The Wave Equation

Lecture 19: The Wave Equation

• The wave equation applies to vibrations of a string, acoustics, and electromagnetism.
• 1D wave equation: $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$, where $c$ represents the wave speed.
• d'Alembert's formula: $u(x,t) = F(x+ct) + G(x-ct)$, $F$ and $G$ denote arbitrary functions.
• Given $u(x,0) = e^{-x^2}$ and $\frac{\partial u}{\partial t}(x,0) = 0$, the solution is $u(x,t) = \frac{1}{2} \left[ e^{-(x+ct)^2} + e^{-(x-ct)^2} \right]$.
• Consider $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$ where $0 < x < L$, $t > 0$, with initial conditions $u(x,0) = f(x)$, $\frac{\partial u}{\partial t}(x,0) = g(x)$, and boundary conditions $u(0,t) = 0$, $u(L,t) = 0$.
• Separation of variables: Let $u(x,t) = X(x)T(t)$, leads to $X(x)T''(t) = c^2 X''(x)T(t)$ and $\frac{T''(t)}{c^2 T(t)} = \frac{X''(x)}{X(x)} = -\lambda$.
• Ordinary differential equations: $X''(x) + \lambda X(x) = 0$, $T''(t) + \lambda c^2 T(t) = 0$.
• Boundary conditions: $X(0) = 0$, $X(L) = 0$.
• Solving for $X(x)$ gives three cases based on $\lambda$. if $\lambda = 0$, $X(x) = Ax + B$, implies $X(x) = 0$.
• If $\lambda < 0$, $X(x) = A e^{\sqrt{-\lambda} x} + B e^{-\sqrt{-\lambda} x}$, implies $X(x) = 0$.
• If $\lambda > 0$, $X(x) = A \cos(\sqrt{\lambda} x) + B \sin(\sqrt{\lambda} x)$, which leads to $\lambda = \left( \frac{n \pi}{L} \right)^2$, and $X(x) = B \sin \left( \frac{n \pi x}{L} \right)$.
• Eigenvalues and Eigenfunctions: $\lambda_n = \left( \frac{n \pi}{L} \right)^2$, $X_n(x) = \sin \left( \frac{n \pi x}{L} \right)$, where $n = 1, 2, 3, \dots$ is proven to be true.

Anatomy and Physiology: Cartilage

• Hyaline cartilage is the most common type, with a glassy appearance, found in articular, costal, respiratory, and nasal cartilage.
• Elastic cartilage contains elastic fibers and is very flexible, found in the external ear and epiglottis.
• Fibrocartilage contains thick collagen fibers; it is strong against tension and compression, found in intervertebral discs, menisci of the knee, and the pubic symphysis.

Bone Cells

• Osteogenic cells are stems cells that differentiate into osteoblasts, found in the periosteum and endosteum.
• Osteoblasts are bone-forming cells that secrete collagen and calcium-binding proteins to make bone matrix (osteoid).
• Osteocytes are mature bone cells residing in lacunae, maintaining bone matrix and acting as stress sensors.
• Osteoclasts are bone-resorbing cells derived from stem cells, which secrete enzymes and acids to break down bone matrix.

Bone Markings

• Condyle: Rounded articular projection.
• Crest: Narrow, prominent ridge of bone.
• Epicondyle: Raised area on or above a condyle.
• Facet: Smooth, nearly flat articular surface.
• Fissure: Narrow, slit-like opening.
• Foramen: Round/oval opening through a bone.
• Fossa: Shallow depression, often articular.
• Head: Bony expansion on a narrow neck.
• Line: Narrow bone ridge, less prominent than a crest.
• Meatus: Canal-like passageway.
• Process: Any bony prominence.
• Ramus: Arm-like bar of bone.
• Sinus: Cavity within a bone, filled with air + mucous membrane.
• Spine: Sharp, slender, often pointed projection.
• Trochanter: Very large, blunt, irregularly shaped process.
• Tubercle: Small rounded projection or process.
• Tuberosity: Large rounded projection, may be roughened.

Chapter 4: Policy Gradient

• Model-free RL learns directly from experiences without a model of the environment
• Policy-based RL learns a policy directly without learning a value function.
• Policy gradient methods optimize the policy by following its gradient.
• Policy gradient methods can learn in continuous action spaces and stochastic policies.

Advantages

• Simplicity: no need to maintain a value function.
• Effectiveness in high-dimensional/continuous action spaces without requiring maximization over actions.
• Stochastic policies: Allows the agent to explore and adapt unlike having value functions always pick one action.

Disadvantages

• Convergence to a local optimum and sensitivity to hyperparameter choice.
• High variance, which can slow learning; variance reduction techniques can be used to mitigate.
• A policy $\pi$ is a function mapping state $s$ to a probability distribution over actions $a$, $\pi(a|s) = P[A_t = a | S_t = s]$.
• Policy parameterization represents the policy using parameters $\theta$, where $\pi_{\theta}(s, a) = P[A_t = a | S_t = s, \theta_t = \theta]$.
• Goal: Adjust $\theta$ to maximize the expected return $J(\theta)$, $J_1(\theta) = V_{\pi_{\theta}}(s_1) = \mathbb{E}{\pi{\theta}}[v_1]$.
• Policy Gradient Theorem
• Policy gradient is: $\nabla_{\theta} J(\theta) = \mathbb{E}{\pi{\theta}}[\nabla_{\theta} \log \pi_{\theta}(s, a)Q^{\pi_{\theta}}(s, a)]$ using the average reward per time step: $J_{avE}(\theta) = \lim_{n \to \infty} \frac{1}{n} \sum_{t=1}^{n} \mathbb{E}[r_t]$
• REINFORCE Monte Carlo Policy Gradient
• stochastic gradient ascent: $\theta \leftarrow \theta + \alpha \nabla_{\theta} \log \pi_{\theta}(s, a) G_t$ where $G_t$ is the return from time step t.
• Algorithm:
  • Initialize policy parameter $\theta$.
  • For each episode ${s_1, a_1, r_2,..., s_{T-1}, a_{T-1}, r_T}$ do:
  • For t = 1 to T-1 do: $\theta \leftarrow \theta + \alpha \nabla_{\theta} \log \pi_{\theta}(s_t, a_t) G_t$
• REINFORCE with Baseline
• Reduces variance by using $\nabla_{\theta} J(\theta) = \mathbb{E}{\pi{\theta}}[\nabla_{\theta} \log \pi_{\theta}(s, a)(Q^{\pi_{\theta}}(s, a) - b(s))]$ subtracting a baseline function.
• Baseline, $b(s)$, does not depend on the action $a$, e.g. value function $V^{\pi_{\theta}}(s)$, $\theta \leftarrow \theta + \alpha \nabla_{\theta} \log \pi_{\theta}(s, a)(G_t - b(s_t))$.

Actor-Critic Methods

• Combine policy gradient methods with value-based methods, uses two networks: an actor and a critic
• Actor uses value to update policies, where $\theta \leftarrow \theta + \alpha_{\theta} \nabla_{\theta} \log \pi_{\theta}(s, a) Q_w(s, a)$
• Critic uses value for values, where $w \leftarrow w + \alpha_w(r + \gamma Q_w(s', a') - Q_w(s, a)) \nabla_w Q_w(s, a)$

Introducción a los sistemas de ecuaciones lineales

• Un sistema de ecuaciones lineales es un conjunto de dos o más ecuaciones lineales con las mismas variables.
• Una solución cumple todas las ecuaciones del sistema.
• Resolver el sistema es encontrar todas las soluciones.
• Los métodos de resolución son sustitución, igualación, reducción (o eliminación), gráfico, Gauss, Regla de Cramer.
• Método de sustitución:
  • Despejamos $x$ en la primera ecuación como $x = 5 - y$
  • Sustituimos en la segunda ecuación $2(5 - y) - y = 1$
  • Resolvemos como $y$: $10 - 2y - y = 1 \Rightarrow -3y = -9 \Rightarrow y = 3$
  • Sustituimos $y = 3$ en $x = 5 - y$: $x = 5 - 3 = 2$, solucion del sistema es $x = 2$, $y = 3$.
• Método de reducción:
  • Sumamos las equations para eliminar $y$: $(3x + 2y) + (4x - 2y) = 7 + 0 \Rightarrow 7x = 7$
  • Resolvemos para $x$: $x = 1$
  • Sustituimos $x = 1$ en la primera ecuación: $3(1) + 2y = 7 \Rightarrow 2y = 4 \Rightarrow y = 2$, la solución del sistema es $x = 1$, $y = 2$.
• Clasificación de sistemas de ecuaciones lineales:
  • Compatible determinado - única solución; compatible indeterminado - infinitas soluciones; incompatible - no tiene solución.

Environmental Science 2010 Free-Response Questions

• The question explores the ecological and economic effects of brown tree snakes in Guam and possible strategies to control the snake population.
  • Includes construction of a graph with years (1968-1988) on the x-axis vs snakes observed on the $y-axis$.
  • The exponential curve is attributed that brown tree snake being an invasive species in Guam, causing their population to rapidly increase.
• Zebra Mussels are introduced via ballast water of ships, depleting phytoplankton/ disrupting food web.
• Kudzu was introduced for erosion control, but overgrows native vegetation/ block sunlight.
• Economic impact of the brown tree snake is the increased costs for controlling the snake population.
  • Suggested control: Introducing a natural predator that targets brown tree snakes.
• Carrying capacity is the max number of individuals of a species in an environment
  • The availability of food/ mammal determines carrying capacity.
• Considers environmental effects of surface coal mining and associated reclamation strategies.
  • Mountaintop removal mining involves explosives to remove mountain tops to access coal seams/debris is pushed into nearby valleys
• Deforestation leading to habitat loss can occur and or stream water quality degrades because of sediment/ chemical runoff.
• The Surface Mining Control and Reclamation Act of 1977 is a US federal law that requires restoration of mined land
• Land Restoration: Reshaping topography to prevent erosion/ planting plant species to stabilize soil
• Benefit: Vegetative buffers can filter sediment. Problem: Buffers may not be effective in heavy rainfall.
• The company can mitigate affects by performing erosion control/reduce sediment runoff. Reason not to mitigate is that the cost is too great.

Lead In Water Questions

• Lead enters drinking water because the pipes are corroded.
• Young children are more vulernable and systems develop more easily to them.
• One way to lower lead concnentrations is to adjust the pH in water. pH is still not perfect due to pipes.
• Lead paint is a source/ kids eat it.
• Lead causes environmental pollution can contaminate soil/water harming plants/animals.
• Lead removal from a contaminate site is excavation and disposal of contaminated soil.

ÁLGEBRA LINEAL: EJERCICIOS 5 DIAGONALIZACIÓN

Consists of ten linear algebra problems

Resumen Ejecutivo

• This is an environmental impact analysis regarding road improvements of the highway PE-3N in San Martin.
• The project consist of improving the existing road via asphalt with 75km length.
• Includes: earth movement; construction of water drainage/sewage system; granular foundation; street lighting; road safety.
• Methodology: analyze climatic information; identify and evaluate impacts; public consultations; mitigation measure; plan of environmental management.
• Results: Air and water qualities are altered; an increase in noise during construction; loss of flora and fauna; risks of erosion and sedimentation.
• Potential Measures: implement pollutant control systems; machinery use; construction drainage; implement re-locations programs.
• Plan: Monitory quality; flora/fauna surveillance; personal training.
• Conclusion: Managed via measures; approval contingent with the implementation.

Thermodynamics

• An isolated system does not exchange energy or matter, a closed system can exchange energy, and an open system can exchange both energy and matter with its surrounding.
• Intensive properties does not depend on mass volume. (Examples: Pressure, temperature, density)
• Extensive Properties is depends on mass volume. (Examples: Mass, volume. Energy)
• Equilibrium requirements involve thermal, mechanical phase and chemical equalibilty.
• Isothermal - constant temperate; Isobaric - constant pressure; adiabatic- no heat transfer; isentropic- constant entropy.
• First Law of Thermodynamics $\Delta E = Q - W$

The 5 Themes of Geography

• Geography is the study of the Earth's surface and relationship between people and the environment.
• Location is the position of anything on Earth's surface, includes: absolute and relative location.
• Formal Region: An area with a common characteristic, such as language or climate like the sahara desert.
• Functional Region: Areas organized around a central point of transportation like dallas and fort worth.
• Vernacular Regions: Areas of inhabitence like the rust belt.