Zhang–Zhang Polynomials of Ribbons

We report a closed-form formula for the Zhang–Zhang polynomial (also known as ZZ polynomial or Clar covering polynomial) of an important class of elementary peri-condensed benzenoids Rbn1,n2,m1,m2, usually referred to as ribbons. A straightforward derivation is based on the recently developed interface theory of benzenoids [Langner and Witek, MATCH Commun. Math. Comput. Chem.2020, 84, 143–176]. The discovered formula provides compact expressions for various topological invariants of Rbn1,n2,m1,m2: the number of Kekulé structures, the number of Clar covers, its Clar number, and the number of Clar structures. The last two classes of elementary benzenoids, for which closed-form ZZ polynomial formulas remain to be found, are hexagonal flakes Ok,m,n and oblate rectangles Orm,n.

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ZZ Polynomials for Isomers of (5,6)-Fullerenes Cn with n = 20–50

Symmetry ◽

10.3390/sym12091483 ◽

2020 ◽

Vol 12 (9) ◽

pp. 1483 ◽

Cited By ~ 2

Author(s):

Henryk A. Witek ◽

Jin-Su Kang

Keyword(s):

Quantum Chemical Calculations ◽

Quantum Chemical ◽

Topological Analysis ◽

Valence Bond ◽

Topological Invariants ◽

Kekulé Structures ◽

Resonance Structures ◽

Clar Number ◽

Benchmark Data ◽

Clar Structures

A compilation of ZZ polynomials (aka Zhang–Zhang polynomials or Clar covering polynomials) for all isomers of small (5,6)-fullerenes Cn with n = 20–50 is presented. The ZZ polynomials concisely summarize the most important topological invariants of the fullerene isomers: the number of Kekulé structures K, the Clar number Cl, the first Herndon number h1, the total number of Clar covers C, and the number of Clar structures. The presented results should be useful as benchmark data for designing algorithms and computer programs aiming at topological analysis of fullerenes and at generation of resonance structures for valence-bond quantum-chemical calculations.

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Clar Covers of Overlapping Benzenoids: Case of Two Identically-Oriented Parallelograms

Symmetry ◽

10.3390/sym12101599 ◽

2020 ◽

Vol 12 (10) ◽

pp. 1599 ◽

Cited By ~ 1

Author(s):

Henryk A. Witek ◽

Johanna Langner

Keyword(s):

Closed Form ◽

Binomial Coefficients ◽

Kekulé Structures ◽

Structural Resemblance ◽

Numerical Tests ◽

Interface Theory ◽

Complete Set

We present a complete set of closed-form formulas for the ZZ polynomials of five classes of composite Kekuléan benzenoids that can be obtained by overlapping two parallelograms: generalized ribbons Rb, parallelograms M, vertically overlapping parallelograms MvM, horizontally overlapping parallelograms MhM, and intersecting parallelograms MxM. All formulas have the form of multiple sums over binomial coefficients. Three of the formulas are given with a proof based on the interface theory of benzenoids, while the remaining two formulas are presented as conjectures verified via extensive numerical tests. Both of the conjectured formulas have the form of a 2×2 determinant bearing close structural resemblance to analogous formulas for the number of Kekulé structures derived from the John-Sachs theory of Kekulé structures.

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Closed Form Formula For Undertaking Specific Parameters Calculation Under Solvency II

SSRN Electronic Journal ◽

10.2139/ssrn.2912153 ◽

2017 ◽

Author(s):

Filippo Siegenthaler

Keyword(s):

Closed Form ◽

Solvency Ii ◽

Closed Form Formula ◽

Parameters Calculation

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A Revision of the Merton Jump-Diffusion Model: A Simple, Closed-Form Formula

SSRN Electronic Journal ◽

10.2139/ssrn.3412520 ◽

2019 ◽

Author(s):

Moawia Alghalith

Keyword(s):

Closed Form ◽

Diffusion Model ◽

Jump Diffusion ◽

Jump Diffusion Model ◽

Closed Form Formula

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ON THE REPRESENTATION OF THE NESTED LOGIT MODEL

Econometric Theory ◽

10.1017/s026646662000047x ◽

2021 ◽

pp. 1-11

Author(s):

Alfred Galichon

Keyword(s):

Closed Form ◽

Logit Model ◽

Random Variables ◽

Nested Logit ◽

Random Utility ◽

Nested Logit Model ◽

Utility Representation ◽

Phd Thesis ◽

Closed Form Formula

In this paper, we give a two-line proof of a long-standing conjecture of Ben-Akiva in his 1973 PhD thesis regarding the random utility representation of the nested logit model, thus providing a renewed and straightforward textbook treatment of that model. As an application, we provide a closed-form formula for the correlation between two Fréchet random variables coupled by a Gumbel copula.

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