Relations between Generalization, Reasoning and Combinatorial Thinking in Solving Mathematical Open-Ended Problems within Mathematical Contest

Algebraic thinking, combinatorial thinking and reasoning skills are considered as playing central roles within teaching and learning in the field of mathematics, particularly in solving complex open-ended mathematical problems Specific relations between these three abilities, manifested in the solving of an open-ended ill-structured problem aimed at mathematical modeling, were investigated. We analyzed solutions received from 33 groups totaling 131 students, who solved a complex assignment within the mathematical contest Mathematics B-day 2018. Such relations were more obvious when solving a complex problem, compared to more structured closed subtasks. Algebraic generalization is an important prerequisite to prove mathematically and to solve combinatorial problem at higher levels, i.e., using expressions and formulas, therefore a special focus should be put on this ability in upper-secondary mathematics education.

Exact solution of a cluster model with next-nearest-neighbor interaction

A 1D cluster model with next-nearest-neighbor interactions and two additional composite interactions is solved; the free energy is obtained and a correlation function is derived exactly. The model is diagonalized by a transformation obtained automatically from its interactions, which is an algebraic generalization of the Jordan–Wigner transformation. The gapless condition is expressed as a condition on the roots of a cubic equation, and the phase diagram is obtained exactly. We find that the distribution of roots for this algebraic equation determines the existence of long-range order, and we again obtain the ground-state phase diagram. We also derive the central charges of the corresponding conformal field theory. Finally, we note that our results are universally valid for an infinite number of solvable spin chains whose interactions obey the same algebraic relations.

A generalization of CHSH and the algebraic structure of optimal strategies

Self-testing has been a rich area of study in quantum information theory. It allows an experimenter to interact classically with a black box quantum system and to test that a specific entangled state was present and a specific set of measurements were performed. Recently, self-testing has been central to high-profile results in complexity theory as seen in the work on entangled games PCP of Natarajan and Vidick \cite{low-degree}, iterated compression by Fitzsimons et al. \cite{iterated-compression}, and NEEXP in MIP* due to Natarajan and Wright \cite{neexp}. The most studied self-test is the CHSH game which features a bipartite system with two isolated devices. This game certifies the presence of a single EPR entangled state and the use of anti-commuting Pauli measurements. Most of the self-testing literature has focused on extending these results to self-test for tensor products of EPR states and tensor products of Pauli measurements.In this work, we introduce an algebraic generalization of CHSH by viewing it as a linear constraint system (LCS) game, exhibiting self-testing properties that are qualitatively different. These provide the first example of LCS games that self-test non-Pauli operators resolving an open questions posed by Coladangelo and Stark \cite{RobustRigidityLCS}. Our games also provide a self-test for states other than the maximally entangled state, and hence resolves the open question posed by Cleve and Mittal \cite{BCSTensor}. Additionally, our games have 1 bit question and log⁡n bit answer lengths making them suitable candidates for complexity theoretic application. This work is the first step towards a general theory of self-testing arbitrary groups. In order to obtain our results, we exploit connections between sum of squares proofs, non-commutative ring theory, and the Gowers-Hatami theorem from approximate representation theory. A crucial part of our analysis is to introduce a sum of squares framework that generalizes the solution group of Cleve, Liu, and Slofstra \cite{BCSCommuting} to the non-pseudo-telepathic regime. Finally, we give the first example of a game that is not a self-test. Our results suggest a richer landscape of self-testing phenomena than previously considered.

The effectiveness of Using Flipped Classroom Strategy in Developing Algebraic Thinking Skills of Secondary Stage Female Students

The aim of the study was to identify The effectiveness of the use of The Flipped Classroom Strategy in The Development of The Algebraic Thinking Skills of high School students and to achieve the objective of the study, the symbol of the study consisted of (56)female Secondary school students in the Secondary stage randomly divided into two groups one which consisted of (29) female students who study the Unit of "Sequences and Series" using Flipped Classroom, in addition the other students were taught the same topics in the usual way from (27) female students. In addition, the measurement test was conducted in the form of the algebraic thinking skill and the preparation of the teacher's guide in the light of the flipping the classroom using "Articulate Storylines (3)", the use of the Educational Platform "Edmodo", and the preparation guide of the activities for the students . The data were processed statistically. The results indicated a statistically Significant differences (0.05) Between that mean scores of the experimental and control groups in the skill of exploring relationships and algebraic functions in the favor of the group’s experimental students attributed To the use of the flipped classroom strategy, and no statistically significant differences (0.50) Between that mean scores of the experimental and control groups in the skill of (patterns and algebraic generalization, use representations and algebraic symbols), and In general there was no statistically significant differences between (0.05) between the average scores of the female students in the experimental and control groups in the post- application to test algebraic thinking skills in the total score. Based on the Findings a number of recommendations and proposals were presented that could contribute to the development and enrichment of mathematics education.

Algebraic generalization of Diffie–Hellman key exchange

The Diffie–Hellman key exchange scheme is one of the earliest and most widely used public-key primitives. Its underlying algebraic structure is a cyclic group and its security is based on the discrete logarithm problem (DLP). The DLP can be solved in polynomial time for any cyclic group in the quantum computation model. Therefore, new key exchange schemes have been sought to prepare for the time when quantum computing becomes a reality. Algebraically, these schemes need to provide some sort of commutativity to enable Alice and Bob to derive a common key on a public channel while keeping it computationally difficult for the adversary to deduce the derived key. We suggest an algebraically generalized Diffie–Hellman scheme (AGDH) that, in general, enables the application of any algebra as the platform for key exchange. We formulate the underlying computational problems in the framework of average-case complexity and show that the scheme is secure if the problem of computing images under an unknown homomorphism is infeasible. We also show that a symmetric encryption scheme possessing homomorphic properties over some algebraic operation can be turned into a public-key primitive with the AGDH, provided that the operation is complex enough. In addition, we present a brief survey on the algebraic properties of existing key exchange schemes and identify the source of commutativity and the family of underlying algebraic structures for each scheme.

General Topos Semantics for Higher-Order Modal Logic

Topos-theoretic semantics for modal logic usually uses structures induced by a surjective geometric morphism between toposes. This talk develops an algebraic generalization of this framework. We take internal adjoints between certain internal frames within a topos, which provides semantics for (intuitionistic) higher-oder modal logic.