*3.1.* Π*-A Isotherm*

Figure 2a shows the Π-*A* isotherm for DPPC in the HKM buffer. DPPC shows a liquid expanded (LE) phase at a very low surface pressure, followed by a minor liquid expanded–liquid condensed (LE–LC) coexistence region at Π ≈ 5 mN·m−1. Further compression yields a LC phase, characterized by a long range molecular order, until it reaches the collapse at Π ≈ 54 mN·m−1. The Π-*A* isotherm does not show a well-defined LE–LC coexistence region as DPPC on water, characterized by a well-defined plateau of coexistence [27,38]. In Figure 2b, we report the corresponding compressional elastic modulus *C*s −1, calculated from the surface pressure isotherm following

$$\text{C}\_{s}\text{ $^{-1}$ } = -A \left(\frac{\partial \Pi}{\partial A}\right) \tag{9}$$

where Π represents the surface pressure and *A* the surface area. The Π-*A* isotherm shown in Figure 2a is quite similar to those previously reported in the literature for DPPC in water, consequently, we interpret them in a similar way [27]. At increasing Π, DPPC molecules are pushed closer and the compressional elastic modulus increases until it reaches a maximum. The low Π region is commonly assigned to a 2D liquid expanded state (LE). A minimum in the compression modulus at Π ≈ 5 mN·m−1, is commonly attributed to the existence of a LE–LC phase transition, which can be more clearly observed than the slight pseudo plateau in the isotherm (Figure 2a). The global maximum value of *C*s −1 is 130 mN·m−1, which corresponds to the LC phase. The values of *C*s −1 in this LC phase are smaller than the ones observed for DPPC at the air/water interface, which present a maximum at *C*s −1 ≈ 230 mN·m−1 [27], indicating a less condensed monolayer. Importantly, Figure 2 shows that a DPPC monolayer in the presence of buffer containing divalent salts exhibits more lateral compressibility due to a less acyl chain compaction, and, therefore, is more permeable [53].

**Figure 2.** (**a**) Π-*A* isotherm of a DPPC monolayer at the air/buffer interface and (**b**) corresponding *C*s −1 as a function of Π.

#### *3.2. NR Results*

Neutron reflectometry measurements were performed to study the structure of DPPC monolayers in the LC phase. In particular, we selected a sample with a surface pressure value of 30 mN·m−1 corresponding to an area per molecule of 53.8 Å2, well above the LE–LC coexistence phase. This guarantees a laterally homogeneous interface on the length scale of the in-plane neutron coherence length, on the order of several microns, and implies that the measured NR can be correlated

with the SLD depth profile averaged across the interfacial area delimited by this coherence length. The reflectivity profiles were recorded over the whole Q-range accessible in three isotopic contrasts: h-DPPC, cm-DPPC, and d62-DPPC in ACMW as shown in Figure 3a. As a reference, the measurement of the bare air/D2O interface is shown also in Figure 3a, including a fit to the data that corresponds to a roughness, *r*0, of 2.8 ± 0.1 Å in agreement with the theoretical value expected for thermally excited capillary waves (∼ *k*B*T*/γ0), with γ0 being the interfacial tension of the bare D2O interface [45,46].

**Figure 3.** (**a**) Neutron reflectivity data of pure D2O interface (black circle), h-DPPC (red squares), cm-DPPC (blue triangles), and d62-DPPC (magenta diamonds) monolayers at Π = 30 mN·m−1 in HKM-ACMW. The fitting curves of the bare D2O interface (orange curve), h-DPPC (red), cm-DPPC (blue), and d62-DPPC (magenta) monolayers are shown. The χ2 value is 2. The figure is displayed on an *R*(Qz)Qz 4 scale to highlight the quality of the fits at high Qz values. (**b**) SLD profiles normal to the interface of monolayers of h-DPPC (red), cm-DPPC (blue), and d62-DPPC (magenta) monolayers at Π = 30 mN·m−1. (**c**) Volume fraction profiles normal to the interface of monolayers to highlight the distribution of tails (violet) and heads (green). (**d**) Cross-sectional area profiles normal to the interface of monolayers to highlight the distribution of tails (violet) and heads (green). Note that the area of the head-groups here does not consider the hydration.

The neutron reflectivity profiles were fitted according to a two-layer model, based on the model recently reported by Campbell et al. [27]. In detail, the model consists of a first layer containing the lipid aliphatic tails in contact with air and, a second one, containing the polar headgroups submerged in the aqueous subphase (see Figure 1a). All parameters used to describe both layers (such as the values of Σ*b* and molecular volumes) are included in Table 1. The best fit of the reflectivity profiles measured is also included in Figure 3a as solid lines. The resulting SLD profiles across the interface are plotted in Figure 3b.

The structural parameters obtained from the fits are summarized in Table 2. The roughness of the three interfaces (air/tails-layer, tails-layer/heads-layer, and heads-layer/subphase) was constrained to be equal. The value obtained from the fitting is 3.0 ± 0.5 Å, which is perfectly consistent with the presence of capillary waves due to thermal fluctuations (usually estimated through the relation: *r* ≈ *r*0 γ0/(γ0 − Π) [54,55]). The value of the thickness of DPPC monolayer is 23.5 Å, with 15.0 Å corresponding to the aliphatic tails in contact with air. Using the parameters from Tables 1 and 2, the variation of the volume fraction, *f* DPPC(*z*), with the distance to the interface, was calculated using the difference of two error functions as follows

$$f\_{\text{DPPC}}(z) = \begin{cases} \frac{1}{2} f\_{\text{tails}} \left[ \text{ERF} \left( \frac{z}{\sqrt{\gamma\_2}} \right) - \text{ERF} \left( \frac{z - t\_{\text{tails}}}{\gamma/\sqrt{2}} \right) \right], & 0 \le z \le t\_{\text{tails}}\\\frac{1}{2} f\_{\text{hads}} \left[ \text{ERF} \left( \frac{z}{\gamma/\sqrt{2}} \right) - \text{ERF} \left( \frac{z - t\_{\text{today}}}{\gamma/\sqrt{2}} \right) \right], & t\_{\text{tails}} \le z \le t\_{\text{tails}} + t\_{\text{heads}} \end{cases} \tag{10}$$

The structural information elucidated by NR on DPPC monolayers can be better interpreted, therefore, by the volume fraction profiles and the corresponding cross-sectional area profiles as a function of the distance from the interface, which are shown in Figure 3c,d, respectively. Such as the volume fraction, the cross-sectional area profile is modulated by the same error function and it is calculated as follows:

$$A\_{\rm DPPC}(z) = \begin{cases} \frac{V\_{\rm m, t \rm tils}}{2 \int\_{t \rm tils/t\_{\rm tils}} \left[ \rm{ERF} \left( \frac{z}{r/\sqrt{2}} \right) - \rm{ERF} \left( \frac{z - t\_{\rm tils}}{r/\sqrt{2}} \right) \right], & 0 \le z \le t\_{\rm tils} \\\frac{V\_{\rm m, b\rm bads}}{2 \int\_{t \rm bends} \int\_{b\rm bads} \left[ \rm{ERF} \left( \frac{z}{r/\sqrt{2}} \right) - \rm{ERF} \left( \frac{z - t\_{\rm bends}}{r/\sqrt{2}} \right) \right], & t\_{\rm tils} \le z \le t\_{\rm tils} + t\_{\rm bends} \end{cases} \tag{11}$$

The values used to calculate the cross-sectional area from Equation (11) are collected from Tables 1 and 2.

**Table 2.** Parameters resulting from the data modeling. \* The hydration degree of the headgroups are considered here.


**4. Economic evaluation**

The evaluation of the economic feasibility of the process of obtaining of the microalgas *D. salina* was carried out on the base of the experience gathered until that moment with the one made for the Chlorella sp. [12, 13], and keeping in mind the pilot plant capacity installed at "Industrial de alimentos" (INDAL).

In this calculations was considerate:

*a—a volume of cultivation of 3 m3 .*

*b—a yield of 0.8 kg of dry alga for m3 of culture medium.*

*c—a productivity of humid alga that represents a biomass from 1.6 to 2.1 kg/day.*

*d—a retention time of 15–20 days for each harvesting, representing a crop cycle every two weeks.*

*e—an available time of 300 days troops, foreseeing 60 days of bottom of technological requirement and other causes.*

The methodology for these calculations were carries out using the pattern of economic evaluation created by the ABL of the University of Murdoch, Western Australia, being calculated the cost of a kg of alga for the pilot plant installed at INDAL industry, as well as for an industrial plant. In the **Table 2** a summarized information of the calculation about the unitary cost is presented. The cost obtained to produce 1 kg of alga, as it has been calculated with the pattern in question it was among \$6.4 and \$4.0 USD.


**5. Conclusions**

In this paper, I argued that we need to read Romans not within the antithetical framework of Judaism–Christianity but as a text reflecting the conflicts between a cultic tendency represented by the gentile members in the Roman church and a Jewish sectarian tendency shown in Paul's arguments. Based on Romans, we learn that some gentile members were assimilating the Jewish law as a means of attaining the Greco–Roman virtue of self-mastery. With this erroneous understanding, they were thinking that they have already attained salvation and were standing in a superior position compared to others. For Paul, this understanding of the law was seriously problematic because it misunderstands the Jewish tradition centered around faith, which is also in line with the gospel of Jesus Christ. Another problem was the assumption that gentile believers have now replaced Israel and that unbelieving Jews are destined for destruction. For Paul, this was also an incorrect understanding of God's faithfulness and mercy that saved gentiles will save "all Israel" at the time of eschaton.

As seeing the danger of gentile believers being drifted away from the root of Judaism, Paul, throughout the letter, strives to reinstate the sectarian boundary and the symbolic unity with Israel. For this goal, he first corrects the gentile misunderstanding of law and salvation by stressing the futuristic aspect of salvation and God's sovereignty in judging humanity. Paul also corrects the wrong idea about the fate of unbelieving Jews by elaborating on God's faithfulness promised to Israel and the overarching divine plan for salvation. Paul also reminds his gentile audience of the Jewish root of their faith, while stressing the separateness of the church from the Roman world. Throughout these efforts, Paul tries to create a united community consisted of believing Jews and gentile, which functions as the group of elects in the time of eschaton, but in no way separated from Judaism.

Did Paul succeed? From the late first-century C.E., we begin to see more churches in different regions of the empire establishing their identities separated from contemporaneous Jewish groups. We also observe the emergence of Christian anti-Jewish polemics, which continued throughout the history of Christianity. In this sense, it might be argued that Paul, ultimately, have failed in uniting two separating groups together and setting up a model for later Jewish–Christian relations. Reflecting the history of opposition between Christians and Jews throughout centuries, however, Paul's ardent appeal for unity in Romans seems to be more meaningful than ever.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The author declares no conflict of interest.

19 For Paul's other efforts to demarcate a firmer boundary of the Roman church from its cultural context, see (Rock 2012).


**Chapter 7 123**

The Dual Nature of Mesenchymal Stem Cells (MSCs): Yin and Yang of the Inflammatory Process *by Carmen Ciavarella and Gianandrea Pasquinelli*

Preface

Since the first description or discovery of mesenchymal stem cells and induced pluripotent stem cells in the 1960s and 2006 respectively, these types of stem cells have witnessed enormous progress not only in their manufacture or preparation but also in their clinical utilization to treat several medical diseases. Many experts in the field believe that, in the near future, these specific types of stem cell may reshape the field of medical therapeutics and may eventually become potentially curative

The book presents the latest information on both stem cell types. It is divided into three sections that cover several topics including: update on both cell types with a focus on their clinical utilizations; the use of various types of mesenchymal stem cells in the treatment of infectious diseases, osteoarthritis, and immunological disorders; in addition to the nature as well as the angiogenic paracrine properties of mesenchymal stem cells. Each chapter was written by scientists and clinicians with

I would like to thank the authors for their valuable contributions, as well as the publishing manager Ms. Sara Debeuc and IntechOpen staff for their remarkable

Department of Hematology and Hematopoietic Stem Cell Transplantation,

Oncology Center, King Fahad Specialist Hospital,

**Khalid A. Al-Anazi**

Dammam, Saudi Arabia

for several chronic and intractable medical conditions.

efforts that ultimately made this book project a reality.

expertise in the field.


*Edited by Yongcun Feng*

Worldwide oil and gas development has shifted from conventional reservoirs to unconventional and deepwater reservoirs, characterized by high pressure, high temperature, ultra-low permeability, and extensively developed natural fractures. This transition to increasingly hostile environments introduces new challenges in well drilling and completion, such as downhole drilling issues, formation damage, and reduced productivity. Aiming to solve the challenges, drilling and completion technologies have excelled greatly in the past two decades. This book covers managed pressure drilling (MPD), the role of artificial intelligence (AI) in refining drilling processes, and the transformative effects of digitalization and automation. Emphasizing efficiency, safety, and environmental responsibility, the book also integrates methods like casing while drilling for improved efficiency, advanced diagnostics for rig safety, stabilization techniques for wellbores in fractured reservoirs, cement sheath integrity maintenance, and the optimization of continuous gas lift. Bridging theoretical concepts with practical applications, the narrative offers insights into both operational techniques and safety strategies, drawing from past experiences. The current state-of-the-art theories, technologies, and practices are covered, bridging the gaps between fundamental theories and engineering applications.

ISBN 978-1-83768-316-1

Advances in Oil and Gas Well Engineering

Published in London, UK © 2024 IntechOpen © bashta / iStock


**3. Entropy Generation**

The entropy generation under the aforementioned assumptions is given as below:

$$E\_{\rm gren}^{\prime\prime\prime} = \frac{k\_f}{T\_0} \left[ \frac{k\_{nf}}{k\_f} + \frac{16T\_{\rm ox} \overline{\alpha}^\* \sigma^\*}{3k^\* k\_f} \right] \left( \frac{\partial T}{\partial y} \right)^2 + \frac{\mu\_{nf}}{T\_0} \left( \frac{\partial u}{\partial y} \right)^2 + \frac{\sigma\_{nf}}{T\_0} B^2(t) u^2 \cos^2 \varepsilon \tag{18}$$

where all terms defined in Equation (15) portray the usual meaning. The entropy generation *NG* is defined as

$$N\_{\rm G} = \left(\frac{k\_{nf}}{k\_f} + \frac{4}{3}R\right) \text{Re}\,\varepsilon \boldsymbol{\theta}^{\prime} + \frac{1}{\left(1-\phi\right)^{2.5}} \frac{\text{Br}\,\text{Re}\_{\rm x}}{\alpha} \text{f}^{\prime\prime 2} + \frac{\text{BrM}}{\alpha} \frac{\sigma\_{nf}}{\sigma\_f} \cos^2\varepsilon f^{22} \tag{19}$$

where *S*0 -- and *Sgen*-- are the characteristic entropy generation rate and the entropy generation rate. The parameters defined in the above equation are given as

$$\alpha = \frac{\Delta T}{T\_{\text{w0}}}, Br = \frac{\mu\_f \mu\_w}{k\_f \Delta T}, Re\_x = \frac{\mu\_w \chi}{\mathbf{v}\_f} \tag{20}$$

#### **4. Results and Discussion**

This section is devoted to witnessing the impression of numerous parameters on the involved profiles whilst keeping in view their physical significance. The MATLAB built-in function bvp4c is utilized to address the differential Equations (9), (10), and (16) with the associated boundary conditions of Equation (11). To solve these, first we have converted the 2nd and 3rd order differential equations to the 1st order by introducing new parameters. The tolerance for the existing problem is fixed as 10−6. The initial guess we yield must satisfy the boundary conditions asymptotically and the solution as well. The results show the influence of solid volume fraction (ϕ), dimensionless film thickness (λ), magnetic parameter (*M*), unsteadiness parameter (*S*), radiation parameter (*R*), thermal relaxation parameter (γ), and non-uniform heat source/sink parameter on the velocity, temperature and entropy generation profiles. Further, the numerical values for the Skin friction and Nusselt number are given in Tables 3 and 4 for different parameters. The numerical values of the parameters are fixed as ϕ = 0.1, *A*\* = *B*\* = λ = γ = 0.5 = *S*, *R* = 1.0 = *M*, and *Pr* = 6.2. Figures 2 and 3 display the impact of solid volume fraction (ϕ) on axial velocity and temperature distribution. For incremented values of the solid volume fraction (ϕ), the velocity and temperature profiles enhance in case of both SWCNTs and MWCNTs. Actually, the convective flow and the solid volume fraction are directly proportionate with each other and this is the main reason behind the enhancement of axial velocity and the temperature of the fluid.

**Table 3.** The comparison table of –θ'(0) with Sandeep [17] for varied estimates of *S* when *R* = *M* = γ = 0, *Pr* = 1.0.

**Figure 2.** The illustration of ϕ versus *f*'(η).

**Figure 3.** The illustration of ϕ versus θ(η).

Figures 4 and 5 depict the behavior of axial velocity and the temperature field for the growth estimates of the film thickness parameter λ. It is found that both velocity and temperature profiles diminish for increasing values of the film thickness parameter λ. In fact, the more the film thickness, the lesser the fluid motion. This is because of the fact that higher values of film thickness dominate the viscous forces, eventually diminishing the fluid velocity. Similar behavior is observed for the temperature field.

**Figure 4.** The illustration t of λ versus *f*'(η).

**Figure 5.** The illustration t of λ versus θ(η).

The effect of the magnetic parameter *M* on the velocity and temperature fields can be visualized in Figures 6 and 7. Figure 6 displays the impact of the magnetic parameter *M* on axial velocity. The is the axial velocity of the declining function of the magnetic parameter *M*. Physically, by enhancing the magnetic parameter *M*, the Lorentz force is strengthened in the flow, which has a tendency to resist the fluid's motion and slow it down. This force also creates heat energy in the flow. Consequently, the temperature distribution increases both the SWCNTs and MWCNTs, which is displayed in Figure 7.

**Figure 7.** The illustration of *M* versus θ(η).

Figures 8 and 9 show the effect of the unsteadiness parameter *S* on the velocity and temperature distributions. It is found that with the increase of the unsteadiness parameter *S*, the axial velocity diminishes. Physically, the bouncy effect acts on the flow and diminishes it due to the increase in the unsteadiness parameter *S*. Therefore, the thermal and momentum boundary layer thicknesses decrease.

**Figure 8.** The illustration of *S* versus *f*'(η).

**Figure 9.** The illustration of *S* versus θ(η).

Figure 10 determines the consequence of the thermal relaxation parameter γ on the temperature of the fluid. It is concluded that the temperature diminishes for increased values of the thermal relaxation parameter γ. The temperature tends to be sharper near the boundary as the value of γ is higher than the points on the growth in the wall slope of the temperature profile.

**Figure 10.** The illustration of γ versus θ(η).

Figure 11 demonstrates the impact of the radiation parameter *R* on the temperature profile. It is comprehended that the temperature field is an increasing function of the radiation parameter *R*. It is also concluded that the thermal boundary layer thickness for both carbon nanotubes is increased. In fact, larger estimates of the radiation parameter reduce the mean absorption coefficient and enhance the radiative heat flux's divergence. Due to this, the temperature of the fluid is upsurged.

**Figure 11.** The illustration of *R* versus θ(η).

The influence of non-uniform heat source/sink parameters *A*\* and *B*\* on the temperature distribution is shown in Figures 12 and 13. It can be understood that the temperature profile augments the boosted estimates of non-uniform heat source/sink parameters.

**Figure 12.** The illustration of *A*\* versus θ(η).

**Figure 13.** The illustration of *B*\* versus θ(η).

The effect of Brinkman number (*Br*), magnetic parameter (*M*) and Reynolds number (*Rex*) on the averaged entropy generation number is demonstrated in Figures 14–16. It is concluded that the entropy generation number increases for mounting estimations of Brinkman number (*Br*), magnetic parameter (*M*) and Reynolds number (*Rex*) for both SWCNT and MWCNT.

**Figure 14.** The illustration of *Br* versus *NG*(η).

**Figure 15.** The illustration of *M* versus *NG*(η).

**Figure 16.** The illustration of *Rex* versus *NG*(η).

Table 3 is erected to envision the precision of the presented model by comparing it with Sandeep [17] who discusses the flow of nanofluids past a thin film under the influence of the magnetic field. To make a comparison, we have neglected the impacts of the volume fraction, electrical conductivity, and thermal relaxation parameters. Excellent alignment is achieved between both results.

Table 4 shows the estimates of the Skin friction coefficient for different parameters. It is seen that the Skin friction coefficient increases for growing values of the magnetic parameter, solid volume fraction, unsteadiness parameter, and film thickness. Table 5 demonstrates the numerical values of Nusselt number for numerous parameters. It is determined that the Nusselt number increases with augmented values of the dimensionless film thickness, radiation parameter, solid volume fraction, and unsteadiness parameter, while it diminishes for growing values of non-uniform heat source/sink.

**Table 5.** The numerical value of the Nusselt number with γ = 0.1, *Pr* = 6.2.

#### **5. Conclusions**

The thin film flow of nanofluid comprising of CNTs of both types (SWCNTs/MWCNTs) is studied whilst keeping in view the important applications of CNTs in many engineering applications. The flow is supported by the additional effects like C-C heat flux and entropy generation. The model is solved numerically with the support of the MATLAB software function bvp4c. The highlights of the existing study are

**Author Contributions:** Data Curation, D.L.; Funding Acquisition, F.H.; Investigation, M.M.; Project Administration, M.R.; F.H.; Resources, D.L.; Software, M.M.; Supervision, M.M.; Validation, F.H.; Visualization, J.D.C.; Writing—Original Draft, J.D.C.

**Funding:** This research was funded by Zayed University research fund, Abu Dhabi, UAE.

**Conflicts of Interest:** The authors declare no conflict of interest.


**1. Introduction**

It is expected that the health risks of climate change will increase globally, with increases in morbidity and mortality from selected climate-sensitive health outcomes, putting additional pressures on health systems [1,2]. The effectiveness of adaptation is influenced by a country's physical location and exposure to climate shocks, levels of socioeconomic development, and healthcare capacity to prepare for and manage climate-related shocks [3]. Tailoring effective adaptation measures requires a clear understanding of which factors led to today's preparedness to manage climate change. Researchers have been urged to pay more attention to the role of the nation state in climate governance, as these actors have the legitimacy and resources to develop long-term visions, stimulate and oversee local approaches, and carry forward adaptation programs, e.g., [4]. In their analysis of health adaptation initiatives in ten Organization for Economic Co-operation and Development (OECD) countries, Austin et al. concluded that national governments play a key role in health adaptation to climate change, but there are competing views on what responsibilities and obligations this will—or

should—include [5]. One view suggests increasing investments in existing public health infrastructure to manage the projected health risks [6]. The second argues climate change will likely affect health by, for instance, destabilising supporting systems or threatening infrastructure, thus necessitating new and innovative responses. Suggested factors for successful adaptation include adopting legislation, ensuring interdepartmental coordination, and increasing self-governance [7–9]. We take an in-depth look at health and rescue systems in Estonia, because they will play a key role in climate adaptation policies in connection with the EU climate change strategy [10].

Informed by the research on risk governance [11,12], we analyse the health and rescue system as a control system for achieving policy goals; assess factors shaping those goals; and assess the factors shaping the process of information-gathering and implementation of protective measures, such as early warning systems or emergency preparations. We examine the extent to which outside (e.g., issue salience) and inside (e.g., institutional capacities and rules) drivers of the health systems shape the success of climate adaptation. We apply the World Health Organization's Operational Framework for Building Climate Resilient Health Systems [13] to understand how the quality of leadership and governance, health workforce, health information systems, medical technologies, service delivery, and climate and health financing shape the quality, efficiency, equity, accountability, and resilience of the health systems [14].


Abstract

Whispering gallery modes (WGMs) are surface modes that propagate azimuthally around resonators with rotational symmetry (toroidal, spherical, or, as in our case, cylindrical shaped, since the optical fiber itself plays the role of the microresonator). These modes are resonant in optical wavelength, and the spectral position of the resonances depends on the radius and the refractive index of the microresonator material. Due to the high-quality factor of the resonances (as high as 107 in cylindrical microresonators), they allow measuring different parameters with high sensitivities and very low detection limits. Here, we report the use of WGMs to characterize the properties of the material that forms the microresonator. In particular, we highlight the use of this technique to measure temperature profiles along conventional and special fibers (such as photosensitive or doped fibers), elastooptic coefficients, and UV-induced absorption loss coefficients of different photosensitive fibers. These parameters of the fibers set the optical response of fiberbased components and may change when the device is in use in an optical system; thus, this technique allows an accurate characterization of the devices and leads to proper designs of components with specific optical responses.

Keywords: whispering gallery modes, surface modes, microresonators, optical fibers, fiber Bragg gratings, elasto-optic effect, thermo-optic effect


**8. Conclusions**

This paper is an outcome of the deep analysis made in understanding the ranking measures of DHFS using CoCf. In the present paper, a deep mathematical analysis is made to study the CoCf of a DHFS and it's concluded that there exist certain limitations in the existing CoCf [29] for DHFS. These shortcomings are pointed out with a detailed mathematical derivation which suggests that there are some mathematical incorrect assumptions involved hence, it is not appropriate to apply the existing CoCf of a DHFS in its present form. This limitation encouraged to propose a valid mathematical expression for ranking DHFSs in terms of CoCfs. Therefore, a new CoCf given by expression 3ð Þ is proposed for DHFSS which is a modified form of expression 1ð Þ [29]. To validate the claim of the modified expressions of the CoCf for DHFSs the detailed mathematical derivation is stated and the results of the reallife problems considered in existing paper [29] are obtained and to validate the obtained results a systematic comparison between the results are made.


**Abstract**

Electrocardiography (ECG) has been a reliable method for monitoring the proper functioning of the cardiovascular system for decades. Recently, there has been a lot of research focusing on accurately analyzing the heart condition through ECG. In recent days, numerous attempts are being made to analyze these signals using deep learning algorithms, including the implementation of artificial neural networks like convolutional neural networks, recurrent neural networks, and the like. In this context, this chapter intends to present some important techniques for classifying heartbeats based on deep neural networks with 1D CNN. Five ECG signals (N, S, V, F, and Q) standardization are based on the AAMI EC57 standard. The primary focus of this chapter is to discuss the techniques to classify ECG signals in those classes with promising accuracy and draw a clear picture of the current state-of-the-art in this sphere of study.

**Keywords:** signal processing, electrocardiography, deep learning, 1D convolutional neural network, recurrent neural network