and odd parts

    Hi, I understand how any function could be decomposed into even and odd parts assuming the function isn't a purely even or odd to start with. It's just like saying that any vector in x-y plane could be decomposed into its x- and y-component assuming it doesn't lie parallel to x- or y-axis...
  34. B

    C/C++ What's the problem with my Cholesky decomposition C++ code?

    Hi at all, I have to calculate the Cholesky decomposition of a symmetric matrix and this is the C ++ code I wrote: boost::numeric::ublas::matrix<double> Math::cholesky(const boost::numeric::ublas::matrix<double> &MatrixA) { int dim = MatrixA.size1()...
  35. Wrichik Basu

    B Clarification of Notation - Fourier decomposition of fields in QFT

    I am studying QFT from A First Book of QFT. It is a very well-written book. However, due to some personal reasons, I cannot buy the printed book at this moment. So I borrowed this book from a person (who, in turn, borrowed it from his university library), and scanned it. Everything is fine...
  36. K

    Decomposition reaction for theophylline, hexobarbital natrium

    Hello! There is a problem to write chemical reactions that goes with substances if they are not stored properly. For example theophylline should be saved from light and though I am trying to find its’ reaction with hv(light) but failed. Please help with some good reference Many thanks in advance
  37. N

    Static equilibrium force decomposition problem

    If you see the $\sum \tau_0 = L\cdot N_1 \cdot cos \theta - LF_1 sin \theta - L/2 \cdot G cos \theta$, all the trigonemetric parts are all opposite of what i can understand, given the angle as drawed in the Picture/url. Please help me :)https://pasteboard.co/IiXr8qA.png
  38. S

    Prove the decomposition of a graph w/ even edges produce a 2-path set

    For my base case I just used a graph with three vertices and 2 edges. Decomposing this would just give us the same graph, which has a path length of 2. The inductive step is where I'm having some trouble: One idea I have is that we take a graph G then inductively remove an edge to create two...
  39. Haorong Wu

    I The Schmidt decomposition in QC

    Hi, there. I have some problems when learning Schmidt decomposition in Nielsen's QC. The statement of Schmidt decomposition is simple and clear, however, the book doesn't give a clear procedure to do the Schmidt decomposition. I don't know whether the proof under the theorem is the the one I...
  40. M

    Linear Algebra: LU Decomposition

    Here is the initial matrix M: M = \begin{bmatrix} 3 & 1 & 6 \\ -6 & 0 & -16 \\ 0 & 8 & -17 \end{bmatrix} I have used the shortcut method outlined in this youtube video: LU Decomposition Shortcut Method. Here are the row reductions that I went through in order to get my U matrix: 1. R_3 -...
  41. J

    A Is the cluster decomposition equivalent to no phase transitions?

    I think the cluster decomposition states that products of space like separated observable decouple when sandwiched with states. An analogy with statistical mechanics seems to suggest that we are stating there are no phase transitions. For example, in the Ising model all spins are correlated in...
  42. J

    A Question about Lorenz invariance and cluster decomposition

    From Weinberg, The Quantum Theory of Fields, Vol. 1, there is the statement that "the only way" to merge Lorentz invariance with the cluster decomposition property (a.k.a. locality) is through a field theory. He uses this argument basically to justify that any quantum theory at low energies...
  43. C

    I Relation between tensor decomposition and helicity amplitude

    It is common to write e.g photon two point function in terms of manifest transverse and longitudinal form factors with lorentz structure factored out, e.g $$\Pi^{\mu \nu} = (g^{\mu \nu} - q^{\mu} q^{\nu}/q^2)T_T + q^{\mu} q^{\nu}T_L,$$ where mu and nu are polarisation indices. How do I relate...
  44. A

    Velocity gradient decomposition of a fluid flow

    If the velocity gradient decomposition is done by symmetric and antisymmetric parts then ##\frac{\partial v^i}{\partial x^j}=\sigma_{ij}+\omega_{ij}## where ##\sigma _{ij}=\frac{1}{2}(\frac{\partial v^i}{\partial x^j}+\frac{\partial v^j}{\partial x^i})## and...
  45. M

    I Matrix Decomposition Explained: Simple Illustration

    Can anyone illustrate for me matrix decomposition in a simple way?
  46. M

    MHB LU decomposition: Total pivoting

    Hey! :o I want to determine the LU decomposition of $A=\begin{pmatrix}0 & 2 & 1\\1 & 10 & 1 \\1 & 1 & 1\end{pmatrix}$ with total pivoting. I have done the following: The biggest element of the whole matrix is $10$, so we exchange the first two rows and the first two columns and then we...
  47. B

    Can Scalar Fields Be Decomposed Similar to Vector Fields?

    If a vector field can be decomposed into a curl field and a gradient field, is there a similar decomposition for scalar fields, say into a divergence field plus some other scalar field?
  48. M

    MHB The decomposition for a symmetric positiv definite matrix is unique

    Hey! :o We have the matrix \begin{equation*}A=\begin{pmatrix}1/2 & 1/5 & 1/10 & 1/17 \\ 1/5 & 1/2 & 1/5 & 1/10 \\ 1/10 & 1/5 & 1/2 & 1/5 \\ 1/17 & 1/10 & 1/5 & 1/10\end{pmatrix}\end{equation*} I have applied the Cholesky decomposition and found that $A=\tilde{L}\cdot \tilde{L}^T$ where...
  49. S

    I Can't understand a step in an LU decomposition proof

    I'm reading about the LU decomposition on this page and cannot understand one of the final steps in the proof to the following: ---------------- Let ##A## be a ##K\times K## matrix. Then, there exists a permutation matrix ##P## such that ##PA## has an LU decomposition: $$PA=LU$$ where ##L## is a...
  50. chwala

    Solving a first order ODE using the Adomian Decomposition method

    Homework Statement how do we solve the ode ## y'+y^2=-2, y(0)=0## using adomian decomposition method?Homework EquationsThe Attempt at a Solution ##Ly = -2-y^2## ## y= 0 + L^{-1}[-2-y^2]## ##y_{0}= -2t## ##y_{1}= -L^{-1}[4t^2] = -4t^3/3## are my steps correct so far in trying to get the Adomian...