Waves of maximal height for a class of nonlocal equations with homogeneous symbols

We discuss the existence and regularity of periodic traveling-wave solutions of a class of nonlocal equations with homogeneous symbol of order $-r$, where $r>1$. Based on the properties of the nonlocal convolution operator, we apply analytic bifurcation theory and show that a highest, peaked, periodic traveling-wave solution is reached as the limiting case at the end of the main bifurcation curve. The regularity of the highest wave is proved to be exactly Lipschitz. As an application of our analysis, we reformulate the steady reduced Ostrovsky equation in a nonlocal form in terms of a Fourier multiplier operator with symbol $m(k)=k^{-2}$. Thereby we recover its unique highest $2\pi$-periodic, peaked traveling-wave solution, having the property of being exactly Lipschitz at the crest.Comment: 25 page

Existence and regularity theory in weighted Sobolev spaces and applications

In the thesis we discuss several questions related to the study of degenerate, possibly nonlinear PDEs of elliptic type. At first we discuss the equivalent conditions between the validity of weighted Poincar\'e inequalities, structure of the functionals on weighted Sobolev spaces, isoperimetric inequalities and the existence and uniqueness of solutions to the degenerate nonlinear elliptic PDEs with nonhomogeneous boundary condition, having the form:\begin{eqnarray}\label{eqn:abs}\left\{\begin{array}{lll}{\rm div} \left( \rho (x)|\nabla u|^{p-2}\nabla u\right) =x^*,\\~~~~~~~~~~~~u-w \in W^{1,p}_{\rho,0} (\Omega),\end{array}\right.\end{eqnarray}involving any given $x^*\in (W^{1,p}_{\rho,0} (\Omega))^*$ and $w\in W^{1,p}_{\rho} (\Omega)$, where $u\in W^{1,p}_{\rho} (\Omega)$ and $W^{1,p}_{\rho} (\Omega)$ denotes certain weighted Sobolev space, $W^{1,p}_{\rho,0} (\Omega)$ is the completion of $\mathcal{C}_{0}^{\infty}(\Omega)$. As a next step, we undertake a natural question how to interpret the nonhomogenous boundary conditions in weighted Sobolev spaces, when the natural analytical tools, like trace embedding theorems, are missing. Our further goal is to contribute to solvability and uniqueness for degenerate elliptic PDEs with nonhomogenous boundary condition being the extension of~\eqref{eqn:abs}. In addition to the monotonicity method used in the first step of our discussion for the problem~\eqref{eqn:abs}, we also exploit Lax-Miligram theorem to treat the linear problem like:\begin{equation*}\begin{cases}-{\rm div} (A(x)\nabla u(x)) + B(x)\cdot\nabla u(x) + C(x)u(x) = x^{*}\ \ \text{for a.e.}\ x\in \Omega, \\~~~~~~~~~~~~~~~~~~ u(x) = g(x) \ \ \text{for a.e. }\ x\in \partial\Omega ,\end{cases}\end{equation*}as well as Ekeland's Variational Principle and Boccardo-Murat techniques to consider problem like:\begin{align*} \begin{cases} - {\rm div} \left( \rho (x)|\nabla u|^{p-2}\nabla u\right) - \lambda\, b(x)| u|^{p-2} u = x^*,\\~~~~~~~~~~~~~~~~~~~u-z \in X , \end{cases}\end{align*}where $p>1,\ \lambda>0$, and the operator $\mathcal{L}_{\lambda} u:= - {\rm div} \left( \rho (x)|\nabla u|^{p-2}\nabla u\right) - \lambda\, b(x)| u|^{p-2} u$ is non-monotone.For the study of the nonhomogeneous BVPs, we apply recent results due to Ka\l{}amajska and myself, where we constructed trace extension operator from weighted Orlicz-Slobodetskii spaces defined on the boundary of the domain to weighted Orlicz-Sobolev spaces in the domain. Information on the spectrum of the corresponding differential operator is also derived. Moreover, some nonexistence and nonuniqueness results are also analyzed

Effect of climate change in paddy production and adaptation strategies in Tharu communities of Dang district, Nepal

A study on effects of climate change on rice (Oryza sativa) production in Tharu communities of Dang district of Nepal was conducted in 2018A.D to investigate the perception and major adaptation strategies followed by Tharu farmers. The study areas were selected purposively. Cross-sectional data was collected using a household survey of 120 households by applying simple random sampling technique with lottery method for sample selection. Primary data were collected using semi-structured and pretested interview schedule, focus group discussion and key informants interview whereas monthly and annual time series data on temperature and precipitation over 21years (1996-2016) were collected from Department of Hydrology and Meteorology, Kathmandu as secondary data. Descriptive statistics and trend analysis were used to analyze the data. The ratio of male and female was found to be equal with higher literacy rate at study area than district. Most of the farmers depended on agriculture only for their livelihood where there was large variation in land distribution. Farmers had better access to FM/radio for agricultural extension information sources. The study resulted that Tharu farmers of Dang perceived all parameters of climate. Temperature and rainfall were the most changing component of climate perceived by farmers. The trend analysis of temperature data of Dang over 21 years showed that maximum, minimum and average temperature were increasing at the rate of 0.031°C, 0.021°C and 0.072°C per year respectively which supports the farmers perception whereas trend of rainfall was decreased with 7.56mm per year. The yearly maximum rainfall amount was increased by 1.15mm. The production of local indigenous rice varieties were decreasing while hybrid and improved rice varieties were increasing. The district rice production trend was increasing which support the farmer’s perception. The study revealed that there were climate change effects on paddy production and using various adaptation strategies to cope in Dang district

2D Unsteady Routing and Flood Inundation Mapping for Lower Region of Brazos River Watershed

Present study uses two dimensional flow routing capabilities of hydrologic engineering center\u27s river analysis system (HEC-RAS) for flood inundation mapping in lower region of Brazo River watershed subjected to frequent flooding. For analysis, river reach length of 20 km located at Richmond, Texas, was considered. Detailed underlying terrain information available from digital elevation model of 1/9-arc second resolution was used to generate the two-dimensional (2D) flow area and flow geometrics. Streamflow data available from gauging station USGS08114000 was used for the full unsteady flow hydraulic modeling along the reach. Developed hydraulic model was then calibrated based on the manning\u27s roughness coefficient for the river reach by comparison with the downstream rating curve. Corresponding water surface elevation and velocity distribution obtained after 2D hydraulic simulation were used to determine the extent of flooding. For this, RAS mapper\u27s capabilities of inundation mapping in HEC-RAS itself were used. Mapping of the flooded areas based on inflow hydrograph on each time step were done in RAS mapper, which provided the spatial distribution of flow. The results from this study can be used for flood management as well as for making land use and infrastructure development decisions

Probability Models For Explaining Migration Process From Eastern Uttar Pradesh, India

This paper aims to examine the pattern of male out-migration and explains the distribution of households according to number of male migrants aged fifteen and above. The suitability of proposed model is tested with primary data collected from remote and semi-urban areas of Varanasi, 2012. Findings highlight that the average number of clusters from the remote households is higher and the average number of individuals per clusters is lower in comparison to the semi-urban villages. The average number of migrants per household has increased with increasing size of households in the remote as well as in semi-urban villages. The average number of migrants per household is higher among upper caste followed by middle caste, Muslims and scheduled caste from the study area. Average number of migrants per household has increased over six times in the low economic status of the households. In the medium and high economic status of the households, average number of migrants per household is found to be around three and two times more respectively, over the last three decades. The increasing average number of migrants per household portray that an increasing propensity of adult male migration from the study area. Over 2.7 times increase in the average number of migrants per household may be primarily due to increasing man-land ratio in the absence of relative growth in employment opportunities. Thus, the existing imbalances in demand and supply of gainful employment opportunities in the region may be the key to continuously increasing the number of migrants per households from the region

Waves of maximal height for a class of nonlocal equations with homogeneous symbols

We discuss the existence and regularity of periodic traveling-wave solutions of a class of nonlocal equations with homogeneous symbol of order -r, where r > 1. Based on the properties of the nonlocal convolution operator, we apply analytic bifurcation theory and show that a highest, peaked, periodic traveling-wave solution is reached as the limiting case at the end of the main bifurcation curve. The regularity of the highest wave is proved to be exactly Lipschitz. As an application of our analysis, we reformulate the steady reduced Ostrovsky equation in a nonlocal form in terms of a Fourier multiplier operator with symbol m(k) = k$^{-2}$. Thereby we recover its unique highest 2π-periodic, peaked traveling-wave solution, having the property of being exactly Lipschitz at the crest

Comprehensive review of key parameters for improving the performance of Solar Air Heaters

Due to declining supplies of once-affordable fossil fuels, research into renewable energy sources is being prioritized. Using solar energy for passive cooling and heating can significantly reduce the demand for primary energy sources. In this survey, we look at how changing a few variables might affect how well a solar air heater functions. It helps researchers learn more about these systems’ development, properties, and potential uses. This research analyses the current literature to emphasize the value of thermal properties. Focusing on laminar sublayer creation and increasing the heat transfer coefficient, it explores ways to increase the efficiency of solar air heaters. After reviewing several articles, it has been determined that rib roughness elements and their geometric characteristics play a significant role in enhancing the thermal performance of solar air heaters. The relative roughness height, pitch, angle of attack, and width are all important factors to consider

Prevalence of multidrug resistant bacteria in causing community acquired urinary tract infection among the patients attending outpatient department of Seti Zonal Hospital, Dhangadi, Nepal

Involvement of multidrug resistant bacteria in causing community acquired infection is very serious public health issue. The main objective of our study was to determine the prevalence of multidrug resistant bacteria in causing community acquired urinary tract infection. In this study we cultured the 384 mid stream urine samples collected aseptically from the patients attending outpatient department of Seti zonal hospital and having no past history of hospitalization. The organisms isolated were identified by using conventional biochemical tests and antimicrobial susceptibility testing was performed by Kirby Bauer disc diffusion technique. Out of total 384 samples 98 (25.52%) samples showed significant bacterial growth. The most prevalent bacterium isolated was Escherichia coli. 42.86% of the bacteria isolated were found to be multidrug resistant (MDR). The involvement of such large numbers of multidrug resistant bacteria in causing community acquired urinary tract infection is very serious issue and cannot be neglected. And some abrupt initiatives should be taken by the responsible authorities to improve or at least avoid the further worsening of the situation.Nepal Journal of Biotechnology. Dec. 2015 Vol. 3, No. 1: 55-5