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Q: Inverse problem of path graph Problem: Let $(G,\sigma)$ be a path graph and let $p$ be an edge of $G$ such that $\sigma(p) = 1$. Let $G' = G /p$ and $\sigma' = \sigma/p$. If $\sigma'(p) = 1$, prove that $G'$ is an induced subgraph of $G$. I have no idea how to prove this. A: A subgraph of a graph is an induced subgraph if and only if every edge is in the subgraph. We shall use this fact, as well as the fact that the path graph you are given has no vertices of even degree. Let $p=v_1v_2$ with $\sigma(p)=1$. Case 1: If $\sigma'(v_1)=1$, then $\sigma(v_2)=1$ or $\sigma(v_2)=0$. If $\sigma(v_2)=1$ the result follows, as $\sigma'(v_2)=1$. If $\sigma(v_2)=0$, then $v_2$ is isolated and so no vertex of $G'$ has degree 1. Hence the subgraph induced by $v_1v_2$ is a subgraph of $G$, and so $G'$ is an induced subgraph of $G$ Case 2: If $\sigma'(v_2)=1$, then $\sigma(v_1)=1$ or $\sigma(v_1)=0$. If $\sigma(v_1)=1$, then $\sigma(v_2)=1$ and we are done. If $\sigma(v_1)=0$, then $v_1$ is isolated and so no vertex of $G'$ has degree 1. Hence $G'$ is an induced subgraph of $G$. A Computer Generated Illustration of the Turbulent Flow Field Around an Airplane Wing. A turbulence model is essential for the accurate prediction of atmospheric wind and turbulence fields, particularly when these fields are at large scales. To assist in the development of turbulence models of large-scale atmospheric flows, we herein present a CG illustration of the flow field around an airfoil at the scale of

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﻿ netcoreapp3.1 [Separation of Biomass and Produced Water from Farmland Irrigation System Using Reverse Osmosis Membrane and Compression Separation Method]. Farmland irrigation systems consist of a header tank and a pipe network connecting a series of headers that can be up to 200 m long. Typically, farm ponds are an extension of the pipe network and are designed to absorb surface runoff as well as volumetric water flux from the pipe network into the pond. With the introduction of desalination technology, volumetric water flux is reduced in the production ponds and many farm ponds need to be replaced with larger holding ponds. In addition, water is also discharged from the pipes connecting the headers to the holding ponds, resulting in excess water discharge into the atmosphere. In this study, a new agricultural wastewater treatment process is proposed in which the excess water is recaptured using a reverse osmosis (RO) membrane connected to a compression separation tower. The combined effect of header tank and pipe network is considered and a model of pipe network is built. The simulation shows that the amount of water discharged to atmosphere is reduced by up to 44% with only a small reduction in water recovery rate. To demonstrate the feasibility of the proposed process, a field test was conducted in September 2015 at Korea's southern island of Jeju. A holding pond system was constructed and its performance was compared with that of a traditional holding pond. The results show that the discharge of water to atmosphere is reduced by approximately 50%, and the water recovery rate is 90%.﻿using System.Net.Http; using Ocelot.Response; using System.Text.Json.Serialization; using System.Threading.Tasks; namespace Ocelot.Configuration.Debug.Middleware.Response { public class HttpDebugMiddlewareResponseConverter : HttpResponseMiddlewareConverter { public HttpDebugMiddlewareResponse