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- https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_(Orloff)/09%3A_Residue_TheoremThumbnail: Illustration of the setting. (Public Domain; Ben pcc via Wikipedia)
- https://math.libretexts.org/Courses/Angelo_State_University/Mathematical_Computing_with_Python/02%3A_Stochastic_Simulation/2.02%3A_Monte_Carlo_SimulationIntroduction to the Monte Carlo simulation as a method of predicting outcome probability when there is interference from random variables.
- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Seven_Sketches_in_Compositionality%3A_An_Invitation_to_Applied_Category_Theory_(Fong_and_Spivak)/07%3A_Logic_of_Behavior_-_Sheaves_Toposes_Languages
- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Seven_Sketches_in_Compositionality%3A_An_Invitation_to_Applied_Category_Theory_(Fong_and_Spivak)/08%3A_Exercise_solutions
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Applied_Geometric_Algebra_(Tisza)/01%3A_Algebraic_Preliminaries/1.06%3A_The_unimodular_group_SL(n_R)_and_the_invariance_of_volumeThis result can be justified also in a more elegant way: The geometrical operations in figures b and c consist of adding the multiple of the vector \(\vec{y}\) to the vector \(\vec{x}\), or adding the...This result can be justified also in a more elegant way: The geometrical operations in figures b and c consist of adding the multiple of the vector \(\vec{y}\) to the vector \(\vec{x}\), or adding the multiple of the second row of the determinant to the first row, and we know that such operations leave the value of the determinant unchanged.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Applied_Geometric_Algebra_(Tisza)/04%3A_Spinor_Calculus/4.02%3A_Rigid_Body_Rotation\[\begin{array}{c} {V(t) = U(\hat{x}_{3}, \frac{\dot{\alpha} t}{2}) V(0) U(\hat{e}_{3}, \frac{\dot{\gamma} t}{2})}\\ {\begin{pmatrix} {e^{-i \dot{\alpha} t/2}}&{0}\\ {0}&{e^{i \dot{\alpha} t/2}} \end{...\[\begin{array}{c} {V(t) = U(\hat{x}_{3}, \frac{\dot{\alpha} t}{2}) V(0) U(\hat{e}_{3}, \frac{\dot{\gamma} t}{2})}\\ {\begin{pmatrix} {e^{-i \dot{\alpha} t/2}}&{0}\\ {0}&{e^{i \dot{\alpha} t/2}} \end{pmatrix} \begin{pmatrix} {e^{-i \alpha (0)/2} \cos (\beta/2) e^{-i \gamma (0)/2}}&{-e^{-i \alpha (0)/2} \sin (\beta/2) e^{i \gamma (0)/2}}\\ {e^{i \alpha (0)/2} \sin (\beta/2) e^{-i \gamma (0)/2}}&{e^{i \alpha (0)/2} \cos (\beta/2) e^{i \gamma (0)/2}} \end{pmatrix} \times \begin{pmatrix} {e^{-i \do…
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Applied_Geometric_Algebra_(Tisza)/02%3A_The_Lorentz_Group_and_the_Pauli_Algebra/2.01%3A_Introduction_to_Lorentz_Group_and_the_Pauli_AlgebraIn the quantitative development of this idea we have to make a choice, whether to start with the classical wave concept and build in the corpscular aspects, or else start with the classical concept of...In the quantitative development of this idea we have to make a choice, whether to start with the classical wave concept and build in the corpscular aspects, or else start with the classical concept of the point particle, endowed with a constant and invariant mass, and modify these properties by means of the wave concept. The course of the present developments is set by the decision of following up the Einsteinian departure.
- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Seven_Sketches_in_Compositionality%3A_An_Invitation_to_Applied_Category_Theory_(Fong_and_Spivak)/05%3A_Signal_flow_graphs-_Props_presentations_and_proofs/5.03%3A_Simplified_Signal_Flow_GraphsWe now return to signal flow graphs, expressing them in terms of props. We will discuss a simplified form without feedback (the only sort we have discussed so far), and then extend to the usual form o...We now return to signal flow graphs, expressing them in terms of props. We will discuss a simplified form without feedback (the only sort we have discussed so far), and then extend to the usual form of signal flow graphs. But before we can do that, we must say what we mean by signals; this gets us into the algebraic structure of “rigs.”
- https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_(Orloff)/10%3A_Definite_Integrals_Using_the_Residue_Theorem/10.08%3A_Solving_DEs_using_the_Fourier_transform\(\begin{array} {ccl} {\lim_{R \to \infty} \int_{C_R} e^{izt} g(z)\ dz = 0} & \ \ \ \ \ \ & {\text{(Theorem 10.2.2(b))}} \\ {\lim_{R \to \infty, r \to 0} \int_{C_2} e^{izt} g(z) \ dz = \pi i \text{Res...\(\begin{array} {ccl} {\lim_{R \to \infty} \int_{C_R} e^{izt} g(z)\ dz = 0} & \ \ \ \ \ \ & {\text{(Theorem 10.2.2(b))}} \\ {\lim_{R \to \infty, r \to 0} \int_{C_2} e^{izt} g(z) \ dz = \pi i \text{Res} (e^{izt} g(z), -1)} & \ \ \ \ \ \ & {\text{(Theorem 10.7.2)}} \\ {\lim_{R \to \infty, r \to 0} \int_{C_4} e^{izt} g(z)\ dz = \pi i \text{Res} (e^{izt} g(z), 1)} & \ \ \ \ \ \ & {\text{(Theorem 10.7.2)}} \\ {\lim_{R \to \infty, r \to 0} \int_{C_1 + C_3 + C_5} e^{izt} g(z) \ dz = \text{p.v.} \hat{y…
- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Street-Fighting_Mathematics%3A_The_Art_of_Educated_Guessing_and_Opportunistic_Problem_Solving_(Mahajan)/06%3A_Analogy/6.02%3A_Topology-_How_many_regionsMust all the regions created by the lines be convex? (A region is convex if and only if a line segment connecting any two points inside the region lies entirely inside the region.) What about the thre...Must all the regions created by the lines be convex? (A region is convex if and only if a line segment connecting any two points inside the region lies entirely inside the region.) What about the three-dimensional regions created by placing planes in space? Hint: Explain first why the pattern generates the \(R_{2}\) row from the \(R_{1}\) row; then generalize the reason to explain the \(R_{3}\) row.
- https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_(Orloff)/02%3A_Analytic_Functions/2.09%3A_Branch_Cuts_and_Function_CompositionWe often compose functions, i.e. f(g(z)) . In general in this case we have the chain rule to compute the derivative. However we need to specify the domain for z where the function is analytic. And ...We often compose functions, i.e. f(g(z)) . In general in this case we have the chain rule to compute the derivative. However we need to specify the domain for z where the function is analytic. And when branches and branch cuts are involved we need to take care.
