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

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1599 ◽  
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.

scholarly journals Zhang–Zhang Polynomials of Ribbons

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 2060
Author(s):  
Bing-Hau He ◽  
Chien-Pin Chou ◽  
Johanna Langner ◽  
Henryk A. Witek
Keyword(s):  
Closed Form ◽  
Topological Invariants ◽  
Important Class ◽  
Kekulé Structures ◽  
Clar Number ◽  
Interface Theory ◽  
Clar Structures ◽  
Closed Form Formula ◽  
Clar Covering Polynomial

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.

scholarly journals Harmonic Numbers and Cubed Binomial Coefficients

2011 ◽  
Vol 2011 ◽  
pp. 1-14
Author(s):  
Anthony Sofo
Keyword(s):  
Closed Form ◽  
Binomial Coefficients ◽  
Harmonic Numbers ◽  
Polygamma Functions ◽  
Form Representation ◽  
The Given

Euler related results on the sum of the ratio of harmonic numbers and cubed binomial coefficients are investigated in this paper. Integral and closed-form representation of sums are developed in terms of zeta and polygamma functions. The given representations are new.

Successive Synthesis of Substructure Modes

1991 ◽  
Vol 58 (3) ◽  
pp. 759-765 ◽  
Author(s):  
Luis E. Suarez ◽  
Mahendra P. Singh
Keyword(s):  
Closed Form ◽  
Characteristic Equation ◽  
Degrees Of Freedom ◽  
Coupled System ◽  
Closed Form Expression ◽  
Element Discretization ◽  
Root Finding ◽  
Complete Set ◽  
Conventional Solution ◽  
Second Eigenvalue

A mode synthesis approach is presented to calculate the eigenproperties of a structure from the eigenproperties of its substructures. The approach consists of synthesizing the substructures sequentially, one degree-of-freedom at a time. At each coupling stage, the eigenvalue is obtained as the solution of a characteristic equation, defined in closed form in terms of the eigenproperties obtained in the preceding coupling stage. The roots of the characteristic equation can be obtained by a simple Newton-Raphson root finding scheme. For each calculated eigenvalue, the eigenvector is defined by a simple closed-form expression. The eigenproperties obtained in the final coupling stage provide the desired eigenproperties of the coupled system. Thus, the approach avoids a conventional solution of the second eigenvalue problem. The approach can be implemented with the complete set or a truncated number of substructure modes; if the complete set of modes is used, the calculated eigenproperties would be exact. The approach can be used with any finite element discretization of structures. It requires only the free interface eigenproperties of the substructures. Successful application of the approach to a moderate size problem (255 degrees-of-freedom) on a microcomputer is also demonstrated.

A Velocity Approach to Elasto-Plastic and Elasto-Viscoplastic Calculation by the Finite Element Method

1990 ◽  
Vol 112 (2) ◽  
pp. 150-154 ◽  
Author(s):  
J. L. Chenot ◽  
M. Bellet
Keyword(s):  
Finite Element ◽  
Metal Forming ◽  
Time Discretization ◽  
The Finite Element Method ◽  
Forming Process ◽  
Element Discretization ◽  
Numerical Tests ◽  
Metal Forming Process ◽  
Order Scheme ◽  
Complete Set

A second order scheme for the time discretization of the elasto-plastic or elasto-viscoplastic behavior is proposed, based on a velocity approach. The complete set of equations is given for the evolution problem in the case of small rotations approximation. The method is quite general and may be applied to a large class of constitutive equations. The finite element discretization is briefly outlined and it is shown that the procedure is quite similar to that of previous displacement formulations. A numerical example concerning the sheet metal forming process, with an elasto-viscoplastic behavior and a membrane approximation, is presented. The numerical tests show a considerable improvement in accuracy for a given increment of time.

SPECTRAL NORMS OF CIRCULANT-TYPE MATRICES WITH BINOMIAL COEFFICIENTS AND HARMONIC NUMBERS

2014 ◽  
Vol 11 (05) ◽  
pp. 1350076 ◽  
Author(s):  
JIANWEI ZHOU ◽  
ZHAOLIN JIANG
Keyword(s):  
Circulant Matrices ◽  
Binomial Coefficients ◽  
Harmonic Numbers ◽  
Numerical Tests ◽  
Skew Circulant

In this paper, we investigate spectral norms for circulant-type matrices, including circulant, skew-circulant, and g-circulant matrices. The entries are product of Binomial coefficients with Harmonic numbers. We obtain explicit identities for these spectral norms. Employing these approaches, we list some numerical tests to verify our results.

scholarly journals Second order alternating harmonic number sums

Filomat ◽  
2016 ◽  
Vol 30 (13) ◽  
pp. 3511-3524 ◽  
Author(s):  
Anthony Sofo
Keyword(s):  
Closed Form ◽  
Second Order ◽  
Integral Representations ◽  
Harmonic Number ◽  
Binomial Coefficients ◽  
Harmonic Numbers

We develop new closed form representations of sums of alternating harmonic numbers of order two and reciprocal binomial coefficients. Moreover we develop new integral representations in terms of harmonic numbers of order two.

scholarly journals The Identical Estimates of Spectral Norms for Circulant Matrices with Binomial Coefficients Combined with Fibonacci Numbers and Lucas Numbers Entries

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Jianwei Zhou
Keyword(s):  
Fibonacci Numbers ◽  
Circulant Matrices ◽  
Binomial Coefficients ◽  
Lucas Numbers ◽  
Numerical Tests

Improved estimates for spectral norms of circulant matrices are investigated, and the entries are binomial coefficients combined with either Fibonacci numbers or Lucas numbers. Employing the properties of given circulant matrices, this paper improves the inequalities for their spectral norms, and gets corresponding identities of spectral norms. Moreover, by some well-known identities, the explicit identities for spectral norms are obtained. Some numerical tests are listed to verify the results.

Exact Solutions for Starting and Oscillatory Flows in an Equilateral Triangular Duct

2016 ◽  
Vol 138 (8) ◽  
Author(s):  
C. Y. Wang
Keyword(s):  
Helmholtz Equation ◽  
Exact Solutions ◽  
Closed Form ◽  
Oscillatory Flow ◽  
Series Solutions ◽  
Eigenvalues And Eigenfunctions ◽  
Approximate Methods ◽  
Oscillatory Flows ◽  
Complete Set ◽  
Starting Flow

Exact series solutions, some in closed-form, for starting flow and oscillatory flow in an equilateral triangular duct are presented. The complete set of eigenvalues and eigenfunctions of the Helmholtz equation is derived, and the method of eigenfunction superposition is used. Exact solutions are rare, fundamental, and serve as accuracy standards for approximate methods.

scholarly journals The Explicit Identities for Spectral Norms of Circulant-Type Matrices Involving Binomial Coefficients and Harmonic Numbers

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Jianwei Zhou ◽  
Xiangyong Chen ◽  
Zhaolin Jiang
Keyword(s):  
Circulant Matrix ◽  
Binomial Coefficients ◽  
Harmonic Numbers ◽  
Numerical Tests ◽  
Skew Circulant ◽  
Explicit Formulae

The explicit formulae of spectral norms for circulant-type matrices are investigated; the matrices are circulant matrix, skew-circulant matrix, andg-circulant matrix, respectively. The entries are products of binomial coefficients with harmonic numbers. Explicit identities for these spectral norms are obtained. Employing these approaches, some numerical tests are listed to verify the results.

scholarly journals When can the sum of $(1/p)$th of the binomial coefficients have closed form?

10.37236/1336 ◽  
1996 ◽  
Vol 4 (2) ◽  
Author(s):  
Marko Petkovšek ◽  
Herbert S. Wilf
Keyword(s):  
Closed Form ◽  
Nonnegative Integer ◽  
Binomial Coefficients

We find all nonnegative integer $r,s,p$ for which the sum $\sum_{k=rn}^{sn}{pn\choose k}$ has closed form.

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