We introduce the notion of relative convergence by means of a four dimensional matrix in the sense of the power series method, which includes Abel's as well as Borel's methods, to prove a Korovkin type approximation theorem by using the test functions {1, y, z, y 2 + z 2 } and a double sequence of positive linear operators defined on modular spaces. We also endeavor to examine some applications related to this new type of approximation.2010 Mathematics Subject Classification. 40A05; 40A30.

In this paper, we introduce and study some strongly almost convergent double sequence spaces by Riesz mean associated with four-dimensional bounded regular matrix and a Musielak–Orlicz function over [Formula: see text]-normed spaces. We make an effort to study some topological and algebraic properties of these sequence spaces. We also study some inclusion relations between the spaces. Finally, we establish some relation between weighted lacunary statistical sequence spaces and Riesz lacunary almost statistical convergent sequence spaces over [Formula: see text]-normed spaces.

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Tauberian theorem serves the purpose to recuperate Pringsheim’s convergence of a double sequence from its (C, 1, 1) summability under some additional conditions known as Tauberian conditions. In this article, we intend to introduce some Tauberian theorems for fuzzy number sequences by using the de la Vallée Poussin mean and double difference operator of order r . We prove that a bounded double sequence of fuzzy number which is Δ u r - convergent is ( C , 1 , 1 ) Δ u r - summable to the same fuzzy number L . We make an effort to develop some new slowly oscillating and Hardy-type Tauberian conditions in certain senses employing de la Vallée Poussin mean. We establish a connection between the Δ u r - Hardy type and Δ u r - slowly oscillating Tauberian condition. Finally by using these new slowly oscillating and Hardy-type Tauberian conditions, we explore some relations between ( C , 1 , 1 ) Δ u r - summable and Δ u r - convergent double fuzzy number sequences.

The purpose of this paper is to discuss emissions from geothermal power plants in New Zealand. Geothermal energy has been considered as a clean and sustainable source of energy. However, the geothermal fluid also contains the greenhouse gases which are emitted to the atmosphere after the electricity generation process. Releasing the gases to the environment after the electricity generation process is neither environmentally suitable and may not be economically feasible if the price of carbon increases. These emissions have become an issue of major concern with the draft advice of the climate change commission recommending that high emitting power stations be closed by 2030. Reinjection of the fluid and the gases back to the reservoir or using CO2 for industrial and agriculture purpose seems to be a viable solution towards achieving a zero-carbon emission target. Reinjection of separated brine is routinely carried out and plays an important role in the geothermal system as it provides pressure support which acts as a barrier to cold water recharge to the reservoir and reduces environmental impacts by deposing water back to the reservoir. Though reinjection provides a feasible solution to the water disposal problem, it can also harm to the reservoir by creating problem such as thermal breakthrough. Therefore, we need well planned and modelled strategies for reinjection systems, that are site-specific as the geologic setting differs from site to site. This paper will provide a qualitative understanding of the various reinjection strategies in New Zealand and the response of the geothermal system to those strategies. If the CO2 gas from a geothermal power plant is reinjected into the reservoir there may be impacts on surface features which may impact economic and cultural factors. This work will outline a study that will be undertaken to optimize the strategy for reinjecting CO2 gas into a geothermal reservoir. The optimization will consider the impacts on power generation, surface features and other factors.