29 research outputs found

    Three results on cycle-wheel Ramsey numbers

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    Given two graphs G1 and G2, the Ramsey number R(G1,G2) is the smallest integer N such that, for any graph G of order N, either G1 is a subgraph of G, or G2 is a subgraph of the complement of G. We consider the case that G1 is a cycle and G2 is a (generalized) wheel. We expand the knowledge on exact values of Ramsey numbers in three directions: large cycles versus wheels of odd order; large wheels versus cycles of even order; and large cycles versus generalized odd wheels

    THE ELECTRONIC JOURNAL OF COMBINATORICS (2014), DS1.14 References

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    and Computing 11. The results of 143 references depend on computer algorithms. The references are ordered alphabetically by the last name of the first author, and where multiple papers have the same first author they are ordered by the last name of the second author, etc. We preferred that all work by the same author be in consecutive positions. Unfortunately, this causes that some of the abbreviations are not in alphabetical order. For example, [BaRT] is earlier on the list than [BaLS]. We also wish to explain a possible confusion with respect to the order of parts and spelling of Chinese names. We put them without any abbreviations, often with the last name written first as is customary in original. Sometimes this is different from the citations in other sources. One can obtain all variations of writing any specific name by consulting the authors database of Mathematical Reviews a

    On the Ramsey number of 4-cycle versus wheel

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    For any fixed graphs GG and HH, the Ramsey number R(G,H)R(G,H) is the smallest positive integer nn such that for every graph FF on nn vertices must contain GG or the complement of FF contains HH. The girth of graph GG is a length of the shortest cycle. A kk-regular graph with the girth gg is called a (k,g)(k,g)-graph. If the number of of vertices in (k,g)(k,g)-graph is minimized then we call this graph a (k,g)(k,g)-cage. In this paper, we derive the bounds of Ramsey number R(C4,Wn)R(C_4,W_n) for some values of nn. By modifying (k,5)(k, 5)-graphs, for k=7k = 7 or 99, we construct these corresponding (C4,Wn)(C_4,W_n)-good graphs. </div

    Wheel and Star-critical Ramsey Numbers for Quadrilateral

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    The star-critical Ramsey number r∗(H1, H2) is the smallest integer k such that every red/blue coloring of the edges of Kn − K1,n−k−1 contains either a red copy of H1 or a blue copy of H2, where n is the graph Ramsey number R(H1, H2). We study the cases of r∗(C4, Cn) and R(C4, Wn). In particular, we prove that r∗(C4, Cn) = 5 for all n \u3e 4, obtain a general characterization of Ramsey-critical (C4, Wn)-graphs, and establish the exact values of R(C4, Wn) for 9 cases of n between 18 and 44

    Picturing Number in the Central Middle Ages.

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    Numeracy was as highly valued as literacy in the schools of Latin-speaking Europe around the year 1000, and the skills inculcated by masters, engendering specific modes of seeing and imagining, had demonstrable impact on contemporary visual culture. The trivium—grammar, rhetoric, and dialectic—continued to be taught as the foundation of learning, but the quadrivium, the four disciplines of number—arithmetic, geometry, astronomy, and music—received new emphasis. Two of the era’s greatest intellects, Gerbert of Aurillac (Pope Sylvester II; c.940–1003) and Abbo of Fleury (c.944–1004), gained renown for their mathematical prowess and charismatic teaching. They educated a generation of Europe's powerful elites—including Emperor Otto III—and a host of anonymous clerics, monks, and priests. In the closed economy of the central middle ages, these men were also the primary patrons, makers, and viewers of objects. Works of the time, like the Pericope Book of Henry II, reveal new qualities when examined through the lens of number. This project is located at the cathedral school of Reims and the monastery school of Saint-Benoüt-sur-Loire (Fleury)—where Gerbert and Abbo were masters, epicenters of a pan-European network of exchange linking monastic, episcopal, and lay institutions. Numeric knowledge was drawn from late antique and early medieval tracts by such figures as Boethius, Calcidius, Macrobius, Martianus Capella, Cassiodorus, Isidore of Seville, and Bede. Manuscript copies of these works produced and used at Reims and Fleury c.1000 give evidence of active engagement with their content, visual as well as verbal. Diagrammatic images earlier devised to explicate numeric concepts were now adapted and artfully elaborated for classroom use. This is evident in important introductions to the quadrivial disciplines prepared by Abbo (Explanatio in Calculo Victorii), Abbo’s student Byrhtferth of Ramsey (Enchiridion), and Gerbert (Isagoge geometriae). Accompanying images to these tracts are witness to contemporary notions of materiality, sight, and the limits of representation. Students of arithmetic became freshly attuned to placement and order. Computistic study developed an active, agile, and "curious" eye, while the practice of geometry exercised the intellectual eye, sharpening it, according to Gerbert, "for contemplating spiritual things and truths."PHDHistory of ArtUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/116774/1/mcnameme_1.pd

    Proceedings of the tenth international conference Models in developing mathematics education: September 11 - 17, 2009, Dresden, Saxony, Germany

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    This volume contains the papers presented at the International Conference on “Models in Developing Mathematics Education” held from September 11-17, 2009 at The University of Applied Sciences, Dresden, Germany. The Conference was organized jointly by The University of Applied Sciences and The Mathematics Education into the 21st Century Project - a non-commercial international educational project founded in 1986. The Mathematics Education into the 21st Century Project is dedicated to the improvement of mathematics education world-wide through the publication and dissemination of innovative ideas. Many prominent mathematics educators have supported and contributed to the project, including the late Hans Freudental, Andrejs Dunkels and Hilary Shuard, as well as Bruce Meserve and Marilyn Suydam, Alan Osborne and Margaret Kasten, Mogens Niss, Tibor Nemetz, Ubi D’Ambrosio, Brian Wilson, Tatsuro Miwa, Henry Pollack, Werner Blum, Roberto Baldino, Waclaw Zawadowski, and many others throughout the world. Information on our project and its future work can be found on Our Project Home Page http://math.unipa.it/~grim/21project.htm It has been our pleasure to edit all of the papers for these Proceedings. Not all papers are about research in mathematics education, a number of them report on innovative experiences in the classroom and on new technology. We believe that “mathematics education” is fundamentally a “practicum” and in order to be “successful” all new materials, new ideas and new research must be tested and implemented in the classroom, the real “chalk face” of our discipline, and of our profession as mathematics educators. These Proceedings begin with a Plenary Paper and then the contributions of the Principal Authors in alphabetical name order. We sincerely thank all of the contributors for their time and creative effort. It is clear from the variety and quality of the papers that the conference has attracted many innovative mathematics educators from around the world. These Proceedings will therefore be useful in reviewing past work and looking ahead to the future
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