Floating Point Numbers

Floating Point Numbers

• Computers can accurately represent integers and floating-point numbers.
• Rationals and reals must be approximated by computers.

Number Representation

• Integers are represented exactly but are limited by the number of bits.
• Rational numbers are ratios of two integers.
• Real numbers possess infinite, non-repeating decimal expansions.

Fixed-Point Numbers

• Use a set amount of digits for the integer and fractional components
• Limited range and precision

Floating-Point Numbers

• Sign, mantissa (significand), exponent, and base are its components.

IEEE 754 Floating Point Standard

• Single precision uses 32 bits, with 1 bit for sign, 8 for exponent, and 23 for mantissa.
• Double precision uses 64 bits, with 1 bit for sign, 11 for exponent and 52 for mantissa.

Normalization

• The mantissa is normalized to contain a single non-zero digit before the radix point.
• For base 2, the format is $1.xxxx \times 2^{yyyy}$.
• The leading 1 is implicit and not stored, optimizing space.

Exponent Bias

• The exponent is adjusted by adding a bias to represent positive and negative exponents.
• Single precision bias is 127.
• Double precision bias is 1023.
• If $E$ is the actual exponent, the stored exponent is $E + \text{Bias}$.

Example: Single Precision

• For the number -0.75:
• The binary is -0.11
• The normalized form is $-1.1 \times 2^{-1}$.
• Sign: 1 (negative)
• Mantissa: 10000000000000000000000 (23 bits)
• Exponent: -1 + 127 = 126 = 01111110 (8 bits)
• The resulting floating point representation: 10111111010000000000000000000000

Special Values

• Zero: Exponent and mantissa are zero; both +0 and -0 exist.
• Infinity: Exponent is all ones, mantissa is zero; both +∞ and -∞ exist.
• NaN (Not a Number): Exponent is all ones, mantissa is non-zero; used for undefined results like 0/0 or ∞ - ∞.

Floating Point Operations

• Operations are performed in hardware and may introduce rounding errors.

Rounding Errors

• Rounding errors happen when an operation's outcome is not exactly representable.
  • Round to nearest (default)
  • Round towards zero
  • Round towards positive infinity
  • Round towards negative infinity

Catastrophic Cancellation

• Occurs when significant digits are lost during the subtraction of nearly identical numbers.

Floating Point in Python

• The float type is usually double precision.
• Precision is limited by the number of bits.
• Libraries like decimal offer arbitrary precision for increased accuracy.

Arbitrary Precision

• Employs software for representing numbers with customizable precision.
• It is useful where precision is paramount, though slower than hardware FP operations.

decimal Module

• Provides arbitrary precision decimal arithmetic.
• Precision and rounding are controllable.

The Endocrine System

• The endocrine system uses hormones for communication.

Hormones

• Hormones are regulatory substances that stimulate specific cells or tissues.

Qualities

• Hormones can be proteins, peptides, steroids, amino acid derivatives, or fatty acid derivatives.
• Produced by glands
• Travel through the bloodstream to act on target tissue expressing a receptor
• High affinity (hormone receptor)
• Low capacity (hormone receptor)

Hormone Types

• Endocrine: released into the bloodstream, travel throughout the body
• Paracrine: act locally by diffusing from its source to target cells nearby
• Autocrine: affect the cells that produce them
• One hormone can signal in multiple ways.

The Hypothalamus and Pituitary Gland

• These work together to control other endocrine glands and bodily functions.

Hypothalamus

• It is located in the brain
• Connected to the pituitary gland by the pituitary stalk
• It secretes hormones that regulate the anterior pituitary
• Synthesizes hormones that are stored in the posterior pituitary

Pituitary Gland

• It is located below the hypothalamus
• Features two parts:
• The anterior pituitary synthesizes and secretes hormones.
• The posterior pituitary stores and secretes hormones produced in the hypothalamus.

Posterior Pituitary Hormones

• Hormones of the posterior pituitary are synthesized in the hypothalamus.
• Transported along axons to posterior pituitary Stored in posterior pituitary
• It releases hormones into the bloodstream
• Examples:*
• Antidiuretic hormone (ADH) stimulates water reabsorption by the kidneys.
• Oxytocin stimulates uterine contractions during childbirth and milk ejection during breastfeeding.

Anterior Pituitary Hormones

• Tropic hormones: TSH, ACTH, FSH, and LH
• Direct hormones: Prolactin, GH, and MSH | Hormone | Target | Effect | | ------------ | --------------- | ---------------------------------------------------------------------- | | Thyroid-stimulating hormone (TSH) | Thyroid |Synthesis and secretion of thyroid hormones. | | Adrenocorticotropic hormone (ACTH)| Adrenal Cortex | Synthesis and secretion of glucocorticoids | | Follicle-stimulating hormone (FSH)| Testes/Ovaries| Gamete production and hormone production | | Luteinizing hormone (LH) |Testes/Ovaries| Hormone production | | Prolactin |Mammary Glands| Milk production. | | Growth hormone (GH) |Liver, Bone etc| Protein synthesis, cell growth. | | Melanocyte-stimulating hormone (MSH)| Melanocytes | Melanin production |

The Thyroid Gland

• The thyroid gland regulates metabolism through its hormones.

Thyroid Hormones

• Thyroxine ($T_4$) and triiodothyronine ($T_3$):
• Stimulate metabolism
• Iodine is required for production.
• Calcitonin:
• Lowers blood calcium levels by stimulating calcium deposition into bone.
• Secreted when blood-calcium levels are high.

Control of Metabolism

• Hypothyroidism:
• It is an underactive thyroid that may cause weight gain, fatigue, and sensitivity to cold.
• Hyperthyroidism:
• It is an overactive thyroid which may cause weight loss, anxiety, and increased heart rate.

The Parathyroid Glands

• Parathyroid Hormone (PTH) regulates blood calcium levels.

Parathyroid Hormone (PTH)

• Increases blood calcium levels
• Stimulates calcium release from bone
• Increases calcium reabsorption in kidneys and calcium absorption in intestines.
• Secreted when blood calcium levels drop.

The Adrenal Glands

• The adrenal glands are important for stress response and hormone regulation.

Adrenal Cortex (Outer Layer)

• Mineralocorticoids (e.g., aldosterone):
• Promote sodium and water reabsorption in kidneys, increasing blood volume and pressure.
• Glucocorticoids (e.g., cortisol): Increase blood glucose levels and suppress the immune system.
• Sex Hormones (e.g., androgens):
• Promote the formation of secondary sexual characteristics.

Adrenal Medulla (Inner Layer)

• Epinephrine (adrenaline) and norepinephrine (noradrenaline):
• Mediate “fight or flight” response in the body.
• Increase heart rate, blood pressure, and blood glucose levels.
• Vasoconstriction of blood vessels to digestive system and vasodilation of blood vessels to skeletal muscles.

The Pancreas

• Pancreatic Hormones, like insulin and glucagon, regulate blood glucose levels.

Pancreatic Hormones

• Insulin:
• Lowers blood glucose levels
• Stimulates glucose uptake by cells and glycogen storage in the liver.
• Glucagon: Increases blood glucose levels.
• Stimulates glycogen breakdown in the liver.
• Increases glucose release into the bloodstream.

Diabetes

• Type 1 Diabetes:
• Autoimmune destruction of insulin-producing cells which requires insulin injections.
• Type 2 Diabetes:
• Insulin resistance that is often associated with obesity.
• Managed with diet, exercise, and medication.

The Gonads

• The gonads are glands that regulate reproduction, sex drive and secondary sex characteristics.

Testes

• Testosterone:
• Promotes sperm production, as well as controlling for maintenance of the male secondary sexual characteristics.

Ovaries

• Estrogen:
• Promotes the generation of female secondary sexual properties and regulates the menstrual cycle.
• Progesterone:
• Prepares uterus for pregnancy and maintains the integrity of the pregnancy.

Propositions and Truth Tables

• An overview of propositions, logical connectives, truth tables, and exercises in logic and reasoning.

Propositions

• A proposition (or assertion) is a statement that is either true or false, but not both.
• Examples: "It is raining", "2 + 2 = 4", For all integers $n \ge 0$, we have $n^2 \ge n$"

Logical Connectives

• Propositions are often used to construct new ones using logical connectives.

• The negation of P, denoted $\neg P$ (or sometimes $\bar{P}$), is true if P is false, and false if P is true. Example: The negation of “It is raining” is “It is not raining”.

• The conjunction of P and Q, denoted $P \land Q$, is true if P and Q are both true, and false otherwise. Example: “It is raining and I am taking my umbrella”.

• The disjunction of P and Q, denoted $P \lor Q$, is true if at least one of the two propositions P or Q is true, and false if both are false. Example: “It is raining or I am taking my umbrella”.

• The implication $P \Rightarrow Q$ is false only when P is true and Q is false. Example: “If it is raining, then I take my umbrella”.

• The equivalence $P \Leftrightarrow Q$ is true when P and Q are both true or when P and Q are both false. Example: “I take my umbrella if and only if it is raining”.

Truth Tables

• Logical connectives can be summarized in truth tables.

Quantifiers

• An overview of quantifiers, including universal, existential, and negation of quantifiers, with related exercises.

Universal Quantifier

• The universal quantifier, denoted $\forall$, means "for all".
• Example: “All cats are gray” is written as: $\forall x \in E, P(x)$, where $E$ is the set of cats and $P(x)$ is the property “x is gray”.

Existential Quantifier

• The existential quantifier, denoted $\exists$, means “there exists at least one”.
• Examples: “There exists a black cat” is written as: $\exists x \in E, P(x)$, where $E$ is the set of cats and $P(x)$ is the property “x is black”.
• “There exists a unique black cat” is written as: $\exists! x \in E, P(x)$.

Negation of Quantifiers

• The negation of $\forall x \in E, P(x)$ is $\exists x \in E, \neg P(x)$.
• The negation of $\exists x \in E, P(x)$ is $\forall x \in E, \neg P(x)$.

Reasoning

Different types of reasoning (direct, contraposition, absurd, disjunction of cases, recurrence) used to prove mathematical statements are introduced.

Types of Reasoning

• Direct Reasoning: Showing that if P is true, then Q is true.
• Reasoning by Contraposition: Showing that if Q is false, then P is false. This shows $(\neg Q) \Rightarrow (\neg P)$ to prove $P \Rightarrow Q$.
• Reasoning by Absurd: Assuming P is true and Q is false, then showing that this leads to a contradiction.
• Reasoning by Disjunction of Cases: Dividing the problem into several cases and showing that the proposition is true in each case.
• Reasoning by Recurrence: Used to show that a property is true for all natural numbers.

Reasoning by Recurrence

• To show that a property $P(n)$ is true for all integers $n \ge n_0$, use these tree steps:
• Initializing shows that is $P(n_0)$ is true.
• Heredity is where it is assumed that $P(n)$ is true for a certain integer $n \ge n_0$, and then is demonstrated that this is $P(n + 1)$ is true.
• Come to the conclusion that the property $P(n)$ is true for every single integer $n \ge n_0$.

Statistical Treatment of Data

• Overview of statistics, the study of data collection, analysis, and interpretation.

What is Statistics?

• Statistics is the field dealing with data collection, analysis, interpretation, presentation, and organization.

Types of Statistics

• Descriptive Statistics summarizes and organizes data using measures of central tendency like mean, median, and mode, and variability like variance and standard deviation.
• Inferential Statistics makes inferences about a population based on a sample, using hypothesis testing and confidence intervals.

Population vs. Sample

• Population is the entire group of individuals or items being studied.
• Sample is a subset of the population selected for analysis.

Variables

Types of Variables

• Qualitative (Categorical) Variables are divided into categories, such as nominal (e.g., colors, genders) and ordinal (e.g., education levels, satisfaction ratings).
• Quantitative (Numerical) Variables are measured numerically and can be discrete (e.g., number of students) or continuous (e.g., height, temperature).

Levels of Measurement

• Nominal is data categorized without any order or ranking (e.g., types of cars).
• Ordinal is data categorized with a meaningful order but inconsistent intervals (e.g., rankings).
• Interval is consistent intervals but no true zero point (e.g., temperature in Celsius).
• Ratio is consistent intervals and a true zero point (e.g., height, weight).

Descriptive Statistics

Measures of Central Tendency

• Mean is calculated by the below formula: $$ \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} $$
• Median is the middle value when data are arranged in order.
• Mode is the most frequent value in the data set.

Inferential Statistics

Hypothesis Testing

• Null Hypothesis ($H_0$) is a statement of no effect or no difference.
• Alternative Hypothesis ($H_1$ or $H_a$) contradicts the null hypothesis.

Steps in Hypothesis Testing

• State the null and alternative hypotheses
• Choose a significance level ($\alpha$)
• Compute the test statistic
• Determine the p-value
• Make a decision:
  • reject $H_0$, if $p \leq \alpha$.
  • fail to reject $H_0$ if, $p > \alpha$

Common Hypothesis Tests

• T-test: Used to compare means of one or two groups.
• ANOVA (Analysis of Variance): Used to compare means of three or more groups.
• Chi-Square Test: Used to test for associations between categorical variables.

Confidence Intervals

• Estimate of a population parameter with a specified confidence level. $$ CI = \bar{x} \pm z \cdot \frac{\sigma}{\sqrt{n}} $$

Data Visualization

• Common charts used to visulaize data include Bar charts and Histograms

Statistical Software

Common Tools

• R is a programming language and environment for statistical computing and graphics.
• Python (with libraries like NumPy, Pandas, SciPy) is a versatile programming language with extensive libraries for data analysis.
• SPSS is a statistical software package used for data analysis and reporting.
• Excel is a spreadsheet software with basic statistical functions.

Algorithmic Game Theory

• Overview of game theory, key concepts like Nash equilibrium, and algorithmic game theory with its applications.

What is Game Theory?

• Game theory is the study of mathematical models of strategic interactions of rational agents.
• Modern theory began with John Nash, who proved that any game with finite players that are also rational possesses equilibrium points in non-cooperative contexts.

Key Concepts

Self-interested Agents

• Tend to have their own descriptions that they prefer to see happen.
• Agents tend to try to achieve the states that they like to see happen.

Rationality

• Rationality is only acting to maximize your own preferences based on your utility.
• Rational does not mean agents are perfect, have unlimited computational power, have complete information.

Strategies

• A strategy is a complete contingency plan and specifies what an agent will do under each possible circumstance.
• Strategy is different than move.
• A move represents an action taken by a particular actor

Solution Concepts

• Predicts what strategies will be chosen by the agents.
• The Nash equilibrium is the most famous solution concept.

Nash Equilibrium

Definition

• Strategies where no agent has an incentive to unilaterally deviate from its chosen strategy.

Example: Prisoner's Dilemma

• Two suspects are arrested for a crime.
• The police have enough evidence to convict them of a minor offense, but not enough evidence to convict them of the major crime.
• If you do not confess, but your accomplice agrees to confess, you will be convicted of the major crime and serving a long sentence.
• If the suspect confesses but the accomplice is silent, then the confessing suspect is released

The payoff matrix is as follows:

• The Nash equilibrium is for both suspects to confess.

Computation

• Finding Nash equilibria can be computationally hard.

Algorithmic Game Theory

• Algorithmic game theory is the intersection of game theory and computer science. Uses game theory to design algorithms. Examines computational complexity of game-theoretic problems.

Examples of Topics in Algorithmic Game Theory

• Mechanism design
• Social choice
• Coalitional game theory
• Network games
• Auctions
• Elections
• Fair division
• Price of anarchy

Mechanism Design

• Designing a game so that agents act that is desirable to the designer.
• Example:
  • Auctions
  • Voting

Social Choice

• Social choice aggregates the preferences of individuals into a collective decision.
• Example: -Voting -Resource allocation

Coalitional Game Theory

• Studies groups of agents who can cooperate to achieve a common goal. Examples:
  • Team formation
  • Cost sharing

Network Games

• Games played on networks.
• Example:
  • Routing -Social networks

Price of Anarchy

• Measures inefficiency due to selfish agents.
• $PoA = \frac{cost(equilibrium)}{cost(optimum)}$
• Example:
  • Traffic routing