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Group theory, the ultimate theory for symmetry, is a powerful tool that has a direct impact on research in robotics, computer vision, computer graphics and medical image analysis. Symmetry is very important in chemistry research and group theory is the tool that is used to determine symmetry. Usually, it is not only the symmetry of molecule but also the symmetries of some local atoms, molecular orbitals, rotations and vibrations of bonds, etc. that are important. Harada-Norton group is an example of a sporadic simple group. There are 14 maximal subgroups of Harada-Norton group. Generators (also known as words) of 11 maximal subgroups are already known. The aim of this note is to give generators of the remaining 3 maximal subgroups, which is an open problem mentioned on A World-wide-web Atlas of Group Representations (http://brauer.maths.qmul.ac.uk/Atlas) [1]. In this report we compute the generators of A6 × A6.D8, 23+2+6.(3 × L3(2)) and 34 : 2.(A4 × A4).4. Moreover we also compute the generators for the Maximal subgroups of some linear groups.

In this publication, we establish a suitable symmetry structure for orthogonal (α − y − G) -contractive mappings and prove fixed point results for an orthogonal (α − y − G) -contractive via orthogonal metric spaces. We give an application to strengthen our main results from the existing literature to prove the existence of a unique analytical solution to the differential equation by converting it into an integral equation through fixed point analysis. [ABSTRACT FROM AUTHOR]

The recent progress in nanotechnology developed more stable nanofluid models with ultra-high thermal performances. The immersion of hybrid nanoparticles in the base fluid presents novel applications in the thermal transport processes, solar energy, engineering devices, and heating systems. This continuation identified the thermal outcomes of the hybrid nanofluid model with the water base fluid. The thermal investigation is encountered by utilizing the copper oxide (CuO), silver (Ag), and titanium dioxide (TiO2) nanoparticles. The thermal selection of these three types of nanoparticles is associated with impressive characteristics and stability. To attain the more progressive thermal aspects, thermal radiation consequences are also focused. The inclined infinite configuration is assumed where these nanoparticles and base fluid are specified. The comparative results for the hybrid nanofluid and modified hybrid models are pronounced. The fractional model of dimensionless partial differential leading equations is formulated by following the recent definitions of fractional derivatives, namely, Atangana–Baleanu and Caputo–Fabrizio fractional derivatives. The semi-analytical simulations are performed via the Laplace technique for nanoparticles temperature and velocity inspection. The significances of the fractional parameter and volume fraction are observed for the velocity profile. [ABSTRACT FROM AUTHOR]

This paper aims to investigate free convection heat transmission in hybrid nanofluids across an inclined pours plate, which characterizes an asymmetrical hybrid nanofluid flow and heat transfer behavior. With an angled magnetic field applied, sliding on the border of walls is also considered with sinusoidal heat transfer boundary conditions. The non-dimensional leading equations are converted into a fractional model using an effective mathematical fractional approach known as the Prabhakar time fractional derivative. Silver (Ag) and titanium dioxide (TiO2) are both considered nanoparticles, with water (H2O) and sodium alginate (C6H9NaO7) serving as the base fluids. The solution of the momentum, concentration, and energy equation is found by utilizing the Laplace scheme, and different numerical algorithms are considered for the inverse of Laplace, i.e., Stehfest and Tzou's. The graphical analysis investigates the impact and symmetry of significant physical and fractional parameters. Consequently, we surmise that water-based hybrid nanofluid has a somewhat higher velocity than sodium alginate-based hybrid nanofluid. Furthermore, the Casson parameter has a dual effect on the momentum profile. Furthermore, the memory effect reduces as fractional restriction increases for both the velocity and temperature layers. The results demonstrate that increasing the heat transmission in the solid nanoparticle volume fractions enhanced the heat transmission. In addition, the numerical assessment examined the increase in mass and heat transmission, while shear stress was increased with an increase in the Prabhakar fractional parameter α. [ABSTRACT FROM AUTHOR]

The idea of symmetry is a built-in feature of the metric function. In this paper, we investigate the existence and uniqueness of a fixed point of certain contraction via orthogonal triangular α-orbital admissible mapping in the context of orthogonal complete Branciari metric spaces endowed with a transitive binary relation. Our results generalize and extend some pioneering results in the literature. Furthermore, the existence criteria of the solutions to fractional integro-differential equations are established to demonstrate the applicability of our results. [ABSTRACT FROM AUTHOR]

With improved characteristics and thermal stability, the nanoparticles show a significant interest in many engineering and industrial processes. The applications of nanomaterials include applications in solar energy systems, chemical reactors, heating and cooling processes, and many bio-medical applications. With such motivations, this investigation aims to address the mixed convection flow of different nanoparticles due to inclined surface in presence of Newtonian heating and slip effects. Two types of nanoparticles like titanium dioxide and aluminum oxide with water base fluid are used to improve the heat transfer rate. The dimensionless governing partial differential equations for this fractional mathematical modal are accomplished by using Caputo-time fractional derivative and Fourier's law of thermal conductivity. The generalized solution of energy and momentum equations is conquered by applying the Laplace transformation (LT) scheme. To illustrate some more physical insight of the problem and to develop the novelty of this article some special cases of velocity field are reflected whose physical clarification is well known in the literature. Furthermore, to investigate the physical implication of the under-deliberation problem, the graphical illustrations are performed by using math software MATHEMATICA. The interesting observations claimed from this investigation revealed that the interaction of thermal rate of change is more impressive for aluminum oxide. The temperature and velocity profiles are enhanced by varying the numerical value of the fractional constraint at large time scale. [ABSTRACT FROM AUTHOR]

The class of harmonic convex functions has acquired a very useful and significant placement among the non-convex functions, since this class not only reinforces some major results of the class of convex functions, but also has supported the development of some remarkable results in analysis where the class of convex functions is silent. Therefore, many researchers have deployed themselves to explore valuable results for this class of non-convex functions. This paper obtains new discrete inequalities for univariate harmonic convex functions on linear spaces related to a Jensen-type and a variant of the Jensen-type results. Our results are refinements of very important recent inequalities presented by Dragomir and Baloch et al. Furthermore, we provide the natural applications of our results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of tricomplex valued metric space and prove some common fixed point theorems. The presented results generalize and expand some of the literature well-known results. We also explore some applications of our key results. [ABSTRACT FROM AUTHOR]

The study of convex functions is one of the most researched of the classical fields. Analysis of the geometric characteristics of these functions is a core area of research in this field; however, a paradigm shift in this research is the application of convexity in optimization theory. The Jensen-Mercer type inequalities are studied extensively in recent years. In the present paper, we extend Jensen-Mercer type inequalities for strong convex function. Some improved inequalities in Hölder sense are also derived. The previously established results are generalized and strengthened by our results. [ABSTRACT FROM AUTHOR]

In this article, we present an orthogonal L -contraction mapping concepts and prove a fixed point theorem on orthogonal complete Branciari metric spaces. As an application, we apply our major results to solving integral equations. [ABSTRACT FROM AUTHOR]

Fractional integral inequalities have a wide range of applications in pure and applied mathematics. In the present research, we establish generalized fractional integral inequalities for MT-non-convex functions and p q -convex functions. Our results extended many inequalities already existing in the literature. [ABSTRACT FROM AUTHOR]

The purpose of this manuscript is to obtain some fixed point results under mild contractive conditions in fuzzy bipolar metric spaces. Our results generalize and extend many of the previous findings in the same approach. Moreover, two examples to support our theorems are obtained. Finally, to examine and strengthen the theoretical results, the existence and uniqueness of the solution to a nonlinear integral equation was studied as a kind of applications. [ABSTRACT FROM AUTHOR]

In this paper, we establish the generalized viscosity implicit rules of asymptotically nonexpansive mappings in CAT(0) spaces. The strong convergence theorems of the implicit rules proposed are proved under certain assumptions imposed on the control parameters. The results presented in this paper improve and extend some recent corresponding results announced. [ABSTRACT FROM AUTHOR]

In this paper, we present a explicit viscosity technique of nonexpansive mappings in the framework of CAT(0) spaces. The strong convergence theorem of the proposed technique is proved under certain assumptions imposed on the sequence of parameters. The results presented in this paper extend and improve some recent announced in the current literature. [ABSTRACT FROM AUTHOR]

We shall discuss three generalized moduli such as generalized modulus of convexity, modulus of smoothness, and modulus of Zou-Cui of quasi-Banach spaces and give some important properties of these moduli. Furthermore, we establish relationships of these generalized moduli with each other. [ABSTRACT FROM AUTHOR]

The aim of this report is to study modulus of convexity δB of a quasi-Banach space B. We prove that δB is convex, continuous, nondecreasing and for arbitrary uniformly convex quasi-Banach space B, δB ... We also prove that a quasi-Banach space B is uniformly convex if and only if δB(∈) ≥ 0. Moreover we prove that a non-trivial quasi-Banach space B is uniformly non-square if and only if δB(∈) > 0. [ABSTRACT FROM AUTHOR]

The aim of this report is to study modulus of convexity δB of a quasi-Banach space B. We prove that δB is convex, continuous, nondecreasing and for arbitrary uniformly convex quasi-Banach space B, δB(∊) = 1 - 1/C√1 - ∊2C2/4 . We also prove that a quasi-Banach space B is uniformly convex if and only if δB(∊) ≥ 0. Moreover we prove that a non-trivial quasi-Banach space B is uniformly non-square if and only if δB(∊) > 0. [ABSTRACT FROM AUTHOR]

In the present note we will introduce a Binomial distribution series and obtain necessary and sufficient conditions for this series belonging to the classes T(λ, α) and C(λ, α). An integral operator related to this series is also considered. [ABSTRACT FROM AUTHOR]

The generalized von Neumann-Jordan constant C(p)NJ (B) and the James constant J(B) for a quasi-Banach space were introduced in [7]. In this note, it is shown that C(p)NJ (B) ≥ 2 for any quasi-Banach space B andC(p)NJ (B) < 2 if and only if B is uniformly non-square. Along with relationship between J(B) and C(p)NJ (B), the criterion for the uniformily smooth quasi-Banach space is also established. [ABSTRACT FROM AUTHOR]