Recursion is a technique for solving complex problems by breaking them down into smaller subproblems. It involves defining a problem in terms of simpler instances of the same problem. Some key examples of using recursion to solve problems include computing factorials, calculating sums, generating Fibonacci numbers, and traversing directory structures. Recursion uses repetition with abstraction - defining functions that call themselves on smaller inputs until they arrive at a base case.
Ensembles of Many Diverse Weak Defenses can be Strong: Defending Deep Neural ...Pooyan Jamshidi
Despite achieving state-of-the-art performance across many domains, machine learning systems are highly vulnerable to subtle adversarial perturbations. Although defense approaches have been proposed in recent years, many have been bypassed by even weak adversarial attacks. Previous studies showed that ensembles created by combining multiple weak defenses (i.e., input data transformations) are still weak. In this talk, I will show that it is indeed possible to construct effective ensembles using weak defenses to block adversarial attacks. However, to do so requires a diverse set of such weak defenses. Based on this motivation, I will present Athena, an extensible framework for building effective defenses to adversarial attacks against machine learning systems. I will talk about the effectiveness of ensemble strategies with a diverse set of many weak defenses that comprise transforming the inputs (e.g., rotation, shifting, noising, denoising, and many more) before feeding them to target deep neural network classifiers. I will also discuss the effectiveness of the ensembles with adversarial examples generated by various adversaries in different threat models. In the second half of the talk, I will explain why building defenses based on the idea of many diverse weak defenses works, when it is most effective, and what its inherent limitations and overhead are.
This document discusses differentiation and defines the derivative. It begins by formally defining the derivative as a limit and then provides formulas to find the derivatives of simple functions like constants, linear functions, and power functions. It also covers numerical derivatives, implicit differentiation, and higher-order derivatives. Examples are provided to illustrate each concept.
The document discusses inverse functions. It defines a one-to-one function as a function where the horizontal line test shows that every horizontal line intersects the graph at most one point. This ensures that each input is mapped to a single output. An inverse function undoes the original function - if f(x) is the original function, its inverse f^-1(x) satisfies f^-1(f(x)) = x.
This document describes the process of using natural and clamped cubic splines to approximate functions based on data points. It presents the mathematical formulas for natural and clamped cubic splines and their derivatives. Code functions are provided to calculate the splines and plot the results. The document demonstrates applying this process to example functions and data, showing the natural and clamped cubic splines accurately fit the original functions.
This document provides an overview of TensorFlow and how to implement machine learning models using TensorFlow. It discusses:
1) How to install TensorFlow either directly or within a virtual environment.
2) The key concepts of TensorFlow including computational graphs, sessions, placeholders, variables and how they are used to define and run computations.
3) An example one-layer perceptron model for MNIST image classification to demonstrate these concepts in action.
This document summarizes key concepts in neural sequence modeling including recurrent neural networks, long short-term memory networks, and neural Turing machines. It outlines recurrent neural networks and how they can be used for sequence modeling. It then describes long short-term memory networks and how they address the vanishing gradient problem in recurrent neural networks using gating mechanisms. Finally, it provides an overview of neural Turing machines and how they use an external memory component with addressing and reading/writing mechanisms controlled by a neural network controller.
Ensembles of Many Diverse Weak Defenses can be Strong: Defending Deep Neural ...Pooyan Jamshidi
Despite achieving state-of-the-art performance across many domains, machine learning systems are highly vulnerable to subtle adversarial perturbations. Although defense approaches have been proposed in recent years, many have been bypassed by even weak adversarial attacks. Previous studies showed that ensembles created by combining multiple weak defenses (i.e., input data transformations) are still weak. In this talk, I will show that it is indeed possible to construct effective ensembles using weak defenses to block adversarial attacks. However, to do so requires a diverse set of such weak defenses. Based on this motivation, I will present Athena, an extensible framework for building effective defenses to adversarial attacks against machine learning systems. I will talk about the effectiveness of ensemble strategies with a diverse set of many weak defenses that comprise transforming the inputs (e.g., rotation, shifting, noising, denoising, and many more) before feeding them to target deep neural network classifiers. I will also discuss the effectiveness of the ensembles with adversarial examples generated by various adversaries in different threat models. In the second half of the talk, I will explain why building defenses based on the idea of many diverse weak defenses works, when it is most effective, and what its inherent limitations and overhead are.
This document discusses differentiation and defines the derivative. It begins by formally defining the derivative as a limit and then provides formulas to find the derivatives of simple functions like constants, linear functions, and power functions. It also covers numerical derivatives, implicit differentiation, and higher-order derivatives. Examples are provided to illustrate each concept.
The document discusses inverse functions. It defines a one-to-one function as a function where the horizontal line test shows that every horizontal line intersects the graph at most one point. This ensures that each input is mapped to a single output. An inverse function undoes the original function - if f(x) is the original function, its inverse f^-1(x) satisfies f^-1(f(x)) = x.
This document describes the process of using natural and clamped cubic splines to approximate functions based on data points. It presents the mathematical formulas for natural and clamped cubic splines and their derivatives. Code functions are provided to calculate the splines and plot the results. The document demonstrates applying this process to example functions and data, showing the natural and clamped cubic splines accurately fit the original functions.
This document provides an overview of TensorFlow and how to implement machine learning models using TensorFlow. It discusses:
1) How to install TensorFlow either directly or within a virtual environment.
2) The key concepts of TensorFlow including computational graphs, sessions, placeholders, variables and how they are used to define and run computations.
3) An example one-layer perceptron model for MNIST image classification to demonstrate these concepts in action.
This document summarizes key concepts in neural sequence modeling including recurrent neural networks, long short-term memory networks, and neural Turing machines. It outlines recurrent neural networks and how they can be used for sequence modeling. It then describes long short-term memory networks and how they address the vanishing gradient problem in recurrent neural networks using gating mechanisms. Finally, it provides an overview of neural Turing machines and how they use an external memory component with addressing and reading/writing mechanisms controlled by a neural network controller.
An introduction to quantum stochastic calculusSpringer
The document discusses tensor products of Hilbert spaces. It defines positive definite kernels on sets and shows how they can be used to define tensor products. Given Hilbert spaces H1, ..., Hn, it constructs a kernel on the cartesian product of the spaces and shows that its Gelfand pair (H,φ) gives a tensor product of the Hilbert spaces. The map φ from the product space into H is multilinear and H is the completion of the algebraic tensor product of the vector spaces H1, ..., Hn.
The document discusses ordered dictionaries and binary search trees (BSTs). It describes how to implement ordered dictionaries using BSTs to support operations like minimum, maximum, predecessor, and successor in logarithmic time. The key aspects are:
1) BSTs allow searching, minimum, maximum, predecessor, and successor operations in O(h) time where h is the height of the tree.
2) Insertion and deletion in a BST takes O(h) time in the worst case, as repeatedly inserting keys in ascending or descending order yields a skewed tree of height n for n nodes.
3) Search works by comparing the search key to the root node's key, and recursively searching the left or right
The document discusses tangent lines to functions. It provides examples of finding the equation of a tangent line with a given slope to specific functions. It also discusses finding the average and instantaneous velocity of an object given its position function.
This document provides an introduction to symbolic math in MATLAB. It discusses differentiation and integration of functions using symbolic operators. Differentiation is defined as finding the rate of change of a function with respect to a variable. Integration finds the original function given its derivative. The document provides examples of differentiating and integrating simple functions in MATLAB's symbolic toolbox and exercises for the reader to practice.
Stefan Kanev: Clojure, ClojureScript and Why They're Awesome at I T.A.K.E. Un...Mozaic Works
This document summarizes a presentation about Clojure and ClojureScript. It describes Clojure as a modern Lisp dialect that runs on the JVM with a focus on concurrency. It notes advantages like stable platforms, access to Java libraries, and immutable persistent data structures that facilitate parallelism. The document discusses Clojure concepts like homoiconicity, concurrency vs parallelism, syntax using s-expressions, macros for metaprogramming, and data structures like maps, vectors, and sets. It also mentions ClojureScript and domain-specific languages.
The document discusses solving the 8 queens problem using backtracking. It begins by explaining backtracking as an algorithm that builds partial candidates for solutions incrementally and abandons any partial candidate that cannot be completed to a valid solution. It then provides more details on the 8 queens problem itself - the goal is to place 8 queens on a chessboard so that no two queens attack each other. Backtracking is well-suited for solving this problem by attempting to place queens one by one and backtracking when an invalid placement is found.
This document discusses various methods for minimizing switching functions, including:
1. The Karnaugh map method, which represents truth tables graphically to find logically adjacent terms that can be combined.
2. Prime implicants and essential prime implicants, which are product terms that cover minterms. The essential ones must be included in the minimal expression.
3. Don't care conditions, which allow further simplification by treating unspecified minterms as don't cares.
4. The Quine-McCluskey tabulation method, which systematically generates prime implicants and finds the essential ones and minimal cover.
The document discusses binary search trees (BSTs). A BST is a binary tree where the key in each internal node is greater than all keys in its left subtree and less than those in its right subtree. The document describes properties of BSTs like traversal orders (inorder, preorder, postorder), operations like searching, insertion, deletion, finding minimum/maximum keys, successors/predecessors. It analyzes the time complexity of these operations as O(h) where h is the height of the tree. Finally, it shows that for a randomly built BST on n keys, the expected height is O(log n) using a probabilistic analysis and Jensen's inequality.
Program verification involves formally proving that a program satisfies certain properties, such as having no defects, by establishing that the program meets its specification. This is done by defining preconditions and postconditions and using weakest preconditions to reason about how the program transforms states. Testing involves running a program with sample inputs and checking that the outputs are as expected to find defects empirically.
The document provides an introduction to TensorFlow and neural networks. It discusses perceptron classifiers for MNIST data, convolutional neural networks for image recognition, recurrent neural networks for language modeling, and generating poems with RNNs. It also introduces Keras as an easier way to build neural networks and provides contact information for the author and upcoming machine learning conferences.
This document provides an overview of basic data structures concepts. It discusses bits and data types, different numeric representations like binary and decimal. It introduces common data types used in programming like integers and floats. It also covers abstract data types and provides examples like stacks, queues and lists. The document describes iterative and recursive algorithms for problems like factorial calculation, binary search and the Towers of Hanoi. It analyzes the time complexity of algorithms like selection sort.
Disk-based data structures are needed to store large datasets that exceed main memory capacity. Hard disks allow large storage but have slow random access. B-trees address this by storing data in pages that can be read sequentially, keeping the tree shallow. Keys in B-tree nodes separate the ranges of keys in subtrees, and nodes have between t-1 and 2t-1 keys. Operations on B-trees like search, insert, and delete involve reading and writing nodes to disk and splitting or merging nodes as needed to maintain the B-tree properties.
The document discusses digital logic design topics including:
1. Boolean algebra concepts such as binary operators, postulates, theorems, and switching functions.
2. Logic minimization techniques for reducing switching functions to canonical forms such as sum of products.
3. Combinational logic circuits including implementation of Boolean functions using gates.
4. Sequential logic circuits and algorithmic state machines.
The document contains code snippets demonstrating the use of TensorFlow for building and training neural networks. It shows how to:
1. Define operations like convolutions, max pooling, fully connected layers using common TensorFlow functions like tf.nn.conv2d and tf.nn.max_pool.
2. Create and initialize variables using tf.Variable and initialize them using tf.global_variables_initializer.
3. Construct a multi-layer perceptron model for MNIST classification with convolutional and fully connected layers.
4. Train the model using tf.train.AdamOptimizer by running optimization steps and calculating loss over batches of data.
5. Evaluate the trained model on test data to calculate accuracy.
Lesson 27: Integration by Substitution (Section 041 slides)Matthew Leingang
The document contains notes from a Calculus I class at New York University on December 13, 2010. It discusses using the substitution method for indefinite and definite integrals. Examples are provided to demonstrate how to use substitutions to evaluate integrals involving trigonometric, exponential, and polynomial functions. The key steps are to make a substitution for the variable in terms of a new variable, determine the differential of the substitution, and substitute into the integral to transform it into an integral involving only the new variable.
This document discusses cryptography and attacks on cryptosystems. It begins with an introduction to classical ciphers like the Caesar cipher and transposition cipher. It then covers the mathematics behind public key cryptography schemes like RSA, including number theory concepts like modulo arithmetic, discrete logarithms, and the Chinese Remainder Theorem. The document discusses the computational basis for RSA security in factoring large numbers and discusses different types of attacks on RSA like timing attacks, implementation attacks, and mathematical attacks using lattice basis reduction. It concludes with recommendations for building a secure RSA system, such as using large random paddings, large secret exponents, and generating strong prime numbers.
Gradient descent is an optimization algorithm used in supervised learning problems to minimize a loss function. It works by taking steps in the negative direction of the gradient of the loss function with respect to the weights at each iteration. Stochastic gradient descent is a variant that updates the weights for each training example, rather than for the full batch. The perceptron is an algorithm that updates the weights only when the prediction is incorrect. It converges when the weights correctly classify all training examples.
The document discusses several topics in natural language processing including distributional semantics, language models, word embeddings, and neural network models like word2vec. It introduces techniques for distributional semantics using distributional properties of words from large datasets. Language models are discussed including n-gram models and language class models that incorporate word classes. Word embedding techniques like word2vec are introduced for generating word vectors using neural networks.
Dynamic programming can be used to solve optimization problems involving overlapping subproblems, such as finding the most valuable subset of items that fit in a knapsack. The knapsack problem is solved by considering all possible subsets incrementally, storing the optimal values in a table. Warshall's and Floyd's algorithms also use dynamic programming to find the transitive closure and shortest paths in graphs by iteratively building up the solution from smaller subsets. Optimal binary search trees can also be constructed using dynamic programming by considering optimal substructures.
The document describes several algorithms that use dynamic programming techniques. It discusses the coin changing problem, computing binomial coefficients, Floyd's algorithm for finding all-pairs shortest paths, optimal binary search trees, the knapsack problem, and multistage graphs. For each problem, it provides the key recurrence relation used to build the dynamic programming solution in a bottom-up manner, often using a table to store intermediate results. It also analyzes the time and space complexity of the different approaches.
An introduction to quantum stochastic calculusSpringer
The document discusses tensor products of Hilbert spaces. It defines positive definite kernels on sets and shows how they can be used to define tensor products. Given Hilbert spaces H1, ..., Hn, it constructs a kernel on the cartesian product of the spaces and shows that its Gelfand pair (H,φ) gives a tensor product of the Hilbert spaces. The map φ from the product space into H is multilinear and H is the completion of the algebraic tensor product of the vector spaces H1, ..., Hn.
The document discusses ordered dictionaries and binary search trees (BSTs). It describes how to implement ordered dictionaries using BSTs to support operations like minimum, maximum, predecessor, and successor in logarithmic time. The key aspects are:
1) BSTs allow searching, minimum, maximum, predecessor, and successor operations in O(h) time where h is the height of the tree.
2) Insertion and deletion in a BST takes O(h) time in the worst case, as repeatedly inserting keys in ascending or descending order yields a skewed tree of height n for n nodes.
3) Search works by comparing the search key to the root node's key, and recursively searching the left or right
The document discusses tangent lines to functions. It provides examples of finding the equation of a tangent line with a given slope to specific functions. It also discusses finding the average and instantaneous velocity of an object given its position function.
This document provides an introduction to symbolic math in MATLAB. It discusses differentiation and integration of functions using symbolic operators. Differentiation is defined as finding the rate of change of a function with respect to a variable. Integration finds the original function given its derivative. The document provides examples of differentiating and integrating simple functions in MATLAB's symbolic toolbox and exercises for the reader to practice.
Stefan Kanev: Clojure, ClojureScript and Why They're Awesome at I T.A.K.E. Un...Mozaic Works
This document summarizes a presentation about Clojure and ClojureScript. It describes Clojure as a modern Lisp dialect that runs on the JVM with a focus on concurrency. It notes advantages like stable platforms, access to Java libraries, and immutable persistent data structures that facilitate parallelism. The document discusses Clojure concepts like homoiconicity, concurrency vs parallelism, syntax using s-expressions, macros for metaprogramming, and data structures like maps, vectors, and sets. It also mentions ClojureScript and domain-specific languages.
The document discusses solving the 8 queens problem using backtracking. It begins by explaining backtracking as an algorithm that builds partial candidates for solutions incrementally and abandons any partial candidate that cannot be completed to a valid solution. It then provides more details on the 8 queens problem itself - the goal is to place 8 queens on a chessboard so that no two queens attack each other. Backtracking is well-suited for solving this problem by attempting to place queens one by one and backtracking when an invalid placement is found.
This document discusses various methods for minimizing switching functions, including:
1. The Karnaugh map method, which represents truth tables graphically to find logically adjacent terms that can be combined.
2. Prime implicants and essential prime implicants, which are product terms that cover minterms. The essential ones must be included in the minimal expression.
3. Don't care conditions, which allow further simplification by treating unspecified minterms as don't cares.
4. The Quine-McCluskey tabulation method, which systematically generates prime implicants and finds the essential ones and minimal cover.
The document discusses binary search trees (BSTs). A BST is a binary tree where the key in each internal node is greater than all keys in its left subtree and less than those in its right subtree. The document describes properties of BSTs like traversal orders (inorder, preorder, postorder), operations like searching, insertion, deletion, finding minimum/maximum keys, successors/predecessors. It analyzes the time complexity of these operations as O(h) where h is the height of the tree. Finally, it shows that for a randomly built BST on n keys, the expected height is O(log n) using a probabilistic analysis and Jensen's inequality.
Program verification involves formally proving that a program satisfies certain properties, such as having no defects, by establishing that the program meets its specification. This is done by defining preconditions and postconditions and using weakest preconditions to reason about how the program transforms states. Testing involves running a program with sample inputs and checking that the outputs are as expected to find defects empirically.
The document provides an introduction to TensorFlow and neural networks. It discusses perceptron classifiers for MNIST data, convolutional neural networks for image recognition, recurrent neural networks for language modeling, and generating poems with RNNs. It also introduces Keras as an easier way to build neural networks and provides contact information for the author and upcoming machine learning conferences.
This document provides an overview of basic data structures concepts. It discusses bits and data types, different numeric representations like binary and decimal. It introduces common data types used in programming like integers and floats. It also covers abstract data types and provides examples like stacks, queues and lists. The document describes iterative and recursive algorithms for problems like factorial calculation, binary search and the Towers of Hanoi. It analyzes the time complexity of algorithms like selection sort.
Disk-based data structures are needed to store large datasets that exceed main memory capacity. Hard disks allow large storage but have slow random access. B-trees address this by storing data in pages that can be read sequentially, keeping the tree shallow. Keys in B-tree nodes separate the ranges of keys in subtrees, and nodes have between t-1 and 2t-1 keys. Operations on B-trees like search, insert, and delete involve reading and writing nodes to disk and splitting or merging nodes as needed to maintain the B-tree properties.
The document discusses digital logic design topics including:
1. Boolean algebra concepts such as binary operators, postulates, theorems, and switching functions.
2. Logic minimization techniques for reducing switching functions to canonical forms such as sum of products.
3. Combinational logic circuits including implementation of Boolean functions using gates.
4. Sequential logic circuits and algorithmic state machines.
The document contains code snippets demonstrating the use of TensorFlow for building and training neural networks. It shows how to:
1. Define operations like convolutions, max pooling, fully connected layers using common TensorFlow functions like tf.nn.conv2d and tf.nn.max_pool.
2. Create and initialize variables using tf.Variable and initialize them using tf.global_variables_initializer.
3. Construct a multi-layer perceptron model for MNIST classification with convolutional and fully connected layers.
4. Train the model using tf.train.AdamOptimizer by running optimization steps and calculating loss over batches of data.
5. Evaluate the trained model on test data to calculate accuracy.
Lesson 27: Integration by Substitution (Section 041 slides)Matthew Leingang
The document contains notes from a Calculus I class at New York University on December 13, 2010. It discusses using the substitution method for indefinite and definite integrals. Examples are provided to demonstrate how to use substitutions to evaluate integrals involving trigonometric, exponential, and polynomial functions. The key steps are to make a substitution for the variable in terms of a new variable, determine the differential of the substitution, and substitute into the integral to transform it into an integral involving only the new variable.
This document discusses cryptography and attacks on cryptosystems. It begins with an introduction to classical ciphers like the Caesar cipher and transposition cipher. It then covers the mathematics behind public key cryptography schemes like RSA, including number theory concepts like modulo arithmetic, discrete logarithms, and the Chinese Remainder Theorem. The document discusses the computational basis for RSA security in factoring large numbers and discusses different types of attacks on RSA like timing attacks, implementation attacks, and mathematical attacks using lattice basis reduction. It concludes with recommendations for building a secure RSA system, such as using large random paddings, large secret exponents, and generating strong prime numbers.
Gradient descent is an optimization algorithm used in supervised learning problems to minimize a loss function. It works by taking steps in the negative direction of the gradient of the loss function with respect to the weights at each iteration. Stochastic gradient descent is a variant that updates the weights for each training example, rather than for the full batch. The perceptron is an algorithm that updates the weights only when the prediction is incorrect. It converges when the weights correctly classify all training examples.
The document discusses several topics in natural language processing including distributional semantics, language models, word embeddings, and neural network models like word2vec. It introduces techniques for distributional semantics using distributional properties of words from large datasets. Language models are discussed including n-gram models and language class models that incorporate word classes. Word embedding techniques like word2vec are introduced for generating word vectors using neural networks.
Dynamic programming can be used to solve optimization problems involving overlapping subproblems, such as finding the most valuable subset of items that fit in a knapsack. The knapsack problem is solved by considering all possible subsets incrementally, storing the optimal values in a table. Warshall's and Floyd's algorithms also use dynamic programming to find the transitive closure and shortest paths in graphs by iteratively building up the solution from smaller subsets. Optimal binary search trees can also be constructed using dynamic programming by considering optimal substructures.
The document describes several algorithms that use dynamic programming techniques. It discusses the coin changing problem, computing binomial coefficients, Floyd's algorithm for finding all-pairs shortest paths, optimal binary search trees, the knapsack problem, and multistage graphs. For each problem, it provides the key recurrence relation used to build the dynamic programming solution in a bottom-up manner, often using a table to store intermediate results. It also analyzes the time and space complexity of the different approaches.
This document discusses dynamic programming and greedy algorithms. It begins by defining dynamic programming as a technique for solving problems with overlapping subproblems. It provides examples of dynamic programming approaches to computing Fibonacci numbers, binomial coefficients, the knapsack problem, and other problems. It also discusses greedy algorithms and provides examples of their application to problems like the change-making problem, minimum spanning trees, and single-source shortest paths.
This document discusses dynamic programming and greedy algorithms. It begins by defining dynamic programming as a technique for solving problems with overlapping subproblems. Examples provided include computing the Fibonacci numbers and binomial coefficients. Greedy algorithms are introduced as constructing solutions piece by piece through locally optimal choices. Applications discussed are the change-making problem, minimum spanning trees using Prim's and Kruskal's algorithms, and single-source shortest paths. Floyd's algorithm for all pairs shortest paths and optimal binary search trees are also summarized.
The document discusses recursive algorithms and divide-and-conquer approaches. It provides examples of recursively defining problems like calculating factorials and solving the Tower of Hanoi puzzle. It also discusses analyzing the time complexity of recursive algorithms by modeling the running time as a recursive function and solving the recursion. The document suggests three methods for solving recurrences: substitution, recursion trees, and the master method.
10.2 using combinations and the binomial theoremhartcher
The document discusses combinations and the binomial theorem. It provides formulas for combinations and explains how to use Pascal's triangle to determine the coefficients in binomial expansions. Examples are worked out expanding (2x + 1)^4, (x - 2y)^3, and finding the coefficient of x^4 in (2x - 7)^9. The key points are that combinations involve choosing objects without regard to order, while permutations consider order; Pascal's triangle determines the coefficients in binomial expansions; and the powers of the terms follow a pattern of the first term decreasing and the second increasing.
This document discusses two algorithms: divide-and-conquer and dynamic programming. Divide-and-conquer breaks problems into independent subproblems, solves the subproblems, and combines their solutions. Dynamic programming solves subproblems once and saves their solutions in a table to solve the original problem more efficiently. Examples include computing the Fibonacci sequence and matrix chain multiplication.
These questions are prepared by Classical Programming Experts and are asked during job interviews.The Solution to the given programs are prepared by Programming Experts and are often asked in job interviews. Knowing solution to these problems will help you clear your concepts.
Recursion is a technique where a method calls itself directly or indirectly. It is useful for solving problems that involve repeating patterns or combinatorial algorithms. The document provides examples of calculating factorials, generating all binary vectors, and finding all paths in a labyrinth recursively. It discusses how to avoid harmful recursion that uses excessive memory and discusses when recursion is preferable to iteration, such as for problems that require exploring multiple continuations at each step. Exercises are provided to help practice implementing various recursive algorithms.
Cuckoo Search Algorithm: An IntroductionXin-She Yang
This presentation explains the fundamental ideas of the standard Cuckoo Search (CS) algorithm, which also contains the links to the free Matlab codes at Mathswork file exchanges and the animations of numerical simulations (video at Youtube). An example of multi-objective cuckoo search (MOCS) is also given with link to the Matlab code.
The document contains 16 multiple choice questions about algorithms, data structures, and graph theory. Each question has 4 possible answers and the correct answer is provided. The maximum number of comparisons needed to merge sorted sequences is 358, and depth first search on a graph represented with an adjacency matrix has a worst case time complexity of O(n^2).
This document contains 52 math and statistics problems assigned to students over three assignments. The problems cover topics like numerical methods, probability, interpolation, and integration. Students are asked to define terms, perform iterative calculations, derive formulas, prove statements, and evaluate integrals using techniques like the trapezoidal rule, Simpson's rule, and Newton's divided difference interpolation.
The document discusses various backtracking techniques including bounding functions, promising functions, and pruning to avoid exploring unnecessary paths. It provides examples of problems that can be solved using backtracking including n-queens, graph coloring, Hamiltonian circuits, sum-of-subsets, 0-1 knapsack. Search techniques for backtracking problems include depth-first search (DFS), breadth-first search (BFS), and best-first search combined with branch-and-bound pruning.
This document provides examples and explanations of finding nth derivatives of various functions. It begins with examples of the nth derivatives of ln(x), ex, sin(x), and xsin(x). Common patterns are identified, such as the nth derivative of ln(x) being (-1)n-1(n-1)!/xn. Exercises are then provided to find formulas for the nth derivatives of additional functions like cos(x), xcos(x), xln(x), e5x, xn, and polynomials.
Recurrent Problem in Discrete Mathmatics.pptxgbikorno
The document discusses three recurrent problems: the Tower of Hanoi problem, the lines in the plane problem, and the Josephus problem. It explains that recurrent problems can be solved using recursion, where the solution to a problem relies on previously solved smaller instances of the same problem. It provides details on the recursive solutions and recurrence relations for calculating the number of moves in the Tower of Hanoi problem and the maximum number of regions created by lines in a plane. It also outlines the general steps to solve a recurrent problem by finding a closed-form expression through a recurrence relation and mathematical induction.
Unit i trigonometry satyabama univesitySelvaraj John
The document provides an overview of topics in trigonometry and matrices. It covers expansions of trigonometric functions like sin nθ and cos nθ in terms of multiples of θ. It also covers separation of complex trigonometric functions into real and imaginary parts. For matrices, it discusses eigen values, eigen vectors, characteristic equations, diagonalization using similarity transformations, and reduction of quadratic forms to canonical forms using orthogonal transformations. It provides examples of finding ranks, sums and products of eigen values, and inverse and powers of matrices using Cayley-Hamilton theorem.
This document summarizes different methods for solving the celebrity problem of identifying a celebrity (if one exists) among a group of people by asking questions of the form "does A know B?". It begins by outlining a naive method that asks all n(n-1) possible questions. It then presents a better solution that iteratively eliminates people, requiring only 3n-3 questions. Finally, it describes the best solution that groups people into pairs, requiring only n-1 questions by treating the problem as a binary tree. The document explains this best solution in detail and notes that no algorithm can solve the problem using fewer than n-1 questions.
The document discusses the theory of NP-completeness. It begins by defining the complexity classes P, NP, NP-hard, and NP-complete. It then explains the concepts of reduction and how none of the NP-complete problems can be solved in polynomial time deterministically. The document provides examples of NP-complete problems like satisfiability (SAT), vertex cover, and the traveling salesman problem. It shows how nondeterministic algorithms can solve these problems and how they can be transformed into SAT instances. Finally, it proves that SAT is the first NP-complete problem by showing it is in NP and NP-hard.
DSA (Data Structure and Algorithm) QuestionsRESHAN FARAZ
The document contains questions related to data structures and algorithms. Some questions ask to perform operations on data structures like stacks, queues, trees, graphs and hash tables. Other questions involve algorithms like topological sorting, shortest path algorithms, minimum spanning tree algorithms, and binary search tree operations. The document tests understanding of concepts taught in an algorithms course through examples and problem solving questions.
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
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.
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.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
Physiology and chemistry of skin and pigmentation, hairs, scalp, lips and nail, Cleansing cream, Lotions, Face powders, Face packs, Lipsticks, Bath products, soaps and baby product,
Preparation and standardization of the following : Tonic, Bleaches, Dentifrices and Mouth washes & Tooth Pastes, Cosmetics for Nails.
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.
2. 1. Motivation
2
Suppose you want to find all the files under a directory that
contains a particular word. How do you solve this problem?
There are several ways to solve this problem.
An intuitive solution is to
use recursion by searching
the files in the subdirectories
recursively.
3. 2. Motivation
3
Try to solve the following
word game.
Reference:
https://www.pennydellpuzzles.com/Upload/PuzzleFiles/Sample/FillIns/fi001
_d0508_freepuz.pdf
4. 3. Motivation – Solving Complex Problems
4
The Eight Queens puzzle consists on placing eight queens on a chessboard
such that no two queens are on the same row, same column, or same
diagonal, as shown in figure below.
Your turn: Try placing n Queens on an n*n board
13. Fibonnaci Numbers.
13
Fibonacci Numbers
1 (Jan) C1
1 (Feb) C1
2 (Mar) C1 C1,1
3 (Apr) C1 C1,2 C1,1
5 (May) C1 C1,3 C1,2 C1,1 C1,1,1
8 (Jun) C1 C1,4 C1,3 C1,2 C1,2,1 C1,1 C1,1,2 C1,1,1
13 (Jul)
21 (Aug)
34 (Sep)
55 (Oct)
89 (Nov)
144 (Dec)
C x,y
Represents a couple of male & female rabbits.
C1,2 is the second offspring pair descending
from couple 1.
Fibonacci sequence first appears in the book Liber Abaci (1202) by Leonardo of Pisa, known as Fibonacci.
Fibonacci considers the growth of an idealized rabbit population, assuming that: a newly born pair of rabbits, one male, one
female, are put in a field; rabbits are able to mate at the age of one month so that at the end of its second month a female can
produce another pair of rabbits; rabbits never die and a mating pair always produces one new pair (one male, one female) every
month from the second month on. The puzzle that Fibonacci posed was: how many pairs will there be in one year?
Taken from: http://en.wikipedia.org/wiki/Fibonacci_number
15. Characteristics of Recursion
15
All recursive methods have the following characteristics:
One or more base cases (the simplest case) are used to stop recursion
(boundary conditions).
Every recursive call reduces the original problem, bringing it
increasingly closer to a base case until it becomes that case.
In general, to solve a problem using recursion,
1. you break it into subproblems.
2. If a subproblem resembles the original problem, you can apply the same approach to
solve the subproblem recursively.
3. This subproblem is almost the same as the original problem in nature with a smaller
size.
16. Problem Solving Using Recursion
16
Let us consider a simple problem of printing a message for n times.
You can break the problem into two subproblems:
1. One is to print the message one time and the other is to print the message for
n-1 times.
2. The second problem is the same as the original problem with a smaller size.
3. The base case for the problem is n==0.
private static void printMsg(String msg, int times) {
// printing a message a number of times
if (times==0)
return;
System.out.println(times + " " + msg);
printMsg(msg, times-1);
}
18. Version2: Recursive Helper Methods
18
Examples: 100a001, MADAMIMADAM, abccba are palindromes.
public static boolean isPalindrome(String s) {
return isPalindrome(s, 0, s.length() - 1);
}
public static boolean isPalindrome(String s, int low, int high)
{
if (high <= low) // Base case
return true;
else if (s.charAt(low) != s.charAt(high)) // Base case
return false;
else
return isPalindrome(s, low + 1, high - 1);
}
19. Directory Size
19
This section presents a problem that is difficult to solve without using
recursion.
The problem is to find the size of a directory.
The size of a directory is the sum of the sizes of all files in the directory.
A directory may contain subdirectories.
directory
...1f
1
2f
1
mf
1
1d
1
2d
1
nd
1
...
)(...)()()(...)()()( 2121 nm dsizedsizedsizefsizefsizefsizedsize
20. Directory Size
20
public class DirectorySize {
public static void main(String[] args) {
System.out.print("Enter a file or a directory: ");
java.util.Scanner input = new java.util.Scanner(System.in);
String s = input.nextLine();
try {
System.out.println( directorySize( new File(s) ) );
}
catch (IOException ex) {
System.out.println(ex.toString());
}
}
public static long directorySize(File file) throws java.io.FileNotFoundException{
if (!file.exists())
throw new java.io.FileNotFoundException(file + " not found");
if (file.isFile()) {
return file.length();
}
else {
File[] files = file.listFiles();
long size = 0;
for (int i = 0; i < files.length; i++)
size += directorySize(files[i]);
return size;
}
}
}
21. Towers of Hanoi
21
There are n disks labeled 1, 2, 3, . . ., n, and three towers labeled
A, B, and C.
No disk can be on top of a smaller disk at any time.
All the disks are initially placed on tower A.
Only one disk can be moved at a time, and it must be the top disk
on the tower.
Invented by french
mathematician
Edouard Lucas (1842)
Image fromWikipedia 7/8/2011
22. Towers of Hanoi, cont.
22
A B
Original position
C
A B
Step 4: Move disk 3 from A to B
C
A B
Step 5: Move disk 1 from C to A
C
A B
Step 1: Move disk 1 from A to B
C
A CB
Step 2: Move disk 2 from A to C
A B
Step 3: Move disk 1 from B to C
C
A B
Step 7: Mve disk 1 from A to B
C
A B
Step 6: Move disk 2 from C to B
C
23. Solution to Towers of Hanoi
23
TheTowers of Hanoi problem can be decomposed into three subproblems.
A B
Original position
C
.
.
.
A B
Step 1: Move the first n-1 disks from A to C recursively
C
.
.
.
A B
Step2: Move disk n from A to C
C
.
.
.
A B
Step3: Move n-1 disks from C to B recursively
C
.
.
.
n-1 disks
n-1 disks
n-1 disks
n-1 disks
Step 2. Move disk n from A to B
24. Solution to Towers of Hanoi
24
1. Move the first n - 1 disks from A to C with the
assistance of tower B.
2. Move disk n from A to B.
3. Move n - 1 disks from C to B with the
assistance of tower A.
25. Solution to Towers of Hanoi
25
public static void main(String[] args) {
int n = 4;
// moving disks from A to B using C as auxiliary tower
System.out.println("Towers of Hanoi. The moves are:");
moveDisks(n, 'A', 'B', 'C');
}
private static void moveDisks( int n, char fromTower, char toTower, char auxTower) {
if (n == 1)
System.out.println("Move disk " + n + " from "
+ fromTower + " to " + toTower);
else {
moveDisks(n-1, fromTower, auxTower, toTower);
System.out.println("Move disk " + n + " from "
+ fromTower + " to " + toTower);
moveDisks(n-1, auxTower, toTower, fromTower);
}
26. Greatest Common Divisor - GCD
26
gcd(2, 3) = 1
gcd(2, 10) = 2
gcd(25, 35) = 5
gcd(14, 28) = 14
Approach 1: Brute-force, start from min(n,m) down to 1, to check if a
number is common divisor for both m and n, if so, it is the greatest
common divisor.
Approach 2: Euclid’s algorithm
Approach 3: Recursive method
GCD
is the largest positive integer that divides
The numbers without a remainder
27. Approach 2: Euclid’s algorithm
27
// Get absolute value of m and n;
t1 = Math.abs(m);
t2 = Math.abs(n);
// r is the remainder of t1 divided by t2;
r = t1 % t2;
while (r != 0) {
t1 = t2;
t2 = r;
r = t1 % t2;
}
// When r is 0, t2 is the greatest common
// divisor between t1 and t2
return t2;
28. Approach 3: Recursive Method
28
gcd(m, n) = n if ( m % n = 0 );
gcd(m, n) = gcd(n, m % n); otherwise;
private static int gcd(int m, int n) {
if (m % n == 0)
return n;
else
return gcd(n, m % n);
}
29. 29
Knapsack Problem
Load a knapsack with any number of items without exceeding its capacity.
//knapsack problem. Simple case - no optimization
ArrayList<Integer> items = new ArrayList<Integer> (
Arrays.asList(8, 2, 9, 4, 1, 5, 3, 4, 2, 6, 7));
ArrayList<Integer> solution = new ArrayList<Integer>();
knapsack(16, solution, items);
for (int n : solution){
System.out.println("Knapsack: " + n );
}
Picture taken from: http://en.wikipedia.org/wiki/Knapsack_problem
30. 30
Knapsack Problem
Load a knapsack with any number of items without exceeding its capacity.
private static void knapsack(int maxWeight,
ArrayList<Integer> solution,
ArrayList<Integer> items) {
if ((maxWeight >= 0 ) && (items.size()>0) ) {
int item = items.remove(0);
solution.add(0, item);
knapsack(maxWeight-item, solution, items);
}
else {
int item = solution.remove(0);
if (items.size()==0)
return;
else{
knapsack(maxWeight+item, solution, items);
}
}
}
Picture taken from: http://en.wikipedia.org/wiki/Knapsack_problem
31. Fractals?
31
A fractal is a geometrical figure
just like triangles, circles, and
rectangles, but fractals can be
divided into parts, each of which is
a reduced-size copy of the whole.
32. Sierpiński Triangle
32
1. It begins with an equilateral triangle, which is considered to be the Sierpinski fractal of
order (or level) 0, as shown in Figure (a).
2. Connect the midpoints of the sides of the triangle of order 0 to create a Sierpinski
triangle of order 1, as shown in Figure (b).
3. Leave the center triangle intact. Connect the midpoints of the sides of the three other
triangles to create a Sierpinski of order 2, as shown in Figure (c).
4. You can repeat the same process recursively to create a Sierpinski triangle of order 3,
4, ..., and so on, as shown in Figure (d).