5.1.1 Proof by Mathematical Induction.pptx
•Download as PPTX, PDF•
0 likes•6 views
Mathematical induction is a method of proof that involves first proving a statement is true for the base case (usually n=1), and then proving that if the statement holds true for an integer k, it also holds true for k+1. This implies the statement is true for all positive integers n. The well-ordering principle states that any non-empty subset of positive integers that is bounded below, has a least element. Mathematical induction can be used to prove statements involving the positive integers, but not necessarily for rational or real numbers. An example proof using induction demonstrates proving the sum of the first n odd integers equals n^2.
Report
Share
Report
Share
Recommended
MATHEMATICAL INDUCTION.ppt
This document provides an introduction to number theory and mathematical induction. It discusses the following key points:
- Number theory studies properties and relationships between numbers like integers and primes. The primes are important building blocks.
- Mathematical induction is a method of proof that allows proving statements about infinite sets of numbers. It involves proving a base case and an induction step to show if the statement is true for one number, it is true for the next.
- As an example, induction is used to prove that the sum of the first n odd numbers equals n^2 for all positive integers n. This involves proving the base case of n=1 and assuming the statement is true for an arbitrary n before showing it holds for
DIGITAL TEXT BOOK
This document is the preface to a textbook on number theory. It discusses the goals of the textbook, which are to encourage independent thinking and problem solving rather than rote memorization. Number theory is well-suited for this purpose as patterns in the natural numbers can be discerned through observation and experimentation, but proving theorems requires rigorous demonstration. The textbook was originally written for a course at Brown University designed to attract non-science majors to mathematics. The prerequisites are few, requiring only high school algebra and a willingness to experiment, make mistakes, learn from them, and persevere.
Math induction
The document describes the principle of mathematical induction and how it is used to prove statements about natural numbers. It contains the following key points:
1. Mathematical induction has two steps: the basis step proves the statement is true for the first natural number, and the inductive step assumes the statement is true for some natural number n, and uses this to prove it is true for n+1.
2. Examples are provided to demonstrate how to apply mathematical induction to prove formulas and inequalities for all natural numbers. The examples show setting up the basis and inductive steps, and using the induction hypothesis within the proof.
3. Inequalities can sometimes be proved using induction by adding or subtracting quantities to both
Mathematical induction by Animesh Sarkar
The document discusses various mathematical induction concepts including weak induction, strong induction, and examples of proofs using induction. It begins with an overview of weak induction and its two steps: proving a statement is true for the base case, and proving if it is true for some step k then it is true for k+1. Strong induction is then introduced which proves a statement is true for all steps up to k implies it is true for k+1. Examples are provided of proofs using induction, such as proving 3n-1 is divisible by 2 and that sums of odd numbers equal squares. Reference sources are listed at the end.
Per4 induction
The document discusses mathematical induction and recursive definitions. It provides examples of using induction to prove statements for all natural numbers, like n < 2n. It also gives examples of recursively defined sequences, functions, and sets, such as the Fibonacci numbers defined by f(n) = f(n-1) + f(n-2). Recursive definitions define an object in terms of itself, similar to induction which proves statements by showing that if true for n, then true for n+1.
Number theory
Number theory, is the course designed for under graduate student in Ethiopia, for mathematics department.
PROOF TECHNIQUES
Proof Techniques
There are some of the most common proof techniques.
1. Direct Proof
2. Proof by Contradiction
3. Proof by Contapositive
4. Proof by Cases
Mathematical Induction DM
This document discusses mathematical induction, which is a method for proving that a proposition is true for all natural numbers. It involves showing that the proposition is true for the base case, usually 0 or 1, and assuming the proposition is true for some value n to prove it is also true for n+1. Two examples are provided: proving n < 2n for all positive integers n, and proving the formula for summing the first n positive integers. The document also introduces the second principle of mathematical induction, which involves showing the proposition is true for values up to n before proving it for n+1.
Recommended
MATHEMATICAL INDUCTION.ppt
This document provides an introduction to number theory and mathematical induction. It discusses the following key points:
- Number theory studies properties and relationships between numbers like integers and primes. The primes are important building blocks.
- Mathematical induction is a method of proof that allows proving statements about infinite sets of numbers. It involves proving a base case and an induction step to show if the statement is true for one number, it is true for the next.
- As an example, induction is used to prove that the sum of the first n odd numbers equals n^2 for all positive integers n. This involves proving the base case of n=1 and assuming the statement is true for an arbitrary n before showing it holds for
DIGITAL TEXT BOOK
This document is the preface to a textbook on number theory. It discusses the goals of the textbook, which are to encourage independent thinking and problem solving rather than rote memorization. Number theory is well-suited for this purpose as patterns in the natural numbers can be discerned through observation and experimentation, but proving theorems requires rigorous demonstration. The textbook was originally written for a course at Brown University designed to attract non-science majors to mathematics. The prerequisites are few, requiring only high school algebra and a willingness to experiment, make mistakes, learn from them, and persevere.
Math induction
The document describes the principle of mathematical induction and how it is used to prove statements about natural numbers. It contains the following key points:
1. Mathematical induction has two steps: the basis step proves the statement is true for the first natural number, and the inductive step assumes the statement is true for some natural number n, and uses this to prove it is true for n+1.
2. Examples are provided to demonstrate how to apply mathematical induction to prove formulas and inequalities for all natural numbers. The examples show setting up the basis and inductive steps, and using the induction hypothesis within the proof.
3. Inequalities can sometimes be proved using induction by adding or subtracting quantities to both
Mathematical induction by Animesh Sarkar
The document discusses various mathematical induction concepts including weak induction, strong induction, and examples of proofs using induction. It begins with an overview of weak induction and its two steps: proving a statement is true for the base case, and proving if it is true for some step k then it is true for k+1. Strong induction is then introduced which proves a statement is true for all steps up to k implies it is true for k+1. Examples are provided of proofs using induction, such as proving 3n-1 is divisible by 2 and that sums of odd numbers equal squares. Reference sources are listed at the end.
Per4 induction
The document discusses mathematical induction and recursive definitions. It provides examples of using induction to prove statements for all natural numbers, like n < 2n. It also gives examples of recursively defined sequences, functions, and sets, such as the Fibonacci numbers defined by f(n) = f(n-1) + f(n-2). Recursive definitions define an object in terms of itself, similar to induction which proves statements by showing that if true for n, then true for n+1.
Number theory
Number theory, is the course designed for under graduate student in Ethiopia, for mathematics department.
PROOF TECHNIQUES
Proof Techniques
There are some of the most common proof techniques.
1. Direct Proof
2. Proof by Contradiction
3. Proof by Contapositive
4. Proof by Cases
Mathematical Induction DM
This document discusses mathematical induction, which is a method for proving that a proposition is true for all natural numbers. It involves showing that the proposition is true for the base case, usually 0 or 1, and assuming the proposition is true for some value n to prove it is also true for n+1. Two examples are provided: proving n < 2n for all positive integers n, and proving the formula for summing the first n positive integers. The document also introduces the second principle of mathematical induction, which involves showing the proposition is true for values up to n before proving it for n+1.
Number Theory.pptx
This document discusses several number theory concepts:
- Euler's totient function φ(n) counts the numbers less than or equal to n that are relatively prime to n.
- Euler's theorem states that if n and a are coprime, then aφ(n) ≡ 1 (mod n). This can be used to reduce large powers modulo n.
- Fermat's little theorem is a special case of Euler's theorem where n is a prime number.
- Wilson's theorem relates a number n and its totient φ(n).
Examples are provided to demonstrate calculating the totient function and using Euler's theorem to find remainders.
Induction
The document discusses different mathematical proofs using induction. It provides examples of proving statements about odd numbers, divisibility by primes, equalities, properties, and inequalities using induction. The key steps are to first prove the base case, then assume the statement is true for some value n and use that to prove it is true for n+1.
CC107 - Chapter 1 The real Number System.pdf
This document provides an overview of the real number system. It defines sets, subsets, integers, rational numbers, and irrational numbers. It establishes the field axioms for real numbers, including the commutative, associative, distributive, identity, negatives, and reciprocals properties. It also describes the order axioms and properties of inequalities. Key theorems are presented on the uniqueness of factorizations of integers into prime numbers and the existence of greatest common divisors. Intervals and geometric representations of real numbers on the real number line are also introduced.
Analysis.pptx
- The document discusses some fundamental concepts in analysis such as natural numbers, zero, infinity, series, and L'Hôpital's rule.
- It provides examples to illustrate issues that can arise from improperly handling concepts like division by zero, divergent series, and interchange of sums.
- The key points are that operations should not be performed on divergent series and sums can only be interchanged if the series is absolutely convergent. L'Hôpital's rule also has specific requirements that must be checked before applying it.
The axiomatic power of Kolmogorov complexity
1. The document discusses random axioms and probabilistic proofs in Peano arithmetic. It describes a proof strategy where one could randomly select an integer n that satisfies some formula φ and add it as a new axiom.
2. While this intuition of probabilistic proofs makes sense, it is not really useful since any statement provable with sufficiently high probability is already provable in PA. However, probabilistic proofs can be exponentially more concise than deterministic proofs.
3. The document also discusses Kolmogorov complexity and how statements about it relate to the provability of PA. It can be shown that if C(x) is less than some value, PA will prove it, but PA will never prove a
CMSC 56 | Lecture 11: Mathematical Induction
Mathematical Induction
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 18, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
mathematicalinductionanddivisibilityrules-160711105713.pdf
Mathematical induction and divisibility rules are methods for proving statements about numbers.
Mathematical induction has two steps: 1) proving the statement is true for the base case, usually n=1. 2) Assuming the statement is true for n=k, proving it is true for n=k+1. Divisibility rules transform numbers into smaller ones while preserving divisibility by certain divisors. Rules exist to test for divisibility by 2, 3, 4, 5, 6, 7, 9, 10 and 13.
Mathematical induction and divisibility rules
A complete and enhanced presentation on mathematical induction and divisibility rules with out any calculation.
Here are some defined formulas and techniques to find the divisibility of numbers.
4AMT122-PART 1-SLIDES.pptx
This document provides guidance on writing mathematical proofs. It defines key terms like direct proof and counterexample. It also outlines the standard format for proofs, including stating the theorem, clearly marking the beginning and end, and providing justification for each step. Common mistakes to avoid are arguing from examples rather than a general case, using the same variable to mean different things, and jumping to a conclusion without proper justification.
1 chapter1 introduction
The document provides an introduction to recursion and mathematical concepts relevant to analyzing algorithms. It begins by motivating selection and word puzzle problems to illustrate the importance of how algorithms scale to large inputs. It then reviews necessary mathematical topics like exponents, logarithms, and series. Finally, it defines recursion, provides examples of both correct and incorrect recursive functions, and discusses techniques for proving properties of recursive algorithms like proof by induction.
Crt
This document discusses applications of the Chinese Remainder Theorem in mathematical olympiads. It begins by providing three different perspectives on the Chinese Remainder Theorem: construction, lifting, and destruction. It then analyzes three example problems to illustrate how each perspective can be applied. The first uses the construction perspective to build integers satisfying certain properties. The second uses the lifting perspective to derive a size result by showing integers are divisible by large primes. The third uses destruction to reduce a statement to prime powers.
Lecture1
The real number system contains rational numbers like integers and fractions, as well as irrational numbers like the square root of 2. It is a complete system where every non-empty set that is bounded above has a least upper bound. This property of completeness is what allows real numbers to correspond precisely to points on a number line. The Archimedean property, which follows from completeness, states that between any real number and zero there is a positive integer multiple. This implies that rational numbers are dense within the real numbers.
Math induction principle (slides)
The document discusses propositions, logical operations, predicates, quantification, and mathematical induction. It provides:
1) Definitions of predicates, propositions, universal and existential quantification, and the principle of mathematical induction.
2) Examples of applying predicates, quantification, and induction to prove statements about integers and sums.
3) The process of proving statements by mathematical induction, which involves showing the basis step and inductive step. It also introduces strong mathematical induction.
Properties of real numbers
The document discusses several properties of real numbers including:
1) The closure properties of addition and multiplication - the sum and product of any two real numbers is a real number.
2) Commutative, associative, and distributive properties of addition and multiplication.
3) Identity properties of addition and multiplication with 0 and 1 being the identity elements.
4) Inverse properties of addition and multiplication where adding/multiplying a number and its inverse results in the identity element.
Arithmetic sequence in elementary and HS
This document defines and provides examples of multiplicative functions in number theory. It states that a multiplicative function f(mn) is equal to f(m)f(n) when m and n are relatively prime. Examples given are the Euler totient function φ and the function n2. It also proves that the sum of multiplicative functions and the sum of divisors of a multiplicative function are also multiplicative. Other concepts defined include the Möbius function, Euler phi function, Carmichael conjecture, perfect numbers, and the sum-of-divisors and number-of-divisors functions.
1
This document discusses mathematical induction. It provides two key steps of mathematical induction: 1) the statement P(1) is true, and 2) if P(k) is true, then P(k+1) is also true. It then gives an example of using mathematical induction to prove that the formula 1 + 2 + 3 + ... + n = n(n+1)/2 is true for all positive integers n. Specifically, it shows that P(1) is true, and that if P(k) is true then P(k+1) is also true, thereby proving the formula using mathematical induction.
6300 solutionsmanual free
The document is a solutions manual providing answers to even-numbered exercises from the textbook "A First Look at Rigorous Probability Theory" by Jeffrey S. Rosenthal. It was compiled by Mohsen Soltanifar, Longhai Li, and Jeffrey S. Rosenthal, and published by World Scientific Publishing Co. in 2010 under a Creative Commons license. The preface explains that providing solutions to only the even exercises aims to help students while still allowing instructors to assign odd exercises for coursework. It notes that errors may be present and should be reported.
11 ma1aexpandednoteswk3
This document discusses sequences and series of numbers. It begins by introducing sequences and the concept of convergence for sequences. A sequence converges to a limit L if, given any positive number ε, there exists an N such that the terms an of the sequence satisfy |L - an| < ε for all n > N.
It then proves some basic properties of convergence, including that a convergent sequence must be bounded, and that limits are preserved under operations. Cauchy's criterion for convergence is introduced - a sequence is Cauchy if given any ε, there exists an N such that |am - an| < ε for all m, n > N. Every convergent sequence is Cauchy, and C
Optimal Estimating Sequence for a Hilbert Space Valued Parameter
Some optimality criteria used in estimation of parameters in finite dimensional space has been extended to a separable Hilbert space. Different optimality criteria and their equivalence are established for estimating sequence rather than estimator. An illustrious example is provided with the estimation of the mean of a Gaussian process
Euler phi
The document discusses the Euler Phi function, which calculates the number of numbers less than n that are relatively prime to n. It provides examples of calculating phi(n) for different types of numbers n. For prime numbers, phi(n) = n-1. For numbers that are a power of a prime like 8, phi(n) = n - n/p, where n is the power of p. For numbers that can be expressed as a product of different primes, phi(n) is the product of phi(n) for each prime factor.
Mind map of terminologies used in context of Generative AI
Mind map of common terms used in context of Generative AI.
More Related Content
Similar to 5.1.1 Proof by Mathematical Induction.pptx
Number Theory.pptx
This document discusses several number theory concepts:
- Euler's totient function φ(n) counts the numbers less than or equal to n that are relatively prime to n.
- Euler's theorem states that if n and a are coprime, then aφ(n) ≡ 1 (mod n). This can be used to reduce large powers modulo n.
- Fermat's little theorem is a special case of Euler's theorem where n is a prime number.
- Wilson's theorem relates a number n and its totient φ(n).
Examples are provided to demonstrate calculating the totient function and using Euler's theorem to find remainders.
Induction
The document discusses different mathematical proofs using induction. It provides examples of proving statements about odd numbers, divisibility by primes, equalities, properties, and inequalities using induction. The key steps are to first prove the base case, then assume the statement is true for some value n and use that to prove it is true for n+1.
CC107 - Chapter 1 The real Number System.pdf
This document provides an overview of the real number system. It defines sets, subsets, integers, rational numbers, and irrational numbers. It establishes the field axioms for real numbers, including the commutative, associative, distributive, identity, negatives, and reciprocals properties. It also describes the order axioms and properties of inequalities. Key theorems are presented on the uniqueness of factorizations of integers into prime numbers and the existence of greatest common divisors. Intervals and geometric representations of real numbers on the real number line are also introduced.
Analysis.pptx
- The document discusses some fundamental concepts in analysis such as natural numbers, zero, infinity, series, and L'Hôpital's rule.
- It provides examples to illustrate issues that can arise from improperly handling concepts like division by zero, divergent series, and interchange of sums.
- The key points are that operations should not be performed on divergent series and sums can only be interchanged if the series is absolutely convergent. L'Hôpital's rule also has specific requirements that must be checked before applying it.
The axiomatic power of Kolmogorov complexity
1. The document discusses random axioms and probabilistic proofs in Peano arithmetic. It describes a proof strategy where one could randomly select an integer n that satisfies some formula φ and add it as a new axiom.
2. While this intuition of probabilistic proofs makes sense, it is not really useful since any statement provable with sufficiently high probability is already provable in PA. However, probabilistic proofs can be exponentially more concise than deterministic proofs.
3. The document also discusses Kolmogorov complexity and how statements about it relate to the provability of PA. It can be shown that if C(x) is less than some value, PA will prove it, but PA will never prove a
CMSC 56 | Lecture 11: Mathematical Induction
Mathematical Induction
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 18, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
mathematicalinductionanddivisibilityrules-160711105713.pdf
Mathematical induction and divisibility rules are methods for proving statements about numbers.
Mathematical induction has two steps: 1) proving the statement is true for the base case, usually n=1. 2) Assuming the statement is true for n=k, proving it is true for n=k+1. Divisibility rules transform numbers into smaller ones while preserving divisibility by certain divisors. Rules exist to test for divisibility by 2, 3, 4, 5, 6, 7, 9, 10 and 13.
Mathematical induction and divisibility rules
A complete and enhanced presentation on mathematical induction and divisibility rules with out any calculation.
Here are some defined formulas and techniques to find the divisibility of numbers.
4AMT122-PART 1-SLIDES.pptx
This document provides guidance on writing mathematical proofs. It defines key terms like direct proof and counterexample. It also outlines the standard format for proofs, including stating the theorem, clearly marking the beginning and end, and providing justification for each step. Common mistakes to avoid are arguing from examples rather than a general case, using the same variable to mean different things, and jumping to a conclusion without proper justification.
1 chapter1 introduction
The document provides an introduction to recursion and mathematical concepts relevant to analyzing algorithms. It begins by motivating selection and word puzzle problems to illustrate the importance of how algorithms scale to large inputs. It then reviews necessary mathematical topics like exponents, logarithms, and series. Finally, it defines recursion, provides examples of both correct and incorrect recursive functions, and discusses techniques for proving properties of recursive algorithms like proof by induction.
Crt
This document discusses applications of the Chinese Remainder Theorem in mathematical olympiads. It begins by providing three different perspectives on the Chinese Remainder Theorem: construction, lifting, and destruction. It then analyzes three example problems to illustrate how each perspective can be applied. The first uses the construction perspective to build integers satisfying certain properties. The second uses the lifting perspective to derive a size result by showing integers are divisible by large primes. The third uses destruction to reduce a statement to prime powers.
Lecture1
The real number system contains rational numbers like integers and fractions, as well as irrational numbers like the square root of 2. It is a complete system where every non-empty set that is bounded above has a least upper bound. This property of completeness is what allows real numbers to correspond precisely to points on a number line. The Archimedean property, which follows from completeness, states that between any real number and zero there is a positive integer multiple. This implies that rational numbers are dense within the real numbers.
Math induction principle (slides)
The document discusses propositions, logical operations, predicates, quantification, and mathematical induction. It provides:
1) Definitions of predicates, propositions, universal and existential quantification, and the principle of mathematical induction.
2) Examples of applying predicates, quantification, and induction to prove statements about integers and sums.
3) The process of proving statements by mathematical induction, which involves showing the basis step and inductive step. It also introduces strong mathematical induction.
Properties of real numbers
The document discusses several properties of real numbers including:
1) The closure properties of addition and multiplication - the sum and product of any two real numbers is a real number.
2) Commutative, associative, and distributive properties of addition and multiplication.
3) Identity properties of addition and multiplication with 0 and 1 being the identity elements.
4) Inverse properties of addition and multiplication where adding/multiplying a number and its inverse results in the identity element.
Arithmetic sequence in elementary and HS
This document defines and provides examples of multiplicative functions in number theory. It states that a multiplicative function f(mn) is equal to f(m)f(n) when m and n are relatively prime. Examples given are the Euler totient function φ and the function n2. It also proves that the sum of multiplicative functions and the sum of divisors of a multiplicative function are also multiplicative. Other concepts defined include the Möbius function, Euler phi function, Carmichael conjecture, perfect numbers, and the sum-of-divisors and number-of-divisors functions.
1
This document discusses mathematical induction. It provides two key steps of mathematical induction: 1) the statement P(1) is true, and 2) if P(k) is true, then P(k+1) is also true. It then gives an example of using mathematical induction to prove that the formula 1 + 2 + 3 + ... + n = n(n+1)/2 is true for all positive integers n. Specifically, it shows that P(1) is true, and that if P(k) is true then P(k+1) is also true, thereby proving the formula using mathematical induction.
6300 solutionsmanual free
The document is a solutions manual providing answers to even-numbered exercises from the textbook "A First Look at Rigorous Probability Theory" by Jeffrey S. Rosenthal. It was compiled by Mohsen Soltanifar, Longhai Li, and Jeffrey S. Rosenthal, and published by World Scientific Publishing Co. in 2010 under a Creative Commons license. The preface explains that providing solutions to only the even exercises aims to help students while still allowing instructors to assign odd exercises for coursework. It notes that errors may be present and should be reported.
11 ma1aexpandednoteswk3
This document discusses sequences and series of numbers. It begins by introducing sequences and the concept of convergence for sequences. A sequence converges to a limit L if, given any positive number ε, there exists an N such that the terms an of the sequence satisfy |L - an| < ε for all n > N.
It then proves some basic properties of convergence, including that a convergent sequence must be bounded, and that limits are preserved under operations. Cauchy's criterion for convergence is introduced - a sequence is Cauchy if given any ε, there exists an N such that |am - an| < ε for all m, n > N. Every convergent sequence is Cauchy, and C
Optimal Estimating Sequence for a Hilbert Space Valued Parameter
Some optimality criteria used in estimation of parameters in finite dimensional space has been extended to a separable Hilbert space. Different optimality criteria and their equivalence are established for estimating sequence rather than estimator. An illustrious example is provided with the estimation of the mean of a Gaussian process
Euler phi
The document discusses the Euler Phi function, which calculates the number of numbers less than n that are relatively prime to n. It provides examples of calculating phi(n) for different types of numbers n. For prime numbers, phi(n) = n-1. For numbers that are a power of a prime like 8, phi(n) = n - n/p, where n is the power of p. For numbers that can be expressed as a product of different primes, phi(n) is the product of phi(n) for each prime factor.
Similar to 5.1.1 Proof by Mathematical Induction.pptx (20)
mathematicalinductionanddivisibilityrules-160711105713.pdf
mathematicalinductionanddivisibilityrules-160711105713.pdf
Optimal Estimating Sequence for a Hilbert Space Valued Parameter
Optimal Estimating Sequence for a Hilbert Space Valued Parameter
Recently uploaded
Mind map of terminologies used in context of Generative AI
Mind map of common terms used in context of Generative AI.
Building RAG with self-deployed Milvus vector database and Snowpark Container...
This talk will give hands-on advice on building RAG applications with an open-source Milvus database deployed as a docker container. We will also introduce the integration of Milvus with Snowpark Container Services.
“Building and Scaling AI Applications with the Nx AI Manager,” a Presentation...
“Building and Scaling AI Applications with the Nx AI Manager,” a Presentation...Edge AI and Vision Alliance
For the full video of this presentation, please visit: https://www.edge-ai-vision.com/2024/06/building-and-scaling-ai-applications-with-the-nx-ai-manager-a-presentation-from-network-optix/
Robin van Emden, Senior Director of Data Science at Network Optix, presents the “Building and Scaling AI Applications with the Nx AI Manager,” tutorial at the May 2024 Embedded Vision Summit.
In this presentation, van Emden covers the basics of scaling edge AI solutions using the Nx tool kit. He emphasizes the process of developing AI models and deploying them globally. He also showcases the conversion of AI models and the creation of effective edge AI pipelines, with a focus on pre-processing, model conversion, selecting the appropriate inference engine for the target hardware and post-processing.
van Emden shows how Nx can simplify the developer’s life and facilitate a rapid transition from concept to production-ready applications.He provides valuable insights into developing scalable and efficient edge AI solutions, with a strong focus on practical implementation.20240605 QFM017 Machine Intelligence Reading List May 2024
Everything I found interesting about machines behaving intelligently during May 2024
Unlock the Future of Search with MongoDB Atlas_ Vector Search Unleashed.pdf
Discover how MongoDB Atlas and vector search technology can revolutionize your application's search capabilities. This comprehensive presentation covers:
* What is Vector Search?
* Importance and benefits of vector search
* Practical use cases across various industries
* Step-by-step implementation guide
* Live demos with code snippets
* Enhancing LLM capabilities with vector search
* Best practices and optimization strategies
Perfect for developers, AI enthusiasts, and tech leaders. Learn how to leverage MongoDB Atlas to deliver highly relevant, context-aware search results, transforming your data retrieval process. Stay ahead in tech innovation and maximize the potential of your applications.
#MongoDB #VectorSearch #AI #SemanticSearch #TechInnovation #DataScience #LLM #MachineLearning #SearchTechnology
UiPath Test Automation using UiPath Test Suite series, part 5
Welcome to UiPath Test Automation using UiPath Test Suite series part 5. In this session, we will cover CI/CD with devops.
Topics covered:
CI/CD with in UiPath
End-to-end overview of CI/CD pipeline with Azure devops
Speaker:
Lyndsey Byblow, Test Suite Sales Engineer @ UiPath, Inc.
Essentials of Automations: The Art of Triggers and Actions in FME
In this second installment of our Essentials of Automations webinar series, we’ll explore the landscape of triggers and actions, guiding you through the nuances of authoring and adapting workspaces for seamless automations. Gain an understanding of the full spectrum of triggers and actions available in FME, empowering you to enhance your workspaces for efficient automation.
We’ll kick things off by showcasing the most commonly used event-based triggers, introducing you to various automation workflows like manual triggers, schedules, directory watchers, and more. Plus, see how these elements play out in real scenarios.
Whether you’re tweaking your current setup or building from the ground up, this session will arm you with the tools and insights needed to transform your FME usage into a powerhouse of productivity. Join us to discover effective strategies that simplify complex processes, enhancing your productivity and transforming your data management practices with FME. Let’s turn complexity into clarity and make your workspaces work wonders!
Pushing the limits of ePRTC: 100ns holdover for 100 days
At WSTS 2024, Alon Stern explored the topic of parametric holdover and explained how recent research findings can be implemented in real-world PNT networks to achieve 100 nanoseconds of accuracy for up to 100 days.
“I’m still / I’m still / Chaining from the Block”
“An Outlook of the Ongoing and Future Relationship between Blockchain Technologies and Process-aware Information Systems.” Invited talk at the joint workshop on Blockchain for Information Systems (BC4IS) and Blockchain for Trusted Data Sharing (B4TDS), co-located with with the 36th International Conference on Advanced Information Systems Engineering (CAiSE), 3 June 2024, Limassol, Cyprus.
Observability Concepts EVERY Developer Should Know -- DeveloperWeek Europe.pdf
Monitoring and observability aren’t traditionally found in software curriculums and many of us cobble this knowledge together from whatever vendor or ecosystem we were first introduced to and whatever is a part of your current company’s observability stack.
While the dev and ops silo continues to crumble….many organizations still relegate monitoring & observability as the purview of ops, infra and SRE teams. This is a mistake - achieving a highly observable system requires collaboration up and down the stack.
I, a former op, would like to extend an invitation to all application developers to join the observability party will share these foundational concepts to build on:
GraphSummit Singapore | Enhancing Changi Airport Group's Passenger Experience...
Dr. Sean Tan, Head of Data Science, Changi Airport Group
Discover how Changi Airport Group (CAG) leverages graph technologies and generative AI to revolutionize their search capabilities. This session delves into the unique search needs of CAG’s diverse passengers and customers, showcasing how graph data structures enhance the accuracy and relevance of AI-generated search results, mitigating the risk of “hallucinations” and improving the overall customer journey.
Enchancing adoption of Open Source Libraries. A case study on Albumentations.AI
Enchancing adoption of Open Source Libraries. A case study on Albumentations.AIVladimir Iglovikov, Ph.D.
Presented by Vladimir Iglovikov:
- https://www.linkedin.com/in/iglovikov/
- https://x.com/viglovikov
- https://www.instagram.com/ternaus/
This presentation delves into the journey of Albumentations.ai, a highly successful open-source library for data augmentation.
Created out of a necessity for superior performance in Kaggle competitions, Albumentations has grown to become a widely used tool among data scientists and machine learning practitioners.
This case study covers various aspects, including:
People: The contributors and community that have supported Albumentations.
Metrics: The success indicators such as downloads, daily active users, GitHub stars, and financial contributions.
Challenges: The hurdles in monetizing open-source projects and measuring user engagement.
Development Practices: Best practices for creating, maintaining, and scaling open-source libraries, including code hygiene, CI/CD, and fast iteration.
Community Building: Strategies for making adoption easy, iterating quickly, and fostering a vibrant, engaged community.
Marketing: Both online and offline marketing tactics, focusing on real, impactful interactions and collaborations.
Mental Health: Maintaining balance and not feeling pressured by user demands.
Key insights include the importance of automation, making the adoption process seamless, and leveraging offline interactions for marketing. The presentation also emphasizes the need for continuous small improvements and building a friendly, inclusive community that contributes to the project's growth.
Vladimir Iglovikov brings his extensive experience as a Kaggle Grandmaster, ex-Staff ML Engineer at Lyft, sharing valuable lessons and practical advice for anyone looking to enhance the adoption of their open-source projects.
Explore more about Albumentations and join the community at:
GitHub: https://github.com/albumentations-team/albumentations
Website: https://albumentations.ai/
LinkedIn: https://www.linkedin.com/company/100504475
Twitter: https://x.com/albumentationsClimate Impact of Software Testing at Nordic Testing Days
My slides at Nordic Testing Days 6.6.2024
Climate impact / sustainability of software testing discussed on the talk. ICT and testing must carry their part of global responsibility to help with the climat warming. We can minimize the carbon footprint but we can also have a carbon handprint, a positive impact on the climate. Quality characteristics can be added with sustainability, and then measured continuously. Test environments can be used less, and in smaller scale and on demand. Test techniques can be used in optimizing or minimizing number of tests. Test automation can be used to speed up testing.
Communications Mining Series - Zero to Hero - Session 1
This session provides introduction to UiPath Communication Mining, importance and platform overview. You will acquire a good understand of the phases in Communication Mining as we go over the platform with you. Topics covered:
• Communication Mining Overview
• Why is it important?
• How can it help today’s business and the benefits
• Phases in Communication Mining
• Demo on Platform overview
• Q/A
Video Streaming: Then, Now, and in the Future
In his public lecture, Christian Timmerer provides insights into the fascinating history of video streaming, starting from its humble beginnings before YouTube to the groundbreaking technologies that now dominate platforms like Netflix and ORF ON. Timmerer also presents provocative contributions of his own that have significantly influenced the industry. He concludes by looking at future challenges and invites the audience to join in a discussion.
Uni Systems Copilot event_05062024_C.Vlachos.pdf
Unlocking Productivity: Leveraging the Potential of Copilot in Microsoft 365, a presentation by Christoforos Vlachos, Senior Solutions Manager – Modern Workplace, Uni Systems
Encryption in Microsoft 365 - ExpertsLive Netherlands 2024
In this session I delve into the encryption technology used in Microsoft 365 and Microsoft Purview. Including the concepts of Customer Key and Double Key Encryption.
Recently uploaded (20)
Monitoring Java Application Security with JDK Tools and JFR Events
Monitoring Java Application Security with JDK Tools and JFR Events
Mind map of terminologies used in context of Generative AI
Mind map of terminologies used in context of Generative AI
Building RAG with self-deployed Milvus vector database and Snowpark Container...
Building RAG with self-deployed Milvus vector database and Snowpark Container...
“Building and Scaling AI Applications with the Nx AI Manager,” a Presentation...
“Building and Scaling AI Applications with the Nx AI Manager,” a Presentation...
20240605 QFM017 Machine Intelligence Reading List May 2024
20240605 QFM017 Machine Intelligence Reading List May 2024
Unlock the Future of Search with MongoDB Atlas_ Vector Search Unleashed.pdf
Unlock the Future of Search with MongoDB Atlas_ Vector Search Unleashed.pdf
UiPath Test Automation using UiPath Test Suite series, part 5
UiPath Test Automation using UiPath Test Suite series, part 5
Essentials of Automations: The Art of Triggers and Actions in FME
Essentials of Automations: The Art of Triggers and Actions in FME
Pushing the limits of ePRTC: 100ns holdover for 100 days
Pushing the limits of ePRTC: 100ns holdover for 100 days
Observability Concepts EVERY Developer Should Know -- DeveloperWeek Europe.pdf
Observability Concepts EVERY Developer Should Know -- DeveloperWeek Europe.pdf
GraphSummit Singapore | Enhancing Changi Airport Group's Passenger Experience...
GraphSummit Singapore | Enhancing Changi Airport Group's Passenger Experience...
Enchancing adoption of Open Source Libraries. A case study on Albumentations.AI
Enchancing adoption of Open Source Libraries. A case study on Albumentations.AI
Climate Impact of Software Testing at Nordic Testing Days
Climate Impact of Software Testing at Nordic Testing Days
Communications Mining Series - Zero to Hero - Session 1
Communications Mining Series - Zero to Hero - Session 1
Encryption in Microsoft 365 - ExpertsLive Netherlands 2024
Encryption in Microsoft 365 - ExpertsLive Netherlands 2024
5.1.1 Proof by Mathematical Induction.pptx
- 1. Proof by Mathematical Induction DISCRETE MATH II 5.1.1
- 2. What is Mathematical Induction? Mathematical induction is a method of proof that will be used throughout your study. The basic theory is this: If we can prove that P(1) is true, and we can prove that whenever P(𝑘) is true, when 𝑘 ∈ ℤ+ (a positive integer), then P 𝑘 + 1 is true, then P(𝑛) is true for all 𝑛 ∈ ℤ+ . * Keep in mind that P 1 represents the least element in the set. More on this…
- 3. The Well-Ordering Principle Before we get too involved in mathematical induction, we must address the issue with ℤ+ . ℤ+ refers to the set of numbers that include the positive integers. You will sometimes see this as ℕ, which is the set of natural numbers including the counting numbers; 1, 2, 3, … So, let’s suppose that 𝑆 ∈ ℤ+, 𝑆 ≠ ∅, and that 𝑆 is bounded below. The well-ordering principle says that 𝑆 has a least element. These seems obvious. We have a subset of a discrete (countable) set, it isn’t empty, and it is bounded below. Therefore, there must be a least element, or an element in the set whose value is less than all other values. Can we use mathematical induction with 𝑆 ∈ ℚ+ or 𝑆 ∈ ℝ+ ?
- 4. Back to Mathematical Induction Let P 𝑛 denote an open mathematical statement that involves one or more occurrences of the variable 𝑛, which represents a positive integer: If 𝑃(1) is true; and If whenever 𝑃(𝑘) is true (for some particular, but arbitrarily chosen 𝑘 ∈ ℤ+), then 𝑃(𝑘 + 1) is true; then P 𝑛 is true for all 𝑛 ∈ ℤ+. Keep in mind that 𝑃(1) just represents the “least element”
- 5. Guided Practice Prove 1 + 3 + 5 + 7 + ⋯ + (2𝑛 − 1) = 𝑛2 for all 𝑛 ≥ 1 𝑖=1 𝑛 (2𝑛 − 1) = 𝑛2