E-polynomial of the SL(3,C)-character variety of free groups.

We compute the E-polynomial of the character variety of representations of a rank r free group in SL(3,C). Expanding upon techniques of Logares, Muñoz and Newstead (Rev. Mat. Complut. 26:2 (2013), 635-703), we stratify the space of representations and compute the E-polynomial of each geometrically described stratum using fibrations. Consequently, we also determine the E-polynomial of its smooth, singular, and abelian loci and the corresponding Euler characteristic in each case. Along the way, we give a new proof of results of Cavazos and Lawton (Int. J. Math. 25:6 (2014), 1450058).

Coordination of aproximations in secondary school students’ understanding of limit concept

The aim of this research is to characterize the coordination of the processes of approximation related to the understanding of the limit of a function. We analyze the answers of 64 post-secondary school students to 7 problems considering the dynamic and metric conception of limit of a function. Results indicate that the metric understanding of the limit in terms of inequality supports that the student is capable of coordinating the approximations in the domain and in the range when lateral approximations coincide. However, the student is not capable of this coordination when lateral approximations do not coincide. This indicates that the metric understanding of the limit begins with the previous construction of the dynamic conception in case of coincidence of the lateral approximations in the range.

Mathematics Game e-Library for Elementary School, study case: Mexico

One of the most relevant subjects for the intellectual formation of elementary school students is Mathematics where its importance goes back to ancient civilizations and which its importance is underestimated nowadays. This phenomenon occurs in Mexico, where 63.1% of the total population of elementary school students between the third and sixth grade have insufficient/elemental level of mathematics knowledge. This has resulted in the need to use a new mechanism to complement student’s classroom learning. With the rapid growth of wireless and mobile technologies, the mobile learning has been gradually considered as a novel and effective form of learning due to it inherits all the advantages of e-learning as well as breaks the limitations of learning time and space occurring in the traditional classroom teaching. This project proposes the use of a Mathematics Game e-Library integrated by a set of games for mobile devices and a distribution/management tool. The games are developed for running on mobile devices and for cover the six competencies related with the mathematics learning approach established in Mexico. The distribution/management tool allows students to reach contents according to their needs; this is achieved through a core engine that infers, from an initial profile, the games that cover the user’s knowledge gaps.

3D visual sensing of human hand for remote operation of a robotic hand

New low cost sensors and open free libraries for 3D image processing are making important advances in robot vision applications possible, such as three-dimensional object recognition, semantic mapping, navigation and localization of robots, human detection and/or gesture recognition for human-machine interaction. In this paper, a novel method for recognizing and tracking the fingers of a human hand is presented. This method is based on point clouds from range images captured by a RGBD sensor. It works in real time and it does not require visual marks, camera calibration or previous knowledge of the environment. Moreover, it works successfully even when multiple objects appear in the scene or when the ambient light is changed. Furthermore, this method was designed to develop a human interface to control domestic or industrial devices, remotely. In this paper, the method was tested by operating a robotic hand. Firstly, the human hand was recognized and the fingers were detected. Secondly, the movement of the fingers was analysed and mapped to be imitated by a robotic hand.

Primary school teacher’s noticing of students’ mathematical thinking in problem solving

Professional noticing of students’ mathematical thinking in problem solving involves the identification of noteworthy mathematical ideas of students’ mathematical thinking and its interpretation to make decisions in the teaching of mathematics. The goal of this study is to begin to characterize pre-service primary school teachers’ noticing of students’ mathematical thinking when students solve tasks that involve proportional and non-proportional reasoning. From the analysis of how pre-service primary school teachers notice students’ mathematical thinking, we have identified an initial framework with four levels of development. This framework indicates a possible trajectory in the development of primary teachers’ professional noticing.

The Zeros of Riemann Zeta Partial Sums Yield Solutions to f(x) + f(2x) + · · · + f(nx) = 0

This paper proves that every zero of any n th , n ≥ 2, partial sum of the Riemann zeta function provides a vector space of basic solutions of the functional equation f(x)+f(2x)+⋯+f(nx)=0,x∈R . The continuity of the solutions depends on the sign of the real part of each zero.

On the Analytic Solutions of the Functional Equations w 1 f(a 1 z) + w 2 f(a 2 z) + ... + w n f(a n z) = 0

In this paper, it is showed that, given an integer number n ≥ 2, each zero of an exponential polynomial of the form w1az1+w2az2+⋯+wnazn, with non-null complex numbers w 1,w 2,…,w n and a 1,a 2,…,a n , produces analytic solutions of the functional equation w 1 f(a 1 z) + w 2 f(a 2 z) + ... + w n f(a n z) = 0 on certain domains of C, which represents an extension of some existing results in the literature on this functional equation for the case of positive coefficients a j and w j.

Deposition of Andrew Eliot against Nathan Prince, Nov. 26, 1741

In this deposition, Eliot describes Prince's anger at John Winthrop's selection as Hollis Professor of Mathematics and Natural Philosophy, which he believed was done "to vex and torture" him. Eliot claims that Prince said: "they have chosen that Boy Winthrop professor, I could teach him his A. B. C. in the Mathematicks, they want to get me away from College."

Rules and orders relating to a Professor of the Mathematicks, of Natural and Experimental Philosophy, in Harvard College, 18 January 1726

This folder contains a single document describing the "rules and orders" of the Hollis Professor of Mathematics and Natural Philosophy. The document begins by defining the subjects to be taught by the Hollis Professor including natural and experimental philosophy, elements of geometry, and the principles of astronomy and geography. It then outlines the number of public and private lectures to be given to students, how much extra time the professor should spend with students reviewing any difficulties they may encounter understanding class subject matter discussed, and stipulates that the professor's duties shall be restricted solely to his teaching activities and not involve him in any religious activities at the College or oblige him to teach any additional studies other than those specified for the Hollis Professor of Mathematics and Natural Philosophy. Furthermore, the rules establish the professor's salary at £80 per year and allow the professor to receive from students, except those students studying theology under the Hollis Professor of Divinity, an additional fee as determined by the Corporation and Board of Overseers, to supplement his income. Moreover, the rules assert that all professorship candidates selected by the Harvard Corporation must be approved by Thomas Hollis during his lifetime or by his executor after his death. Finally, the rules state that the Hollis professor take an oath to the civil government and declare himself a member of the Protestant reformed religion. This document is signed by Thomas Hollis and four witnesses, John Hollis, Joshua Hollis, Richard Solly, and John Williams.

The summary of a course of experimental lectures in natural philosophy by Dr. John Winthrop (manuscript copy), 1772-1774

Notebook with a handwritten copy of lecture summaries for a physics course given by Harvard Professor John Winthrop. The notes were made by Timothy Foster in 1772 and 1773. The volume contains twenty-five lectures with some diagrams.

Student lecture notes of Benjamin Peirce, 1797-1798

This collection contains approximately twenty-three handwritten lecture summaries on six leaves made by Harvard undergraduate Benjamin Peirce between September 1797 and November 22, 1798. The summaries generally provide a few sentences describing the topic covered and primarily pertain to lectures on English grammar delivered by Eliphalet Pearson, the Hollis Professor of Hebrew and other Oriental Languages. There are also summaries for single lectures by David Tappan, the Hollis Professor Divinity; Samuel Webber, the Hollis Professor of Mathematics and Natural Philosophy; and John Snelling Popkin, the Greek Tutor from 1795 to 1798, and later the Eliot Professor of Greek Literature. There is also an undated summary of a lecture by Benjamin Waterhouse, the Hersey Professor of Theory and Practice of Physic.

Rubbing of the gravestone of Harvard librarian Nathaniel Ward made by David S. Ferriero, October 15, 1972

Large rubbing of the gravestone of Nathaniel Ward, librarian of Harvard college for one week in 1768. The rubbing was made by David S. Ferriero, and is signed and dated October 15, 1972 in the lower right corner.

Macerals in surface sediments of the Laptev Sea (Arctic Ocean)

Surface sediments from the Laptev Sea and adjacent continental slope were studied for their composition of particulate organic matter (OM) by means of maceral analysis. The composition of macerals in sediments gives information about the environment, terrigenous supply from the hinterland, and marine OM. With reference to their biological sources, we distinguish between terrigenous and marine macerals. We found that the particulate OM in the surface sediments of the Laptev Sea is predominantly of terrigenous origin (mean: 78%). However, distinct variations exist when looking in detail. In the shelf area, sediments may contain up to 99% terrigenous OM. Freshwater algae occur directly north of the river mouths, reflecting the strong fluvial influence. Relatively high amounts of marine OM (20-40%) are restricted to the upper continental slope, the Vilkitsky Strait and west of the New Siberian Islands, explained by increased surface-water productivity due to increased fluvial nutrient supply, open-water conditions, and phytoplankton blooms at the ice-edge.