Statics: Equilibrium and Reactions

Statics - Chapter 3: Equilibrium

• Equilibrium occurs when the sum of forces and moments is zero: $\sum \overrightarrow{F} = 0$ and $\sum \overrightarrow{M_0} = 0$.

Reactions at Supports and Connections for a 2-D Structure

• Cable connections result in tension with a known line of action.
• Roller connections result in a force with a known line of action.
• Smooth surface connections result in a force perpendicular to the surface.
• Hinge connections result in a force with unknown direction.
• Fixed support connections result in a force and couple with unknown direction and magnitude.

Two- and Three- Force Members

• A Two-Force Member in equilibrium has two forces with the same magnitude, acting along the same line, and opposing sense.
• A Three-Force Member in equilibrium must have three forces that are either concurrent or parallel.

Sample Problem 3.1: Concrete Block Suspended by Cables

• Find the tension in each cable supporting a 200-kg concrete block.
• Equilibrium equations are:
  • $\sum F_x = 0$: $-T_{AC}\cos 60^\circ + T_{AB} = 0$
  • $\sum F_y = 0$: $T_{AC}\sin 60^\circ - 1962 N = 0$.
• Solving these yields $T_{AC} = 2266 N$ and $T_{AB} = 1133 N$.

Sample Problem 3.2: Crane Lifting a Machine

• Determine the force in each cable and the reaction at point C for a crane lifting a 2400-lb machine.
• Relevant equilibrium equations include:
  • $\sum M_c = 0$: $A_B (36 ft) - 2400 lb (5 ft + 14ft) = 0$
  • $\sum F_x = 0$: $C_x - A_B \cos 30^\circ = 0$
  • $\sum F_y = 0$: $C_y + A_B \sin 30^\circ - 2400 lb = 0$.
• Solving these equations yields $A_B = 1267 lb$, $C_x = 1097 lb$, and $C_y = 1767 lb$.

Sample Problem 3.3: Frame Supporting a Load

• A frame with members $AD$, $CFE$ and $DB$ supports a 500-N load at A; find the reactions at $B$ and $E$.
• Analyze member $AD$, using $\sum M_D = 0$, $500N (0.2m) - A_B (0.15m) = 0$, to find $A_B = 666.7 N$.
• Also for AD: $\sum F_x = 0$, $D_x - 500 N = 0$ resulting in $D_x = 500 N$, and $\sum F_y = 0$, $D_y - A_B = 0$ resulting in $D_y = 666.7 N$.
• Analyze member $CFE$, with $\sum M_E = 0$, $-C_x (0.4m) - D_x (0.2m) + D_y (0.2m) = 0$, to find $C_x = 83.35 N$.
• Also for CFE: $\sum F_x = 0$, $E_x + C_x - D_x = 0$ resulting in $E_x = 416.65 N$, and $\sum F_y = 0$, $E_y - D_y = 0$ resulting in $E_y = 666.7 N$.
• The reactions at B and E are then $B_x = -A_B = -666.7 N$ and $B_y = -83.35 N$.

Understanding Deep Learning - Backpropagation - Chapter 4

• The backpropagation algorithm is a cornerstone of deep learning.

The Forward Pass

• Input vector: $\mathbf{x} \in \mathbb{R}^{D}$
• Weights: $\mathbf{W}^{(l)} \in \mathbb{R}^{M \times N}$
• Bias: $\mathbf{b}^{(l)} \in \mathbb{R}^{M}$
• Activation function: $\sigma$
• Forward propagation equations: $\mathbf{a}^{(l)} = \mathbf{W}^{(l)} \mathbf{h}^{(l-1)} + \mathbf{b}^{(l)}$ and $\mathbf{h}^{(l)} = \sigma(\mathbf{a}^{(l)})$, where $\mathbf{h}^{(0)} = \mathbf{x}$ and $\mathbf{y} = \mathbf{h}^{(L)}$.

Computing the Gradients

• Loss function: $J(\mathbf{\theta})$, where $\mathbf{\theta}$ are the parameters.
• Gradient descent update rule: $\mathbf{\theta} \leftarrow \mathbf{\theta} - \alpha \nabla_{\mathbf{\theta}} J(\mathbf{\theta})$, with learning rate $\alpha$.

Backpropagating the Error

• $\mathbf{\delta}^{(l)} = \frac{\partial J}{\partial \mathbf{a}^{(l)}}$
• Backpropagation equations:
  • $\mathbf{\delta}^{(L)} = \nabla_{\mathbf{y}} J \odot \sigma'(\mathbf{a}^{(L)})$
  • $\mathbf{\delta}^{(l)} = ((\mathbf{W}^{(l+1)})^T \mathbf{\delta}^{(l+1)}) \odot \sigma'(\mathbf{a}^{(l)})$

Computing the Parameter Gradients

• $\frac{\partial J}{\partial \mathbf{W}^{(l)}} = \mathbf{\delta}^{(l)} (\mathbf{h}^{(l-1)})^T$
• $\frac{\partial J}{\partial \mathbf{b}^{(l)}} = \mathbf{\delta}^{(l)}$.

Backpropagation Algorithm

• Perform forward pass to get activations for all layers
• Compute error term $\mathbf{\delta}^{(L)}$ for output layer.
• For $l = L-1, L-2,..., 1$, compute $\mathbf{\delta}^{(l)} = ((\mathbf{W}^{(l+1)})^T \mathbf{\delta}^{(l+1)}) \odot \sigma'(\mathbf{a}^{(l)})$.
• Compute gradients $\frac{\partial J}{\partial \mathbf{W}^{(l)}} = \mathbf{\delta}^{(l)} (\mathbf{h}^{(l-1)})^T$ and $\frac{\partial J}{\partial \mathbf{b}^{(l)}} = \mathbf{\delta}^{(l)}$.

Algorithmic Complexity

• Complexity represents the resources required by an algorithm.
• Time complexity refers to execution time.
• Space complexity refers to memory space.
• Both time and space complexity are expressed as functions of the size of the input.

How to Measure Complexity

• Experimental analysis involves implementing and running the algorithm with varying-sized inputs.
• Theoretical analysis characterizes running time as a function of the input size $n$.

Experimental Analysis Limitations

• It requires implementing the algorithm.
• Results rely on hardware and software.
• Experiments use a limited set of inputs.

Theoretical Analysis

• Is based on a pseudo-code description.
• Evaluates time as a function of input size.

Primitive Operations

• These are basic computations.
• They are identifiable in pseudocode.
• They have constent time cost.
• Examples include assigning values, comparing numbers, indexing into arrays, etc.

Counting Primitive Operations

• Determine the maximum number of primitive operations as a function of input size.
• Example: in Algorithm arrayMax(A, n) $7n - 2$ primitive operations are executed in the worst case arrayMax.

Growth Rate of Running Time

• Changing the hardware/software environment affects running time by a constant factor.
• The number of primitive operations is related to actual running time by a constant factor.

Focus on Growth Rate

• The growth rate of running time is most important for algorithm efficiency for LARGE input sizes.

Asymptotic Notation

• Includes Big-Oh, Big-Omega, and Big-Theta notation.

Big-Oh Notation

• Given functions $f(n)$ and $g(n)$, $f(n)$ is $O(g(n))$ if there are positive constants $c$ and $n_0$ such that $f(n) \leq cg(n)$ for $n \geq n_0$.
• Describes the asymptotic upper bound.

Big-Omega Notation

• Given functions $f(n)$ and $g(n)$, $f(n)$ is $\Omega(g(n))$ if there is a positive constant $c$ such that $f(n) \geq cg(n)$ for infinitely many values of $n$.
• Describes the asymptotic lower bound.

Big-Theta Notation

• Given functions $f(n)$ and $g(n)$, $f(n)$ is $\Theta(g(n))$ if $f(n)$ is $O(g(n))$ and $f(n)$ is $\Omega(g(n))$.
• Used to describe the asymptotic tight bound.

Properties of Asymptotic Notation

• Constant Factors: $O(cf(n)) = O(f(n))$, $\Omega(cf(n)) = \Omega(f(n))$, and $\Theta(cf(n)) = \Theta(f(n))$.
• Summation: $O(f(n)) + O(g(n)) = O(max(f(n), g(n)))$, etc.
• Product: $O(f(n)) * O(g(n)) = O(f(n) * g(n))$, etc.

Big-Oh Rules

• If $f(n)$ is a polynomial of degree $d$, then $f(n)$ is $O(n^d)$.
• Use the smallest possible class of functions.
• Use the simplest expression of the class.

Examples

• Examples of determining asymptotic complexity
  • $f(n) = 2n^3 + 4n^2 + 10n$ is $O(n^3)$.
  • $f(n) = 2^n + n^2$ is $O(2^n)$.
  • $f(n) = n \log n + n$ is $O(n \log n)$.
  • $f(n) = \log_2 n + \log_{10}n$ is $O(\log n)$.

Comparing Growth Rates

• This is based on $lim_{n \to \infty} \frac{f(n)}{g(n)}$.
• A limit of 0 indicates $f(n)$ grows slower than $g(n)$.
• A limit of $c > 0$ indicates $f(n)$ and $g(n)$ grow at the same rate.
• A limit of $\infty$ indicates $f(n)$ grows faster than $g(n)$.

Common Functions (Growth Rates)

• From smallest to largest: Constant $O(1)$, Logarithmic $O(\log n)$, Log-Linear $O(n \log n)$, Linear $O(n)$, Quadratic $O(n^2)$, Cubic $O(n^3)$, Exponential $O(2^n)$.

Prefix Averages Example (pseudo-code)

• Prefix averages algorithm 1 is $O(n^2)$.
• Prefix averages algorithm 2 is $O(n)$.

Algorithmes gloutons (Greedy Algorithms)

• These algorithms solve optimization problems.
• At each step, make the locally best choice.
• Intention is to find the globally optimal solution.
• In other words: make a sequence of locally optimal choices, without ever going back on a choice.

Exemple introductif (Introductory example)

• The problem is to make change for an amount c with the fewest coins given a coin system.
• An example coin system is euro coins {1, 2, 5, 10, 20, 50, 100, 200}, in centimes.
• A greedy algorithm might be: choose the largest coin <= amount, add it to the solution, then decrease the amount left to make change for.
• To make change for 37 centimes: choose 20, leaving 17; choose 10, leaving 7; choose 5, leaving 2; choose 2, leaving 0.
• The solution is therefore: 1 piece of 20, 1 piece of 10, 1 piece of 5 and 1 piece of 2.

Optimalité? (Optimality)

• Is the greedy algorithm always optimal? Does it always lead to the fewest coins in the solution?
• In euros, yes.
• But not for all coin systems.
• For example: coins of {1, 3, 4} making change for 6. Greedy chooses 4, 1, 1 for 3 coins; but optimal is 3, 3 for 2 coins.

Schéma général (General Scheme)

fonction glouton(données)
    solution = solution_vide
    tant que solution pas complète
        choix = choisir(données)  # optimal local choice
        si choix valide
            ajouter choix à solution
            données = mettre_à_jour(données, choix)
        sinon
            retourner "pas de solution"
    retourner solution

Quand utiliser un algorithme glouton? (When to use a greedy algorithm?)

• Greedy algorithms are simple to implement and often efficient, but they do not always guarantee optimality of the solution.
• Necessary conditions include: the problem admits an optimal solution that is built step by step by locally optimal choices, and choices never compromise possible later choices.
• Satisfying these conditions does not guarantee optimality, which requires proof.

Exemples d’applications (Examples of applications)

• Making change (in euros).
• Fractional knapsack (sac à dos fractionnaire) problem.
• Minimum spanning tree (arbre couvrant de poids minimum).
• Shortest paths (plus courts chemins).
• Task scheduling (ordonnancement de tâches).
• Data compression (compression de données).

Descriptive Statistics

• These are definitions of basic concepts in statistics.

Population

• The entire set of individuals or objects of interest.

Sample

• A subset of the population.

Variable

• An attribute subject to variations from one individual to another.

Data

• Observed or measured values of a variable.

Types de variables (Types of variables)

• Qualitative and quantitative variables.

Variables qualitatives (Qualitative Variables)

• Qualitative variables are non-numeric, either nominal or ordinal.
• Nominal variables cannot be ordered (e.g. eye color, gender).
• Ordinal variables can be ordered (e.g satisfaction level, level of education).

Variables quantitatives (Quantitative Variables)

• Quantitative variables can be measured numerically, and are either discrete or continuous.
• Discrete variables take a limited count (e.g. number of children, number of cars in parking).
• Continuous variables have any value in a range (e.g. height, weight).

Mesures de tendance centrale (Measures of Central Tendency)

• Three measures of central tendency: mean, median, mode.

Moyenne (Mean)

• Population $\qquad \mu = \frac{\sum_{i=1}^{N} x_i}{N}$.
• Sample $\qquad \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$.

Médiane (Median)

• The value separating a data set into two equal parts.

Mode

• The value that appears the most often in a data set.

Mesures de dispersion (Measures of Dispersion)

• Range, variance, standard deviation, coefficient of variation.

Étendue (Range)

• The difference between the largest and smallest values.

Variance

• Population $\qquad \sigma^2 = \frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N}$.
• Sample $\qquad s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}$.

Écart-type (Standard Deviation)

• Population $\qquad \sigma = \sqrt{\sigma^2}$.
• Sample $\qquad s = \sqrt{s^2}$.

Coefficient de variation (Coefficient of Variation)

• Population $\qquad CV = \frac{\sigma}{\mu}$.
• Sample $\qquad CV = \frac{s}{\bar{x}}$.

Fonction Exponentielle (The Exponential Function)

• The exponential function, exp(x) or $e^x$, is the unique function that satisfies the differential equation $f'(x) = f(x)$ with $f(0) = 1$.

Propriétés (Properties)

• $e^0 = 1$
• $e^1 = e \approx 2.718$
• $e^{a+b} = e^a \cdot e^b$
• $e^{a-b} = \frac{e^a}{e^b}$
• $(e^a)^b = e^{ab}$
• $e^x > 0$ for all $x \in \mathbb{R}$
• $\lim_{x \to +\infty} e^x = +\infty$
• $\lim_{x \to -\infty} e^x = 0$

Dérivée (Derivative)

• The derivative of the exponential function is itself, $\frac{d}{dx} e^x = e^x$.
• If $u(x)$ is a differentiable function of $x$, then $\frac{d}{dx} e^{u(x)} = u'(x) e^{u(x)}$.

Représentation Graphique (Graphical Representation)

• Passes through (0, 1).
• Always positive.
• Always increasing.
• X-axis is horizontal asymptote as $x \to -\infty$.

Tableau de variation (Variation Table)

Shows function behavior as limited by $\pm \infty$.

Applications

• Has uses in: mathematics, physics, engineering, finance, and in modeling exponential growth and decay.