The document discusses various numerical methods for analyzing mechanical components under applied loads, including the finite element method. It describes weighted residual methods like the Galerkin method and collocation method which approximate solutions by minimizing residuals. The variational or Rayleigh-Ritz method selects displacement fields to minimize total potential energy. The finite element method divides a structure into small elements and applies these methods to obtain approximate solutions for displacements and stresses at discrete points.
The document provides an introduction to finite element methods. It discusses how finite element analysis (FEA) is used to solve complex engineering problems that cannot be solved through closed-form analytical methods. The key steps of FEA include discretizing the domain into finite elements, deriving element equations, assembling equations into a global system of equations, and solving the system of equations. Common element types include line, triangular, and brick elements. The accuracy of FEA depends on factors like element size and shape, approximation functions, and numerical integration.
The document provides an introduction to finite element methods. It discusses how finite element analysis (FEA) is used to solve engineering and science problems involving complex geometries and conditions. FEA works by dividing a body into finite elements and approximating variable fields within each element. This discretization process sets up algebraic equations that approximate the continuous solution. FEA can model problems with complex shapes, loads, materials and include time-dependent effects. It has advantages over closed-form solutions and its accuracy can be improved by refining the mesh. Examples of 1D and 2D elements and approximations are presented.
Linear regression [Theory and Application (In physics point of view) using py...ANIRBANMAJUMDAR18
Machine-learning models are behind many recent technological advances, including high-accuracy translations of the text and self-driving cars. They are also increasingly used by researchers to help in solving physics problems, like Finding new phases of matter, Detecting interesting outliers
in data from high-energy physics experiments, Founding astronomical objects are known as gravitational lenses in maps of the night sky etc. The rudimentary algorithm that every Machine Learning enthusiast starts with is a linear regression algorithm. In statistics, linear regression is a linear approach to modelling the relationship between a scalar response (or dependent variable) and one or more explanatory variables (or independent
variables). Linear regression analysis (least squares) is used in a physics lab to prepare the computer-aided report and to fit data. In this article, the application is made to experiment: 'DETERMINATION OF DIELECTRIC CONSTANT OF NON-CONDUCTING LIQUIDS'. The entire computation is made through Python 3.6 programming language in this article.
- The document details a state space solver approach for analog mixed-signal simulations using SystemC. It models analog circuits as sets of linear differential equations and solves them using the Runge-Kutta method of numerical integration.
- Two examples are provided: a digital voltage regulator simulation and a digital phase locked loop simulation. Both analog circuits are modeled in state space and simulated alongside a digital design to verify mixed-signal behavior.
- The state space approach allows modeling analog circuits without transistor-level details, improving simulation speed over traditional mixed-mode simulations while still capturing system-level behavior.
The document provides an introduction to finite element analysis. It discusses the need for computational methods to solve problems involving complex geometries and boundary conditions that cannot be solved through closed-form analytical methods. The finite element method is introduced as a numerical technique that involves discretizing a continuous domain into discrete subdomains called elements, and approximating variations in dependent variables within each element. This allows setting up algebraic equations that can be solved to approximate the continuous solution. Advantages of the finite element method include its ability to model complex shapes and behaviors, and refine solutions through mesh refinement. Basic concepts such as element types, discretization, and derivation of element equations are described.
The document discusses dimensional analysis and Buckingham Pi theorem. It begins by defining dimensions, units, and fundamental vs. derived dimensions. It then discusses dimensional homogeneity and uses examples to show how dimensional analysis can be used to identify non-dimensional parameters and reduce the number of variables in equations. The Buckingham Pi theorem is introduced as a method to systematically create dimensionless pi terms from physical variables. Steps of the theorem and examples applying it are provided. Overall, the document provides an overview of dimensional analysis and Buckingham Pi theorem as tools for understanding relationships between physical quantities and reducing complexity in experimental modeling.
The document discusses various numerical methods for analyzing mechanical components under applied loads, including the finite element method. It describes weighted residual methods like the Galerkin method and collocation method which approximate solutions by minimizing residuals. The variational or Rayleigh-Ritz method selects displacement fields to minimize total potential energy. The finite element method divides a structure into small elements and applies these methods to obtain approximate solutions for displacements and stresses at discrete points.
The document provides an introduction to finite element methods. It discusses how finite element analysis (FEA) is used to solve complex engineering problems that cannot be solved through closed-form analytical methods. The key steps of FEA include discretizing the domain into finite elements, deriving element equations, assembling equations into a global system of equations, and solving the system of equations. Common element types include line, triangular, and brick elements. The accuracy of FEA depends on factors like element size and shape, approximation functions, and numerical integration.
The document provides an introduction to finite element methods. It discusses how finite element analysis (FEA) is used to solve engineering and science problems involving complex geometries and conditions. FEA works by dividing a body into finite elements and approximating variable fields within each element. This discretization process sets up algebraic equations that approximate the continuous solution. FEA can model problems with complex shapes, loads, materials and include time-dependent effects. It has advantages over closed-form solutions and its accuracy can be improved by refining the mesh. Examples of 1D and 2D elements and approximations are presented.
Linear regression [Theory and Application (In physics point of view) using py...ANIRBANMAJUMDAR18
Machine-learning models are behind many recent technological advances, including high-accuracy translations of the text and self-driving cars. They are also increasingly used by researchers to help in solving physics problems, like Finding new phases of matter, Detecting interesting outliers
in data from high-energy physics experiments, Founding astronomical objects are known as gravitational lenses in maps of the night sky etc. The rudimentary algorithm that every Machine Learning enthusiast starts with is a linear regression algorithm. In statistics, linear regression is a linear approach to modelling the relationship between a scalar response (or dependent variable) and one or more explanatory variables (or independent
variables). Linear regression analysis (least squares) is used in a physics lab to prepare the computer-aided report and to fit data. In this article, the application is made to experiment: 'DETERMINATION OF DIELECTRIC CONSTANT OF NON-CONDUCTING LIQUIDS'. The entire computation is made through Python 3.6 programming language in this article.
- The document details a state space solver approach for analog mixed-signal simulations using SystemC. It models analog circuits as sets of linear differential equations and solves them using the Runge-Kutta method of numerical integration.
- Two examples are provided: a digital voltage regulator simulation and a digital phase locked loop simulation. Both analog circuits are modeled in state space and simulated alongside a digital design to verify mixed-signal behavior.
- The state space approach allows modeling analog circuits without transistor-level details, improving simulation speed over traditional mixed-mode simulations while still capturing system-level behavior.
The document provides an introduction to finite element analysis. It discusses the need for computational methods to solve problems involving complex geometries and boundary conditions that cannot be solved through closed-form analytical methods. The finite element method is introduced as a numerical technique that involves discretizing a continuous domain into discrete subdomains called elements, and approximating variations in dependent variables within each element. This allows setting up algebraic equations that can be solved to approximate the continuous solution. Advantages of the finite element method include its ability to model complex shapes and behaviors, and refine solutions through mesh refinement. Basic concepts such as element types, discretization, and derivation of element equations are described.
The document discusses dimensional analysis and Buckingham Pi theorem. It begins by defining dimensions, units, and fundamental vs. derived dimensions. It then discusses dimensional homogeneity and uses examples to show how dimensional analysis can be used to identify non-dimensional parameters and reduce the number of variables in equations. The Buckingham Pi theorem is introduced as a method to systematically create dimensionless pi terms from physical variables. Steps of the theorem and examples applying it are provided. Overall, the document provides an overview of dimensional analysis and Buckingham Pi theorem as tools for understanding relationships between physical quantities and reducing complexity in experimental modeling.
This document provides an overview of machine learning concepts including feature selection, dimensionality reduction techniques like principal component analysis and singular value decomposition, feature encoding, normalization and scaling, dataset construction, feature engineering, data exploration, machine learning types and categories, model selection criteria, popular Python libraries, tuning techniques like cross-validation and hyperparameters, and performance analysis metrics like confusion matrix, accuracy, F1 score, ROC curve, and bias-variance tradeoff.
This document describes specification tests that can be used after estimating dynamic panel data models using the generalized method of moments (GMM) estimator. It presents GMM estimators for first-order autoregressive models with individual fixed effects that exploit moment restrictions from assuming serially uncorrelated errors. Monte Carlo simulations are used to evaluate the small-sample performance of tests of serial correlation based on GMM residuals, Sargan tests, and Hausman tests. The tests are also applied to estimated employment equations using an unbalanced panel of UK firms.
Finite element method have many techniques that are used to design the structural elements like automobiles and building materials as well. we use different design software to get our simulated results at ansys, pro-e and matlab.we use these results for our real value problems.
Development, Optimization, and Analysis of Cellular Automaton Algorithms to S...IRJET Journal
This document summarizes research on using cellular automaton algorithms to solve stochastic partial differential equations (SPDEs). It proposes a finite-difference method to approximate an SPDE modeling a random walk with angular diffusion. A Monte Carlo algorithm is also developed for comparison. Analysis finds a moderate correlation between the two methods, suggesting the finite-difference approach is reasonably accurate. It also identifies an inverse-square relationship between variables, linking to a foundational stochastic analysis concept. The research concludes the finite-difference method shows promise for approximating SPDEs while considering boundary conditions.
Finite Element Analysis (FEA) is a numerical method for solving complex engineering problems. The document discusses conducting FEA on a fixed-free cantilever beam to study the effect of mesh density on solution accuracy. Analytical solutions are derived and used to validate FEA results. A beam model is created in ABAQUS with varying element sizes. As element count increases, FEA results converge towards analytical solutions, though with increased computation time. An element count of 4125 provided an optimal balance between accuracy and cost.
This document discusses molecular dynamics simulation methods. It begins by introducing molecular dynamics and Newton's laws of motion. It then describes how molecular dynamics simulations work by integrating Newton's equations of motion to obtain successive molecular configurations over time. Different algorithms for performing the integration are discussed, including the Verlet algorithm and leap-frog method. Predictor-corrector algorithms are also covered. The document provides guidance on best practices for setting up and running molecular dynamics simulations.
Urban strategies to promote resilient cities The case of enhancing Historic C...inventionjournals
This research tackles disaster prevention problems in dense urban areas, concentrating on the urban fire challenge in Historic Cairo district, Egypt, through disaster risk management approach. The study area suffers from the strike of several urban fire outbreaks, that resulted in disfiguring historic monuments and destroying unregulated traditional markets. Therefore, the study investigates the significance of hazard management and how can urban strategies improve the city resilient through reducing the impact of natural and man-made threats. The main findings of the research are the determination of the vulnerability factors in Historic Cairo district, either regarding management deficiency or issues related to the existing urban form. It is found that the absence of the mitigation and preparedness phases is the main problem in the risk management cycle in the case study. Additionally, the coping initiatives adopted by local authorities to address risks are random and insufficient. The study concludes with recommendations which invoke incorporating hazard management stages (pre disaster, during disaster and post disaster) into the process of evolving development planning. Finally, solutions are offered to mitigate, prepare, respond and recover from fire disasters in the case study. The solutions include urban policies, land-use planning, urban design outlines, safety regulation and public awareness and training.
Master of Computer Application (MCA) – Semester 4 MC0079Aravind NC
The document describes mathematical models and provides examples of different types of models. It discusses linear vs nonlinear models, deterministic vs probabilistic models, static vs dynamic models, discrete vs continuous models, and deductive vs inductive vs floating models. It also explains the Erlang family of distributions used in queuing systems and provides the probability density function and cumulative distribution function. Finally, it outlines the graphical method algorithm for solving a linear programming problem with two variables in 8 steps.
This document summarizes research on edge-preserving image reconstruction methods for electrical impedance tomography (EIT) applied to lung imaging. It presents three main contributions:
1. A level set-based reconstruction algorithm that was tested on clinical EIT data from lung patients, showing improved results over conventional methods.
2. An algorithm using level sets and L1 norms that better preserves edges and is more robust to noise and outliers.
3. A generalized inverse problem formulation using weighted L1 and L2 norms on data and regularization terms, improving robustness.
Evaluation of the methods showed they produced more accurate shapes and were more robust to uncertainties compared to traditional techniques. Future work is proposed on combining level
A Bibliography on the Numerical Solution of Delay Differential Equations.pdfJackie Gold
This document is a bibliography containing references to papers and technical reports on numerical methods for solving delay differential equations. It includes 58 references divided into sections on dense-output methods for ordinary differential equations, numerical methods for delay differential equations, dynamics and stability of delay differential equations, applications of delay differential equations, and functional differential equations. The bibliography aims to provide an introduction to earlier works in the field as well as more recent publications that are available through online search tools.
This document discusses numerical methods for solving differential equations. It begins with an introduction to differential equations and why numerical methods are useful for complex systems that cannot be solved analytically. It then describes Euler's method, the simplest numerical method for approximating solutions to differential equations. Specifically, it uses the example of the differential equation dy/dt = λy to demonstrate how Euler's method works by taking small time steps and extrapolating the solution forward. The document provides MATLAB code to demonstrate Euler's method visually.
Alexander Litvinenko's research interests include developing efficient numerical methods for solving stochastic PDEs using low-rank tensor approximations. He has made contributions in areas such as fast techniques for solving stochastic PDEs using tensor approximations, inexpensive functional approximations of Bayesian updating formulas, and modeling uncertainties in parameters, coefficients, and computational geometry using probabilistic methods. His current research focuses on uncertainty quantification, Bayesian updating techniques, and developing scalable and parallel methods using hierarchical matrices.
The document provides an introduction to the finite element method (FEM). It discusses how FEM can be used to obtain approximate solutions to boundary value problems in engineering. It outlines the general steps involved, including preprocessing (defining the model), solution/processing (computing unknown values), and postprocessing (analyzing results). Examples of FEM applications include structural analysis, fluid flow, heat transfer, and more. The key aspects of FEM include discretizing the domain into simple elements, choosing shape functions to approximate variations within each element, and assembling the element equations into a global system of equations to solve.
Deep learning models can be improved for physical processes by incorporating prior scientific knowledge. The paper proposes a method where a neural network predicts parameters like motion fields in governing equations, rather than directly predicting outputs. It applies this to ocean surface temperature prediction. The model predicts a motion field from past temperature images using a CNN. It then uses this motion field in a warping scheme based on the advection-diffusion equation to forecast future temperature. This outperforms comparison methods by leveraging physics knowledge without requiring manual specification of equations.
Finite Element Analysis Lecture Notes Anna University 2013 Regulation NAVEEN UTHANDI
One of the most Simple and Interesting topics in Engineering is FEA. My work will guide average students to score good marks. I have given you full package which includes 2 Marks and Question Banks of previous year. All the Best
For Guidance : Comment Below Happy to Teach and Learn along with you guys
Some Engg. Applications of Matrices and Partial DerivativesSanjaySingh011996
This document contains a submission by three students to Dr. Sona Raj Mam regarding partial differentiation, matrices and determinants, and eigenvectors and eigenvalues. It provides examples of how these mathematical concepts are applied in fields like engineering. Partial differentiation is used in economics to analyze demand and in image processing for edge detection. Matrices and determinants allow representing linear transformations in graphics software. Eigenvalues and eigenvectors have applications in areas like computer science, smartphone apps, and modeling structures in civil engineering. The document also provides real-world examples and references textbooks and websites for further information.
Design of Resonators for Coupled Magnetic Resonance-based Wireless Power Tran...Quang-Trung Luu
Quang-Trung Luu, Duc-Hung Tran, Bao-Huy Nguyen, Yem Vu-Van, and Cao-Minh Ta, "Design of Resonators for Coupled Magnetic Resonance-based Wireless Power Transmission System," In Proc. Vietnam Conference on Control and Automation, Da Nang, Nov. 2013.
1) Mathematical modeling formulates problems or physical systems into mathematical language using equations that can then be solved using numerical methods, graphics, or analytical solutions.
2) The key components of a mathematical model are dependent variables, independent variables, parameters, and forcing functions.
3) Mathematical models can be classified as linear or nonlinear, deterministic or probabilistic, static or dynamic, and lumped or distributed parameters.
The Sample Average Approximation Method for Stochastic Programs with Integer ...SSA KPI
The document describes a sample average approximation method for solving stochastic programs with integer recourse. It approximates the expected recourse cost function using a sample average based on a sample of scenarios. It shows that as the sample size increases, the solution to the sample average approximation problem converges exponentially fast to the optimal solution of the true stochastic program. It also describes statistical and deterministic techniques for validating candidate solutions. Preliminary computational results applying this method are also mentioned.
CHAPTER 3Understanding Regulations, Accreditation Criteria, and .docxtiffanyd4
CHAPTER 3
Understanding Regulations, Accreditation Criteria, and Other Standards ofPractice
NAEYC Administrator Competencies Addressed in This Chapter:
Management Knowledge and Skills
2. Legal and Fiscal Management
· Knowledge and application of the advantages and disadvantages of different legal structures
· Knowledge of different codes and regulations as they relate to the delivery of early childhood program services
· Knowledge of child custody, child abuse, special education, confidentiality, anti-discrimination, insurance liability, contract, and laborlaws pertaining to program management
5. Program Operations and Facilities Management
· Knowledge and application of policies and procedures that meet state/local regulations and professional standards pertaining to thehealth and safety of young children
7. Marketing and public relations
· Skill in developing a business plan and effective promotional literature, handbooks, newsletters, and press releases
Early Childhood Knowledge and Skills
5. Children with Special Needs
· Knowledge of licensing standards, state and federal laws (e.g., ADA, IDEA) as they relate to services and accommodations for childrenwith special needs
10. Professionalism
· Knowledge of laws, regulations, and policies that impact professional conduct with children and families
· Knowledge of center accreditation criteria
Learning Outcomes
After studying this chapter, you will be able to:
1. Describe the purpose of regulations that apply to programs of early care and education and list several topics they address.
2. Identify several ways accreditation standards are different from child care regulations.
3. State the purpose of Quality Rating and Improvement Systems (QRIS).
4. List some ways qualifications for administrators and teachers are different for licensure, for accreditation, and in QRIS systems.
5. Identify laws that apply to the childcare workplace, such as those that govern the program’s financial management and employees’well-being.
Marie’s Experience
Marie has been successful over the years in keeping her center in compliance with all licensing regulations. She is proud of her teachers andconfident that the center consistently goes above and beyond licensing provisions designed simply to keep children healthy and safe. She knowsthat the center provides high-quality care to the children it serves, but has never pursued accreditation or participated in her state’s optionalQuality Rating and Improvement System (QRIS) because of the time and effort it would require. Her families have confidence in her program anddo not seem to need this additional assurance that it provides high-quality services day in and day out.
Large numbers of families rely on out-of-home care for their infants, toddlers, preschoolers, and school-age children during the workday. In2011, there were 312,254 licensed child care facilities with a capacity to serve almost 10.2 million children. About 34% of these facilitieswere child care center.
Chapter 3 Human RightsINTERNATIONAL HUMAN RIGHTS–BASED ORGANIZ.docxtiffanyd4
Chapter 3 Human Rights
INTERNATIONAL HUMAN RIGHTS–BASED ORGANIZATIONS LIKE THE UN COMMISSION ON HUMAN RIGHTS HAVE MADE MONITORING HUMAN RIGHTS A GLOBAL ISSUE. The United Nations is headquartered in New York City.
Learning Objectives
1. 3.1Review the expansion of and the commitment to the human rights agenda
2. 3.2Evaluate the milestones that led to the current concerns around human rights
3. 3.3Evaluate some of the philosophical controversies over human rights
4. 3.4Recognize global, regional, national, and local institutions and rules designed to protect human rights across the globe
5. 3.5Report the efforts made globally in bringing violators of human rights to justice
6. 3.6Relate the need for stricter laws to protect women’s human rights across the globe.
7. 3.7Recognize the need to protect the human rights of the disabled
8. 3.8Distinguish between the Western and the Islamic beliefs on individual and community rights
9. 3.9Review the balancing act that needs to be played while fighting terrorism and protecting human rights
10. 3.10Report the controversy around issuing death penalty as punishment
When Muammar Qaddafi used military force to suppress people demonstrating in Libya for a transition to democracy, there was a general consensus that there was a global responsibility to protect civilians. However, when Bashar Assad used fighter jets, tanks, barrel bombs, chemical weapons, and a wide range of brutal methods, including torture, to crush the popular uprising against his rule in Syria, the world did not respond forcefully to protect civilians. The basic reason given for allowing Syria to descend into brutality and chaos was that it was difficult to separate Syrians favoring human rights from those who embraced terrorism. Although cultural values differ significantly from one society to another, our common humanity has equipped us with many shared ideas about how human beings should treat each other. Aspects of globalization, especially communications and migration, reinforce perceptions of a common humanity. In general, there is global agreement that human beings, simply because we exist, are entitled to at least three types of rights. First is civil rights, which include personal liberties such as freedom of speech, religion, and thought; the right to own property; and the right to equal treatment under the law. Second is political rights, including the right to vote, to voice political opinions, and to participate in the political process. Third is social rights, including the right to be secure from violence and other physical danger, the right to a decent standard of living, and the right to health care and education. Societies differ in terms of which rights they emphasize. Four types of human rights claims that dominate global politics are
1. The abuse of individual rights by governments
2. Demands for autonomy or independence by various groups
3. Demands for equality and privacy by groups with unconventional lifestyles
4. Cla.
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This document describes specification tests that can be used after estimating dynamic panel data models using the generalized method of moments (GMM) estimator. It presents GMM estimators for first-order autoregressive models with individual fixed effects that exploit moment restrictions from assuming serially uncorrelated errors. Monte Carlo simulations are used to evaluate the small-sample performance of tests of serial correlation based on GMM residuals, Sargan tests, and Hausman tests. The tests are also applied to estimated employment equations using an unbalanced panel of UK firms.
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This document discusses molecular dynamics simulation methods. It begins by introducing molecular dynamics and Newton's laws of motion. It then describes how molecular dynamics simulations work by integrating Newton's equations of motion to obtain successive molecular configurations over time. Different algorithms for performing the integration are discussed, including the Verlet algorithm and leap-frog method. Predictor-corrector algorithms are also covered. The document provides guidance on best practices for setting up and running molecular dynamics simulations.
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This document summarizes research on edge-preserving image reconstruction methods for electrical impedance tomography (EIT) applied to lung imaging. It presents three main contributions:
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2. An algorithm using level sets and L1 norms that better preserves edges and is more robust to noise and outliers.
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Finite Element Analysis Lecture Notes Anna University 2013 Regulation NAVEEN UTHANDI
One of the most Simple and Interesting topics in Engineering is FEA. My work will guide average students to score good marks. I have given you full package which includes 2 Marks and Question Banks of previous year. All the Best
For Guidance : Comment Below Happy to Teach and Learn along with you guys
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1) Mathematical modeling formulates problems or physical systems into mathematical language using equations that can then be solved using numerical methods, graphics, or analytical solutions.
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3) Mathematical models can be classified as linear or nonlinear, deterministic or probabilistic, static or dynamic, and lumped or distributed parameters.
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Similar to Chapter24rev1.pptPart 6Chapter 24Boundary-Valu.docx (20)
CHAPTER 3Understanding Regulations, Accreditation Criteria, and .docxtiffanyd4
CHAPTER 3
Understanding Regulations, Accreditation Criteria, and Other Standards ofPractice
NAEYC Administrator Competencies Addressed in This Chapter:
Management Knowledge and Skills
2. Legal and Fiscal Management
· Knowledge and application of the advantages and disadvantages of different legal structures
· Knowledge of different codes and regulations as they relate to the delivery of early childhood program services
· Knowledge of child custody, child abuse, special education, confidentiality, anti-discrimination, insurance liability, contract, and laborlaws pertaining to program management
5. Program Operations and Facilities Management
· Knowledge and application of policies and procedures that meet state/local regulations and professional standards pertaining to thehealth and safety of young children
7. Marketing and public relations
· Skill in developing a business plan and effective promotional literature, handbooks, newsletters, and press releases
Early Childhood Knowledge and Skills
5. Children with Special Needs
· Knowledge of licensing standards, state and federal laws (e.g., ADA, IDEA) as they relate to services and accommodations for childrenwith special needs
10. Professionalism
· Knowledge of laws, regulations, and policies that impact professional conduct with children and families
· Knowledge of center accreditation criteria
Learning Outcomes
After studying this chapter, you will be able to:
1. Describe the purpose of regulations that apply to programs of early care and education and list several topics they address.
2. Identify several ways accreditation standards are different from child care regulations.
3. State the purpose of Quality Rating and Improvement Systems (QRIS).
4. List some ways qualifications for administrators and teachers are different for licensure, for accreditation, and in QRIS systems.
5. Identify laws that apply to the childcare workplace, such as those that govern the program’s financial management and employees’well-being.
Marie’s Experience
Marie has been successful over the years in keeping her center in compliance with all licensing regulations. She is proud of her teachers andconfident that the center consistently goes above and beyond licensing provisions designed simply to keep children healthy and safe. She knowsthat the center provides high-quality care to the children it serves, but has never pursued accreditation or participated in her state’s optionalQuality Rating and Improvement System (QRIS) because of the time and effort it would require. Her families have confidence in her program anddo not seem to need this additional assurance that it provides high-quality services day in and day out.
Large numbers of families rely on out-of-home care for their infants, toddlers, preschoolers, and school-age children during the workday. In2011, there were 312,254 licensed child care facilities with a capacity to serve almost 10.2 million children. About 34% of these facilitieswere child care center.
Chapter 3 Human RightsINTERNATIONAL HUMAN RIGHTS–BASED ORGANIZ.docxtiffanyd4
Chapter 3 Human Rights
INTERNATIONAL HUMAN RIGHTS–BASED ORGANIZATIONS LIKE THE UN COMMISSION ON HUMAN RIGHTS HAVE MADE MONITORING HUMAN RIGHTS A GLOBAL ISSUE. The United Nations is headquartered in New York City.
Learning Objectives
1. 3.1Review the expansion of and the commitment to the human rights agenda
2. 3.2Evaluate the milestones that led to the current concerns around human rights
3. 3.3Evaluate some of the philosophical controversies over human rights
4. 3.4Recognize global, regional, national, and local institutions and rules designed to protect human rights across the globe
5. 3.5Report the efforts made globally in bringing violators of human rights to justice
6. 3.6Relate the need for stricter laws to protect women’s human rights across the globe.
7. 3.7Recognize the need to protect the human rights of the disabled
8. 3.8Distinguish between the Western and the Islamic beliefs on individual and community rights
9. 3.9Review the balancing act that needs to be played while fighting terrorism and protecting human rights
10. 3.10Report the controversy around issuing death penalty as punishment
When Muammar Qaddafi used military force to suppress people demonstrating in Libya for a transition to democracy, there was a general consensus that there was a global responsibility to protect civilians. However, when Bashar Assad used fighter jets, tanks, barrel bombs, chemical weapons, and a wide range of brutal methods, including torture, to crush the popular uprising against his rule in Syria, the world did not respond forcefully to protect civilians. The basic reason given for allowing Syria to descend into brutality and chaos was that it was difficult to separate Syrians favoring human rights from those who embraced terrorism. Although cultural values differ significantly from one society to another, our common humanity has equipped us with many shared ideas about how human beings should treat each other. Aspects of globalization, especially communications and migration, reinforce perceptions of a common humanity. In general, there is global agreement that human beings, simply because we exist, are entitled to at least three types of rights. First is civil rights, which include personal liberties such as freedom of speech, religion, and thought; the right to own property; and the right to equal treatment under the law. Second is political rights, including the right to vote, to voice political opinions, and to participate in the political process. Third is social rights, including the right to be secure from violence and other physical danger, the right to a decent standard of living, and the right to health care and education. Societies differ in terms of which rights they emphasize. Four types of human rights claims that dominate global politics are
1. The abuse of individual rights by governments
2. Demands for autonomy or independence by various groups
3. Demands for equality and privacy by groups with unconventional lifestyles
4. Cla.
CHAPTER 13Contributing to the ProfessionNAEYC Administrator Co.docxtiffanyd4
CHAPTER 13
Contributing to the Profession
NAEYC Administrator Competencies Addressed in This Chapter:
Management Knowledge and Skills
1. Personal and Professional Self-Awareness
· The ability to evaluate ethical and moral dilemmas based on a professional code of ethics
8. Leadership and Advocacy
· Knowledge of the legislative process, social issues, and public policy affecting young children and their families
· The ability to advocate on behalf of young children, their families and the profession
Early Childhood Knowledge and Skills
1. Historical and Philosophical Foundations
· Knowledge of research methodologies
10. Professionalism
· Knowledge of different professional organizations, resources, and issues impacting the welfare of early childhood practitioners
· Ability to make professional judgments based on the NAEYC “Code of Ethical Conduct and Statement of Commitment”
· Ability to work as part of a professional team and supervise support staff or volunteers
Learning Outcomes
After studying this chapter, you will be able to:
1. Describe how the field of early childhood education has made progress achieving two of the eight criteria of professional status.
2. Identify the advocacy tools that early childhood advocates should have at their disposal.
3. Discuss opportunities that program administrators have to contribute to the field’s future.
Grace’s Experience
Grace had found that working with children came naturally, and she considered herself to be a gifted teacher after only a short time in theclassroom. She thought she would spend her entire career working directly with children. She is now somewhat surprised how much she isenjoying the new responsibilities that come with being a program director. She is gaining confidence that she can work effectively with allfamilies, even when faced with difficult conversations; and her skills as a supervisor, coach, and mentor are increasing as well. She is nowcomfortable as a leader in her own center and is considering volunteering to fill a leadership role in the local early childhood professionalorganization. That would give her opportunities to refine her leadership skills while contributing to the quality of care provided for childrenthroughout her community.
Early childhood administrators are leaders. They contribute to the profession by making the public aware of the field’s emergingprofessionalism, including its reliance on a code of ethics; engaging in informed advocacy; becoming involved in research to increase whatwe know about how children learn, grow, and develop; and coaching and mentoring novices, experienced practitioners, and emergingleaders.
13.1 PROMOTING PROFESSIONALIZATION1
Lilian Katz, one of the most influential voices in the field of early care and education, began discussions about the professionalism of thefield in the mid-1980s. Her work extended a foundation that had been laid by sociologists, philosophers, and other scholars and continuesto influence how early childhoo.
Chapter 2 The Law of EducationIntroductionThis chapter describ.docxtiffanyd4
Chapter 2 The Law of Education
Introduction
This chapter describes the various agencies and types of law that affect education. It also discusses the organization and functions of the various judicial bodies that have an impact on education. School leadership candidates are introduced to standards of review, significant federal civil rights laws, the contents of legal decisions, and a sample legal brief.
Focus Questions
1. How are federal courts organized, and what kind of decisions do they make?
2. What is law? How is law different from policy?
3. From what source does the authority of local boards of education emanate?
4. How can campus and district leaders remain current with changes in law and policy at the national and state level?
Key Terms
1.
2.
3.
4. En banc
5.
6.
7.
8.
9.
10.
11. Stare decisis
12.
13.
14.
15.
Case Study Confused Yet?
As far as Elise Daniels was concerned, the monthly meeting of the 20 River County middle school principals was the most informative and relaxing activity in her school year. Twice per year, the principals invited a guest to speak to the group. Elise was particularly interested in the fall special guest speaker, the attorney for the state school boards association. Elise had heard him speak several times, so she was aware of his deep knowledge of school law and emerging issues. As the attorney, spoke Elise found herself becoming more anxious. It was as if the attorney was speaking a foreign language. Tinker rules, due process, Title IX, Office of Civil Rights, and the state bullying law. Elise found herself thinking, “The Americans with Disabilities Act has been amended? How am I supposed to keep up with all of this?”
Leadership Perspectives
Middle School Principal Elise Daniels in the case study “Confused Yet?” is correct. School law can be confusing. Educators work in a highly regulated environment directly and indirectly impacted by a wide variety of local, state, and federal authorities. When P–12 educators refer to “the law,” they are often referring to state and/or federal statutes enacted by legislatures (). This understanding is correct. The U.S. Congress and 50 state legislatures are active in the law-making business. To make matters more difficult, the law is constantly changing and evolving as new situations arise. For example, 10 years ago few if any states had passed antibullying laws. By 2008, however, almost every state had some form of antibullying legislation on the books. Soon after, the phenomenon of cyberbullying emerged, and state legislators rushed to add cyberbullying and/or electronic bullying to their state education laws. One can only guess at what new real or perceived problem affecting public P–12 schools will be next.
P–12 educators also refer to school board policy as “law.” However, law and policy are not necessarily identical. , p. 4) defines policy as “one way through which a political system handles a public problem. It includes a government’s expressed inten.
CHAPTER 1 Legal Heritage and the Digital AgeStatue of Liberty,.docxtiffanyd4
CHAPTER 1 Legal Heritage and the Digital Age
Statue of Liberty, New York Harbor
The Statue of Liberty stands majestically in New York Harbor. During the American Revolution, France gave the colonial patriots substantial support in the form of money for equipment and supplies, officers and soldiers who fought in the war, and ships and sailors who fought on the seas. Without the assistance of France, it is unlikely that the American colonists would have won their independence from Britain. In 1886, the people of France gave the Statue of Liberty to the people of the United States in recognition of friendship that was established during the American Revolution. Since then, the Statue of Liberty has become a symbol of liberty and democracy throughout the world.
Learning Objectives
After studying this chapter, you should be able to:
1. Define law.
2. Describe the functions of law.
3. Explain the development of the U.S. legal system.
4. List and describe the sources of law in the United States.
5. Discuss the importance of the U.S. Supreme Court’s decision in Brown v. Board of Education.
Chapter Outline
1. Introduction to Legal Heritage and the Digital Age
2. What Is Law?
1. Landmark U.S. Supreme Court Case • Brown v. Board of Education
3. Schools of Jurisprudential Thought
1. CASE 1.1 • U.S. Supreme Court Case • POM Wonderful LLC v. Coca-Cola Company
2. Global Law • Command School of Jurisprudence of Cuba
4. History of American Law
1. Landmark Law • Adoption of English Common Law in the United States
2. Global Law • Civil Law System of France and Germany
5. Sources of Law in the United States
1. Contemporary Environment • How a Bill Becomes Law
2. Digital Law • Law of the Digital Age
6. Critical Legal Thinking
1. CASE 1.2 • U.S. Supreme Court Case • Shelby County, Texas v. Holder
“ Where there is no law, there is no freedom.”
—John Locke Second Treatise of Government, Sec. 57
Introduction to Legal Heritage and the Digital Age
In the words of Judge Learned Hand, “Without law we cannot live; only with it can we insure the future which by right is ours. The best of men’s hopes are enmeshed in its success.”1 Every society makes and enforces laws that govern the conduct of the individuals, businesses, and other organizations that function within it.
Although the law of the United States is based primarily on English common law, other legal systems, such as Spanish and French civil law, also influence it. The sources of law in this country are the U.S. Constitution, state constitutions, federal and state statutes, ordinances, administrative agency rules and regulations, executive orders, and judicial decisions by federal and state courts.
Human beings do not ever make laws; it is the accidents and catastrophes of all kinds happening in every conceivable way that make law for us.
Plato
Laws IV, 709
Businesses that are organized in the United States are subject to its laws. They are also subject to the laws of other countries in which they operate. Busin.
CHAPTER 1 BASIC CONCEPTS AND DEFINITIONS OF HUMAN SERVICESPAUL F.docxtiffanyd4
This chapter provides definitions and concepts related to the field of human services. It discusses how human services aims to help individuals, families, and communities cope with problems and promote well-being. The chapter outlines three basic concepts in human services: intervention, professionalism, and education. It also discusses the generalist roles of human service workers in helping clients and delivering services. Finally, the chapter examines the social ideology of human services and how it relates to ideas about individual rights and responsibilities in society.
CHAPTER 20 Employment Law and Worker ProtectionWashington DC.docxtiffanyd4
CHAPTER 20 Employment Law and Worker Protection
Washington DC
Federal and state laws provide workers’ compensation and occupational safety laws to protect workers in the United States.
Learning Objectives
After studying this chapter, you should be able to:
1. Explain how state workers’ compensation programs work and describe the benefits available.
2. Describe employers’ duty to provide safe working conditions under the Occupational Safety and Health Act.
3. Describe the minimum wage and overtime pay rules of the Fair Labor Standards Act.
4. Describe the protections afforded by the Family and Medical Leave Act.
5. Describe unemployment insurance and Social Security.
Chapter Outline
1. Introduction to Employment Law and Worker Protection
2. Workers’ Compensation
1. Case 20.1 • Kelley v. Coca-Cola Enterprises, Inc.
3. Occupational Safety
1. Case 20.2 • R. Williams Construction Company v. Occupational Safety and Health Review Commission
4. Fair Labor Standards Act
1. Case 20.3 U.S. SUPREME COURT Case • IBP, Inc. v. Alvarez
5. Family and Medical Leave Act
6. Consolidated Omnibus Budget Reconciliation Act and Employee Retirement Income Security Act
7. Government Programs
“ It is difficult to imagine any grounds, other than our own personal economic predilections, for saying that the contract of employment is any the less an appropriate subject of legislation than are scores of others, in dealing with which this Court has held that legislatures may curtail individual freedom in the public interest.”
—Stone, Justice Dissenting opinion, Morehead v. New York (1936)
Introduction to Employment Law and Worker Protection
Generally, the employer–employee relationship is subject to the common law of contracts and agency law. This relationship is also highly regulated by federal and state governments that have enacted myriad laws that protect workers from unsafe working conditions, require employers to provide workers’ compensation to employers injured on the job, prohibit child labor, require minimum wages and overtime pay to be paid to workers, require employers to provide time off to employees with certain family and medical emergencies, and provide other employee protections and rights.
Poorly paid labor is inefficient labor, the world over.
Henry George
This chapter discusses employment law, workers’ compensation, occupational safety, pay and hour rules, and other laws affecting employment.
Workers’ Compensation
Many types of employment are dangerous, and many workers are injured on the job each year. Under common law, employees who were injured on the job could sue their employers for negligence. This time-consuming process placed the employee at odds with his or her employer. In addition, there was no guarantee that the employee would win the case. Ultimately, many injured workers—or the heirs of deceased workers—were left uncompensated.
Workers’ compensation acts were enacted by states in response to the unfairness of that result. These acts crea.
Chapter 1 Global Issues Challenges of GlobalizationA GROWING .docxtiffanyd4
Chapter 1 Global Issues: Challenges of Globalization
A GROWING WORLDWIDE CONNECTEDNESS IN THE AGE OF GLOBALIZATION HAS GIVEN CITIZENS MORE OF A VOICE TO EXPRESS THEIR DISSATISFACTION. In Brazil, Protestors calling for a wide range of reforms marched toward the soccer stadium where a match would be played between Brazil and Uruguay.
Learning Objectives
1. 1.1Identify important terms in international relations
2. 1.2Report the need to adopt an interdisciplinary approach in understanding the impact of new world events
3. 1.3Examine the formation of the modern states with respect to the thirty years’ war in 1618
4. 1.4Recall the challenges to the four types of sovereignty
5. 1.5Report that the European Union was created by redefining the sovereignty of its nations for lasting peace and security
6. 1.6Recall the influence exerted by the Catholic church, transnational companies, and other NGOs in dictating world events
7. 1.7Examine how globalization has brought about greater interdependence between states
8. 1.8Record the major causes of globalization
9. 1.9Review the most important forms of globalization
10. 1.10Recount the five waves of globalization
11. 1.11Recognize reasons as to why France and the US resist globalization
12. 1.12Examine the three dominant views of the extent to which globalization exists
Revolutions in technology, finance, transportation, and communications and different ways of thinking that characterize interdependence and globalization have eroded the power and significance of nation-states and profoundly altered international relations. Countries share power with nonstate actors that have proliferated as states have failed to deal effectively with major global problems.
Many governments have subcontracted several traditional responsibilities to private companies and have created public-private partnerships in some areas. This is exemplified by the hundreds of special economic zones in China, Dubai, and elsewhere. Contracting out traditional functions of government, combined with the centralization of massive amounts of data, facilitated Edward Snowden’s ability to leak what seems to be an almost unlimited amount of information on America’s spying activities.
The connections between states and citizens, a cornerstone of international relations, have been weakened partly by global communications and migration. Social media enable people around the world to challenge governments and to participate in global governance. The prevalence of mass protests globally demonstrates growing frustration with governments’ inability to meet the demands of the people, especially the global middle class.
The growth of multiple national identities, citizenships, and passports challenges traditional international relations. States that played dominant roles in international affairs must now deal with their declining power as global power is more diffused with the rise of China, India, Brazil, and other emerging market countries. States are i.
CHAPTER 23 Consumer ProtectionRestaurantFederal and state go.docxtiffanyd4
This chapter discusses various laws and government regulations regarding consumer protection. It covers regulations of food and drug safety, including the Food, Drug, and Cosmetic Act which is enforced by the Food and Drug Administration. The chapter also discusses laws providing protections for consumers in regards to products, automobiles, healthcare, unfair business practices, and consumer finances. The overall goal of consumer protection laws is to promote safety and prohibit abusive practices against consumers.
Chapter 18 When looking further into the EU’s Energy Security and.docxtiffanyd4
Chapter 18
: When looking further into the EU’s Energy Security and ICT sustainable urban development, and government policy efforts:
Q2
– What are the five ICT enablers of energy efficiency identified by European strategic research Road map to ICT enabled Energy-Efficiency in Buildings and constructions, (REEB, 2010)?
identify and name those
five ICT enablers
,
provide a brief narrative for each enabler,
note:
Need 400 words. Need references
Please find the attached
.
CHAPTER 17 Investor Protection and E-Securities TransactionsNe.docxtiffanyd4
CHAPTER 17 Investor Protection and E-Securities Transactions
New York Stock Exchange
This is the home of the New York Stock Exchange (NYSE) in New York City. The NYSE, nicknamed the Big Board, is the premier stock exchange in the world. It lists the stocks and securities of approximately 3,000 of the world’s largest companies for trading. The origin of the NYSE dates to 1792, when several stockbrokers met under a buttonwood tree on Wall Street. The NYSE is located at 11 Wall Street, which has been designated a National Historic Landmark. The NYSE is now operated by NYSE Euronext, which was formed when the NYSE merged with the fully electronic stock exchange Euronext.
Learning Objectives
After studying this chapter, you should be able to:
1. Describe the procedure for going public and how securities are registered with the Securities and Exchange Commission (SEC).
2. Describe e-securities transactions and public offerings.
3. Describe the requirements for qualifying for private placement, intrastate, and small offering exemptions from registration.
4. Describe insider trading that violates Section 10(b) of the Securities Exchange Act of 1934.
5. Describe the changes made to securities law by the Jumpstart Our Business Startups (JOBS) Act and its effect on raising capital by small businesses.
Chapter Outline
1. Introduction to Investor Protection and E-Securities Transactions
2. Securities Law
1. LANDMARK LAW • Federal Securities Laws
3. Definition of Security
4. Initial Public Offering: Securities Act of 1933
1. BUSINESS ENVIRONMENT • Facebook’s Initial Public Offering
2. CONTEMPORARY ENVIRONMENT • Jumpstart Our Business Startups (JOBS) Act: Emerging Growth Company
5. E-Securities Transactions
1. DIGITAL LAW • Crowdfunding and Funding Portals
6. Exempt Securities
7. Exempt Transactions
8. Trading in Securities: Securities Exchange Act of 1934
9. Insider Trading
1. Case 17.1 • United States v. Bhagat
2. Case 17.2 • United States v. Kluger
3. ETHICS • Stop Trading on Congressional Knowledge Act
10. Short-Swing Profits
11. State “Blue-Sky” Laws
“The insiders here were not trading on an equal footing with the outside investors.”
—Judge Waterman Securities and Exchange Commission v. Texas Gulf Sulphur Company 401 F.2d 833, 1968 U.S. App. Lexis 5796 (1968)
Introduction to Investor Protection and E-Securities Transactions
Prior to the 1920s and 1930s, the securities markets in this country were not regulated by the federal government. Securities were issued and sold to investors with little, if any, disclosure. Fraud in these transactions was common. To respond to this lack of regulation, in the early 1930s Congress enacted federal securities statutes to regulate the securities markets, including the Securities Act of 1933 and the Securities Exchange Act of 1934. The federal securities statutes were designed to require disclosure of information to investors, provide for the regulation of securities issues and trading, and prevent fraud. Today, many .
Chapter 13 Law, Ethics, and Educational Leadership Making the Con.docxtiffanyd4
Chapter 13 Law, Ethics, and Educational Leadership: Making the Connection
Introduction
This chapter presents examples from the ISLLC standards of the relationship between law and ethics. The chapter also provides examples of how knowledge of law and the application of ethical principles to decision making helps guide school leaders through the sometimes treacherous waters of educational leadership.
Focus Questions
1. How may ethical considerations and legal knowledge guide school leader decision making?
2. Why is it important to consider a balance between these two sometimes competing concepts?
Case Study So Many Detentions, So Little Time
Jefferson Middle School (JMS) was the most racially and culturally diverse of the three middle schools in Riverboat School District, a relatively affluent bedroom community within commuter distance of Capital City. Unfortunately, the culture of Jefferson Middle School was not going well. Over the past 5 years, assistant superintendent Sharon Grey had seen JMS become a school divided by an underlying animosity along racial and socioeconomic lines. This animosity was characterized by numerous clashes between student groups, between teachers and students, between campus administrators and teachers, and between teachers and parents. Sharon finally concluded that JMS was a “mess.”
After much thought and a few sleepless nights, Sharon as part of her job description made the recommendation to the Riverboat school board to not reemploy Jeremy Smith as principal of JMS. Immediately after the board decision, Sharon organized a search committee of teachers, parents, and campus administrators and began the process of finding the right principal for JMS. The committee finally agreed on Charleston Jones. Charleston was a relatively inexperienced campus administrator but had impressed the committee with his instructional leadership knowledge, intelligence, and youthful energy. However, the job of stabilizing JMS was proving to be more of a challenge than anyone had anticipated.
Charleston had instituted a schoolwide discipline plan and had insisted that teachers and school administrators not deviate from the plan. However, he could sense that things were still not right. Animosity among student and parent groups remained just below the surface, ready to erupt at the slightest provocation. Clashes between teachers and students were still relatively frequent. Teachers still blamed one another, school administrators, and the school resource officer for a lack of order in the school. Change was not coming quickly to RMS, and Charleston understood that although school management had improved, several aspects of school culture were less than desirable. Student suspension rates remained high, and parental support was waning. As one of the assistant principals remarked after the umpteenth student referral, “So many detentions, so little time!”
Charleston felt the need to talk. He reached for the phone and made an appointment with.
Chapter 12 presented strategic planning and performance with Int.docxtiffanyd4
Chapter 12 presented strategic planning and performance with Intuit. Define Key Performance Indicators (KPI) and Key Risk Indicators (KRI)? How does an organization come up with these key indicators? Do you know of any top-down indicators? Do you know of any bottom-up indicators? Give some examples of both. In what way does identifying these indicators help an organization? Are there any other key indicators that would help an organization?
Requirements:
Initial posting by Wednesday
Reply to at least 2 other classmates by Sunday (Post a response on different days throughout the week)
Provide a minimum of 2 references on the initial post and one reference any response posts.
Proper APA Format (References & Citations)/No plagiarism
.
ChapterTool KitChapter 7102715Corporate Valuation and Stock Valu.docxtiffanyd4
ChapterTool KitChapter 710/27/15Corporate Valuation and Stock Valuation7-4 Valuing Common Stocks—Introducing the Free Cash Flow (FCF) Valuation ModelData for B&B Corporation (Millions)Constant free cash flow (FCF) =$10Weighted average cost of capital (WACC) =10%Short-term investments =$2Debt =$28Preferred stock =$4Number of shares of common stock =5The first step is to estimate the value of operations, which is the present value of all expected free cash flows. Because the FCF's are expected to be constant, this is a perpetuity. The present value of a perpetuity is the cash flow divided by the cost of capital:Value of operations (Vop) =FCF/WACCValue of operations (Vop) =$100.00millionB&B's total value is the sum of value of operations and the short-term investments: Value of operations$100+ ST investments$2Estimated total intrinsic value$102The next step is to estimate the intrinsic value of equity, which is the remaining total value after accounting for the claims of debtholders and preferred stockholders: Value of operations$100+ ST investments$2Estimated total intrinsic value$102− All debt$28− Preferred stock$4Estimated intrinsic value of equity$70The final step is to estimate the intrinsic common stock price per share, which is the estimated intrinsic value of equity divided by the number of shares of common stock: Value of operations$100+ ST investments$2Estimated total intrinsic value$102− All debt$28− Preferred stock$4Estimated intrinsic value of equity$70÷ Number of shares5Estimated intrinsic stock price =$14.00The figure below shows a summary of the previous calculations.Figure 7-2B&B Corporation's Sources of Value and Claims on Value (Millions of Dollars except Per Share Data)Inputs:Valuation AnalysisConstant free cash flow (FCF) =$10Value of operations$100Weighted average cost of capital (WACC) =10%+ ST investments$2Short-term investments =$2Estimated total intrinsic value$102Debt =$28− All debt$28Preferred stock =$4− Preferred stock$4Number of shares of common stock =5Estimated intrinsic value of equity$70÷ Number of shares5Estimated intrinsic stock price$14.00Data for Pie ChartsShort-term investments =$2Value of operations =$100Total =$102Debt =$28Preferred stock =$4Estimated equity value =$70Total =$1027-5 The Constant Growth Model: Valuation when Expected Free Cash Flow Grows at a Constant RateCase 1: The expected free cash flow at t=1 and the expected constant growth rate after t=1 are known.First expected free cash flow (FCF1) =$105Weighted average cost of capital (WACC) =9%Constant growth rate (gL) =5%When free cash flows are expected to grow at a constant rate, the value of operations is:Value of operations (Vop) =FCF1 / [WACC-gL]Value of operations (Vop) =$2,625Case 2: Constant growth is expected to begin immediately.Most recent free cash flow (FCF0) =$200Weighted average cost of capital (WACC) =12%Constant growth rate (gL) =7%When free cash flows are expected to grow at a constant rate, the value of operations is:.
CHAPTER 12Working with Families and CommunitiesNAEYC Administr.docxtiffanyd4
CHAPTER 12
Working with Families and Communities
NAEYC Administrator Competencies Addressed in This Chapter:
Management Knowledge and Skills
6. Family Support
· Knowledge and application of family systems and different parenting styles
· The ability to implement program practices that support families of diverse cultural, ethnic, linguistic, and socio-economic backgrounds
· The ability to support families as valued partners in the educational process
3. Staff Management and Human Relations
· The ability to relate to staff and board members of diverse racial, cultural, and ethnic backgrounds
7. Marketing and Public Relations
· The ability to promote linkages with local schools
9. Oral and Written Communication
· Knowledge of oral communication techniques, including establishing rapport, preparing the environment, active listening, and voicecontrol
· The ability to communicate ideas effectively in a formal presentation
Early Childhood Knowledge and Skills
6. Family and Community Relationships
· Knowledge of the diversity of family systems, traditional, non-traditional and alternative family structures, family life styles, and thedynamics of family life on the development of young children
· Knowledge of socio-cultural factors influencing contemporary families including the impact of language, religion, poverty, race,technology, and the media
· Knowledge of different community resources, assistance, and support available to children and families
· Knowledge of different strategies to promote reciprocal partnerships between home and center
· Ability to communicate effectively with parents through written and oral communication
· Ability to demonstrate awareness and appreciation of different cultural and familial practices and customs
· Knowledge of child rearing patterns in other countries
10. Professionalism
· Ability to make professional judgments based on the NAEYC “Code of Ethical Conduct and Statement of Commitment”
Learning Outcomes
After studying this chapter, you will be able to:
1. Explain three approaches that programs of early care and education might take to working with families.
2. Identify some of the benefits enjoyed by children, families, and programs when families are engaged with the programs serving theiryoung children.
3. Describe some effective strategies for building trusting relationships with all families.
4. Identify the stakeholder groups and the kinds of expertise that should be represented on programs’ advisory committees and boardsof directors.
Grace’s Experience
The program that Grace directs has been an important part of the neighborhood for more than 20 years. She knows she is benefiting from thegoodwill it has earned over the years. It is respected because of its tradition of high-quality outreach projects, such as the sing-along the childrenpresent at the senior center in the spring. The program’s tradition of community involvement has meant that local businesses have always beenwilling to help out when asked fo.
Chapter 10. Political Socialization The Making of a CitizenLear.docxtiffanyd4
Chapter 10. Political Socialization: The Making of a Citizen
Learning Objectives
· 1Describe the model citizen in democratic theory and explain the concept.
· 2Define socialization and explain the relevance of this concept in the study of politics.
· 3Explain how a disparate population of individuals and groups (families, clans, and tribes) can be forged into a cohesive society.
· 4Demonstrate how socialization affects political behavior and analyze what happens when socialization fails.
· 5Characterize the role of television and the Internet in influencing people’s political beliefs and behavior, and evaluate their impact on the quality of citizenship in contemporary society.
The year is 1932. The Soviet Union is suffering a severe shortage of food, and millions go hungry. Joseph Stalin, leader of the Communist Party and head of the Soviet government, has undertaken a vast reordering of Soviet agriculture that eliminates a whole class of landholders (the kulaks) and collectivizes all farmland. Henceforth, every farm and all farm products belong to the state. To deter theft of what is now considered state property, the Soviet government enacts a law prohibiting individual farmers from appropriating any grain for their own private use. Acting under this law, a young boy reports his father to the authorities for concealing grain. The father is shot for stealing state property. Soon after, the boy is killed by a group of peasants, led by his uncle, who are outraged that he would betray his own father. The government, taking a radically different view of the affair, extols the boy as a patriotic martyr.
Stalin considered the little boy in this story a model citizen, a hero. How citizenship is defined says a lot about a government and the philosophy or ideology that underpins it.
The Good Citizen
Stalin’s celebration of a child’s act of betrayal as heroic points to a distinction Aristotle originally made: The good citizen is defined by laws, regimes, and rulers, but the moral fiber (and universal characteristics) of a good person is fixed, and it transcends the expectations of any particular political regime.*
Good citizenship includes behaving in accordance with the rules, norms, and expectations of our own state and society. Thus, the actual requirements vary widely. A good citizen in Soviet Russia of the 1930s was a person whose first loyalty was to the Communist Party. The test of good citizenship in a totalitarian state is this: Are you willing to subordinate all personal convictions and even family loyalties to the dictates of political authority, and to follow the dictator’s whims no matter where they may lead? In marked contrast are the standards of citizenship in constitutional democracies, which prize and protect freedom of conscience and speech.
Where the requirements of the abstract good citizen—always defined by the state—come into conflict with the moral compass of actual citizens, and where the state seeks to obscure or obliterate t.
Chapters one and twoAnswer the questions in complete paragraphs .docxtiffanyd4
Chapters one and two
Answer the questions in complete paragraphs (at least 3), APA style (citations/references) and make sure to separate/number the answers
1. Explain the differences between Classic Autism and Asperger Disorder according to the DSM-V (Diagnostic Statistical Manual of the American Psychiatric Association).
2. How is ASD identified and diagnosed? Name and describe some of the measurement tools.
3. Describe the characteristics of ASD under each criterion: a) language deficits, b) social differences, c) behavior, and d) motor deficits.
4. List and describe the evidence-base practices for educating ASD children discussed in chapter 2.
5. Describe the differences between a focused intervention and comprehensive treatment models.
6. What are the components of effective instruction for students with ASD?
.
ChapterTool KitChapter 1212912Corporate Valuation and Financial .docxtiffanyd4
ChapterTool KitChapter 1212/9/12Corporate Valuation and Financial Planning12-2 Financial Planning at MicroDrive, Inc.The process used by MicroDrive to forecast the free cash flows from its operating plan is described in the sections below.Setting Up the Model to Forecast OperationsWe begin with MicroDrive's most recent financial statements and selected additional data.Figure 12-1 MicroDrive’s Most Recent Financial Statements (Millions, Except for Per Share Data)INCOME STATEMENTSBALANCE SHEETS20122013Assets20122013Net sales$ 4,760$ 5,000Cash$ 60$ 50COGS (excl. depr.)3,5603,800ST Investments40-Depreciation170200Accounts receivable380500Other operating expenses480500Inventories8201,000EBIT$ 550$ 500Total CA$ 1,300$ 1,550Interest expense100120Net PP&E1,7002,000Pre-tax earnings$ 450$ 380Total assets$ 3,000$ 3,550Taxes (40%)180152NI before pref. div.$ 270$ 228Liabilities and equityPreferred div.88Accounts payable$ 190$ 200Net income$ 262$ 220Accruals280300Notes payable130280Other DataTotal CL$ 600$ 780Common dividends$48$50Long-term bonds1,0001,200Addition to RE$214$170Total liabilities$ 1,600$ 1,980Tax rate40%40%Preferred stock100100Shares of common stock5050Common stock500500Earnings per share$5.24$4.40Retained earnings800970Dividends per share$0.96$1.00Total common equity$ 1,300$ 1,470Price per share$40.00$27.00Total liabs. & equity$ 3,000$ 3,550The figure below shows all the inputs required to project the financial statements for the scenario that has been selected with the Scenario Manager: Data, What-If Analysis, Scenario Manager. There are two scenarios. The first is named Status Quo because all operating ratios except the sales growth rate are assumed to remain unchanged. The initial sales growth rate was chosen by MicroDrive's managers based on the existing product lines. The growth rate declines over time until it eventually levels off at a sustainable rate. The other scenario is named Final because it is the set of inputs chosen by MicroDrive's management team.Section 1 shows the inputs required to estimate the items in an operating plan. For each of these inputs, Section 1 shows the industry averages, the actual values for the past two years for MicroDrive, and the forecasted values for the next five years. The managers assumed the inputs for future years (except the sales growth rate) would be equal to the inputs in the first projected year.MicroDrive's managers assume that sales will eventually level off at a sustaniable constant rate.Sections 2 and 3 show the data required to estimate the weighted average cost of capital. Section 4 shows the forecasted growth rate in dividends.Note: These inputs are linked throughout the model. If you want to change an input, do it here and not other places in the model.Figure 12-2MicroDrive's Forecast: Inputs for the Selected ScenarioStatus QuoIndustryMicroDriveMicroDriveInputsActualActualForecast1. Operating Ratios2013201220132014201520162017201.
Chapters 4-6 Preparing Written MessagesPrepari.docxtiffanyd4
Chapters 4-6: Preparing Written Messages
Preparing Written Messages
Lesson Outline
Seven Steps to Preparing Written Messages
Effective Sentences and Coherent Paragraphs
Revise to Grab Your Audience’s Attention
Improve Readability
Proofread and Revise
Seven Steps to Preparing
Written Messages
Seven Preparation Steps
Step 1: Consider Contextual Forces
Step 2: Determine Purpose, Channel, and Medium
Step 3: Envision Audience
Step 4: Adapt Message to Audience Needs and Concerns
Step 5: Organize the Message
Step 6: Prepare First Draft
Step 7: Revise, Edit, and Proofread
Effective Sentences and
Coherent Paragraphs
Step 6: Prepare the First Draft
Proceed Deductively or Inductively
Know Logical Sequence of Minor Points
Write rapidly with Intent to Rewrite
Use Active More Than Passive Voice
Craft Powerful Sentences
Rely on Active Voice—Subject Doer of Action
(Passive—Subject Receiver of Action Sentence Is Less Emphatic)
Passive Voice Uses
Conceal the Doer/Avoid Finger Pointing
Doer Is Unknown
Place More Emphasis on What Was Done
(Receiver of Action)
5
Emphasize Important Ideas
Techniques
Sentence Structure—place important ideas in simple sentences/place in independent clauses (emphasis)
Repetition—repeat a word in a sentence
Labeling Words—use words that signal important
Position—position it first or last in a clause, sentence, paragraph, or presentation
Space and Format—use extraordinary amount of space for important items or use headings
Develop Coherent Paragraphs
Develop Deductive/Inductive Paragraphs Consistently
Link Ideas to Achieve Coherence
Keep Paragraphs Unified
Vary Sentence and Paragraph Length
Position Topic Sentences and
Link Ideas
Deductive—topic sentence precedes details
Inductive—topic sentence follows details
Link Ideas to Achieve Coherence (Cohesion)
Repeat Word from Preceding Sentence
Use a Pronoun for a Noun in Preceding Sentence
Use Connecting Words (e.g., Conjunctive Adverbs)
Link Paragraphs by Using Transition Words
Use Transition Sentences before Headings,
But Not Subheadings
Paragraph Unity
Keep Paragraphs Unified—support must be focused on topic sentences
Ensure Paragraphs Cover Topic Sentence, But Do Not Write Extraneous Materials
Arrange Paragraphs in a Logical and Systematic Sequence
Vary Sentence and
Paragraph Length
Vary Sentence Length (Average—Short)
Vary Sentence Structure (Sentence Variety)
Vary Paragraph Length (Average—Short
8-10 Lines)
Changes in Tense, Voice, and Person in Paragraphs Are Discouraged
Revise to Grab
Reader’s Attention
Cultivate a Frame of Mind (Mind-set) for Revising and Proofreading
Have Your Revising/Editing Space/Room
View from Audience Perspective (You Attitude)
Revise until No More Changes Would Improve the Document
Be Willing to Allow Others to Make Suggestions (Writer’s Pride of Ownership?)
Ensure Error-Free Messages
Use Visual Enhancements for More Readability
Add Only When They Aid Comprehension
Create an A.
How to Setup Warehouse & Location in Odoo 17 InventoryCeline George
In this slide, we'll explore how to set up warehouses and locations in Odoo 17 Inventory. This will help us manage our stock effectively, track inventory levels, and streamline warehouse operations.
How to Build a Module in Odoo 17 Using the Scaffold MethodCeline George
Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
The simplified electron and muon model, Oscillating Spacetime: The Foundation...RitikBhardwaj56
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
4. function dy=dydxn(x,y)
dy=[y(2);…
-0.05*(200-y(1))-2.7e-9*(1.6e9-y(1)^4)];Code for residual:
function r=res(za)
[x,y]=ode45(@dydxn, [0 10], [300 za]);
r=y(length(x),1)-400;Code for finding root of residual:
fzero(@res, -50) Code for solving system:
[x,y]=ode45(@dydxn, [0 10], [300 fzero(@res, -50) ]);
Finite-Difference MethodsThe most common alternatives to the
shooting method are finite-difference approaches.In these
techniques, finite differences are substituted for the derivatives
in the original equation, transforming a linear differential
equation into a set of simultaneous algebraic equations.
Finite-Difference ExampleConvert:
5. into n-1 simultaneous equations at each interior point using
centered difference equations:
Finite-Difference Example (cont)Since T0 and Tn are known,
they will be on the right-hand-side of the linear algebra system
(in this case, in the first and last entries, respectively):
Derivative Boundary ConditionsNeumann boundary conditions
are resolved by solving the centered difference equation at the
point and rewriting the system equation accordingly.For
example, if there is a Neumann condition at the T0 point,
Finite-Difference Method for Nonlinear ODEsRoot location
methods for systems of equations may be used to solve
nonlinear ODEs.Another method is to adapt a successive
substitution algorithm to calculate the values of the interior
points.
d
2
T
dx
20. behavior of a simple physical system.Understanding how
numerical methods afford a means to generalize solutions in a
manner that can be implemented on a digital
computer.Understanding the different types of conservation
laws that lie beneath the models used in the various engineering
disciplines and appreciating the difference between steady-state
and dynamic solutions of these models.Learning about the
different types of numerical methods we will cover in this book.
A Simple Mathematical ModelA mathematical model can be
broadly defined as a formulation or equation that expresses the
essential features of a physical system or process in
mathematical terms.Models can be represented by a functional
relationship between dependent variables, independent
variables, parameters, and forcing functions.
Model Function
Dependent variable - a characteristic that usually reflects the
behavior or state of the system Independent variables -
dimensions, such as time and space, along which the system’s
behavior is being determined Parameters - constants reflective
of the system’s properties or composition Forcing functions -
external influences acting upon the system
Model Function ExampleAssuming a bungee jumper is in mid-
flight, an analytical model for the jumper’s velocity, accounting
for drag, is
21. Dependent variable - velocity vIndependent variables - time
tParameters - mass m, drag coefficient cdForcing function -
gravitational acceleration g
Model ResultsUsing a computer (or a calculator), the model can
be used to generate a graphical representation of the system.
For example, the graph below represents the velocity of a 68.1
kg jumper, assuming a drag coefficient of 0.25 kg/m
Numerical ModelingSome system models will be given as
implicit functions or as differential equations - these can be
solved either using analytical methods or numerical
methods.Example - the bungee jumper velocity equation from
before is the analytical solution to the differential equation
where the change in velocity is determined by the gravitational
forces acting on the jumper versus the drag force.
Numerical MethodsTo solve the problem using a numerical
method, note that the time rate of change of velocity can be
approximated as:
22. Euler's MethodSubstituting the finite difference into the
differential equation gives
Solve for
Numerical ResultsApplying Euler's method in 2 s intervals
yields:
How do we improve the solution?Smaller steps
Bases for Numerical ModelsConservation laws provide the
foundation for many model functions. Different fields of
engineering and science apply these laws to different paradigms
within the field.Among these laws are:
Conservation of mass
Conservation of momentum
Conservation of charge
Conservation of energy
23. Summary of Numerical MethodsThe book is divided into five
categories of numerical methods:
Dependent
variable
=
f
independent
variables
,
parameters,
forcing
functions
æ
è
ç
ö
ø
÷
v
t
(
)
=
gm
c
d
tanh
gc
d
m
t
26. create a simple line plot based on an equation.
The MATLAB EnvironmentMATLAB uses three primary
windows-
Command window - used to enter commands and data
Graphics window(s) - used to display plots and graphics
Edit window - used to create and edit M-files
(programs)Depending on your computer platform and the
version of MATLAB used, these windows may have different
looks and feels.
Calculator ModeThe MATLAB command widow can be used as
a calculator where you can type in commands line by line.
Whenever a calculation is performed, MATLAB will assign the
result to the built-in variable ansExample:
>> 55 - 16
ans =
39
MATLAB VariablesWhile using the ans variable may be useful
for performing quick calculations, its transient nature makes it
less useful for programming.MATLAB allows you to assign
values to variable names. This results in the storage of values
to memory locations corresponding to the variable
27. name.MATLAB can store individual values as well as arrays; it
can store numerical data and text (which is actually stored
numerically as well).MATLAB does not require that you pre-
initialize a variable; if it does not exist, MATLAB will create it
for you.
ScalarsTo assign a single value to a variable, simply type the
variable name, the = sign, and the value:
>> a = 4
a =
4Note that variable names must start with a letter, though they
can contain letters, numbers, and the underscore (_) symbol
Scalars (cont)You can tell MATLAB not to report the result of a
calculation by appending the semi-solon (;) to the end of a line.
The calculation is still performed.You can ask MATLAB to
report the value stored in a variable by typing its name:
>> a
a =
4
28. Scalars (cont)You can use the complex variable i (or j) to
represent the unit imaginary number.You can tell MATLAB to
report the values back using several different formats using the
format command. Note that the values are still stored the same
way, they are just displayed on the screen differently. Some
examples are:short - scaled fixed-point format with 5 digitslong
- scaled fixed-point format with 15 digits for double and 7
digits for singleshort eng - engineering format with at least 5
digits and a power that is a multiple of 3 (useful for SI prefixes)
Format Examples>> format short; pi
ans =
3.1416
>> format long; pi
ans =
3.14159265358979
>> format short eng; pi
29. ans =
3.1416e+000
>> pi*10000
ans =
31.4159e+003Note - the format remains the same unless
another format command is issued.
Arrays, Vectors, and MatricesMATLAB can automatically
handle rectangular arrays of data - one-dimensional arrays are
called vectors and two-dimensional arrays are called
matrices.Arrays are set off using square brackets [ and ] in
MATLABEntries within a row are separated by spaces or
commasRows are separated by semicolons
Array Examples>> a = [1 2 3 4 5 ]
a =
1 2 3 4 5
>> b = [2;4;6;8;10]
30. b =
2
4
6
8
10Note 1 - MATLAB does not display the bracketsNote 2 - if
you are using a monospaced font, such as Courier, the displayed
values should line up properly
MatricesA 2-D array, or matrix, of data is entered row by row,
with spaces (or commas) separating entries within the row and
semicolons separating the rows:
>> A = [1 2 3; 4 5 6; 7 8 9]
A =
1 2 3
4 5 6
31. 7 8 9
Useful Array CommandsThe transpose operator (apostrophe)
can be used to flip an array over its own diagonal. For example,
if b is a row vector, b’ is a column vector containing the
complex conjugate of b.The command window will allow you to
separate rows by hitting the Enter key - script files and
functions will allow you to put rows on new lines as well.The
who command will report back used variable names; whos will
also give you the size, memory, and data types for the arrays.
Accessing Array EntriesIndividual entries within a array can be
both read and set using either the index of the location in the
array or the row and column.The index value starts with 1 for
the entry in the top left corner of an array and increases down a
column - the following shows the indices for a 4 row, 3 column
matrix:
1 5 9
2 6 10
3 7 11
4 8 12
32. Accessing Array Entries (cont)Assuming some matrix C:
C =
2 4 9
3 3 16
3 0 8
10 13 17C(2) would report 3C(4) would report 10C(13)
would report an error!Entries can also be access using the row
and column:C(2,1) would report 3C(3,2) would report 0C(5,1)
would report an error!
Array Creation - Built InThere are several built-in functions to
create arrays:zeros(r,c) will create an r row by c column matrix
of zeroszeros(n) will create an n by n matrix of zerosones(r,c)
will create an r row by c column matrix of onesones(n) will
create an n by n matrix one oneshelp elmat has, among other
things, a list of the elementary matrices
Array Creation - Colon OperatorThe colon operator : is useful
in several contexts. It can be used to create a linearly spaced
array of points using the notation
33. start:diffval:limit
where start is the first value in the array, diffval is the
difference between successive values in the array, and limit is
the boundary for the last value (though not necessarily the last
value).
>>1:0.6:3
ans =
1.0000 1.6000 2.2000 2.8000
Colon Operator - NotesIf diffval is omitted, the default value is
1:
>>3:6
ans =
3 4 5 6To create a decreasing series, diffval must be
negative:
>> 5:-1.2:2
34. ans =
5.0000 3.8000 2.6000If start+diffval>limit for an
increasing series or start+diffval<limit for a decreasing series,
an empty matrix is returned:
>>5:2
ans =
Empty matrix: 1-by-0To create a column, transpose the output
of the colon operator, not the limit value; that is, (3:6)’ not 3:6’
Array Creation - linspaceTo create a row vector with a specific
number of linearly spaced points between two numbers, use the
linspace command.linspace(x1, x2, n) will create a linearly
spaced array of n points between x1 and x2
>>linspace(0, 1, 6)
ans =
0 0.2000 0.4000 0.6000 0.8000 1.0000If n is omitted, 100
points are created.To generate a column, transpose the output of
the linspace command.
35. Array Creation - logspaceTo create a row vector with a specific
number of logarithmically spaced points between two numbers,
use the logspace command.logspace(x1, x2, n) will create a
logarithmically spaced array of n points between 10x1 and 10x2
>>logspace(-1, 2, 4)
ans =
0.1000 1.0000 10.0000 100.0000If n is omitted, 100 points
are created.To generate a column, transpose the output of the
logspace command.
Character Strings & EllipsisAlphanumeric constants are
enclosed by apostrophes (')
>> f = 'Miles ';
>> s = 'Davis'Concatenation: pasting together of strings
>> x = [f s]
x =
Miles DavisEllipsis (...): Used to continue long lines
>> a = [1 2 3 4 5 ...
6 7 8]
a =
1 2 3 4 5 6 7 8You cannot use an ellipsis
within single quotes to continue a string. But you can piece
together shorter strings with ellipsis
>> quote = ['Any fool can make a rule,' ...
' and any fool will mind it']
quote =
36. Any fool can make a rule, and any fool will mind it
Mathematical OperationsMathematical operations in MATLAB
can be performed on both scalars and arrays.The common
operators, in order of priority, are:^Exponentiation4^2 = 8-
Negation (unary operation)-8 = -8* /Multiplication and
Division2*pi = 6.2832 pi/4 = 0.7854Left Division62 = 0.3333
+ -Addition and Subtraction3+5 = 8 3-5 = -2
Order of OperationsThe order of operations is set first by
parentheses, then by the default order given above:
y = -4 ^ 2 gives y = -16
since the exponentiation happens first due to its higher default
priority, but
y = (-4) ^ 2 gives y = 16
since the negation operation on the 4 takes place first
Complex NumbersAll the operations above can be used with
complex quantities (i.e. values containing an imaginary part
entered using i or j and displayed using i)
>> x = 2+i*4; (or 2+4i, or 2+j*4, or 2+4j)
>> y = 16;
>> 3 * x
37. ans =
6.0000 +12.0000i
>> x+y
ans =
18.0000 + 4.0000i
>> x'
ans =
2.0000 - 4.0000i
Vector-Matrix CalculationsMATLAB can also perform
operations on vectors and matrices.The * operator for matrices
is defined as the outer product or what is commonly called
“matrix multiplication.”
The number of columns of the first matrix must match the
number of rows in the second matrix.
The size of the result will have as many rows as the first matrix
and as many columns as the second matrix.
The exception to this is multiplication by a 1x1 matrix, which is
actually an array operation.The ^ operator for matrices results
in the matrix being matrix-multiplied by itself a specified
number of times.
Note - in this case, the matrix must be square!
38. Element-by-Element CalculationsAt times, you will want to
carry out calculations item by item in a matrix or vector. The
MATLAB manual calls these array operations. They are also
often referred to as element-by-element operations. MATLAB
defines .* and ./ (note the dots) as the array multiplication and
array division operators.
For array operations, both matrices must be the same size or one
of the matrices must be 1x1Array exponentiation (raising each
element to a corresponding power in another matrix) is
performed with .^
Again, for array operations, both matrices must be the same size
or one of the matrices must be 1x1
Built-In FunctionsThere are several built-in functions you can
use to create and manipulate data.The built-in help function can
give you information about both what exists and how those
functions are used:
help elmat will list the elementary matrix creation and
manipulation functions, including functions to get information
about matrices.
help elfun will list the elementary math functions, including
trig, exponential, complex, rounding, and remainder
functions.The built-in lookfor command will search help files
for occurrences of text and can be useful if you know a
function’s purpose but not its name
GraphicsMATLAB has a powerful suite of built-in graphics
functions.Two of the primary functions are plot (for plotting 2-
D data) and plot3 (for plotting 3-D data).In addition to the
39. plotting commands, MATLAB allows you to label and annotate
your graphs using the title, xlabel, ylabel, and legend
commands.
Plotting Example
t = [0:2:20]’;
g = 9.81; m = 68.1; cd = 0.25;
v = sqrt(g*m/cd) * tanh(sqrt(g*cd/m)*t);
plot(t, v)
Plotting Annotation Example
title('Plot of v versus t')
xlabel('Values of t')
ylabel('Values of v')
grid
Plotting OptionsWhen plotting data, MATLAB can use several
different colors, point styles, and line styles. These are
40. specified at the end of the plot command using plot specifiers as
found in Table 2.2.The default case for a single data set is to
create a blue line with no points. If a line style is specified
with no point style, no point will be drawn at the individual
points; similarly, if a point style is specified with no point
style, no line will be drawn.Examples of plot specifiers:‘ro:’ -
red dotted line with circles at the points‘gd’ - green diamonds at
the points with no line‘m--’ - magenta dashed line with no point
symbols
Other Plotting Commands
hold on and hold off
hold on tells MATLAB to keep the current data plotted and add
the results of any further plot commands to the graph. This
continues until the hold off command, which tells MATLAB to
clear the graph and start over if another plotting command is
given. hold on should be used after the first plot in a series is
made.
subplot(m, n, p)
subplot splits the figure window into an mxn array of small axes
and makes the pth one active. Note - the first subplot is at the
top left, then the numbering continues across the row. This is
different from how elements are numbered within a matrix!
Chapter3rev1.ppt
Part 1
Chapter 3
42. stored with a .m extension.There are two main kinds of M-file
Script files
Function files
Script FilesA script file is merely a set of MATLAB commands
that are saved on a file - when MATLAB runs a script file, it is
as if you typed the characters stored in the file on the command
window.Scripts can be executed either by typing their name
(without the .m) in the command window, by selecting the
Debug, Run (or Save and Run) command in the editing window,
or by hitting the F5 key while in the editing window. Note that
the latter two options will save any edits you have made, while
the former will run the file as it exists on the drive.
Function FilesFunction files serve an entirely different purpose
from script files. Function files can accept input arguments
from and return outputs to the command window, but variables
created and manipulated within the function do not impact the
command window.
Function File SyntaxThe general syntax for a function is:
function outvar = funcname(arglist)
% helpcomments
statements
43. outvar = value;
whereoutvar: output variable namefuncname: function’s name
arglist: input argument list; comma-delimited list of what the
function calls values passed to ithelpcomments: text to show
with help funcnamestatements: MATLAB commands for the
function
SubfunctionsA function file can contain a single function, but it
can also contain a primary function and one or more
subfunctionsThe primary function is whatever function is listed
first in the M-file - its function name should be the same as the
file name.Subfunctions are listed below the primary function.
Note that they are only accessible by the main function and
subfunctions within the same M-file and not by the command
window or any other functions or scripts.
InputThe easiest way to get a value from the user is the input
command:
n = input('promptstring')
MATLAB will display the characters in promptstring, and
whatever value is typed is stored in n. For example, if you type
pi, n will store 3.1416…
n = input('promptstring', 's')
MATLAB will display the characters in promptstring, and
44. whatever characters are typed will be stored as a string in n.
For example, if you type pi, n will store the letters p and i in a
2x1 char array.
OutputThe easiest way to display the value of a matrix is to
type its name, but that will not work in function or script files.
Instead, use the disp command
disp(value)
will show the value on the screen.If value is a string, enclose
it in single quotes.
Formatted OutputFor formatted output, or for output generated
by combining variable values with literal text, use the fprintf
command:
fprintf('format', x, y,...)
where format is a string specifying how you want the value
of the variables x, y, and more to be displayed - including
literal text to be printed along with the values.The values in the
variables are formatted based on format codes.
Format and Control CodesWithin the format string, the
following format codes define how a numerical value is
45. displayed:
%d - integer format
%e - scientific format with lowercase e
%E - scientific format with uppercase E
%f - decidmal format
%g - the more compact of %e or %fThe following control codes
produce special results within the format string:
n - start a new line
t - tab
- print the characterTo print a ' put a pair of ' in the format
string
Creating and Accessing FilesMATLAB has a built-in file format
that may be used to save and load the values in variables.
save filename var1 var2 ... varn
saves the listed variables into a file named filename.mat. If no
46. variable is listed, all variables are saved.
load filename var1 var2 ...varn
loads the listed variables from a file named filename.mat. If no
variable is listed, all variables in the file are loaded. Note -
these are not text files!
ASCII Files To create user-readable files, append the flag
-ascii to the end of a save command. This will save the data to
a text file in the same way that disp sends the data to a
screen.Note that in this case, MATLAB does not append
anything to the file name so you may want to add an extension
such as .txt or .dat.To load a rectangular array from a text file,
simply use the load command and the file name. The data will
be stored in a matrix with the same name as the file (but without
any extension).
Structured ProgrammingStructured programming allows
MATLAB to make decisions or selections based on conditions
of the program.Decisions in MATLAB are based on the result of
logical and relational operations and are implemented with if,
if…else, and if…elseif structures.Selections in MATLAB are
based on comparisons with a test expression and are
implemented with switch structures.
Relational OperatorsFrom Table 3.2: Summary of relational
operators in MATLAB:ExampleOperatorRelationshipx ==
47. 0==Equalunit ~= ‘m’~=Not equala < 0<Less thans > t>Greater
than3.9 <= a/3<=Less than or equal tor >= 0>=Greater than or
equal to
Logical Operators
~x (Not): true if x is false (or zero); false otherwise
x & y (And): true if both x and y are true (or non-zero)
x | y (Or): true if either x or y are true (or non-zero)
Order of OperationsPriority can be set using parentheses. After
that, Mathematical expressions are highest priority, followed by
relational operators, followed by logical operators. All things
being equal, expressions are performed from left to right.Not is
the highest priority logical operator, followed by And and
finally OrGenerally, do not combine two relational operators!
If x=5, 3<x<4 should be false (mathematically), but it is
calculated as an expression in MATLAB as:
3<5<4, which leads to true<4 at which point true is converted to
1, and 1<4 is true!Use (3<x)&(x<4) to properly evaluate.
DecisionsDecisions are made in MATLAB using if structures,
which may also include several elseif branches and possibly a
catch-all else branch.Deciding which branch runs is based on
the result of conditions which are either true or false.
If an if tree hits a true condition, that branch (and that branch
only) runs, then the tree terminates.
48. If an if tree gets to an else statement without running any prior
branch, that branch will run.Note - if the condition is a matrix,
it is considered true if and only if all entries are true (or non-
zero).
SelectionsSelections are made in MATLAB using switch
structures, which may also include a catch-all otherwise
choice.Deciding which branch runs is based on comparing the
value in some test expression with values attached to different
cases.
If the test expression matches the value attached to a case, that
case’s branch will run.
If no cases match and there is an otherwise statement, that
branch will run.
LoopsAnother programming structure involves loops, where the
same lines of code are run several times. There are two types of
loop:
A for loop ends after a specified number of repetitions
established by the number of columns given to an index
variable.
A while loop ends on the basis of a logical condition.
for LoopsOne common way to use a for…end structure is:
for index = start:step:finish
statements
49. end
where the index variable takes on successive values in the
vector created using the : operator.
VectorizationSometimes, it is more efficient to have MATLAB
perform calculations on an entire array rather than processing
an array element by element. This can be done through
vectorization.for loopVectorizationi = 0; for t = 0:0.02:50 i = i
+ 1; y(i) = cos(t); endt = 0:0.02:50;
y = cos(t);
while LoopsA while loop is fundamentally different from a for
loop since while loops can run an indeterminate number of
times. The general syntax is
while condition
statements
end
where the condition is a logical expression. If the condition is
true, the statements will run and when that is finished, the loop
will again check on the condition.Note - though the condition
may become false as the statements are running, the only time
it matters is after all the statements have run.
50. Early TerminationSometimes it will be useful to break out of a
for or while loop early - this can be done using a break
statement, generally in conjunction with an if
structure.Example:
x = 24
while (1)
x = x - 5
if x < 0, break, end
end
will produce x values of 24, 19, 14, 9, 4, and -1, then stop.
AnimationTwo ways to animate plots in MATLAB:
Using looping with simple plotting functions
This approach merely replots the graph over and over again.
Important to use the axis command so that the plots scales are
fixed.
Using special function: getframe and movie
This allows you to capture a sequence of plots (getframe) and
then play them back (movie).
51. ExampleThe (x, y) coordinates of a projectile can be generated
as a function of time, t,with the following parametric equations
- 0.5 gt2
where v0 = initial velocity (m/s)
q0 = initial angle (radians)
g = gravitational constant (= 9.81 m/s2)
ScriptThe following code illustrates both approaches:
clc,clf,clear
g=9.81; theta0=45*pi/180; v0=5;
t(1)=0;x=0;y=0;
plot(x,y,'o','MarkerFaceColor','b','MarkerSize',8)
axis([0 3 0 0.8])
M(1)=getframe;
dt=1/128;
for j = 2:1000
t(j)=t(j-1)+dt;
x=v0*cos(theta0)*t(j);
y=v0*sin(theta0)*t(j)-0.5*g*t(j)^2;
plot(x,y,'o','MarkerFaceColor','b','MarkerSize',8)
axis([0 3 0 0.8])
M(j)=getframe;
if y<=0, break, end
end
pause
movie(M,1)
52. Result
Nesting and IndentationStructures can be placed within other
structures. For example, the statements portion of a for loop
can be comprised of an if...elseif...else structure.For clarity of
reading, the statements of a structure are generally indented to
show which lines of controlled are under the control of which
structure.
Anonymous & Inline FunctionsAnonymous functions are simple
one-line functions created without the need for an M-file
fhandle = @(arg1, arg2, ...) expressionInline functions are
essentially the same as anonymous functions, but with a
different syntax:
fhandle = inline('expression', 'arg1', 'arg2',...)Anonymous
functions can access the values of variables in the workspace
upon creation, while inlines cannot.
Function FunctionsFunction functions are functions that operate
on other functions which are passed to it as input arguments.
The input argument may be the handle of an anonymous or
inline function, the name of a built-in function, or the name of a
M-file function.Using function functions will allow for more
dynamic programming.
54. exact mathematical formulations are represented by
approximations.Knowing how to use the Taylor series to
estimate truncation errors.Understanding how to write forward,
backward, and centered finite-difference approximations of the
first and second derivatives.Recognizing that efforts to
minimize truncation errors can sometimes increase roundoff
errors.
Accuracy and PrecisionAccuracy refers to how closely a
computed or measured value agrees with the true value, while
precision refers to how closely individual computed or
measured values agree with each other.
inaccurate and imprecise
accurate and imprecise
inaccurate and precise
accurate and precise
Error DefinitionsTrue error (Et): the difference between the true
value and the approximation.Absolute error (|Et|): the absolute
difference between the true value and the approximation.True
fractional relative error: the true error divided by the true
expressed as a percentage.
55. Error Definitions (cont)The previous definitions of error relied
on knowing a true value. If that is not the case, approximations
can be made to the error.The approximate percent relative error
can be given as the approximate error divided by the
approximation, expressed as a percentage - though this presents
the challenge of finding the approximate error!For iterative
processes, the error can be approximated as the difference in
values between sucessive iterations.
Using Error EstimatesOften, when performing calculations, we
may not be concerned with the sign of the error but are
interested in whether the absolute value of the percent relative
referred to as a stopping criterion.
Roundoff ErrorsRoundoff errors arise because digital computers
cannot represent some quantities exactly. There are two major
facets of roundoff errors involved in numerical
calculations:Digital computers have size and precision limits on
their ability to represent numbers.Certain numerical
manipulations are highly sensitive to roundoff errors.
Computer Number RepresentationBy default, MATLAB has
adopted the IEEE double-precision format in which eight bytes
(64 bits) are used to represent floating-point numbers:
n = ±(1+f) x 2eThe sign is determined by a sign bitThe mantissa
f is determined by a 52-bit binary numberThe exponent e is
56. determined by an 11-bit binary number, from which 1023 is
subtracted to get e
Floating Point RangesValues of -1023 and +1024 for e are
reserved for special meanings, so the exponent range is -1022 to
1023.The largest possible number MATLAB can store has
f of all 1’s, giving a significand of 2 - 2-52, or approximately 2
e of 111111111102, giving an exponent of 2046 - 1023 = 1023
10308The smallest
possible number MATLAB can store with full precision has
f of all 0’s, giving a significand of 1
e of 000000000012, giving an exponent of 1-1023 = -1022
This yields 2- -308
Floating Point PrecisionThe 52 bits for the mantissa f
correspond to about 15 to 16 base-10 digits. The machine
epsilon - the maximum relative error between a number and
MATLAB’s representation of that number, is thus
2- -16
Roundoff Errors with
Arithmetic ManipulationsRoundoff error can happen in several
circumstances other than just storing numbers - for
example:Large computations - if a process performs a large
number of computations, roundoff errors may build up to
become significantAdding a Large and a Small Number - Since
57. the small number’s mantissa is shifted to the right to be the
same scale as the large number, digits are lostSmearing -
Smearing occurs whenever the individual terms in a summation
are larger than the summation itself. (x + 10-20) - x = 10-20
mathematically, but
x = 1; (x + 10-20) - x gives a 0 in MATLAB!
Truncation ErrorsTruncation errors are those that result from
using an approximation in place of an exact mathematical
procedure.Example 1: approximation to a derivative using a
finite-difference equation:
Example 2: The Taylor Series
The Taylor Theorem and SeriesThe Taylor theorem states that
any smooth function can be approximated as a polynomial.The
Taylor series provides a means to express this idea
mathematically.
The Taylor Series
Truncation ErrorIn general, the nth order Taylor series
58. expansion will be exact for an nth order polynomial.In other
cases, the remainder term Rn is of the order of hn+1, meaning:
The more terms are used, the smaller the error, and
The smaller the spacing, the smaller the error for a given
number of terms.
Numerical DifferentiationThe first order Taylor series can be
used to calculate approximations to derivatives:
Given:
Then:
This is termed a “forward” difference because it utilizes data at
i and i+1 to estimate the derivative.
Differentiation (cont)There are also backward and centered
difference approximations, depending on the points
used:Forward:
Backward:
Centered:
Total Numerical ErrorThe total numerical error is the
summation of the truncation and roundoff errors.The truncation
error generally increases as the step size increases, while the
59. roundoff error decreases as the step size increases - this leads to
a point of diminishing returns for step size.
Other ErrorsBlunders - errors caused by malfunctions of the
computer or human imperfection.Model errors - errors resulting
from incomplete mathematical models.Data uncertainty - errors
resulting from the accuracy and/or precision of the data.
dv
dt
@
D
v
D
t
=
v
(
t
i
+
1
)
-
v
(
t
i
66. Roots“Roots” problems occur when some function f can be
written in terms of one or more dependent variables x, where
the solutions to f(x)=0 yields the solution to the problem.These
problems often occur when a design problem presents an
implicit equation for a required parameter.
Graphical MethodsA simple method for obtaining the estimate
of the root of the equation f(x)=0 is to make a plot of the
function and observe where it crosses the x-axis.Graphing the
function can also indicate where roots may be and where some
root-finding methods may fail:
Same sign, no roots
Different sign, one root
Same sign, two roots
Different sign, three roots
Bracketing MethodsBracketing methods are based on making
two initial guesses that “bracket” the root - that is, are on either
side of the root.Brackets are formed by finding two guesses xl
and xu where the sign of the function changes; that is, where
f(xl ) f(xu ) < 0The incremental search method tests the value of
the function at evenly spaced intervals and finds brackets by
identifying function sign changes between neighboring points.
Incremental Search HazardsIf the spacing between the points of
an incremental search are too far apart, brackets may be missed
due to capturing an even number of roots within two
points.Incremental searches cannot find brackets containing
even-multiplicity roots regardless of spacing.
67. BisectionThe bisection method is a variation of the incremental
search method in which the interval is always divided in half.If
a function changes sign over an interval, the function value at
the midpoint is evaluated.The location of the root is then
determined as lying within the subinterval where the sign
change occurs.The absolute error is reduced by a factor of 2 for
each iteration.
Programming Bisection
Bisection ErrorThe absolute error of the bisection method is
solely dependent on the absolute error at the start of the process
(the space between the two guesses) and the number of
iterations:
The required number of iterations to obtain a particular absolute
error can be calculated based on the initial guesses:
False PositionThe false position method is another bracketing
method.It determines the next guess not by splitting the bracket
in half but by connecting the endpoints with a straight line and
determining the location of the intercept of the straight line
(xr).The value of xr then replaces whichever of the two initial
guesses yields a function value with the same sign as f(xr).
68. False Position Illustration
Bisection vs. False PositionBisection does not take into account
the shape of the function; this can be good or bad depending on
the function!Bad:
E
a
n
=
D
x
0
2
n
n
=
log
2
D
x
0
E
a
,
d
æ
è
ç
71. Open MethodsOpen methods differ from bracketing methods, in
that open methods require only a single starting value or two
starting values that do not necessarily bracket a root.Open
methods may diverge as the computation progresses, but when
they do converge, they usually do so much faster than
bracketing methods.
Graphical Comparison of Methods
Bracketing method
Diverging open method
Converging open method - note speed!
Simple Fixed-Point IterationRearrange the function f(x)=0 so
that x is on the left-hand side of the equation: x=g(x)Use the
new function g to predict a new value of x - that is,
xi+1=g(xi)The approximate error is given by:
ExampleSolve f(x)=e-x-xRe-write as x=g(x) by isolating x
(example: x=e-x)Start with an initial guess (here, 0)
Continue until some tolerance
72. -
100.0000100.00011.0000100.000 76.3220.76320.3679171.828
35.1350.46030.6922 46.854 22.0500.62840.5005 38.309
11.7550.533
ConvergenceConvergence of the simple fixed-point iteration
method requires that the derivative of g(x) near the root has a
magnitude less than 1.
Convergent, 0≤g’<1
Convergent, -1<g’≤0
Divergent, g’>1
Divergent, g’<-1
Newton-Raphson MethodBased on forming the tangent line to
the f(x) curve at some guess x, then following the tangent line
to where it crosses the x-axis.
Pros and ConsPro: The error of the i+1th iteration is roughly
proportional to the square of the error of the ith iteration - this
is called quadratic convergenceCon: Some functions show slow
or poor convergence
Secant MethodsA potential problem in implementing the
Newton-Raphson method is the evaluation of the derivative -
there are certain functions whose derivatives may be difficult or
inconvenient to evaluate.For these cases, the derivative can be
73. approximated by a backward finite divided difference:
Secant Methods (cont)Substitution of this approximation for the
derivative to the Newton-Raphson method equation gives:
Note - this method requires two initial estimates of x but does
not require an analytical expression of the derivative.
MATLAB’s fzero FunctionMATLAB’s fzero provides the best
qualities of both bracketing methods and open methods.Using
an initial guess:
x = fzero(function, x0)
[x, fx] = fzero(function, x0)function is a function handle to the
function being evaluatedx0 is the initial guessx is the location
of the rootfx is the function evaluated at that rootUsing an
initial bracket:
x = fzero(function, [x0 x1])
[x, fx] = fzero(function, [x0 x1])As above, except x0 and x1 are
guesses that must bracket a sign change
74. fzero OptionsOptions may be passed to fzero as a third input
argument - the options are a data structure created by the
optimset commandoptions = optimset(‘par1’, val1, ‘par2’,
val2,…)parn is the name of the parameter to be setvaln is the
value to which to set that parameterThe parameters commonly
used with fzero are:display: when set to ‘iter’ displays a
detailed record of all the iterationstolx: A positive scalar that
sets a termination tolerance on x.
fzero Exampleoptions = optimset(‘display’, ‘iter’);Sets options
to display each iteration of root finding process[x, fx] =
fzero(@(x) x^10-1, 0.5, options)Uses fzero to find roots of
f(x)=x10-1 starting with an initial guess of x=0.5.MATLAB
reports x=1, fx=0 after 35 function counts
PolynomialsMATLAB has a built in program called roots to
determine all the roots of a polynomial - including imaginary
and complex ones.x = roots(c)x is a column vector containing
the rootsc is a row vector containing the polynomial
coefficientsExample:Find the roots of
f(x)=x5-3.5x4+2.75x3+2.125x2-3.875x+1.25x = roots([1 -3.5
2.75 2.125 -3.875 1.25])
Polynomials (cont)MATLAB’s poly function can be used to
determine polynomial coefficients if roots are given:b =
poly([0.5 -1])Finds f(x) where f(x) =0 for x=0.5 and x=-
1MATLAB reports b = [1.000 0.5000 -0.5000]This corresponds
to f(x)=x2+0.5x-0.5MATLAB’s polyval function can evaluate a
75. polynomial at one or more points:a = [1 -3.5 2.75 2.125 -3.875
1.25];If used as coefficients of a polynomial, this corresponds
to f(x)=x5-3.5x4+2.75x3+2.125x2-3.875x+1.25polyval(a, 1)This
calculates f(1), which MATLAB reports as -0.2500
e
a
=
x
i
+
1
-
x
i
x
i
+
1
100
%
f
'
(
x
i
)
=
f
(
x
i
)
79. Chapter ObjectivesUnderstanding why and where optimization
occurs in engineering and scientific problem
solving.Recognizing the difference between one-dimensional
and multi-dimensional optimization.Distinguishing between
global and local optima.Knowing how to recast a maximization
problem so that it can be solved with a minimizing
algorithm.Being able to define the golden ratio and understand
why it makes one-dimensional optimization efficient.
Objectives (cont)Locating the optimum of a single-variable
function with the golden-section search.Locating the optimum
of a single-variable function with parabolic
interpolation.Knowing how to apply the fminbnd function to
determine the minimum of a one-dimensional function.Being
able to develop MATLAB contours and surface plots to
visualize two-dimensional functions.Knowing how to apply the
fminsearch function to determine the minimum of a
multidimensional function.
OptimizationOptimization is the process of creating something
that is as effective as possible.From a mathematical perspective,
optimization deals with finding the maxima and minima of a
function that depends on one or more variables.
Multidimensional OptimizationOne-dimensional problems
involve functions that depend on a single dependent variable -
for example, f(x).Multidimensional problems involve functions
that depend on two or more dependent variables - for example,
80. f(x,y)
Global vs. LocalA global optimum represents the very best
solution while a local optimum is better than its immediate
neighbors. Cases that include local optima are called
multimodal.Generally desire to find the global optimum.
Golden-Section SearchSearch algorithm for finding a minimum
on an interval [xl xu] with a single minimum (unimodal
of two interior points x1 and x2; by using the golden ratio, one
of the interior points can be re-used in the next iteration.
Golden-Section Search (cont)
If f(x1)<f(x2), x2 becomes the new lower limit and x1 becomes
the new x2 (as in figure).If f(x2)<f(x1), x1 becomes the new
upper limit and x2 becomes the new x1.In either case, only one
new interior point is needed and the function is only evaluated
one more time.
Code for Golden-Section Search
Parabolic InterpolationAnother algorithm uses parabolic
81. interpolation of three points to estimate optimum location.The
location of the maximum/minimum of a parabola defined as the
interpolation of three points (x1, x2, and x3) is:
The new point x4 and the two
surrounding it (either x1 and x2
or x2 and x3) are used for the
next iteration of the algorithm.
fminbnd FunctionMATLAB has a built-in function, fminbnd,
which combines the golden-section search and the parabolic
interpolation.[xmin, fval] = fminbnd(function, x1, x2)Options
may be passed through a fourth argument using optimset,
similar to fzero.
Multidimensional VisualizationFunctions of two-dimensions
may be visualized using contour or surface/mesh plots.
fminsearch FunctionMATLAB has a built-in function,
fminsearch, that can be used to determine the minimum of a
multidimensional function.[xmin, fval] = fminsearch(function,
x0)xmin in this case will be a row vector containing the location
82. of the minimum, while x0 is an initial guess. Note that x0 must
contain as many entries as the function expects of it.The
function must be written in terms of a single variable, where
different dimensions are represented by different indices of that
variable.
fminsearch FunctionTo minimize
f(x,y)=2+x-y+2x2+2xy+y2
rewrite as
f(x1, x2)=2+x1-x2+2(x1)2+2x1x2+(x2)2[email protected](x)
2+x(1)-x(2)+2*x(1)^2+2*x(1)*x(2)+x(2)^2
[x, fval] = fminsearch(f, [-0.5, 0.5])Note that x0 has two entries
- f is expecting it to contain two values.MATLAB reports the
minimum value is 0.7500 at a location of [-1.000 1.5000]
d
=
(
f
-
1
)(
x
u
-
87. Overview (cont)A horizontal set of elements is called a row and
a vertical set of elements is called a column.The first subscript
of an element indicates the row while the second indicates the
column.The size of a matrix is given as m rows by n columns,
or simply m by n (or m x n).1 x n matrices are row vectors.m x
1 matrices are column vectors.
Special MatricesMatrices where m=n are called square
matrices.There are a number of special forms of square
matrices:
Matrix OperationsTwo matrices are considered equal if and only
if every element in the first matrix is equal to every
corresponding element in the second. This means the two
matrices must be the same size.Matrix addition and subtraction
are performed by adding or subtracting the corresponding
elements. This requires that the two matrices be the same
size.Scalar matrix multiplication is performed by multiplying
each element by the same scalar.
Matrix MultiplicationThe elements in the matrix [C] that results
from multiplying matrices [A] and [B] are calculated using:
Matrix Inverse and TransposeThe inverse of a square,
nonsingular matrix [A] is that matrix which, when multiplied by
[A], yields the identity matrix.[A][A]-1=[A]-1[A]=[I] The
88. transpose of a matrix involves transforming its rows into
columns and its columns into rows.(aij)T=aji
Representing Linear AlgebraMatrices provide a concise notation
for representing and solving simultaneous linear equations:
Solving With MATLABMATLAB provides two direct ways to
solve systems of linear algebraic equations [A]{x}={b}:Left-
division
x = AbMatrix inversion
x = inv(A)*bThe matrix inverse is less efficient than left-
division and also only works for square, non-singular systems.
Symmetric
A
[
]
=
5
1
2
1
3
98. DeterminantsThe determinant D=|A| of a matrix is formed from
the coefficients of [A].Determinants for small matrices are:
Determinants for matrices larger than 3 x 3 can be very
complicated.
Cramer’s RuleCramer’s Rule states that each unknown in a
system of linear algebraic equations may be expressed as a
fraction of two determinants with denominator D and with the
numerator obtained from D by replacing the column of
coefficients of the unknown in question by the constants b1, b2,
…, bn.
Cramer’s Rule ExampleFind x2 in the following system of
equations:
Find the determinant D
Find determinant D2 by replacing D’s second column with b
Divide
99. Naïve Gauss EliminationFor larger systems, Cramer’s Rule can
become unwieldy.Instead, a sequential process of removing
unknowns from equations using forward elimination followed
by back substitution may be used - this is Gauss
elimination.“Naïve” Gauss elimination simply means the
process does not check for potential problems resulting from
division by zero.
Naïve Gauss Elimination (cont)Forward eliminationStarting
with the first row, add or subtract multiples of that row to
eliminate the first coefficient from the second row and
beyond.Continue this process with the second row to remove the
second coefficient from the third row and beyond.Stop when an
upper triangular matrix remains.Back substitutionStarting with
the last row, solve for the unknown, then substitute that value
into the next highest row.Because of the upper-triangular nature
of the matrix, each row will contain only one more unknown.
Naïve Gauss Elimination Program
Gauss Program EfficiencyThe execution of Gauss elimination
depends on the amount of floating-point operations (or flops).
The flop count for an n x n system is:
100. Conclusions:As the system gets larger, the computation time
increases greatly.Most of the effort is incurred in the
elimination step.
PivotingProblems arise with naïve Gauss elimination if a
coefficient along the diagonal is 0 (problem: division by 0) or
close to 0 (problem: round-off error)One way to combat these
issues is to determine the coefficient with the largest absolute
value in the column below the pivot element. The rows can
then be switched so that the largest element is the pivot
element. This is called partial pivoting.If the rows to the right
of the pivot element are also checked and columns switched,
this is called complete pivoting.
Partial Pivoting Program
Tridiagonal SystemsA tridiagonal system is a banded system
with a bandwidth of 3:
Tridiagonal systems can be solved using the same method as
Gauss elimination, but with much less effort because most of
the matrix elements are already 0.
115. Chapter ObjectivesUnderstanding that LU factorization involves
decomposing the coefficient matrix into two triangular matrices
that can then be used to efficiently evaluate different right-
hand-side vectors.Knowing how to express Gauss elimination as
an LU factorization.Given an LU factorization, knowing how to
evaluate multiple right-hand-side vectors.Recognizing that
Cholesky’s method provides an efficient way to decompose a
symmetric matrix and that the resulting triangular matrix and its
transpose can be used to evaluate right-hand-side vectors
efficiently.Understanding in general terms what happens when
MATLAB’s backslash operator is used to solve linear systems.
LU FactorizationRecall that the forward-elimination step of
Gauss elimination comprises the bulk of the computational
effort.LU factorization methods separate the time-consuming
elimination of the matrix [A] from the manipulations of the
right-hand-side [b].Once [A] has been factored (or
decomposed), multiple right-hand-side vectors can be evaluated
in an efficient manner.
LU FactorizationLU factorization involves two
steps:Factorization to decompose the [A] matrix into a product
of a lower triangular matrix [L] and an upper triangular matrix
[U]. [L] has 1 for each entry on the diagonal.Substitution to
solve for {x}Gauss elimination can be implemented using LU
factorization
Gauss Elimination as
116. LU Factorization[A]{x}={b} can be rewritten as [L][U]{x}={b}
using LU factorization.The LU factorization algorithm requires
the same total flops as for Gauss elimination.The main
advantage is once [A] is decomposed, the same [L] and [U] can
be used for multiple {b} vectors.MATLAB’s lu function can be
used to generate the [L] and [U] matrices:
[L, U] = lu(A)
Gauss Elimination as
LU Factorization (cont)To solve [A]{x}={b}, first decompose
[A] to get [L][U]{x}={b}Set up and solve [L]{d}={b}, where
{d} can be found using forward substitution.Set up and solve
[U]{x}={d}, where {x} can be found using backward
substitution.In MATLAB:
[L, U] = lu(A)
d = Lb
x = Ud
Cholesky FactorizationSymmetric systems occur commonly in
both mathematical and engineering/science problem contexts,
118. Chapter ObjectivesKnowing how to determine the matrix
inverse in an efficient manner based on LU
factorization.Understanding how the matrix inverse can be used
to assess stimulus-response characteristics of engineering
systems.Understanding the meaning of matrix and vector norms
and how they are computed.Knowing how to use norms to
compute the matrix condition number.Understanding how the
magnitude of the condition number can be used to estimate the
precision of solutions of linear algebraic equations.
Matrix InverseRecall that if a matrix [A] is square, there is
another matrix [A]-1, called the inverse of [A], for which
[A][A]-1=[A]-1[A]=[I] The inverse can be computed in a
column by column fashion by generating solutions with unit
vectors as the right-hand-side constants:
Matrix Inverse (cont)Recall that LU factorization can be used to
efficiently evaluate a system for multiple right-hand-side
vectors - thus, it is ideal for evaluating the multiple unit vectors
needed to compute the inverse.
Stimulus-Response Computations