Young children’s multimodal mathematical explanations

This paper investigates how three children provided mathematical explanations whilst playing with a set of glass jars in a Swedish preschool. Using the idea of semiotic bundles combined with the work on multimodal interactions, the different semiotic resources used individually and in combinations by the children are described. Given that the children were developing their verbal fluency, it was not surprising to find that they also included physical arrangements of the jars and actions to support their explanations. Hence, to produce their explanations of different attributes such as thin and sameness, the children drew on each other’s gestures and actions with the jars. This research has implications for how the relationship between verbal language and gestures can be viewed in regard to young children’s explanations.     

Tropical decomposition of Young’s partition lattice

Young’s partition lattice L(m,n) consists of integer partitions having m parts where each part is at most n. Using methods from complex algebraic geometry, R. Stanley proved that this poset is rank-symmetric, unimodal, and strongly Sperner. Moreover, he conjectured that it has a symmetric chain decomposition, which is a stronger property. Despite many efforts, this conjecture has only been proved for min(m,n)≤4. In this paper, we decompose L(m,n) into level sets for certain tropical polynomials derived from the secant varieties of the rational normal curve in projective space, and we find that the resulting subposets have an elementary raising and lowering algorithm. As a corollary, we obtain a symmetric chain decomposition for the subposet of L(m,n) consisting of “sufficiently generic” partitions.     

Refined Young inequalities with Specht’s ratio

In this paper, we show that the ν-weighted arithmetic mean is greater than the product of the ν-weighted geometric mean and Specht’s ratio. As a corollary, we also show that the ν-weighted geometric mean is greater than the product of the ν-weighted harmonic mean and Specht’s ratio. These results give the improvements for the classical Young inequalities, since Specht’s ratio is generally greater than 1. In addition, we give an operator inequality for positive operators, applying our refined Young inequality.     

Monotonic core solutions: beyond Young’s theorem

Young’s theorem implies that every core concept violates monotonicity. In this paper, we investigate when such a violation of monotonicity by a given core concept is justified. We introduce a new monotonicity property for core concepts. We pose several open questions for this new property. The open questions arise because the most important core concepts (the nucleolus and the per capita nucleolus) do not satisfy the property even in the class of convex games.     

Young’s Seminormal Form and Simple Modules for S n in Characteristic p

We realize the integral Specht modules for the symmetric group S _n as induced modules from the subalgebra of the group algebra generated by the Jucys–Murphy elements. We deduce from this that the simple modules for ${{\mathbb F}_p} S_n $ are generated by reductions modulo p of the corresponding Jucys–Murphy idempotents.     

Order Ideals in Weak Subposets of Young’s Lattice and Associated Unimodality Conjectures

The k-Young lattice Y^k is a weak subposet of the Young lattice containing partitions whose first part is bounded by an integer k > 0. The Y^k poset was introduced in connection with generalized Schur functions and later shown to be isomorphic to the weak order on the quotient of the affine symmetric group S_{k + 1} by a maximal parabolic subgroup. We prove a number of properties for Y^k including that the covering relation is preserved when elements are translated by rectangular partitions with hook-length k. We highlight the order ideal generated by an m x n rectangular shape. This order ideal, L^k(m, n), reduces to L(m, n) for large k, and we prove it is isomorphic to the induced subposet of L(m, n) whose vertex set is restricted to elements with no more than k - m + 1 parts smaller than m. We provide explicit formulas for the number of elements and the rank-generating function of L^k(m, n). We conclude with unimodality conjectures involving q-binomial coefficients and discuss how implications connect to recent work on sieved q-binomial coefficients.AMS Subject Classification: 06A06, 05A17, 05A10, 05E05.     

Periods and Young’s diagram of Fermat–Euler’s geometrical progressions of residues

The Gibbs phenomenon is described for the Fourier series of a function at its jump, the function being defined along the finite circle ℤ/pℤ.     

Children’s unit concepts in measurement: a teaching experiment spanning grades 2 through 5

We examined ways of improving students’ unit concepts across spatial measurement situations. We report data from our teaching experiment during a six-semester longitudinal study from grade 2 through grade 5. Data include instructional task sequences designed to help children (a) integrate multiple representations of unit, (b) coordinate and group units into higher-order units, and (c) recognize the arbitrary nature of unit in comparison contexts and student’s responses to tasks. Our results suggest reflection on multiplicative relations among quantities prompted a more fully-developed unit concept. This research extends prior work addressing the growth of unit concepts in the contexts of length, area, and volume by demonstrating the viability of level-specific instructional actions as a means for promoting an informal theory of measurement.     

Goldbach’s conjecture in max-algebra

The Goldbach conjecture is one of the best known open problems in number theory. It claims that every even integer greater than 2 can be written as the sum of two primes. The present paper formulates a max-algebraic claim that is equivalent to Goldbach’s conjecture. The max-algebraic analogue allows examination of the conjecture by the methods of max-algebra. A max-algebra is an algebraic structure in which classical addition \(+\) and multiplication \(\times \) are replaced by the operations maximum \(\oplus \) and addition \(\otimes \), in other words \(a\oplus b=\max \{a,b\}\) and \(a\otimes b=a+b\).     

Generalizations of Young’s Theorem to real functions of several variables

In 1907 W. H. Young classified the real-valued Baire one functions on the line which have the Darboux (intermediate-value) property as those which are bilaterally approachable. Here we investigate generalizations of this theorem to the setting of real-valued Baire one functions of several variables which possess various “Darboux-like” properties.