Physical-chemical properties studying of molecular structures via topological index calculating
-
Jianzhang Wu
,
Mohammad Reza Farahani
Abstract
It’s revealed from the earlier researches that many physical-chemical properties depend heavily on the structure of corresponding moleculars. This fact provides us an approach to measure the physical-chemical characteristics of substances and materials. In our article, we report the eccentricity related indices of certain important molecular structures from mathematical standpoint. The eccentricity version indices of nanostar dendrimers are determined and the reverse eccentric connectivity index for V-phenylenic nanotorus is discussed. The conclusions we obtained mainly use the trick of distance computation and mathematical derivation, and the results can be applied in physics engineering.
1 Introduction
Many experiments reveal the evidence that the physical-chemical properties of pounds and materials are closely related to their molecular structures. For example, the structure-dependency of total π-electron energy Eπ heavily relies on the sum of squares of the vertex degrees of the molecular graph. Topological indices, as a result, are introduced as numerical parameters of molecular structures, and they play a key role in realizing the physical-chemical properties of substances.
In theoretical computational model, a molecular structure can be represented as a molecular graph G in which each vertex is expressed as an atom and each bound between atoms is denoted as an edge. Then, a topological index can be regarded as a score operator f : G → ℝ+ which maps each molecular graph to a positive real number.
In the past four decades, scientists introduced many indices from the application perspective, such as Zagreb index, Wiener index, sum connectivity index and harmonic index which reflect certain physical-chemical characteristics of molecular structure. There were many contributions to report these distance-based or degree-based indices of special molecular structures (See Farahani et al. [1], Jamil et al. [2], Gao and Wang [3], Gao et al. [4–7] and Gao and Wang [8, 9] for more details). The notations and terminologies that were used (but undefined in our paper) can be found in Bondy and Murty [10].
For a fixed vertex u ∈ V(G), the eccentricity ec(u) of vertex u is defined as the largest distance between u and any other vertex v in G. There are several eccentricity related indices introduced for the engineering purpose.
The first atom-bond connectivity index (in short, ABC index) is introduced by Estrada et al. [11] as
where d(v) is the degree of vertex v in the molecular graph G. Correspondingly, the fifth atom bond connectivity index is denoted as
The first geometric-arithmetic index (shortly, GA index) rased by Vukić ević and Furtula [12] as
Several results on GA index can refer to Zhou et al. [13], Rodríguez and Sigarreta [14], [15] and [16], Husin et al. [17], Bahrami and Alaeiyan [18], Sigarreta [19], Divnic et al. [20], Das et al. [21], Mahmiani et al. [22], Fath-Tabar et al. [23] and [24], Das et al. [25], Gutman and Furtula [26], Furtula and Gutman [27], and Shabani et al. [28]. The fourth geometric-arithmetic index is defined by Lee et al. [29] which is stated as
Moreover, the first and the second multiplicative eccentricity index are defined as
and
respectively. And, the fourth and sixth Zagreb polynomials are denoted as
and
respectively.
As an important index, the eccentric connectivity index (ECI) of molecular graph G is introduced as
Ranjini and Lokesha [30] determined the eccentric connectivity index of the subdivision graph of tadpole graphs, complete graphs and wheel graphs. Morgan et al. [31] yielded the sharp lower bound on ξc(G) for a given graph order. Furthermore, the tight upper and lower bounds are manifested for trees with given vertex number and diameter. Hua and Das [32] presented the relationship between the eccentric connectivity index and Zagreb indices. De [33] deduced the eccentric connectivity index and polynomial of the thorn graphs. Eskender and Vumar [34] studied the eccentric distance sum and eccentric connectivity index of generalized hierarchical product graphs. In addition, the eccentric connectivity index of F-sum graphs by virtue of certain invariants of the factors are presented as well. Ilić and Gutman [35] proposed that the broom reaches the maximum ξc(G) among trees with fixed maximum vertex degree, and the trees with minimum ξc(G) are also characterized. Iranmanesh and Hafezieh [36] computed the eccentric connectivity index of some special graph families. Dankelmann et al. [37] raised the sharp upper bound for eccentric connectivity index and presented the molecular graphs which asymptotically attain this bound. Morgan et al. [38] presented the tight lower bound of eccentric connectivity index for a tree with given order and diameter. Rao and Lakshmi [39] determined the eccentric connectivity index of phenylenic nanotubes.
Ediz [40] introduced a new distance-based topological index called reverse eccentric connectivity index which is stated as
where
Although there have been several contributions in distance-based and degree-based indices of molecular structures, the study of eccentricity related indices for certain special compounds is still largely limited. On the other hand, as critical and widespread molecular structures, nanostar dendrimers and V-phenylenic nanotorus are widely used in physics, chemistry, biology, medical and material science. For these reasons, we study the exact expressions of eccentricity related indices for nanostar dendrimers and V-phenylenic nanotorus.
2 Main Results and Proofs
In this paper, we mainly study the eccentricity related indices for nanostar dendrimers (Subsection 2.1) and the reverse eccentric connectivity index of V-phenylenic nanotorus (Subsection 2.2).
2.1 Eccentricity related index of nanostar dendrimers
D1[n] is the first class of nanostar dendrimer family in which |V(D1[n])| = 36n – 12 and |E(D1[n])| = 42n – 15, and D2[n] is the second class of nanostar dendrimer family in which |V(D2[n])| = 120 · 2n – 108 and |E(D2[n])| = 140 · 2n – 127. The example of basic structure of D1[n] and D2[n] can refer to Figure 1 and Figure 2, respectively.
![Figure 1 D1[1] and D1[2]](/document/doi/10.1515/phys-2017-0029/asset/graphic/j_phys-2017-0029_fig_001.jpg)
D1[1] and D1[2]
![Figure 2 D2[1] and D2[2]](/document/doi/10.1515/phys-2017-0029/asset/graphic/j_phys-2017-0029_fig_002.jpg)
D2[1] and D2[2]
Theorem 1
For n = 1, we have
For n ≥ 2, we get
Proof
Since D1[n] is symmetrical, we can mark the vertices some representative symbols which are showed in Figure 1. Next, we only present the detailed proof of GA4 index, and other parts of result can be obtained in the similar way.
By the analysis of molecular structure, the edge set of D1[n] can be divided into six subsets which are described as follows:
(u, v): with eccentricities 3n+4 and 3n+3, and there are six edges in this class;
(a1, v): with eccentricities 3n + 4 and 3n + 3, and there are three edges in this class;
(ai, bi): with eccentricities 3n + 3i+1 and 3n+3i+2, and there are 3 · 2i edges in i-th generation of this class;
(bi, ci): with eccentricities 3n+3i+2 and 3n+3i+3, and there are 3 · 2i edges in i-th generation of this class;
(ci, di) : with eccentricities 3n+3i+3 and 3n+3i+4, and there are 3 · 2i edges in i-th generation of this class;
(ci, ai+1): with eccentricities 3n+3i+3 and 3n+3i+4, and there are 3 · 2i edges in i-th generation of this class.
Thus, by the definition of the fourth GA index, we infer
and for n ≥ 2
Therefore, we get the expected result. □
Theorem 2
For n = 1, we have
For n ≥ 2, we get
Proof
Since D2[n] is also symmetrical, we can mark the vertices some representative symbols which are showed in Figure 2. We only present the detailed proof of GA4 index, and other parts of result can be obtained in the similar way.
By the analysis of molecular structure, the edge set of D2[n] can be divided into 17 subsets which are described as follows:
(u, v):with eccentricities 10n – 3 and 10n – 3, and there are two edges in this class;
(v, w): with eccentricities 10n – 3 and 10n – 2, and there are four edges in this class;
(w, b1):with eccentricities 10n – 2 and 10n – 1, and there are two edges in this class;
(ai, bi): with eccentricities 10n+10i–10 and 10n + 10i– 11, and there are 2i edges in i-th generation of this class;
(bi, ci): with eccentricities 10n + 10i–11 and 10n +10i– 10, and there are 2i edges in i-th generation of this class;
(ci, di): with eccentricities 10n+10i–10 and 10n+10i–9, and there are 2i+1 edges in i-th generation of this class;
(di, ei): with eccentricities 10n+10i–9 and 10n+10i–8, and there are 2i+1 edges in i-th generation of this class;
(ei, fi): with eccentricities 10n+10i–8 and 10n+10i–7, and there are 2i+1 edges in i-th (i ≠ n)generation of this class;
(en, fn): with eccentricities 10n+10i–8 and 10n+10i–7, and there are 2n+2 edges in this class;
(ei, gi): with eccentricities 10n+10i–8 and 10n +10i–8, and there are 2i edges in i-th generation of this class;
(ai′, bi′):with eccentricities 10n+10i–5 and 10n+10i– 6, and there are 2i+1 edges in i-th (1 ≤ i ≤ n –1 and n ≥ 2) generation of this class;
(bi′, ci′): with eccentricities 10n+10i–6 and 10n+10i–5, and there are 2i+1 edges in i-th (1 ≤ i ≤ n –1 and n ≥ 2) generation of this class;
(ci′, di′): with eccentricities 10n+10i–5 and 10n+10i–4, and there are 2i+2 edges in i-th (1 ≤ i ≤ n –1 and n ≥ 2) generation of this class;
(di′, ei′): with eccentricities 10n +10i–4 and 10n + 10i– 3, and there are 2i+2 edges in i-th (1 ≤ i ≤ n – 1 and n ≥ 2) generation of this class;
(ei′, fi′): with eccentricities 10n+10i–3 and 10n+10i–2, and there are 2i+2 edges in i-th (1 ≤ i ≤ n – 1 and n ≥ 2) generation of this class;
(fi, bi′):with eccentricities 10n+10i–7 and 10n+10i–6, and there are 2i+1 edges in i-th (1 ≤ i ≤ n –1 and n ≥ 2) generation of this class;
(fi′, bi+1): with eccentricities 10n + 10i–2 and 10n+ 10i – 1, and there are 2i+1 edges in i-th (1 ≤ i ≤ n – 1 and n ≥ 2) generation of this class.
Thus, in view of the definition of the fourth GA index, we have
and for n ≥ 2
Thus, we yield the expected result. □
2.2 Reverse eccentric connectivity index for V-phenylenic nanotorus
In this subsection, we consider the V-phenylenic nanotorus TO[p, q] (here p is the number of hexagons in each row and q is the number of hexagons in each column) which is a widely used nano structure. For the example of TO[4, 5], please refer to Figure 3.
![Figure 3 V-phenylenic nanotorus TO[4, 5]](/document/doi/10.1515/phys-2017-0029/asset/graphic/j_phys-2017-0029_fig_003.jpg)
V-phenylenic nanotorus TO[4, 5]
By the molecular structure analysis, we check that |V(TO[p, q])| = 6pq and |E(TO[p, q])| = 9pq. Furthermore, TO[p, q] is a cubic such that d(v) = 3 for any v ∈ V(G) and ec(v) = ec(u) for any two vertices u and v.
Now, we present the main result of this part.
Theorem 3
The reverse eccentric connectivity index for V-phenylenic nanotorus is sated as follows:
If p ≡ 0(mod2), then
If p ≡ 1(mod2), then
Proof
Since TO[p, q] is a cubic, we have S(v) = 9 for any v ∈ V(G).
The whole proof process can be divided into four parts.
p ≡ 0(mod2) and q ≡ 0(mod2).
If q ≥ p, then
Then
If q ≤ p, then
Then
p ≡ 1(mod2) and q ≡ 1(mod2).
If q ≥ p, then
Then
If q ≤ p, then
Then
p ≡ 0(mod2) and q ≡ 1(mod2).
If q ≥ p, then
Then
If q ≤ p, then
Then
p ≡ 1(mod2) and q ≡ 0(mod2).
If q ≥ p, then
Then
If q ≤p, then
Then
Hence, we get the desired conclusion. □
3 Conclusion
In this paper, we mainly report the eccentricity related indices for nanostar dendrimers and the reverse eccentric connectivity index of V-phenylenic nanotorus. Since these indices are widely used in the analysis of physical-chemical properties, they possess a promising prospect of application in physical, chemical, medical and material engineering.
Acknowledgement
The authors thank the reviewers for their constructive comments in improving the quality of this paper. This work was supported in part by the National Natural Science Foundation of China [grant number 60903131]; Science and Technology of Jiangsu Province [grant number BE2011173]; and Key Laboratory of Computer Network and Information Integration Founding in Southeast University.
Conflicts of interest: The authors declare that there is no conflict of interests regarding the publication of this paper.
References
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- Comparisons of feature extraction algorithm based on unmanned aerial vehicle image
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- Research on identification method of heavy vehicle rollover based on hidden Markov model
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- Classifying BCI signals from novice users with extreme learning machine
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- Topics on data transmission problem in software definition network
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- Statistical inferences with jointly type-II censored samples from two Pareto distributions
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- Estimation for coefficient of variation of an extension of the exponential distribution under type-II censoring scheme
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- Analysis on trust influencing factors and trust model from multiple perspectives of online Auction
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- Coupling of two-phase flow in fractured-vuggy reservoir with filling medium
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- Production decline type curves analysis of a finite conductivity fractured well in coalbed methane reservoirs
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- Flow Characteristic and Heat Transfer for Non-Newtonian Nanofluid in Rectangular Microchannels with Teardrop Dimples/Protrusions
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- The size prediction of potential inclusions embedded in the sub-surface of fused silica by damage morphology
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- Research on carbonate reservoir interwell connectivity based on a modified diffusivity filter model
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- The method of the spatial locating of macroscopic throats based-on the inversion of dynamic interwell connectivity
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- Unsteady mixed convection flow through a permeable stretching flat surface with partial slip effects through MHD nanofluid using spectral relaxation method
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- A volumetric ablation model of EPDM considering complex physicochemical process in porous structure of char layer
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- Numerical simulation on ferrofluid flow in fractured porous media based on discrete-fracture model
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- Macroscopic lattice Boltzmann model for heat and moisture transfer process with phase transformation in unsaturated porous media during freezing process
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- Modelling of intermittent microwave convective drying: parameter sensitivity
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- Simulating gas-water relative permeabilities for nanoscale porous media with interfacial effects
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- Simulation of counter-current imbibition in water-wet fractured reservoirs based on discrete-fracture model
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- Investigation effect of wettability and heterogeneity in water flooding and on microscopic residual oil distribution in tight sandstone cores with NMR technique
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- Analytical modeling of coupled flow and geomechanics for vertical fractured well in tight gas reservoirs
- Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
- Special Issue: Ever New "Loopholes" in Bell’s Argument and Experimental Tests
- Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
- The ultimate loophole in Bell’s theorem: The inequality is identically satisfied by data sets composed of ±1′s assuming merely that they exist
- Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
- Erratum to: The ultimate loophole in Bell’s theorem: The inequality is identically satisfied by data sets composed of ±1′s assuming merely that they exist
- Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
- Rhetoric, logic, and experiment in the quantum nonlocality debate
- Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
- What If Quantum Theory Violates All Mathematics?
- Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
- Relativity, anomalies and objectivity loophole in recent tests of local realism
- Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
- The photon identification loophole in EPRB experiments: computer models with single-wing selection
- Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
- Bohr against Bell: complementarity versus nonlocality
- Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
- Is Einsteinian no-signalling violated in Bell tests?
- Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
- Bell’s “Theorem”: loopholes vs. conceptual flaws
- Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
- Nonrecurrence and Bell-like inequalities
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Three-dimensional computer models of electrospinning systems
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Electric field computation and measurements in the electroporation of inhomogeneous samples
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Modelling of magnetostriction of transformer magnetic core for vibration analysis
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Comparison of the fractional power motor with cores made of various magnetic materials
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Dynamics of the line-start reluctance motor with rotor made of SMC material
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Inhomogeneous dielectrics: conformal mapping and finite-element models
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Topology optimization of induction heating model using sequential linear programming based on move limit with adaptive relaxation
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Detection of inter-turn short-circuit at start-up of induction machine based on torque analysis
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Current superimposition variable flux reluctance motor with 8 salient poles
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Modelling axial vibration in windings of power transformers
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Field analysis & eddy current losses calculation in five-phase tubular actuator
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Hybrid excited claw pole generator with skewed and non-skewed permanent magnets
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Electromagnetic phenomena analysis in brushless DC motor with speed control using PWM method
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Field-circuit analysis and measurements of a single-phase self-excited induction generator
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- A comparative analysis between classical and modified approach of description of the electrical machine windings by means of T0 method
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Field-based optimal-design of an electric motor: a new sensitivity formulation
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Application of the parametric proper generalized decomposition to the frequency-dependent calculation of the impedance of an AC line with rectangular conductors
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Virtual reality as a new trend in mechanical and electrical engineering education
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Holonomicity analysis of electromechanical systems
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- An accurate reactive power control study in virtual flux droop control
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Localized probability of improvement for kriging based multi-objective optimization
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Research of influence of open-winding faults on properties of brushless permanent magnets motor
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Optimal design of the rotor geometry of line-start permanent magnet synchronous motor using the bat algorithm
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Model of depositing layer on cylindrical surface produced by induction-assisted laser cladding process
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Detection of inter-turn faults in transformer winding using the capacitor discharge method
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- A novel hybrid genetic algorithm for optimal design of IPM machines for electric vehicle
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Lamination effects on a 3D model of the magnetic core of power transformers
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Detection of vertical disparity in three-dimensional visualizations
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Calculations of magnetic field in dynamo sheets taking into account their texture
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- 3-dimensional computer model of electrospinning multicapillary unit used for electrostatic field analysis
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Optimization of wearable microwave antenna with simplified electromagnetic model of the human body
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Induction heating process of ferromagnetic filled carbon nanotubes based on 3-D model
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Speed control of an induction motor by 6-switched 3-level inverter
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- Application of ANNs approach for wave-like and heat-like equations
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- Study on node importance evaluation of the high-speed passenger traffic complex network based on the Structural Hole Theory
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- A mathematical/physics model to measure the role of information and communication technology in some economies: the Chinese case
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- Numerical modeling of the thermoelectric cooler with a complementary equation for heat circulation in air gaps
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- On the libration collinear points in the restricted three – body problem
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- Research on Critical Nodes Algorithm in Social Complex Networks
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- A simulation based research on chance constrained programming in robust facility location problem
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- A mathematical/physics carbon emission reduction strategy for building supply chain network based on carbon tax policy
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- Mathematical analysis of the impact mechanism of information platform on agro-product supply chain and agro-product competitiveness
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- A real negative selection algorithm with evolutionary preference for anomaly detection
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- A privacy-preserving parallel and homomorphic encryption scheme
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- Random walk-based similarity measure method for patterns in complex object
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- A Mathematical Study of Accessibility and Cohesion Degree in a High-Speed Rail Station Connected to an Urban Bus Transport Network
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- Design and Simulation of the Integrated Navigation System based on Extended Kalman Filter
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- Oil exploration oriented multi-sensor image fusion algorithm
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- Analysis of Product Distribution Strategy in Digital Publishing Industry Based on Game-Theory
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- Expanded Study on the accumulation effect of tourism under the constraint of structure
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- Unstructured P2P Network Load Balance Strategy Based on Multilevel Partitioning of Hypergraph
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- Research on the method of information system risk state estimation based on clustering particle filter
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- Demand forecasting and information platform in tourism
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- Physical-chemical properties studying of molecular structures via topological index calculating
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- Local kernel nonparametric discriminant analysis for adaptive extraction of complex structures
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- City traffic flow breakdown prediction based on fuzzy rough set
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- Conservation laws for a strongly damped wave equation
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- Blending type approximation by Stancu-Kantorovich operators based on Pólya-Eggenberger distribution
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- Computing the Ediz eccentric connectivity index of discrete dynamic structures
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- A discrete epidemic model for bovine Babesiosis disease and tick populations
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- Study on maintaining formations during satellite formation flying based on SDRE and LQR
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- Relationship between solitary pulmonary nodule lung cancer and CT image features based on gradual clustering
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- A novel fast target tracking method for UAV aerial image
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- Fuzzy comprehensive evaluation model of interuniversity collaborative learning based on network
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- Conservation laws, classical symmetries and exact solutions of the generalized KdV-Burgers-Kuramoto equation
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- After notes on self-similarity exponent for fractal structures
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- Excitation probability and effective temperature in the stationary regime of conductivity for Coulomb Glasses
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- Comparisons of feature extraction algorithm based on unmanned aerial vehicle image
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- Research on identification method of heavy vehicle rollover based on hidden Markov model
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- Classifying BCI signals from novice users with extreme learning machine
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- Topics on data transmission problem in software definition network
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- Statistical inferences with jointly type-II censored samples from two Pareto distributions
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- Estimation for coefficient of variation of an extension of the exponential distribution under type-II censoring scheme
- Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
- Analysis on trust influencing factors and trust model from multiple perspectives of online Auction
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- Coupling of two-phase flow in fractured-vuggy reservoir with filling medium
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- Production decline type curves analysis of a finite conductivity fractured well in coalbed methane reservoirs
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- Flow Characteristic and Heat Transfer for Non-Newtonian Nanofluid in Rectangular Microchannels with Teardrop Dimples/Protrusions
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- The size prediction of potential inclusions embedded in the sub-surface of fused silica by damage morphology
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- Research on carbonate reservoir interwell connectivity based on a modified diffusivity filter model
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- The method of the spatial locating of macroscopic throats based-on the inversion of dynamic interwell connectivity
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- Unsteady mixed convection flow through a permeable stretching flat surface with partial slip effects through MHD nanofluid using spectral relaxation method
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- A volumetric ablation model of EPDM considering complex physicochemical process in porous structure of char layer
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- Numerical simulation on ferrofluid flow in fractured porous media based on discrete-fracture model
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- Macroscopic lattice Boltzmann model for heat and moisture transfer process with phase transformation in unsaturated porous media during freezing process
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- Modelling of intermittent microwave convective drying: parameter sensitivity
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- Simulating gas-water relative permeabilities for nanoscale porous media with interfacial effects
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- Simulation of counter-current imbibition in water-wet fractured reservoirs based on discrete-fracture model
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- Investigation effect of wettability and heterogeneity in water flooding and on microscopic residual oil distribution in tight sandstone cores with NMR technique
- Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
- Analytical modeling of coupled flow and geomechanics for vertical fractured well in tight gas reservoirs
- Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
- Special Issue: Ever New "Loopholes" in Bell’s Argument and Experimental Tests
- Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
- The ultimate loophole in Bell’s theorem: The inequality is identically satisfied by data sets composed of ±1′s assuming merely that they exist
- Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
- Erratum to: The ultimate loophole in Bell’s theorem: The inequality is identically satisfied by data sets composed of ±1′s assuming merely that they exist
- Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
- Rhetoric, logic, and experiment in the quantum nonlocality debate
- Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
- What If Quantum Theory Violates All Mathematics?
- Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
- Relativity, anomalies and objectivity loophole in recent tests of local realism
- Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
- The photon identification loophole in EPRB experiments: computer models with single-wing selection
- Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
- Bohr against Bell: complementarity versus nonlocality
- Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
- Is Einsteinian no-signalling violated in Bell tests?
- Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
- Bell’s “Theorem”: loopholes vs. conceptual flaws
- Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
- Nonrecurrence and Bell-like inequalities
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Three-dimensional computer models of electrospinning systems
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Electric field computation and measurements in the electroporation of inhomogeneous samples
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Modelling of magnetostriction of transformer magnetic core for vibration analysis
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Comparison of the fractional power motor with cores made of various magnetic materials
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Dynamics of the line-start reluctance motor with rotor made of SMC material
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Inhomogeneous dielectrics: conformal mapping and finite-element models
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Topology optimization of induction heating model using sequential linear programming based on move limit with adaptive relaxation
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Detection of inter-turn short-circuit at start-up of induction machine based on torque analysis
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Current superimposition variable flux reluctance motor with 8 salient poles
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Modelling axial vibration in windings of power transformers
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Field analysis & eddy current losses calculation in five-phase tubular actuator
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Hybrid excited claw pole generator with skewed and non-skewed permanent magnets
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Electromagnetic phenomena analysis in brushless DC motor with speed control using PWM method
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Field-circuit analysis and measurements of a single-phase self-excited induction generator
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- A comparative analysis between classical and modified approach of description of the electrical machine windings by means of T0 method
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Field-based optimal-design of an electric motor: a new sensitivity formulation
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Application of the parametric proper generalized decomposition to the frequency-dependent calculation of the impedance of an AC line with rectangular conductors
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Virtual reality as a new trend in mechanical and electrical engineering education
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Holonomicity analysis of electromechanical systems
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- An accurate reactive power control study in virtual flux droop control
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Localized probability of improvement for kriging based multi-objective optimization
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Research of influence of open-winding faults on properties of brushless permanent magnets motor
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Optimal design of the rotor geometry of line-start permanent magnet synchronous motor using the bat algorithm
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Model of depositing layer on cylindrical surface produced by induction-assisted laser cladding process
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Detection of inter-turn faults in transformer winding using the capacitor discharge method
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- A novel hybrid genetic algorithm for optimal design of IPM machines for electric vehicle
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Lamination effects on a 3D model of the magnetic core of power transformers
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Detection of vertical disparity in three-dimensional visualizations
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Calculations of magnetic field in dynamo sheets taking into account their texture
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- 3-dimensional computer model of electrospinning multicapillary unit used for electrostatic field analysis
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Optimization of wearable microwave antenna with simplified electromagnetic model of the human body
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Induction heating process of ferromagnetic filled carbon nanotubes based on 3-D model
- Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
- Speed control of an induction motor by 6-switched 3-level inverter
