In this work, a weighted H lder function that approximates a Jacobi polynomial which solves the second order singular Sturm-Liouville equation is discussed. This is generally equivalent to the Jacobean translations and the moduli of smoothness. This paper aims to focus on improving methods of approximation and finding the upper and lower estimates for the degree of approximation in weighted H lder spaces by modifying the modulus of continuity and smoothness. Moreover, some properties for the moduli of smoothness with direct and inverse results are considered.

Idiomatic expressions in Russian journalism make one important borrowed means for making a dialogue with the receiver's intellect in so far as it has the distinct feature of  having clarity and exactness of meaning. The meaning is seen as a shortcut for covering a series of concepts and details so as to arrive at the intended meaning. This is done by stimulating the reader by the use of certain clear idioms. The use of such idioms in a journalistic text is not for a linguistic purpose only, but it is a cultural and social phenomenon reflecting the type of current changes in the society and it aims at discoursing with the reader's mind. This paper is a practi

Soil improvement has developed as a realistic solution for enhancing soil properties so that structures can be constructed to meet project engineering requirements due to the limited availability of construction land in urban centers. The jet grouting method for soil improvement is a novel geotechnical alternative for problematic soils for which conventional foundation designs cannot provide acceptable and lasting solutions. The paper's methodology was based on constructing pile models using a low-pressure injection laboratory setup built and made locally to simulate the operation of field equipment. The setup design was based on previous research that systematically conducted unconfined compression testing (U.C.Ts.). Th

The present study dealt with the morphological, anatomical,trichomespollen grains,and ecological characteristics of Caroxylon jordanicola (EigAkhani & Roalson (Amaranthaceae) in Al-Tar Caves, Karbala, Iraq which belongs to the Amaranthaceae family. The results of the present study demonstrated that There are distinctive characteristics of the studied species distinguish it from other species and facilitate its diagnosis. The sample was diagnosed using the taxonomic keys of the Iraqi flora and the flora of neighboring countriesIn addition to some available research. The results of the morphological and anatomical features investigation provide really significant taxonomical value to distinguish the species. The results that showe

Markov chains are an application of stochastic models in operation research, helping the analysis and optimization of processes with random events and transitions. The method that will be deployed to obtain the transient solution to a Markov chain problem is an important part of this process. The present paper introduces a novel Ordinary Differential Equation (ODE) approach to solve the Markov chain problem. The probability distribution of a continuous-time Markov chain with an infinitesimal generator at a given time is considered, which is a resulting solution of the Chapman-Kolmogorov differential equation. This study presents a one-step second-derivative method with better accuracy in solving the first-order Initial Value Problem