Exponential Index Research Articles

Exponential Wiener index of some silicate networks

Any graph that depicts a particular molecular structure can be given a topological graph index, also known as a molecular descriptor. This index can be used to examine numerical data and discover more about specific physical properties of molecules. Silicates are hard crystals that can be sliced into little pieces. Therefore, silicon play a vital role in many real time applications. It is the authentic material to make microchips which run in PDA, PC and other game tools. In the current study, we have determined the new topological index for a few silicate structures, including cyclic, single-chain, and Pyro silicates by grouping the vertices of the corresponding molecular graphs based on their distance sums. This index is known as the exponential Wiener index. It is expanded to include their line graphs as well. The numerical values of exponential Wiener index and multiplicative exponential Wiener index for the molecular graphs of cyclic silicates, single chain silicates and their line graphs with n vertices where n = 3, 4, 5 are explored.

Controllable three-dimensional helical energy backflow in an optical focusing field.

The energy of light generally moves forward along the light propagation direction. However, in recent years, a counter-intuitive effect has been discovered that energy backflow can occur in certain special light fields. Presently, controlling the energy backflow of light for application requirements is becoming a significant challenge. Here we propose a method to generate and control a three-dimensional helical energy backflow (HEB) optical field. By designing an exponential helical conical phase mask, the energy backflow region in a common tightly focused field can be stretched into a three-dimensional helical shape with an ultra-long longitudinal length. The length of the HEB region can be flexibly extended up to 47.6λ by varying the exponential phase index and the initial topological charge. Furthermore, we demonstrate that nanoparticles within the HEB field are subjected to optical pulling forces, thereby drawing the nanoparticles back to the light source. This work presents a new, to the best of our knowledge, approach for the three-dimensional manipulation of the energy backflow field with potential applications in nanoparticle manipulation and detection.

Extremal Trees for Logarithmic VDB Topological Indices

Vertex-degree-based (VDB) topological indices have been applied in the study of molecular structures and chemical properties. At present, the exponential VDB index has been studied extensively. Naturally, we began to consider the logarithmic VDB index lnTf. In this paper, we first discuss the necessity of a logarithmic VDB index, and then present sufficient conditions so that Pn and Sn are the only trees with the smallest and greatest values of lnTf(T). As applications, the minimal and maximal trees of some logarithmic VDB indices are determined. Through our work, we found that the logarithmic VDB index lnTf has excellent discriminability, but the relevant results are not completely opposite to the exponential VDB index. The study of logarithmic VDB indices is an interesting but difficult task that requires further resolution.

Preparation, mechanical analysis and investigation of swelling behavior of boron nitride reinforced hydrogel polymer composite films

AbstractThis study presents the preparation, hydrogel kinetics, and mechanical analysis of Boron Nitride (BN) reinforced PVA/PVP/PEO‐BN hydrogel composite films using Polyvinyl alcohol (PVA), Polyvinyl pyrrolidone (PVP), and Polyethylene oxide (PEO) commercial polymers. Dynamic mechanical analysis tests reveal that PVA90PEO5PVP5‐BN composite films exhibit plastic‐viscoelastic behavior under a maximum force of 18 N. The Young's Modulus values for PVA90PEO5PVP5, PVA90PEO5PVP5‐BN%10, and PVA90PEO5PVP5‐BN%20 are 0.22, 0.32, and 0.44 MPa, respectively. The highest % Strain value is observed in PVA90PEO5PVP5‐BN%20, reaching 279.80%. In the hydrogel kinetics study, Schott's and Fickian models are utilized. The regression from the Fickian model is quite low, with exponential diffusion index, n, values lower than 0.5, indicating a classical Fickian water diffusion mechanism. Schott's model provides graphs with significantly higher regression compared to the Fickian model. The results indicate compatibility with the other model and confirm the presence of water‐based diffusion. Equilibrium swelling values (Se, g H2O/g gel) are 6.71, 7.03, and 7.91 for PVA90PEO5PVP5, PVA90PEO5PVP5, PVA90PEO5PVP5‐BN%10, and PVA90PEO5PVP5‐BN%20, respectively. Differential Scanning Calorimetry (DSC) analysis results show that the glass transition temperatures, Tg, are 52.37, 60.20, and 63.05°C for PVA90PEO5PVP5, PVA90PEO5PVP5‐BN%10, and PVA90PEO5PVP5‐BN%20, respectively.

Complete solution to open problems on exponential augmented Zagreb index of chemical trees

One of the crucial problems in combinatorics and graph theory is characterizing extremal structures with respect to graph invariants from the family of chemical trees. Cruz et al. (2020) [7] presented a unified approach to identify extremal chemical trees for degree-based graph invariants in terms of graph order. The exponential augmented Zagreb index (EAZ) is a well-established graph invariant formulated for a graph G asEAZ(G)=∑vivj∈E(G)e(didjdi+dj−2)3, where di signifies the degree of vertex vi, and E(G) is the edge set. Due to some special counting features of EAZ, it was not covered by the aforementioned unified approach. As a result, the exploration of extremal chemical trees for this invariant was posed as an open problem in the same article. The present work focuses on generating a complete solution to this problem. Our findings offer maximal and minimal chemical trees of EAZ in terms of the graph order n.

Open problem on the maximum exponential augmented Zagreb index of unicyclic graphs

Open problem on the maximum exponential augmented Zagreb index of unicyclic graphs

Study of Air Ordering Procedure for Information Broadcasting

Wireless data broadcasting is a new way to share public information with many mobile users. To measure how well these systems work, we look at two main things: how long it takes to access the information (access latency) and how long the device has to stay active (tuning time). Indexing technology helps reduce tuning time by letting devices find information quickly and then switch to a low-power mode while waiting. Different indexing methods work better or worse, so comparing them can be hard. In this paper, I look at and compare some popular indexing methods for wireless data broadcasting, such as distributed index, exponential index, hash table, and Huffman tree index. I use probability theory to analyze how each method performs. DOI: https://doi.org/10.52783/tjjpt.v45.i02.6393

Analysis of boundary layer characteristics in supergravitational turbulent thermal convection

We investigate the boundary layer characteristics within annular centrifugal Rayleigh–Bénard convection (ACRBC) considering a Rayleigh number Ra∈[108,1011], a Prandtl number Pr = 10.7, and an inverse Rossby number Ro−1=16. Our study is based on the temperature and velocity data obtained from direct numerical simulations. Different from the flow over a flat plate, the ACRBC system bifurcates into three regions: the plume-impacting regions, plume-ejecting regions, and plume-sweeping regions, and all three regions are moving with the zonal flow. Our focus is primarily on the temperature dynamics within the plume-sweeping region, where the wind of large-scale circulation shears the boundary. We determine the transient thermal boundary layer thickness over time using the slope method, relying on the temperature curve's orientation relative to the wall. Notably, the probability density function distribution of the thermal boundary layer thickness is reminiscent of traditional RBC systems, albeit with a more extended exponential tail. Employing a dynamic frame based on time resampling, we discern that the temperature boundary layer traits align with the Prandtl–Blasius boundary layer theory. In conclusion, we show that the exponential decay index for the thermal boundary layer thickness harmonizes with the system's heat transfer scaling law. It is found that the ratio between the inner and outer boundary layer thickness remains stable, providing theoretical guidance for the design and control of the internal flow field of high-speed rotating machinery.

On the exponential augmented Zagreb index of graphs

On the exponential augmented Zagreb index of graphs

Analyzing activity and injury risk in elite curling athletes: seven workload monitoring metrics from session-RPE.

The study aimed to compare the differences in the performance of seven session-rating of perceived exertion (RPE)-derived metrics (coupled and uncoupled acute: chronic workload ratio (ACWR), weekly ratio of workload change, monotony, standard deviation of weekly workload change, exponentially weighted moving average (EWMA), and robust exponential decreasing index (REDI)) in classifying the performance of an injury prediction model after taking into account the time series (no latency, 5-day latency, and 10-day latency). The study documented the RPE of eight curlers in their daily training routine for 211 days prior to the Olympic Games. Seven Session-RPE (sRPE)-derived metrics were used to build models at three time series nodes using logistic regression and multilayer perceptron. Receiver operating characteristic plots were plotted to evaluate the model's performance. Among the seven sRPE-derived metrics multilayer perceptron models, the model without time delay (same-day load corresponding to same-day injury) exhibited the highest average classification performance (86.5%, AUC = 0.773). EMWA and REDI demonstrated the best classification performance (84.4%, p < 0.001). Notably, EMWA achieved the highest classifying accuracy in the no-delay time series (90.0%, AUC = 0.899), followed by the weekly load change rate under the 5-day delay time series (88.9%, AUC = 0.841). EWMA without delay is a more sensitive indicator for detecting injury risk.

Formula omitted]-graphs with maximum exponential second Zagreb index

For a graph G=(V,E), the exponential second Zagreb index is defined as eM2(G)=∑uv∈Eed(u)d(v), where d(v) is the degree of vertex v∈V. Recently, Eliasi (2022) posed a conjecture about (n,m)-graphs with maximum eM2 when n≤m≤2n−3. In this paper, we show that this conjecture is true. Furthermore, we determine the graphs with maximum eM2 among all (n,m)-graphs when 2n−3<m.

Application of Lorentz geometry to refractive optical systems

Lorentz geometry represents the mathematical foundation of Einstein’s general relativity. From the concept of the cone of edge rays, it was introduced successfully in the study of homogeneous media nonimaging optics, Gutiérrez et al. (1996), Winston et al. (2005). In this paper we present a new Lorentzian geometry formalism to analyze optical systems with refractive elements. We have obtained the general system of partial differential equations and the integrability conditions to be fulfilled. This new mathematical formalism is the basis for a new optical technique to analyze the irradiance pattern produced by refractive optical systems. We apply this Lorentzian formalism to an optical media with exponential refraction index. Comparisons between irradiance patterns obtained by this formalism and by raytrace simulations are also presented.

Computational analysis of melting radiative heat transfer for solar Riga trough collectors of Jeffrey hybrid-nanofluid flow: A new stochastic approach

Current investigation deals with the melting heat transfer for the Jeffrey hybrid-nanofluid flow in parabolic trough solar collectors through Darcy Forchheimer porous media over a variable thick vertical elongation Riga surface under the effect of solar radiation. The impacts of viscous dissipation and Joule heating are also investigated. Equations governing the Jeffrey’s hybrid nanofluid flow are higher-order non-linear partial differential equations (PDEs). These governing PDEs are transformed into the non-linear ordinary differential equations (ODEs) by introducing appropriate similarity transformations and dimensionless parameters. Runga Kutta’s fourth-order numerical scheme is implemented with the shooting technique to solve coupled higher-order ODEs. Results for velocity profile, temperature profile, drag coefficient and Nusselt number are discussed for various influential parameters. Artificial neural networking is also performed to predict Nusselt numbers in different cases and scenarios. The Artificial Neural Network gives the desired outputs with the highest possible accuracy. It is observed that the temperature profile rises with increase in Hartmann number, porosity parameter, and Forchheimer number. However thermal profile worsens with escalates in the exponential index. Radiant energy of the Sun is a renewable energy source available in considerable amounts in our living environment. A parabolic trough solar collector is an efficient solar collector that stores the concentrated incoming radiant energy from the Sun to fulfill the requirement of high temperatures in thermal energy storage systems. It has various applications in solar-powered appliances for cooking and air conditioning, solar-powered systems for treating wastewater, photovoltaic lighting, solar-powered cars, aircraft, etc.

Breakup characteristics of ultra-high-pressure GDI spray of a single-hole injector under various thermodynamic conditions

In this study, breakup characteristics of a single-hole GDI injector were investigated at various fuel temperatures and injection pressures. High-speed Mie-scattering imaging technique was employed. Injection pressure varied from 100 to 1000 bar, fuel temperature ranged from 21 to 120 °C, and ambient pressure was adjusted from 0.5 to 4.5 bar to simulate various actual engine conditions. Three semi-empirical models of spray penetration were evaluated. Results showed that the revised Hiroyasu and Arai model demonstrated the best match with an average R2 value of 0.986 at fuel temperature of 21 °C. For evaporating conditions, fuel temperature to ambient temperature ratio T/T0 with an exponential index of −0.47 was incorporated into the model, and it provided excellent penetration prediction under all conditions with an average R2 value of 0.981. Results demonstrated that it is inapplicable to rise temperature to compensate penetration increment induced by increased injection pressure. Furthermore, the normalized variation in spray penetration resulted by increased injection pressure or fuel temperature was nearly independent of other injection parameters. Breakup time and length derived from the revised model closely matched the experimental data, although the accuracy decreased slightly at higher ambient pressures.

A Unified Approach for Extremal General Exponential Multiplicative Zagreb Indices

The study of the maximum and minimal characteristics of graphs is the focus of the significant field of mathematics known as extreme graph theory. Finding the biggest or smallest graphs that meet specified criteria is the main goal of this discipline. There are several applications of extremal graph theory in various fields, including computer science, physics, and chemistry. Some of the important applications include: Computer networking, social networking, chemistry and physics as well. Recently, in 2021 exponential multiplicative Zagreb indices were introduced. In generalization, we introduce the generalized form of exponential multiplicative Zagreb indices for α∈R+\{1}. Furthermore, to see the behaviour of generalized first and second exponential Zagreb indices for α∈R+\{1}, we used a transformation method. In term of the two newly developed generalized exponential multiplicative Zagreb indices, we will investigate the extremal bicyclic, uni-cyclic and trees graphs. Four graph transformations are used and some bounds are presented in terms of generalized exponential multiplicative Zagreb indices.

Nonlinear waves in a composite optical structure containing a dielectric layer between a photorefractive crystal and a medium with an exponential index profile

Three-layered asymmetric structure with different optical properties of the layers is considered. The structure consist of a photorefractive crystal with diffusion nonlinearity and very weak dark intensity, which is separated by optically linear dielectric layer from medium with spatially distributed refractive index according with an exponential profile. The paper objective is a theoretical analysis of wave disturbances arising from the introduction of a dielectric waveguide layer between photorefractive and graded-index media. Waves of new four kinds are described theoretically by exact solutions to the wave equation with spatial inhomogeneous and intensity-dependent refractive index. The waves differ from each other in the distribution profile and the range of existence. Waves with both oscillating and non-oscillating field profiles inside the dielectric layer are obtained. It is found that the dielectric layer significantly affects the main characteristics of the field distribution in the waveguide structure in the direction transverse to the layers. It is shown controlling the values of the characteristics of the layers allows selecting the required field profile, which can be used to suppress oscillations or increase the field intensity in certain regions of a layered structure.

Quantifying the Scale Effect of the Relationship between Land Surface Temperature and Landscape Pattern

The spatial scaling of patterns and processes is a hot topic of research in landscape ecology, and different scales may yield completely inconsistent results. Therefore, to understand the impact of the scale effect on urban heat island effect, this study analyzes the correlation between surface temperature and landscape index at different spatial scales over Nanjing. The scale effect is calculated thorough curve fitting of the Pearson’s correlation coefficient between ten landscape indices and land surface temperature at different window sizes, and the optimal one is determined. We have found that landscape indices can be divided into exponential and Gaussian landscape indices whose correlation with land surface temperature at different windows conforms to binomial exponential or multi-Gaussian functions, respectively. The optimal window size is approximately 4000–5100 m for exponential landscape indices, 1000–2000 m for aggregation index (AI) and percentage of like adjacencies (PLADJ), 6330 m for contagion (CONTAG) and 4380 m for total edge contrast index (TECI). Moreover, CONTAG and TECI have a high correlation coefficient plateau where the Pearson correlation coefficient is high and changes by less than 0.03 as the window size changes by more than 3000 m, which makes it possible to decrease the window size in order to save the calculation time without an obvious decrease in the Pearson correlation coefficient. To achieve this, we proposed a suitable window selection function so that the window size becomes 4260 m and 2070 m, respectively. The window sizes obtained in this study are just suitable in Nanjing, but the window sizes in other cities can also be obtained by the method in this study. This study provides a reference for future research on the relationship between landscape pattern and land surface temperature and its driving mechanisms, as well as for the impact of urban land use planning on the heat island effect.

Adaptive memory-based event-triggering resilient LFC for power system under DoS attack

Considering denial-of-service (DoS) attacks in the communication network, this work develops the adaptive memory-based event triggering resilient LFC approach for multi-area power systems with wind power. First of all, the impact of each DoS attack is evaluated by the amount of continual information loss it causes. Meanwhile, an adaptive memory event triggering mechanism (AMETM) is proposed, where the dynamic event triggering parameter and the triggering mechanism are all related to the historical transmitted states. Then, based on the proposed AMETM and DoS attacks, a sampling-data LFC model is established. In order to construct the LFC scheme with the desired faster state response, the exponential decay index is introduced as a performance index. Next the exponential stability problem of LFC sampling model is transformed into the asymptotic stability problem of polynomial uncertain system. Secondly, the exponential stability criterion and the memory controllers design method are obtained by bilateral closed-loop Lyapunov–Krasovskii functional (LKF) and LMI technology, which ensure the proposed memory control scheme can quickly stabilize the LFC system under a longer DoS attack. Lastly, a one-area typical LFC system is presented to verify the superiority and effectiveness of the proposed AMETM. The resilient control with fast state response and DoS attack is studied by two-areas LFC systems with wind power.

Impact of convective tip condition on the transient thermal behavior of wet porous convective radiative exponential profiled inclined longitudinal fin structures subject to heat generation

AbstractA convective‐radiative fully wet porous inclined longitudinal fin of exponential profile is the focus of the current work. The thermal behavior of the fin under unsteady conditions has been analyzed for adiabatic and convective boundary conditions. The exponential fin and its counterpart inverted exponential fin are simultaneously investigated by considering temperature‐relevant thermal conductivity and heat generation. The modeled governing equation upon nondimensionalization reduces to a partial differential equation which is computed by employing the finite difference approach. The impact of relevant parameters like the convective parameter, radiative parameter, wet porous parameter, dimensionless time, exponential index, dimensionless ambient temperature, generation number, thermal conductivity parameter, and angle of inclination on thermal characteristics of exponential and inverted exponential fin structures with adiabatic and convective boundary restrictions have been examined. One of the main outcomes was that the inverted exponential fin with an adiabatic tip gives rise to the highest thermal curve, and the tapered exponential fin with a convective tip resulted in the lowest thermal curve.

Nonlinear guided wave propagation along layers with the exponential index profile, constant index and the Kerr nonlinearity

Layered planar waveguide consisting of the layer with an exponential profile of the dielectric permittivity, optical linear layer, and nonlinear medium of Kerr-type is considered. Exact analytical solutions describing the nonlinear guided waves in such waveguide are found. The profiles of the waves in perpendicular to the layers direction are analyzed. The distinctions between layered waveguide structures with a linear optical substrate and a nonlinear one and the impact of dielectric interlayer separating the exponentially graded-index main waveguide layer and nonlinear material of Kerr-type are discussed. The presence of a dielectric interlayer leads to the possibility of the appearance of field oscillations inside it. The possibility of controlling the field localization length across the layers and its amplitude by changing the interlayer width is shown.