# 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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...

20. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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...

49. ### 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. ### 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...