1. We just write out (i). Recall that X1 ∪ X2 = {x : x ∈ X1 or x ∈ X2}. Hence x ∈ X1 =⇒ x ∈ X1 ∪ X2, so that X1 ⊆ X1 ∪ X2. By symmetry, X1 ⊆ X1 ∪ X2. Finally, suppose that Y is a set such that X1 ⊆ Y and X2 ⊆ Y . Let x ∈ X1 ∪X2. Then x ∈ X1 or x ∈ X2. If x ∈ X1, then x ∈ Y since X1 ⊆ Y . Similarly, if x ∈ X2, then x ∈ Y . Thus in either case x ∈ Y , so that X1 ∪X2 ⊆ Y . 2. (i) Injective, not su...

for each n ∈ N. Note that each On is open since it is a union of open sets (cf. Theorem 2.24(a) in Rudin). We claim that A = ⋂∞n=1 On. First, if a ∈ A, then a ∈ (a− 1/n, a+ 1/n) for each n ∈ N, so a ∈ On for each n ∈ N, and hence a ∈ ⋂∞n=1 On. Thus, A ⊆ ⋂∞n=1 On. Conversely, assume x ∈ R and x / ∈ A. Since A is closed, it follows that x is not a limit point of A, so there exists an ε > 0 such t...

action potential through the heart. During the cardiac action potential, Ca2+ enters each cardiac myocyte through L-type Ca2+ channels [as the Ca2+ current (ICa)] and activates adjacent Ca2+ release channels [ryanodine receptors (RyR)] in the membrane of the intracellular Ca2+ store, the sarcoplasmic reticulum (SR), causing Ca2+ release from the SR (5). Ventricular myocytes, which cause contrac...

Data aggregation scheme is a technique which reduces the large amount of transmission in wireless sensor networks. In the existing system, homomorphic encryptions are used to conceal communication during aggregation. The enciphered data can be aggregated algebraically without decryption, hence compromising the security issues and adversaries are able to collect valuable data. Hence these scheme...

Our approach of rule-based reenement 1 provides a formal description for the stepwise system development based on Petri nets. Rules with a left-hand and a right-hand side allow replacing subnets in a given algebraic high-level net system. The extension of these rules supports the preservation of system properties. In this paper we extend the preservation of safety properties signiicantly. We de...

action potential through the heart. During the cardiac action potential, Ca2+ enters each cardiac myocyte through L-type Ca2+ channels [as the Ca2+ current (ICa)] and activates adjacent Ca2+ release channels [ryanodine receptors (RyR)] in the membrane of the intracellular Ca2+ store, the sarcoplasmic reticulum (SR), causing Ca2+ release from the SR (5). Ventricular myocytes, which cause contrac...

Renewable energy sources are gaining high prominence in today’s world. However, these sources do not supply energy throughout the year and hence efficiency is required when extracting energy from them. Wind energy is a recently developing area of common interest. Efficiency of a wind turbine is however, very low. Hence, detection of fault in the system becomes very essential so as to increase t...

In this paper, a new structure for reactive power market is presented. Considering reactive power losses caused by active power flow in the reactive power market is the main purpose of this paper. Hence, this study tries to improve reactive power market and create fair competition in reactive power generation.

In the present paper, we give a new approach to Caristi's fixed pointtheorem on non-Archimedean fuzzy metric spaces. For this we define anordinary metric $d$ using the non-Archimedean fuzzy metric $M$ on a nonemptyset $X$ and we establish some relationship between $(X,d)$ and $(X,M,ast )$%. Hence, we prove our result by considering the original Caristi's fixedpoint theorem.