Counting Fixed Points and Rooted Closed Walks of the Singular Map $$x mapsto {x^{{x^n}}}$$x↦xxn Modulo Powers of a Prime

The “self-power” map x ⟼ xx modulo m and its generalized form $$x mapsto {x^{{x^n}}}$$ modulo m are of considerable interest for both theoretical reasons and for potential applications to cryptography. In this paper, we use p-adic methods, primarily p-adic interpolation, Hensel’s lemma, and lifting singular points modulo p, to count fixed points and rooted closed walks of equations related to these maps when m is a prime power. In particular, we introduce a new technique for lifting singular solutions of several congruences in several unknowns using the left kernel of the Jacobian matrix.     

Mathematical literacy in undergraduates: role of gender,emotional intelligence and emotional self-efficacy

This empirical study explores the roles that Emotional Intelligence (EI) and Emotional Self-Efficacy (ESE) play in undergraduates’ mathematical literacy, and the influence of EI and ESE on students’ attitudes towards and beliefs about mathematics. A convenience sample of 93 female and 82 male first-year undergraduates completed a test of mathematical literacy, followed by an online survey designed to measure the students’ EI, ESE and factors associated with mathematical literacy. Analysis of the data revealed significant gender differences. Males attained a higher mean test score than females and out-performed the females on most of the individual questions and the associated mathematical tasks. Overall, males expressed greater confidence in their mathematical skills, although both males’ and females’ confidence outweighed their actual mathematical proficiency. Correlation analyses revealed that males and females attaining higher mathematical literacy test scores were more confident and persistent, exhibited lower levels of mathematics anxiety and possessed higher mathematics qualifications. Correlation analyses also revealed that in male students, aspects of ESE were associated with beliefs concerning the learning of mathematics (i.e. that intelligence is malleable and that persistence can facilitate success), but not with confidence or actual performance. Both EI and ESE play a greater role with regard to test performance and attitudes/beliefs regarding mathematics amongst female undergraduates; higher EI and ESE scores were associated with higher test scores, while females exhibiting higher levels of ESE were also more confident and less anxious about mathematics, believed intelligence to be malleable, were more persistent and were learning goal oriented. Moderated regression analyses confirmed mathematics anxiety as a negative predictor of test performance in males and females, but also revealed that in females EI and ESE moderate the effects of anxiety on test performance, with the relationship between anxiety and test performance linked more to emotional management (EI) than to ESE.     

Elementary teachers’ mathematical beliefs and mathematics anxiety: How do they shape instructional practices?

This quantitative study investigated the relationships among practicing elementary teachers’ (N = 153) beliefs about mathematics and its teaching and learning, mathematics anxiety, and instructional practices in mathematics. When viewed singly, the findings reveal the teachers with higher levels of mathematics anxiety tend to use less standards‐based instruction and those with beliefs oriented toward a problem‐solving view of mathematics reported more standards‐based teaching. A combined analysis shows that after controlling for mathematical beliefs, teaching longevity, and educational degree attainment, there is no relationship between teachers’ mathematics anxiety and instructional practices. These findings suggest a spurious relationship between anxiety and practices, with beliefs having the strongest relationship with practices. Several suggestions for positively influencing the mathematical beliefs and affect in general of elementary teachers while learning mathematics are offered.     

Thin sequences and the Gram matrix

We provide a new proof of Volberg’s Theorem characterizing thin interpolating sequences as those for which the Gram matrix associated to the normalized reproducing kernels is a compact perturbation of the identity. In the same paper, Volberg characterized sequences for which the Gram matrix is a compact perturbation of a unitary as well as those for which the Gram matrix is a Schatten-2 class perturbation of a unitary operator. We extend this characterization from 2 to p, where 2 ≤ p ≤ ∞.     

Solving binary cutting stock problems by column generation and branch-and-bound

We present an algorithm for the binary cutting stock problem that employs both column generation and branch-and-bound to obtain optimal integer solutions. We formulate a branching rule that can be incorporated into the subproblem to allow column generation at any node in the branch-and-bound tree. Implementation details and computational experience are discussed.This research was supported by NSF and AFOSR grant DDM-9115768     

Structure elucidation with lanthanide-induced shifts 5. Evaluation of the binding ability of various functional groups

The equilibrium binding constants have been evaluated for the association in CCl₄ of the lanthanide shift reagent. Eu(fod)₃, with a series of adamantane derivatives containing a variety of different functional groups. It is shown that if steric effects are held constant the binding abilities of the adamantane derivatives exhibit a good correlation with the proton affinities of the methyl analogs containing the same functional groups. The results permit the prediction of binding strength of other functional groups with Eu(dpm)₃ as well as with Eu(fod)₃ and thus can be of great assistance in planning and interpreting the results of experiments with lanthanide shift reagents.Part 4:D. J. Raber, M. D. Johnston, jr., C. M. Campbell, C. M. Janks, andP. Sutton, Org. Mag. Res.11, 323 (1978).     

Synthesis and structure of ditetrabutylammonium tetrachlorodioxoneptunium(VI)

The reaction of a stock solution of NpO₂² + in 2 M HCl with two equivalents of tetrabutylammonium chloride in 6 M HCl results in the preparation of the title compound. This compound, [NBu₄]₂[NpO₂Cl₄], is isostructural with the uranium analogue, [NBu₄]₂[UO₂Cl₄]. The anion of the complex, [NpO₂Cl₄]²–, adopts a pseudooctahedral geometry common to six-coordinate actinyl complexes in which the neptunium metal is coordinated by two trans axial oxo groups with NpO(oxo) bond lengths of 1.733(5)Å, and four equatorial chloride ligands with Np–Cl bond lengths in the range 2.637(2)–2.676(2)Å. The neptunyl ONpO bond angle is 178.6(3)°. Charge balance of the neptunyl tetrachloride anion is maintained by two tetrabutylammonium cations. This complex crystallizes in the monoclinic space group P2₁/n (a = 15.416(4), b = 15.362(4), c = 18.479(5)Å, = 108.147(6)°, V = 4158.4(19)ų, Z = 4).