Experienced provers’ uses of contexts while engaging in combinatorial proof of binomial identities

We provide a combinatorial proof of a symplectic character identity relating the sum of a product of symplectic Schur functions to the product . This formula owes its origin to the existence of a dual pair of symplectic groups acting on spinors, as pointed out by Hasegawa. The first combinatorial proof, based on symplectic tableaux and a variation of the Robinson–Schensted–Knuth correspondence, was due to Terada. Here we use Schützenberger’s jeu de taquin, augmented by two simple zero weight transformations. The identity itself generalizes a well-known identity expressing as a sum of products of Schur functions that was due to Littlewood and proved combinatorially by Remmel. We offer an alternative combinatorial proof of this identity by means of the jeu de taquin, as a precursor to the proof of the symplectic identity.

A combinatorial proof of the equivalence of the classical and combinatorial definitions of Schur function

Combinatorial proof is an important topic both for combinatorics education and proof education researchers, but relatively little has been studied about the teaching and learning of combinatorial proof. In this paper, we focus on one specific phenomenon that emerged during interviews with mathematicians and students who were experienced provers as they discussed and engaged in combinatorial proof. In particular, participants used a wide variety of cognitive models to interpret multiplication by a constant when reasoning about binomial identities, some of which seemed to be more (or less) effective in helping produce a combinatorial proof. We present these cognitive models and describe episodes that illustrate implications of these cognitive models for our participants’ work on proving binomial identities. Our findings both inform research on combinatorial proof and highlight the importance of understanding subtleties of the familiar operation of multiplication.

Catalysis and catalytic cycles are widely used in both research and industry. The addition of a catalysis experiment into the undergraduate laboratory curriculum is important and necessary training to expand student learning. Here, an experiment demonstrating iron-based tandem catalysis was developed and implemented in an organic chemistry course for second-year and upper-division undergraduate students at Mississippi State University. The experiment uses TLC and 1H NMR to track the reaction and identify the core pharmaceutical structure produced during the experiment. Reactions contained inside the catalytic cycles include iminium generation, imino-ene condensation, and aza-Prins reaction. The catalytic cycle approach allows for a decrease in time for reaction, chemicals, and waste produced. This experiment was successfully tested over two semesters (110 students) at a major university. Learning objectives were met to support student understanding of catalytic cycles and reaction mechanisms. This laboratory experiment provided hands-on experience using TLC and 1H NMR to identify a core structure of pharmaceutical compounds produced from petroleum.

Investigating undergraduate students’ proof schemes and perspectives about combinatorial proof

Some Observations on Dyson's New Symmetries of Partitions

Increasingly, genocide scholars question whether the Rwandan catastrophe would have occurred were it not for policies shaped in Paris. From the beginning of the slaughter, France was proactive in supporting the regime that promoted Hutu superiority and opposing the Rwandan Patriotic Front (RPF) that finally did end the genocide. Unlike the US (see Jared Cohen's One Hundred Days of Silence, CH, Aug'07, 44-7088), France did send troops into the country with its Operation Turquoise in June 1994. But they chose to conceptualize the conflict as a civil war, remaining neutral, rather than prioritize ending the genocide. They failed to save many Tutsis who were defenseless. French troops even refused to arrest or detain the genocidaires and facilitated their escape. Many argue that the subsequent bloodbath in the Congo-including atrocities on the part of the RPF-would have been less likely to occur were it not for those policies. Most of this book is based on Kroslak's 2002 dissertation. Her criteria for judging France's responsibility were knowledge, involvement, and capacity. Altogether, her verdict is reasonable. This book significantly contributes to literature that documents duplicitous and disreputable policies still denied in Paris. Summing Up: Essential. General readers, upper-division undergraduate students, graduate students, and research faculty.

Combinatorial proofs of identities of Calkin and Hirschhorn

We give combinatorial proofs of q-Stirling identities using restricted growth words. This includes a poset theoretic proof of Carlitz's identity, a new proof of the q-Frobenius identity of Garsia and Remmel and of Ehrenborg's Hankel q-Stirling determinantal identity. We also develop a two parameter generalization to unify identities of Mercier and include a symmetric function version.

Ramanujan's lost notebook: Combinatorial proofs of identities associated with Heine's transformation or partial theta functions

Recently, Andrews and Yee studied two-variable generalizations of two identities involving partition functions $p_\omega(n)$ and $p_\nu(n)$ introduced by Andrews, Dixit and Yee. In this paper, we present a combinatorial proof of an interesting identity in their work.

Using sequences of finite length with positive integer entries and the inversion statistic on such sequences, a collection of binomial and multinomial identities are extended to their q-analog form via combinatorial proofs. Using the major index statistic on sequences, a connection between finite differences of the coefficients of generalized Galois numbers and integer partitions with kinds is established.

In this article, we present data from 2,151 upper division undergraduate students regarding the frequency of cheating, reasons for cheating, and influence of penalties on cheating. We also discuss how a model that develops an internalized code of ethics will counteract academic dishonesty.

In electrical and computer engineering, communications is an interesting but challenging area of study for undergraduate students. To improve the quality of learning and understanding of the basic principles of communications, we have been developing a proof-of-concept prototype of a 2.4 GHz communication system as a framework of laboratory experiments suitable for undergraduate students. The framework primarily consists of a small mesh network of mobile robots realized in hardware and software and analog beamforming realized with off-the-shelf hardware components. In this Work-in-Progress paper, we describe our prototype under development and highlight key functionalities of the prototype that will eventually appear as specific contents of laboratory experiments in the near future. From the progress so far, we have observed that development of our proof-of-concept communication system, initially for research in indoor positioning technologies, is suitable for laboratory experiment for a semester-long communications course for upper-division undergraduate students.

The Internet today lacks an identity protocol for identifying people and organizations. As a result, service providers needed to build and maintain their own databases of user information. This solution is costly to the service providers, inefficient as much of the information is duplicated across different providers, difficult to secure as evidenced by recent large-scale personal data breaches around the world, and cumbersome to the users who need to remember different sets of credentials for different services. Furthermore, personal information could be collected for data mining, profiling and exploitation without users' knowledge or consent. The ideal solution would be self-sovereign identity, a new form of identity management that is owned and controlled entirely by each individual user. This solution would include the individual's consolidated digital identity as well as their set of verified attributes that have been cryptographically signed by various trusted issuers. The individual provides proof of identity and membership by sharing relevant parts of their identity with the service providers. Consent for access may also be revoked hence giving the individual full control over its own data. This survey critically investigates different blockchain based identity management and authentication frameworks. A summary of the state-of-the-art blockchain based identity management and authentication solutions from year 2014 to 2018 is presented. The paper concludes with the open issues, main challenges and directions highlighted for future work in this area. In a nutshell, the discovery of this new mechanism disrupted the existing identity management and authentication solutions and by providing a more promising secure platform.