Math Project Presentation New
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The document discusses the ballot problem of counting votes for multiple candidates. It defines the ballot problem for two candidates, where the formula to count the number of ways ballots can be arranged such that one candidate is always ahead is (a+b choose a). It notes this has been proved for two candidates. The objective is to find the formula and proof for many candidates. It provides examples and formulas for counting ballots for three candidates, and conjectures a formula for n candidates proved by strong mathematical induction. Applications in biology and cryptography are also mentioned.
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109
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H0.
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Guest Speaker Segment with Hannah Barrington: Dive into the world of dynamic real estate marketing with Hannah, the Marketing Manager at Workspace Group. Hear firsthand how their team generates engaging descriptions for thousands of office units by integrating diverse data sources—from PDF floorplans to web pages—using FME transformers, like OpenAIVisionConnector and AnthropicVisionConnector. This use case will show you how GenAI can streamline content creation for marketing across the board.
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With Skybuffer AI, various AI models can be integrated into a single communication channel such as Microsoft Teams. This integration empowers business users with insights drawn from SAP backend systems, enterprise documents, and the expansive knowledge of Generative AI. And the best part of it is that it is all managed through our intuitive no-code Action Server interface, requiring no extensive coding knowledge and making the advanced AI accessible to more users.
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Inconsistent user experience and siloed data, high costs, and changing customer expectations – Citizens Bank was experiencing these challenges while it was attempting to deliver a superior digital banking experience for its clients. Its core banking applications run on the mainframe and Citizens was using legacy utilities to get the critical mainframe data to feed customer-facing channels, like call centers, web, and mobile. Ultimately, this led to higher operating costs (MIPS), delayed response times, and longer time to market.
Ever-changing customer expectations demand more modern digital experiences, and the bank needed to find a solution that could provide real-time data to its customer channels with low latency and operating costs. Join this session to learn how Citizens is leveraging Precisely to replicate mainframe data to its customer channels and deliver on their “modern digital bank” experiences.
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Read Taking AI to the Next Level in Manufacturing to gain insights on AI adoption in the manufacturing industry, such as:
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Math Project Presentation New
- 1. THE BALLOT PROBLEM OF MANY CANDIDATES
- 5. Objective To find the formula and proof of Ballot problem for many candidates.
- 6. In the case of two candidates (The Original Ballot problem)
- 7. Suppose is the ballots of the 1 st candidate. is the ballots of the 2 nd candidate, when . Define “1” as the ballot given to 1 st candidate. “ -1” as the ballot given to 2 nd candidate. In the case of two Candidates
- 8. The number of ways to count the ballots for required condition. The number of permutation of the sequence: such that the partial sum is always positive. The number of ways to walk on the lattice plane starting at (0,0) and finish at (a,b), and not allow pass through the line y=x except (0,0) = =
- 9. In the case of two Candidates
- 10. Reflection Principle “ Reflection principle” is use to count the number of path, one can show that the number of illegal ways which begin at (1,0) is equal to the number of ways begin at (0,1). It implies that, if we denote as the number of ways as required:
- 11. Reflection Principle Figure 1 Figure 2
- 12. In the case of three Candidates
- 13. In the case of three Candidates
- 14. In the case of three Candidates Figure 3 Figure 4
- 15. How to count ?
- 16. How to count ?
- 18. Example Counting Front View F(1) F(2) F(3) F(4) F(5)
- 19. Example Counting Side View S(1) S(2)
- 20. Example Counting Matching 2 2 1 1 1 S(2) 1 1 1 1 1 S(1) F(5) F(4) F(3) F(2) F(1) F(i) * S(j) F(1) F(2) F(3) F(4) F(5) S(1) S(2) 5 7 12 Total
- 22. Counting (5,4) with D.P. y x x+y 1 1 1 1 1 1 1 4 3 2 1 0 0 9 5 2 0 0 0 14 5 0 0 0 0 14 0 0 0 0 0
- 23. Formula for three candidates
- 24. Definition is the number of ways to count the ballot that correspond to the required condition Lemma 1.1 Lemma 1.2 Formula for three candidates
- 25. Formula for three candidates Conjecture
- 26. Let Consider Use strong induction; given is the “base” therefore Proof Hence, the base case is true. Formula for three candidates
- 28. Formula for three candidates
- 29. By strong induction, we get that, Formula for three candidates
- 30. Formula for n candidates
- 31. is the number of ways to count the ballots of the n candidates such that, while the ballots are being counted, the winner will always get more ballots than the loser. Definition Lemma 3 Formula for n candidates
- 32. (base) n=2 n=k Mathematical Induction Sub - Strong Mathematical Induction (base) n=k-1 Proof
- 33. Formula for n candidates
- 34. Formula for n candidates
- 35. Formula for n candidates
- 36. By mathematical induction, Formula for n candidates
- 37. 1. Think about the condition “never less than” instead of “always more than.” Development 2. What if we suppose that the ballots of the 3 rd candidate don’t relate to anyone else?
- 39. Application In Cryptography Define the plaintext (code) used to send the data Increases the security of the system.
- 40. Reference Chen Chuan-Chong and Koh Khee-Meng, Principles and Techniques in Combinatorics , World Scientific, 3rd ed., 1999. Marc Renault, Four Proofs of the Ballot Theorem, U.S.A., 2007. Michael L. GARGANO, Lorraine L. LURIE Louis V. QUINTAS, and Eric M. WAHL, The Ballot Problem, U.S.A.,2005. Miklos Bona, Unimodality, Introduction to Enumerative Combinatorics, McGrawHill, 2007. Sriram V. Pemmaraju, Steven S. Skienay, A System for Exploring Combinatorics and Graph Theory in Mathematica, U.S.A., 2004.