15 Fullerenes are spherical carbon molecules consisting of n carbon atoms. Their surface consists of twelve pentagons and n/2-10 hexagons only. Mathematically, we can model them as three-dimensional convex polyhedra with restricted facets.
Using the Schlege terms
Nowadays, graphs are a ubiquitous mathematical model to represent complex networks, maps, or molecules.
In my work with my supervisors, Prof. Viktor Buchstaber and Prof. Dr. Evgeny Spodarev, I use graph-theoretical techniques and spectral properties of fullerene graphs to analyze their structure. Using these abstract mathematical properties, we can predict real-world properties of fullerenes, like their ability to conduct heat and electricity. Even fundamental properties, like the technical realization of specific fullerenes, are still unknown. I want to find extraordinary examples of fullerenes worth analyzing in more detail with my work.
(Unbounded) Hexagonal lattice
noun
The tiling of an indefinitely two-dimensional plane in regular hexagons.
Example: (i) "We consider a discrete-time random walk on the nodes of an unbounded hexagonal lattice."
(ii) "Stimulated by potential applications in many fields of science and engineering, in this paper we aim
to study a discrete-time random walk on the nodes of an unbounded hexagonal lattice."
(iii) "Moreover, hexagonal lattices have been adopted to characterize landscapes in suitable spatial
models of bird populations since hexagons allow better packing of territories in space, see Pulliam
et al. [1992]"
de: Sechseckgitter
cumbersome
adjective
large or heavy and therefore difficult to carry or use; unwieldy
Example: The proof follows after very cumbersome and tedious calculations, and then are omitted.
de: sperrig; schwerfällig
customary
adjective
according to the customs or usual practices associated with a particular society, place, or set of circumstances.
Example: The validity of a customary convergence to a 2-dimensional Brownian motion is also shown.
de: üblich, gebräuchlich
First-Passage-Time Problem
noun
Mathematics: The amount of time required for a stochastic process, starting from some initial state, to encounter a threshold for the first time is referred to variously as a first-passage-time.
Example: (i) "In this section we study the first-passage-time problem of the randomwalk (Xn,Yn) through suitable boundaries, under specific symmetry conditions."
(ii) "In conclusion, we also face the first-passage-time problem of the considered random walk through straight-line boundaries."
(iii) "Finally, we investigate the first-passage-time problem of the random walk through a vertical straight-line."
de: Stoppzeit-Problem
Fullerene (molecule)
noun
A fullerene is an allotrope of carbon whose molecule consists of carbon atoms connected by single and double bonds to form a closed mesh, with rings of five or six atoms.
Example: (i) "Fullerene graphs are mathematical models of
fullerene molecules, i.e., molecules comprised only by carbon atoms different than graphites
and diamonds."
(ii)"The discovery of the Buckminsterfullerene marked the birth of fullerene
chemistry and nanotechnology, but at the same time the fullerenes were studied from different
perspectives."
(iii) "Fullerenes can also be seen as graphs, vertices represent atoms, and edges represent bonds
between atoms. A fullerene graph is a 3-connected 3-regular planar graph with only pentagonal
and hexagonal faces. In what follows fullerene graphs will be also called fullerenes.
Due to Whitney’s Theorem (1933) simple planar 3-connected graphs have a unique planar
embedding, and therefore the same holds for fullerene graphs. By Euler’s formula follows
the next property of fullerene graphs."
de: Fulleren
Isosceles triangle
noun
An isosceles triangle is a triangle that has two sides of equal length. Sometimes, it is specified as having exactly two sides of equal length, and sometimes as having at least two sides of equal length. The last version thus includes the equilateral triangle as a special case.
Example: (i) "When icosahedral fullerenes are
constructed all 20 triangles are the same and equilateral, but in the case of fullerenes with
lower symmetries the triangles can be isosceles or scalene."
(ii) "Every isosceles triangle has an axis of symmetry along the perpendicular bisector of its base."
(iii) "Euclid defined an isosceles triangle as a triangle with exactly two equal sides, but modern treatments prefer to define isosceles triangles as having at least two equal sides."
de: Gleichschenkliges Dreieck
Nanotube
noun
Carbon nanotubes are tubes made of carbon atoms with diameters typically measured in nanometers.
Example: (i) "Carbon nanotubes can exhibit remarkable electrical conductivity, while others are semiconductors."
(ii) "This result confirms Conjecture 7.1 for (0,5)-nanotubes."
(iii) "Those pairs of pentagonal faces can be distributed in a widely
varied list of possible configurations, leading to molecules of different shapes, including
nanotubes of all possible types."
de: Kohlenstoffnanoröhre
peculiar
adjective
unusual and strange, sometimes in an unpleasant way
Example: (i) "Fullerenes excluding direct pentagon fusion are produced by the overlap of 12
nanocones and show a peculiar thermodynamic stability following the isolated pentagon rule (IPR) early observed by Kroto."
(ii) "It's peculiar that they didn't tell us they were going away."
(iii) "The copy editor will check type size and technical details to see if anything looks peculiar."
de: Sonderbar, eigentümlich, eigenartig
polymer science
noun
Polymer science is a subfield of materials science concerned with polymers, primarily synthetic polymers such as plastics and elastomers. The field of polymer science includes researchers in multiple disciplines including chemistry, physics, and engineering.
Example: An example of the usefulness of these results in the context of polymer science is provided
at the end of Section 5.
de: Polymerwissenschaft
tedious
adjective
boring and tiring, esp. because long or often repeated
Example: (i) "... the proof then follows after very cumbersome and tedious calculations."
(ii) "After one year they had finally completed the tedious process."
(iii) "The development of innovative pharmaceutical drugs is a very complex, risky and tedious process"
de: Deutsch: Langweilig, langwierig
Русский: утомительный, скучный
to accompany
verb
to go with someone or to be provided or exist at the same time as something;
Example: (i) "An accompanying fullerene database includes all general isomers up to C150 (for IPR isomers up to C200), including the number of Hamiltonian cycles for general isomers up to C110 and for IPR isomers up to C122"
(ii) "Each abstract is accompanied by full bibliographic information."
(iii) "Economic change was accompanied by changes to laws and institutions."
de: begleiten, beifügen
to exploit so./sth.
verb
Exploit commonly means to selfishly take advantage of someone in order to profit from them or otherwise benefit oneself. It can also be used in a more neutral way that doesn’t imply selfishness: to make the best use of something, especially an opportunity, to create a profit or other benefit.
Example: Hereafter we exploit various symmetry properties for the transition probabilities given in Proposition 3.2.
de: jmdn./etw. ausbeuten
jmdn./etw. ausnutzen
to pose sth.
verb
to cause something, especially a problem or difficulty;
to ask a question, especially in a formal situation such as a meeting;
Example: (i) "The generation of these fascinating hollow polyhedra poses therefore a challenge to mathematicians and chemists,
which found some guidance in the publication of an atlas of fullerenes by Fowler, Manolopoulos almost two decades ago containing the first database of fullerene structures
up to C50 and IPR cages up to C100."
(ii) "The specific pose subjects used in this study required them to position their fingers accurately along the edges of the coin."
(iii) "The processes that pose the greatest risk to the embryo are photothermal and photochemical in nature."
de: darstellen, posieren
to reverberate
verb
If an event or idea reverberates somewhere, it has an effect on everyone or everything in a place or group;
(If a loud deep sound reverberates, it continues to be heard around an area, so that the area seems to shake)
Example: (i) "this fact naturally reverberates on the overall molecular structure of the two involved Cn isomers: generally the action of the 2-switch sequence results in non-isomorphic graphs for the two interconnected Cn fullerenes."
(ii) "Two specific issues have reverberated in this debate."
(iii) "The aftereffects of its defeat reverberated in state and national politics for years."
de: ausstrahlen, einen Effekt haben
(zurückstrahlen)
to reward
verb
give something to (someone) in recognition of their services, efforts, or achievements.
Example: In 1991, the Science magazine pronounced the buckminsterfullerene for the “Molecule of the year”, and later in 1996 the
discovery of C60 was rewarded with the Nobel price for chemistry.
