This document shows the calculation of the change in p and q values (Δp and Δq) between an initial and final value. It then uses these changes and initial values to calculate the value of an equation (Eq(p)) which equals -0.173.
Performing Manual and Automated Iterations in Engineering Equation Solver (EES) - Examples from Heat Transfer.
All the EES codes shown in the examples are available at: https://goo.gl/KExGFi
The document provides an overview of key concepts in set theory, including:
- A set is a collection of distinct objects called elements, with no significance to element order.
- Sets are equal if they contain the same elements. Subsets contain all elements of another set.
- Other concepts covered include cardinality, power sets, Cartesian products, set operations like union and intersection, and proving set identities.
This document discusses Gaussian quadrature formulas, which approximate definite integrals of functions by using weighted sums of function values at specified points. It presents the one-point, two-point, and three-point Gaussian quadrature formulas. The one-point formula is exact for polynomials up to degree 1, the two-point formula is exact for polynomials up to degree 3, and the three-point formula is exact for polynomials up to degree 5. Examples are provided to demonstrate applying the formulas.
Gaussian Quadrature Formulas, which are simple and will help learners learn about Gauss's One, Two and Three Point Formulas, I have also included sums so that learning can be easy and the method can be understood.
1. The document lists various mathematical formulae related to algebra, quadratic equations, arithmetic and geometric progressions, factorials, and logarithms. Some key formulae include: (a + b)2 = a2 + 2ab + b2, the quadratic formula for solving ax2 + bx + c = 0, the formulas for the nth term and sum of terms for arithmetic and geometric progressions, the factorial formula n! = n(n − 1)! , and logarithm properties such as loga mn = loga m + loga n.
This document provides an overview of MATLAB and image processing. It discusses basic MATLAB commands and functions like arithmetic operations, if/else statements, for/while loops, functions, plotting and graphing, loading and saving data, multidimensional arrays, and effective MATLAB coding techniques like vectorization. The document also mentions commands for plotting, legend, axis, loading and saving data, and reserving memory space using zeros.
Performing Manual and Automated Iterations in Engineering Equation Solver (EES) - Examples from Heat Transfer.
All the EES codes shown in the examples are available at: https://goo.gl/KExGFi
The document provides an overview of key concepts in set theory, including:
- A set is a collection of distinct objects called elements, with no significance to element order.
- Sets are equal if they contain the same elements. Subsets contain all elements of another set.
- Other concepts covered include cardinality, power sets, Cartesian products, set operations like union and intersection, and proving set identities.
This document discusses Gaussian quadrature formulas, which approximate definite integrals of functions by using weighted sums of function values at specified points. It presents the one-point, two-point, and three-point Gaussian quadrature formulas. The one-point formula is exact for polynomials up to degree 1, the two-point formula is exact for polynomials up to degree 3, and the three-point formula is exact for polynomials up to degree 5. Examples are provided to demonstrate applying the formulas.
Gaussian Quadrature Formulas, which are simple and will help learners learn about Gauss's One, Two and Three Point Formulas, I have also included sums so that learning can be easy and the method can be understood.
1. The document lists various mathematical formulae related to algebra, quadratic equations, arithmetic and geometric progressions, factorials, and logarithms. Some key formulae include: (a + b)2 = a2 + 2ab + b2, the quadratic formula for solving ax2 + bx + c = 0, the formulas for the nth term and sum of terms for arithmetic and geometric progressions, the factorial formula n! = n(n − 1)! , and logarithm properties such as loga mn = loga m + loga n.
This document provides an overview of MATLAB and image processing. It discusses basic MATLAB commands and functions like arithmetic operations, if/else statements, for/while loops, functions, plotting and graphing, loading and saving data, multidimensional arrays, and effective MATLAB coding techniques like vectorization. The document also mentions commands for plotting, legend, axis, loading and saving data, and reserving memory space using zeros.
This document describes how to solve the traveling salesperson problem (TSP) using dynamic programming. It defines g(i,S) as the length of the shortest path from vertex i through all vertices in S to vertex 1. It shows that g(1,V-{1}) gives the optimal tour length, and that g(i,S) can be calculated using g values for smaller sets S via an equation. The complexity is O(n22n) time and O(n2n) space, which is better than brute force but still prohibitive for large n due to the exponential space needed.
This document contains information about sets and set operations. It defines what a set is, provides examples of standard sets like natural numbers and real numbers, and discusses concepts like subsets, power sets, Cartesian products, unions, intersections, differences and complements of sets. It also presents methods to prove equations involving set operations.
Extended network and algorithm finding maximal flows IJECEIAES
Graph is a powerful mathematical tool applied in many fields as transportation, communication, informatics, economy, in ordinary graph the weights of edges and vertexes are considered independently where the length of a path is the sum of weights of the edges and the vertexes on this path. However, in many practical problems, weights at a vertex are not the same for all paths passing this vertex, but depend on coming and leaving edges. The paper develops a model of extended network that can be applied to modelling many practical problems more exactly and effectively. The main contribution of this paper is algorithm finding maximal flows on extended networks.
This document provides the questions for a mathematics assignment covering chapters 1-8 of class 12. There are 40 questions in total, ranging from proving trigonometric identities and evaluating definite integrals to solving equations using matrices and checking continuity of functions. The questions cover a wide range of calculus, trigonometry and algebra topics.
This document discusses multiplying and factoring algebraic expressions. It explains factoring expressions of the form ax^2 + bx + c by finding two numbers whose product is c and sum is b. Examples are provided of factoring various algebraic expressions, including x^3 + 6x^2 + 9x, x^2 - x - 12, and x^2 + 2x - 35. Factoring strategies like finding the product and sum of the factors to match the b and c terms are outlined.
Numerical integration;Gaussian integration one point, two point and three poi...vaibhav tailor
The document discusses numerical integration using Gaussian quadrature. It describes one-point, two-point, and three-point Gaussian quadrature rules. For each rule, it provides the formula used to approximate a definite integral of a function over an interval by calculating a weighted sum of the function values at specified points. Examples are included to demonstrate applying the one-point, two-point, and three-point rules to evaluate definite integrals.
1. The document discusses various vector algebra concepts including vector projection, work done by a force, compound angle formulas, and vector cross and dot products.
2. Vector projection and work done formulas are derived. The compound angle formulas for cosine and sine are proved using vector diagrams.
3. Cross and dot products are explained. Various applications are covered including proving trigonometric identities like the sine rule and finding areas and volumes.
The document contains 46 mathematical formulae related to algebra, quadratic equations, arithmetic progressions, geometric progressions, factorials, and binomial expansions. Some key formulae include:
1) (a + b)2 = a2 + 2ab + b2 for expanding a binomial square.
2) The quadratic formula for solving ax2 + bx + c = 0 is x = (-b ± √(b2 - 4ac))/2a.
3) The nth term of an arithmetic progression with first term a and common difference d is an = a + (n - 1)d.
4) The nth term of a geometric progression with first term a and common ratio
- Elliptic curves are algebraic structures used in cryptography and defined by cubic equations over finite fields. They form groups where the group operation is point addition.
- A digital signature algorithm can be based on elliptic curves by using the discrete logarithm problem over elliptic curve groups.
- The seminar discusses definitions of groups, elliptic curves, and their use in cryptography including defining the group operation of point addition on elliptic curves and calculating parameters like the group order.
Least Square Optimization and Sparse-Linear SolverJi-yong Kwon
The document discusses least-square optimization and sparse linear systems. It introduces least-square optimization as a technique to find approximate solutions when exact solutions do not exist. It provides an example of using least-squares to find the line of best fit through three points. The objective is to minimize the sum of squared distances between the line and points. Solving the optimization problem yields a set of linear equations that can be solved using techniques like pseudo-inverse or conjugate gradient. Sparse linear systems with many zero entries can be solved more efficiently than dense systems.
This document provides mathematical preliminaries for automata, including:
- Sets, functions, relations, graphs, and proof techniques like induction and proof by contradiction.
- It defines sets, set operations, functions, relations, graphs, trees, and binary trees.
- It also covers topics like equivalence relations, equivalence classes, Cartesian products, and power sets.
This document contains 6 multi-part calculus problems involving functions, derivatives, integrals, areas, volumes and series. The problems cover topics like finding derivatives and integrals of functions, analyzing graphs of functions, calculating areas and volumes of revolved regions, determining convergence of series, and relating functions to their derivatives.
The document contains a test with 9 problems involving calculus concepts like double and triple integrals in Cartesian, polar, and spherical coordinates, line integrals, vector fields, and surface integrals. Students are instructed to show all work algebraically and given reference sheets on derivatives and anti-derivatives. The test covers multiple topics within multivariate calculus.
The document contains information about graphing a quadratic equation. It provides the vertex of (-5,-2) and finds the equation -3=-1/9(x-5)^2-2 by using the vertex form. It then compares the graphing calculator's graph of y=-0.473x^2+4.81x-14.33 to the equation -1/9(x-5)^2-2 derived. Both graphs open down and have negative equations, showing the steps were close to the actual equation.
The document contains information about graphing a quadratic equation. It provides the vertex of (-5,-2) and finds the equation by using the vertex form y=a(x-h)^2+k. The calculated equation is y=-1/9(x-5)^2-2. This equation is compared to the graphing calculator's equation of y=-.473x^2+4.81x-14.33, which is similar but not exactly the same. Both equations are negative and open down. The actual equation given is y=-.473x^2+4.81x-14.33.
This document describes the calculation of accelerations and velocities in an exercise where the initial velocity is 12,000 and the final velocity is 3,000. It shows that the acceleration is calculated as the minimum of 1/5 of the current velocity squared or the difference between the current and final velocities. This acceleration is then subtracted from the current velocity to calculate the next velocity, until the final velocity of 3,000 is reached after 5 iterations.
This document describes how to solve the traveling salesperson problem (TSP) using dynamic programming. It defines g(i,S) as the length of the shortest path from vertex i through all vertices in S to vertex 1. It shows that g(1,V-{1}) gives the optimal tour length, and that g(i,S) can be calculated using g values for smaller sets S via an equation. The complexity is O(n22n) time and O(n2n) space, which is better than brute force but still prohibitive for large n due to the exponential space needed.
This document contains information about sets and set operations. It defines what a set is, provides examples of standard sets like natural numbers and real numbers, and discusses concepts like subsets, power sets, Cartesian products, unions, intersections, differences and complements of sets. It also presents methods to prove equations involving set operations.
Extended network and algorithm finding maximal flows IJECEIAES
Graph is a powerful mathematical tool applied in many fields as transportation, communication, informatics, economy, in ordinary graph the weights of edges and vertexes are considered independently where the length of a path is the sum of weights of the edges and the vertexes on this path. However, in many practical problems, weights at a vertex are not the same for all paths passing this vertex, but depend on coming and leaving edges. The paper develops a model of extended network that can be applied to modelling many practical problems more exactly and effectively. The main contribution of this paper is algorithm finding maximal flows on extended networks.
This document provides the questions for a mathematics assignment covering chapters 1-8 of class 12. There are 40 questions in total, ranging from proving trigonometric identities and evaluating definite integrals to solving equations using matrices and checking continuity of functions. The questions cover a wide range of calculus, trigonometry and algebra topics.
This document discusses multiplying and factoring algebraic expressions. It explains factoring expressions of the form ax^2 + bx + c by finding two numbers whose product is c and sum is b. Examples are provided of factoring various algebraic expressions, including x^3 + 6x^2 + 9x, x^2 - x - 12, and x^2 + 2x - 35. Factoring strategies like finding the product and sum of the factors to match the b and c terms are outlined.
Numerical integration;Gaussian integration one point, two point and three poi...vaibhav tailor
The document discusses numerical integration using Gaussian quadrature. It describes one-point, two-point, and three-point Gaussian quadrature rules. For each rule, it provides the formula used to approximate a definite integral of a function over an interval by calculating a weighted sum of the function values at specified points. Examples are included to demonstrate applying the one-point, two-point, and three-point rules to evaluate definite integrals.
1. The document discusses various vector algebra concepts including vector projection, work done by a force, compound angle formulas, and vector cross and dot products.
2. Vector projection and work done formulas are derived. The compound angle formulas for cosine and sine are proved using vector diagrams.
3. Cross and dot products are explained. Various applications are covered including proving trigonometric identities like the sine rule and finding areas and volumes.
The document contains 46 mathematical formulae related to algebra, quadratic equations, arithmetic progressions, geometric progressions, factorials, and binomial expansions. Some key formulae include:
1) (a + b)2 = a2 + 2ab + b2 for expanding a binomial square.
2) The quadratic formula for solving ax2 + bx + c = 0 is x = (-b ± √(b2 - 4ac))/2a.
3) The nth term of an arithmetic progression with first term a and common difference d is an = a + (n - 1)d.
4) The nth term of a geometric progression with first term a and common ratio
- Elliptic curves are algebraic structures used in cryptography and defined by cubic equations over finite fields. They form groups where the group operation is point addition.
- A digital signature algorithm can be based on elliptic curves by using the discrete logarithm problem over elliptic curve groups.
- The seminar discusses definitions of groups, elliptic curves, and their use in cryptography including defining the group operation of point addition on elliptic curves and calculating parameters like the group order.
Least Square Optimization and Sparse-Linear SolverJi-yong Kwon
The document discusses least-square optimization and sparse linear systems. It introduces least-square optimization as a technique to find approximate solutions when exact solutions do not exist. It provides an example of using least-squares to find the line of best fit through three points. The objective is to minimize the sum of squared distances between the line and points. Solving the optimization problem yields a set of linear equations that can be solved using techniques like pseudo-inverse or conjugate gradient. Sparse linear systems with many zero entries can be solved more efficiently than dense systems.
This document provides mathematical preliminaries for automata, including:
- Sets, functions, relations, graphs, and proof techniques like induction and proof by contradiction.
- It defines sets, set operations, functions, relations, graphs, trees, and binary trees.
- It also covers topics like equivalence relations, equivalence classes, Cartesian products, and power sets.
This document contains 6 multi-part calculus problems involving functions, derivatives, integrals, areas, volumes and series. The problems cover topics like finding derivatives and integrals of functions, analyzing graphs of functions, calculating areas and volumes of revolved regions, determining convergence of series, and relating functions to their derivatives.
The document contains a test with 9 problems involving calculus concepts like double and triple integrals in Cartesian, polar, and spherical coordinates, line integrals, vector fields, and surface integrals. Students are instructed to show all work algebraically and given reference sheets on derivatives and anti-derivatives. The test covers multiple topics within multivariate calculus.
The document contains information about graphing a quadratic equation. It provides the vertex of (-5,-2) and finds the equation -3=-1/9(x-5)^2-2 by using the vertex form. It then compares the graphing calculator's graph of y=-0.473x^2+4.81x-14.33 to the equation -1/9(x-5)^2-2 derived. Both graphs open down and have negative equations, showing the steps were close to the actual equation.
The document contains information about graphing a quadratic equation. It provides the vertex of (-5,-2) and finds the equation by using the vertex form y=a(x-h)^2+k. The calculated equation is y=-1/9(x-5)^2-2. This equation is compared to the graphing calculator's equation of y=-.473x^2+4.81x-14.33, which is similar but not exactly the same. Both equations are negative and open down. The actual equation given is y=-.473x^2+4.81x-14.33.
This document describes the calculation of accelerations and velocities in an exercise where the initial velocity is 12,000 and the final velocity is 3,000. It shows that the acceleration is calculated as the minimum of 1/5 of the current velocity squared or the difference between the current and final velocities. This acceleration is then subtracted from the current velocity to calculate the next velocity, until the final velocity of 3,000 is reached after 5 iterations.
This document calculates the price (P) of a bond that has a face value of 5000, a coupon rate of 6% yielding an annual coupon of 300, and is currently trading at 92.5% of face value resulting in a current price of 4625. It further calculates that there are 15 days of accrued interest since the last coupon payment using the formula f=number of days/360. The final price of the bond is 4637.50, which is the sum of the current price and the accrued interest.
The document shows a mathematical calculation to determine an interest rate of 5% on a loan amount of 12,988 over 5 years where the principal is 3000 and the future value is 4,329.33.