This document provides an introduction to networks and matrices. It defines key network terminology such as nodes, edges, degrees, paths, circuits, trees and graphs. It discusses using networks to model real-world systems and solve problems. Examples of network problems presented include the shortest path problem, shortest connection problem, Chinese postman problem and travelling salesman problem. The historical origins and applications of network theory are also summarized.
The document defines different types of angles and how to measure them using a protractor. It explains that angles can be measured in degrees from 0° to 360° and defines right angles as being 90°, acute angles as less than 90°, and obtuse angles as greater than 90° but less than 180°. A straight angle is 180°. Examples are given measuring various angles and identifying their type.
The document provides a lesson plan for teaching algebraic expressions and identities to 8th grade students. It outlines objectives to help students understand identities in algebraic expressions, the relationship between algebra, geometry and arithmetic, and how to apply identities to solve problems. Example activities are presented to show representing algebraic expressions geometrically and applying identities to evaluate expressions and arithmetic problems. Key identities introduced are (a + b)2, (a - b)2, and (a + b)(a - b). Students are given practice problems to solve using the identities.
Trigonometry refers to measuring the sides of a triangle. The word comes from the Greek words 'trigon' meaning triangle and 'metron' meaning measurement. Radian is a unit used to measure angles, where one radian is equal to the central angle that cuts off an arc equal in length to the radius of a circle.
- A particle starts from the point with position vector (3i + 7j) m and then moves with constant velocity (2i – j) ms-1. The question asks to find the position vector of the particle 4 seconds later.
- Substituting the values into the displacement equation gives the final position vector as (12i + 3j) m.
- A second particle is given a position vector of (2i + 4j) m at time t = 0 and a position vector of (12i + 16j) m five seconds later. Using the displacement equation gives the velocity of the particle as (2i + 4j) ms-1.
- For a third particle
Applications of trigonometry in real lifeChloe Cheney
Trigonometry finds its applications in various important fields that make our life easier. We have prepared a list of Applications of Trigonometry in Life.
Vectors have both magnitude and direction, while scalars only have magnitude; displacement is a vector that describes the change in position between an initial and final location. The document covers vector addition and subtraction using diagrams, as well as multiplying, dividing, and finding the x- and y-components of vectors.
This document discusses three topics related to triangles: the Basic Proportionality Theorem, areas of similar triangles, and trigonometry. It explains the Basic Proportionality Theorem and its proof. It then discusses that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Finally, it defines trigonometry as the measurement of triangles and describes the ratios of the sides of a right triangle and some real-life applications of trigonometry such as periodic functions and architectural design.
basic & secondary parts of triangles.pptJoeyDeLuna1
This document defines various geometric terms and concepts related to triangles. It defines angles, perpendicular lines, degrees, triangles, and their basic and secondary parts such as sides, vertices, angles, angle bisectors, altitudes, and medians. It also discusses how triangles can be classified based on the number of congruent sides as scalene, isosceles, or equilateral triangles, and based on angle measures as acute, right, obtuse, or equiangular triangles. Some examples and properties of triangles are provided, followed by true/false questions to test understanding. The next topic to be covered is quadrilaterals.
The document defines different types of angles and how to measure them using a protractor. It explains that angles can be measured in degrees from 0° to 360° and defines right angles as being 90°, acute angles as less than 90°, and obtuse angles as greater than 90° but less than 180°. A straight angle is 180°. Examples are given measuring various angles and identifying their type.
The document provides a lesson plan for teaching algebraic expressions and identities to 8th grade students. It outlines objectives to help students understand identities in algebraic expressions, the relationship between algebra, geometry and arithmetic, and how to apply identities to solve problems. Example activities are presented to show representing algebraic expressions geometrically and applying identities to evaluate expressions and arithmetic problems. Key identities introduced are (a + b)2, (a - b)2, and (a + b)(a - b). Students are given practice problems to solve using the identities.
Trigonometry refers to measuring the sides of a triangle. The word comes from the Greek words 'trigon' meaning triangle and 'metron' meaning measurement. Radian is a unit used to measure angles, where one radian is equal to the central angle that cuts off an arc equal in length to the radius of a circle.
- A particle starts from the point with position vector (3i + 7j) m and then moves with constant velocity (2i – j) ms-1. The question asks to find the position vector of the particle 4 seconds later.
- Substituting the values into the displacement equation gives the final position vector as (12i + 3j) m.
- A second particle is given a position vector of (2i + 4j) m at time t = 0 and a position vector of (12i + 16j) m five seconds later. Using the displacement equation gives the velocity of the particle as (2i + 4j) ms-1.
- For a third particle
Applications of trigonometry in real lifeChloe Cheney
Trigonometry finds its applications in various important fields that make our life easier. We have prepared a list of Applications of Trigonometry in Life.
Vectors have both magnitude and direction, while scalars only have magnitude; displacement is a vector that describes the change in position between an initial and final location. The document covers vector addition and subtraction using diagrams, as well as multiplying, dividing, and finding the x- and y-components of vectors.
This document discusses three topics related to triangles: the Basic Proportionality Theorem, areas of similar triangles, and trigonometry. It explains the Basic Proportionality Theorem and its proof. It then discusses that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Finally, it defines trigonometry as the measurement of triangles and describes the ratios of the sides of a right triangle and some real-life applications of trigonometry such as periodic functions and architectural design.
basic & secondary parts of triangles.pptJoeyDeLuna1
This document defines various geometric terms and concepts related to triangles. It defines angles, perpendicular lines, degrees, triangles, and their basic and secondary parts such as sides, vertices, angles, angle bisectors, altitudes, and medians. It also discusses how triangles can be classified based on the number of congruent sides as scalene, isosceles, or equilateral triangles, and based on angle measures as acute, right, obtuse, or equiangular triangles. Some examples and properties of triangles are provided, followed by true/false questions to test understanding. The next topic to be covered is quadrilaterals.
Let f(x) =
and g(x) =
1 x
x
that fog is not defined.
19.
If f : A B is bijective, then write n(A) in terms of n(B).
20.
If f : A B is onto and n(A) = 5, n(B) = 3. Then write n(A).
SHORT ANSWER TYPE QUESTIONS (2 MARKS)
1.
Define reflexive, symmetric
The document discusses different types of number systems including natural numbers, whole numbers, rational numbers, integers, and irrational numbers. It provides definitions and examples of each type of number system. Natural numbers are used for counting and include zero depending on the system. Whole numbers include all natural numbers and zero. Rational numbers include integers and fractions. Integers include both positive and negative whole numbers. Irrational numbers have decimal values that continue forever without repeating patterns, such as pi.
1) Romberg integration is a numerical method for approximating definite integrals based on Richardson extrapolation of the trapezoidal rule. It provides better approximations than the trapezoidal rule by reducing the true error through recursive calculations.
2) The derivation of Romberg integration involves applying Richardson's extrapolation to the error estimation of the trapezoidal rule. This allows computing a more accurate integral using the results from two less accurate integrals.
3) An example application calculates the volume of water in a tank using Romberg integration, Composite Simpson's rule, and Gaussian quadrature. Romberg integration provided the most accurate result with less computation time compared to the other methods.
This document provides an overview of trigonometry presented by Vijay. It begins by listing the materials needed and encouraging note taking. The presentation then defines trigonometric ratios like sine, cosine and tangent using a right triangle. It also covers trigonometric ratios of specific angles like 45 and 30 degrees as well as complementary angles. The document concludes by explaining several trigonometric identities and providing a short summary of key points.
Trigonometry is a branch of mathematics that studies triangles and their relationships. The document discusses how trigonometry is used in the fields of architecture, astronomy, and geology. In architecture, trigonometry is used to calculate angles to ensure structural stability and safety. Astronomers use trigonometry and concepts like parallax to calculate distances between stars. Geologists use trigonometry to estimate true dip angles of bedding to determine slope stability important for building foundations.
The document discusses the history and importance of numerical algorithms. It began with early papers in the 1940s and projects like ENIAC for calculating trajectories. Over time, important people and projects advanced areas like linear algebra, differential equations, and approximation methods. Today, numerical algorithms are widely used in applications such as digital imaging, engineering, finance, and more. Understanding numerical methods is crucial for teaching mathematics with technology, as it ensures students have proper conceptual knowledge to analyze problems.
We cover the inverses to the trigonometric functions sine, cosine, tangent, cotangent, secant, cosecant, and their derivatives. The remarkable fact is that although these functions and their inverses are transcendental (complicated) functions, the derivatives are algebraic functions. Also, we meet my all-time favorite function: arctan.
This document introduces different number systems, including natural numbers, whole numbers, integers, rational numbers, and irrational numbers. It defines each type of number and provides examples. Rational numbers can be written as fractions using integers, while irrational numbers cannot be expressed as ratios of integers. Together, rational and irrational numbers make up the set of real numbers, which correspond one-to-one with points on the real number line. Famous mathematicians like Pythagoras, Cantor, and Dedekind contributed to the understanding and classification of these different number systems.
This document is a textbook on elementary linear algebra. Chapter 1 introduces linear equations in n unknowns and systems of m linear equations in n unknowns. It discusses solving linear equations and systems for solutions. Geometrically, solving systems of linear equations in two or three unknowns corresponds to finding the common point of intersection of lines or planes. Examples are provided to illustrate solving single equations, systems of equations, and finding polynomials that satisfy given points.
Trigonometry is the branch of mathematics dealing with relationships between sides and angles of triangles. The document traces the origins of common trigonometric functions like sine, cosine, and tangent from ancient Indian and Greek mathematicians. It then explains how trigonometric ratios are used to calculate the angle of elevation of a room by measuring the wall length and diagonal distance and applying the sine ratio formula.
Methods of variation of parameters- advance engineering mathe mathematicsKaushal Patel
The method of variation of parameters can be used to find the particular integral of a second order linear differential equation with constant coefficients. This method involves:
1) Finding the complementary function which is the general solution to the associated homogeneous equation.
2) Assuming the particular integral is of the form u times the first term in the complementary function plus v times the second term, where u and v are functions of x.
3) Differentiating this and using the differential equation to determine expressions for u' and v' in terms of the complementary functions and the function being integrated (X).
4) Integrating u' and v' to find u and v and the particular integral.
This lesson plan is for a 9th grade mathematics class on lines and angles. It aims to teach students about angle sum property of triangles. The lesson will review lines, rays, segments and angles, as well as adjacent, complementary, and supplementary angles. Students will learn about angles formed by a transversal cutting parallel lines. An activity involves cutting angles from a paper triangle and arranging them to show the sum is 180 degrees. The lesson aims to help students describe geometric terms, explain the linear pair axiom, and solve problems involving angle types and relationships in triangles. A worksheet assesses students' ability to calculate unknown angles using the angle sum property theorem.
Trigonometry is the branch of mathematics dealing with triangles and trigonometric functions of angles. It is derived from Greek words meaning "three angles" and "measure". Trigonometry specifically studies relationships between sides and angles of triangles, and calculations based on trigonometric functions like sine, cosine, and tangent. Trigonometry has many applications in fields like astronomy, navigation, architecture, engineering, and more.
This document provides an overview of triangles, including definitions, types, properties, secondary parts, congruency, and area calculations. It defines a triangle as a 3-sided polygon with three angles and vertices. Triangles are classified by side lengths as equilateral, isosceles, or scalene, and by angle measures as acute, obtuse, or right. Key properties discussed include the angle sum theorem, exterior angle theorem, and Pythagorean theorem. Secondary parts like medians, altitudes, perpendicular bisectors, and angle bisectors are also defined. Tests for triangle congruency such as SSS, SAS, ASA, and RHS are outlined. Formulas are provided for calculating the areas of
Production Engineering - Laplace TransformationEkeedaPvtLtd
Production Engineering is a specialization of Mechanical Engineering. This Engineering focuses mainly on Materials Science, Machine Tools, and Quality Control. Professional Production Engineer design, develop, implement, operate, and manage manufacturing systems. Production Engineering combines the knowledge of management science with manufacturing tech. As a Production Engineer, you are given deeper insight into the various sectors on how to produce and resolved the shortcoming with the goal of providing your customers with satisfactory service in a budget production. The manufactured products range from turbines, engines and pumps, airplanes, robotic equipment, and integrated circuits.
Trigonometry is a branch of mathematics used to define relationships between sides and angles of triangles, especially right triangles. It has applications in fields like architecture, astronomy, geology, navigation, and oceanography. Trigonometric functions like sine, cosine, and tangent are ratios that relate sides and angles, and trigonometry allows distances, heights, and depths to be easily calculated. Architects use trigonometry to design buildings, astronomers use it to measure distances to stars, and geologists use it to determine slope stability.
Trigonometry is used to calculate unknown heights, distances, and angles using relationships between sides and angles of triangles. It was developed by ancient Greek mathematicians like Thales and Hipparchus to solve problems in astronomy and geography. Some key applications include using trigonometric ratios like tangent and cotangent along with known distances and angles of elevation/depression to determine the height of objects like towers, buildings, and mountains when direct measurement is not possible. The document provides historical context and examples to illustrate how trigonometric concepts have been applied to problems involving finding heights, distances, and other unknown measurements through the use of triangles and their properties.
1. The document is an introduction to statistical machine learning by Christfried Webers from NICTA and The Australian National University in 2011.
2. It covers basic concepts in linear algebra that are important for statistical machine learning such as linear transformations, matrices, vectors, and matrix-vector multiplication.
3. The document provides code examples and visual explanations of concepts like how a matrix A multiplies a vector V to produce a result vector R.
The document discusses the TRAPATT diode, which is a type of microwave generator diode. It operates by forming a trapped space charge plasma within the p-n junction region when a high electric field propagates through and depresses the field. This allows the diode to switch high currents with nanosecond transition times. Typical parameters include a power of 1.2 kW at 1.2 GHz and maximum efficiency of 75% at 0.6 GHz, within an operating frequency range of 0.5-50 GHz. It has advantages for pulsed operation but a high noise figure. Applications include low power Doppler radars, radios, and radar transmitters.
This document provides information about two-port network parameters including Z, Y, H, and ABCD parameters. It defines a two-port network as having two ports for input and output with two terminals pairs. The document explains that the parameters relate the terminal voltages and currents and can be determined by setting the input or output port to open or short circuit conditions. Examples are given to show how to calculate the parameters for simple circuits. Key points are summarized in less than 3 sentences.
Let f(x) =
and g(x) =
1 x
x
that fog is not defined.
19.
If f : A B is bijective, then write n(A) in terms of n(B).
20.
If f : A B is onto and n(A) = 5, n(B) = 3. Then write n(A).
SHORT ANSWER TYPE QUESTIONS (2 MARKS)
1.
Define reflexive, symmetric
The document discusses different types of number systems including natural numbers, whole numbers, rational numbers, integers, and irrational numbers. It provides definitions and examples of each type of number system. Natural numbers are used for counting and include zero depending on the system. Whole numbers include all natural numbers and zero. Rational numbers include integers and fractions. Integers include both positive and negative whole numbers. Irrational numbers have decimal values that continue forever without repeating patterns, such as pi.
1) Romberg integration is a numerical method for approximating definite integrals based on Richardson extrapolation of the trapezoidal rule. It provides better approximations than the trapezoidal rule by reducing the true error through recursive calculations.
2) The derivation of Romberg integration involves applying Richardson's extrapolation to the error estimation of the trapezoidal rule. This allows computing a more accurate integral using the results from two less accurate integrals.
3) An example application calculates the volume of water in a tank using Romberg integration, Composite Simpson's rule, and Gaussian quadrature. Romberg integration provided the most accurate result with less computation time compared to the other methods.
This document provides an overview of trigonometry presented by Vijay. It begins by listing the materials needed and encouraging note taking. The presentation then defines trigonometric ratios like sine, cosine and tangent using a right triangle. It also covers trigonometric ratios of specific angles like 45 and 30 degrees as well as complementary angles. The document concludes by explaining several trigonometric identities and providing a short summary of key points.
Trigonometry is a branch of mathematics that studies triangles and their relationships. The document discusses how trigonometry is used in the fields of architecture, astronomy, and geology. In architecture, trigonometry is used to calculate angles to ensure structural stability and safety. Astronomers use trigonometry and concepts like parallax to calculate distances between stars. Geologists use trigonometry to estimate true dip angles of bedding to determine slope stability important for building foundations.
The document discusses the history and importance of numerical algorithms. It began with early papers in the 1940s and projects like ENIAC for calculating trajectories. Over time, important people and projects advanced areas like linear algebra, differential equations, and approximation methods. Today, numerical algorithms are widely used in applications such as digital imaging, engineering, finance, and more. Understanding numerical methods is crucial for teaching mathematics with technology, as it ensures students have proper conceptual knowledge to analyze problems.
We cover the inverses to the trigonometric functions sine, cosine, tangent, cotangent, secant, cosecant, and their derivatives. The remarkable fact is that although these functions and their inverses are transcendental (complicated) functions, the derivatives are algebraic functions. Also, we meet my all-time favorite function: arctan.
This document introduces different number systems, including natural numbers, whole numbers, integers, rational numbers, and irrational numbers. It defines each type of number and provides examples. Rational numbers can be written as fractions using integers, while irrational numbers cannot be expressed as ratios of integers. Together, rational and irrational numbers make up the set of real numbers, which correspond one-to-one with points on the real number line. Famous mathematicians like Pythagoras, Cantor, and Dedekind contributed to the understanding and classification of these different number systems.
This document is a textbook on elementary linear algebra. Chapter 1 introduces linear equations in n unknowns and systems of m linear equations in n unknowns. It discusses solving linear equations and systems for solutions. Geometrically, solving systems of linear equations in two or three unknowns corresponds to finding the common point of intersection of lines or planes. Examples are provided to illustrate solving single equations, systems of equations, and finding polynomials that satisfy given points.
Trigonometry is the branch of mathematics dealing with relationships between sides and angles of triangles. The document traces the origins of common trigonometric functions like sine, cosine, and tangent from ancient Indian and Greek mathematicians. It then explains how trigonometric ratios are used to calculate the angle of elevation of a room by measuring the wall length and diagonal distance and applying the sine ratio formula.
Methods of variation of parameters- advance engineering mathe mathematicsKaushal Patel
The method of variation of parameters can be used to find the particular integral of a second order linear differential equation with constant coefficients. This method involves:
1) Finding the complementary function which is the general solution to the associated homogeneous equation.
2) Assuming the particular integral is of the form u times the first term in the complementary function plus v times the second term, where u and v are functions of x.
3) Differentiating this and using the differential equation to determine expressions for u' and v' in terms of the complementary functions and the function being integrated (X).
4) Integrating u' and v' to find u and v and the particular integral.
This lesson plan is for a 9th grade mathematics class on lines and angles. It aims to teach students about angle sum property of triangles. The lesson will review lines, rays, segments and angles, as well as adjacent, complementary, and supplementary angles. Students will learn about angles formed by a transversal cutting parallel lines. An activity involves cutting angles from a paper triangle and arranging them to show the sum is 180 degrees. The lesson aims to help students describe geometric terms, explain the linear pair axiom, and solve problems involving angle types and relationships in triangles. A worksheet assesses students' ability to calculate unknown angles using the angle sum property theorem.
Trigonometry is the branch of mathematics dealing with triangles and trigonometric functions of angles. It is derived from Greek words meaning "three angles" and "measure". Trigonometry specifically studies relationships between sides and angles of triangles, and calculations based on trigonometric functions like sine, cosine, and tangent. Trigonometry has many applications in fields like astronomy, navigation, architecture, engineering, and more.
This document provides an overview of triangles, including definitions, types, properties, secondary parts, congruency, and area calculations. It defines a triangle as a 3-sided polygon with three angles and vertices. Triangles are classified by side lengths as equilateral, isosceles, or scalene, and by angle measures as acute, obtuse, or right. Key properties discussed include the angle sum theorem, exterior angle theorem, and Pythagorean theorem. Secondary parts like medians, altitudes, perpendicular bisectors, and angle bisectors are also defined. Tests for triangle congruency such as SSS, SAS, ASA, and RHS are outlined. Formulas are provided for calculating the areas of
Production Engineering - Laplace TransformationEkeedaPvtLtd
Production Engineering is a specialization of Mechanical Engineering. This Engineering focuses mainly on Materials Science, Machine Tools, and Quality Control. Professional Production Engineer design, develop, implement, operate, and manage manufacturing systems. Production Engineering combines the knowledge of management science with manufacturing tech. As a Production Engineer, you are given deeper insight into the various sectors on how to produce and resolved the shortcoming with the goal of providing your customers with satisfactory service in a budget production. The manufactured products range from turbines, engines and pumps, airplanes, robotic equipment, and integrated circuits.
Trigonometry is a branch of mathematics used to define relationships between sides and angles of triangles, especially right triangles. It has applications in fields like architecture, astronomy, geology, navigation, and oceanography. Trigonometric functions like sine, cosine, and tangent are ratios that relate sides and angles, and trigonometry allows distances, heights, and depths to be easily calculated. Architects use trigonometry to design buildings, astronomers use it to measure distances to stars, and geologists use it to determine slope stability.
Trigonometry is used to calculate unknown heights, distances, and angles using relationships between sides and angles of triangles. It was developed by ancient Greek mathematicians like Thales and Hipparchus to solve problems in astronomy and geography. Some key applications include using trigonometric ratios like tangent and cotangent along with known distances and angles of elevation/depression to determine the height of objects like towers, buildings, and mountains when direct measurement is not possible. The document provides historical context and examples to illustrate how trigonometric concepts have been applied to problems involving finding heights, distances, and other unknown measurements through the use of triangles and their properties.
1. The document is an introduction to statistical machine learning by Christfried Webers from NICTA and The Australian National University in 2011.
2. It covers basic concepts in linear algebra that are important for statistical machine learning such as linear transformations, matrices, vectors, and matrix-vector multiplication.
3. The document provides code examples and visual explanations of concepts like how a matrix A multiplies a vector V to produce a result vector R.
The document discusses the TRAPATT diode, which is a type of microwave generator diode. It operates by forming a trapped space charge plasma within the p-n junction region when a high electric field propagates through and depresses the field. This allows the diode to switch high currents with nanosecond transition times. Typical parameters include a power of 1.2 kW at 1.2 GHz and maximum efficiency of 75% at 0.6 GHz, within an operating frequency range of 0.5-50 GHz. It has advantages for pulsed operation but a high noise figure. Applications include low power Doppler radars, radios, and radar transmitters.
This document provides information about two-port network parameters including Z, Y, H, and ABCD parameters. It defines a two-port network as having two ports for input and output with two terminals pairs. The document explains that the parameters relate the terminal voltages and currents and can be determined by setting the input or output port to open or short circuit conditions. Examples are given to show how to calculate the parameters for simple circuits. Key points are summarized in less than 3 sentences.
This document discusses different parameter representations for two-port networks:
- Z-parameters (open-circuit impedance parameters) relate terminal voltages to currents.
- Y-parameters (short-circuit admittance parameters) relate terminal currents to voltages.
- T-equivalent and Pi-equivalent models provide lumped-element representations of reciprocal networks.
- h-parameters and g-parameters describe the terminal characteristics of some electronic devices.
- Transmission (ABCD) parameters express source variables in terms of destination variables.
Conversion formulas are provided between the different parameter sets. Examples are included to illustrate determining the parameters for simple circuits.
This document provides an introduction to two-port networks. It defines the standard configuration of a two-port network with two ports, an input and an output. It then describes various two-port parameter sets - including Z, Y, transmission, and hybrid parameters - and provides examples of calculating the parameters for simple resistive networks. It also demonstrates how the parameters change when elements are added outside the two ports of the original network.
Orthogonal frequency-division multiplexing (OFDM) is a digital multi-carrier modulation technique that partitions the available bandwidth into multiple orthogonal sub-carriers. Each sub-carrier is modulated with a conventional modulation scheme at a low symbol rate and maintains similar data rates as single-carrier modulation in the same bandwidth. OFDM provides advantages like easy adaptation to severe channel conditions, robustness against interference and fading, and high spectral efficiency through FFT implementation. Some disadvantages include sensitivity to Doppler shift and frequency synchronization errors. OFDM has been used in various wireless and wired communication systems.
The TRAPATT diode is a microwave semiconductor device that operates as both an amplifier and oscillator below 10 GHz. It consists of an N-type semiconductor with a depletion region width of 2.5-12.5 micrometers and a P+ region of 2.5-7.5 micrometers in diameter. When a high voltage is applied, an avalanche zone propagates through the diode, trapping electrons and holes in the depletion region and allowing the diode to efficiently deliver RF power to an external load. The TRAPATT diode has advantages of high efficiency between 15-40% and low power dissipation, making it suitable for pulsed applications like low power Doppler radars between 3-50 GHz.
This document discusses microwave junctions and S-parameters. It provides information on:
1) Power dividers and directional couplers which are passive microwave components used for power division or combining. S-parameters are used to define the power relationships between ports.
2) The scattering matrix (S-matrix) is a matrix that defines the power relationships between ports in terms of incident and reflected voltage waves. It is commonly used for microwave analysis since direct voltage and current measurements are difficult at high frequencies.
3) Examples are provided to demonstrate calculating S-matrix coefficients for different microwave junction configurations like E-plane and H-plane tee junctions. Properties of reciprocal and lossless networks in relation to the S
This document discusses Y or admittance parameters, which describe the behavior of linear two-port electrical systems using equations that relate currents and voltages. The Y parameters - Y11, Y12, Y21, and Y22 - are short circuit admittances that can be used to find the currents at each port if the voltages are known and one port is short circuited. Y parameters are useful for analyzing two-port networks and assessing the small signal stability of electrical systems.
Microwaves are electromagnetic waves with frequencies between 1-30 GHz. They have wavelengths between 1-30 cm. Microwaves allow for greater bandwidth and transmission of high-speed information. While they require specialized components, microwaves are used widely in modern communication systems. Transmission lines like waveguides and coaxial cables precisely guide microwave signals. Advanced transistors and integrated circuits enable microwave signal amplification.
1) IMPATT (impact ionization avalanche transit time) diodes rely on carrier impact ionization and drift in the high electric field region of a semiconductor junction to produce negative resistance at microwave frequencies.
2) An IMPATT diode consists of an avalanche region where impact ionization occurs, generating electron-hole pairs, and a drift region where holes drift toward the contact, introducing a transit time delay.
3) When reverse biased above the breakdown voltage, the electric field causes impact ionization, producing a pulsed carrier current that lags the external current by 90 degrees, resulting in negative conductance that allows the diode to oscillate at microwave frequencies.
This document discusses the development of a new type of battery that could revolutionize energy storage. It describes how the battery uses a solid electrolyte material that conducts ions quickly without using liquid electrolytes. This leads to a battery that charges faster, lasts longer, and poses less risk of fire. The solid-state design could enable electric vehicles to match gas vehicles in range and charging time. Mass production of this solid-state battery technology would help accelerate the transition to renewable energy on a global scale.
The document provides information on microwave engineering and transmission lines. It discusses topics such as the frequency range of microwave engineering (1 GHz and above), transmission line theory, characteristic impedance, propagation velocity, lossless transmission lines, and formulas for calculating the characteristic impedance of different transmission line structures including coaxial cable, striplines, and dual striplines.
The document discusses the TRAPATT diode, which is a type of p-n junction diode that generates microwaves. It operates by forming a trapped plasma within the junction region when a high electric field propagates through. Key points:
- It was first reported in 1967 and can generate power over 1 kW at frequencies up to 50 GHz with efficiencies up to 75%
- It operates by inducing avalanche breakdown to generate a dense plasma of electrons and holes within the depletion region, which becomes trapped and oscillates the voltage and current
- Applications include low power Doppler radars, radio altimeters, and radar transmitters due to its pulsed operation capabilities between 3-50 GHz
This document discusses various parameters used to characterize two-port networks, including Z, Y, S, and ABCD parameters. It explains that Z parameters relate voltages and currents using an impedance matrix, while Y parameters use an admittance matrix. S parameters describe the scattering of waves at the ports in terms of reflection and transmission coefficients. The ABCD matrix relates voltages and currents at the ports and allows for easy cascading of networks. Measurements are typically made using a vector network analyzer to determine the S-parameter scattering matrix.
1935 – Heil oscillator
1939 – klystron amplifier
1944 – Helix type TWT
In the early 1950s – low power output of linear beam tubes to high power levels
Finally invention of Magnetrons
Several devices were developed – two significant devices among them are
1) extended interaction klystron
2) Twystron hybrid amplifier
CYLINDRICAL
LINEAR
COAXIAL
VOLTAGE-TUNABLE
INVERTED COAXIAL
FREQUENCY-AGILE COAXIAL
The document provides an overview of the TCP/IP model, describing each layer from application to network. The application layer allows programs access to networked services and contains high-level protocols like TCP and UDP. The transport layer handles reliable delivery via protocols like TCP and UDP. The internet layer organizes routing with the IP protocol. The network layer consists of device drivers and network interface cards that communicate with the physical transmission media.
This document summarizes key concepts from Chapter 1 of an introduction to data communication systems textbook. It provides definitions for the five components of a data communication system, the three criteria for evaluating systems, and two types of line configurations. It also summarizes different network topologies including the number of cables required for each, defines an internet as an interconnection of networks, and notes that standards are important to ensure compatibility. Sample exercises are provided assessing different network failures and applications' sensitivity to delay.
This document contains answers to questions about data communication systems and network topologies. It defines the five components of a data communication system as the sender, receiver, transmission medium, message, and protocol. It also discusses the advantages of distributed processing and different network topologies like mesh, star, bus, and ring. It provides information on the number of cable links required for each topology and factors that determine if a communication system is a LAN or WAN.
This document summarizes previous work on comparing the network costs of different interconnection network topologies. It discusses mesh, hexagonal mesh, honeycomb mesh, torus, hexagonal torus, and honeycomb torus networks. Previous studies have calculated the degree, diameter, number of nodes, and network cost (defined as the product of degree and diameter) for each network as the number of nodes or network dimension increases. This paper aims to compare the network costs of the Spidergon and Honeycomb Torus networks based on these parameters.
This document provides an outline for a textbook on computer networks. It begins with an introduction to computer networks, including their uses in business and home applications. It then discusses network hardware, software, and reference models. Several example networks are described, including the Internet, ATM, Ethernet, and wireless LANs. It concludes with a section on network standardization bodies.
This document contains information about misleading graphs including:
1) An upcoming quiz, midterm, and placement test dates as well as a reminder to bring pencils.
2) Examples of misleading line graphs about temperature and spending that manipulate scales or leave out key information.
3) A student activity to analyze misleading graphs and create their own, trading with a partner to explain why each is misleading.
4) Homework assigned on analyzing scales to identify misleading graphs and indicators that a graph may be misleading.
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Networks and Matrices
1. 7
Networks and matrices
Contents: A Network diagrams
B Constructing networks
C Problem solving with networks
D Modelling with networks
E Interpreting information with matrices
F Further matrices
G Addition and subtraction of matrices
H Scalar multiplication
I Matrix multiplication
J Using technology for matrix operations
Review set 7
2. 358 NETWORKS AND MATRICES (Chapter 7) (T4)
Networks are used to show how things are connected. They can be used to help solve
problems in train scheduling, traffic flow, bed usage in hospitals, and project management.
The construction of networks belongs to the branch of mathematics known as Topology.
Various procedures known as algorithms are applied to the networks to find maximum or
minimum solutions. This further study of the constructed networks belongs to the branch of
mathematics known as Operations Research.
HISTORICAL NOTE
The theory of networks was developed centuries ago, but the application of the
theoretical ideas is relatively recent.¡
Real progress was made during and after the Second World War as
mathematicians, industrial technicians, and members of the armed services worked
together to improve military operations.¡
Since then the range of applications has extended,
from delivering mail to a neighbourhood to landing
man on the moon.¡
Businesses and governments are increasingly using
networks as a planning tool.¡ They usually wish to
find the optimum solution to a practical problem.¡ This
may be the solution which costs the least, uses the
least time, or requires the smallest amount of material.
OPENING PROBLEMS
There are several common types of networking problem that we will be
investigating in this chapter.¡ For some types there are well-known algorithms
that provide quick solutions.¡ A few problems, however, remain a great
challenge for 21st century mathematicians.
1 Shortest Path Problem
2 2
Dien and Phan are good friends who like to
visit each other.¡ The network alongside shows 1
3 3 1
the roads between their houses.¡ The times 2 Phan’s
taken to walk each road are shown in minutes.¡ Dien’s 2 house
house 3 1 3
a Assuming there is no back-tracking, how 3
many different paths are there from Dien’s 1 2
house to Phan’s house?¡ 3
1
b Which is the quickest route? 2
2 Shortest Connection Problem
A computer network relies on every computer being 5
connected to every other.¡ The connections do not have to be 4 6
direct, i.e., a connection via another computer is fine.¡ What 6
4 9 8
is the minimum length of cable required to connect all 5
computers to the network?¡ The solid line on the network 6
shows one possible solution, but not the shortest one. 8
3. NETWORKS AND MATRICES (Chapter 7) (T4) 359
3 Chinese Postman Problem P
2
2
A postman needs to walk down every street in his district 1
in order to deliver the mail. The numbers indicate the 1
distances along the roads in hundreds of metres. The 3
1 2
postman starts at the post office P, and returns there after 3
delivering the mail. What route should the postman take
4
to minimise the distance travelled?
A
4 Travelling Salesman Problem 11
7 B
A salesman leaves his home H in the morning, and over 12
the course of his day he needs to attend meetings in towns H 10 5
13
A, B, C and D. At the end of the day he returns home. In 8
9
which order should he schedule his meetings so that the
6 C
distance he must travel is minimised? D
In addition to the problems posed above, we will investigate how networks can be used to
model projects with many steps. We will see how networks are related to matrices and learn
how operations with matrices can be performed.
A NETWORK DIAGRAMS
A network diagram or finite graph is a structure or model where things of interest are
linked by physical connections or relationships.¡
The things of interest are represented by dots or circles and the connections or relationships
represented by connecting lines.¡
For example, in the Travelling Salesman Problem (Opening Problem question 4) the dots
represent the salesman’s house and the four towns.¡ The connecting lines represent the roads
between them.
TERMINOLOGY
² A graph or network is a set of points, some or all of which
are connected by a set of lines. node or
vertex
² The points are known as nodes or vertices (singular vertex).
² In a graph, we can move from node to node along the lines.
If we are allowed to move in either direction along the lines, adjacent nodes
the lines are called edges and the graph is undirected. Pairs
of nodes that are directly connected by edges are said to be
adjacent. edge
arc
If we are only allowed to move
in one direction along the lines,
the lines are called arcs.¡
arc
The graph is then known as a
digraph or directed graph.
4. 360 NETWORKS AND MATRICES (Chapter 7) (T4)
² An edge or arc may sometimes be assigned a number called 5
its weight. This may represent the cost, time, or distance 3
required to travel along that line. 2 3
Note that the lengths of the arcs and edges are not drawn
3 2
to scale, i.e., they are not in proportion to their weight. 4
² The degree or valence of a vertex is the A loop counts as one edge, but adds
two to the degree of the vertex.
number of edges or arcs which touch it.¡ So this vertex has degree 3
A loop starts and ends at the same vertex.¡
degree 3 isolated
It counts as one edge, but it contributes vertex,
two to the degree of a vertex.¡ degree 0
An isolated vertex has degree zero. degree 2 degree 2
² A connected graph is a graph in which it is
possible to travel from every vertex to every
other vertex by following edges. The graph
is otherwise said to be disconnected.
a connected graph a disconnected graph
² A complete graph is a graph in which every vertex is adjacent to every other vertex.
This graph is complete. This graph is not complete. The addition of the ‘dotted’ edge
would make it complete.
² A simple graph is a graph in which no vertex connects to itself, and every pair of vertices
is directly connected by at most one edge.
a simple graph non-simple graphs
² A path is a connected sequence of edges or E D
arcs showing a route that starts at one vertex
and ends at another. A
For example: A - B - E - D E D
B C
² A circuit is a path that starts and ends at the A
same vertex.
For example: A - B - C - E - A B C
5. NETWORKS AND MATRICES (Chapter 7) (T4) 361
² An Eulerian circuit is a circuit that traverses each E D
edge (or arc) exactly once.
For example: A
A-B-C-B-E-C-D-E-A
B C
E D
² A Hamiltonian path is a path which starts at
one vertex and visits each vertex once and
only once. It may not finish where it started. A
For example: A - E - B - C - D B C
E D
² A Hamiltonian circuit is a circuit which starts
at one vertex, visits each vertex exactly once,
and returns to where it started. A
For example: A - B - C - D - E - A B C
² A spanning tree is a simple graph with no circuits which connects all vertices.
For example:
is a spanning tree.
EXERCISE 7A.1
1 For the networks shown, state:
i the number of vertices
ii the number of edges
iii the degree of vertex B
iv whether the graph is simple or non-simple
v whether the graph is complete or not complete
vi whether the graph is connected or not connected.
a B
b c A
d
B B
A
E B E
A C C D
A
C C
D E D
D
2 For the graph shown:
B C
a name a path starting at A and ending at E
A E
that does not “visit” any vertex more than once
b name a Hamiltonian path starting at A and finishing at E
D
c name a Hamiltonian circuit from A
d name a Hamiltonian circuit from D
e draw a spanning tree which includes the edges AB and AF. F
6. 362 NETWORKS AND MATRICES (Chapter 7) (T4)
3 For the network shown, draw as many B
spanning trees as you can find.
A C
D
4 For the graph shown, list all possible B
Hamiltonian circuits starting at A.
A E C
D
5 A gas pipeline is to be constructed to link several towns in the country. Assuming the
pipeline construction costs are the same everywhere in the region, the cheapest network
is:
A a Hamiltonian circuit B an Eulerian circuit
C a spanning tree D a complete graph
6 Give real life examples of situations where a person may be interested in using
a a Hamiltonian path b a Hamiltonian circuit
INVESTIGATION 1 LEONHARD EULER AND
THE BRIDGES OF KÖNIGSBERG
The 18th century Swiss mathematician Leonhard Euler is famous in many
areas of mathematics and science, including calculus, mechanics, optics,
and astronomy.
In fact, Euler published more research papers than any other mathematician in history.
However, nowhere was Euler’s contribution more significant than in topology. You have
already met Eulerian circuits that were named in his honour. In this investigation we will
meet another of his most famous contributions which is closely related to Eulerian circuits.
The town of Königsberg or Kaliningrad as it is now
known, was situated on the river Pregel in Prussia.¡ It had
seven bridges linking two islands and the north and south
banks of the river.
What to do:
1 The question is: could a tour be made of the town, re-
turning to the original point, that crosses all of the
bridges exactly once?¡ A simplified map of
Kaliningrad is shown alongside.¡ Euler answered this
question - can you?
2 Apparently, such a circuit is not possible.¡ However, river
it would be possible if either one bridge was removed
or one was added.¡ Which bridge would you remove?¡
Where on the diagram would you add a bridge?
7. NETWORKS AND MATRICES (Chapter 7) (T4) 363
3 For each of the following graphs determine:
i the number of vertices with even degree
ii the number of vertices with odd degree
iii whether or not the graph has an Eulerian circuit.
(Try to find an Eulerian circuit by trial and error.)
Summarise your results in a table:
Network No of vertices No of vertices Eulerian circuit (Yes/No)
with even degree with odd degree
a
.
.
.
a b c A
B B
B D
C
A C
A
C
d A e f
A
B C
B E
D
C D
E F
g A h A i A
D
B F B E
G
E
C E F
C D B D
D
4 Can you see a simple rule to determine whether or not a graph has an Eulerian circuit?
State the rule. Design your own networks or graphs to confirm the rule.
From Investigation 1 you should have determined that:
An undirected network has an Eulerian circuit if and only if there are no vertices
of odd degree.
8. 364 NETWORKS AND MATRICES (Chapter 7) (T4)
EXAMINING NETWORK DIAGRAMS
Networks can be used to display a wide variety of information. They can be used as:
² flow charts showing the flow of fluid, traffic, or people
² precedence diagrams showing the order in which jobs must occur to complete a
complex task
² maps showing the distances or times between locations
² family trees showing how people are related
² displays for sports results and many more things besides.
EXERCISE 7A.2
1 Consider the network showing which
people are friends in a table tennis club.
Mark
Alan Sarah
Paula Michael
Sally Gerry
a What do the nodes of this graph represent? b What do the edges represent?
c Who is Michael friendly with? d Who has the most friends?
e How could we indicate that while Alan really likes Paula, Paula thinks Alan is
a bit of a creep?
2 Alex is planning to water the garden with an sprinkler head joining clamp
underground watering system which has pop- T-piece
up sprinkers. A network is to be drawn to
show where the sprinklers are to be located. joiner
a Could the network be considered to be
elbow elbow
directed? with
b What are the nodes of the diagram? thread PVC T-piece PVC
c What are the edges? pipe with pipe
thread
d What problem might Alex be interested in solving?
3 At a school sports day, a round-robin ‘tug of
war’ competition is held between the houses. blue red
a What are the nodes of this graph?
b What do the edges represent?
c What is the significance of the degree of yellow green
each node?
d Why, in this context, must we have a complete graph?
e How many matches must be played in total?
9. NETWORKS AND MATRICES (Chapter 7) (T4) 365
f Using B ´ Blue, R ´ Red, etc. BR is the match between the Blues and the Reds.
List all possible games using this method.
g How could you indicate, on the network, who won each match?
4 Jon constructed a network model of his morning activities from waking up to leaving
for school.
eat cereal
dress make toast eat toast
shower clean teeth
wake up
drink juice leave
listen to the radio
Write a brief account of Jon’s morning activities indicating the order in which events
occur.
5 A roughly drawn map of the Riverland of SA
Waikerie Renmark
is shown alongside. The distances between 34
towns in kilometres are shown in black. The 22 15
times in minutes required to travel between Barmera 20
Kingston
the towns are shown in red. 10
9 12 Berri
a What are the nodes of the network? 10
b What do we call the red and black
numbers? 38 27 17 24
c What is the shortest distance by road Loxton
from Waikerie to Renmark according to
the map?
d Is it further to travel from Renmark to Kingston or Renmark to Loxton?
e Is it quicker to travel from Renmark to Kingston or Renmark to Loxton?
6 The network alongside describes
the distribution of water in a home. sink
kitchen
If we added weights to this net- dishwasher
work, describe three things they sink
could represent. water
bathroom shower
mains
toilet
sink
laundry
washing machine
DISCUSSION
Discuss the following:
² What types of computer network could be used in an office or school?
² What type of network would be suitable for a telephone system?
² What would the vertices and edges represent in each case?
10. 366 NETWORKS AND MATRICES (Chapter 7) (T4)
B CONSTRUCTING NETWORKS
Often we are given data in the form of a table, a list of jobs, or a verbal description. We need
to use this information to construct a network that accurately displays the situation.
Example 1 Self Tutor
computer
one printer
Model a Local Area Network computer
(LAN) of three computers two
connected through a server server
computer
to a printer and scanner. scanner
three
Example 2 Self Tutor
Model the access to rooms on the first floor of this two-storey house as a network.
lounge dining kitchen
stairs hall
verandah
bedroom bath bedroom
1 2
lounge dining kitchen
hall
verandah
bed 1 bath bed 2
Note: The rooms are the nodes, the doorways are the edges.
EXERCISE 7B.1
1 Draw a network diagram to represent the roads between towns P, Q, R, S and T if:
Town P is 23 km from town Q and 17 km from town T
Town Q is 20 km from town R and 38 km from town S
Town R is 31 km from town S.
2 Draw a network diagram of friendships if: A has friends B, D and F;
B has friends A, C and E;
C has friends D, E and F.
11. NETWORKS AND MATRICES (Chapter 7) (T4) 367
3 Draw a network diagram to represent how Australia’s states and territories share their
borders. Represent each state as a node and each border as an edge.
Australia’s states and territories
Darwin
Brisbane
Perth Adelaide
Canberra Sydney
1 : 50 000 000 Melbourne
0 500 1000 km
Hobart
4 a Model the room access on the first floor of the house plan below as a network
diagram.
kitchen dining bedroom 3
ensuite
hall
family bedroom 1 bedroom 2 bathroom
plan: first floor
b Model access to rooms and outside for the ground floor of the house plan below as
a network diagram. Consider outside as an external room represented by a single
node.
entry garage roller
door
living
laundry
study bathroom rumpus
plan: ground floor
12. 368 NETWORKS AND MATRICES (Chapter 7) (T4)
INVESTIGATION 2 TOPOLOGICAL EQUIVALENCE
My house has ducted air conditioning in all of the main rooms.¡ There are
three ducts that run out from the main unit.¡ One duct goes to the lounge
room and on to the family room.¡ The second duct goes to the study and on
to the kitchen.¡ The third duct goes upstairs, where it divides into three sub-
ducts that service the three bedrooms.
What to do:
1 Draw a network to represent the ducting in my house.
2 Do you think your network provides a good idea of the layout of my house?
3 Compare your network to those drawn by your classmates. Check each network to
make sure it correctly represents the description of my ducting. Do all of the correct
networks look the same?
4 For a given networking situation, is there always a unique way to draw the network?
In Investigation 2 you should have found that there can be many ways to represent the
same information.
Networks that look different but represent the same information are said to be topologically
equivalent or isomorphic.
For example, the following networks A
A D D
are topogically equivalent:
C
C
B B
Check to make sure each node has the correct connections.
EXERCISE 7B.2
1 Which of the networks in the diagrams following are topologically equivalent?
a b c d
e f g h
i j k l
13. NETWORKS AND MATRICES (Chapter 7) (T4) 369
2 Label corresponding vertices of the following networks to show their topological equiv-
alence:
3 Draw a network diagram with the following specifications:
² there are five nodes (or vertices)
² two of the nodes each have two edges
² one node has three edges and one node has four.
Remember when comparing your solution with others that networks may appear different
but be topologically equivalent.
PRECEDENCE NETWORKS
Networks may be used to represent the steps involved in a project.
The building of a house, the construction of a newsletter, and cooking an evening meal all
require many separate tasks to be completed.
Some of the tasks may happen concurrently (at the same time) while others are dependent
upon the completion of another task.
If task B cannot begin until task A is completed, then task A is a prerequisite for task B.
For example, putting water in the kettle is a prerequisite to boiling it.
If we are given a list of tasks necessary to complete a project, we need to
² write the tasks in the order in which they must be performed
² determine any prerequisite tasks
² construct a network to accurately represent the project.
Consider the tasks involved in making a cup of tea. They are listed below, along with their
respective times (in seconds) for completion. A table like this is called a precedence table
or an activity table.
A Retrieve the cups. 15
B Place tea bags into cups. 10
C Fill the kettle with water. 20
D Boil the water in the kettle. 100
E Pour the boiling water into the cups. 10
F Add the milk and sugar. 15
G Stir the tea. 10
14. 370 NETWORKS AND MATRICES (Chapter 7) (T4)
Example 3 Self Tutor
The steps involved in preparing a home-made pizza are listed below.
A Defrost the pizza base.
B Prepare the toppings.
C Place the sauce and toppings on the pizza.
D Heat the oven.
E Cook the pizza.
a Which tasks can be performed concurrently?
b Which tasks are prerequisite tasks?
c Draw a precedence table for the project.
d Draw a network diagram for the project.
a Tasks A, B and D may be performed concurrently, i.e., the pizza base could be
defrosting and the oven could be heating up while the toppings are prepared.
b The toppings cannot be placed on the pizza until after the toppings have been
prepared.
) task B is a prerequisite for task C.
The pizza cannot be cooked until everything else is done.
) tasks A, B, C and D are all prerequisites for task E.
c A precedence table shows the tasks and any Task Prerequisite Tasks
prerequisite tasks.
A
B
C B
d The network diagram may now be drawn. D
E A, B, C, D
A
B C E
start
finish
D
EXERCISE 7B.3
1 The tasks for the following projects are not in the correct order. For each project write
the correct order in which the tasks should be completed.
a Preparing an evening meal:
1 find a recipe 2 clean up 3 prepare ingredients
4 cook casserole 5 set table 6 serve meals
b Planting a garden:
1 dig the holes 2 purchase the trees 3 water the trees
4 plant the trees 5 decide on the trees required
2 Which tasks in question 1 could be performed concurrently?
15. NETWORKS AND MATRICES (Chapter 7) (T4) 371
3 The activities involved in Task Prerequisite
preparing, barbecuing and A Gather ingredients -
serving a meal of ham- B Pre-heat barbecue -
burgers are given in the C Mix and shape hamburgers A
table alongside. D Cook hamburgers B, C
Draw a network diagram E Prepare salad and rolls A
to represent this project. F Assemble hamburgers and serve D, E
4 Your friend’s birthday is approaching and you
decide to bake a cake. The individual tasks
are listed below:
A Mix the ingredients.
B Heat the oven.
C Place the cake mixture into the cake tin.
D Bake the cake.
E Cool the cake.
F Mix the icing.
G Ice the cake.
a Draw a precedence table for the project.
b Which tasks may be performed concurrently?
c Draw a network diagram for the project.
5 The construction of a back yard shed includes the following steps:
A Prepare the area for the shed.
B Prepare the formwork for the concrete.
C Lay the concrete.
D Let concrete dry.
E Purchase the timber and iron sheeting.
F Build the timber frame.
G Fix iron sheeting to frame.
H Add window, door and flashing.
a Draw a precedence table for the project.
b Which tasks may be performed concurrently?
c Draw a network diagram for the project.
6 The separate steps involved in hosting a party are listed below:
A Decide on a date to hold the party.
B Prepare invitations.
C Post the invitations.
D Wait for RSVPs.
E Clean and tidy the house.
F Organise food and drinks.
G Organise entertainment.
16. 372 NETWORKS AND MATRICES (Chapter 7) (T4)
Draw a network diagram to model the project, indicating which tasks may be performed
concurrently and any prerequisites that exist.
7 The tasks for cooking a breakfast of bacon and eggs, tea and toast are listed below:
boil kettle; toast bread; collect plate and utensils; serve breakfast; gather ingredients;
cook bacon and eggs; make tea; gather cooking utensils.
a Rearrange the tasks so that they are in the order that they should be completed.
b Which tasks can be performed concurrently?
c Construct a precedence table and network diagram for the project.
DISCUSSION CONCURRENCE
We have already seen how some tasks may be performed concurrently.¡
What factors may determine whether the tasks are performed
concurrently in practice?¡ How might the number of available workers
have an effect?
C PROBLEM SOLVING WITH NETWORKS
In this section we will investigate a number of network problems, including:
² the number of paths through a network
² the shortest and longest paths through a network
² the shortest connection problem
² the Chinese Postman problem
² the Travelling Salesman Problem.
NUMBER OF PATHS PROBLEMS
It is often useful to know how many paths there are from one point on a network to another
point. Such problems are usually associated with directed networks or ones for which we are
not allowed to back-track.
INVESTIGATION 3 NUMBER OF PATHS
Many suburbs of Adelaide are laid out as rectangular F
grids.¡ Suppose we start in one corner of a suburb and
wish to travel to the opposite corner.¡ A property of the
rectangular grid is that all paths from the start S to the
finish F have equal length so long as we never move away
from the finish F.¡ But how many of these paths are there? S
What to do:
1 For each network below, how many a F b F
‘shortest’ paths are there from the
start S to the finish F? S S
17. NETWORKS AND MATRICES (Chapter 7) (T4) 373
2 Can you find a method which will help you F
count the number of paths in a network such as:
S ?
Consider the network in Investigation 3 question 1 a. F
Since no back-tracking is allowed, the network is
essentially directed, as shown:
S
NUMBER OF PATHS ALGORITHM
In ‘number of paths’ problems we are interested in finding the total number of paths which
go from one node to another node without backtracking. Each connection on the graph allows
traffic in one direction only.
Consider the directed network:
B
How many paths are there from A to B?
A
Step 1: From A, there are two possible nodes we can go B
to. There is one possible path to each of them so
we write 1 near each. 1
A
1
1 B one path from
Step 2: We then look at the next step in the journey. At
below + one path
each successive node we add the numbers from from the left
1
the previous nodes which lead to that point. 2
A
1 1
Step 3: Repeat Step 2 until we finish at B. So, there are 1 3 6B
6 possible paths for getting from A to B.
1 3
2
A
1 1
Note: ² If we must avoid a given node, start by putting a zero (0) at that node.
² If we must pass through a particular node, start by “avoiding” nodes that do
not allow passage through it.
18. 374 NETWORKS AND MATRICES (Chapter 7) (T4)
Example 4 Self Tutor
Alvin (A) wishes to visit Barbara (B) across town. B
Alvin’s other friend Sally lives at the intersection S.
S
Find how many different pathways are possible in go-
ing from Alvin’s place to Barbara’s place:
a if there are no restrictions
b if Alvin must visit Sally along the way
c if Alvin must avoid Sally in case he gets stuck A
talking.
a 1
4 10 20 35
B Starting from A, we write numbers on the
diagram indicating the number of paths
S P from the previous vertices. For example,
1 15
3 6 10 to get to P we must come from S or T.
1
T
5 So, 6 + 4 = 10.
2 3 4
The total number of paths is 35.
A
1 1 1 1
b 0 0 6 12 18
B We must go through S, so there are some
vertices which are no longer useful to us.
S We start by marking the places we cannot
1
3 6 6 6
go with zeros.
1 The total number of
2 3 0 0 Can you see
paths from A to B is 18. how to find the
A 1 1 0 0 answer to c
from a and b
4 4 8 17
c 1 B As S must be avoided, only?
the total number of paths
S
1
3 0 4
9 is 17.
1 5
2 3 4
A 1 1 1 1
EXERCISE 7C.1
In each question, assume that back-tracking is not allowed.
1 Calculate the number of paths from S to F in each of the following networks:
a b A
S F
S A F
B
c d F
S F
S
19. NETWORKS AND MATRICES (Chapter 7) (T4) 375
2 a How many paths exist from A to L in the J K L
network shown?
G H
b How many of the paths go through G? I
c How many of the paths do not go through H? E F
d How many paths would there be if they must
go through F and H? A B C D
e What assumption have you made in your answers?
3 Solve the Opening Problem question 1a for the number of paths from Dien’s house to
Phan’s house.
SHORTEST PATH PROBLEMS
A shortest path problem is a problem where we must find the shortest path
between one vertex (or node) and another in a network.
The shortest path does not necessarily imply that all vertices are visited. They need not be.
Note also that by ‘shortest’ path we are not always referring to the path of shortest distance.
We may also be referring to the quickest route that takes the shortest time, or the cheapest
route of lowest cost.
Adelaide
A typical example is the problem
of travelling from Adelaide to Mt 96 Tailem Bend
Gambier. A network of the road 54 130
Meningie
route is shown alongside. The dis-
Keith
tances between towns are shown in 146
kilometres. 109
Kingston SE
Which route from Adelaide to Mt 41 Naracoorte
109
Gambier has the shortest distance? Robe 101
80
Find the answer by trial and error. Milicent 51
Mt Gambier
INVESTIGATION 4 SHORTEST PATH BY ‘TRIAL AND ERROR’
Eric has a choice of several paths home 5 B 4 C
to walk to school.¡ The arcs of the A
directed network not only show 4 4 3
the direction he must follow to 3 G 1 1 E 2
H D
take him closer to the school but F
also the time it takes (in minutes) 4 3 4 4
3 4 1
to walk that section of the journey. I L
J K school
What to do:
1 How many different paths are possible for Eric to travel to school?
2 Describe the shortest path that he may travel from home to school.
3 What is the minimum time in which he can walk to school?
20. 376 NETWORKS AND MATRICES (Chapter 7) (T4)
INVESTIGATION 5 SHORTEST PATH BY ‘TIGHT STRING’
B
Note: You will require string to 4 7
cut into varying lengths C E
for this investigation. A
5
6 6
What to do: D
Construct the network shown using lengths of string proportional to the lengths marked on
the edges, for example the value 5 could be 5 cm in length, the value 4 could be 4 cm.
Tie the string at the nodes where edges meet.
While holding your network at the knots representing the vertices at either end of the path
(A and E in this case), carefully pull the string tight.
The path that has been pulled tight is the shortest path.
EXERCISE 7C.2
1 Several street networks for students travelling from home to school are shown below.
The numbers indicate the times in minutes. In each case, describe the quickest path and
give the minimum travelling time.
a A 2 B 4 C b A 2 B 3 C
home home
3 3 2 1 1 2
D 3 E 2 F D 2 E 5 F
4 3 4 4
3 3 3
G 3 H 4 I G 2 H 4 I
school school
c d home
A 8 B A 3 B
home
7 5 4 4 2
4 D 3 E
5 C C D
6 3 2 2 4 6
F 9 G 4 H F
school E 3 school
2 George has just arrived at
Heathrow airport in Epping
London.¡ Can you help Ealing
4
him work out the cheapest Paddington 3
Broadway
way to get to Epping 5
2.4
using the Tube and 2.6 Liverpool
2
Street
overground trains?¡ Prices Notting Hill
2.1 2.6
for each section of the Holborn
4.9
route are given in pounds Heathrow
sterling.
Although the trial and error and tight string methods are valid ways to find shortest paths,
they become incredibly time-consuming when the networks are large and complicated.
The mathematician Dijkstra devised a much more efficient algorithm for large networks.
21. NETWORKS AND MATRICES (Chapter 7) (T4) 377
DIJKSTRA’S SHORTEST PATH ALGORITHM
Step 1: Assign a value of 0 to the starting vertex. Draw a box around the vertex label
and the 0 to show the label is permanent.
Step 2: Consider all unboxed vertices connected to the vertex you have just boxed.
Label them with the minimum weight from the starting vertex via the set of
boxed vertices.
Step 3: Choose the least of all of the unboxed labels on the whole graph, and make it
permanent by boxing it.
Step 4: Repeat steps 2 and 3 until the destination vertex has been boxed.
Step 5: Back-track through the set of boxed vertices to find the shortest path through
the graph.
At each stage we try to find the shortest path from a given vertex to the starting vertex. We
can therefore discard previously found shortest paths as we proceed, until we have obtained
the actual shortest path from start to finish.
Example 5 Self Tutor
Find the shortest path from A to B
E in the network alongside. 4 5
2
C
A E
3
3 2 7
D
Step 1: Assign 0 to the starting vertex and 4B
draw a box around it.
4 2 5
0 C
Step 2: The nodes B and D are connected to A E
3
A. Label them with their distances 2
3 7
from A.
D
Step 3: The unboxed vertex with the smallest 3
label is D. Draw a box around it.
Step 2: Consider the unboxed nodes connected 4
to D. B
D to C has length 2, so we label C 4 5
with 3 + 2 = 5. 0
2
C
D to E has length 7, so we label E A 5
3
E 10
with 3 + 7 = 10: 3 2 7
Step 3: The unboxed vertex with the smallest D
3
label is B. Draw a box around it.
22. 378 NETWORKS AND MATRICES (Chapter 7) (T4)
Step 2: Consider the unboxed nodes connected 4
B
to B.
For C we would have 4 + 2 = 6. 4 2 5
This is more than 5 so the label is 0
A 5 C E 10 9
3
unchanged. 2
For E we have 4 + 5 = 9, so we 3 7
replace the label with 9. D
3
Step 3: The unboxed vertex with the smallest
4
label is C. Draw a box around it. B
Step 2: The only unboxed node connected to 4 2 5
8
C is E. 5 + 3 = 8, and this is less 0
A 5 C E 10 9
than 9 so we change the label. 3
3 2 7
Step 3: The unboxed vertex with the smallest
label is the finishing vertex E. D
3
We now backtrack through the network to find the shortest path: E Ã C Ã D Ã A.
So, the shortest path is A ! D ! C ! E and its length is 8 units.
Note: ² In practice, we do not usually write out all of the steps and numerous diagrams.
These have only been included to help you understand the procedure.
² Suppose in Example 5 above that we are forced to pass through B on the way
to E. To solve this problem we now find the minimum length from A to B and
then add to it the minimum length from B to E.
Example 6 Self Tutor
A 2 B 3 C 5 D 2 E
For the network alongside, find
1 3 6 2 3
the shortest path from A to O
2 3 2 1
that passes through M. What is F G H I J
the length of this path? 4 5 1 2 4
2 2 4 2
K L M N O
Since we must pass through M, we find the shortest path from A to M and then
the shortest path from M to O.
0 2 5 10 7 7 5 7
A 2 B 3 C 5 D 2 E A 2 B 3 C 5 D 2 E
1 3 6 2 3 1 3 6 2 3
1F 2 3 2 1 6F 2 3 2 1
G H I J G H I J
4 5 3 1 6 2 8 4 4 5 4 1 1 2 3 4 4
2 2 4 2 2 2 4 2
K L M N O K L M N O
5 7 7 4 2 0 4 6
8
Working backwards we get O Ã N Ã M Ã H Ã G Ã F Ã A.
) the shortest path is A ! F ! G ! H ! M ! N ! O with length
7 + 6 = 13 units.
23. NETWORKS AND MATRICES (Chapter 7) (T4) 379
EXERCISE 7C.3
1 Find the shortest path from A to Z and state its value:
a B 3 b A B
C 6
4 4 4
1 4 6
A 5 C 5 E
D Z
3 7 8
3 5
E
D Z
c d
2 B 5
A C
G 15 H 10 B
3 3 2
3 9 12
D F 13 14
4 E 14 12
5 6 A F C
6 1 12 Z
G I 12 15
H 10
2 4 35
J Z E 14
2 D
Adelaide
2 Consider again the problem on page 375.
Find the shortest route from Adelaide to 96 Tailem Bend
Mt Gambier using the routes shown. 54 130
Meningie
Keith
146
109
Kingston SE
41 Naracoorte
109
Robe 101
80
Milicent 51
Mt Gambier
U 8 Q
3 The time to run various tracks in an orienteering
8 3 3
event is shown. Times are in minutes.
6 T R
P 3 5 a Find the quickest way to get from P to Q.
8 7 b Which is the quickest way if the track PS
S
V 7 becomes impassable due to a flood?
4 Solve Opening Problem question 1b for the quickest route between Dien’s house and
Phan’s house.
A
5 A construction company has a warehouse 10 4
at W in the diagram shown. Often several 7 F
trips per day are made from the warehouse 9
W
to each of the construction sites. Find the 6
shortest route to each of the construction B 7
16
sites from the warehouse. Distances are 8
in km. 13 E
6
C
9
D
24. 380 NETWORKS AND MATRICES (Chapter 7) (T4)
6 The network alongside shows the 32 10 38
5
connecting roads between three ma- 13
25 41
jor towns, A, B and C. The weights 21 18
9 7
on the edges represent distances, in B
19
C
kilometres. A 16
12 15
32
a Find the length of the shortest 18
30
29
18
20 24
10
path between towns A and C 51
that goes through town B. 19 16
b Find the length of the shortest path between towns A and C.
7 A salesman is based in city T.¡ The table shows the T U V W X Y
airfares in dollars for direct flights between the cities.
T ¡ 80 ¡ 30 ¡ 80
a Put the information onto a network diagram.¡ U 80 ¡ 80 ¡ ¡ ¡
b Determine the minimum cost of flights from T V ¡ 80 ¡ ¡ 40 50
to each of the other cities.¡
W 30 ¡ ¡ ¡ 80 60
c Determine the cheapest cost between T and Y if X ¡ ¡ 40 80 ¡ 50
he must travel via V.
Y 80 ¡ 50 60 50 ¡
DISCUSSION SHORTEST PATH PROBLEMS
What other situations involving ‘shortest’ paths can you think of?
LONGEST PATH PROBLEMS
In some situations we may need to find the longest path through a network. This may be to
maximise profits or points collected on the way. Clearly we cannot allow back-tracking in
the network since otherwise we could continually go to and from between nodes to increase
the length forever. We can therefore only find longest paths if the network is directed or if
we are not allowed to return to a node previously visited.
Unfortunately, we cannot find longest paths by simply changing “minimum” to “maximum”
in Dijkstra’s Shortest Path Algorithm. There are other more complex algorithms that can be
used, but they are beyond the scope of this course. We therefore present examples to be
solved by trial and error.
Adelaide
EXERCISE 7C.4
6 Tailem Bend
1 A hardware salesman needs to travel from
4
Adelaide to Mt Gambier for a meeting.¡ He has
six hours to make the journey, which gives him 5 Keith
enough spare time that he can visit some clients 4
along the way, but not enough that we can back-
Kingston SE
track to pick up other routes.¡ Based on previous 3 Naracoorte
4
sales he estimates the amount of money (in Robe
5
hundreds of dollars) he can earn along each 2
section of the route.¡ Which way should the Milicent 4
salesman go to maximise his earnings? Mt Gambier
25. NETWORKS AND MATRICES (Chapter 7) (T4) 381
2 The network shows the time in minutes 3 3 3
S A B C
to drive between intersections in a street
network. 1 2 2 1
4 4 3 2
a Determine the route from S to M D E F G H
which would take the shortest time.
2 2 1 2 2
b Determine the route from S to M 5 3 4 3
which would take the longest time. I J K L M
SHORTEST CONNECTION PROBLEMS
In shortest connection problems we seek the most efficient way of joining vertices
with edges so that paths exist from any vertex to any other vertex. Vertices do not
need to be linked directly to each other, but they all must be somehow linked.
Situations requiring such a solution include computers and printers in a network (as in
Opening Problem question 2), connecting a set of pop-up sprinklers to a tap, or connecting
a group of towns with telephone or power lines.
You should recall from the beginning of this chapter that a spanning tree is a simple graph
with no circuits which connects all vertices. The optimum solution requires a spanning tree
for which the sum of the lengths of the edges is minimised. We call this the minimum length
spanning tree.
INVESTIGATION 6 SCHOOL FOUNTAINS
A school has decided to add four new
drinking fountains for its students.¡ The C
water is to come from an existing
purification system at point A; the points B
B, C, D and E mark the locations of the new fountains.¡
Clearly all fountains must be connected to the
D
purification system, though not necessarily directly. A
What to do:
1 Using the scale on the diagram, measure the
E
length of pipe required for all fountains to be
connected directly to the purification system.¡ scale 1¡:¡5000
2 By trial and error, find the minimum length spanning tree for the network which
gives the shortest length of pipe to complete the project.¡ What is this length of pipe?
EXERCISE 7C.5
Use trial and error to solve the following shortest connection problems:
1 a 7 b
6
3
5
5 6 4
7 6 6
7
5
8 4
26. 382 NETWORKS AND MATRICES (Chapter 7) (T4)
2 Joseph the electrician is doing the wiring for a power point
6 7 existing
house extension. The new room is to have two power
powerpoints and a light in the centre of the ceiling. 5 supply
The numbers on the diagram indicate the distances power 6 7
point 2
in metres of cable needed between those points. 1
8
What is the shortest length of cable required? light
11 switch
Just like the shortest path problem, the shortest connection problem becomes very time-
consuming and awkward for large networks. However, we can again use algorithms to
rapidly solve the problem.
PRIM’S ALGORITHM
In this algorithm we begin with any vertex. We add vertices one at a time according to which
is nearest. For this reason Prim’s algorithm is quite effective on scale diagrams where edges
are not already drawn in. It also has the advantage that because we add a new vertex with
each step, we are certain that no circuits are formed. The algorithm is:
Step 1: Start with any vertex.
Step 2: Join this vertex to the nearest vertex.
Step 3: Join on the vertex which is nearest to either of those already connected.
Step 4: Repeat until all vertices are connected.
Step 5: Add the lengths of all edges included in the minimum length spanning tree.
Example 7 Self Tutor
Consider the example of connecting the
school fountains in Investigation 5. C
No distances between the purification B
system and the fountains were given, but
the diagram was drawn to scale.
D
Scale 1 : 5000 A
E
scale 1¡:¡5000
Step 1: We choose to start at E.
C
Step 2: Clearly D is closest to E so we
include DE in the solution. B
(Length DE is 123 m using the
scale diagram.)
D
Step 3: Consider distances from D and A
E. B is closest to D so we in-
clude BD in the solution.
(Length BD is 62 m.) E
27. NETWORKS AND MATRICES (Chapter 7) (T4) 383
Step 4: Consider distances from B, D or E. A is the next closest vertex (to B),
so we include AB in the solution. (Length AB is 75 m.)
Consider distances from A, B, D or E. C is the only vertex remaining.
It is closest to B, so we include BC in the solution.
(Length BC is 140 m.)
All vertices are connected so we stop.
Step 5: The minimum length is 400 m.
NEAREST NEIGHBOUR ALGORITHM
In this algorithm we start with all edges of the network in place. We remove the longest
edges one by one without disconnecting the graph until only the minimum length spanning
tree remains. The algorithm is:
Step 1: Locate the longest length edge which can be removed without disconnecting the
graph.¡
Step 2: Remove this edge.¡
Step 3: Repeat Steps 1 and 2 until the removal of any more edges will disconnect the
graph.¡
Step 4: Add the lengths of all edges in the minimum length spanning tree.
Example 8 Self Tutor
Use the Nearest Neighbour algorithm to find B 8 D
10
the minimum length spanning tree for the 10 7 E
graph alongside. 12
A C 15 13
12
13
F
14
H 11 G
B D
The longest length edge is DG (length 15), so 8
10
this is removed first.¡ 10 7 E
12
The next longest edge is FG (length 14) and it A C 13
can also be removed without disconnecting 12
the graph.¡ 13
F
There are two edges of weight 13.¡ The edge H 11 G
that joins AH can be removed.¡
B 8 D
The edge EF cannot be removed because that 10
E
would disconnect node F.¡ 10 7
12
There are two edges of weight 12.¡ The edge A C 13
CD can be removed, but EG cannot be 12
removed without disconnecting G and H. F
H 11 G
28. 384 NETWORKS AND MATRICES (Chapter 7) (T4)
The removal of any more edges would D
B 8
make the graph disconnected, so this 10
final graph is the minimum length 10 7 E
spanning tree. A C 13
The total weight of the minimum length 12
spanning tree is F
10 + 8 + 7 + 10 + 13 + 12 + 11 = 71 H 11 G
EXERCISE 7C.6
1 Use Prim’s algorithm to find the minimum length spanning tree for the following net-
works. In each case state the minimum length.
a b A c A
11 10 11 10
A 10 B 12 B B
O 14
H
14
15 10 E 12 16
10 12 15 15
10 16
D F
13
12
D C D 10
11 13 C C
16
E
2 Use the Nearest Neighbour algorithm to find the minimum length spanning tree for the
following networks. In each case state the minimum length.
a b c
G
9
A 12 B R 4 3
5
15 Q A
O 12 12 9 4 F 5
14 E
14 C 12
4 B
12
12 T 16 6 5
12 15 5
F 14 7
E 14 D 7
5
12
P 5 C
S D
17
DISCUSSION MINIMUM CONNECTION ALGORITHMS
Discuss the advantages and disadvantages of Prim’s algorithm and the
Nearest Neighbour algorithm.¡
What determines which is better for a given problem?
Hint: One algorithm adds edges each step; the other removes edges each step.
For each of the following problems, use whichever algorithm is most appropriate to help
find the solution:
3 Solve the Opening Problem question 2 for the minimum length of cable required for
the computer network. Assume the lengths given are in metres.