Minimum-Weight Edge Discriminator in Hypergraphs

In this paper we introduce the concept of minimum-weight edge-discriminators in hypergraphs, and study its various properties. For a hypergraph $\mathcal H=(\mathcal V, \mathcal E)$, a function $\lambda: \mathcal V\rightarrow \mathbb Z^{+}\cup\{0\}$ is said to be an {\it edge-discriminator} on $\mathcal H$ if $\sum_{v\in E_i}{\lambda(v)}>0$, for all hyperedges $E_i\in \mathcal E$, and $\sum_{v\in E_i}{\lambda(v)}\ne \sum_{v\in E_j}{\lambda(v)}$, for every two distinct hyperedges $E_i, E_j \in \mathcal E$. An {\it optimal edge-discriminator} on $\mathcal H$, to be denoted by $\lambda_\mathcal H$, is an edge-discriminator on $\mathcal H$ satisfying $\sum_{v\in \mathcal V}\lambda_\mathcal H (v)=\min_\lambda\sum_{v\in \mathcal V}{\lambda(v)}$, where the minimum is taken over all edge-discriminators on $\mathcal H$. We prove that any hypergraph $\mathcal H=(\mathcal V, \mathcal E)$, with $|\mathcal E|=n$, satisfies $\sum_{v\in \mathcal V} \lambda_\mathcal H(v)\leq n(n+1)/2$, and equality holds if and only if the elements of $\mathcal E$ are mutually disjoint. For $r$-uniform hypergraphs $\mathcal H=(\mathcal V, \mathcal E)$, it follows from results on Sidon sequences that $\sum_{v\in \mathcal V}\lambda_{\mathcal H}(v)\leq |\mathcal V|^{r+1}+o(|\mathcal V|^{r+1})$, and the bound is attained up to a constant factor by the complete $r$-uniform hypergraph. Next, we construct optimal edge-discriminators for some special hypergraphs, which include paths, cycles, and complete $r$-partite hypergraphs. Finally, we show that no optimal edge-discriminator on any hypergraph $\mathcal H=(\mathcal V, \mathcal E)$, with $|\mathcal E|=n (\geq 3)$, satisfies $\sum_{v\in \mathcal V} \lambda_\mathcal H (v)=n(n+1)/2-1$, which, in turn, raises many other interesting combinatorial questions.Comment: 22 pages, 5 figure

Minimum-Weight Edge Discriminators in Hypergraphs

In this paper we introduce the notion of minimum-weight edge-discriminators in hypergraphs, and study their various properties. For a hypergraph H = (V , E), a function λ : V → Z+∪{0} is said to be an edge-discriminator on H if ∑v∈Eiλ(v)\u3e0, for all hyperedges Ei ∈ E and ∑v∈Eiλ(v) ≠ ∑v∈Ejλ(v), for every two distinct hyperedges Ei,Ej, ∈ E. An optimal edge-discriminator on H, to be denoted by λH, is an edge-discriminator on H satisfying ∑v∈VλH(v) = minλ ∑v∈Vλ(v), where the minimum is taken over all edge-discriminators on H. We prove that any hypergraph H = (V , E), with |E| = m, satisfies ∑v∈VλH(v) ≤ m(m+1)/2, and the equality holds if and only if the elements of E are mutually disjoint. For r-uniform hypergraphs H = (V,E), it follows from earlier results on Sidon sequences that ∑v∈VλH(v) ≤ |V|r+1+o(|V|r+1), and the bound is attained up to a constant factor by the complete r-uniform hypergraph. Finally, we show that no optimal edge-discriminator on any hypergraph H = (V,E), with |E| = m (≥3), satisfies ∑v∈VλH(v) = m(m+1)/2−1. This shows that all integer values between m and m(m+1)/2 cannot be the weight of an optimal edge-discriminator of a hypergraph, and this raises many other interesting combinatorial questions

Numerical Modeling and Analytical Validation for Transient Temperature Distribution in a Heterogeneous Geothermal Reservoir due to Cold-Water Reinjection

ABSTRACT Reinjection of cooled geothermal fluid after extraction of heat is a common practice in order to maintain the geothermal reservoir pressure, which gradually declines due to continuous extraction of geothermal fluid. Reinjection of geothermal fluid into the geothermal reservoir ensures its safe disposal and enhances the heat recovery the efficiency of the geothermal reservoir for extracting heat energy. But since the injected geothermal fluid is cooler than the geothermal reservoir it generates a cold front near the injection well which propagates through the reservoir domain. Heterogeneity of the geothermal aquifer is also an important factor to consider since homogeneous medium is practically very rare in nature and the thermo-hydrogeological properties of the medium varies in an aquifer. The present study deals with the modeling of the transient temperature distribution in a heterogeneous geothermal reservoir in response to injection of cold geothermal water. The heterogeneous geothermal aquifer considered here is a confined aquifer with homogeneous layers of finite length and overlain and underlain by impermeable rock media. All the different layers in the aquifer and the overlying and underlying rocks are of different thermo-hydrogeological properties. The numerical modeling for the transient temperature distribution in the porous aquifer is modeled here using a software code DuMu x . The heat transport modes considered are the advection, conduction and the heat loss to the confining rock media. Results show that heterogeneity plays a very significant role in determining the transient temperature distribution and controlling the advancement of the thermal front in the reservoir. The numerical model developed here is validated in this study using an analytical model. Temperature distribution derived by both methods match with each other quite well. INTRODUCTION In a geothermal power plant the heat energy of the geothermal water is extracted for power production. The waste-water which is produced after heat extraction is then reinjected back into the geothermal reservoir. One purpose of the reinjection is safe disposal of the thermal wastewater which otherwise could have created thermal pollution if disposed on surface. Moreover reinjection helps in keeping the reservoir pressure intact which gradually declines due to continuous extraction of geothermal fluid. Also according t

A Review on the Research Advances in Groundwater–Surface Water Interaction with an Overview of the Phenomenon

Groundwater and surface water, though thought to be different entities in the past, are connected throughout the different landforms of the world. Despite being studied for quite some time, the interaction between groundwater and surface water (GW–SW) has received attention recently because of the heavy exploitation of both of these resources. This interaction is responsible for a phenomenon like contaminant transport, and understanding it helps to estimate the effects of climate change, land use on chemical behavior, and the nature of water. Hence, knowledge of GW–SW interactions is required for hydrologists to optimize resources and analyze the related processes. In this review article, different aspects of the interaction are discussed. Starting from the basics of the phenomenon, this work highlights the importance of GW–SW interactions in the hydrological cycle. Different mechanisms of GW–SW interactions are briefly examined to describe the phenomenon. The scales of interaction are also elucidated where the classification is addressed along with a brief introduction to the large scale and sediment reach scales. The study then moves on to the investigation methodologies used for the process of SW–GW interaction and their classifications based on whether they are field methods or modeling techniques. Various literature is then explored in terms of research approaches. Finally, we highlight the applicability of the methods for different scenarios. This work is aimed to summarize advances made in the field, finding research gaps and suggest the way forward, which would be helpful for hydrologists, policymakers and practicing engineers for planning water resources development and management

Analytical solutions for movement of cold water thermal front in a heterogeneous geothermal reservoir

In the present study an analytical model has been presented to describe the transient temperature distribution and advancement of the thermal front generated due to the reinjection of heat depleted water in a heterogeneous geothermal reservoir. One dimensional heat transport equation in porous media with advection and longitudinal heat conduction has been solved analytically using Laplace transform technique in a semi infinite medium. The heterogeneity of the porous medium is expressed by the spatial variation of the flow velocity and the longitudinal effective thermal conductivity of the medium. A simpler solution is also derived afterwards neglecting the longitudinal conduction depending on the situation where the contribution to the transient heat transport phenomenon in the porous media is negligible. Solution for a homogeneous aquifer with constant values of the rock and fluid parameters is also derived with an aim to compare the results with that of the heterogeneous one. The effect of some of the parameters involved, on the transient heat transport phenomenon is assessed by observing the variation of the results with different magnitudes of those parameters. Results prove the heterogeneity of the medium, the flow velocity and the longitudinal conductivity to have great influence and porosity to have negligible effect on the transient temperature distribution. (C) 2013 Elsevier Inc. All rights reserved

Geothermal reservoirs - A brief review

A brief discussion and review of the geothermal reservoir systems, geothermal energy and modeling and simulation of the geothermal reservoirs has been presented here. Different types of geothermal reservoirs and their governing equations have been discussed first. The conceptual and numerical modeling along with the representation of flow though fractured media, some issues related to non isothermal flow through fractured media, the efficiency of the geothermal reservoir, structure of the numerical models, boundary conditions and calibration procedures have been illustrated. A brief picture of the Indian scenario and some barriers related with geothermal power are discussed and presented thereafter. Finally some gaps of the existing knowledge and recent focuses of research are discussed

Analytical solutions for transient temperature distribution in a geothermal reservoir due to cold water injection

An analytical solution to describe the transient temperature distribution in a geothermal reservoir in response to injection of cold water is presented. The reservoir is composed of a confined aquifer, sandwiched between rocks of different thermo-geological properties. The heat transport processes considered are advection, longitudinal conduction in the geothermal aquifer, and the conductive heat transfer to the underlying and overlying rocks of different geological properties. The one-dimensional heat transfer equation has been solved using the Laplace transform with the assumption of constant density and thermal properties of both rock and fluid. Two simple solutions are derived afterwards, first neglecting the longitudinal conductive heat transport and then heat transport to confining rocks. Results show that heat loss to the confining rock layers plays a vital role in slowing down the cooling of the reservoir. The influence of some parameters, e.g. the volumetric injection rate, the longitudinal thermal conductivity and the porosity of the porous media, on the transient heat transport phenomenon is judged by observing the variation of the transient temperature distribution with different values of the parameters. The effects of injection rate and thermal conductivity have been found to be profound on the results

Forecasting future groundwater recharge from rainfall under different climate change scenarios using comparative analysis of deep learning and ensemble learning techniques

<p>The codes and data provide a glimpse of the work done by the authors to study the groundwater recharge phenomenon based on predominantly the rainfall recharge. The factors considered and their dataset from 1986 to 2019 are included in this repository in .xlsx format which were used to develop and train the models in this study. The codes for different AI/ML/DL models are also included in this repository.   </p&gt