A combinatorial proof of the Dense Hindman’s Theorem

The Dense Hindman’s Theorem states that, in any finite coloring of the natural numbers, one may find a single color and a “dense” set B₁, for each b₁∈B₁ a “dense” set (depending on b₁), for each a “dense” set (depending on b₁,b₂), and so on, such that for any such sequence of b_i, all finite sums belong to the chosen color. (Here density is often taken to be “piecewise syndetic”, but the proof is unchanged for any notion of density satisfying certain properties.) This theorem is an example of a combinatorial statement for which the only known proof requires the use of ultrafilters or a similar infinitary formalism. Here we give a direct combinatorial proof of the theorem.     

Student Quantifications as Meanings for Quantified Variables in Complex Mathematical Statements

This study investigates undergraduate students’ meanings for quantified variables in mathematical statements involving multiple quantifiers. Clinical interviews with nine undergraduate students were conducted to explore students’ meanings for quantified variables. More specifically, students were asked to interpret and evaluate the Intermediate Value Theorem (IVT) and three other statements with similar logical structure to the IVT. In this paper, we provide our definitions for a quantified variable and student quantification as theoretical constructs from a constructivist viewpoint. Using these theoretical constructs, we interpret our main results, five categories of students’ meanings for quantified variables, which emerged from our qualitative analysis. We also discuss how our findings of students’ various meanings for quantified variables contribute to literature on theories of meaning and address issues in curriculum and instruction.     

M.Riesz定理的注记

设u(z)是单位圆内的实值调和函数 ,若 p_平均Mp(r ,u) =12π∫2π0｜u(reiθ) ｜pdθ1 p <∞ ,则称u(z) ∈hp( 1

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Some multivariate inequalities with applications

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