Gas Laws: Boyle's, Charles's and Avogadro's Law

5.1 Gas Characteristics

• Gases are easily compressed.
• Gases expand indefinitely to occupy available space.
• Gases form uniform mixtures with each other irrespective of their identities.

5.2 Pressure

• Pressure is calculated as force divided by area: $P = \frac{F}{A}$.
• Pascals (Pa) are the SI units for pressure: $1 Pa = 1 \frac{N}{m^2}$.
• Common pressure units and their conversions: $1 atm = 760 mmHg = 760 torr = 1.01325 \times 10^5 Pa = 101.325 kPa$.

5.3 The Gas Laws

Boyle's Law

• For a fixed amount of gas at constant temperature, volume is inversely proportional to pressure.
• Boyle's Law Formula: $P_1V_1 = P_2V_2$

Charles's Law

• The volume of a fixed amount of gas at constant pressure is directly proportional to its absolute temperature.
• Charles's Law Formula: $\frac{V_1}{T_1} = \frac{V_2}{T_2}$

Avogadro's Law

• At constant temperature and pressure, the volume of a gas is directly proportional to the number of moles of gas.
• Avogadro's Law Formula: $\frac{V_1}{n_1} = \frac{V_2}{n_2}$

5.4 The Ideal-Gas Equation

• Ideal Gas Law Formula: $PV = nRT$
• $R = 0.08206 \frac{L \cdot atm}{mol \cdot K} = 8.314 \frac{J}{mol \cdot K}$ (Ideal gas constant)

5.5 Further Applications of the Ideal-Gas Equation

Gas Densities and Molar Mass

• Density is calculated as mass divided by volume ($d = \frac{m}{V}$).
• Density formula in relation to $PM$: $d = \frac{PM}{RT}$.
• Molar mass is derived from density by: $M = \frac{dRT}{P}$

Volumes of Gases in Chemical Reactions

• Ideal gas law is useful for converting between moles and volume in chemical reactions.

5.6 Gas Mixtures and Partial Pressures

• Total pressure of a gas mixture is the sum of the partial pressures of each gas: $P_t = P_1 + P_2 + P_3 +...$
• Partial pressure is calculated using mole fractions: $P_i = X_iP_t$.
• Mole fraction is the ratio of moles of one component to the total moles: $X_i = \frac{n_i}{n_t}$

5.7 Kinetic-Molecular Theory

• Gases consist of large numbers of molecules in continuous, random motion.
• The volume of the molecules is negligible compared to the total volume of the gas.
• Attractive and repulsive forces between gas molecules are negligible.
• Energy can be transferred between molecules but average kinetic energy remains constant at a constant temperature.
• The average kinetic energy of the molecules is proportional to absolute temperature, and at any given temperature, all gases have same average kinetic energy.
• Average kinetic energy: $\epsilon = \frac{1}{2}mv^2$

Root-Mean-Square Speed

• Root-mean-square speed formula: $u = \sqrt{\frac{3RT}{M}}$

5.8 Molecular Effusion and Diffusion

• Effusion is the process of gas escaping through a small hole.
• Diffusion is the spread of one substance through another.

Graham's Law

• The rate of effusion or diffusion of a gas is inversely proportional to the square root of its molar mass: $\frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}}$

5.9 Real Gases: Deviations from Ideal Behavior

• Real gases deviate from ideal behavior under high pressure and low temperature.

Van der Waals Equation

• van der Waals Equation: $(P + \frac{an^2}{V^2})(V - nb) = nRT$.
• Constants $a$ and $b$ are specific to each gas and correct for non-ideal behavior.

Chapter 4: The Building Blocks: Variables, Types, and Operators

4.1 Variables

4.1.1. Declaration and Assignment

• Variables are named storage locations.
• Variables must be declared before use, specifying name and data type.
  • Example for int: int age;
  • Example for double: double salary;
• Assignment uses the equals sign (=).
  • Example for int: age = 30;
  • Example for double: salary = 50000.00;

4.1.2. Naming Conventions

• Variable symbol must start with a letter, underscore (_), or dollar sign ($).
• Variable Names cannot start with a digit.
• Variable Names are case-sensitive.
• Descriptive names are important.
• Camel case for variables (firstName) is ideal
• Pascal case for classes (MyClass).
• Upper case for constants (MAX_VALUE).

4.2. Data Types

4.2.1. Primitive Data Types

• Primitive Data Types represent single values directly.
  • Integer Types:
    • byte (8-bit)
    • short (16-bit)
    • int (32-bit)
    • long (64-bit)
  • Floating-Point Types:
    • float (32-bit)
    • double (64-bit)
  • Character Type:
    • char (16-bit Unicode character)
  • Boolean Type:
    • boolean (true or false)

4.2.2. Non-Primitive Data Types (Reference Types)

• Non-Primitive Data Types store references to memory locations.
  • String
  • Arrays
  • Classes
  • Interfaces
• Non-Primitive Data Types Default value is null.

4.3. Operators

4.3.1. Arithmetic Operators

• Perform mathematical calculations:

4.3.2. Assignment Operators

• Assign values to variables.
• | Operator | Description | Example | | :------- | :---------------------------- | :-------- | | = | Assignment | x = y | | += | Add and assign | x += y | | -= | Subtract and assign | x -= y | | *= | Multiply and assign | x *= y | | /= | Divide and assign | x /= y | | %= | Modulus and assign | x %= y |

4.3.3. Comparison Operators

• Compare two values.

4.3.5. Bitwise Operators

• Perform operations on individual bits.

4.3.6. Other Operators

4.4. Type Conversion

4.4.1. Implicit Conversion (Widening)

• Automatically done by the compiler when no data loss occurs.
  • int to long
  • float to double

4.4.2. Explicit Conversion (Casting)

• Requires a cast operator (type).
• May result in data loss.
  • double to int

4.5. Constants

• Values cannot be changed after initialization.
• Declared using the final keyword.
  • final double PI = 3.14159;
• Naming convention: uppercase with underscores.

4.6. Literals

• Represent fixed values in the code.
• Integer literals: 42, -10
• Floating-point literals: 3.14, 2.0f
• Character literals: 'A', '4'
• String literals: "Hello, World!"
• Boolean literals: true, false
• Null literal: null

Chapter 14: The Laws of Thermodynamics

I. Thermodynamics

• Thermodynamics studies energy, its transformations, and its relation to matter.
  • Thermal Physics: Study of heat, temperature, related phenomena.
  • Thermodynamics deals with macroscopic changes; what happens, not why.
  • Statistical Mechanics: Relates macroscopic observations to microscopic parameters; why things happen.

II. Zeroth Law

• If objects A and B are separately in thermal equilibrium with object C, then A and B are in thermal equilibrium with each other (Thermal Equilibrium).
• Temperature determines if an object is in thermal equilibrium with others (no energy exchange).

III. Temperature Scales

• Celsius: $T_C = T - 273.15$
• Fahrenheit: $T_F = \frac{9}{5}T_C + 32$
• Thermal Expansion:
  • Linear Expansion: $\Delta L = \alpha L_0 \Delta T$
  • Volume Expansion: $\Delta V = \beta V_0 \Delta T$
    • Where $\beta = 3\alpha$
• Water's density is greatest at $4^\circ C$.

IV. Heat and Internal Energy

• Internal Energy (U): Energy associated with microscopic components of a system, including KE and PE of atoms/molecules.

• Heat (Q): Transfer of energy across a system boundary due to a temperature difference.

  • Units: Joule, Calorie, kcal ($1 cal = 4.186 J$)

  • Specific Heat (c): Energy to change the temperature of 1 kg of a substance by $1^\circ C$; $Q = mc\Delta T$

  • Calorimetry: Technique to measure specific heat.

  • Latent Heat (L): Heat required for a phase change.

    $Q = mL$

• Lf = heat of fusion

• Lv = heat of vaporization

• | Material | Melting Point ($^\circ C$) | $L_f$ (J/kg) | Boiling Point ($^\circ C$) | $L_v$ (J/kg) | | :--------- | :-------------------------: | :----------: | :-------------------------: | :----------: | | Helium | –269.65 | 5.23E+03 | –268.93 | 2.09E+04 | | Nitrogen | –209.97 | 2.55E+04 | –195.79 | 2.01E+05 | | Oxygen | –218.79 | 1.39E+04 | –182.97 | 2.13E+05 | | Ethyl | –114 | 1.04E+05 | 78 | 8.54E+05 | | Alcohol | | | | | | Water | 0 | 3.33E+05 | 100 | 2.26E+06 | | Lead | 327.3 | 2.45E+04 | 1750 | 8.70E+05 | | Aluminum | 660.4 | 3.97E+05 | 2450 | 1.14E+07 | | Silver | 960.8 | 8.82E+04 | 2193 | 2.33E+06 | | Copper | 1083 | 1.34E+05 | 2595 | 5.06E+06 | | Gold | 1063 | 6.44E+04 | 2660 | 1.58E+06 |

  • Work: transfer of energy that is not due to a temperature difference.

V. The First Law of Thermodynamics

• The change in internal energy of a system ($\Delta U$) is equal to the heat added to the system (Q) minus the work done by the system (W).

  $\Delta U = Q - W$

• Types of Processes*: - Adiabatic: no heat transfer ($Q = 0$) - a. $\Delta U = -W$

  • Constant Volume (Isochoric): no change in volume ($W = 0$)
    • a. $\Delta U = Q$ - Closed Cycle: process that returns to its starting point ($\Delta U = 0$) -a. $Q = W$
    • Free Expansion: adiabatic process where gas expands into a vacuum ($Q = W = 0$)

      • a. $\Delta U = 0$

      • Constant Pressure (Isobaric): process that occurs at constant pressure

        • $W = P\Delta V$

        • $PV$ diagrams describe state of gas.

        • Work done is area under the curve

        • $W = \int_{V_i}^{V_f} P dV$

        • Consequences of the First Law

1. Energy is conserved.
2. The internal energy of an isolated system remains constant.
3. Perpetual motion machines are impossible.

VI. Heat Transfer

• A. Conduction*: transfer of energy through a material via molecular collisions.

$P = kA\frac{T_H - T_C}{L}$

• Constant Thermal Conductivity (k): measure of how well a material conducts heat.
• Convection: transfer of energy by the movement of a fluid.
• Radiation: transfer of energy by electromagnetic waves.

$P = \sigma A e T^4$

• $\sigma = 5.6696 \times 10^{-8} W/m^2 \cdot K^4$ (Stefan-Boltzmann constant)
• e = emissivity (0 to 1)

Lecture 16 Singular Value Decomposition (SVD) continued

Recap

• Any matrix $A \in \mathbb{R}^{m \times n}$ can be factorized as: $$A = U \Sigma V^T$$
• $U \in \mathbb{R}^{m \times m}$ is an orthogonal matrix whose columns are the left singular vectors of A.
• $V \in \mathbb{R}^{n \times n}$ is an orthogonal matrix whose columns are the right singular vectors of A.
• $\Sigma \in \mathbb{R}^{m \times n}$ is a diagonal matrix with non-negative real numbers on the diagonal, called the singular values of A.

How to compute SVD

• Let's derive the SVD of A.
• $A^T A = (U \Sigma V^T)^T U \Sigma V^T = V \Sigma^T U^T U \Sigma V^T = V \Sigma^T \Sigma V^T$
• $A^T A$ is symmetric
• $V$ columns are the eigenvectors of $A^T A$ that diagonalize A
• The eigenvalues of $A^T A$ are the singular values squared, i.e., $\sigma_i^2$.
• $A A^T$ can be diagonalized with an orthogonal matrix $U$, which columns are the eigenvectors of $A A^T$. The eigenvalues of $A A^T$ are also the singular values squared, i.e. $\sigma_i^2$.
• Steps to compute SVD:
  • Compute the eigenvectors of $A^T A$. These are the columns of $V$.
• Compute the eigenvectors of $A A^T$. These are the columns of $U$.
• Compute the singular values of A. These are the square roots of the eigenvalues of $A^T A$ or $A A^T$. Compute Σ with singular values on the diagonal.

Example

$$ A = \begin{bmatrix} 1 & 2 \ 3 & 4 \ \end{bmatrix} $$

• Compute $A^T A$

$$ A^T A = \begin{bmatrix} 1 & 3 \ 2 & 4 \ \end{bmatrix} \begin{bmatrix} 1 & 2 \ 3 & 4 \ \end{bmatrix} = \begin{bmatrix} 10 & 14 \ 14 & 20 \ \end{bmatrix} $$

• Compute the eigenvalues of $A^T A$

$$ det(A^T A - \lambda I) = 0 $$

$$ det(\begin{bmatrix} 10 - \lambda & 14 \ 14 & 20 - \lambda \ \end{bmatrix}) = (10 - \lambda)(20 - \lambda) - 14^2 = 0 $$

$$ \lambda^2 - 30\lambda + 200 - 196 = 0 $$

$$ \lambda^2 - 30\lambda + 4 = 0 $$

$$ \lambda = \frac{30 \pm \sqrt{30^2 - 4 \cdot 4}}{2} = \frac{30 \pm \sqrt{884}}{2} = 15 \pm \sqrt{221} $$

• Therefore, $\lambda_1 = 15 + \sqrt{221} \approx 29.866$ and $\lambda_2 = 15 - \sqrt{221} \approx 0.134$

• Compute the eigenvectors of $A^T A$

For $\lambda_1 = 15 + \sqrt{221}$:

$$ \begin{bmatrix} 10 - (15 + \sqrt{221}) & 14 \ 14 & 20 - (15 + \sqrt{221}) \ \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \ \end{bmatrix} = 0 $$

$$ \begin{bmatrix}

• 5 - \sqrt{221} & 14 \ 14 & 5 - \sqrt{221} \ \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \ \end{bmatrix} = 0 $$

$$ (-5 - \sqrt{221})x_1 + 14x_2 = 0 $$

$$ x_2 = \frac{5 + \sqrt{221}}{14} x_1 $$

• Let $x_1 = 14$, then $x_2 = 5 + \sqrt{221}$, so the eigenvector is

$$ v_1 = \begin{bmatrix} 14 \ 5 + \sqrt{221} \ \end{bmatrix} $$

Noramilize $v_1$: $$ ||v_1|| = \sqrt{14^2 + (5 + \sqrt{221})^2} = \sqrt{196 + 25 + 10\sqrt{221} + 221} = \sqrt{442 + 10\sqrt{221}} \approx 21.17 $$

$$ v_1 = \frac{1}{21.17} \begin{bmatrix} 14 \ 5 + \sqrt{221} \ \end{bmatrix} \approx \begin{bmatrix} 0.66 \ 0.75 \ \end{bmatrix} $$

For $\lambda_2 = 15 - \sqrt{221}$:

$$ \begin{bmatrix}

• 5 + \sqrt{221} & 14 \ 14 & 5 + \sqrt{221} \ \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \ \end{bmatrix} = 0 $$

$$ (-5 + \sqrt{221})x_1 + 14x_2 = 0 $$

$$ x_2 = \frac{5 - \sqrt{221}}{14} x_1 $$

• Let $x_1 = 14$, then $x_2 = 5 - \sqrt{221}$, so the eigenvector is

$$ v_2 = \begin{bmatrix} 14 \ 5 - \sqrt{221} \ \end{bmatrix} $$

Normalize $v_2$:

$$ ||v_2|| = \sqrt{14^2 + (5 - \sqrt{221})^2} = \sqrt{196 + 25 - 10\sqrt{221} + 221} = \sqrt{442 - 10\sqrt{221}} \approx 1.887 $$

$$ v_2 = \frac{1}{1.887} \begin{bmatrix} 14 \ 5 - \sqrt{221} \ \end{bmatrix} \approx \begin{bmatrix} 0.75 \

• 0.66 \ \end{bmatrix} $$

Therefore:

$$ V = \begin{bmatrix} 0.66 & 0.75 \ 0.75 & -0.66 \ \end{bmatrix} $$

$$ \Sigma = \begin{bmatrix} \sqrt{29.866} & 0 \ 0 & \sqrt{0.134} \ \end{bmatrix} = \begin{bmatrix} 5.465 & 0 \ 0 & 0.366 \ \end{bmatrix} $$

• Compute U
• Since $A = U \Sigma V^T$, then $AV = U \Sigma$, so $U = AV \Sigma^{-1}$ $$ U = \begin{bmatrix} 1 & 2 \ 3 & 4 \ \end{bmatrix} \begin{bmatrix} 0.66 & 0.75 \ 0.75 & -0.66 \ \end{bmatrix} \begin{bmatrix} 1/5.465 & 0 \ 0 & 1/0.366 \ \end{bmatrix} $$

$$ U = \begin{bmatrix} 2.16 & -0.57 \ 4.98 & -0.09 \ \end{bmatrix} \begin{bmatrix} 0.183 & 0 \ 0 & 2.732 \ \end{bmatrix} = \begin{bmatrix} 0.395 & -1.557 \ 0.911 & -0.246 \ \end{bmatrix} $$

However, the columns of U are not orthogonal, so we need to normalize them.

$$ u_1 = \frac{1}{\sqrt{0.395^2 + 0.911^2}} \begin{bmatrix} 0.395 \ 0.911 \ \end{bmatrix} = \begin{bmatrix} 0.405 \ 0.914 \ \end{bmatrix} $$

$$ u_2 = \frac{1}{\sqrt{(-1.557)^2 + (-0.246)^2} }\begin{bmatrix}

• 1.557 \
• 0.246 \ \end{bmatrix} = \begin{bmatrix}
• 0.985 \
• 0.156 \ \end{bmatrix} $$

Therefore:

$$ U = \begin{bmatrix} 0.405 & -0.985 \ 0.914 & -0.156 \ \end{bmatrix} $$

Chapitre 2: Corps des nombres

§2.1 Extensions de corps

Définition 2.1.1

• A field extension $L/K$ is defined when $K$ is a field and $L$ is a superfield of $K$, meaning $K \subset L$ and the operations of $K$ are restrictions of those in $L$.

Définition 2.1.2

• For a field extension $L/K$, the map $K \times L \rightarrow L$ given by $(a, x) \mapsto ax$ makes $L$ a $K$-vector space.
• The degree of $L$ over $K$, denoted by $[L:K]$, is the dimension of $L$ as a $K$-vector space.
• An extension $L/K$ is finite if $[L:K] < \infty$, and infinite otherwise.

Proposition 2.1.3

• If $K \subset L \subset M$ are fields, then $[M:K] = [M:L] \cdot [L:K]$.

Définition 2.1.4

• In a field extension $L/K$, an element $\alpha \in L$ is algebraic over $K$ if there exists a non-zero polynomial $f(x) \in K[x]$ such that $f(\alpha) = 0$. Otherwise, $\alpha$ is said to be transcendental.
• The extension $L/K$ is algebraic if every $\alpha \in L$ is algebraic over $K$.

Proposition 2.1.5

• Every finite extension $L/K$ is also algebraic.

Définition 2.1.6

• For a field extension $L/K$ with $\alpha \in L$ algebraic over $K$, the minimal polynomial of $\alpha$ over $K$, denoted $\text{irr}(\alpha, K)$, is the unique monic polynomial $f(x) \in K[x]$ of minimal degree such that $f(\alpha) = 0$.

Proposition 2.1.7

• Let $L/K$ be a field extension and $\alpha \in L$ algebraic over $K$.
• The minimal polynomial $\text{irr}(\alpha, K)$ is unique.
• If $f(x) \in K[x]$ and $f(\alpha) = 0$, then $\text{irr}(\alpha, K) \mid f(x)$.
• The polynomial $\text{irr}(\alpha, K)$ is irreducible in $K[x]$.
• If $f(x) \in K[x]$ is monic and irreducible with $f(\alpha) = 0$, then $f(x) = \text{irr}(\alpha, K)$.

Définition 2.1.8

• For a field extension $L/K$ and $\alpha \in L$ algebraic over $K$, the field $K(\alpha)$ is defined as the smallest subfield of $L$ containing $K$ and $\alpha$, given by $K(\alpha) = {f(\alpha) \mid f(x) \in K[x]}$.

Partial Differential Equations

Introduction

Partial Differential Equations

A PDE is a differential equation involving multivariable functions and their partial derivatives, used for problems in sound, heat, electrostatics, electrodynamics, fluid flow, and elasticity.

General Form

$F(x, y, u, u_x, u_y, u_{xx}, u_{xy}, u_{yy},...) = 0$

Example

$u_{xx} + u_{yy} = 0$ (Laplace Equation)

Classifications

Order

• The order of a PDE is the highest order derivative in the equation.
• $u_t = c^2 u_{xx}$ is a second-order PDE.

Linearity

• A PDE is linear if the dependent variable and its derivatives appear linearly. Otherwise, it is non-linear.
• $u_t = c^2 u_{xx}$ is linear, while $u_t = u u_x$ is non-linear.

Homogeneity

• A linear PDE is homogeneous if each term contains the dependent variable or its derivatives.
• $u_t = c^2 u_{xx}$ is homogeneous, while $u_t = c^2 u_{xx} + f(x, t)$ is non-homogeneous.

Common PDEs

Heat Equation

• $u_t = \alpha u_{xx}$
• Describes temperature changes over time.

Wave Equation

• $u_{tt} = c^2 u_{xx}$
• Describes wave propagation.

Laplace Equation

• $u_{xx} + u_{yy} = 0$
• Describes steady-state phenomena.

Boundary and Initial Conditions

Boundary Conditions

• Conditions specifying the values of the unknown function on the boundary of the domain.
  • Dirichlet: $u = f(x)$ (function value).
  • Neumann: $\frac{\partial u}{\partial n} = g(x)$ (normal derivative value).
  • Robin: $a u + b \frac{\partial u}{\partial n} = h(x)$ (combination).

Initial Conditions

• Conditions specifying the values of the unknown function at an initial time.
  • $u(x, 0) = f(x)$ (initial state).
  • $u_t(x, 0) = g(x)$ (initial rate of change).

Methods of Solving PDEs

Analytical Methods

- Separation of Variables: Converts PDE to ODEs.
- Method of Characteristics: For first-order PDEs.
- Integral Transforms: Transforms PDE into an algebraic form.

Numerical Methods

- Finite Difference Method (FDM): Approximates derivatives.
- Finite Element Method (FEM): Divides domain into elements.
- Finite Volume Method (FVM): Integrates PDE over control volumes.

Software

- MATLAB
- COMSOL
      - FEATool Multiphysics
      - Python with NumPy, SciPy, and FEniCS.

Algorithmic Trading & Order Execution

1. Algorithmic Trading

Algorithm Functions

• Breaks orders into smaller ones.
• Places orders automatically.
• Adapts to market conditions.

Rationale

• Reduced market impact & transaction costs, increased execution speed & quality.

Types of Algorithms

Execution Algorithms

• Execute orders efficiently while minimizing market impact.
  • VWAP (Volume Weighted Average Price)
    • Executes orders at the volume-weighted average price.
  • TWAP (Time Weighted Average Price)
    • Executes orders at the time-weighted average price.
  • Percentage of Volume (POV)
    • Participates in trading at a % of market volume.
  • Implementation Shortfall
    • Minimizes the difference between the execution price & the theoretical price.

Tactical Algorithms

• Exploit short-term market opportunities.
• Pairs Trading
  • Identifies mispricings between related assets.
• Statistical Arbitrage
  • Employs statistical models to exploit short-term price discrepancies.
• Market Making
  • Provides liquidity with both buy and sell orders.

High-Frequency Trading (HFT)

• Extremely high speeds, short-term investment horizons, & high order-to-trade ratios.

2. Order Placement

Dark Pools

• Private exchanges/forums for trading securities.
  • Allows institutional investors to trade large blocks of shares without revealing their intentions to the market, reducing market impact.

Characteristics

• Non-displayed orders.
• Potential price improvement but lacks transparency.

Broker Responsibilities

Best Execution:

• Brokers seek the most favorable terms for clients.

Order Routing:

• Orders routed to favorable markets.

Disclosure:

• Brokers have disclosure responsibilities for order routing practices.

3. Order Execution

Market Order

• Buy/sell immediately at best available price.
• Guarantees execution but not price.

Limit Order

• Buy/sell at a specific price or better.
• Guarantees price but not execution.

Stop Order

Becomes a market order when the stop price is reached. Buy order is placed above and a sell order is placed below the current market price.

Stop-Limit Order

• Becomes a limit order when the stop price is reached, combines stop and limit features.
  • Provides price certainty but lacks guaranteed execution.

4. Trading Costs

Types of Costs

Explicit Costs

• Direct costs such as brokerage commissions, exchange fees, and taxes.

Implicit Costs

Indirect costs consisting of: Market Impact, Adverse Selection, Opportunity Cost, and Delay Costs.

Measuring Trading Costs

$$ Implementation \ Shortfall = Explicit \ Costs + Implicit \ Costs - Decision \ Price $$

Note:

• Decision Price is the price of the security when the decision to trade was made.

Algorithmic Trading - Summary

Heat Treatment Furnaces

Heat Treatment

• Heat treatment involves heating or cooling a metal (or alloy) to change the microstructure and properties.

Furnace Types

• Batch Furnaces: Box, Pit, Bell, Car-bottom