Complexity Theory: Polynomial Time

• This lecture introduces complexity theory

Polynomial Time

• An algorithm is polynomial time if its runtime is $O(n^k)$ with constant $k$, where $n$ is input size
• The complexity class P includes decision problems solvable by an algorithm in polynomial time
• SORTING is an example: arranging n numbers in order
• CONNECTIVITY is an example: finding path between 2 nodes (s and t) in graph G = (V, E)
• PRIMALITY is an example: determining integer n is prime or not

Nondeterministic Polynomial Time (NP)

• Complexity class NP includes problems where a solution can be verified in polynomial time
• HAMPATH is an example: determining if Hamiltonian path from s to t exists in graph G = (V, E)
• CLIQUE is an example: determining if clique of size k exists in graph G = (V, E)

Polynomial-Time Reducibility

• Problem A is polynomial-time reducible to problem B ($A \leq_P B$) if polynomial-time algorithm f exists
• Algorithm f transforms instance x of A into instance f(x) of B
  • If x is YES for A, f(x) is YES for B
  • If x is NO for A, f(x) is NO for B
• If $A \leq_P B$, A "no harder" than B
• A polynomial-time algorithm for B enables solving A in polynomial time

NP-Completeness

• Decision problem B is NP-complete if:
  • $B \in NP$
  • $A \leq_P B$ for every $A \in NP$
• NP-complete problems are the "hardest" in NP
• Finding polynomial-time algorithm for NP-complete implies all problems in NP solvable in polynomial time and P = NP
• The SAT decision problem is NP-complete due to Cook-Levin Theorem

SAT (Satisfiability)

• Given boolean formula in Conjunctive Normal Form (CNF), can variables be assigned to make it true?

3-SAT

• Special case of SAT: each clause has 3 literals
• Example: $(x_1 \lor \neg x_2 \lor x_3) \land (\neg x_1 \lor x_4 \lor x_5) \land (x_2 \lor \neg x_3 \lor \neg x_4)$
• 3-SAT is NP-complete
  • 3-SAT $\in$ NP
  • SAT $\leq_P$ 3-SAT
• Convert any SAT clause into set of clauses with 3 literals each:
  • $(x) \rightarrow (x \lor x \lor x)$
  • $(x \lor y) \rightarrow (x \lor y \lor y)$
  • $(x_1 \lor x_2 \lor... \lor x_k) \rightarrow (x_1 \lor x_2 \lor y_1) \land (\neg y_1 \lor x_3 \lor y_2) \land... \land (\neg y_{k-3} \lor x_{k-2} \lor y_{k-2}) \land (\neg y_{k-2} \lor x_{k-1} \lor x_k)$

Implications of NP-Completeness

• If P $\neq$ NP, NP-complete problems are not solvable in polynomial time
• Many important problems are NP-complete, like HAMPATH, CLIQUE, 3-SAT, VERTEX-COVER, SUBSET-SUM
• Consider approximation algorithms, heuristics or exponential-time algorithms when NP-complete problem arises

Transistors

• Semiconductor devices used for amplify/switch electrical signals and power
• Key building blocks of electronics today

Transistor Types

• Bipolar Junction Transistors (BJTs)
• Field-Effect Transistors (FETs)

Bipolar Junction Transistors(BJTs)

• Current-controlled transistors: the flow of current depends on the flow of current into its base
• Used in amplifiers, switches and regulators

Structure

• Three layers of semiconductor material are doped differently. The three terminals are emitter, base, and collector.

Types

• NPN: Emitter/collector doped with n-type material, base with p-type
• PNP: Emitter/collector doped with p-type material, base with n-type

Function

• Current is applied applied to BJT base, causes a larger current to flow from collector to the emitter

Field-Effect Transistors (FETs)

• Voltage-controlled transistors: flow of current controlled by voltage to the gate
• Used in amplifiers, switches, and variable resistors

Structure

• Three terminals: source, drain, and gate. The source and drain are connected by a channel and the gate allows the channel to be controlled.

Types

• Junction FET (JFET): Gate is p-n union
• Metal-Oxide-Semiconductor FET (MOSFET): Gate is isolated from the channel

Function

• Gate voltage creates an electric field which impacts channel conductance, affecting the flow of current

Transistor Applications

• Amplifiers
• Switches
• Oscillators
• Digital circuits
• Voltage regulators
• Microprocessors
• Memory

Transistor Advantages

• High Reliability
• Low Cost
• Small size
• Low power

Transistor Conclusion

• Versatile devices, key building block in electronic devices

Clinical Trial Protocol

• Phase 3, Randomized, Double-Blind, Placebo-Controlled, Parallel-Group, Multi-Center Study
• Evaluates Efficacy and Safety of MEDI4736
• Applies to Patients: With Unresectable Stage III Non-Small Cell Lung Cancer (NSCLC); Have Not Progressed Following Concurrent Chemoradiation

Table of Contents Include

• Protocol Summary
• Introduction
  • Background Information
  • Rationale for Study
• Objectives
  • Primary: Target
  • Secondary: Target
• Study Design
  • Overall Plan
  • Randomization and Blinding
  • Population -
    • Inclusion: Criteria
    • Exclusion: Criteria
  • Treatment
    • Investigational Product:
    • Method of Assignment
    • Dosage and Administration
    • DurationConcomitant Medications
    • Management of Overdose
  • Assessments
    • Efficacy: Assessments
    • Safety: Assessments
    • Pharmacokinetic (Optional)
    • Biomarker (Optional)
  • Procedures
  • Visits
  • Discontinuation
  • End of Study
• Statistical Considerations
  • Sample Size
  • Statistical Analysis
    • Efficacy: Analysis
    • Safety: Analysis
  • Interim Analysis (If Applicable)
• Data Managment
  • Data Collection
  • Quality Assurance
• Ethical Considerations
  • Ethical Principles
  • Informed Consent
  • Regulatory Compliance
• Publication Policy
• References
• Appendices

Chemical Principles

Quantitative Description of Chemical Reactions

Stoichiometry - Reaction Relationships

• Stoichiometric Coefficients: Balanced equation describes the relative number of moles per reactant/product

  • $2H_2(g) + O_2(g) \longrightarrow 2H_2O(g)$
  • 2 mol $H_2$ react with 1 mol $O_2$ to produce 2 mol $H_2O$

• Stoichiometric Ratios: Relates amounts of two species

  • $2H_2(g) + O_2(g) \longrightarrow 2H_2O(g)$
  • $\frac{n(H_2)}{n(O_2)} = \frac{2}{1}$
  • $\frac{n(H_2O)}{n(H_2)} = \frac{2}{2} = 1$

Reaction Yield

• Actual yield of a reaction is often less than the theoretical yield
  • Side reactions
  • Incomplete reaction
  • Losses during recovery of product
• $$\text{Percent yield} = \frac{\text{Actual yield}}{\text{Theoretical yield}} \times 100%$$

Limiting Reactant

• The limiting reactant is consumed completely in a chemical reaction while other reactants are excess
• $N_2(g) + 3H_2(g) \longrightarrow 2NH_3(g)$
  • 1 mol $N_2$ and 2 mol $H_2$ allowed to react
  • $H_2$ is the limiting reactant
  • $N_2$ is in excess

Adipic Acid Example

• $H_2C_6H_8O_4$ is used to produce nylon & made by oxidating cyclohexane ($C_6H_{12}$)
• 25.0 kg cyclohexane react with 25.0kg $O_2$
• $2C_6H_{12}(l) + 5O_2(g) \longrightarrow 2H_2C_6H_8O_4(s) + 2H_2O(g)$

What is the limiting reactant?

• $n(C_6H_{12}) = \frac{25.0 \times 10^3 g}{84.16 \frac{g}{mol}} = 297 mol$
• $n(O_2) = \frac{25.0 \times 10^3 g}{32.00 \frac{g}{mol}} = 781 mol$
• $\frac{n(O_2)}{n(C_6H_{12})} = \frac {781}{297} = 2.63 < \frac{5}{2}$, therefore $O_2$ is the Limiting Reactant

What is theoretical yield of adipic acid?

• $n(H_2C_6H_8O_4) = \frac{2}{5} \times n(O_2) = \frac{2}{5} \times 781 mol = 312 mol$
• $m(H_2C_6H_8O_4) = 312 mol \times 146.1 \frac{g}{mol} = 45.6 \times 10^3 g = 45.6 kg$

Data Structures and Algorithms

Objectives

• Explain common data structures and algorithms
• Select appropriate data structures and algorithms for problem solving
• Write programs that use data structures and algorithms for problem solving

Data Structures

• A way of organizing and storing data in a computer so that it can be accessed and used efficiently

Types

• Arrays: Collection of the same typed elements, stored contiguously
• Linked Lists: Sequences of nodes, with data and a pointer to the next node
• Stacks: LIFI (Last-In-First-Out)
• Queues: FIFO (First-In-First-Out)
• Trees: Hierarchical data structure, nodes connected by edges
• Graphs: Nodes (vertices) & edges. Edges connect pairs of nodes
• Hash Tables: Uses a hash function to map keys to values

Algorithms

• A step-by-step procedure for solving a problem

Types

• Searching Algorithms
  • Algorithms for finding an element
    • Linear Search
    • Binary Search
• Sorting Algorithms
  • Algorithms for arranging elements
    • Bubble Sort
    • Insertion Sort
    • Merge Sort
    • Quick Sort
• Graph Algorithms
  • Algorithms for graphs
    • Depth-First Search (DFS)
    • Breadth-First Search (BFS)
    • Dijkstra's Algorithm
    • Prim's Algorithm
    • Kruskal's Algorithm
• Dynamic Programming
  • Solves optimization problems breaking them down into smaller problems
• Greedy Algorithms
  • Makes locally optimal choices, looking for global optimum

Algorithm Analysis

• Time Complexity: The run time as a function of the input size
• Space Complexity: The memory amount as a function of the input size
• Big O Notation: Asymptotic behavior of functions; use to type time/space

Big O Notation types

• O(1): constant time
• O(log n): Logrithemic time
• O(n): Linear Time
• O(n log n) - Linearithmic Time
• O($n^2$) - Quadratic Time
• O($2^n$) - Exponential Time
• O(n!) - Factorial Time

Key Concepts

• Abstract Data Type: Theoretical model defined by it operations
• Recursion: A function calls itself to solve smaller subproblems
• Divide and conquer: Break a problem into smaller subproblems, solve and combine

Applications

• Database Management Systems: Uses data structures and algorithms for data
• Computer Graphics: Used in simulating the images and 3-D models
• Operating Systems: file system organization, process scheduling, memory
• Arificial Intelligence: In machine language, and computer visions
• Networking: Routing, congestion control, and security

Chapitre 2 : Algèbre relationnelle (Chapter 2: Relational Algebra)

Introduction

• Relational algebra is a theoretical query language that operates on relations and produces other relationships.
• Provides a formalism for manipulating data stored in relational databases.

Opérateurs de base (Basic Operators)

• Sélection ($\sigma$) (Selection): Filters tuples of a relation based on a condition
  • Example: $\sigma_{age > 25}(Employés)$ (Employees)
• Projection ($\Pi$) (Projection): Selects certain columns from a relation
  • Example: $\Pi_{nom, salaire}(Employés)$ (Employees)
• Union ($\cup$) (Union): Combines the tuples of two compatible relations
  • Example: $Employés \cup Clients$ (Employees or Clients)
• Intersection ($\cap$) (Intersection): Returns tuples common to 2 compatible relations
  • Example: $Employés \cap Clients$ (Employees or Clients)
• Différence (-) (Difference): Returns tuples present in 1 relation, but not in the other
  • Example: $Employés - Clients$ (Employees, not Clients)
• Produit cartésien ($\times$) (Cartesian Product): Combines each tuple with others
  • Example: $Employés \times Départements$ (Employees with Departments)
• Renommage ($\rho$) (Renaming): Renames a relation or its attributes
  • Example: $\rho_{NouveauxEmployés}(Employés)$ (New Employees or Employees)

Opérateurs dérivés (Derived Operators)

• Jointure ($\Join$) (Join): Combines tuples based on condition
  • Example: $Employés \Join_{Employés.département = Départements.id} Départements$ (Join Employees with Departments)
• Division ($\div$) (Division): Returns tuples of 1 relation that corresponds to all tuples of another.
  • Example: Achats \div Clients (Purchases and Clients)

Exemples de requêtes (Examples of Queries

• Etudiants (Students): (id, nom, age, adresse)
• Cours (courses): (id, titre, crédits)
• Inscriptions (registrations): (id_etudiant, id_cours, date)

Requêtes (Queries)

1. Student names over 20: $\Pi_{nom}(\sigma_{age > 20}(Etudiants))$
2. Course tiles for students: $\Pi_{titre}(\Cours \Join_{Cours.id = Inscriptions.id_cours}(\sigma_{id_etudiant = 123}(Inscriptions)))$
3. Student names: $\Pi_{nom}(Etudiants \Join_{Etudiants.id = Inscriptions.id_etudiant} Inscriptions) \div \Pi_{id}(\Cours)$

Avantages et inconvénients (Advantages and Disadvantages

Avantages (Advantages)

• Formalisme mathématique précis (Precise mathematical formalism)
• Base pour l'optimisation des requêtes (Basis for query optimization)

Inconvénients (Disadvantages)

• Peu intuitif pour les utilisateurs non techniques (Not intuitive for users)
• Verbeux pour les requêtes complexes (Verbose)

Conclusion

• Fundamental tool for understanding/manipulating databases
• Underlies languages like SQL
• Allows efficient query optimiation

Lab 1: Introduction to DOS

Objectives

• Become familiar with Linux environment and DOS
• Learn how to use basic DOS commands
• Practice creating simple DOS nodes

Deliverables

• A written report including:
  • Answers to all questions
  • Code of DOS Nodes
  • Screen Capture of Turtlesim Simulation
  • A brief discussion and conclusions about the lab
• All of your nodes

Introduction

What is DOS?

• DOS is a flexible framework for writing robot software
• DOS is a collection of tools, libraries, and conventions
  • Complex
  • Robust robot behavior
  • A wide variety of robotiic platforms

Why DOS?

• Facilitates code reuse in robotics research and development
• Large and Active Community
• Provides Simulation, Visualization, and Debugging
• Abstraction from underlying Hardware

DOS Concepts

• Nodes: Executable per specific tasks
• Topics: Named bases which nods exchange messages, to publish and reciever
• Messages: Data used to communicate, with standard and custom sets
• Services: Request/Reply functions to be called with certain inputs
• Paramters: Stored configuration values

DOS Architecture

• DOS master tracks, topics, and services, which finds all of connections

Preliminaries

• Installation: If not, use http://wiki.ros.org/Installation
• Environment Setup: Sourcing is setup for the bash script

DOS Basic Commands

• roscore: starts the DOS master
• rosnode: displays listing for DOS Nodes
• rostopic: displays listing for topics, lists, and pulishing.
• rosservice: lists and calls DOS services
• roslaunch: command line for launching and configuring

DOS programming

• Creating: Before creating a DOS node, create a package
  • catkin_create_pkg: Creates a ROS program and set dependencies
    • my_package.roscpp, rospy, std_msg
• CD to catkin_ws.src
• Building a ROS node: C++ or Python
  • Add executable to CMakeLists.txt

Building

• Target libraries inside of cmakeLists.txt
• Build from catkin_make

Running DOS Node

• Need to source with the bash script

Turtlesim simulation

• DOS simulation

DOS Installation: Follow installation instructions on ROS website

• Command: sudo apt-get install ros-noetic-turtlesim

DOS Running command: rosrun turtlesim turtlesim_node

DOS Control

• Run command: Command: rosrun turtlesim turtle_teleop_key
• DOS turtles can be given position and orientation

Questions

1.  What is a ROS node?
2.  What is a ROS topic?
3.  What is a ROS message?
4.  What is the difference between a ROS topic and a ROS service?
5.  What is a ROS package?
6.  How do you create a ROS package?
7.  How do you build a ROS node?
8.  How do you run a ROS node?
9.  How do you find out the name of the topics that are currently active?
10. How do you display the messages being published to a topic?
11. How do you publish a message to a topic?
12. How do you create a message definition?

Extra Excercises

• 1: Writes a DOS to publish turle position
• 2: Write a DOS, that can use to rest the turtle and services
• 3: Create a launch file,

Chapter 14: Social Psychology

Social Psychology Defined

• The study of how an individual's thoughts, feelings, and behaviors are affected by the presence of others

Social Cognition

• Studies processing of information related to people and social situations. Understanding cognitive processes
  • Attribution: Explains why people give certain behaviors
    • Internal attribution: Cause due to trait/personality -External attribution: Cause due to external factor
      • Fundamental attribution error: Overestimate role of dispositional factors
      • Actor-observer bias: Attributing own behavior situational causes
      • Self-serving bias: Taking credit for our successes and denying failures
  • Attitudes: evaluation of people, objects, and ideas
    • Cognitive: thoughts/beliefs
    • behavioral: emotions
      • Behavioral: actions
    • cognitive dissonance: uncomfortablity felt behavior

    •       Persuasion: Changing attitude throught the transmisiion of messages
      

Social influenve

• Ways people are affected by others Conformity: One feeling, matched those to a goup of people - Group polarization: the tendency for a group to make decisions that are more extreme than the initial inclination of its members. Informational/Social

Social Relations

• How to maintain relations
• Prejudice*. negative attitude Stenotype, generalization of beilfs Discriminations - Negative Behaviour
• *Aggregations: Verval Actions .Bystander
• *Bystander, .Proximity:
• *Intraposition,**.Share common and values .Companionate Passionate
• **
• **

This lecture introduces complexity theory. It covers polynomial time, nondeterministic polynomial time (NP), and polynomial-time reducibility.