Tavas and pashmaks
•Download as PPTX, PDF•
1 like•504 views
The document describes finding the largest subset of winners from a set of competitors where each competitor has values for r and s. A competitor can be a winner if there exist values R and S such that the ratio R/r + S/s is minimum. It notes that a competitor cannot win if another has both lower r and s values. The analysis shows that the winners are the subset of competitors forming the lower-left convex hull when plotted as (1/r, 1/s) points. Implementation requires care with precision issues when dealing with fractions and handling duplicated competitors for the convex hull algorithm.
Report
Share
Report
Share
Recommended
A Simplified Approach to Calculating Truss Forces
A simplified approach to calculating truss forces. This presentation is built upon a wonderful presentation which comes from Fazirah Abdul Ghafar. This presentation uses Power Point Animations and is best viewed using Power Point Slide Show to present.
Coursera 2week
This document summarizes key concepts in machine learning linear regression including:
1) Expressing the hypothesis for multiple features using a multivariate linear function.
2) Applying gradient descent to minimize the cost function for multivariate linear regression.
3) The importance of feature scaling to ensure features are on a similar scale.
4) Debugging techniques such as evaluating multiple learning rates to address issues with gradient descent convergence.
5) Polynomial regression which models the dependent variable as a polynomial function of independent variables.
6) The normal equation approach for analytically solving for the optimal parameter values without iteration.
Merge sort
Merge sort is a sorting algorithm that uses a divide and conquer approach. It divides an array into halves, recursively sorts the halves, and then merges the sorted halves into a single sorted array. The key steps are dividing, conquering by recursively sorting subarrays, and combining the sorted subarrays through a merging process. It has advantages of being a stable sort and having no worst-case scenario, though it requires more memory than other sorts.
State feedback example
The document finds the state feedback gain matrix K to place the closed loop poles at specific locations for a controllable system. It first transforms the system into controllable canonical form, then calculates K using the characteristic equation. The closed loop transfer function is determined to be a 3rd order function divided by a 3rd order polynomial with coefficients matching the desired closed loop poles.
Basic calculus (i)
The document provides an overview of basic calculus concepts including:
- Exponents and exponent rules for multiplying, dividing, and raising to powers.
- Algebraic expressions including monomials, binomials, polynomials, and equations.
- Common identities for exponents, polynomials, trigonometric functions.
- The definition of a function as a correspondence between variables where each input has a single output.
- Examples of basic functions including power, exponential, logarithmic, and trigonometric functions.
Normal approximation to the binomial distribution
This document discusses using the normal distribution to approximate the binomial distribution when n is large and p is around 0.5. It provides continuity correction rules for using the normal approximation. It then works through an example calculating the probability of different outcomes when sampling 400 seeds with a 0.35 probability of being ryegrass. It calculates the probabilities using the normal approximation and continuity corrections.
K to 12 math
Basic Mathematics (non-calculus) for k-12 students in B.C. Canada. Intended as a guide for teaching basic math to young learners, and uploaded as a personal favor to my friend Oliver Cougur. This is a supplement teaching/learning material, and functions as a 'cheat sheet' for instructors and/or students.
This is not intended as curriculum material. I guarantee nothing. I claim no ownership or discovery of any of the material in this document, however I reserve my right of creative expression for materials contained. This document may not be sold, copied or altered in anyway by anyone.
Please report any errors to s.grantwilliam@ieee.org
Ks
This document discusses the composition of forces and moments. It defines key terms like resultant force and moment of a force. It describes the parallelogram, triangle and polygon laws for combining concurrent coplanar forces into a single resultant force. It also explains Varignon's principle of moments, which states that the algebraic sum of the moments of individual forces equals the moment of the resultant force about the same point. Several example problems are provided to illustrate how to use these principles to find the magnitude and direction of resultant forces and moments in systems of coplanar concurrent and non-concurrent forces.
Recommended
A Simplified Approach to Calculating Truss Forces
A simplified approach to calculating truss forces. This presentation is built upon a wonderful presentation which comes from Fazirah Abdul Ghafar. This presentation uses Power Point Animations and is best viewed using Power Point Slide Show to present.
Coursera 2week
This document summarizes key concepts in machine learning linear regression including:
1) Expressing the hypothesis for multiple features using a multivariate linear function.
2) Applying gradient descent to minimize the cost function for multivariate linear regression.
3) The importance of feature scaling to ensure features are on a similar scale.
4) Debugging techniques such as evaluating multiple learning rates to address issues with gradient descent convergence.
5) Polynomial regression which models the dependent variable as a polynomial function of independent variables.
6) The normal equation approach for analytically solving for the optimal parameter values without iteration.
Merge sort
Merge sort is a sorting algorithm that uses a divide and conquer approach. It divides an array into halves, recursively sorts the halves, and then merges the sorted halves into a single sorted array. The key steps are dividing, conquering by recursively sorting subarrays, and combining the sorted subarrays through a merging process. It has advantages of being a stable sort and having no worst-case scenario, though it requires more memory than other sorts.
State feedback example
The document finds the state feedback gain matrix K to place the closed loop poles at specific locations for a controllable system. It first transforms the system into controllable canonical form, then calculates K using the characteristic equation. The closed loop transfer function is determined to be a 3rd order function divided by a 3rd order polynomial with coefficients matching the desired closed loop poles.
Basic calculus (i)
The document provides an overview of basic calculus concepts including:
- Exponents and exponent rules for multiplying, dividing, and raising to powers.
- Algebraic expressions including monomials, binomials, polynomials, and equations.
- Common identities for exponents, polynomials, trigonometric functions.
- The definition of a function as a correspondence between variables where each input has a single output.
- Examples of basic functions including power, exponential, logarithmic, and trigonometric functions.
Normal approximation to the binomial distribution
This document discusses using the normal distribution to approximate the binomial distribution when n is large and p is around 0.5. It provides continuity correction rules for using the normal approximation. It then works through an example calculating the probability of different outcomes when sampling 400 seeds with a 0.35 probability of being ryegrass. It calculates the probabilities using the normal approximation and continuity corrections.
K to 12 math
Basic Mathematics (non-calculus) for k-12 students in B.C. Canada. Intended as a guide for teaching basic math to young learners, and uploaded as a personal favor to my friend Oliver Cougur. This is a supplement teaching/learning material, and functions as a 'cheat sheet' for instructors and/or students.
This is not intended as curriculum material. I guarantee nothing. I claim no ownership or discovery of any of the material in this document, however I reserve my right of creative expression for materials contained. This document may not be sold, copied or altered in anyway by anyone.
Please report any errors to s.grantwilliam@ieee.org
Ks
This document discusses the composition of forces and moments. It defines key terms like resultant force and moment of a force. It describes the parallelogram, triangle and polygon laws for combining concurrent coplanar forces into a single resultant force. It also explains Varignon's principle of moments, which states that the algebraic sum of the moments of individual forces equals the moment of the resultant force about the same point. Several example problems are provided to illustrate how to use these principles to find the magnitude and direction of resultant forces and moments in systems of coplanar concurrent and non-concurrent forces.
On ranges and null spaces of a special type of operator named 𝝀 − 𝒋𝒆𝒄𝒕𝒊𝒐𝒏. – ...
In this article, 𝜆 − 𝑗𝑒𝑐𝑡𝑖𝑜𝑛 has been introduced which is a generalization of trijection
operator as introduced in P.Chandra’s Ph. D. thesis titled “Investigation into the theory of operators
and linear spaces” (Patna University,1977). We obtain relation between ranges and null spaces of two
given 𝜆 − 𝑗𝑒𝑐𝑡𝑖𝑜𝑛𝑠 under suitable conditions.
Differentiation
This document summarizes differentiation and the concept of the derivative. It defines secant and tangent lines, and explains that the derivative at a point equals the slope of the tangent line at that point. It provides the definition of the derivative as the limit of the difference quotient. Examples are given of finding derivatives using this limit definition and applying power rule theorems of differentiation. The document also demonstrates using the product rule to take the derivative of a sum of terms.
Taller 1 parcial 3
This document discusses three exercises involving linear transformations. The exercises ask the reader to determine if given functions define linear transformations and to determine the output of linear transformations given their behavior on sample inputs. The document provides the definitions, inputs, and step-by-step workings to solve each exercise. It concludes that exercises 1 and 3 define linear transformations while exercise 2 does not and determines the output of two other linear transformations.
A uniform distribution has density function find n (1)
A uniform distribution has density function
f(x)=1/8 0≤x≤8
Find how large the value of ‘n’ must be in order
P(3.5≤x ̅≤4.5)≥0.8
Linear sort
Linear sorting algorithms like counting sort, bucket sort, and radix sort can sort arrays of numbers in linear O(n) time by exploiting properties of the data. Counting sort works for integers within a range [0,r] by counting the frequency of each number and using the frequencies to place numbers in the correct output positions. Bucket sort places numbers uniformly distributed between 0 and 1 into buckets and sorts the buckets. Radix sort treats multi-digit numbers as strings by sorting based on individual digit positions from least to most significant.
Mechanics engineering statics forces analysis 3D
This document provides an introduction to engineering mechanics and statics. It discusses types of loads, force analysis, moments, and force resultants. It also describes how to resolve forces into three rectangular components in three situations: 1) when given the force and coordinate dimensions, 2) when given coordinates and two angles, and 3) when given the starting and ending force coordinates. Examples are provided for resolving forces into x, y, and z components.
三角関数の加法定理と関連公式（人間科学のための基礎数学 補足資料）
The document summarizes trigonometric addition formulas and related formulas. It provides proofs of the formulas using properties of coordinates on the unit circle. Specifically, it proves formulas for cosine, sine, and tangent of the sum or difference of two angles α and β using the x-y coordinates of points on two superimposed unit circles with angles of α, β, α+β, and -β.
Piii taller transformaciones lineales
The document analyzes linear transformations in R2 and R3. It determines whether three given functions define linear transformations based on whether they satisfy the property T(αu + βv) = αT(u) + βT(v).
The first function f(x,y) = (3x - y, x + y) is determined to define a linear transformation in R2 since it satisfies the property.
The second function f(x,y,z) = (x,y,z2) is determined not to define a linear transformation in R3 since z2 is not a linear term.
The third function f(x,y,z) = (x +
Properties of coefficient of correlation
The document discusses properties of the coefficient of correlation (r) including:
1) r always lies between -1 and 1
2) r is the geometric mean of the two regression coefficients
3) Several examples are shown calculating r from regression coefficients and comparing to Pearson's coefficient of correlation.
Length contraction - Special Relativity
Deduction of the Contraction of Length in Special Relativity using Lorentz Transformation Equation and an Animation to Exhibit the Effect
The derivatives module03
The document discusses the definition and calculation of the tangent line and derivative of functions. It provides examples of finding the equation of the tangent line to graphs at given points, as well as calculating the derivative of functions. The key points are:
- The slope of the tangent line at a point (a, f(a)) on a graph y=f(x) is the limit of the slope of secant lines as they approach the point.
- Examples show how to apply the definition of the tangent line to find the equation of the line for various functions at given points.
- The derivative of a function f(x) is defined as the limit of the difference quotient, and represents the slope of the tangent
Z transforms
1. The document discusses using Z-transforms to solve linear difference equations with constant coefficients.
2. It provides the working procedure which involves taking the Z-transform of both sides of the difference equation, rearranging to isolate the Z-transform of the unknown function U(z), and then taking the inverse Z-transform to find the solution in terms of n.
3. As an example, it shows the step-by-step solution of the difference equation un+2 - 2un+1 + un = 3n + 5 using this method. The solution is found to be 1/2n(n-1)(n+3) + c0 + (c1-c0)n
taller transformaciones lineales
The document discusses linear transformations. It provides examples of determining if functions define linear transformations by checking if they satisfy the property that T(αu + βv) = αT(u) + βT(v). It then gives an example of using a system of equations to determine the output of a linear transformation T for a given input, when the outputs of T for two other example inputs are given.
Gcse Maths Resources
The document discusses function notation and transformations of functions. It provides examples of writing quadratic, cubic, and rational functions in function notation. It also demonstrates how functions are translated vertically by adding or subtracting a number from the original function. Translating a function f(x) by a results in the new function f(x)+a or f(x)-a.
On ranges and null spaces of a special type of operator named 𝝀 − 𝒋𝒆𝒄𝒕𝒊𝒐𝒏. – ...
In this article, 𝜆 − 𝑗𝑒𝑐𝑡𝑖𝑜𝑛 has been introduced which is a generalization of trijection
operator as introduced in P.Chandra’s Ph. D. thesis titled “Investigation into the theory of operators
and linear spaces” (Patna University,1977). We obtain relation between ranges and null spaces of two
given 𝜆 − 𝑗𝑒𝑐𝑡𝑖𝑜𝑛𝑠 under suitable conditions
Module9 the pearson correlation
The Pearson Product Moment Correlation Coefficient (r) measures the strength of the linear relationship between two variables. The r value is calculated using a formula involving the sum of the products of paired values and sum of squares for the two variables. r values range from +1 to -1, with values closer to these extremes indicating a stronger correlation and values closer to 0 indicating weaker or no correlation. A positive r represents a positive correlation and a negative r represents a negative correlation. The example calculates r = 0.962 for time spent studying and test scores, indicating a strong positive correlation.
6 28 18_hack_hunterdon_meetup_deep_rl
The document discusses the potential risks of advanced artificial intelligence and deep reinforcement learning. It notes Elon Musk's concern that AI could become an "immortal dictator from which we would never escape." It then provides an overview of reinforcement learning concepts like the optimal action-value function and using deep neural networks to approximate this function. It lists several resources for learning more about reinforcement learning, including books, courses, code implementations, and influential papers that demonstrate techniques like deep Q-learning.
Integral calculus
1) The document discusses basic rules and concepts of integration, including that integration is the inverse process of differentiation and that the indefinite integral of a function f(x) is notated as ∫f(x) dx = F(x) + c, where F(x) is the primitive function and c is the constant of integration.
2) Methods of integration discussed include the substitution method, where a function is substituted for the variable, and integration by parts, which uses the product rule in reverse to solve integrals involving products.
3) Finding the constant of integration c requires knowing the value of the primitive function F(x) at a specific point, which eliminates the family of functions and isolates a
Tugas Aljabar Linear
1) The document presents the work of a student named Nurkhlifah Anwar solving a system of linear equations using Gaussian elimination and substitution methods.
2) Both methods resulted in the same solutions of x1 = -15/2, x2 = 15, x3 = 23/2.
3) Gaussian elimination uses row operations to transform the matrix into reduced row echelon form and directly obtain the variable values. Substitution sequentially substitutes values into the original equations to solve for each variable.
On ranges and null spaces of a special type of operator named 𝝀 − 𝒋𝒆𝒄𝒕𝒊𝒐𝒏. – ...
This document discusses λ-jections, a generalization of projection operators. It shows that if E is a λ-jection, then expressions of the form aE^2 + bE are projections under certain conditions on a and b. Three specific projections - A, B, and C - are obtained. It is shown that A and B are orthogonal, and that the range and null space of E are equal to those of C. The ranges of A and B are also characterized in terms of the operator E.
Optimum engineering design - Day 5. Clasical optimization methods
The document discusses various numerical optimization methods for solving unconstrained nonlinear problems. It covers iterative methods for finding the optimal solution, including techniques for determining a suitable search direction and performing a line search to minimize the objective function along that direction. Specific methods covered include the steepest descent method, conjugate gradient method, Newton's method, trust region methods, and line search techniques like the golden section search and quadratic approximation.
Optimum Engineering Design - Day 2b. Classical Optimization methods
This document provides an overview of an optimization methods course, including its objectives, prerequisites, and materials. The course covers topics such as linear programming, nonlinear programming, and mixed integer programming problems. It also includes mathematical preliminaries on topics like convex sets and functions, gradients, Hessians, and Taylor series expansions. Methods for solving systems of linear equations and examples are presented.
More Related Content
What's hot
On ranges and null spaces of a special type of operator named 𝝀 − 𝒋𝒆𝒄𝒕𝒊𝒐𝒏. – ...
In this article, 𝜆 − 𝑗𝑒𝑐𝑡𝑖𝑜𝑛 has been introduced which is a generalization of trijection
operator as introduced in P.Chandra’s Ph. D. thesis titled “Investigation into the theory of operators
and linear spaces” (Patna University,1977). We obtain relation between ranges and null spaces of two
given 𝜆 − 𝑗𝑒𝑐𝑡𝑖𝑜𝑛𝑠 under suitable conditions.
Differentiation
This document summarizes differentiation and the concept of the derivative. It defines secant and tangent lines, and explains that the derivative at a point equals the slope of the tangent line at that point. It provides the definition of the derivative as the limit of the difference quotient. Examples are given of finding derivatives using this limit definition and applying power rule theorems of differentiation. The document also demonstrates using the product rule to take the derivative of a sum of terms.
Taller 1 parcial 3
This document discusses three exercises involving linear transformations. The exercises ask the reader to determine if given functions define linear transformations and to determine the output of linear transformations given their behavior on sample inputs. The document provides the definitions, inputs, and step-by-step workings to solve each exercise. It concludes that exercises 1 and 3 define linear transformations while exercise 2 does not and determines the output of two other linear transformations.
A uniform distribution has density function find n (1)
A uniform distribution has density function
f(x)=1/8 0≤x≤8
Find how large the value of ‘n’ must be in order
P(3.5≤x ̅≤4.5)≥0.8
Linear sort
Linear sorting algorithms like counting sort, bucket sort, and radix sort can sort arrays of numbers in linear O(n) time by exploiting properties of the data. Counting sort works for integers within a range [0,r] by counting the frequency of each number and using the frequencies to place numbers in the correct output positions. Bucket sort places numbers uniformly distributed between 0 and 1 into buckets and sorts the buckets. Radix sort treats multi-digit numbers as strings by sorting based on individual digit positions from least to most significant.
Mechanics engineering statics forces analysis 3D
This document provides an introduction to engineering mechanics and statics. It discusses types of loads, force analysis, moments, and force resultants. It also describes how to resolve forces into three rectangular components in three situations: 1) when given the force and coordinate dimensions, 2) when given coordinates and two angles, and 3) when given the starting and ending force coordinates. Examples are provided for resolving forces into x, y, and z components.
三角関数の加法定理と関連公式（人間科学のための基礎数学 補足資料）
The document summarizes trigonometric addition formulas and related formulas. It provides proofs of the formulas using properties of coordinates on the unit circle. Specifically, it proves formulas for cosine, sine, and tangent of the sum or difference of two angles α and β using the x-y coordinates of points on two superimposed unit circles with angles of α, β, α+β, and -β.
Piii taller transformaciones lineales
The document analyzes linear transformations in R2 and R3. It determines whether three given functions define linear transformations based on whether they satisfy the property T(αu + βv) = αT(u) + βT(v).
The first function f(x,y) = (3x - y, x + y) is determined to define a linear transformation in R2 since it satisfies the property.
The second function f(x,y,z) = (x,y,z2) is determined not to define a linear transformation in R3 since z2 is not a linear term.
The third function f(x,y,z) = (x +
Properties of coefficient of correlation
The document discusses properties of the coefficient of correlation (r) including:
1) r always lies between -1 and 1
2) r is the geometric mean of the two regression coefficients
3) Several examples are shown calculating r from regression coefficients and comparing to Pearson's coefficient of correlation.
Length contraction - Special Relativity
Deduction of the Contraction of Length in Special Relativity using Lorentz Transformation Equation and an Animation to Exhibit the Effect
The derivatives module03
The document discusses the definition and calculation of the tangent line and derivative of functions. It provides examples of finding the equation of the tangent line to graphs at given points, as well as calculating the derivative of functions. The key points are:
- The slope of the tangent line at a point (a, f(a)) on a graph y=f(x) is the limit of the slope of secant lines as they approach the point.
- Examples show how to apply the definition of the tangent line to find the equation of the line for various functions at given points.
- The derivative of a function f(x) is defined as the limit of the difference quotient, and represents the slope of the tangent
Z transforms
1. The document discusses using Z-transforms to solve linear difference equations with constant coefficients.
2. It provides the working procedure which involves taking the Z-transform of both sides of the difference equation, rearranging to isolate the Z-transform of the unknown function U(z), and then taking the inverse Z-transform to find the solution in terms of n.
3. As an example, it shows the step-by-step solution of the difference equation un+2 - 2un+1 + un = 3n + 5 using this method. The solution is found to be 1/2n(n-1)(n+3) + c0 + (c1-c0)n
taller transformaciones lineales
The document discusses linear transformations. It provides examples of determining if functions define linear transformations by checking if they satisfy the property that T(αu + βv) = αT(u) + βT(v). It then gives an example of using a system of equations to determine the output of a linear transformation T for a given input, when the outputs of T for two other example inputs are given.
Gcse Maths Resources
The document discusses function notation and transformations of functions. It provides examples of writing quadratic, cubic, and rational functions in function notation. It also demonstrates how functions are translated vertically by adding or subtracting a number from the original function. Translating a function f(x) by a results in the new function f(x)+a or f(x)-a.
On ranges and null spaces of a special type of operator named 𝝀 − 𝒋𝒆𝒄𝒕𝒊𝒐𝒏. – ...
In this article, 𝜆 − 𝑗𝑒𝑐𝑡𝑖𝑜𝑛 has been introduced which is a generalization of trijection
operator as introduced in P.Chandra’s Ph. D. thesis titled “Investigation into the theory of operators
and linear spaces” (Patna University,1977). We obtain relation between ranges and null spaces of two
given 𝜆 − 𝑗𝑒𝑐𝑡𝑖𝑜𝑛𝑠 under suitable conditions
Module9 the pearson correlation
The Pearson Product Moment Correlation Coefficient (r) measures the strength of the linear relationship between two variables. The r value is calculated using a formula involving the sum of the products of paired values and sum of squares for the two variables. r values range from +1 to -1, with values closer to these extremes indicating a stronger correlation and values closer to 0 indicating weaker or no correlation. A positive r represents a positive correlation and a negative r represents a negative correlation. The example calculates r = 0.962 for time spent studying and test scores, indicating a strong positive correlation.
6 28 18_hack_hunterdon_meetup_deep_rl
The document discusses the potential risks of advanced artificial intelligence and deep reinforcement learning. It notes Elon Musk's concern that AI could become an "immortal dictator from which we would never escape." It then provides an overview of reinforcement learning concepts like the optimal action-value function and using deep neural networks to approximate this function. It lists several resources for learning more about reinforcement learning, including books, courses, code implementations, and influential papers that demonstrate techniques like deep Q-learning.
Integral calculus
1) The document discusses basic rules and concepts of integration, including that integration is the inverse process of differentiation and that the indefinite integral of a function f(x) is notated as ∫f(x) dx = F(x) + c, where F(x) is the primitive function and c is the constant of integration.
2) Methods of integration discussed include the substitution method, where a function is substituted for the variable, and integration by parts, which uses the product rule in reverse to solve integrals involving products.
3) Finding the constant of integration c requires knowing the value of the primitive function F(x) at a specific point, which eliminates the family of functions and isolates a
Tugas Aljabar Linear
1) The document presents the work of a student named Nurkhlifah Anwar solving a system of linear equations using Gaussian elimination and substitution methods.
2) Both methods resulted in the same solutions of x1 = -15/2, x2 = 15, x3 = 23/2.
3) Gaussian elimination uses row operations to transform the matrix into reduced row echelon form and directly obtain the variable values. Substitution sequentially substitutes values into the original equations to solve for each variable.
On ranges and null spaces of a special type of operator named 𝝀 − 𝒋𝒆𝒄𝒕𝒊𝒐𝒏. – ...
This document discusses λ-jections, a generalization of projection operators. It shows that if E is a λ-jection, then expressions of the form aE^2 + bE are projections under certain conditions on a and b. Three specific projections - A, B, and C - are obtained. It is shown that A and B are orthogonal, and that the range and null space of E are equal to those of C. The ranges of A and B are also characterized in terms of the operator E.
What's hot (20)
On ranges and null spaces of a special type of operator named 𝝀 − 𝒋𝒆𝒄𝒕𝒊𝒐𝒏. – ...
On ranges and null spaces of a special type of operator named 𝝀 − 𝒋𝒆𝒄𝒕𝒊𝒐𝒏. – ...
A uniform distribution has density function find n (1)
A uniform distribution has density function find n (1)
On ranges and null spaces of a special type of operator named 𝝀 − 𝒋𝒆𝒄𝒕𝒊𝒐𝒏. – ...
On ranges and null spaces of a special type of operator named 𝝀 − 𝒋𝒆𝒄𝒕𝒊𝒐𝒏. – ...
On ranges and null spaces of a special type of operator named 𝝀 − 𝒋𝒆𝒄𝒕𝒊𝒐𝒏. – ...
On ranges and null spaces of a special type of operator named 𝝀 − 𝒋𝒆𝒄𝒕𝒊𝒐𝒏. – ...
Similar to Tavas and pashmaks
Optimum engineering design - Day 5. Clasical optimization methods
The document discusses various numerical optimization methods for solving unconstrained nonlinear problems. It covers iterative methods for finding the optimal solution, including techniques for determining a suitable search direction and performing a line search to minimize the objective function along that direction. Specific methods covered include the steepest descent method, conjugate gradient method, Newton's method, trust region methods, and line search techniques like the golden section search and quadratic approximation.
Optimum Engineering Design - Day 2b. Classical Optimization methods
This document provides an overview of an optimization methods course, including its objectives, prerequisites, and materials. The course covers topics such as linear programming, nonlinear programming, and mixed integer programming problems. It also includes mathematical preliminaries on topics like convex sets and functions, gradients, Hessians, and Taylor series expansions. Methods for solving systems of linear equations and examples are presented.
Vehicle Routing Problem using PSO (Particle Swarm Optimization)
The document describes using particle swarm optimization to solve a capacitated vehicle routing problem. The objective is to minimize the total distance traveled by a fleet of vehicles to service customers while meeting vehicle capacity constraints. It provides details on initializing particle positions and velocities, calculating fitness based on route distance, and iteratively updating particles to find the optimal solution. The proposed approach is demonstrated on an example problem with 10 customers, 4 vehicles of capacity 200, and distances given between all locations. Pseudocode and equations are included for implementing the particle swarm optimization algorithm to find high-quality routes for the vehicle routing problem.
Mathematics of nyquist plot [autosaved] [autosaved]
This document provides a summary of a lecture on the mathematics of Nyquist plots. Some key topics covered include:
- Complex calculus concepts like Cauchy's theorem and the principle of argument.
- Derivation of the Cauchy-Riemann equations.
- Analytic functions and Cauchy's integral formula.
- Residue theorem and its application to determining stability using encirclements in the Nyquist plot.
- Construction of the Nyquist path and an example application to determine stability of a closed-loop system.
Max flows via electrical flows (long talk)
1) The document describes algorithms for solving the maximum flow and electrical flow problems on graphs.
2) It introduces the multiplicative weight update method, which can be used to find an approximate maximum flow in Oε(m3/2) time by reducing the problem to approximating electrical flows.
3) The algorithm works by having the "follower" maintain a distribution over edges using MWU based on "money" or congestion values revealed by approximate electrical flow computations.
Paper study: Attention, learn to solve routing problems!
This is the paper from ICLR 2019
The authors aim to use the attention mechanism (especially Transfomer) to solve routing problem including TSP.
Support vector machines
This document discusses support vector machines (SVMs). It explains that SVMs are supervised learning models that can be used for classification or regression tasks. The document outlines hard and soft margin SVMs, describing how soft margin SVMs allow for some errors in the classification. It presents the mathematical formulations of linear SVMs, including defining the decision boundary, maximizing the margin between classes, and deriving the dual and primal optimization problems. Finally, it introduces kernel methods that can extend linear SVMs to handle nonlinear decision boundaries using kernel tricks.
Lec05.pptx
This document provides an overview of logistic regression. It discusses the hypothesis representation using a sigmoid function to output probabilities between 0 and 1. It describes using maximum likelihood estimation to learn the parameters θ by minimizing the cost function. Gradient descent is used to optimize the cost function. The document also briefly mentions regularization and multi-class classification extensions.
Applied Algorithms and Structures week999
The document discusses algorithms and data structures. It begins with two quotes about programming and algorithms. It then provides pseudocode for naive and optimized recursive Fibonacci algorithms, as well as an iterative dynamic programming version. It also covers dynamic programming approaches for calculating Fibonacci numbers, Catalan numbers, the chessboard traversal problem, the rod cutting problem, longest common subsequence, and assembly line traversal. The key ideas are introducing dynamic programming techniques like memoization and bottom-up iteration to improve the time complexity of recursive algorithms from exponential to polynomial.
LINEAR ALGEBRA, WITH OPTIMIZATION
LINEAR ALGEBRA, WITH OPTIMIZATION; VECTORS & MATRICES, LINEAR SUB-SPACES, LINEAR INDEPENDENCE & DIMENSION; MATRIX SUB-SPACES, MATRIX RANK, MATRIX INVERSE, SYSTEMS OF EQUATIONS, etc.
Page rank - from theory to application
Part 1: Brief introduction of Markov chain theory and how Page rank works with intuitive explanation of linear algebra.
Part 2: Applications of Perron-Frobenius theorem. MCMC and simulated annealing.
Lecture_10_SVD.pptx
The summary is:
1. The singular value decomposition (SVD) provides a factorization of any matrix A as A = UΣVT, where U and V are orthogonal matrices and Σ is a diagonal matrix of singular values.
2. The SVD exists for any matrix A by considering the eigendecompositions of the positive semidefinite matrices ATA and AAT.
3. The SVD is useful for finding least squares solutions to overdetermined or underdetermined systems of linear equations, by using the pseudoinverse of A.
Quadratic form and functional optimization
This slideshow describes a mathematics topic on quadratic form in English.
I made the slideshow to understand this topic on a deeper way.
I like linear algebra very much!!
2Multi_armed_bandits.pptx
This document discusses reinforcement learning and the multi-armed bandit problem. It introduces concepts like the epsilon-greedy algorithm, which balances exploiting the best known action and exploring potentially better actions. It also discusses updating action values using temporal difference learning and the exploration strategy of using upper confidence bounds. Policy-based algorithms are introduced, which assign preference values to actions and select probabilistically using the softmax function. The preferences can then be updated using policy gradient ascent based on observed rewards.
Rational function 11
- A rational expression is a ratio of two polynomial expressions, where the denominator is not equal to zero.
- To find the domain of a rational expression, set the denominator equal to zero and solve for values of x that make the denominator equal to zero. These values are excluded from the domain.
- To find the range, find the horizontal asymptote by comparing the degrees of the numerator and denominator. The range is all real numbers except the constant value of the horizontal asymptote.
Bounded arithmetic in free logic
This document discusses Buss's theories of bounded arithmetic (S2i) and how they relate to the polynomial hierarchy (PH). It proposes using the separation of Buss's theories to approach the separation of levels of PH. It presents a consistency proof for S2i inside S2i+2 by introducing a predicate E for term existence and defining a bounded truth definition. The goal is to separate S2i and S2i+2 through Gödel's incompleteness theorem by showing S2i+2 can prove the i-consistency of S2i-E but S2i cannot prove its own consistency. Future work aims to simplify S2i-E and formally prove S2i can derive the i
Solving Poisson Equation using Conjugate Gradient Method and its implementation
General description of solving poisson equation using conjugate gradient Method and its implementation. i present this material on my lab meeting
Paper Study: OptNet: Differentiable Optimization as a Layer in Neural Networks
Paper from ICML 2017.
In this slide, show the full math derivation but talk less about interior-point method.
DL_lecture3_regularization_I.pdf
Regularization helps prevent overfitting in machine learning models. It can be viewed as adding extra constraints or terms to the training objective function. Common regularization methods include l2 regularization, which adds a penalty that is proportional to the square of the weights, and l1 regularization, which uses an absolute value penalty. These help prevent overfitting by constraining the complexity of the model. l2 regularization scales down the weights, while l1 regularization induces sparsity by driving small weights to exactly zero.
Unit-1 Basic Concept of Algorithm.pptx
The document discusses various topics related to algorithms including algorithm design, real-life applications, analysis, and implementation. It specifically covers four algorithms - the taxi algorithm, rent-a-car algorithm, call-me algorithm, and bus algorithm - for getting from an airport to a house. It also provides examples of simple multiplication methods like the American, English, and Russian approaches as well as the divide and conquer method.
Similar to Tavas and pashmaks (20)
Optimum engineering design - Day 5. Clasical optimization methods
Optimum engineering design - Day 5. Clasical optimization methods
Optimum Engineering Design - Day 2b. Classical Optimization methods
Optimum Engineering Design - Day 2b. Classical Optimization methods
Vehicle Routing Problem using PSO (Particle Swarm Optimization)
Vehicle Routing Problem using PSO (Particle Swarm Optimization)
Mathematics of nyquist plot [autosaved] [autosaved]
Mathematics of nyquist plot [autosaved] [autosaved]
Paper study: Attention, learn to solve routing problems!
Paper study: Attention, learn to solve routing problems!
Solving Poisson Equation using Conjugate Gradient Method and its implementation
Solving Poisson Equation using Conjugate Gradient Method and its implementation
Paper Study: OptNet: Differentiable Optimization as a Layer in Neural Networks
Paper Study: OptNet: Differentiable Optimization as a Layer in Neural Networks
Recently uploaded
Null Bangalore | Pentesters Approach to AWS IAM
#Abstract:
- Learn more about the real-world methods for auditing AWS IAM (Identity and Access Management) as a pentester. So let us proceed with a brief discussion of IAM as well as some typical misconfigurations and their potential exploits in order to reinforce the understanding of IAM security best practices.
- Gain actionable insights into AWS IAM policies and roles, using hands on approach.
#Prerequisites:
- Basic understanding of AWS services and architecture
- Familiarity with cloud security concepts
- Experience using the AWS Management Console or AWS CLI.
- For hands on lab create account on [killercoda.com](https://killercoda.com/cloudsecurity-scenario/)
# Scenario Covered:
- Basics of IAM in AWS
- Implementing IAM Policies with Least Privilege to Manage S3 Bucket
- Objective: Create an S3 bucket with least privilege IAM policy and validate access.
- Steps:
- Create S3 bucket.
- Attach least privilege policy to IAM user.
- Validate access.
- Exploiting IAM PassRole Misconfiguration
-Allows a user to pass a specific IAM role to an AWS service (ec2), typically used for service access delegation. Then exploit PassRole Misconfiguration granting unauthorized access to sensitive resources.
- Objective: Demonstrate how a PassRole misconfiguration can grant unauthorized access.
- Steps:
- Allow user to pass IAM role to EC2.
- Exploit misconfiguration for unauthorized access.
- Access sensitive resources.
- Exploiting IAM AssumeRole Misconfiguration with Overly Permissive Role
- An overly permissive IAM role configuration can lead to privilege escalation by creating a role with administrative privileges and allow a user to assume this role.
- Objective: Show how overly permissive IAM roles can lead to privilege escalation.
- Steps:
- Create role with administrative privileges.
- Allow user to assume the role.
- Perform administrative actions.
- Differentiation between PassRole vs AssumeRole
Try at [killercoda.com](https://killercoda.com/cloudsecurity-scenario/)
Blood finder application project report (1).pdf
Blood Finder is an emergency time app where a user can search for the blood banks as
well as the registered blood donors around Mumbai. This application also provide an
opportunity for the user of this application to become a registered donor for this user have
to enroll for the donor request from the application itself. If the admin wish to make user
a registered donor, with some of the formalities with the organization it can be done.
Specialization of this application is that the user will not have to register on sign-in for
searching the blood banks and blood donors it can be just done by installing the
application to the mobile.
The purpose of making this application is to save the user’s time for searching blood of
needed blood group during the time of the emergency.
This is an android application developed in Java and XML with the connectivity of
SQLite database. This application will provide most of basic functionality required for an
emergency time application. All the details of Blood banks and Blood donors are stored
in the database i.e. SQLite.
This application allowed the user to get all the information regarding blood banks and
blood donors such as Name, Number, Address, Blood Group, rather than searching it on
the different websites and wasting the precious time. This application is effective and
user friendly.
一比一原版(osu毕业证书)美国俄勒冈州立大学毕业证如何办理
原版一模一样【微信：741003700 】【(osu毕业证书)美国俄勒冈州立大学毕业证成绩单】【微信：741003700 】学位证，留信认证（真实可查，永久存档）原件一模一样纸张工艺/offer、雅思、外壳等材料/诚信可靠,可直接看成品样本，帮您解决无法毕业带来的各种难题！外壳，原版制作，诚信可靠，可直接看成品样本。行业标杆！精益求精，诚心合作，真诚制作！多年品质 ,按需精细制作，24小时接单,全套进口原装设备。十五年致力于帮助留学生解决难题，包您满意。
本公司拥有海外各大学样板无数，能完美还原。
1:1完美还原海外各大学毕业材料上的工艺：水印，阴影底纹，钢印LOGO烫金烫银，LOGO烫金烫银复合重叠。文字图案浮雕、激光镭射、紫外荧光、温感、复印防伪等防伪工艺。材料咨询办理、认证咨询办理请加学历顾问Q/微741003700
【主营项目】
一.毕业证【q微741003700】成绩单、使馆认证、教育部认证、雅思托福成绩单、学生卡等！
二.真实使馆公证(即留学回国人员证明,不成功不收费)
三.真实教育部学历学位认证（教育部存档！教育部留服网站永久可查）
四.办理各国各大学文凭(一对一专业服务,可全程监控跟踪进度)
如果您处于以下几种情况：
◇在校期间，因各种原因未能顺利毕业……拿不到官方毕业证【q/微741003700】
◇面对父母的压力，希望尽快拿到；
◇不清楚认证流程以及材料该如何准备；
◇回国时间很长，忘记办理；
◇回国马上就要找工作，办给用人单位看；
◇企事业单位必须要求办理的
◇需要报考公务员、购买免税车、落转户口
◇申请留学生创业基金
留信网认证的作用:
1:该专业认证可证明留学生真实身份
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
6:个人职称评审加20分
7:个人信誉贷款加10分
8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才
办理(osu毕业证书)美国俄勒冈州立大学毕业证【微信：741003700 】外观非常简单，由纸质材料制成，上面印有校徽、校名、毕业生姓名、专业等信息。
办理(osu毕业证书)美国俄勒冈州立大学毕业证【微信：741003700 】格式相对统一，各专业都有相应的模板。通常包括以下部分：
校徽：象征着学校的荣誉和传承。
校名:学校英文全称
授予学位：本部分将注明获得的具体学位名称。
毕业生姓名：这是最重要的信息之一，标志着该证书是由特定人员获得的。
颁发日期：这是毕业正式生效的时间，也代表着毕业生学业的结束。
其他信息：根据不同的专业和学位，可能会有一些特定的信息或章节。
办理(osu毕业证书)美国俄勒冈州立大学毕业证【微信：741003700 】价值很高，需要妥善保管。一般来说，应放置在安全、干燥、防潮的地方，避免长时间暴露在阳光下。如需使用，最好使用复印件而不是原件，以免丢失。
综上所述，办理(osu毕业证书)美国俄勒冈州立大学毕业证【微信：741003700 】是证明身份和学历的高价值文件。外观简单庄重，格式统一，包括重要的个人信息和发布日期。对持有人来说，妥善保管是非常重要的。
Butterfly Valves Manufacturer (LBF Series).pdf
We have designed & manufacture the Lubi Valves LBF series type of Butterfly Valves for General Utility Water applications as well as for HVAC applications.
一比一原版(爱大毕业证书)爱荷华大学毕业证如何办理
原版一模一样【微信：741003700 】【(爱大毕业证书)爱荷华大学毕业证成绩单】【微信：741003700 】学位证，留信认证（真实可查，永久存档）原件一模一样纸张工艺/offer、雅思、外壳等材料/诚信可靠,可直接看成品样本，帮您解决无法毕业带来的各种难题！外壳，原版制作，诚信可靠，可直接看成品样本。行业标杆！精益求精，诚心合作，真诚制作！多年品质 ,按需精细制作，24小时接单,全套进口原装设备。十五年致力于帮助留学生解决难题，包您满意。
本公司拥有海外各大学样板无数，能完美还原。
1:1完美还原海外各大学毕业材料上的工艺：水印，阴影底纹，钢印LOGO烫金烫银，LOGO烫金烫银复合重叠。文字图案浮雕、激光镭射、紫外荧光、温感、复印防伪等防伪工艺。材料咨询办理、认证咨询办理请加学历顾问Q/微741003700
【主营项目】
一.毕业证【q微741003700】成绩单、使馆认证、教育部认证、雅思托福成绩单、学生卡等！
二.真实使馆公证(即留学回国人员证明,不成功不收费)
三.真实教育部学历学位认证（教育部存档！教育部留服网站永久可查）
四.办理各国各大学文凭(一对一专业服务,可全程监控跟踪进度)
如果您处于以下几种情况：
◇在校期间，因各种原因未能顺利毕业……拿不到官方毕业证【q/微741003700】
◇面对父母的压力，希望尽快拿到；
◇不清楚认证流程以及材料该如何准备；
◇回国时间很长，忘记办理；
◇回国马上就要找工作，办给用人单位看；
◇企事业单位必须要求办理的
◇需要报考公务员、购买免税车、落转户口
◇申请留学生创业基金
留信网认证的作用:
1:该专业认证可证明留学生真实身份
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
6:个人职称评审加20分
7:个人信誉贷款加10分
8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才
办理(爱大毕业证书)爱荷华大学毕业证【微信：741003700 】外观非常简单，由纸质材料制成，上面印有校徽、校名、毕业生姓名、专业等信息。
办理(爱大毕业证书)爱荷华大学毕业证【微信：741003700 】格式相对统一，各专业都有相应的模板。通常包括以下部分：
校徽：象征着学校的荣誉和传承。
校名:学校英文全称
授予学位：本部分将注明获得的具体学位名称。
毕业生姓名：这是最重要的信息之一，标志着该证书是由特定人员获得的。
颁发日期：这是毕业正式生效的时间，也代表着毕业生学业的结束。
其他信息：根据不同的专业和学位，可能会有一些特定的信息或章节。
办理(爱大毕业证书)爱荷华大学毕业证【微信：741003700 】价值很高，需要妥善保管。一般来说，应放置在安全、干燥、防潮的地方，避免长时间暴露在阳光下。如需使用，最好使用复印件而不是原件，以免丢失。
综上所述，办理(爱大毕业证书)爱荷华大学毕业证【微信：741003700 】是证明身份和学历的高价值文件。外观简单庄重，格式统一，包括重要的个人信息和发布日期。对持有人来说，妥善保管是非常重要的。
This study Examines the Effectiveness of Talent Procurement through the Imple...
In the world with high technology and fast
forward mindset recruiters are walking/showing interest
towards E-Recruitment. Present most of the HRs of
many companies are choosing E-Recruitment as the best
choice for recruitment. E-Recruitment is being done
through many online platforms like Linkedin, Naukri,
Instagram , Facebook etc. Now with high technology E-
Recruitment has gone through next level by using
Artificial Intelligence too.
Key Words : Talent Management, Talent Acquisition , E-
Recruitment , Artificial Intelligence Introduction
Effectiveness of Talent Acquisition through E-
Recruitment in this topic we will discuss about 4important
and interlinked topics which are
Digital Twins Computer Networking Paper Presentation.pptx
A Digital Twin in computer networking is a virtual representation of a physical network, used to simulate, analyze, and optimize network performance and reliability. It leverages real-time data to enhance network management, predict issues, and improve decision-making processes.
Object Oriented Analysis and Design - OOAD
This ppt gives detailed description of Object Oriented Analysis and design.
Determination of Equivalent Circuit parameters and performance characteristic...
Includes the testing of induction motor to draw the circle diagram of induction motor with step wise procedure and calculation for the same. Also explains the working and application of Induction generator
Tools & Techniques for Commissioning and Maintaining PV Systems W-Animations ...
Join us for this solutions-based webinar on the tools and techniques for commissioning and maintaining PV Systems. In this session, we'll review the process of building and maintaining a solar array, starting with installation and commissioning, then reviewing operations and maintenance of the system. This course will review insulation resistance testing, I-V curve testing, earth-bond continuity, ground resistance testing, performance tests, visual inspections, ground and arc fault testing procedures, and power quality analysis.
Fluke Solar Application Specialist Will White is presenting on this engaging topic:
Will has worked in the renewable energy industry since 2005, first as an installer for a small east coast solar integrator before adding sales, design, and project management to his skillset. In 2022, Will joined Fluke as a solar application specialist, where he supports their renewable energy testing equipment like IV-curve tracers, electrical meters, and thermal imaging cameras. Experienced in wind power, solar thermal, energy storage, and all scales of PV, Will has primarily focused on residential and small commercial systems. He is passionate about implementing high-quality, code-compliant installation techniques.
Applications of artificial Intelligence in Mechanical Engineering.pdf
Historically, mechanical engineering has relied heavily on human expertise and empirical methods to solve complex problems. With the introduction of computer-aided design (CAD) and finite element analysis (FEA), the field took its first steps towards digitization. These tools allowed engineers to simulate and analyze mechanical systems with greater accuracy and efficiency. However, the sheer volume of data generated by modern engineering systems and the increasing complexity of these systems have necessitated more advanced analytical tools, paving the way for AI.
AI offers the capability to process vast amounts of data, identify patterns, and make predictions with a level of speed and accuracy unattainable by traditional methods. This has profound implications for mechanical engineering, enabling more efficient design processes, predictive maintenance strategies, and optimized manufacturing operations. AI-driven tools can learn from historical data, adapt to new information, and continuously improve their performance, making them invaluable in tackling the multifaceted challenges of modern mechanical engineering.
Levelised Cost of Hydrogen (LCOH) Calculator Manual
The aim of this manual is to explain the
methodology behind the Levelized Cost of
Hydrogen (LCOH) calculator. Moreover, this
manual also demonstrates how the calculator
can be used for estimating the expenses associated with hydrogen production in Europe
using low-temperature electrolysis considering different sources of electricity
Prediction of Electrical Energy Efficiency Using Information on Consumer's Ac...
Energy efficiency has been important since the latter part of the last century. The main object of this survey is to determine the energy efficiency knowledge among consumers. Two separate districts in Bangladesh are selected to conduct the survey on households and showrooms about the energy and seller also. The survey uses the data to find some regression equations from which it is easy to predict energy efficiency knowledge. The data is analyzed and calculated based on five important criteria. The initial target was to find some factors that help predict a person's energy efficiency knowledge. From the survey, it is found that the energy efficiency awareness among the people of our country is very low. Relationships between household energy use behaviors are estimated using a unique dataset of about 40 households and 20 showrooms in Bangladesh's Chapainawabganj and Bagerhat districts. Knowledge of energy consumption and energy efficiency technology options is found to be associated with household use of energy conservation practices. Household characteristics also influence household energy use behavior. Younger household cohorts are more likely to adopt energy-efficient technologies and energy conservation practices and place primary importance on energy saving for environmental reasons. Education also influences attitudes toward energy conservation in Bangladesh. Low-education households indicate they primarily save electricity for the environment while high-education households indicate they are motivated by environmental concerns.
SELENIUM CONF -PALLAVI SHARMA - 2024.pdf
Begin your journey to contribute to Selenium - A Talk at the Selenium Conference 2024
Ericsson LTE Throughput Troubleshooting Techniques.ppt
Ericsson LTE Throughput Troubleshooting Techniques
Mechanical Engineering on AAI Summer Training Report-003.pdf
Mechanical Engineering PROJECT REPORT ON SUMMER VOCATIONAL TRAINING
AT MBB AIRPORT
Recently uploaded (20)
This study Examines the Effectiveness of Talent Procurement through the Imple...
This study Examines the Effectiveness of Talent Procurement through the Imple...
Digital Twins Computer Networking Paper Presentation.pptx
Digital Twins Computer Networking Paper Presentation.pptx
Determination of Equivalent Circuit parameters and performance characteristic...
Determination of Equivalent Circuit parameters and performance characteristic...
Tools & Techniques for Commissioning and Maintaining PV Systems W-Animations ...
Tools & Techniques for Commissioning and Maintaining PV Systems W-Animations ...
Applications of artificial Intelligence in Mechanical Engineering.pdf
Applications of artificial Intelligence in Mechanical Engineering.pdf
Levelised Cost of Hydrogen (LCOH) Calculator Manual
Levelised Cost of Hydrogen (LCOH) Calculator Manual
Prediction of Electrical Energy Efficiency Using Information on Consumer's Ac...
Prediction of Electrical Energy Efficiency Using Information on Consumer's Ac...
Ericsson LTE Throughput Troubleshooting Techniques.ppt
Ericsson LTE Throughput Troubleshooting Techniques.ppt
Mechanical Engineering on AAI Summer Training Report-003.pdf
Mechanical Engineering on AAI Summer Training Report-003.pdf
Tavas and pashmaks
- 1. Codeforces Round #299 <Tavas and Pashmaks> Minsu Kim
- 2. Description • Given a set of competitors which have 𝑟𝑖 and 𝑠𝑖 as its value, find the largest subset of possible winners W • a competitor can be a winner if there exists 𝑅, 𝑆 ∈ ℝ+ 2 where 𝑡 = 𝑅 𝑟 𝑖 + 𝑆 𝑠 𝑖 has the minimum value in the set. • Constraints : 1 ≤ 𝑛 ≤ 2 × 105, 1 ≤ 𝑠𝑖, 𝑟𝑖 ≤ 104
- 3. Obvious Facts • A competitor 𝑖 cannot be a winner if another competitor 𝑗 such that 𝑟𝑖 < 𝑟𝑗 𝑎𝑛𝑑 𝑠𝑖 < 𝑠𝑗 exists. 𝑟 𝑠 (𝑟𝑗, 𝑠𝑗) (𝑟𝑖, 𝑠𝑖)
- 4. Geometric Analysis • 𝑡 should be the minimum for 𝑖 to be a winner • min 𝑡 = 𝑅 𝑟 𝑖 + 𝑆 𝑠 𝑖 = R, S ∙ 1 𝑟 𝑖 , 1 𝑠 𝑖 = R, S × | 1 𝑟 𝑖 , 1 𝑠 𝑖 | × cos 𝜃 • Boxed one is the component of 𝟏 𝒓 𝒊 , 𝟏 𝒔 𝒊 in dir. of 𝐑, 𝐒 • Therefore, suppose R, S is any unit vector in the first quadrant and find the possible winners!
- 8. Insight • Winners turned out to be.. the subset of Convex Hull for the points 1 𝑟 𝑖 , 1 𝑠 𝑖 ! • Exactly, the lower-left part of the Convex Hull.
- 9. Implementation • There are some pesky things in implementation.. 1. Precision Problem : repeated cross production of fractions • Use integers. • 𝑓𝑜𝑟 𝑃 1 𝑝 𝑥 , 1 𝑝 𝑦 , 𝐴 1 𝑎 𝑥 , 1 𝑎 𝑦 , 𝐵 1 𝑏 𝑥 , 1 𝑏 𝑦 , 𝑐ℎ𝑒𝑐𝑘𝑖𝑛𝑔 𝑐𝑜𝑢𝑛𝑡𝑒𝑟 − 𝑐𝑙𝑜𝑐𝑘𝑤𝑖𝑠𝑒 𝑤𝑖𝑙𝑙 𝑏𝑒 • 𝐴 − 𝑃 × 𝐵 − 𝑃 = (𝑝 𝑥−𝑎 𝑥)(𝑝 𝑦−𝑏 𝑦) 𝑎 𝑥 𝑏 𝑦 𝑝 𝑥 𝑝 𝑦 − 𝑝 𝑦−𝑎 𝑦 𝑝 𝑥−𝑏 𝑥 𝑎 𝑦 𝑏 𝑥 𝑝 𝑥 𝑝 𝑦 > 0 ⇔ 𝑎 𝑦 𝑏 𝑥 𝑝 𝑥 − 𝑎 𝑥 𝑝 𝑦 − 𝑏 𝑦 − 𝑎 𝑥 𝑏 𝑦 𝑝 𝑦 − 𝑎 𝑦 𝑝 𝑥 − 𝑏 𝑥 > 0 2. Duplicated Competitors : vary for the Convex Hull algorithm • Regard as one and check it later