Related Experiment Video
Updated: Mar 5, 2026
06:35
Construction and Systematical Symmetric Studies of a Series of Supramolecular Clusters with Binary or Ternary Ammonium Triphenylacetates
Published on: February 15, 2016
8.5K
Split-Facets for Balanced Minimal Evolution Polytopes and the Permutoassociahedron
Stefan Forcey1, Logan Keefe2, William Sands2
1Department of Mathematics, The University of Akron, Akron, OH, 44325-4002, USA. sforcey@uakron.edu.
Bulletin of Mathematical Biology
|March 24, 2017
Summary
Researchers discovered a new sublattice within the balanced minimal evolution (BME) polytope, revealing its connection to phylogenetic tree splits. This finding expands our understanding of BME polytope structures and their exponential growth in phylogenetic analysis.
Area of Science:
- Phylogenetic Theory
- Computational Biology
- Discrete Geometry
Background:
- The structure of the balanced minimal evolution (BME) polytope is crucial for understanding phylogenetic relationships.
- Identifying the top-dimensional facets of this polytope is a key challenge in phylogenetic theory.
- Previous work by Hodge, Haws, and Yoshida established a lattice of faces for the BME polytope.
Purpose of the Study:
- To investigate the face structure of the balanced minimal evolution (BME) polytope.
- To identify and characterize the top-dimensional facets of the BME polytope.
- To establish a connection between phylogenetic tree splits and the BME polytope's combinatorial structure.
Main Methods:
- Mathematical analysis of the balanced minimal evolution (BME) polytope.
- Exploration of the poset of faces of the BME polytope.
- Comparison with the structure of the permutoassociahedron and its quotients.
Main Results:
- A sublattice within the BME polytope's face poset was identified, isomorphic to a quotient of the permutoassociahedron.
- This sublattice represents compatible sets of splits from phylogenetic trees.
- Maximal elements in this new poset correspond to single leaf splits, with most being facets of the BME polytope, exhibiting exponential growth.
Conclusions:
- The study reveals a novel combinatorial structure within the BME polytope related to phylogenetic splits.
- This extends previous findings on the BME polytope's face lattice.
- The results provide new insights into the complexity and structure of phylogenetic models.
Related Concept Videos
01:34Woodward–Hoffmann Selection Rules and Microscopic Reversibility
4.1K
Electrocyclic reactions, cycloadditions, and sigmatropic rearrangements are concerted pericyclic reactions that proceed via a cyclic transition state. These reactions are stereospecific and regioselective. The stereochemistry of the products depends on the symmetry characteristics of the interacting orbitals and the reaction conditions. Accordingly, pericyclic reactions are classified as either symmetry-allowed or symmetry-forbidden. Woodward and Hoffmann presented the selection criteria for...
4.1K
01:18Aromatic Hydrocarbon Cations: Structural Overview
4.1K
Cycloheptatriene is a neutral monocyclic unsaturated hydrocarbon that consists of an odd number of carbon atoms and an intervening sp3 carbon in the ring. The three double bonds in the ring correspond to 6 π electrons, which is a Huckel number, and therefore satisfies the criteria of 4n + 2 π electrons. However, the intervening sp3 carbon disrupts the continuous overlap of p orbitals. As a result, cycloheptatriene is not aromatic.
Removing one hydrogen from the intervening CH2 group...
Removing one hydrogen from the intervening CH2 group...
4.1K
01:21Plastic Deformations of Members with a Single Plane of Symmetry
419
When a structural member undergoes plastic deformation due to bending, it is crucial to understand the position of the neutral axis and the stress distribution. This member, characterized by a single plane of symmetry, exhibits a uniform stress distribution, with negative stress above the neutral axis and positive stress below. Notably, the neutral axis does not align with the centroid of the cross-section. This misalignment is typical in cases where the cross-section is not rectangular or...
419
01:16Perpendicular-Axis Theorem
4.7K
The perpendicular-axis theorem states that the moment of inertia of a planar object about an axis perpendicular to its plane is equal to the sum of the moments of inertia about two mutually perpendicular concurrent axes lying in the plane of the body.
Consider a circular disc of mass M and radius R lying along an x-y plane. The origin lies at the center of the disc, and the z-axis is perpendicular to the disc's plane. All three axes coincide at the disc's center. The moment of inertia of this...
Consider a circular disc of mass M and radius R lying along an x-y plane. The origin lies at the center of the disc, and the z-axis is perpendicular to the disc's plane. All three axes coincide at the disc's center. The moment of inertia of this...
4.7K
01:30Geometry of Hyperbolas
562
A hyperbola consists of all points where the absolute difference of distances to two fixed points, called foci, remains constant. The standard equation isEach branch extends infinitely and approaches two asymptotes, which guide the curve’s behavior. The parameters a and b define key features: a measures the distance from the center to each vertex along the transverse axis, while b influences the slopes of the asymptotes. The asymptotes have equationsA rectangle centered at the origin with...
562
02:57Coordination Number and Geometry
19.3K
For transition metal complexes, the coordination number determines the geometry around the central metal ion. Table 1 compares coordination numbers to molecular geometry. The most common structures of the complexes in coordination compounds are octahedral, tetrahedral, and square planar.
19.3K