Mathematics Conference Papers, Posters and Presentations #noDrivingDirections 0 Images from series: Mathematics Conference Papers, Posters and Presentations relativeToGround -77.005923 38.889779 9000000 Sharp bounds for decomposing graphs into edges and triangles <p>Let pi3(G) be the minimum of twice the number of edges plus three times the number of triangles over all edge-decompositions of G into copies of K2 and K3. We are interested in the value of pi3(n), the maximum of pi3(G) over graphs G with n vertices. This specific extremal function was first studied by Gyori and Tuza [Decompositions of graphs into complete subgraphs of given order, Studia Sci. Math. Hungar. 22 (1987), 315--320], who showed that pi3(n)<9n2/16.<br />In a recent advance on this problem, Kral, Lidicky, Martins and Pehova [arXiv:1710:08486] proved via flag algebras that pi3(n)<(1/2+o(1))n2, which is tight up to the o(1) term.<br />We extend their proof by giving the exact value of pi3(n) for large n, and we show that Kn and Kn/2,n/2 are the only extremal examples.</p> ]]>
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Developing Criteria to Design and Assess Mathematical Modeling Problems: From Problems to Social Justice <p>Despite the interest in modeling and the importance of social justice, there has not been much attention to connecting modeling with social justice. To fill this gap, we developed criteria for mathematical modeling problems that embrace the characteristics of problems and social justice through three phases: literature analysis, thematic categories, and piloting. The criteria will help teacher educators when selecting modeling problems to be used in teacher preparation programs and assessing the modeling problems posed by PSTs.</p> ]]>
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How Much Do I Know About Mathematical Modeling? <p>Despite the importance of teachers’ conception of mathematical modeling, limited attention is given to this area in the current literature. In this study we examined 78 preservice teachers’ (PSTs) views of mathematical modeling and how their conceptions are reflected in their performance of mathematical modeling problems. Analyses of survey responses revealed that our PSTs seem to develop narrow views of mathematical modeling. In addition, although a large portion of PSTs mistook mathematical modeling with mathematical models or with traditional word problems, we found a positive association between PSTs’ conceptions of mathematical modeling and their mathematical modeling abilities.</p> ]]>
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Independent Sets near the Lower Bound in Bounded Degree Graphs <p>By Brook’s Theorem, every n-vertex graph of maximum degree at most ∆≥3 and clique number at most ∆ is ∆-colorable, and thus it has an independent set of size at least n/∆. We give an approximate characterization of graphs with independence number close to this bound, and use it to show that the problem of deciding whether such a graph has an independent set of size at least n/∆ +k has a kernel of size O(k).</p> ]]>
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Preservice Teacher Learning to Help English Language Learners Make Sense of Mathematics <p>This study investigates how preservice teachers (PST) help English language learners (ELLs) understand cognitive demanding mathematical problems using complicated language use. Three mathematics PSTs worked with ELLs in one-on-one settings while receiving individual interventions. The strategies they implemented were analyzed based on four categories: mathematical content, culture/life experience, mathematical/cognitive process, and mathematical/contextual language. As time evolved, all of the PSTs began to integrate life connection strategies and various visuals that are closely related to mathematical situations, which they learned during the interventions. This study suggests that PSTs require significant preparation infused with practical experiences and examples in order to design a linguistically and conceptually rich lesson.</p> ]]>
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The Inverse Eigenvalue Problem of a Graph <p>Inverse eigenvalue problems appear in various contexts throughout mathematics and engineering, and refer to determining all possible lists of eigenvalues (spectra) for matrices fitting some description. The inverse eigenvalue problem of a graph refers to determining the possible spectra of real symmetric matrices whose pattern of nonzero off-diagonal entries is described by the edges of a given graph (precise definitions of this and other terms are given in the next paragraph). This problem and related variants have been of interest for many years and were originally approached through the study of ordered multiplicity lists.</p> ]]>
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A Note on the Computable Categoricity of l(p) Spaces <p>Suppose that p is a computable real and that p >= 1. We show that in both the real and complex case, l(p) is computably categorical if and only if p = 2. The proof uses Lamperti's characterization of the isometries of Lebesgue spaces of sigma-finite measure spaces.</p> ]]>
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Eventually Nonnegative Matrices and their Sign Patterns <p>A matrix A ∈ R n×n is eventually nonnegative (respectively, eventually positive) if there exists a positive integer k0 such that for all k ≥ k0, Ak ≥ 0 (respectively, Ak > 0). Here inequalities are entrywise and all matrices are real and square. An eigenvalue of A is dominant if its magnitude is equal to the spectral radius of A. A matrix A has the strong Perron-Frobenius property if A has a unique dominant eigenvalue that is positive, simple, and has a positive eigenvector. It is well known (see, e.g., [10]) that the set of matrices for which both A and AT have the strong Perron-Frobenius property coincides with the set of eventually positive matrices. Eventually nonnegative matrices and eventually positive matrices have applications to positive control theory (see, e.g., [13]).</p> ]]>
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Relationships between the Completion Problems for Various Classes of Matrices ]]>
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