Additive Combinatorics Tao, Terence; Vu, Van H.
09/2006, Letnik:
v.Series Number 105
eBook
Additive combinatorics is the theory of counting additive structures in sets. This theory has seen exciting developments and dramatic changes in direction in recent years thanks to its connections ...with areas such as number theory, ergodic theory and graph theory. This graduate-level 2006 text will allow students and researchers easy entry into this fascinating field. Here, the authors bring together in a self-contained and systematic manner the many different tools and ideas that are used in the modern theory, presenting them in an accessible, coherent, and intuitively clear manner, and providing immediate applications to problems in additive combinatorics. The power of these tools is well demonstrated in the presentation of recent advances such as Szemerédi's theorem on arithmetic progressions, the Kakeya conjecture and Erdos distance problems, and the developing field of sum-product estimates. The text is supplemented by a large number of exercises and new results.
Learning combinatorics material requires a digital learning medium that is able to support students' independent learning. The unavailability of digital media makes students experience difficulties, ...so independent learning cannot be carried out effectively. This research aims to develop a digital-based smart calculator application as a learning medium for combinatorics material that meets the criteria of being valid, effective, and practical. This research applies research and development (R&D) methods whose stages are the preliminary stage, planning stage, formative evaluation stage, prototyping stage, and large class trials (field tests). A total of 31 Bangka Belitung State Manufacturing Polytechnic TRPL students were the subjects of this research. The data collection techniques applied are expert validation sheets, effectiveness trial questionnaires, and practicality trial questionnaires. Expert validation activities provide evidence that the media is valid and ready to be used in learning. Based on effectiveness and practicality trials on 31 students, the results showed that the digital-based smart calculator application was classified as effective and practical. Based on the research conducted, it can be concluded that the development of a digital-based smart-calculator application is valid, effective, and practical, so that the digital-based smart-calculator application is ready to be used and applied to learning combinatorics material.BAHASA INDONESIA ABSTRACT: Pembelajaran materi kombinatorika membutuhkan suatu media pembelajaran digital yang mampu mendukung pembelajaran mandiri mahasiswa. Tidak tersedianya media digital ini membuat mahasiswa mengalami kesulitan sehingga pembelajaran mandiri tidak dapat terlaksana dengan efektif. Penelitian ini bertujuan untuk mengembangkan aplikasi smart – calculator berbasis digital sebagai media pembelajaran materi kombinatorika yang memenuhi kriteria valid, efektif, dan praktis Penelitian ini menerapkan metode penelitian dan pengembangan (R&D) yang tahapannya yaitu tahap preliminary, tahap perencanaan, tahap formative evaluation, tahap prototyping, dan uji coba kelas besar (field test). Sejumlah 31 mahasiswa TRPL Politeknik Manufaktur Negeri Bangka Belitung menjadi subjek dalam penelitian ini. Teknis pengumpulan data yang diterapkan adalah menggunakan lembar validasi ahli, angket uji coba keefektifan, dan angket uji coba kepraktisan. Kegiatan validasi ahli memberikan hasil bahwa media valid dan siap digunakan dalam pembelajaran. Berdasarkan uji coba keefektifan dan kepraktisan terhadap 31 mahasiswa memberikan hasil bahwa aplikasi smart – calculator berbasis digital tergolong efektif dan praktis. Berdasarkan penelitian yang dilakukan dapat disimpulkan bahwa pengembangan aplikasi smart – calculator berbasis digital tergolong valid, efektif, dan praktis sehingga aplikasi smart – calculator berbasis digital siap digunakan dan diterapkan pada pembelajaran materi kombinatorika.
Analytic combinatorics aims to enable precise quantitative predictions of the properties of large combinatorial structures. The theory has emerged over recent decades as essential both for the ...analysis of algorithms and for the study of scientific models in many disciplines, including probability theory, statistical physics, computational biology, and information theory. With a careful combination of symbolic enumeration methods and complex analysis, drawing heavily on generating functions, results of sweeping generality emerge that can be applied in particular to fundamental structures such as permutations, sequences, strings, walks, paths, trees, graphs and maps. This account is the definitive treatment of the topic. The authors give full coverage of the underlying mathematics and a thorough treatment of both classical and modern applications of the theory. The text is complemented with exercises, examples, appendices and notes to aid understanding. The book can be used for an advanced undergraduate or a graduate course, or for self-study.
Článek analyzuje přístupy k vybraným kombinatorickým konceptům v pěti českých středoškolských učebnicích matematiky s cílem zjistit, do jaké míry tyto učebnice poskytují podporu učitelům pro výuku ...založenou na teorii poznávacího procesu v matematice. Analýza učebnic byla vedena na pozadí teorie generických modelů, přičemž důraz byl položen na přítomnost a kvalitu a) izolovaných modelů a nemodelů budoucího poznatku, b) výzev k zobecňování jako nutnému předpokladu tvorby generického modelu, c) grafických reprezentací jako nezbytné podpory rozvoje kombinatorického uvažování. Bylo zjištěno, že učebnice jsou si v mnoha směrech dosti podobné, ale v jistých ohledech se odlišují. V učebnicích jsme identifikovali mj. nedostatečnou motivaci pro kombinatoriku, nízký počet izolovaných modelů budoucího poznatku, absenci explicitních výzev k zobecňování a z toho plynoucí absenci významného konceptu izomorfismus. Navzdory výzkumně prokázanému pozitivnímu vlivu přítomnosti a vlastní tvorby grafických reprezentací na rozvoj žákovského kombinatorického myšlení se v učebnicích vyskytují reprezentace zřídka, a navíc s malou rozmanitostí. Autoři učebnic až na výjimky nevyzývají čtenáře k tvorbě vlastních grafických reprezentací. Od analyzovaných učebnic se odlišuje jedna, která obsahuje časté výzvy k tvorbě vlastních reprezentací, pracuje záměrně s izomorfními úlohami a vyzývá čtenáře k zobecňování konkrétních postupů.
This study described undergraduate students’ gesture in problem solving based on the stage of thinking by Mason which consisted of the stage of entry, attack, and review. The study adopted ...descriptive qualitative approach. Subjects were chosen based on their response in problem solving. Data collection was carried out by administering test, conducting interview, and recording video to explore students’ gesture in problem solving. Results showed that the students that could answer the combinatorical problems given correctly, did pointing and writing gesture in the phases of entry and attack and acted pointing, representational, and writing gesture in review phase.Abstrak: Penelitian ini bertujuan untuk mendeskripsikan gesture mahasiswa dalam menyelesiakan masalah dengan jawaban benar berdasarkan tahap berpikir Mason. Sehingga peneliti dapat mengetahui jenis gesture yang dilakukan mahasiswa pada setiap tahap entry, attack, maupun review. Penelitian ini menggunakan pendekatan kualitatif deskriptif. Subjek penelitian ini diambil berdasarkan jawaban mahasiswa dalam menyelesaikan masalah dengan tepat. Pengumpulan data dilakukan dengan pemberian lembar tes, wawancara, dan video recorder untuk mengetahui gesture mahasiswa tersebut saat menyelesaikan masalah. Hasil menunjukkan mahasiswa yang dapat menjawab masalah kombinatorika dengan benar, pada fase entry dan attack melakukan gesture pointing dan writing. Pada fase review melakukan gesture pointing, representational, dan writing.
The need for variations in learning resources for students of SMK Diponegoro Tumpang and the development of 2stcentury technology are the basis of this development research. The purpose of this paper ...is to describe the development of valid, practical and effective learning media for combinatoric material. This development uses the ADDIE model. The test subjects in the development of learning media consisted of expert validators, practitioner validators, observers, 5 students of class XII as group trials, and 22 students of class XII KPR as field trials. The instruments used were learning media, lesson plans, observation sheets, student response questionnaires, and teacher response questionnaires. The results of the development state that learning media are valid, practical, and very effective.Abstrak: Kebutuhan variasi sumber belajar bagi siswa SMK Diponegoro Tumpang dan perkembangan teknologi abad 21 merupakan dasar dari penelitian pengembangan ini. Tujuan penulisan ini adalah mendeskripsikan pengembangan media pembelajaran materi kombinatorika yang valid, praktis, dan efektif. Pengembangan ini menggunakan model ADDIE. Subjek uji coba dalam pengembangan media pembelajaran ini terdiri dari validator ahli, validator praktisi, observer, 5 siswa kelas XII sebagai uji coba kelompok, dan 22 siswa kelas XII KPR sebagai uji coba lapangan. Instrumen yang digunakan adalah media pembelajaran, RPP, lembar observasi, angket respon siswa, dan angket respon guru. Hasil pengembangan menyatakan bahwa media pembelajaran valid, praktis, dan sangat efektif.
Kombinatorika u kemiji Brueckler, Franka Miriam
Kemija u industriji; časopis kemičara i tehnologa Jugoslavije,
8/2016, Letnik:
65, Številka:
7-8
Journal Article
Recenzirano
Odprti dostop
U ovome se članku ukratko opisuju osnovni kombinatorni principi (permutacije i kombinacije s ponavljanjem i bez ponavljanja) koji se primjenjuju u kemiji, s težištem na metodama enumeracije određenih ...tipova molekula. Posebno, kreće se od osnovnih pojmova teorije grafova te je opisano kako se mogu prebrojati neki tipovi grafova koji odgovaraju pojedinim klasama molekula.
Figurate numbers Deza, Elena; Deza, Michel Marie
2012., 2012, 2012-01-20
eBook
Figurate numbers have a rich history with many applications. The main purpose of this book is to provide a thorough and complete presentation of the theory of figurate numbers, giving much of their ...properties, facts and theorems with full proofs. This book is the first of this topic written in unified systematic way. It also contains many exercises with solutions.
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