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In this paper, we prove the general solution and generalized Hyers–Ulam stability of n-dimensional functional equations of the form ∑ i = 1 i ≠ j ≠ k n f ⁢ ( - x i - x j - x k + ∑ l = 1 l ≠ i ≠ j ≠ k n x l ) = ( n 3 - 9 ⁢ n 2 + 20 ⁢ n - 12 6 ) ⁢ ∑ i = 1 n f ⁢ ( x i ) , \sum_{\begin{subarray}{c}i=1\\ i\neq j\neq k\end{subarray}}^{n}f\biggl{(}-x_{i}-x_{j}-x_{k}+\sum_{% \begin{subarray}{c}l=1\\ l\neq i\neq j\neq k\end{subarray}}^{n}x_{l}\biggr{)}=\biggl{(}\frac{n^{3}-9n^{% 2}+20n-12}{6}\biggr{)}\sum_{i=1}^{n}f(x_{i}), where n is a fixed positive integer with ℕ - { 0 , 1 , 2 , 3 , 4 } \mathbb{N}-\{0,1,2,3,4\} , in a Banach space via direct and fixed point methods.

In this paper, we find the general solution of a Septoicosic functional equation (11) for all \(x,y\in X\) and investigate its general Hyers-Ulam stability in Banach Space using direct and fixed point methods.

Ranunculus wallichianus is a medicinally important plant and an endemic species to Western Ghats of South India. An efficient and reliable indirect regeneration protocol system for R. wallichianus was developed from leaf explants in the present investigation. Leaf explants were cultured on both full-strength and half-strength MS (Murashige & Skoog) medium supplemented with different concentrations (1.0 mg L− 1 to 3.0 mg L− 1) of 2,4-D and NAA. Among the different concentrations tested, the highest percentage of yellowish green compact nodular callus formation was observed on half-strength MS medium with 2.0 mg L− 1 of 2, 4-D. Then, the in vitro raised organogenic callus was cultured on half strength MS medium containing various concentrations (1.0 mg L− 1 to 3.0 mg L− 1) of BA, KIN and TDZ with 0.5 mg L− 1 NAA and 10% CW for in vitro shoot regeneration. The highest percentage of regeneration response (97%) and maximum number of shoots formation (11.1 ± 0.13 shoots/culture with 9.2 ± 0.35 cm mean shoot length) were obtained from MS medium containing 2.5 mg L− 1 BA with 0.5 mg L− 1 NAA and 10% CW. The well elongated in vitro raised shoots were rooted in half strength MS medium with 2.5 mg L− 1 IBA + 250 mg L− 1 activated charcoal shows high frequency of root formation. The well rooted plantlets were successfully hardened and acclimatized with the survival rate of 94%. Clonal fidelity of in vitro raised plantlets was assessed by using DNA based RAPD and ISSR molecular markers. The total of 56 and 47 monomorphic bands were obtained from RAPD and ISSR markers respectively. This present in vitro propagation protocol system could be an effective for the conservation of R. wallichianus with their genetic purity and its further investigations.

In this work, we have to introduce a generalized quadratic functional equation and derive its solution. The main objective of this work is to investigate the Hyers-Ulam stability of quadratic functional equation in non-Archimedean n,β-normed spaces.

In this work, we examine the generalized Hyers-Ulam orthogonal stability of the quartic functional equation in quasi-β-normed spaces. Moreover, we prove that this functional equation is not stable in a special condition by a counterexample. © 2021 Nazek Alessa et al.

The investigation of the stabilities of various types of equations is an interesting and evolving research area in the field of mathematical analysis. Recently, there are many research papers published on this topic, especially additive, quadratic, cubic, and mixed type functional equations. We propose a new functional equation in this study which is quite different from the functional equations already dealt in the literature. The main feature of the equation dealt in this study is that it has three different solutions, namely, additive, quadratic, and mixed type functions. We also prove that the stability results hold good for this equation in intuitionistic random normed space (briefly, IRN-space).

In this paper, we use direct and fixed-point techniques to examine the generalised Ulam-Hyers stability results of the general Euler-Lagrange quadratic mapping in non-Archimedean IFN spaces (briefly, non-Archimedean Intuitionistic Fuzzy Normed spaces) over a field.

The objectives of current investigation are (1) to find out wavelength of maximum absorbance (λmax) for combined cyclosporin A and etodolac solution followed by selection of mobile phase suitable for the RP-HPLC method, (2) to define analytical target profile and critical analytical attributes (CAAs) for the analytical quality by design, (3) to screen critical method parameters with the help of full factorial design followed by optimization with face-centered central composite design (CCD) approach-driven artificial neural network (ANN)-linked with the Levenberg-Marquardt (LM) algorithm for finding the RP-HPLC conditions, (4) to perform validation of analytical procedures (trueness, linearity, precision, robustness, specificity and sensitivity) using combined drug solution, and (5) to determine drug entrapment efficiency value in dual drug-loaded nanocapsules/emulsions, percentage recovery value in human plasma spiked with two drugs and solution state stability analysis at different stress conditions for substantiating the double-stage systematically optimized RP-HPLC method conditions. Through isobestic point and scouting step, 205 nm and ACN:H2O mixture (74:26) were selected respectively as the λmax and mobile phase. The ANN topology (3:10:4) indicating the input, hidden and output layers were generated by taking the 20 trials produced from the face-centered CCD model. The ANN-linked LM model produced minimal differences between predicted and observed values of output parameters (or CAAs), low mean squared error and higher correlation coefficient values in comparison to the respective values produced by face-centered CCD model. The optimized RP-HPLC method could be applied to analyze two drugs concurrently in different formulations, human plasma and solution state stability checking.

This paper deals with the approximate solution of the following functional equation fx7+y77=f(x)+f(y), where f is a mapping from R into a normed vector space. We show stability results of this equation in quasi-&beta, Banach spaces and (&beta, p)-Banach spaces. We also prove the nonstability of the previous functional equation in a relevant case.

In this current work, we introduce the finite variable additive functional equation and we derive its solution. In fact, we investigate the Hyers–Ulam stability results for the finite variable additive functional equation in fuzzy normed space by two different approaches of direct and fixed point methods.

The present experimental investigation is conducted to examine the working characteristics of compression ignition (CI) engine using oxygenated additives such as diethyl ether and ethanol to the blends of Calophyllum inophyllum biodiesel. Experiments are conducted on a water-cooled single-cylinder constant speed DI (direct injection) diesel engine under same operating circumstances. The results indicates that an enhancement of brake thermal efficiency is up to 3.7% and 6.2% with an addition of ethanol and DEE additive in biodiesel blended fuels respectively at maximum load condition. This is attributable to the existence of higher oxygen value and volatile nature of the additives. The combustion characteristics like in-cylinder pressure and net heat release rate of DEE blended fuels are 1.1-5.2% and 0.4-2.7% higher than other blends respectively. This is mainly due to higher cetane number and high volatile nature of DEE. This also results in higher NO

Globally, buildings utilize 35 % of the final energy use and contribute to approximately one-third of CO2 emissions. Hence, reducing the energy use of buildings contributes to a large amount of CO2 emissions to be decreased. The building’s energy use is affected by many parameters, including wind which plays an important role in building energy use. In this thesis, we aim to analyze the impact of wind parameters on building’s energy use on two multi-family building types with natural ventilation at various wind sheltering conditions at different climatic zones in Sweden. Building energy simulation models (BES) of a standalone and an attached building located in Visby, Sweden, were constructed with the use of the dynamic BES IDA ICE. Luleå and Malmö were taken as other two study locations to investigate the impact from different climate zones. The simulations were performed with the constructed calculation models, with the various wind sheltering conditions at the different climatic zones to calculate the energy use of the buildings and ventilation and infiltration losses. The sensitivity analysis was then carried out based on changing the wind profile of the climate file to evaluate the impact of wind on the ventilation and infiltration losses, as well as the heat energy use of the building. The results showed that the energy use for space heating of the attached building was 89 kWh/m2 (38 %) lower than the standalone building. The energy use varies between 9–20 kWh/m2 (3–10 %) considering the exposed, semi-exposed and sheltered wind condition for the two building types. In the different climate zones, Luleå has 47 kWh/m2 higher energy use compared to Visby and Malmö for the standalone building. The corresponding figure for the attached building is 25 kWh/m2. The sensitivity analysis show that when the wind speed is increased by 100 %, the ventilation and infiltration losses increase between 3563–18683 kWh (54–61 %) while the energy use of the building increases between 11–54 kWh/m2 (20–27 %).

In this paper, we introduced m-dimensional quadratic functional equation of the form ∑1≤i

A majority of population living in the villages of Andaman & Nicobar Islands depends not only on the farming but a major role of their livelihood is animal rearing. Due to remote locality, the cure for their cattle from diseases is not easier; therefore, they develop their own methods by using locally available plants in their backyard, nearby forest, and home remedies to cure their animals and poultry. Many of these practices are quite effective and being used by them since long past. In most of the cases plants or plant parts are given as such, seldom in combination with some other plants, plant parts, or home remedies. The old-aged farmers and livestock owners gained the ethnoveterinary knowledge from their own experience or from their ancestors through the words of mouth. Livestock are an important and integrated component of the agricultural production system in developing countries. Antibiotics are indiscriminately used in livestock and poultry production and the phenomenon of the problem of multidrug resistance has evolved with a great concern for an organic livestock production system. The importance of traditional and alternative medicine out of medicinal plants has been realized for sustainable livestock production by the veterinary practitioners and medicinal plants have comprised a significant proportion of veterinary research. Due to isolated spread of 572 islands in remote locations of A&N Islands, rural farmers are dependent on indigenous knowledge for the treatment of their livestock and poultry rather than immediate animal husbandry and veterinary services. A&N Islands are the hot spot of a large number of medicinal plants. A field survey was done to study and document these ethnoveterinary medicinal plants. A total of 41 plants from 27 families were identified which are of ethnoveterinary importance from the different villages of South and North Andaman districts of Andaman & Nicobar Islands. Extensive research works are being carried out to explore medicinal plants as growth promoter and immunomodulator in poultry.

Rapid industrialization and growth in population in urban regions augment the pollution levels from transportation sectors, especially from diesel fleets. A wide array of research activities were carried out to satisfy the energy needs as well as reduce the emission levels, which poses a big challenge to the research community. In this situation, biomass-derived fuels provide a ray of hope to the research community to address the emission problem by adapting closed carbon cycle at low cost. This chapter gives an overview to the readers about the present energy scenario, biomass-based fuel, upgradation techniques for biomass fuel, and engine adaptability of biomass-based fuels. This chapter provides a clear glimpse of biomass energy, one of the potential energy resources in the near future.

In the current work, the intuitionistic fuzzy version of Hypers-Ulam stability for a k-dimensional cubic functional equation $$\begin{aligned} \begin{aligned}&\sum _{j=1}^{k}f\left[ \sum _{i(\ne j)=1}^{k}n^{i}x_{i}-n^{j}x_{j}\right] +(6-k)f\left( \sum _{i=1}^{k}n^{i}x_{i}\right) \\ =&4\left[ \sum _{j=1}^{k}\sum _{i(

Biodiesel is proven to be the best substitute for petroleum-based conventional diesel fuel in existing engines with or without minor engine modifications. The performance characteristics of biodiesel as a fuel in CI engine are slightly lower than that of diesel fuel. The emission characteristics of biodiesel are better than diesel fuel except NOX emission. The thermo-physical properties of biodiesel are improved by suspending the nano metal particles in the biodiesel, which make them an observable choice for the use of nanoparticles-added fuels in CI engine. High surface area of nanoparticles that promotes higher operating pressure and heat transfer rates that further quicken the combustion process by providing better oxidation. Thus, it has been inferred that addition of nanoparticles as an additive to biodiesel fuel blends in diesel engines and its effects on performance, combustion, and emission characteristics are discussed in this chapter.

The purpose of this current work, we introduced the finite dimensional quadratic functional equation of the form φ(∑i=1msi)+∑1≤i

Depletion of non-renewable energy sources are at elevated manner due to the rapid growth of industrialization and transportation sector in last few decades and leads to further energy demand. Biodiesels especially second-generation fuels from non-edible oil resources are alternate sources for replacement of diesel fuel in CI engines due to their considerable environmental benefits. In the present work, non-edible feedstock of Calophyllum inophyllum seed oil (tamanu oil) is used for biodiesel production. Transesterification method is used for preparation of biodiesel in the existence of methanol with NaOH as catalyst. The copper nanoparticles are synthesized by electrochemical method, and it is characterized by using X-ray diffraction analysis (XRD) and scanning electron microscopy (SEM). XRD and SEM results confirm the presence of copper nanoparticle and size of around 30 nm. This paper aims to investigate the effects of the copper additive nanoparticles with biodiesel blends on the engine performance, combustion and emission characteristics of single-cylinder direct-injection diesel engine and compared that with diesel fuel. The results showed that the addition of nano-additives enhances brake thermal efficiency and reduces specific fuel consumption compared to biodiesel blends but slightly lower than diesel. Combustion characteristics also are enhanced by improved oxidation reaction inside the combustion chamber which resulted in higher heat release rate. The emissions of HC, NOx and O2 are significantly reduced for nano-additive blends compared to diesel but increased CO2 emission was observed. It is noticed that higher CO2 emission and substantial reduction of unused O2 emissions from engine fueled with nano-additive are evident for enhanced oxidation and better combustion. Energy and exergy analysis of the diesel engine is carried out to estimate the effect of using nanoparticle additive with biodiesel.

The objectives of the present investigations are (1) to envisage a risk assessment plan for nonphospholipid-based topical ophthalmic emulsions with the help of failure mode and effect analysis (FMEA), (2) to screen the risky formulation and process variables by the Taguchi design, (3) to optimize systematically an emulsion formula by face-centered central composite design (CCD), (4) to incorporate cyclosporin A (0.05 or 0.1% w/w) into the optimized emulsions and predict the in vitro drug release kinetic via a particle diffusion-controlled mathematical model equation, and (5) to assess the emulsion's toxicity using in vitro hemolysis study. Through the risk priority number (RPN) scores of FMEA, half-normal and Pareto charts of the Taguchi design, 3D-response surface graphs, and overlay plots of CCD, the emulsion formula was systematically optimized. Irrespective of the two different drug loadings into optimized emulsions, the drug entrapment efficiency values ranged from 73.20 ± 0.13 to 74.42 ± 0.15%. The film diffusion or ion-exchange process fails to interpret the in vitro drug release kinetic profile. A permissible percentage hemolysis value of above 10% but below 25% guidance was observed for emulsions with or without cyclosporin A. The systematically optimized phospholipidless ophthalmic emulsions could further be exploited commercially for managing dry-eye syndrome.

In this paper, we present the Hyers-Ulam stability of Cubic functional equation. where n is greater than or equal to 4, in Random Normed Space.

Classification algorithms are efficiently utilized in the area of general medical diagnosis applications in order to identify the disorders in advance. One such disease, breast cancer is the most prevalent and earnest quandary with women in most of the developing countries. Many attempts are made in order to identify this problem with the objective of high precision and better accuracy. In this paper, an attempt is made with the most popular and efficient classification algorithms namely Naive Bayes, Multilayer Perceptron, Radial basis function network, nearest neighbour, Conjunctive rule to amend the efficiency of the detection, accuracy for the breast cancer dataset. As an objective of improving accuracy, an efficient dimensionality reduction technique is incorporated in this work. The performances of these approaches are evaluated using the metrics such as the precision, recall, f-measure, roc, Balanced Classification Rate (BCR), Matthews Correlation Coefficient (MCC) and accuracy. From these measures it is clearly observed that Naive Bayes algorithm is able to achieve high accuracy rate along with minimum error rate when compared to other algorithms. The review can be stretched out to draw the execution of other characterization systems on an extended information set with more particular ascribes to get more exact outcomes.

In this paper, we use direct and fixed-point techniques to examine the generalised Ulam–Hyers stability results of the general Euler–Lagrange quadratic mapping in non-Archimedean IFN spaces (briefly, non-Archimedean Intuitionistic Fuzzy Normed spaces) over a field.