vault backup: 2023-05-23 17:05:48
Affected files: .obsidian/graph.json .obsidian/workspace-mobile.json .obsidian/workspace.json STEM/AI/Literature.md STEM/AI/Neural Networks/MLP.md STEM/AI/Properties.md STEM/Quantum/Orbitals.md STEM/Quantum/Schrödinger.md STEM/Quantum/Wave Function.md STEM/Signal Proc/Convolution.md STEM/Signal Proc/Image/Image Processing.md STEM/img/hydrogen-electron-density.png STEM/img/hydrogen-wave-function.png STEM/img/orbitals-radius.png STEM/img/radial-equations.png STEM/img/radius-electron-density-wf.png STEM/img/wave-function-nodes.png STEM/img/wave-function-polar-segment.png STEM/img/wave-function-polar.png
@ -1,7 +1,7 @@
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#lit
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[https://web.stanford.edu/~jurafsky/slp3/A.pdf](https://web.stanford.edu/~jurafsky/slp3/A.pdf)
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[Towards Data Science: 3 Things You Need To Know Before You Train-Test Split](https://towardsdatascience.com/3-things-you-need-to-know-before-you-train-test-split-869dfabb7e50)
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[https://machinelearningmastery.com/train-final-machine-learning-model/](https://machinelearningmastery.com/train-final-machine-learning-model/)
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[https://medium.com/@canerkilinc/hands-on-tensorflow-2-0-multi-label-classifications-with-mlp-88fc97d6a7e6](https://medium.com/@canerkilinc/hands-on-tensorflow-2-0-multi-label-classifications-with-mlp-88fc97d6a7e6)
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[https://stackoverflow.com/questions/61557536/how-tensorflow-keras-go-from-one-hot-encoded-outputs-to-class-predictions-for](https://stackoverflow.com/questions/61557536/how-tensorflow-keras-go-from-one-hot-encoded-outputs-to-class-predictions-for)
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[https://stackoverflow.com/questions/35226198/is-this-one-hot-encoding-in-tensorflow-fast-or-flawed-for-any-reason/35227732#35227732](https://stackoverflow.com/questions/35226198/is-this-one-hot-encoding-in-tensorflow-fast-or-flawed-for-any-reason/35227732#35227732)
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[train-final-machine-learning-model](https://machinelearningmastery.com/train-final-machine-learning-model/)
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[hands-on-tensorflow-2-0-multi-label-classifications-with-mlp](https://medium.com/@canerkilinc/hands-on-tensorflow-2-0-multi-label-classifications-with-mlp-88fc97d6a7e6)
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[stackoverflow - how-tensorflow-keras-go-from-one-hot-encoded-outputs-to-class-predictions-for](https://stackoverflow.com/questions/61557536/how-tensorflow-keras-go-from-one-hot-encoded-outputs-to-class-predictions-for)
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[stackoverflow - is-this-one-hot-encoding-in-tensorflow-fast-or-flawed-for-any-reason](https://stackoverflow.com/questions/35226198/is-this-one-hot-encoding-in-tensorflow-fast-or-flawed-for-any-reason/35227732#35227732)
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@ -19,4 +19,4 @@ A finite feed-forward MLP with 1 hidden layer can in theory approximate any math
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![[tlu.png]]
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- $o_1$ to $o_4$ must all be one to overcome -3.5 bias and force output to 1
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![[mlp-non-linear-decision.png]]
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- Can generate a non-linear decision boundary
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- Can generate a non-linear [[Decision Boundary|decision boundary]]
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@ -36,13 +36,13 @@ Explanation-based learning uses both
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## Level of Explanation
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- Classical has emphasis on building symbolic representations
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- Models cognition as sequential processing of symbolic representations
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- Neural nets emphasis on parallel distributed processing models
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- [[Properties+Capabilities|Neural nets]] emphasis on parallel distributed processing models
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- Models assume information processing takes place through interactions of large numbers of neurons
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## Processing style
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- Classical processing is sequential
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- Von Neumann Machine
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- Neural nets use parallelism everywhere
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- [[Properties+Capabilities|Neural nets]] use parallelism everywhere
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- Source of flexibility
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- Robust
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@ -50,7 +50,7 @@ Explanation-based learning uses both
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- Classical emphasises language of thought
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- Symbolic representation has quasi-linguistic structure
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- New symbols created from compositionality
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- Neural nets have problem describing nature and structure of representation
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- [[Properties+Capabilities|Neural nets]] have problem describing nature and structure of representation
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Symbolic AI is the formal manipulation of a language of algorithms and data representations in a top-down fashion
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@ -1,14 +1,6 @@
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$$\psi(r,\theta,\phi)=R(r)\cdot Y_{ml}(\theta, \phi)$$
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Wave functions are products of
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Radial Function
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- $R_{n,l}(r)$
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Spherical Harmonic
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- $Y_{ml}(\theta, \phi)$
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[[Wave Function]]
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Absolute value of wave function squared gives probability density of finding electron inside differential volume $dV$ centred on $r, \theta, \phi$
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$$|\psi(r,\theta,\phi)|^2$$
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# Quantum Numbers
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## Quantum Numbers
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$$n$$
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Principal quantum number
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- 1, 2, 3...
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@ -24,3 +16,30 @@ $$m$$
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Z-component / Magentic of $l$
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- $-l$ to $+l$
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- ***Orientation*** of orbital
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![[wave-function-polar-segment.png]]
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## Filling
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1. Aufbau
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- Start lowest in energy
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2. Pauli's Exclusion
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- Max two electrons per orbital
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- No two electrons can have same n, l, m, s tuple
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3. Hund's Rule of Maximum Multiplicity
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- Orbitals with same energy filled one at a time
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- Degenerate
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![[orbitals-radius.png]]
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![[wave-function-nodes.png]]
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## Radial
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![[radial-equations.png]]
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- Z = Atomic number
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- Bohr radius
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- $a_0=\frac \hbar {\alpha mc}$
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- Normalisation
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- $\int_0^\infty r^2R_{nl}^*R_{nl}dr=1$
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![[radius-electron-density-wf.png]]
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@ -1,6 +1,6 @@
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$$-\frac{\hbar^2}{2m}\nabla^2\psi+V\psi=E\psi$$
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- Time Independent
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- $\psi$ is the wave function
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- $\psi$ is the [[Wave Function]]
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Quantum counterpart of Newton's second law in classical mechanics
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21
Quantum/Wave Function.md
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@ -0,0 +1,21 @@
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$$\psi(r,\theta,\phi)=R(r)\cdot Y_{ml}(\theta, \phi)$$
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Wave functions are products of
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Radial Function
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- $R_{n,l}(r)$
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Spherical Harmonic
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- $Y_{ml}(\theta, \phi)$
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Forms [[Orbitals]]
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Absolute value of wave function squared gives probability density of finding electron inside differential volume $dV$ centred on $r, \theta, \phi$
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$$|\psi(r,\theta,\phi)|^2$$
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![[wave-function-polar.png]]
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![[hydrogen-wave-function.png]]
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![[wave-function-polar-segment.png]]
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![[wave-function-nodes.png]]
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![[hydrogen-electron-density.png]]
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![[radius-electron-density-wf.png]]
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@ -6,13 +6,13 @@ $$x(t)=x_1(t)\circledast x_2(t)=\int_{-\infty}^\infty x_1(t-\tau)\cdot x_2(\tau)
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# Properties
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1. $x_1(t)\circledast x_2(t)=x_2(t)\circledast x_1(t)$
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1. Commutativity
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- Commutativity
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2. $(x_1(t)\circledast x_2(t))\circledast x_3(t)=x_1(t)\circledast (x_2(t)\circledast x_3(t))$
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1. Associativity
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- Associativity
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3. $x_1(t)\circledast [x_2(t)+x_3(t)]=x_1(t)\circledast x_2(t)+ x_1(t)\circledast x_3(t)$
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1. Distributivity
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- Distributivity
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4. $Ax_1(t)\circledast Bx_2(t)=AB[x_1(t)\circledast x_2(t)]$
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1. Associativity with Scalar
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- Associativity with Scalar
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5. Symmetrical graph about origin
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# Applications
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@ -24,3 +24,7 @@ $$x(t)=x_1(t)\circledast x_2(t)=\int_{-\infty}^\infty x_1(t-\tau)\cdot x_2(\tau)
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# Polynomial Multiplication
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- Convolving coefficients of two poly gives coefficients of product
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# Discrete
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$$G[i,j]=H[u,v]\circledast F[i,j]$$
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$$G[i,j]=\sum^k_{u=-k}\sum^k_{v=-k} H[u,v]F[i-u,j-v]$$
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1
Signal Proc/Image/Image Processing.md
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[[Convolution#Discrete]]
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