An enumeration of equilateral triangle dissections

We enumerate all dissections of an equilateral triangle into smaller equilateral triangles up to size 20, where each triangle has integer side lengths. A perfect dissection has no two triangles of the same side, counting up- and down-oriented triangles as different. We computationally prove W. T. Tutte's conjecture that the smallest perfect dissection has size 15 and we find all perfect dissections up to size 20.Comment: Final version sent to journal

Modelling the powertrain rubber coupling under dynamic conditions

This paper presents a strategy for computational modelling of elastic rubber couplings under dynamic loading. Methods how to determine static and dynamic characteristics of the elastic coupling based on static and dynamic experimental tests of rubber elements are presented. The nonlinear deformation behaviour, frequency and temperature dependent properties of rubber are considered for computational models. The model is applied to the elastic coupling connecting an in-line six-cylinder natural gas engine and an electrical generator. Loading forces are based on in-cylinder pressure measurement. Experimental verification of the computational model results is carried out by measuring the values on a test engine using the non-contact laser measuring technique

Homogeneous toroidal Latin bitrades

Let T be a partial Latin square. Then T is a Latin trade if there exists a partial Latin square T1, called a trade mate of T, with the properties that (i) a cell is filled in T1 if and only if it is filled in T, (ii) no entry occurs in the same cell in T and T1, (iii) in any given row or column, T and T1 contain the same elements. The pair {T, T1} is called a Latin bitrade. A Latin trade T (and T1) is said to be (r, c, e)- homogeneous if each row contains precisely r entries, each column contains precisely c entries, and each entry occurs precisely e times. An (r, c, e)-homogeneous Latin bitrade can be embedded on the torus only for three parameter sets, namely (r, c, e) = (3, 3, 3), (4, 4, 2) or (6, 3, 2). In this talk I will present classifications for all three cases

On Left Conjugacy Closed Loops With A Nucleus Of Index Two

A loop Q is said to be left conjugacy closed (LCC) if the left translations form a set of permutations that is closed under conjugation. Loops in which the left and right nuclei coincide and are of index 2 are necesarilly LCC, and they are constructed in the paper explicitly. LCC loops Q with the right nucleus G of index 2 o#er a larger diversity. A sample of results: if Z(G) = 1, then Q is also right conjugacy closed. For each m 2 one can construct Q of order 2m in such a way that its left nucleus is trivial. If Q is involutorial, then it is a Bol loop

On Distances Of 2-Groups And 3-Groups

This paper is concerned with finite groups G(#) and G(#) of order n that are not isomorphic, and where the size of GG; u#v u#v} is the least possible (with respect to the given n). It surveys the case of 2-groups, discusses the possible generalization of the known results to p-groups, p an odd prime, and establishes the least possible distance in the case when G(#) is an elementary abelian 3-group

Structural Interactions Of Conjugacy Closed Loops

We study conjugacy closed loops by means of their multi- plication groups. Let Q be a conjugacy closed loop, N its nucleus, A the associator subloop, and L and R the left and right multiplication group, respectively. Put M = {a ∈ Q; La ∈ R}. We prove that the cosets of A agree with orbits of [L, R], that Q/M ∼= (Inn Q)/L1 and that one can define an abelian group on Q/N ×L1. We also explain why the study of finite conjugacy closed loops can be restricted to the case of N/A nilpotent. Group [L,R] is shown to be a subgroup of a power of A (which is abelian), and we prove that Q/N can be embedded into Aut ([L, R]). Finally, we describe all conjugacy closed loops of order pq