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33 pages, 3753 KB

Open AccessArticle

Matching Polynomials of Symmetric, Semisymmetric, Double Group Graphs, Polyacenes, Wheels, Fans, and Symmetric Solids in Third and Higher Dimensions

by Krishnan Balasubramanian

Symmetry 2025, 17(1), 133; https://doi.org/10.3390/sym17010133 - 17 Jan 2025

Cited by 1 | Viewed by 2227

Abstract

The primary objective of this study is the computation of the matching polynomials of a number of symmetric, semisymmetric, double group graphs, and solids in third and higher dimensions. Such computations of matching polynomials are extremely challenging problems due to the computational and [...] Read more.

The primary objective of this study is the computation of the matching polynomials of a number of symmetric, semisymmetric, double group graphs, and solids in third and higher dimensions. Such computations of matching polynomials are extremely challenging problems due to the computational and combinatorial complexity of the problem. We also consider a series of recursive graphs possessing symmetries such as D_2h-polyacenes, wheels, and fans. The double group graphs of the Möbius types, which find applications in chemically interesting topologies and stereochemistry, are considered for the matching polynomials. Hence, the present study features a number of vertex- or edge-transitive regular graphs, Archimedean solids, truncated polyhedra, prisms, and 4D and 5D polyhedra. Such polyhedral and Möbius graphs present stereochemically and topologically interesting applications, including in chirality, isomerization reactions, and dynamic stereochemistry. The matching polynomials of these systems are shown to contain interesting combinatorics, including Stirling numbers of both kinds, Lucas polynomials, toroidal tree-rooted map sequences, and Hermite, Laguerre, Chebychev, and other orthogonal polynomials. Full article

(This article belongs to the Collection Feature Papers in Chemistry)

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19 pages, 3611 KB

Open AccessArticle

Delineation of Optimized Single and Multichannel Approximate DA-Based Filter Design Using Influential Single MAC Strategy for Trans-Multiplexer

by Britto Pari James, Leung Man-Fai, Mariammal Karuthapandian and Vaithiyanathan Dhandapani

Sensors 2024, 24(22), 7149; https://doi.org/10.3390/s24227149 - 7 Nov 2024

Cited by 2 | Viewed by 1005

Abstract

In this paper, a multichannel FIR filter design based on the Time Division Multiplex (TDM) approach that incorporates one multiply and add unit, regardless of the variable coefficient length and varying channels, by associating the resource sharing doctrine is suggested. A multiplier based [...] Read more.

In this paper, a multichannel FIR filter design based on the Time Division Multiplex (TDM) approach that incorporates one multiply and add unit, regardless of the variable coefficient length and varying channels, by associating the resource sharing doctrine is suggested. A multiplier based on approximate distributed arithmetic (DA) circuits is employed for effective resource optimization. Although no explicit multiplication was conducted in this realization, the radix-8 and radix-4 Booth algorithms are utilized in the DA framework to curtail and optimize the partial products (PPs). Furthermore, the input stream is truncated with an erratum mending unit to roughly construct the partial products. For an aggregation of PPs, an approximate Wallace tree is taken into consideration to further minimize hardware expenses. Consequently, the suggested design’s latency, utilized area, and power usage are largely reduced. The Xilinx Vertex device is expedited, given the synthesis of the suggested multichannel realization with 16 taps, which is simulated using the Verilog formulary. It is observed that the filter structure with one channel produced the desired results, and the system’s frequency can support up to 429 MHz with a reduced area. Utilizing TSMC 180 nm CMOS technology and the Cadence RC compiler, cell-level performance is also achieved. Full article

(This article belongs to the Special Issue Electronic Test and Measurement Instrumentation: Design and Applications (2nd Edition))

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17 pages, 577 KB

Open AccessArticle

Navier–Stokes Equation in a Cone with Cross-Sections in the Form of 3D Spheres, Depending on Time, and the Corresponding Basis

by Muvasharkhan Jenaliyev, Akerke Serik and Madi Yergaliyev

Mathematics 2024, 12(19), 3137; https://doi.org/10.3390/math12193137 - 7 Oct 2024

Cited by 2 | Viewed by 1157

Abstract

The work establishes the unique solvability of a boundary value problem for a 3D linearized system of Navier–Stokes equations in a degenerate domain represented by a cone. The domain degenerates at the vertex of the cone at the initial moment of time, and, [...] Read more.

The work establishes the unique solvability of a boundary value problem for a 3D linearized system of Navier–Stokes equations in a degenerate domain represented by a cone. The domain degenerates at the vertex of the cone at the initial moment of time, and, as a consequence of this fact, there are no initial conditions in the problem under consideration. First, the unique solvability of the initial-boundary value problem for the 3D linearized Navier–Stokes equations system in a truncated cone is established. Then, the original problem for the cone is approximated by a countable family of initial-boundary value problems in domains represented by truncated cones, which are constructed in a specially chosen manner. In the limit, the truncated cones will tend toward the original cone. The Faedo–Galerkin method is used to prove the unique solvability of initial-boundary value problems in each of the truncated cones. By carrying out the passage to the limit, we obtain the main result regarding the solvability of the boundary value problem in a cone. Full article

(This article belongs to the Section C1: Difference and Differential Equations)

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16 pages, 790 KB

Open AccessArticle

Eigenvalue −1 of the Vertex Quadrangulation of a 4-Regular Graph

by Vladimir R. Rosenfeld

Axioms 2024, 13(1), 72; https://doi.org/10.3390/axioms13010072 - 22 Jan 2024

Viewed by 1944

Abstract

The vertex quadrangulation $Q G$ of a 4-regular graph G visually looks like a graph whose vertices are depicted as empty squares, and the connecting edges are attached to the corners of the squares. In a previous work [JOMC 59, 1551–1569 (2021)], the [...] Read more.

The vertex quadrangulation $Q G$ of a 4-regular graph G visually looks like a graph whose vertices are depicted as empty squares, and the connecting edges are attached to the corners of the squares. In a previous work [JOMC 59, 1551–1569 (2021)], the question was posed: does the spectrum of an arbitrary unweighted graph

Q G

include the full spectrum

{3, {(- 1)}^{3}}

of the tetrahedron graph (complete graph

K_{4}

)? Previously, many bipartite and nonbipartite graphs

Q G

with such a subspectrum have been found; for example, a nonbipartite variant of the graph

Q K_{5}

. Here, we present one of the variants of the nonbipartite vertex quadrangulation

Q O

of the octahedron graph O, which has eigenvalue

(- 1)

of multiplicity 2 in the spectrum, while the spectrum of the bipartite variant

Q O

contains eigenvalue

(- 1)

of multiplicity 3. Thus, in the case of nonbipartite graphs, the answer to the question posed depends on the particular graph

Q G

. Here, we continue to explore the spectrum of graphs

Q G

. Some possible connections of the mathematical theme to chemistry are also noted. Full article

(This article belongs to the Special Issue Spectral Graph Theory, Molecular Graph Theory and Their Applications)

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26 pages, 8516 KB

Open AccessArticle

Transient Thermal Stresses in FG Porous Rotating Truncated Cones Reinforced by Graphene Platelets

by Masoud Babaei, Faraz Kiarasi, Kamran Asemi, Rossana Dimitri and Francesco Tornabene

Appl. Sci. 2022, 12(8), 3932; https://doi.org/10.3390/app12083932 - 13 Apr 2022

Cited by 48 | Viewed by 3288

Abstract

The present work studies an axisymmetric rotating truncated cone made of functionally graded (FG) porous materials reinforced by graphene platelets (GPLs) under a thermal loading. The problem is tackled theoretically based on a classical linear thermoelasticity approach. The truncated cone consists of a [...] Read more.

The present work studies an axisymmetric rotating truncated cone made of functionally graded (FG) porous materials reinforced by graphene platelets (GPLs) under a thermal loading. The problem is tackled theoretically based on a classical linear thermoelasticity approach. The truncated cone consists of a layered material with a uniform or non-uniform dispersion of GPLs in a metal matrix with open-cell internal pores, whose effective properties are determined according to the extended rule of mixture and modified Halpin–Tsai model. A graded finite element method (FEM) based on Rayleigh–Ritz energy formulation and Crank–Nicolson algorithm is here applied to solve the problem both in time and space domain. The thermo-mechanical response is checked for different porosity distributions (uniform and functionally graded), together with different types of GPL patterns across the cone thickness. A parametric study is performed to analyze the effect of porosity coefficients, weight fractions of GPL, semi-vertex angles of cone, and circular velocity, on the thermal, kinematic, and stress response of the structural member. Full article

(This article belongs to the Special Issue 5th Anniversary of Nanotechnology and Applied Nanosciences Section—Recent Advances in Carbon Composites and Complex Materials)

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22 pages, 14856 KB

Open AccessFeature PaperArticle

Spectrum-Adapted Polynomial Approximation for Matrix Functions with Applications in Graph Signal Processing

by Tiffany Fan, David I. Shuman, Shashanka Ubaru and Yousef Saad

Algorithms 2020, 13(11), 295; https://doi.org/10.3390/a13110295 - 13 Nov 2020

Cited by 3 | Viewed by 4765

Abstract

We propose and investigate two new methods to approximate

f (A) b

for large, sparse, Hermitian matrices

A

. Computations of this form play an important role in numerous signal processing and machine learning tasks. The main idea behind both methods [...] Read more.

We propose and investigate two new methods to approximate

f (A) b

for large, sparse, Hermitian matrices

A

. Computations of this form play an important role in numerous signal processing and machine learning tasks. The main idea behind both methods is to first estimate the spectral density of

A

, and then find polynomials of a fixed order that better approximate the function f on areas of the spectrum with a higher density of eigenvalues. Compared to state-of-the-art methods such as the Lanczos method and truncated Chebyshev expansion, the proposed methods tend to provide more accurate approximations of

f (A) b

at lower polynomial orders, and for matrices

A

with a large number of distinct interior eigenvalues and a small spectral width. We also explore the application of these techniques to (i) fast estimation of the norms of localized graph spectral filter dictionary atoms, and (ii) fast filtering of time-vertex signals. Full article

(This article belongs to the Special Issue Efficient Graph Algorithms in Machine Learning)

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27 pages, 473 KB

Open AccessFeature PaperArticle

Convergence Assessment of the Trajectories of a Bioreaction System by Using Asymmetric Truncated Vertex Functions

by Alejandro Rincón, Gloria Yaneth Florez and Gerard Olivar

Symmetry 2020, 12(4), 513; https://doi.org/10.3390/sym12040513 - 2 Apr 2020

Cited by 5 | Viewed by 2197

Abstract

In several open and closed-loop systems, the trajectories converge to a region instead of an equilibrium point. Identifying the convergence region and proving the asymptotic convergence upon arbitrarily large initial values of the state variables are regarded as important issues. In this work, [...] Read more.

In several open and closed-loop systems, the trajectories converge to a region instead of an equilibrium point. Identifying the convergence region and proving the asymptotic convergence upon arbitrarily large initial values of the state variables are regarded as important issues. In this work, the convergence of the trajectories of a biological process is determined and proved via truncated functions and Barbalat’s Lemma, while a simple and systematic procedure is provided. The state variables of the process asymptotically converge to a compact set instead of an equilibrium point, with asymmetrical bounds of the compact sets. This convergence is rigorously proved by using asymmetric forms with vertex truncation for each state variable and the Barbalat’s lemma. This includes the definition of the truncated

V_{i}

functions and the arrangement of its time derivative in terms of truncated functions. The proposed truncated function is different from the common one as it accounts for the model nonlinearities and the asymmetry of the vanishment region. The convergence analysis is valid for arbitrarily large initial values of the state variables, and arbitrarily large size of the convergence regions. The positive invariant nature of the convergence regions is proved. Simulations confirm the findings. Full article

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27 pages, 5528 KB

Open AccessArticle

Nonlinear Stability Analysis of Eccentrically Stiffened Functionally Graded Truncated Conical Sandwich Shells with Porosity

by Duc-Kien Thai, Tran Minh Tu, Le Kha Hoa, Dang Xuan Hung and Nguyen Ngọc Linh

Materials 2018, 11(11), 2200; https://doi.org/10.3390/ma11112200 - 6 Nov 2018

Cited by 22 | Viewed by 4076

Abstract

This paper analyzes the nonlinear buckling and post-buckling characteristics of the porous eccentrically stiffened functionally graded sandwich truncated conical shells resting on the Pasternak elastic foundation subjected to axial compressive loads. The core layer is made of a porous material (metal foam) characterized [...] Read more.

This paper analyzes the nonlinear buckling and post-buckling characteristics of the porous eccentrically stiffened functionally graded sandwich truncated conical shells resting on the Pasternak elastic foundation subjected to axial compressive loads. The core layer is made of a porous material (metal foam) characterized by a porosity coefficient which influences the physical properties of the shells in the form of a harmonic function in the shell’s thickness direction. The physical properties of the functionally graded (FG) coatings and stiffeners depend on the volume fractions of the constituents which play the role of the exponent in the exponential function of the thickness direction coordinate axis. The classical shell theory and the smeared stiffeners technique are applied to derive the governing equations taking the von Kármán geometrical nonlinearity into account. Based on the displacement approach, the explicit expressions of the critical buckling load and the post-buckling load-deflection curves for the sandwich truncated conical shells with simply supported edge conditions are obtained by applying the Galerkin method. The effects of material properties, core layer thickness, number of stiffeners, dimensional parameters, semi vertex angle and elastic foundation on buckling and post-buckling behaviors of the shell are investigated. The obtained results are validated by comparing with those in the literature. Full article

(This article belongs to the Special Issue Advances in Mechanical Problems of Functionally Graded Materials and Structures)

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9 pages, 186 KB

Open AccessArticle

Study of Dynamical Chiral Symmetry Breaking in (2 + 1) Dimensional Abelian Higgs Model

by Jian-Feng Li, Shi-Song Huang, Hong-Tao Feng, Wei-Min Sun and Hong-Shi Zong

Symmetry 2010, 2(2), 907-915; https://doi.org/10.3390/sym2020907 - 19 Apr 2010

Viewed by 7821

Abstract

In this paper, we study the dynamical mass generation in the Abelian Higgs model in 2 + 1 dimensions. Instead of adopting the approximations in [Jiang H et al., J. Phys. A 41 2008 255402.], we numerically solve the coupled Dyson–Schwinger Equations [...] Read more.

In this paper, we study the dynamical mass generation in the Abelian Higgs model in 2 + 1 dimensions. Instead of adopting the approximations in [Jiang H et al., J. Phys. A 41 2008 255402.], we numerically solve the coupled Dyson–Schwinger Equations (DSEs) for the fermion and gauge boson propagators using a specific truncation for the fermion-photon vertex ansatz and compare our results with the corresponding ones in the above mentioned paper. It is found that the results quoted in the above paper remain qualitatively unaffected by refining the truncation scheme of the DSEs, although there exist large quantitative differences between the results presented in the above paper and ours. In addition, our numerical results show that the critical number of fermion flavor N_c decreases steeply with the the gauge boson mass m_a (or the ratio of the Higgs mass m_h to the gauge boson mass m_a, r = m_h/m_a) increasing. It is thus easier to generate a finite fermion mass by the mechanism of DCSB for a small ratio r for a given m_a. Full article

(This article belongs to the Special Issue Symmetry Breaking Phenomena)

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