| Angle trisection with origami and proving its correctness using Gröbner basis | 59-78 |
Abstract
In this paper, trisection of an angle performed by origami is explained in detail, as well as the correctness of obtained construction. Two different correctness conjectures, one based on trigonometry identities for triple angle and another, based on triangle congruence are formulated. All geometric constraints appearing in premises and conclusion of the conjecture are formulated as polynomials over the set of appropriately chosen variables, and the correctness conjectures are proved using Gröbner basis method. For performing calculations over polynomials obtained, the computer tool Singular is used.
Keywords: Origami constructions; angle trisection; Gröbner basis method; Singular.
MSC Subject Classification: 97G40, 97H20
MathEduc Subject Classification: G44, H24
| Mixing problems represented by quasi-digraphs | 79-103 |
Abstract
A notable class of problems often employed in undergraduate courses on differential equations is that of mixing problems: those involving a number of brine-filled tanks equipped with a number of brine-transporting pipes. Closed mixing problems, which feature neither filler nor drainer pipes, have been studied on a general level, where the flow networks are represented by digraphs [A. Slav\'ı k, Mixing problems with many tanks, American Mathematical Monthly, 120 (2013), 806-821]. In this paper, we extend the study to open mixing problems, which may feature filler and/or drainer pipes, representing the flow networks by a generalization of digraphs: quasi-digraphs. We formulate sufficient conditions under which such a mixing problem can be modeled as a system of linear ordinary differential equations whose coefficient matrix is the negative of the transpose of the Laplacian of the associated quasi-digraph. Subsequently, we formulate the analogues for mixing problems represented by weighted quasi-digraphs, and by cascade-type multilayer weighted quasi-digraphs. At the end of this paper, we propose suggestions for instructors on how our materials could be distilled to form a set of taught materials or a mini-project enriching an undergraduate differential equations course.
Keywords: Differential equation; mixing problem; digraph; quasi-digraph; multilayer.
MSC Subject Classification: 97I40
MathEduc Subject Classification: I45
| A generalization of Ptolemy's theorem | 104-111 |
Abstract
In this paper, we introduce a novel generalization of the classic Ptolemy's theorem, focusing on its triangle version. We explore this generalization's implications and provide several applications that illustrate its utility in geometric problem-solving.
Keywords: Ptolemy's theorem; van Schooten's theorem; cyclic quadrilateral.
MSC Subject Classification: 97G40
MathEduc Subject Classification: G44
| Finite generativity of homology and cohomology modules | 112-118 |
Abstract
In this paper, we consider the following question: if all homology groups of a space $X$ are finitely generated, and if $R$ is a commutative ring with identity, is it true that the homology and cohomology $R$-modules $H_i(X;R)$ and $H^i(X;R)$ are also finitely generated? We show that the answer to this question is negative in general, but affirmative if $R$ is an integral domain. In the case when $R$ is a principal ideal domain, and $H_i(X;R)$ is finitely generated for all $i$, we also discuss computing $H_i(X;M)$ and $H^i(X;M)$ for a finitely generated $R$-module $M$.
Keywords: Homology; cohomology; finitely generated module.
MSC Subject Classification: 97H99
MathEduc Subject Classification: H75