This article discusses deterministic and combinatorial algorithms for submodular maximization, focusing on their applications in various fields such as combinatorics, machine learning, image processing, and algorithmic game theory. It covers key concepts like submodularity, examples of submodular op

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Discrete Optimization is a field of applied mathematics that uses techniques from combinatorics, graph theory, linear programming, and algorithms to solve optimization problems over discrete structures. This involves creating mathematical models, defining objective functions, decision variables, and

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The Twelvefold Way in combinatorics classifies enumerative problems related to finite sets, focusing on functions from set N to set X under various conditions like injective or surjective. It considers equivalence relations and orbits under group actions, providing a systematic approach to counting

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Explore the principles of inclusion-exclusion in combinatorics, focusing on scenarios involving sets and intersections. Learn how to calculate the number of strings that contain specific elements by applying these principles effectively, with detailed examples and explanations.

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Explanation of counting principles in combinatorics including permutations, combinations, binomial theorem, and overcounting scenarios with examples like anagrams. Also covers important facts and rules related to combinations. Highlights the importance of starting homework early and accessing option

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Understanding counting principles in enumerative combinatorics is essential for solving mathematical problems involving permutations and combinations. The concepts discussed include calculating probabilities, determining the number of outcomes, and applying counting rules to various scenarios such a

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This research explores an additive combinatorics approach to the log-rank conjecture in communication complexity, addressing the maximum total bits sent on worst-case inputs and known bounds. It discusses the Polynomial Freiman-Ruzsa Conjecture and Approximate Duality, highlighting technical contrib

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Delve into the fascinating world of graph colorings, chromatic polynomials, and notable conjectures in discrete geometry. Explore the impact of June Huh in bringing Hodge theory to combinatorics and his proof of various mathematical conjectures. Uncover the significance of the four-color theorem, co

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An exploration into combinatorics, focusing on arranging objects and counting possibilities. From dividing polygons to listing objects, delve into the world of counting and arrangement. Learn how counting plays a vital role in algorithms and probability, and discover the complexity it adds to variou

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Exploring common quicksort implementations, algorithms with probabilities of failure, and small probabilities of failure in computer science. The content covers concepts like combinatorics, probability, continuous probability, and their applications in computer science and machine learning. Strategi

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Combinatorics, a key facet of discrete mathematics, explores the arrangement of objects and finds applications in various fields like discrete probability and algorithm analysis. The Rule of Sum, a fundamental principle, dictates how tasks can be accomplished when they cannot be done simultaneously.

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Dr. J.Frost introduces the Senior Maths Club, catering to those interested in challenging math beyond A-level curriculum and preparing for competitions like the SMC and BMO. The club offers a broad grounding in math for university interviews, focusing on topics like number theory, combinatorics, and

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Delve into the realm of combinatorics with Dr. J. Frost as your guide. Discover key topics like slot filling, factorial and permutation functions, distinguishable vs. indistinguishable objects, recurrence relations, compositions, and partitions. Uncover the art of counting and arranging objects with

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Igor Faradjev recounts the origin of algebraic combinatorics in the 1968-1990 period at the Institute for Systems Analysis. He reflects on his personal experiences, relationships with key mathematicians, and the innovative work undertaken in the Mathematical Laboratory of the Institute for Theoretic

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Explore combinatorics concepts in Year 8 mathematics through systematic counting methods, including examples of counting right-angled triangles, frog hop sequences, and combinations of starters and main courses. Practice exercises included to enhance understanding of problem-solving strategies in co

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The problem set from the Berkeley Math Tournament Fall 2012 includes questions on combinatorics, algebra, logic, and geometry. It challenges participants with scenarios involving coin probabilities, student major combinations, number selection games, and truth-telling puzzles. Test your problem-solv

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Explanation of combinatorics theorems, such as the Division Rule and Rearranging with Duplicates, along with practical examples like counting anagrams and organizing pairs. The Pigeonhole Principle is also illustrated, showcasing applications in various scenarios with clear steps and outcomes.

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The Generalized Pigeonhole Principle is illustrated through an example involving selecting cards from a deck. By strategically grouping the cards, we determine the minimum number needed to guarantee at least three cards of the same suit are chosen. Additionally, the process is applied to finding the

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