Teoría de Juegos Algorítmica

Algorithmic Game Theory

• Game theory provides mathematical frameworks for analyzing strategic interactions among rational decision-makers.
• Rationality implies that players aim to maximize their expected payoff while acting in their self-interest.
• Strategic interaction means that a player’s decision outcome depends on the decisions of other players.

Applications of Game Theory

• Economics which includes auctions, bargaining, and market equilibrium analysis.
• Political science areas such as voting strategies, lobbying efforts, and dynamics in international relations.
• Computer science applications include designing network routing protocols, mechanism design for fair allocation, and strategies in e-commerce.

Normal-Form Games

• Normal-form games define the game by specifying players, strategies available to each, and payoffs for each strategy combination.
• Example: Prisoner's Dilemma, it outlines cooperation versus defection strategies and their payoff matrix.

Nash Equilibrium

• Nash equilibrium defines a set of strategies where no player benefits from unilaterally changing their strategy.
• Each strategy is the best response considering other players' strategies, preventing improvements by individual deviation.

Finding Nash Equilibria

• Dominant strategy elimination involves iteratively removing strictly dominated strategies to simplify the game.
• Best response analysis identifies optimal strategies for each player based on the strategies of others.
• Mixed strategies involve players randomizing over pure strategies to create unpredictability.

Mechanism Design

• Mechanism design involves creating games to achieve a desired outcome, especially when players possess private information.
• Applications include designing auction formats, voting rules, and matching markets.

Mechanism Design Challenges

• Incentive compatibility guarantees truthful revelation of private information for optimal outcomes.
• Efficiency ensures the mechanism's outcome is socially optimal.
• Budget balance seeks to avoid external subsidies for maintaining the operating mechanism.

Algorithmic Considerations

• Computation complexity assesses the efficiency of computing Nash equilibria or optimal mechanisms.
• Communication complexity measures the communicative demands for players to reach agreements or implement mechanisms.
• Learning focuses on how players adapt and optimize play in repeated games or complex environments.

Current Research

• Mechanism design without money designs mechanisms where monetary transfers are not possible.
• Fair division researches evenly dividing resources among individuals with varying preferences.
• Social networks analyzes the strategic interactions of individuals within social networks.
• Artificial intelligence focuses on using game theory to design smart agents capable of strategic interaction.

Analisi Matematica I (a.a. 2013-2014)

Esercizio 1

• An assertion for every $n \in \mathbb{N}, n \geq 1$, is proven true by induction: $\sum_{k=1}^{n} k^{2}=\frac{n(n+1)(2 n+1)}{6}$
• An assertion refers to $\sum_{k=1}^{n} k^{2}=\frac{n(n+1)(2 n+1)}{6}$

Induction

• A base step is a true case since $\sum_{k=1}^{1} k^{2}=1^{2}=1=\frac{1 \cdot 2 \cdot 3}{6}$.
• An indutive step starts with proving the assertion being true for a certain value, then demonstrating that it is also true for all values.

Esercizio 2

• The sequence for $n \geq 1$ begins from $x_{1}=1$ and $x_{n+1}=\sqrt{2 x_{n}}$.

Lecture 16: Particle Accelerators

Introduction to Particle Accelerators

• Particle accelerators are tools used to explore matter’s building blocks and forces.
• Accelerators speed charged particles to high energies and allow scientists to explore matter.
• This allows recreation of conditions present in the universe's early stages and probing the structure of matter.

History of Accelerators

• 1932: The first accelerator was created by Cockcroft & Walton.
• 1930s: The cyclotron was invented by E.O. Lawrence.
• Post WWII: The synchrotron was developed.
• Today: Large facilities such as LHC at CERN.

Applications of Particle Accelerators

• Fundamental research to explore new particles, probe structure, and test physics theories.
• Medical applications include radiation cancer therapy, medical imaging, and sterilizing medical equipment.
• Industrial applications include ion implantation, material processing, and non-destructive testing.

Accelerator Principles: Acceleration

• Electric fields are for accelerating the charged particles, per the equation $\vec{F} = q\vec{E}$.
• $\vec{F}$ is the force on the particle.
• $q$ is the particle's charge.
• $\vec{E}$ is the electric field.

Beam Focusing

• Magnetic fields are for focusing particle beams per the equation $\vec{F} = q(\vec{v} \times \vec{B})$.
• $\vec{v}$ is the particle velocity.
• $\vec{B}$ is the magnetic field.

Types of Accelerators

• Linear Accelerators accelerate particles in a straight line using radio-frequency waves (example: SLAC at Stanford).
• Circular Accelerators allow particles to travel a circular path using magnets (examples: Cyclotrons, Synchrotrons).

Key Components in Particle Accelerators

• RF Cavities: Generate accelerating electric fields to boost particle energy.
• Magnets:
  • Dipole magnets bend the beam in a circular path.
  • Quadrupole magnets focus the beam to avoid spreading.
  • Sextupole magnets correct for chromatic aberrations within electromagnetic lenses.
• The Vacuum System is a high vacuum that minimizes particle collision w/ gas molecules and particle loss prevention.
• Injector: A source of accelreating particles, usually includes an ion source and pre-accelerator.

Synchrotrons

• Particles accelerate in a circle with a constant radius.
• The magnetic field increases in synchrony with the particle energy.
• RF are used for cavities to provide acceleration

Synchrotron Key Parameters

• Energy represents the maximum energy of the accelerated particles.
• Luminosity measures the collision rate in colliders, $L = \frac{N^2f}{4\pi\sigma_x\sigma_y}$
• $N$ is the number of particles per bunch.
• $f$ is the collision frequency.
• $\sigma_x, \sigma_y$ are the horizontal and vertical beam sizes at the interaction point.

Synchrotron Challenges

• Space Charge Effects: Repulsive forces between charged particles cause beam expansion and instability.
• Synchrotron Radiation:
  • Electromagnetic emission from accelerated charged particles leads to energy loss.
  • Light particles at high energies face the radiation.
• Magnet Technology:
  • High-field superconducting magnets achieve necessary high energies in synchrotrons.
  • High-field superconducting magnets requires advanced materials and cryogenic systems.

Synchrotron Future Directions

• Energy Frontier to explore new physics beyond the Standard Model at the Future Circular Collider (FCC) at CERN.
• Intensity Frontier increases sensitivity to rare processes via beam intensity gains on Project X at Fermilab.
• Advanced Acceleration Techniques with potential for high gradients and compact accelerators, e.g., plasma wakefield acceleration.

Algorithmic Trading and Order Execution

• Algorithmic trading executes orders by automating preprogrammed instructions and trading.
• It entails a variety of strategies, from high-frequency trades to slow execution strategies.

Algorithmic Trading Primary Goal

• Execute significantly large orders without moving the price
• Orders sent to market are smaller with timed instructions than what the trader wanted to execute, thus hiding the order

Use Algorithmic Trading to:

• Find the best price to reduce transaction costs.
• Improves order execution, speeding up time and accuracy.
• Gain faster access, automating across exchanges.
• React quicker to improve trading speed with market chagnes.
• Test backtesting and trading strategies on historical data.
• Reduce and remove human emotional influence that result bias.

VWAP (Volume Weighted Average Price) Strategy

• An algorithmic trading strategy, executes its volume-weighted average price (VWAP) order.
• VWAP finds the average price a stock has been traded at throughout the day, based on both volume and price.
• Reduces market impact by breaking large orders into pieces, releasing them at set intervals with historic trading proportions.

VWAP Implementation

• The total order slices are small tranches dependent on volume patterns gathering accurate historical data.
• Realse tranches based on the continuous adjusment & monitoring of patterns with dynamic real-time conditions

VWAP Pros

• Provides performance benchmark and minimal movement reduces market impact

VWAP Cons

• Can lead to volume prediction errors of thinly traded stocks or all assets

Other Algorithmic Strategies:

• Time Weighted Average Price (TWAP)
• Percentage of Volume (POV)
• Implementation Shortfall
• Pairs Trading
• Mean Reversion
• Delta-Neutral Hedging
• Statistical Arbitrage
• High-Frequency Trading (HFT)

Order Execution: Definition

Completes a security order to buy or sell including order routing, matching, and settlement.

Execution Considerations should have a good:

• Speed - How quickly the trade is executed
• Price must be set well
• Size: Quantity of shares or contracts
• Limited Market Impact on a security's price
• Should prevent Information Leakage where risks of order detailing become public

Key Order Types

• A market order immediately executes at the best price.
• Limite orders execute only at a specific price or better
• Stop orders turn into market orders when a stop price is reached
• Stock-limit order transform to a limit order when a stop price is reached
• Only a section of a hidden order is displayed
• Order must execute completely or cancels fill on kill FOK
• Immediate or cancel IOC can cancel any section of the order that wasn't executed as well as All or None AON executions

Execution Venues

• For centralised locations for trading, Stock Exchanges are key
• Electronic Communication Networks or ECNs are automative systems that match orders
• Dark pools or DTPs do not distplay order information which can be beneficial
• Over the counter markets OTCs are decentralizsated not listed on exchanges and helpful

Bernoulli's Principle

• An increase of speed in fluid corresponds with a decrease in pressure or fluid's potential energy

Explanation of Bernoulli's Principle

• Daniel Bernoulli published "Hydrodynamica" in 1738 which is where the principle stems but Leonhard Euler modified its current form.
• Bernoulli's principle means that when pressure lowers, flow increases or vise versa
• The relationship is given by the equation for flow velocity = absoulte pressure, fluid velocity, and density

Bernoulli’s Equation for Incompressible Fluids

• $P + \frac{1}{2} \rho v^2 + \rho g h =$ constant. The parts break down as: --The absolute pressure of the fluid is written, $P$
• -The fluid velocity is written, $v$ --The Fluid Density is written, $\rho$
• -A point of reference is at height (h)
• -Due to gravity, there was acceleration $g$

Applications of Bernoulli's Principle

• Airplane Wings where shape leads to air flowing faster over the the top rather than under it. Lower the pressure helps lead to a net upwared lift.
• Spray Bottles use air flowing over the top of a tube that is connected to a liquid reservoir. Reduced pressure draws the liquid, where it sprays out.
• Chimneys have increased altitude and wind speed with less pressure at the top.

MIKROE

What is mikroBUS™?

• MikroBus™ standard is an add-on boards' pinout simplifies the process of expanding functionalty to development boards.

MikroBus Motivation

• The problem with having lots variations with different kinds
• In an effort to solve compatibility issues when working with varying standards in different platforms or vendors in the embedded world, microBUS™ offers a solution.

MikroBus Standard Benefits

• Provides open, cost effective easy access to a wide variety of add on boards, which expands hardware configuration.

MicroBus Standard Technical specifications

• In General, there are 2x8 14 pins on 28.6x25.4mm planes that work in -40 to +85°C at 3.3V, 5V with at least 50 mbps and 1Amp or .8mm amps.

Pin Outs of MikroBus Standard

• At a Top VIew image there is top to bottom of a left to right of 1-8 to 1-8 pins with varying specifications
• 1 Reset , 2 AN analog Input, 3PWM for motors Dimming, 4 An interupt signals event has occurred, 5RX from data receival serial , 6Transmit Data Serial TX, 7Clock SCK signal SPI, 8:Master In Slate Out MISO (for Slave to Send Data)
• Left Side Pin Specifications are : --1: Reset Line --2: AN Analoge Input
• -3: PWM output
• -4: Interupt Lines
• -5: Data Recieved, Serial.
• -6: Transmit data
• -7:Clock Signal SPI (Used when you need to use Clock when communicating with an device)
• -8: Master In/ Slave Out SPI (Used to send data from target to host)
• Right Side Pin Specifications:
• -1: Chip Select (Select Device Communication.
• -2: Master Out Slave In (Use Master to send data in SPI).
• -3:I2C Data Line (Data from 2 devices send through this).
• -4: Clock signal I2C (Used when you need to use when clock when communicating with an device).
• -5: Data received Serial.
• -6 Transmit data
• -7: Power in (3.3V)
• -8: Power In (5V)

How to Use MikroBus in Hardware and Software?

• Plug Boards into a MikroBus socket on the development board then, use a Usb cable to connect
• Use software to download libraries, include codes and control board.

More about MikroBus Standard

• Open to use under MIT License.

Matplotlib

What is Matplotlib?

• It is a python library that helps create static, interactive and animated visualizations.

Methods of installations:

• Use pip to install the visualization/ charts library: '''python pip install matplotlib '''

• Or use conda: '''python conda install matplotlib '''

• pyplot makes the library like that of Matlab and provides each function changes to a figure

• Example of plotting: '''python

Plotting a simple line graph

plt.plot([1, 2, 3, 4], [5, 6, 7, 8]) plt.xlabel('X-axis') plt.ylabel('Y-axis') plt.title('Simple Line Graph') plt.show() '''

Kinds of Simple plots:

Plots by Lines:

'''python import matplotlib.pyplot as plt x = [1, 2, 3, 4, 5] y = [2, 4, 6, 8, 10] plt.plot(x, y, label='Linear') '''

• Plots by scatter: '''python import matplotlib.pyplot as plt x = [1, 2, 3, 4, 5] y = [2, 4, 6, 8, 10] plt.scatter(x, y, label='Data Points', color='red') plt.xlabel('X-axis') '''

• Plots by bar chart: '''python import matplotlib.pyplot as plt categories = ['A', 'B', 'C', 'D'] values = [25, 30, 15, 20] plt.bar(categories, values, color='green') plt.xlabel('Categories') plt.ylabel('Values') '''

• Plots by histogram: '''python import matplotlib.pyplot as plt import numpy as np data = np.random.normal(0, 1, 1000) plt.hist(data, bins=30, color='skyblue', edgecolor='black') '''

• Using Pie Chart '''python import matplotlib.pyplot as plt labels = ['A', 'B', 'C', 'D'] sizes = [25, 30, 15, 20] plt.pie(sizes, labels=labels, autopct='%1.1f%%', startangle=90)

'''

Customization

• You can further change and edit a lot of this library through markers and lines:

'''python import matplotlib.pyplot as plt x = [1, 2, 3, 4, 5] y = [2, 4, 6, 8, 10] plt.plot(x, y, color='purple', marker='o', linestyle='--', linewidth=2, markersize=8) '''

• Legends that can be changed by its size, colors with additional information when coding.
• Subplots on a Matplotlib have multiple rows, column, and data by using subplot() method with a figure: '''python import matplotlib.pyplot as plt #plot 1: x = np.array([0, 1, 2, 3]) y = np.array([3, 8, 1, 10]) plt.subplot(1, 2, 1) plt.plot(x,y) plt.title("SALES") #plot 2: '''
• 3D with a mplot:

'''python import matplotlib.pyplot as plt from mpl_toolkits.mplot3d import Axes3D fig = plt.figure() ax = fig.add_subplot(111, projection='3d') x = [1, 2, 3, 4, 5] y = [2, 4, 6, 8, 10] '''

• You can save file using code below :

'''python import matplotlib.pyplot as plt x = [1, 2, 3, 4, 5] y = [2, 4, 6, 8, 10] plt.plot(x, y) plt.xlabel('X-axis') plt.ylabel('Y-axis') plt.title('Simple Line Graph') plt.savefig('line_graph.png') '''

Quantum Mechanics

• Quantum Mechanics is a branch of physics that studies the atomic and subatomic interactions. It explains nature's physical properties on a subatomic level.

Fundamental Principles:

• Physical properties can only have discreet, quantized values.
• Exhibitions of wav and particle
• Simultaneous precision is impossible with property pairings like position and momentum.
• At one time, multiple superpositions are at play at once but at different rates.
• It takes 2 quanta systems share the same fate, but they are split by distance

Key Equations

• The equations of Schrodinger help with quantam mechanical problems.
• The same is true for that of Heisenberg's equation.

Application

• Quantum Computing: Useing quantum for future computing.
• Quantam Cryptography: Making safer communication with quantam methods.
• Materials Sciemce to help desigm new materia.
• Medical imaging: Develop new imaging techs for us later.