Direct sum decomposition of spaces of periodic functions and some connections between shift operators, periodicity of solutions of difference equations, circulant matrices, cyclotomic polynomials, and roots of unity

It was proved that the space $\mathbb{P}_p$ of all periodic function of fundamental period $p$ is a direct sum of the space $\mathbb{P}_{p/2}$ of all periodic functions of fundamental period $p/2$ and the space $\mathbb{AP}_{p/2}$ of all antiperiodic functions of fundamental antiperiod $p/2$. In this paper, we study some connections between periodic functions, cyclotomic polynomials, roots of unity, circulant matrices, and some classes of difference equations. In particular, we state and prove the sufficient condition for the existence of periodic solutions of integer period or arbitrary period of some difference equation. We also show that the space $\mathbb{P}_n$ of all periodic functions of integer period $n$ can be decomposed as the direct sum of operators' kernels $\ker (\Phi_d(E))$, where $\Phi_d(E),\, 1 \leq d \leq n, d|n$ are the cyclotomic polynomials of the shift operator $E$. We state and prove important theorems, state and prove the necessary and sufficient conditions for a linear difference equation with constant coefficients to have periodic solutions.Comment: 10 page

Utilization of Coffee Husk and Pulp Waste as Soil Amendment. A Review

Coffee is one of the most valuable primary products in the world trade, and also a central and popular part of our culture and the crop is cultivated in about 80 countries across the globe and entangles huge business worldwide. Industrial coffee production involves either a dry or a wet processing method for the removal of the shell and mucilaginous parts from the cherries resulting in the production of coffee husk and pulp, respectively. These main by-products generated by the coffee processing plants and are disposed into arable land and surface water causing environmental pollution due to presence of toxic materials such as caffeine and tannins. To contribute to safe disposal of solid biomass waste in coffee processing many research was conducted to assess the opportunities and challenges of expanded use of solid waste from coffee processing in energy and agricultural services in across coffee producing countries. Therefore the objective of this review was to summarize the literatures and the current knowledge on the utilization of coffee husk and pulp waste for soil amendment, for sustainable agricultural systems within small-scale farming in developing countries like Ethiopia, using these easily available materials as good option for developing plant-nutrient management strategies in highly weathered soil area. Keywords: Coffee husk and pulp, coffee processing, soil amendment, waste management DOI: 10.7176/JNSR/12-11-02 Publication date:June 30th 202

Integrated Nutrient Management for Improved Coffee Production at West and Southwest Ethiopia. A Review

Soil degradation is one of Ethiopia's most severe problems, contributing for the country's low coffee production. Many researches have been conducted by Jimma agricultural research and its sub centers to ameliorate the challenges and improve coffee productions across major coffee growing areas of the west and south western Ethiopia. The goal of this review was to summarize and document major research achievements recorded so far in integrated nutrient management on coffee and recommend future research directions. Integrated application of inorganic and organic fertilizers is the main sources for replenishing plant nutrients in agricultural soils. However, selecting the optimum combination of these resources based on soil type and crop species is necessary. In this context, field experiment was conducted at Haru and Agaro research sub center to assess the effect of decomposed coffee husk and NP Fertilizer on soil physicochemical properties and yield of coffee. Research results indicated yield of Limmu coffee increased by application of 50% recommended rate of N and P mineral fertilizer (RMF) (172 and 77 kg ha-1 NP, respectively)  + 50% recommended rate  of decomposed coffee husk (DCH) (10 ton ha-1 ) at Agaro. Similar trial conducted at Haru also showed that application of 50% NP + 75%DCH improved yield of Wollega coffee variety. Besides, application of Desmodium cover crop + 50% NP (172 and 63 kg ha-1 N and P, respectively) and NP mineral fertilizer ensure high coffee yield at Jimma. In conclusion, integrated use of organic and inorganic fertilizers was recommended for coffee production at Agaro, Haru and Jimma and their surrounding areas. Future research should concentrate on evaluating other organic inputs and combined reclamation of lime-mineral fertilizer-compost, frequency of application, economic benefits, and long-term effects on soil physicochemical properties, coffee yield and bean quality, and establish cost effective soil fertility management in coffee-growing areas of the country. Keywords: Coffee, decomposed coffee husk, desmodium and soil fertility DOI: 10.7176/JEES/13-2-01 Publication date:March 31st 202

The initial-boundary value problem for some nonlocal nonlinear wave type problem

In this doctoral thesis we study some nonlinear nonlocal wave type equations. We consider three related problems which have connections with the study of some physical models in the theory of nonlocal elasticity. The nonlocality term is introduced via a convolution type integral operator L with kernel ; de ned on a bounded domain Ωc Rn as Lav(x) = IΩ a(x-y)v(y)dy: The rst problem is an initial value problem for the nonlocal nonlinear integro-partial di erential equation given by, t - △u= Lag(u) x ∈Ω, t >0 For this problem rstly local well-posedness is studied. Further analysis of the solution, like global existence and nite time blow-up, are investigated by various approaches such as assumptions of various smoothness and growth conditions on the nonlinearity. The second problem included in the thesis is the initial boundary value problem for some nonlocal nonlinear wave type equation given by the equation, utt - △u= Lag(u) x ∈Ω, t >0. (0.0.1) u=0 x ∈Ω, t>0. Local well-posedness of this problem is studied in proper Banach space settings. The third problem is variant form of the second problem and is given by the equation, utt - △u= Lau + g(u) x ∈Ω, t >0. (0.0.2) u=0 x ∈Ω, t>0. Unlike the previous problem, here the nonlocality and nonlinearity are expressed with separate terms. While we have imposed general assumptions on g(u) such as that it should be su ciently smooth and g(0) = 0, in the last two problems we have used power type nonlinear function of the form g(u) = ∣ u∣ p-1u, p > 1. The symmetry of the integral operator involved in the last problem enables us to de ne an explicit energy, which is a conserved quantity. For further analysis of solutions of the third problem, we have used the method of Nehari Manifold. Functionals like the total energy, the potential energy and the Nehari functional associated to the equation are de ned and the potential well depth is obtained in terms these functionals. The two subsets of the initial value space, namely the stable set and the unstable set that are invariant under the ow of the solution are obtained accordingly. Based on the initial energy and the sets where the initial data are located in, the blow-up or the global existence conditions for solutions is analysed

Review on Post-harvest and Green Bean Coffee Processing in Ethiopia

Coffee (Coffea Arabica L.) is the major source of currency for Ethiopia and contributes more than 35% of the total export earnings. It is originated in Ethiopia. Coffee belongs to the family Rubiacea and the genus Coffea. The two main species of Coffee cultivated on a world scale are Arabica Coffee (Coffea arabica L.) and Robusta Coffee (Coffea canaphora L.), which account for about 98-99% of the world Coffee production. Even if the high production potential and economic importance of Coffee production and productivity of crop is still limited due to post-harvest handling and processing problems. Coffee processing is aimed to lower the water content of fresh cherries to a level which allows the preservation of beans (about 11-12 %), removing all the covering which surround the beans and preparing the beans according to market requirements. Unintentional or intentional harvesting of cherries at several stages of maturation may have adverse impacts on coffee quality if these materials are processed together. Cherries at different stages of maturation should be separated and processed using different techniques in order to retain as much quality as possible. Cherries must undergo dry, semi-dry or wet processing as soon as possible after harvesting. The storage of fresh cherries in bags, heaps, hoppers or silos should not last more than eight hours. Otherwise unwanted fermentation will generate the dreaded “stinker” beans, which appear when the temperature before pulping exceeds 40-42°C. Even if the temperature does not rise to the point of creating stinkers, uncontrolled fermentation may have other negative impacts on the cup. Coffee cherries should not be kept in plastic bags. The damp airtight atmosphere that develops inside plastic bags favors unwanted fermentation that damages coffee quality. Therefore, the main objective of this paper is to review on post-harvest and green bean coffee processing in Ethiopian. Keywords: Green Coffee, Post harvest, Processing DOI: 10.7176/JNSR/9-9-05 Publication date:May 31st 201

Analysis of the Differential-Difference Equation y(x+1/2)−y(x−1/2)=y′(x)y(x+1/2)-y(x-1/2) = y'(x)

In this paper we study some solution techniques of differential-difference equation $$y'(x) = y(x + 1/2)- y(x- 1/2),$$ first without an initial condition and then with some initial function $h$ defined on the unit interval $[-1/2, 1/2]$. We show some sufficient conditions that an initial function $h$ is admissible, i.e., it yields a unique continuous solution on some symmetric interval about $0$.Comment: 14 pages, 1 tabl

Socio-Economic and Profitability of Fisheries Enterprises the Case of Fincha Amarti Nashe Reservoir of Oromia State, Horo Guduru Wollega zone, Ethiopia

The study focused on the analysis of value chain- The case of Fincha Amarti Nashe reservoir of Oromia state, Horo Guduru Wollega zone, Ethiopia. The socio-economic characteristics of the stake holders along the fisheries enterprises and Analyses of market performance were employed to analyze marketing profit and profit margin of the chain actors. Marketing margin was calculated by taking the difference between fisher men and retail prices. Analyses of fish value chain performance were done using market share and gross margin analysis.  The chain is governed by wholesalers and retailers who have capital advantage over the other chain actors. Therefore, fisher men are forced to capture a lower share of profit margin. The constraints were also described and the factors that influence consumer preferences in the study area were identified. The data were collected from both primary and secondary sources. The primary data for this study were collected from 122 selected by Multi-stage random sampled fisher men, 6 traders and 30 consumers through structured questionnaires and checklist. Descriptive statistics and econometric analysis were used to analyze the qualitative data collected from fish producers, traders and consumers by using multinomial logit model. The data which was collected from the sample fisher men, consumers were and traders were analyzed using descriptive statistics which include mean, standard deviation, chi-square, t-test, frequency table, and percentiles were used to analyze the data collected. Fishing was found to be male dominated (75.25 %), while processing was found to be female dominated (10.75 %) in the three selected kebeles of the district. The Other constraints experienced by fishermen were lack of market, low price of product, lack of transport, lack of market information, distance to the market and Perishability. Consumers also face high cost of fish, deterioration of fish and poor sanitary condition problems. In conclusion, therefore, fisher men are forced to capture a lower share of profit margin. Thus, strategic plan at improves extension services on training of local fisheries, entrepreneurial with management skills and assembly of marketing team provided to the concerning stakeholders, women’s empowerment and gender equity, launch  credit access for the fisheries sector & Introducing improved fishing inputs and accessing market with facilitating means of usual  transport, strengthening the supportive activities such as market information centers would also lift up fish supply, NGO`s , research centers and gov`t concerned sectors attentions are recommended to speed up the chain’s development. Therefore Government should provide loan scheme for the proper increase and boasting of fisher men and processors activities. Keywords: Socio-economics, Fishermen, Fish Consumers, Constraints, Value chain analysis, Fish, Actors, Marketing margin, Market outlet, Multinomial logit Mode