| Rectification of vector fields and the failure of its dual | 1-13 |
Abstract
Although tangent and cotangent bundles are isomorphic vector bundles, the geometric and the dynamic properties of vector fields and differential\br 1-forms differ fundamentally. By contrasting two genuinely different -- though formally dual -- concepts of differential equations, this paper exposes a subtle asymmetry whose careful analysis is pedagogically valuable for advanced undergraduate university students.
Keywords: Rectification; duality; differential equations.
MSC Subject Classification: 97I40, 97I70
MathEduc Subject Classification: I75, I65, I45
| Rethinking calculus instruction for management students: The gradual linearization approach | 14-26 |
Abstract
Traditional calculus instruction, with its reliance on formal limits, often creates unnecessary barriers for management students seeking essential quantitative skills. To address these barriers, we advocate for gradual linearization, an underutilized method that replaces formal limit arguments with local linear approximations. This approach fosters deeper conceptual clarity and offers a rigorous yet accessible pathway to core concepts -- including differentiation rules and Taylor expansions -- central to marginal analysis and optimization. Adopting this limit-free approach enables management educators to equip future managers with quantitative reasoning skills for practical applications, without the detour of formal limits.
Keywords: Gradual linearization; limit-free differentiation; simple proofs in differential calculus.
MSC Subject Classification: 97I40
MathEduc Subject Classification: I45
| The poset of nested means generated by the harmonic, geometric, arithmetic, and quadratic means | 27-41 |
Abstract
Motivated by the classical chain of inequalities $H\le G\le A\le Q$ and recent work by Meštrović [The Teaching of Mathematics 27, 1 (2024), 27--32], we perform a complete structural analysis of the finite family of nested means $$ \mathcal M=\{M_1(M_2,M_3): M_1,M_2,M_3ın\{H,G,A,Q\}\}. $$ We identify the $26$ distinct elements of $\mathcal M$ and determine the full structure of the associated partially ordered set (poset) under the pointwise order. Our investigation shows that the poset has height $19$ and contains exactly $30$ incomparable pairs. Consequently, we construct a maximal chain of length $18$ that contains $8$ nontrivial inequalities, thereby providing affirmative and optimal answers to two open questions posed by Meštrović. The proofs rely on a unified algebraic reduction to a single variable $u=G/A ın (0,1]$, which converts pointwise comparisons into explicit polynomial nonnegativity and factorizations.
Keywords: Nested means; poset; mean inequalities; maximal chain; classical means; polynomial nonnegativity.
MSC Subject Classification: 97H30, 26E60, 06A06, 26D07
MathEduc Subject Classification: H34
| A brief introduction to hypergraph expanders and the coboundary expansion of $\Lambda_n^3$ | 42-56 |
Abstract
Building on the ideas from [P. F. Wild, High-dimensional expansion and crossing numbers of simplicial complexes, Doctoral thesis submitted to Graduate School of the Institute of Science and Technology Austria, 2022], in this paper we obtain an upper bound for the expansion coefficient (Cheeger constant) $\eta_1(\Lambda_n^3)$ of the complete, multipartite complex $\Lambda_n^3$. We take this particular calculation as an opportunity to give the reader a glimpse into the general theory of graph expanders and their higher dimensional analogues.
Keywords: Expander graphs; high dimensional expanders; cohomology.
MSC Subject Classification: 97K30, 97H99, 05C48, 05E45
MathEduc Subject Classification: K35, H75
| A short survey of variations of the game Bulgarian Solitaire | 57-62 |
Abstract
Let $N$ be an arbitrary positive integer and let $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_l)$ be a partition of $N$ of length $l$, i.e., $\sum_{i=1}^{l}\lambda_i=N$ with parts $\lambda_1\ge \lambda_2\ge\ldots\ge \lambda_l\ge 1$. Define $T(\lambda)$ as the partition of $n$ with parts $l,\lambda_1-1,\lambda_2-1\ldots,\lambda_l-1$, ignoring any zeros that might occur. Starting with a partition $\lambda$ of $N$, we describe Bulgarian solitaire by repeatedly applying the shiftoperation $T$ to obtain the sequence of partitions $$\lambda,T(\lambda),\;T^2(\lambda),\;\ldots.$$ Bulgarian solitaire is a dynamical system on integer partition of a positive integer $n$ which converges to a unique fixed point if $n=1+2+\cdots +k$ is a triangular number. In this paper we give a short survey of this popular mathematical card game and several variations of this game.
Keywords: Bulgarian solitaire; cycle of partitions; shift operation; triangular number; partition of an integer.
MSC Subject Classification: 97F60, 97K20, 05A17, 11P81
MathEduc Subject Classification: F64, K24