12 research outputs found

    Quasipolynomial size frege proofs of Frankl's Theorem on the trace of sets

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    We extend results of Bonet, Buss and Pitassi on Bondy's Theorem and of Nozaki, Arai and Arai on Bollobas' Theorem by proving that Frankl's Theorem on the trace of sets has quasipolynomial size Frege proofs. For constant values of the parameter t, we prove that Frankl's Theorem has polynomial size AC(0)-Frege proofs from instances of the pigeonhole principle.Peer ReviewedPostprint (author's final draft

    Characterizing extremal digraphs for identifying codes and extremal cases of Bondy's theorem on induced subsets

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    An identifying code of a (di)graph GG is a dominating subset CC of the vertices of GG such that all distinct vertices of GG have distinct (in)neighbourhoods within CC. In this paper, we classify all finite digraphs which only admit their whole vertex set in any identifying code. We also classify all such infinite oriented graphs. Furthermore, by relating this concept to a well known theorem of A. Bondy on set systems we classify the extremal cases for this theorem

    Set Systems and Families of Permutations with Small Traces

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    We study the maximum size of a set system on nn elements whose trace on any bb elements has size at most kk. We show that if for some bi0b \ge i \ge 0 the shatter function fRf_R of a set system ([n],R)([n],R) satisfies fR(b)<2i(bi+1)f_R(b) < 2^i(b-i+1) then R=O(ni)|R| = O(n^i); this generalizes Sauer's Lemma on the size of set systems with bounded VC-dimension. We use this bound to delineate the main growth rates for the same problem on families of permutations, where the trace corresponds to the inclusion for permutations. This is related to a question of Raz on families of permutations with bounded VC-dimension that generalizes the Stanley-Wilf conjecture on permutations with excluded patterns

    Tractability of Cut-free Gentzen-type propositional calculus with permutation inference II

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    AbstractIn Arai (1996), we introduced a new inference rule called permutation to propositional calculus and showed that cut-free Gentzen system LK (GCNF) with permutation (1) satisfies the feasible subformula property, and (2) proves pigeonhole principle and k-equipartition polynomially. In this paper, we survey more properties of our system. First, we prove that cut-free LK+permutation has polynomial size proofs for nonunique endnode principle, Bondy's theorem. Second, we remark the fact that permutation inference has an advantage over renaming inference in automated theorem proving, since GCNF+renaming does not always satisfy the feasible subformula property. Finally, we discuss on the relative efficiency of our system vs. Frege systems and show that Frege polynomially simulates GCNF+renaming if and only if Frege polynomially simulates extended Frege

    We didn\u27t miss a day : a history in narratives of schooling efforts for Jewish children and youths in German-occupied Europe

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    This is a study of adult and youth narratives about creating and participating in schooling during what has become known as the Holocaust. Jewish narrators created works that described and analyzed their experiences and educational efforts while in hiding, in ghettos, and in concentration camps. The narratives are in the form of diaries, journals, autobiographies, testimonies, and interviews. The narratives were analyzed in order to discover personal and shared themes and are interpreted and presented in ways meant to retain their particular natures and styles. Short pieces from other sources are included to enhance understanding of the roles of education and schooling in the experiences of Jews trapped in the Final Solution . Narrators are introduced through short biographies. Each narrative is offered in segments interlaced with discussion of the contexts and interpretations that enhance understanding of the narrators and their schooling efforts. Following the narratives are discussions of individual and shared themes and of views critical of schooling efforts on behalf of Jewish children. Relationships between social, political, cultural and ideological positions and schooling form a subtext of the analysis of the narratives. Educational efforts, often under fearsome bans on education for Jewish children, ranged from the autodidactic efforts of isolated children to complex, yet often clandestine, school systems. Schooling was an opportunity for resistance to German plans to destroy Judaism—when intellectual resistance was often the only possibility to fight back. Schooling connected youths and adults to each other and to their pasts, while creating possibilities for a future that many did not live to experience. It sent survivors into that future with a sense of having prepared for a new life. Many emerged from hiding places and sites of imprisonment and torture with little else. Their families and communities destroyed, their material resources stolen, no longer welcome in their own lands—only the intellectual growth and the sense of camaraderie. fostered in the educational enterprise, accompanied them into an often hostile and strange post-war world

    A sufficient condition guaranteeing large cycles in graphs

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    AbstractWe generalize Bedrossian-Chen-Schelp's condition (1993) for the existence of large cycles in graphs, and give infinitely many examples of graphs which fulfill the new condition for hamiltonicity, while the related condition by Bedrossian, Chen, and Schelp is not fulfilled

    Pancyclicity of Hamiltonian line graphs

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    Let f(n) be the smallest integer such that for every graph G of order n with minimum degree 3(G)>f(n), the line graph L(G) of G is pancyclic whenever L(G) is hamiltonian. Results are proved showing that f(n) = ®(n 1/3)

    The Peruvian Education Reform of 1968-1980 and Seventh-day Adventist Education at Inca Union College : a Study in Models

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    Problem. The purpose of this study was to compare and contrast two distinct educational models: (1) Peruvian educational reform, which affected both public and private education throughout the country, and (2) the Seventh-day Adventist educational system as represented by Inca Union College. This research was limited geographically to Peru chronologically to the educational reform of 1968 and 1980. After the historical background was established, emphasis was placed on the Peruvian educational reform as a model of innovation as compared with the Seventh-day Adventist model at Inca Union College. Method. This study utilized the historical method of research. Major sources included documents regarding the history and educational philosophy of Peruvian educational reform and Inca Union College. Minutes of the institution, periodicals, and other primary sources were used. Conclusions. Both models had similar outward appearances, especially since they promulgated the need for a holistic education which assumes that people need formation in physical, intellectual, spiritual, vocational, and social aspects; nevertheless, the study of their philosophical foundations demonstrates different meanings for their programs and activities. In conclusions, it may be stated that: (1) The Peruvian educational reform identifies itself with humanism and is anthropocentric, while the Seventh-day Adventist system classifies itself as theocentric. From this observation derive the other conclusions in the various philosophical categories. (2) While the Peruvian system views social change as its ultimate goal, Seventh-day Adventist education seeks man\u27s redemption in both the present and eschatological dimensions. (3) The Peruvian system accepts conscientization as an epistemological means which stimulates creative and critical thinking about social reality.Seventh-day Adventist education amplifies social reality to include the relationship with the rest of humanity and with God. (4) The Peruvian reform recognizes education for work as the source of personal and societal well-being. Adventist education recognizes the importance of societal well-being; in addition to this, it presents work as a means of restoring God\u27s image in man. (5) Both systems promulgate the need for a holistic education but with different meanings. (6) Under the educational reform, religious education received unprecedented support and freedom through participation of all religious confessions in the National Religious Education Council

    Existence of Dλ-cycles and Dλ-paths

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    A cycle of C of a graph G is called a Dλ-cycle if every component of G − V(C) has order less than λ. A Dλ-path is defined analogously. In particular, a D1-cycle is a hamiltonian cycle and a D1-path is a hamiltonian path. Necessary conditions and sufficient conditions are derived for graphs to have a Dλ-cycle or Dλ-path. The results are generalizations of theorems in hamiltonian graph theory. Extensions of notions such as vertex degree and adjacency of vertices to subgraphs of order greater than 1 arise in a natural way
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