Computing a Counting Polynomial of an Infinite Family of Linear Polycene Parallelogram Benzenoid Graph P(a,b)

2007 ◽  
Vol 3 (1) ◽  
pp. 186-190
Author(s):  
Mohammad Reza Farahani
Keyword(s):  
Infinite Family ◽  
Benzenoid Graph ◽  
First Derivative ◽  
Omega Polynomial ◽  
Counting Polynomial ◽  
First Time

Omega polynomial was defined by M.V. Diudea in 2006 as  where the number of edges co-distant with e is denoted by n(e). One can obtain Theta Θ, Sadhana Sd and Pi Π polynomials by replacing xn(e) with n(e)xn(e), x|E|-n(e) and n(e)x|E|-n(e) in Omega polynomial, respectively. Then Theta Θ, Sadhana Sd and Pi Π indices will be the first derivative of Θ(x), Sd(x) and Π(x) evaluated at x=1. In this paper, Pi Π(G,x) polynomial and Pi Π(G) index of an infinite family of linear polycene parallelogram benzenoid graph P(a,b) are computed for the first time.

Computing the Theta Polynomial Θ(G, x) and the Theta Index Θ(G) of Titania Nanotubes TiO2(m, n)

2017 ◽  
Vol 14 (1) ◽  
pp. 715-717
Author(s):  
Yingfang Li ◽  
Li Yan ◽  
Mohammad R Farahani ◽  
Muhammad Imran ◽  
Muhammad K Jamil
Keyword(s):  
Graph Theory ◽  
Molecular Graph ◽  
Infinite Family ◽  
Titania Nanotubes ◽  
Chemical Graph ◽  
First Derivative ◽  
Chemical Graph Theory ◽  
Omega Polynomial ◽  
Definition Of ◽  
First Time

Let G = (V,E) be a simple connected molecular graph in chemical graph theory, where the vertex/atom set and edge/bond set of G denoted by V(G) and E(G), respectively and its vertices correspond to the atoms and the edges correspond to the bonds. Two counting polynomials the “Omega Ω(G,x) and Theta Θ(G,x)” polynomials of a molecular graph G were defined by Diudea as Ω(G,x) = ΣeE(G) xn(E) and Θ(G,x) = ΣeE(G) xn(E), where n(E) denotes the number of edges co-distant with the edge E. From definition of these counting polynomials, we can obtain the Theta polynomial by inserting the coefficient n(E) in the Omega polynomial. Then the Theta index will be the first derivative of the Theta polynomial Θ(G,x) evaluated at x = 1. The goal of this paper is to compute the Theta polynomial Θ(G,x) and the Theta index Θ(G) of an infinite family of the Titania Nanotubes TiO2(m,n) for the first time.

scholarly journals Computing the Omega polynomial of an infinite family of the linear parallelogram P(n,m)

2006 ◽  
Vol 2 (2) ◽  
pp. 106-109
Author(s):  
Mohammad Reza Farahani
Keyword(s):  
Molecular Graph ◽  
Infinite Family ◽  
Benzenoid Graph ◽  
Omega Polynomial

Let G=(V,E) be a molecular graph, where V(G) is a non-empty set of vertices and E(G) is a set of edges. The Omega polynomial Ω(G,x) was introduced by Diudea in 2006 and this defined as  where m(G,c) the number of qoc strips of length c. In this paper, we compute the omega polynomial of an infinite family of the linear parallelogram P(n,n) of benzenoid graph.

scholarly journals Some finite solvable groups with non-trivial lattice endomorphisms

2003 ◽  
Vol 68 (1) ◽  
pp. 141-153
Author(s):  
S. E. Stonehewer ◽  
G. Zacher
Keyword(s):  
Infinite Family ◽  
Solvable Groups ◽  
Nilpotency Class ◽  
Derived Length ◽  
First Time

The main purpose of this paper is to exhibit a doubly-infinite family of examples which are extensions of a p-group by a p′-group, with the action satisfying some conditions of Zappa (1951), arising from his study of dual-standard (meet-distributive) subgroups. The examples show that Zappa's conditions do not bound the nilpotency class (or even the derived length) of the p-group. The key to this work is found in closely related conditions of Hartley (published here for the first time). The examples use some exceptional relationships between primes.

QUANTUM DESIGNS: FOUNDATIONS OF A NONCOMMUTATIVE DESIGN THEORY

2011 ◽  
Vol 09 (01) ◽  
pp. 445-507 ◽  
Author(s):  
GERHARD ZAUNER
Keyword(s):  
Design Theory ◽  
Numerical Solutions ◽  
Infinite Family ◽  
Positive Operator Valued Measures ◽  
Master's Thesis ◽  
The 1960S ◽  
Equiangular Lines ◽  
First Time ◽  
Projection Matrices ◽  
Special Case

This is a one-to-one translation of a German-written Ph.D. thesis from 1999. Quantum designs are sets of orthogonal projection matrices in finite(b)-dimensional Hilbert spaces. A fundamental differentiation is whether all projections have the same rank r, and furthermore the special case r = 1, which contains two important subclasses: Mutually unbiased bases (MUBs) were introduced prior to this thesis and solutions of b + 1 MUBs whenever b is a power of a prime were already given. Unaware of those papers, this concept was generalized here under the notation of regular affine quantum designs. Maximal solutions are given for the general case r ≥ 1, consisting of r(b2 - 1)/(b - r) so-called complete orthogonal classes whenever b is a power of a prime. For b = 6, an infinite family of MUB triples was constructed and it was — as already done in the author's master's thesis (1991) — conjectured that four MUBs do not exist in this dimension. Symmetric informationally complete positive operator-valued measures (SIC POVMs) in this paper are called regular quantum 2-designs with degree 1. The assigned vectors span b2 equiangular lines. These objects had been investigated since the 1960s, but only a few solutions were known in complex vector spaces. In this thesis further maximal analytic and numerical solutions were given (today a lot more solutions are known) and it was (probably for the first time) conjectured that solutions exist in any dimension b (generated by the Weyl–Heisenberg group and with a certain additional symmetry of order 3).

scholarly journals On Connectivity Indices of an Infinite Family of the Linear Parallelogram of Benzenoid Graph

2015 ◽  
Vol 54 ◽  
pp. 131-134
Author(s):  
Mohammad Reza Farahani
Keyword(s):  
Topological Index ◽  
Molecular Graph ◽  
Infinite Family ◽  
Infinite Class ◽  
Benzenoid Graph ◽  
Connectivity Indices ◽  
Atom Bond

A topological index of a graph G is a numeric quantity related to G which is describe molecular graph G. In this paper the Atom Bond Connectivity (ABC) and Geometric-Arithmetic (GA) indices of an infinite class of the linear parallelogram of benzenoid graph.

scholarly journals A Survey of the Hadamard Maximal Determinant Problem

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Patrick Browne ◽  
Ronan Egan ◽  
Fintan Hegarty ◽  
Padraig Ó Catháin
Keyword(s):  
Historical Development ◽  
Design Theory ◽  
Infinite Family ◽  
Upper Bounds ◽  
Quadratic Residues ◽  
Maximum Value ◽  
Gram Matrices ◽  
The Subject ◽  
First Time ◽  
Determinant Theory

In a celebrated paper of 1893, Hadamard established the maximal determinant theorem, which establishes an upper bound on the determinant of a matrix with complex entries of norm at most 1. His paper concludes with the suggestion that mathematicians study the maximum value of the determinant of an $n \times n$ matrix with entries in $\{ \pm 1\}$. This is the Hadamard maximal determinant problem. This survey provides complete proofs of the major results obtained thus far. We focus equally on upper bounds for the determinant (achieved largely via the study of the Gram matrices), and constructive lower bounds (achieved largely via quadratic residues in finite fields and concepts from design theory). To provide an impression of the historical development of the subject, we have attempted to modernise many of the original proofs, while maintaining the underlying ideas. Thus some of the proofs have the flavour of determinant theory, and some appear in print in English for the first time. We survey constructions of matrices in order $n \equiv 3 \mod 4$, giving asymptotic analysis which has not previously appeared in the literature. We prove that there exists an infinite family of matrices achieving at least 0.48 of the maximal determinant bound. Previously the best known constant for a result of this type was 0.34.

scholarly journals New and explicit constructions of unbalanced Ramanujan bipartite graphs

2021 ◽  
Author(s):  
Shantanu Prasad Burnwal ◽  
Kaneenika Sinha ◽  
Mathukumalli Vidyasagar
Keyword(s):  
Infinite Family ◽  
Bipartite Graphs ◽  
Affirmative Answer ◽  
Ramanujan Graphs ◽  
The Third ◽  
Construction Methods ◽  
Explicit Constructions ◽  
Ramanujan Graph ◽  
Computational Work ◽  
First Time

AbstractThe objectives of this article are threefold. Firstly, we present for the first time explicit constructions of an infinite family of unbalanced Ramanujan bigraphs. Secondly, we revisit some of the known methods for constructing Ramanujan graphs and discuss the computational work required in actually implementing the various construction methods. The third goal of this article is to address the following question: can we construct a bipartite Ramanujan graph with specified degrees, but with the restriction that the edge set of this graph must be distinct from a given set of “prohibited” edges? We provide an affirmative answer in many cases, as long as the set of prohibited edges is not too large.

Annulate lamellae in the Sertoli cells of guinea pig testis after unilateral torsion of the spermatic cord

1984 ◽  
Vol 42 ◽  
pp. 196-197
Author(s):  
J. Chakraborty ◽  
A. P. Sinha Hikim ◽  
J. S. Jhunjhunwala
Keyword(s):  
Sertoli Cells ◽  
Guinea Pigs ◽  
Spermatic Cord ◽  
Present Report ◽  
Cell Types ◽  
Annulate Lamellae ◽  
Spermatogenic Cells ◽  
Electron Microscopic ◽  
Structural Details ◽  
First Time

Although the presence of annulate lamellae was noted in many cell types, including the rat spermatogenic cells, this structure was never reported in the Sertoli cells of any rodent species. The present report is based on a part of our project on the effect of torsion of the spermatic cord to the contralateral testis. This paper describes for the first time, the fine structural details of the annulate lamellae in the Sertoli cells of damaged testis from guinea pigs.One side of the spermatic cord of each of six Hartly strain adult guinea pigs was surgically twisted (540°) under pentobarbital anesthesia (1). Four months after induction of torsion, animals were sacrificed, testes were excised and processed for the light and electron microscopic investigations. In the damaged testis, the majority of seminiferous tubule contained a layer of Sertoli cells with occasional spermatogonia (Fig. 1). Nuclei of these Sertoli cells were highly pleomorphic and contained small chromatinic clumps adjacent to the inner aspect of the nuclear envelope (Fig. 2).

Electron spectroscopic imaging and diffraction

1991 ◽  
Vol 49 ◽  
pp. 706-707
Author(s):  
M. Rühle ◽  
J. Mayer ◽  
J.C.H. Spence ◽  
J. Bihr ◽  
W. Probst ◽  
...  
Keyword(s):  
Materials Science ◽  
Electron Spectroscopic Imaging ◽  
Magnetic Filter ◽  
Magnetic Imaging ◽  
Illumination System ◽  
Probe Size ◽  
Diffraction Patterns ◽  
Fritz Haber ◽  
Koehler Illumination ◽  
First Time

A new Zeiss TEM with an imaging Omega filter is a fully digitized, side-entry, 120 kV TEM/STEM instrument for materials science. The machine possesses an Omega magnetic imaging energy filter (see Fig. 1) placed between the third and fourth projector lens. Lanio designed the filter and a prototype was built at the Fritz-Haber-Institut in Berlin, Germany. The imaging magnetic filter allows energy-filtered images or diffraction patterns to be recorded without scanning using efficient area detection. The energy dispersion at the exit slit (Fig. 1) results in ∼ 1.5 μm/eV which allows imaging with energy windows of ≤ 10 eV. The smallest probe size of the microscope is 1.6 nm and the Koehler illumination system is used for the first time in a TEM. Serial recording of EELS spectra with a resolution < 1 eV is possible. The digital control allows X,Y,Z coordinates and tilt settings to be stored and later recalled.

A Novel TEM Technique for Characterizing As-Grown CVD Diamond Films

1991 ◽  
Vol 49 ◽  
pp. 956-957 ◽  
Author(s):  
Z.L. Wang ◽  
J. Bentley ◽  
R.E. Clausing ◽  
L. Heatherly ◽  
L.L. Horton
Keyword(s):  
Diamond Films ◽  
Chemical Vapor ◽  
Growth Direction ◽  
Dark Field ◽  
Growth Surface ◽  
Specimen Preparation ◽  
Transmission Electron ◽  
Diffraction Patterns ◽  
First Time ◽  
Microstructural Studies

Microstructural studies by transmission electron microscopy (TEM) of diamond films grown by chemical vapor deposition (CVD) usually involve tedious specimen preparation. This process has been avoided with a technique that is described in this paper. For the first time, thick as-grown diamond films have been examined directly in a conventional TEM without thinning. With this technique, the important microstructures near the growth surface have been characterized. An as-grown diamond film was fractured on a plane containing the growth direction. It took about 5 min to prepare a sample. For TEM examination, the film was tilted about 30-45° (see Fig. 1). Microstructures of the diamond grains on the top edge of the growth face can be characterized directly by transmitted electron bright-field (BF) and dark-field (DF) images and diffraction patterns.

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