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Geometrical properties of elementary operators form a core area of interest in func-tional analysis. Determining the shape of the boundary of the numerical ranges of algebraic elementary operators still remains interesting. In this note, we deter-mine the shape of the boundary of the numerical ranges of algebraic elementary operators. In particular, we show that the boundaries conform to spheroidality and ellipsoidality criteria. We also show the relationship between these two shapes.

KEYWORDS: Algebraic elementary operator, Boundary, Numerical range, Spheroidality, Ellipsoidality.

This paper deals with a diffusion-advection system modeling  occulation process in a un-stirred chemostat. We prove that there is a finite upper bound on the total bacteria density which enables us to prove existence of global solution. Then, we show existence of non-trivial positive solutions to the corresponding steady-state system. Also, the uniform stability of the constant solution is investigated.

KEYWORDS: occulation, global solution, lower-upper solutions, microbial growth, spectral theory, steady solution.

Many scholars nowadays have proposed numerous integral transformations. Most often, integral transformations are used to solve systems, differential equations, and integral-dierential equations. Rachid and Patil introduced the Taki integral trans-form in (2025). A set of dierential equations with boundary conditions is present in our health sciences medication absorption models. The Taki integral transform is used in this study to solve the drug absorption model.

KEYWORDS: Integral equations, Integral transforms, Differential equations, Taki integral transform, Drug absorption, Health science.

In this paper the total inventory cost and optimum order quantity are obtained in intuitionistic fuzzy sense for deteriorating items. Deterioration rate, holding cost and shortage cost are considered as vague and imprecise in nature. These parameters are considered as fuzzy parameters. Therefore in the proposed model these parameters are considered as fuzzy parameters. The vagueness of these parameters are firstly represented by triangular fuzzy numbers and then triangular intuitionistic fuzzy numbers to obtain total inventory cost. Centroid Method is used to defuzzify the fuzzy parameters to obtain total cost, optimum order quantity and shortage quantity. The model is developed in fuzzy as well as intuitionist fuzzy environment and illustrated by a numerical example to compare the results. The sensitivity analysis of the optimum solution with respect to the changes in the different parameter values is also discussed. The study concludes that intuitionistic fuzzy model is found more realistic than fuzzy and crisp model.

KEYWORDS: Intuitionistic fuzzy number (IFN), Centroid, Defuzzification, Inventory model.

In this paper we study the Cauchy-Dirichlet problem for a modified non-autonomous modified Navier-Stokes equation in a bounded domain. The existence and uniqueness of a weak solution of the problem are proved by Galerkin method. We then show the existence of a unique minimal D-pullback attractor for the process associated to the problem with respect to a large class of non-autonomous forcing terms. Finally, when the force is time-independent and "small", the existence, uniqueness and stability of a stationary solution are also investigated.

KEYWORDS: modified Navier-Stokes equations; weak solution; stationary solution; pullback attractor; stability.

In this article we address a general category of equilibrium problems known as mixed quasi hemiequilibrium problems (MQHEP) on Hadamard manifolds. By using KKM technique, we establish the existence and uniqueness of solutions to MQHEP when the underlying bifunctions are monotone. We provide examples in the Hadamard manifold context to illustrate our results. To tackle mixed quasi hemiequilibrium problems on these nonlinear domains, we additionally look into a few iterative al-gorithms. Some certain cases of MQHEP are provided. These broad types of equi-librium problems are new on Hadamard manifolds. We hope our results and recom-mendations will encourage further research in this interesting and fascinating field of study.

KEYWORDS: Riemannian Geometry, Hadamard manifolds, Mixed Quasi Hemiequilibrium problems, KKM mappings.

In a course in Inferential Statistics, students usually learn point estimation, confidence interval estimation, and hypothesis testing. Although they learn quite a bit about the parameters in a model based on a random sample, they still know very little about how to predict a future random outcome. This paper explains the difficulty involved and proposes a way to predict efficiently to minimize the overall cost associated with the error of prediction.

KEYWORDS: Binomial, Multinomial, Discrete, Continuous, Penalty function.

In this paper, we show by mathematical reasoning that Astrology can have only 4608 predictions not more. This reasoning is based on the pillars of Astrology 12 houses (Bhavs), 12 Zodiac signs (Rashis), 7 house lords (Rashi Swamis), 2 Rahu (north node of the moon and Ketu (south node of the moon) and 27 lunar mansions or Constellations (Nachhetras), in short on 12, 12, 7, 2 and 27 (twelve twelve seven two and (two seven) or 60 pillars of Astrology.

KEYWORDS: Zodiac signs (Rashis),House (Bhav),Planets (Grahas),Shadow Grahas (Rahu,Ketu), Birth chart(Kundali), Constellations(Nachhetras) etc.

Using the estimated coefficient of variation of the study variable, a product estimator is proposed which is found to be less biased than the conventional product estimator. Further, it is shown to be more efficient than the exponential product estimator under conditions that hold good in practice. Empirical studies have been carried out to support theoretical investigations.

Keywords: New product estimator, study variable, estimated coefficient of variation, percent relative bias.