Initiation into Hermetics; last ten pages theory

Matrices - Definition

• Matrices are rectangular arrays comprising numbers or symbols arranged in rows and columns.
• Entries or elements are individual items in a matrix.
• Matrices are denoted using uppercase letters (e.g., $A, B, C$).
• A matrix's size or dimension is defined by its number of rows ($m$) and columns ($n$); e.g., an $m \times n$ matrix.
• Entries are represented by lowercase letters with subscripts showing the row and column (e.g., $a_{ij}$).

Example of a Matrix

• $A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \ a_{21} & a_{22} & \cdots & a_{2n} \ \vdots & \vdots & \ddots & \vdots \ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$
• Rows comprise horizontal lines of elements.
• Columns comprise vertical lines of elements.

Types of Matrices

Square Matrix

• Possesses an equal number of rows and columns ($m = n$).
• Example: $A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$

Row Matrix

• Consists of only one row ($m = 1$).
• Example: $A = \begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$

Column Matrix

• Consists of only one column ($n = 1$).
• Example: $A = \begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix}$

Zero Matrix

• All elements are zero.
• Example: $A = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix}$

Diagonal Matrix

• A square matrix where all non-diagonal elements are zero.
• Example: $A = \begin{bmatrix} 1 & 0 \ 0 & 2 \end{bmatrix}$

Identity Matrix

• A diagonal matrix in which all diagonal elements are one, denoted as $I$.
• Example: $I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}$

Triangular Matrix

• Upper Triangular Matrix: All elements below the main diagonal are zero.
  • Example: $A = \begin{bmatrix} 1 & 2 \ 0 & 3 \end{bmatrix}$
• Lower Triangular Matrix: All elements above the main diagonal are zero.
  • Example: $A = \begin{bmatrix} 1 & 0 \ 2 & 3 \end{bmatrix}$

Matrix Operations

Addition

• Involves matrices with the same dimensions adding corresponding elements.
• $(A + B){ij} = a{ij} + b_{ij}$
• Example: $\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} + \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \ 10 & 12 \end{bmatrix}$

Subtraction

• Involves matrices with the same dimensions subtracting corresponding elements.
• $(A - B){ij} = a{ij} - b_{ij}$
• Example: $\begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix} - \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} = \begin{bmatrix} 4 & 4 \ 4 & 4 \end{bmatrix}$

Scalar Multiplication

• Each matrix element is multiplied by the scalar.
• $(cA){ij} = c \cdot a{ij}$
• Example: $2 \cdot \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} = \begin{bmatrix} 2 & 4 \ 6 & 8 \end{bmatrix}$

Matrix Multiplication

• The product of two matrices $A$ and $B$ is defined if the number of columns in $A$ is equal to the number of rows in $B$.
• If $A$ is an $m \times n$ matrix and $B$ is an $n \times p$ matrix, then the product $AB$ is an $m \times p$ matrix.
• The element in the $i$-th row and $j$-th column of $AB$ is obtained by multiplying the elements of the $i$-th row of $A$ by the corresponding elements of the $j$-th column of $B$ and summing the products.
• $(AB){ij} = \sum{k=1}^{n} a_{ik} \cdot b_{kj}$
• Example: $\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \cdot \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix} = \begin{bmatrix} 1 \cdot 5 + 2 \cdot 7 & 1 \cdot 6 + 2 \cdot 8 \ 3 \cdot 5 + 4 \cdot 7 & 3 \cdot 6 + 4 \cdot 8 \end{bmatrix} = \begin{bmatrix} 19 & 22 \ 43 & 50 \end{bmatrix}$

Properties of Matrix Operations

• Associativity:
  • (A + B) + C = A + (B + C)
  • (AB)C = A(BC)
• Distributivity:
  • A(B + C) = AB + AC
  • (A + B)C = AC + BC
• Identity Matrix:
  • AI = A
  • IA = A
• Scalar Multiplication:
  • c(A + B) = cA + cB
  • (c + d)A = cA + dA
  • (cd)A = c(dA)

Transpose of a Matrix

• $A^T$ (transpose of matrix $A$) is obtained by interchanging rows and columns.
• For an $m \times n$ matrix $A$, the transpose $A^T$ is an $n \times m$ matrix where $(A^T){ij} = a{ji}$.
• Example: If $A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$, then $A^T = \begin{bmatrix} 1 & 3 \ 2 & 4 \end{bmatrix}$.

Properties of Transpose

• (AT)T = A
• (A + B)T = AT + BT
• (cA)T = cAT
• (AB)T = BTAT

Inverse of a Matrix

• Denoted as $A^{-1}$, is a matrix that when multiplied by $A$, yields the identity matrix $I$.
• $AA^{-1} = A^{-1}A = I$
• An invertible/nonsingular matrix has an inverse.
• A non-invertible/singular matrix has a determinant of zero.

Finding the Inverse

• 2x2 Matrix:
  • For a 2x2 matrix $A = \begin{bmatrix} a & b \ c & d \end{bmatrix}$, the inverse is: $$ A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} $$ where (ad - bc) is the determinant of A
• General Method
  • Use Gaussian elimination or other methods to find the inverse of larger matrices.

Properties of Inverse

• (A-1)-1 = A
• (AB)-1 = B-1A-1
• (cA)-1 = (1/c)A-1
• (AT)-1 = (A-1)T

Determinant of a Matrix

• A scalar value computed from the elements of a square matrix.
• Indicates whether the matrix is invertible.
• Denoted as det(A) or |A|.

Calculating the Determinant

• 2x2 Matrix:
  • For a 2x2 matrix $A = \begin{bmatrix} a & b \ c & d \end{bmatrix}$, the determinant is: $$ det(A) = ad - bc $$
• 3x3 Matrix:
  • For a 3x3 matrix, the determinant can be calculated using the rule of Sarrus or cofactor expansion. $$ A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} $$ $$ det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) $$
• Larger Matrices:
  • For larger matrices, cofactor expansion or row reduction methods are used.

Properties of Determinants

• det(AT) = det(A)
• det(AB) = det(A)det(B)
• If A has a row or column of all zeros, then det(A) = 0.
• If A has two identical rows or columns, then det(A) = 0.
• If A is a triangular matrix, then det(A) is the product of the diagonal entries.
• If A has two rows or columns interchanged, the determinant changes sign.
• If a multiple of one row is added to another, the determinant remains unchanged.
• If a row is multiplied by a scalar c, the determinant is multiplied by c.
• det(A-1) = 1/det(A)

Apache Kafka Quick Start Guide

Introduction

• Kafka is a distributed streaming platform enabling real-time data streaming between applications and microservices.
• Key capabilities include publishing/subscribing to record streams, storing streams in a fault-tolerant manner, and processing streams as they occur.

Use Cases for Kafka

• Website activity tracking for measuring, aggregating, and personalizing user activity.
• Metrics collection from distributed applications to produce centralized operational data feeds.
• Log aggregation as a centralized logging solution for serving logs from multiple machines and applications.
• Stream processing involving frameworks like Spark Streaming, Flink, Storm, and Kafka Streams for building real-time data processing applications.

Advantages of Kafka

• Designed as a high-performance distributed commit log system.
• Superior performance, scalability, and durability compared to traditional messaging systems.
• Combines messaging, storage, and stream-processing capabilities.

Key Concepts and Terminology

• Cluster: Kafka runs in a cluster on one or more servers or brokers.
• Topics: Logical categories for organizing records.
• Record: Each record consists of a key, a value, and a timestamp.
• Producers: Publish data to topics.
• Consumers: Subscribe to topics and process data.
• Partitions: Topics are divided into partitions, each being an ordered, immutable sequence of records.
• Offset: A sequential ID identifying records within a partition.
• Replication: Partition replication ensures durability and fault tolerance.
• ZooKeeper: Apache ZooKeeper manages the Kafka cluster.
• Consumer Groups: Consumers organized into groups, each reading from exclusive partitions within a topic.

Getting Started

Required environment

• Java 8+
• Download the latest version of Kafka from the Apache Kafka's web site

Starting the Kafka Environment

• Open separate consoles to run these commands:
  • Start ZooKeeper: bin/zookeeper-server-start.sh config/zookeeper.properties
  • Start Kafka server: bin/kafka-server-start.sh config/server.properties

Creating a Topic

• bin/kafka-topics.sh --create --topic quickstart-events --bootstrap-server localhost:9092
• Describes the topic: bin/kafka-topics.sh --describe --topic quickstart-events --bootstrap-server localhost:9092

Sending Messages

• Use the console producer to write messages to the topic:
  • bin/kafka-console-producer.sh --topic quickstart-events --bootstrap-server localhost:9092

Consuming Messages

• Use the console consumer to read messages:
  • bin/kafka-console-consumer.sh --topic quickstart-events --from-beginning --bootstrap-server localhost:9092

Processing Data with Kafka Streams

• Lightweight client library for building robust streaming applications and microservices, without needing a separate stream-processing cluster.
• To run the wordcount app execute command:
  • bin/kafka-streams-application.sh config/kafka-streams.properties
• To read from the output topic, one should execute:
  • bin/kafka-console-consumer.sh --topic streams-wordcount-output --from-beginning --bootstrap-server localhost:9092 \ --property print.key=true \ --property value.deserializer=org.apache.kafka.common.serialization.LongDeserializer
• To feed the input topic, one should execute
  • bin/kafka-console-producer.sh --topic streams-plaintext-input --bootstrap-server localhost:9092

Connecting Kafka with Kafka Connect

• Tool for reliably streaming data between Kafka and other systems.
• To start the console source connector, one should execute:
  • bin/connect-standalone.sh config/connect-standalone.properties config/connect-file-source.properties
• To start the console sink connector, one should execute:
  • bin/connect-standalone.sh config/connect-standalone.properties config/connect-file-sink.properties

Stopping the Kafka Environment

• Stop the console clients using Ctrl-C.
• Stop the Kafka server by pressing Ctrl-C in the server console.
• Stop the ZooKeeper server by pressing Ctrl-C in the server console.

Chapter 14: Transport in Plants

Means of Transport

Diffusion: Movement by Concentration Gradient

• Slow process.
• Not important for long distance transport.

Facilitated Diffusion

• By proteins (channels, carriers).
• No energy required.

Active Transport

• Uses energy.
• By membrane proteins (pumps).
• Accumulation against concentration gradient.

Plant-water relationship

• Water Potential $\Psi = \Psi_s + \Psi_p$
  • $\Psi_w$ = Water potential
  • $\Psi_s$ = Solute potential
  • $\Psi_p$ = Pressure potential

Plant-water relationship - Related Definitions

• Osmosis: diffusion of water.
• Plasmolysis: Contraction of the protoplast of a plant cell as a result of loss of water from the cell.
• Imbibition: Absorption of water.

Long Distance Transport

Mass Flow

• Bulk movement of substances due to pressure difference.
• Due to:
  • Positive hydrostatic pressure gradient (e.g., garden hose).
  • Negative hydrostatic pressure gradient (suction)

Xylem

• Transports water and minerals.
• Unidirectional (roots to stem)

Phloem

• Transports food (sugars).
• Multidirectional (source to sink).

Definition of matrices

• An $m \times n$ matrix is an arrangement of $m \cdot n$ elements into $m$ horizontal rows and $n$ vertical columns.
• $A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \ a_{21} & a_{22} & \cdots & a_{2n} \ \vdots & \vdots & \ddots & \vdots \ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$
• Notation:
  • $A = (a_{ij})$: Element of matrix $A$ at position $(i, j)$.
  • $A_{m \times n}$: The size of matrix $A$ is $m \times n$.

Example of matrices

• $A = \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \end{bmatrix}$ is a $2 \times 3$ matrix where $a_{12} = 2$ and $a_{23} = 6$.

Squared Matrices

• Matrix $A$ is a square matrix if the number of rows is equal to the number of columns. If $A$ is a square matrix of size $n \times n$, it is an n-th order square matrix.

Example of squared matrices

• $A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$ is a 2nd-order square matrix.

Diagonal Elements

• In a square matrix $A$, the diagonal elements are $a_{11}, a_{22}, \ldots, a_{nn}$.

Diagonal Matrix

• A square matrix $A$ is a diagonal matrix if all its elements except the diagonal elements are zeros.

Example of diagonal matrices

• $A = \begin{bmatrix} 1 & 0 & 0 \ 0 & 2 & 0 \ 0 & 0 & 3 \end{bmatrix}$ is a diagonal matrix.

Triangular Matrix

• A square matrix $A$ is a triangular matrix if all its elements above or below the diagonal are zeros. If the elements above the diagonal are zeros, then $A$ is a lower triangular matrix. Similarly, if the elements below the diagonal are zeros, then $A$ is an upper triangular matrix.

Example of a triangular matrix

• $A = \begin{bmatrix} 1 & 0 & 0 \ 2 & 3 & 0 \ 4 & 5 & 6 \end{bmatrix}$ is a lower triangular matrix.
• $(AB)C = A(BC)$
• $B = \begin{bmatrix} 1 & 2 & 3 \ 0 & 4 & 5 \ 0 & 0 & 6 \end{bmatrix}$ is an upper triangular matrix.

Zero Matrices

• A matrix $A$ whose every element is zero is a zero matrix, denoted by $O$.

The sum and scalar multiplication of matrices

Definition

• Let $A = (a_{ij})$ and $B = (b_{ij})$ be two $m \times n$ matrices. The sum of matrices $A$ and $B$ is an $m \times n$ matrix $C = (c_{ij})$, where $c_{ij} = a_{ij} + b_{ij}$ for all $i$ and $j$.

Example for the sum of matrices

• $\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} + \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \ 10 & 12 \end{bmatrix}$

Multiplication of Matrices

• Let $A = (a_{ij})$ be an $m \times n$ matrix and $c$ a real number. The scalar multiplication of matrix $A$ is an $m \times n$ matrix $C = (c_{ij})$, where $c_{ij} = c \cdot a_{ij}$ for all $i$ and $j$.

Theorem

• Let $A$ and $B$ be two $m \times n$ matrices, and $c$ and $d$ two real numbers. Then:
  • $A + B = B + A$
  • $(A + B) + C = A + (B + C)$
  • $c(A + B) = cA + cB$
  • $(c + d)A = cA + dA$

Scalar Multiplication

• Let $A = (a_{ij})$ be an $m \times n$ matrix and $B = (b_{ij})$ an $n \times p$ matrix. The product of matrices $A$ and $B$ is an $m \times p$ matrix $C = (c_{ij})$, where $c_{ij} = a_{i1}b_{1j} + a_{i2}b_{2j} + \cdots + a_{in}b_{nj}$ for all $i$ and $j$.

Matrix Transpose

• The transpose of a matrix $A$ is the matrix $A^T$ obtained by swapping the rows and columns of matrix $A$. If $A$ is an $m \times n$ matrix, then $A^T$ is an $n \times m$ matrix.

Derivation Rules

Basic Derivation Rules

• Constant Rule: If $f(x) = c$, where $c$ is a constant, then $f'(x) = 0$.
• Power Rule: If $f(x) = x^n$, where $n$ is any real number, then $f'(x) = nx^{n-1}$.
• Sum/Subtraction Rule: If $h(x) = f(x) \pm g(x)$, then $h'(x) = f'(x) \pm g'(x)$.
• Product Rule: If $h(x) = f(x)g(x)$, then $h'(x) = f'(x)g(x) + f(x)g'(x)$.
• Quotient Rule: If $h(x) = \frac{f(x)}{g(x)}$, then $h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$.
• Chain Rule: If $h(x) = f(g(x))$, then $h'(x) = f'(g(x)) \cdot g'(x)$.

Transcendental Functions

• Sine: If $f(x) = \sin(x)$, then $f'(x) = \cos(x)$.
• Cosine: If $f(x) = \cos(x)$, then $f'(x) = -\sin(x)$.
• Tangent: If $f(x) = \tan(x)$, then $f'(x) = \sec^2(x)$.
• Exponential: If $f(x) = e^x$, then $f'(x) = e^x$.
• Natural Logarithm: If $f(x) = \ln(x)$, then $f'(x) = \frac{1}{x}$.

Examples

• Derive $f(x) = 3x^2 + 2x - 1$
  • $f'(x) = 6x + 2$
• Derive $f(x) = \sin(x) \cos(x)$
  • $f'(x) = \cos^2(x) - \sin^2(x)$
• Derive $f(x) = e^{x^2}$
  • $f'(x) = 2xe^{x^2}$

Static

Objectives

• Experimentally determine the static equilibrium conditions for concurrent forces.
• Check the accuracy of the vector decomposition of forces.

Theoretical Foundations

• Statics is the branch of mechanics that analyzes bodies at rest.
• A body is in static equilibrium when the sum of all forces acting on it equals zero (translational equilibrium) and the sum of all torques equals zero (rotational equilibrium).
• Focus on the translational equilibrium of concurrent forces. The resultant of all forces acting on the body should be zero.

Equilibrium Conditions

• $\sum \overrightarrow{F_i} = 0$
• In two dimensions (x, y), this translates to:
  • $\sum F_{ix} = 0$
  • $\sum F_{iy} = 0$

Calculations for Forces

Calculate x and y components of each force:

• $F_x = F \cos(\theta)$
• $F_y = F \sin(\theta)$

Diagram

• A free body diagram is a graphical representation used to visualize the applied forces, moments, and resulting reactions on a body.

Greedy algorithms

General principles

• Greedy algorithms are a method for solving optimization problems.
• They consist of making, step by step, the choice that seems the best at the moment, in the hope of obtaining a global optimal solution.
  • Make a choice.
  • Solve the resulting subproblem.
    • Decisions (a greedy choice).
    • Problem transformations.

Remarks for the use of greedy algorithms

• Algorithms never go back on theirs choices. Not the backtracking.
• It does not compute all possible solutions.
• It does not always provide the optimal solution.
• It is often easy to implement and not expensive in terms of resources.