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ISSN : 1225-8504(Print)
ISSN : 2287-8165(Online)
Journal of the Korean Society of International Agricultue Vol.32 No.2 pp.120-129
DOI : https://doi.org/10.12719/KSIA.2020.32.2.120

# Factors Affecting Agribusiness in Port Harcourt City and Obio/Akpor Local Government Areas in Rivers State, Nigeria

Jaja Samuel Rowland*, Shi-yong Piao**, Dea-seop Lee*, Jong-In Lee**
*Department of International Cooperation Major in Global Agricultural Economics, Kangwon National University, South Korea
**Department of Agricultural and Resource Economics, Kangwon National University, South Korea
Corresponding author (Phone) +82-10-7347-4123 (E-mail) leejongin@kangwon.ac.kr
February 24, 2020 May 26, 2020 June 2, 2020

## Abstract

The study examined the factors affecting agribusiness as well as determined whether inadequate human and technological capacities affect agribusiness in Port Harcourt City and Obio/Akpor Local Government Areas of Rivers State, Nigeria. Data were collected from 300 agribusiness firms in the two Local Government Areas. Data were analyzed with descriptive statistics, factor analysis, and multinomial logistics regression. The multinomial logistic regression result was within 5% significance level, meaning that inadequate human and technological capacities negatively affect agribusiness survival. To increase the income level of the agribusiness firms, on the one hand, the internal management of enterprises should be strengthened, and the collaboration and technical capabilities of staff should be improved. On the other hand, they should pay attention to scientific and technological innovation and use more advanced means to promote enterprise reform.

# 나이지리아 리버 주 포트하코드시 오비오·아코퍼 지역의 농기업에 영향을 미치는 요인

자자 사무엘 롤랜드*, 박 세영**, 이 대섭*, 이 종인**
*강원대학교 대학원 국제협력학과
**강원대학교 농업자원경제학과

## INTRODUCTION

Agriculture, in most developing countries, is both the traditional pursuit and the key to the sustained modern economic growth. The growth of the economy without doubts conjoint with agriculture progress. The contribution of agriculture to economic growth according to Oyakhilomen et al (2014) is in:

• i. Provision of adequate food for an increasing population.

• ii. Supply of numerous raw materials for the growth of the industrial sector.

• iii. Earning foreign exchange through agricultural exports.

• iv. Provision of a market for the industrial sector’s products.

The production of foods and raw materials is a necessary thing for the stimulation and development of all other sectors of every national economy, that is to say, the development of the agricultural sector positively affects every other sector of the national economic growth Oladimeji (2004). As a sector that focuses on primary production, agriculture itself has to be technological and innovative ori-ented in order to enhance the productivity of the sector (Nyienakuna 2010). According to (Mhlanga, 2010), the agricultural sector in Nigeria consists of little competition, large and growing demand for food and beverage products and favorable environmental conditions. With the trade deficit in agricultural products where demands continue to rise, it creates immense opportunity for those willing to invest in revitalizing the sector. Although, the sector is yet to reach its potential production level with crop and livestock production remaining below demand, not enough to support the ever-growing population. As a result, Nigeria once a net exporter of food has now become a net importer spending over $4.2billion on food in 2010 with import rising exponentially to feed the growing population. For instance, with Nigeria’s current fish production output estimated at 0.5 million tons, its true potential is 2.5 million tons annually with opportunities for investments going beyond direct production. Certain drawbacks show imbalances and maladjustment resulting from uneven progress in the growth and development from agriculture to agribusiness, for example, glutted markets, unstable processes, uneconomic farm units, poor managerial training and lack of broadly conceived agribusiness policy (M.C Igbokwuwe et al 2015). Agriculture remains one of the most dynamic and critical sectors in the Nigerian economy. It employs 70% of Nigerians, including many rural women, and contributes up to 22.86 % of the country’s GDP. As in many other African countries, agriculture in Nigeria largely depends on food crops for the home market, given the Nigerian population estimated at 196 million people. In spite of this reality, Nigeria remains a net importer of food, for many reasons. First of all, the majority of the agriculture-focused operations in the country are small-scale, with limited innovation regarding inputs, harvesting, processing, distribution, and access to markets (NBS, 2018). The system and characteristics of farm agribusiness linkages in Nigeria can best be understood within the context of Nigeria's economic situation; obviously, agriculture plays the primary role of supply of raw materials to the agro-industrial processing and manufacturing sector (FAO 2010). Furthermore, it facilitates the other basic functions of agriculture such as food supply, provision of employment opportunities and income generation as well as contribution to foreign exchange earnings through exports; In Nigeria, the pace of accomplishing synergy between agriculture and industrial sector has persistently been very slow. This is partly because of the frequent changes in policy beginning with the import substitution strategies of the pre-1986 era that discouraged industrialists from patronizing locally produced raw materials (FAO 2010). ## Overview of Nigeria Agriculture In Nigeria, there are over 30 million hectares of farmland under cultivation season to season, falling substantially short of the estimated 78.5 million hectares of land that is required for farming to feed Nigeria’s growing population. The country’s agricultural sector is dominated by smallholder farmers who work an average of 4–5 acres each, under rain-fed conditions (NIRSAL, 2017). Most of them lack knowledge of modern practices, have insufficient capital and own little or no equipment of their own. As a result, much of the farm machinery, seeds, and chemicals they require are purchased and distributed to them by the government under various agricultural assistance/subsidy programs. One of such subsidy packages is the e-Wallet system introduced in 2014 under the Agricultural Transformation Agenda (ATA) through which subsidized electronic vouchers for inputs were delivered directly to the farmers' mobile phones and then the vouchers were used as cash to purchase the inputs directly from agrodealers nearest to them (Obayelu 2017). This initiative has proven to be quite efficient and effective in providing poor farmers with much-needed resources and eliminating corruption in the subsidy program (Charles D 2004). There are about ten relatively large commercial scale farms in Nigeria deploying some form of mechanized processes, these and a few construction companies can purchase tractors for agricultural and non-agricultural purposes(APP 2016). During the Structural Adjustment years, the government encouraged backward integration, but inconsistencies in macroeconomic policy initiatives between 1986 and 1995 discouraged farmers from expanding production of suitable agricultural raw materials for local processing and manufacturing (FAO 2004). Backward integration and the privatization of state-owned enterprises are currently emphasized as a desirable policy objective by the new democratic government but growth in the manufacturing and agribusiness sector has changed very little in the past decade, Between 1990 and 1999, manufacturing including agro-industrial output in real terms actually dropped to about 92 percent of the level reached in 1990; Whilst the other productive sectors did register modest growth, this was more than offset – in terms of per capita output – by the annual increase in population (FAO 2004). The one exception was agriculture, there were no significant changes in the structure of output over the decade in Nigeria; in 1999, agriculture accounted for nearly 40 percent of GDP, virtually the same proportion as in 1990; the average share of output originating from manufacturing declined from eight to nine percent (FAO 2004). The crude oil sector share of GDP accounted for 12.7 percent in 1999, virtually the same as nine years ago; Without the adequate capacity to effectively engage in several aspects of agricultural innovation, transforming agriculture in sub-Saharan Africa will be a long, drawn-out process; the development of the agricultural sector will also require a different set of skills in order for agribusiness institutions to adjust and expand in order to become competitive in the regional and international markets, agribusiness firms in sub-Sahara Africa (Nigeria) need sufficiently and appropriately trained human capital (World Bank, 2013). Similarly, Francis, (2017) attributed the reason why most agribusinesses fail in Nigeria to the inability of farm owners to pay attention to managerial skills needed for sustaining such business. The survival rate of agribusiness in Rivers State Nigeria is low due to factors which majority of the agribusiness operators attributed to poor capital. Survival rates of Agribusinesses' output are low mostly after the first year due to a number of factors that sometimes result in borrowing so as to obtain sufficient finance to facilitate their early stages of growth. Asma et al (2015). The important question begging for answer is why the low rate of agribusiness survival in Rivers State, Nigeria? What is the level of managerial and technological capacities of agribusiness operators? It is against this backdrop that this study is being designed to enable us to determine those factors that affect the survival of agribusinesses as well as examine the effectiveness of human and technological capacities development on agribusiness survival in Rivers State. ## DATA AND MODEL ### Data Collection This study adopted a cross-sectional research design. In order to collect as much relevant information on the subject matter as possible, data was collected from both primary and secondary sources. Primary Sources: A set of well-structured questionnaire was used to gather information from agribusiness operators (respondents) and a personal interview was conducted where necessary. Secondary data were also consulted in writing the thesis especially from sources such as Journals, conferences, seminars, textbooks as well as online sources. Information from these sources formed the basic core of the study. A total of 300 agribusiness operators within the two (2) L.G.A.s were randomly sampled for the survey, in each L.G.A, 150 agribusiness operators were sampled. The study relied heavily on primary data obtained from the operators. Section A of the questionnaire was structured to obtain information on the socio-economic characteristics or variables of the respondents including gender, age, marital status, household size, educational qualification, income/ revenue level, years of business experience and the number of employees. The purpose is to extract information on the distinctive factors that affect the survival of agribusiness. Section B highlighted common agribusinesses that operate within the State as well as those possible factors/ variables that could affect the smooth operation of agribusiness. Section C of the questionnaire concentrated on how human and technological capacities development could affect agribusiness survival separately. It also pointed out relevant areas that require capacity development in order to for agribusiness to survival and also some constraint that is mitigating against the effective use of technology for the survival of agribusinesses. ### Description of Study Area This study was conducted in Obio/Akpor and Port Harcourt City Local Government Areas (L. G. As) in Rivers State Nigeria. Nigeria, a country located on the western coast of Africa. Nigeria has a diverse geography, with climates ranging from arid to humid equatorial. Rivers State also known as the treasure base of the nation, one of the 36 States of the Federation lies mainly within the Delta region of Rivers Niger. Rivers State is made up of 23 Local Government Areas (LGAs) and covers a land size of about 21,850 square km with a population of about 5,185,400 according to the 2006 census. The state is bordered in the North by Imo, Anambra and Abia State, in the East by Akwa Ibom State, in the West by Bayelsa and Delta State and in the South by the Atlantic Ocean. Port Harcourt City Local Government Area is located in the heart of Rivers State where the industrial activities take place. It has a total size of 109 square kilometers (42 sq mi) and lies within Latitude: 4°51'N. Longitude: 7°0'E with a total population of 538,558 according to the 2006 population census. Port Harcourt features a tropical monsoon climate with lengthy and heavy rainy seasons and very short dry seasons. Port Harcourt’s heaviest precipitation is seen during the month of September where an average of approximately 370 mm of rain is seen, and average temperature typically between 250C -290C in the city. Obio/Akpor is a local government area in the metropolis of Port Harcourt, one of the major centers of economic activities in Nigeria, and one of the major cities of the Niger Delta. The local government area covers 260 km2 and at the 2006 Census held a population of 464,789. Obio/Akpor is located on latitudes 4°45'N and 4°60'N and longitudes 6°50'E and 8°00'E with its headquarters at Rumuodomaya. ### Theoritical Models Data obtained from the field survey were edited, coded and cleaned to ensure consistency, uniformity, and accuracy. The data collected were processed and analyzed with the use of SPSS version 22. In order to achieve objective one of this study i.e. to determine the factors that affect agribusiness as well as the technological constraint faced by agribusiness operators in Rivers State, descriptive statistics such as percentage and frequency were adopted. To achieve objectives two, that is, inadequate human and technological capacities whether they affect agribusiness in Rivers State. Factor, cluster, and multinomial logistic regression analysis. ### Factor Analysis: Based on Chua's (2009) suggestion, factor analysis is the procedure used by many researchers to organize, identify and minimize big items from the questionnaire to certain constructs under one dependent variable in research. Kaiser- Meyer-Olkin (KMO) test was done to identify whether the data is suitable for factor analysis. The KMO test formula as stated by M. J Norusis (1994). $K M O = ∑ j = 1 n ∑ i = 1 n r i j z ( ∑ j = 1 n ∑ i = 1 n r i j 2 + ∑ j = 1 n ∑ j = 1 n α i j 2 )$ (1) Where: • [rij] = correlation coefficient • ij]= Partial correlation coefficient. According to Johnson and Wichern (2002), the factor analysis model is as stated below: $X 1 = λ 11 F 1 + λ 12 F 2 + … + λ 1 m F m + ε 1 X 2 = λ 21 F 1 + λ 22 F 2 + … + λ 2 m F m + ε 2 X n = λ n 1 F 1 + λ n 2 F 2 + … + λ n m F m + ε n$ (2) Where, • X1-Xn= Standardized scores • F1-Fm = Standardized factor scores • λ11-λnm= Factor loadings • ε1-εn = Error variance ### Multinomial Logistic Regression Model The logistic regression model is an extension of the binomial logistic regression model. It is used to predict a dependent variable given one or more independent variables. A multinomial logistic regression model was used to determine the extent to which the dependent variable (monthly net profit) can be predicted by the independent variables. Multinomial logit models are discrete choice models used for the case of a dependent variable with more than two categories (Gujarati and Sangeetha, 2007 and Greene, 2008). Perhaps the simplest approach to multinomial data is to nominate one of the response categories as a baseline or reference categories, calculate log-odds for all other categories relative to the baseline, and then let the log-odds be a linear function of the predictors. In the multinomial logit model, we assume that the log-odds of each response follow a linear model. $η i j = ln ( π i j 1 − π i j ) = α j + β j x i , j = 1 , 2 , … , j − 1$ (3) Where, • πij = the probability that the i agribusiness firm earn j monthly net profit • αj = intercept or constant • βj is a vector of regression coefficients, for j=1, 2,…, j−1 • x = explanatory variable • j = the number of levels in the categorical response variable, • i = represents the number of explanatory variables. Note that we have written the constant explicitly, so we will assume henceforth that the model matrix X does not include a column of ones. The factors that affect agribusiness, as well as technological constraints faced by agribusiness operators, are pre-sented in the descriptive form in table 1 below. Econometric/empirical results obtained from the factor analysis and a multinomial logistic regression model. The multinomial logistic regression model was used to examine whether inadequate human and technological capacities development affects agribusiness. The results are discussed below. ## ESTIMATION AND RESULTS ### Descriptive Statistics The survey that was randomly carried-out among agribusiness operators, out of the 300 respondents interviewed to obtain the extent to which the above factors affect its agribusiness on the scale of “not very great extent to very great extent”, according to the respondents, 35.7% indicated that inadequate human capacity development affect them to a very great extent, 29.7% indicated great extent, 23% were neutral while not great extent and not very great extent shows 6.3% and 5.3 respectively. Another factor examined was inadequate capital and respondents indicated as follows, 50.7% affected to a very great extent, 31.7 to a great extent, 10.3% were indifferent, 6% says not great extent while 1.3% indicated not very great extent. The majority of the respondents indicated that to a very great extent inadequate capital affects their businesses. Inadequate technological capacity development, an important factor was also examined and the respondents indicated as follows, very great extent 33.3%, and great extent 26.3%, those who indicated neutral was 23.3%, not great extent category was 5.7% while not very great extent was 4.3%. The majority of the respondents (33.3%) were indifferent regarding inadequate technological capacity development. The reason given was a lack of information and the ability to apply the technology. As shown in <Table 1> all the factors examined were above 20% for both “great and very great extent” scale and below 10% for both “not great and not very great extent. This implies that these factors greatly affect agribusiness in Rivers State. Sequel to the results of the survey, out of the 300 agribusiness operators interviewed on the extent to which these constraints affect its application of technology in its agribusinesses. As obtained from the survey, for high cost of purchasing the technology, 7.0% of the respondents stated that it is not to a very great extent, 6.3% indicated not great extent, 23% were neutral, meanwhile 32.0% and 32.0% revealed that high purchasing cost affect them to a great extent and very great extent respectively. Another constraint was lack/inadequate technical knowhow. 39.3% of the respondents stated that this constraint affects them to a very great extent, 26.0% to a great extent, 25.3% were neither here nor there, while 5.3% of the respondents indicated not great extent, and the least was 4.0% responded to not very great extent. The table shows that the majority of the respondents don’t use technology due to the lack of/inadequate technical skills and knowledge required by agribusiness operators. 35.7% of the respondents were neutral, those who lack idea on how to source for the technology affect to a great extent constitute 29.0% while those who are a constraint to a very great extent were 19.7%. Those within the scale of not very great extent and not great extent were 7.7% and 8.0% respectively. This shows that the majority are indifferent about how to source for the technology why 29.0% second highest are a constraint to a great extent due to no knowledge on how to source for the technology that would improve their agribusinesses. The survey also shows that 33.0% of the respondents indicated that to a very great extent the high cost of maintenance is a constraint to them, while 31.3% were a constraint to a great extent. 25.3% were in a neutral position and 5.0% stated that it is not to a great extent while 5.3% of the respondents indicated not very great extent. A majority (43.0%) of the respondents revealed that they are a constraint to a very great extent, 27.7% were of great extent, 20.7% neutral while not great extent and not very great extent scale were of the same 4.3%. From the above table, more agribusiness operators would be willing to adopted and invest in the appropriate technology if the chances of success thereafter are high. ### Empirical Results of Objective two #### (1) Factors extraction and loading analysis Factor analysis was employed to construct new factors (components) that affect agribusiness in the study area. Kaiser–Meyer-Olkin (KMO) measure of sampling adequacy and Bartlett’s Test of Sphericity are both tests that can be used to determine the factorability of the matrix as a whole. Based on the results <Table 3> the Kaiser –Meyer- Olkin (KMO) measure of Sampling Adequacy is 0.894 which is greater than 0.6, this confirmed that the data set was fit to be use subjected to model for data reduction analysis Furthermore, the result value of Bartlett’s Test of Sphericity significant (P<.000). Based on the results, it is ideal to proceed with the factor analysis After the extraction or identification of the preferential factors in the first step, the next step was to subject the factors to rotated component matrix analysis. This was done with the aim of obtaining the covariance and correlations of the rotated factor loadings. The results obtained are presented below in <Table 4> Values below 0.50 were omitted from the loading and just those above 0.50 were retained to identify the loading under each factor. After factor analysis, the following results are obtained. Factor 1 conclude: production, storage, packaging, processing, technology, post-harvest handling. Factor 2 conclude: marketing skills, bood and record-keeping, business planning/development, finance and credit management. Factor 3 conclude: finance and credit management, income, investment analysis. Factor 4 conclude: transportation, supply. #### (2) Multi-collinearity test The factor scores obtained from the factor loading were saved as variables and subjected to a multi-collinearity test to observe if any collinearity existed and if the data set was suitable for the multinomial logistic regression model chosen. The variance inflation factors (VIF) values which are all less than 10 show the there are no perfect multicollinearity, <Table 6> shows, therefore, it is fit for the analysis. The model tested significant (P<.000) at a 5% confidence interval, therefore, it was appropriate to use for the analysis of the factor scores. In addition, the VIF values of the selected variables that were subjected to analysis were all within an acceptable range. (4) Result of Multinomial Logistic Regression Analysis: Null Hypothesis (H0): There is no significant relationship between inadequate human & technological capacities development and agribusiness. Alternative Hypothesis (Ha): There is a significant relationship between human & technological capacities development and agribusiness. Model Fitting Information in <Table 7>, describes the relationship between inadequate human & technological capacities development and agribusiness survival and reveals that probability of the model chi-square 240.897 was 0.000, less than the level of significance of 0.05 (i.e p<0.05) which means that the model is statistically significant to predict the dependent variable(monthly net profit). In this survey, the model summary is given as follows: $ln ( p ( m n p = < 55 ) p ( m n p = > 140 ) ) = 13.458 + 0.517 ( Factor1 ) +0.525 ( Factor2 ) - 0.501 ( Factor3 ) - 4. 892 ( Factor4 ) ln ( p ( m n p = < 56 − 85 ) p ( m n p = > 140 ) ) = 10.725 + 0.439 ( Factor1 ) +0.422 ( Factor2 ) - 0.435 ( Factor3 ) -3 .933 ( Factor4 ) ln ( p ( m n p = 86 − 111 ) p ( m n p = > 140 ) ) = 9.352 + 0.130 ( Factor1 ) +0.226 ( Factor2 ) + 0.608 ( Factor3 ) -2 .730 ( Factor4 ) ln ( p ( m n p = > 111 − 140 ) p ( m n p = > 140 ) ) = 3.982 − 0.368 ( Factor1 ) +0.896 ( Factor2 ) - 0.132 ( Factor3 ) -1 .406 ( Factor4 )$ Monthly net profit <$55 compared to >$140: For one unit increase in agribusiness operator with inadequate technological capacity whose monthly net profit is <$55 compared to >$140, the multinomial log-odds of business survival would increase by 0.517 unit while other variables are held constant. The variable was statistically significant at p<0.036, and the odds ratio from the result is 1.68 meaning the odd increase by 68% that is, a unit increase in agribusiness operator with inadequate technological capacity whose monthly net profit is <$55 increases the odds by 68% holding all other factors constant. The second variable of interest, holding other variables constant, one unit increase in agribusiness operator with inadequate human capacity development whose monthly net profit is <$55 compared to >$140, the log-odds of business survival would increase by 0.525 units. The result shows that its odd ratio is 1.69 which means that if all factors are held constant, as agribusiness operators without human capacity development whose monthly net profit of <$55 increases by one unit, the odd of making <$55 monthly net profit compared to >$140 increases 1.69 times (i.e. increases by 69%) compared to making >$140 monthly net profit. The variable was also statistically significant at p<0.027, implying that we would reject the null hypothesis.

Monthly net profit of $56-$85 compared to >$140. Factor 1, 2 and 3 were not statistically significant when the month net profit of agribusiness operators within the range of$56-$85 was compared with the reference category. However, the odd ratios show that a unit increase in agribusiness operator with inadequate technological and human capacity whose monthly net profit is within the range of$56-$85 increases the odds by 55% and 52% respectively compared to making >$140 while holding all other factors constant.

Monthly net profit of $86 -$111 compared to >$140. Factor 1 is statistically significant at p<0.001, factor 2 at p<0.049, this also means that we can reject the null hypothesis. The odds for factor 1 and factor 2 are 1.14 and 1.25, meaning that if other factors are held constant, a unit increase in agribusiness operators with inadequate technological and human capacity whose monthly net profit falls within the range of$86 - $111 increases the odds by 14% and 25% respectively compared to making >$140. Factor 3 was also statistically significant at p<0.018. For the coefficient for factor 3 is 0.608, this means that for one unit agribusiness operators with inadequate capital whose monthly net profit is within $86 -$111 compared to monthly net profit of >$140, holding other factors constant, the multinomial log-odds for making <$55 is expected to increase by 0.608. This literarily means that there is a 61% probability of earning monthly net profit within the range of $86 -$111 rather than the monthly net profit of >$140 due to inadequate capital to effectively run the firm. Monthly net profit within the range of$112-$140 compared to >$140: The result indicates that factor 2 was statistically significant at p<0.035. The odd ratio also shows that inadequate human capacity development gross affects agribusiness. Factor 4 in all the categories of monthly net profit, factor 3 in the categories of <$55,$112-$140, factor 1 in the category of$112-\$140 showed negative coefficients. This means that these factors when compared with the reference category relatively have less effect on agribusiness and high effect in the categories where its coefficients are positive. The implication of the result is that there is less likely effect of the dependent variable, unlike inadequate technological and human capacities development that strongly shows the effect on agribusiness. This result also reflects the result of the descriptive statistics which indicated that a high percentage of agribusiness firms is to a great and very great extent are affected inadequate human capacity development, inadequate technological capacity development, and inadequate capital among other factors. In addition, the result shows that the inadequate technological capacity development was due to some constraint faced by agribusiness firms such as Business environment, high cost of purchase/maintenance, chances of success if adopted. The results are inconsonant with the study of ChegeNg’ang’a and Gichira (2017) which revealed that technological capacities and human resource capacities affected the growth of Agribusiness Micro and Small Enterprises in Embu County. The result is also in line with the findings of Suresh et al (2016) which identified that human and technological capacities are needed for agribusiness development and management in sub-Saharan Africa at the individual, organization, and policy process levels.

## CONCLUSIONS

The results of the factor analysis in this paper shows, factor 1 conclude: production, storage, packaging, processing, technology, post-harvest handling. Factor 2 conclude: marketing skills, bood and record-keeping, business planning/ development, finance and credit management. Factor 3 conclude: finance and credit management, income, investment analysis. Factor 4 conclude: transportation, supply.

The totality of the multinomial logistic results shows the percentage decrease in the odds of the independent variables with respect to the different categories of monthly net profits (dependent variable). The result revealed that the odd or probability of earning low net profit by firms with inadequate technological and human capacity development in the study areas is high, which means that inadequate human and technological capacities negatively affect agribusiness negatively. This implies that the increase in the number of agribusinesses with inadequate technological and human capacity development will continue to cause a decrease in their income level and as such, the chances of agribusiness firms will continue to decline causing little or no competition in the agricultural sector. The lowest monthly net profit category indicated higher odds or probability; implying that the increase in the number of agribusiness firms with inadequate technological and human capacity development will continue to cause a decrease in their income level and as such, the chances of agribusiness will continue to decline.

## 적 요

본 연구에서는 포트하코트 시와 나이지리아 리버스주 오비 오/아크박 지역의 인력과 기술력 부족이 농업 종합 기업 발전 에 영향을 주는지 검토하고, 또한 이외 어떠한 요인이 농업 종합 기업에 영향을 미치는지 분석을 해보았다. 데이터는 두 지방 정부에서 300개의 농업기업에 대해 확보한 자료이다. 분 석방법은 인구적 통계, 요인분석 및 다항로지스틱회귀분석을 이용하였다. 다항로지스틱 분석결과 유의성이 5% 이내이며, 이는 인력과 기술력 부족이 농업 종합 기업 발전에 영향을 주 는 것을 의미한다. 이에 농업 종합 기업의 수익을 높이기 위 해 기업 내부의 관리를 철저히 해야되고 내부 인원의 협작 및 기술능력을 강화시켜야 한다. 또한 과학기술혁신을 통해 기업 의 개혁을 촉진해야 한다.

## Figure

Map of Selected Study Areas

## Table

Factors that Affect Agribusiness in Rivers State

Constraints faced by agribusiness operators in the use of technology

KMO and Bartlett's Test

Rotated Component Matrixa of the extracted factors

ANOVAa test of the model

Model Fitting Information

Coefficients of factor scores after multinomial logistics regression analysis

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13. Nigerian Institute of Social and Economic Research (NISER).1999. A Characterization of industrial demand for major agricultural commodities in Nigeria, Ajakaiye D.O. and Akande S.O. (eds) Nigerian Institute of Social and Economic Research, Ibadan, Nigeria.
14. Nyienakuna M. G. and Udoh, A. J. 2010. Adoption of improved poultry technologies by poor resource farmers in Nigeria: implications to meat protein availability in the 21st century.
15. Obayelu, A. 2017. What Does Literature Say about the Determinants jig of Adoption of Agricultural Technologies by Smallholders Farmers? Agricultural Research & Technology. VL 06.555676. 2017/04/21
16. Oladimeji Idowu Oladele.2004. Effect of World Bank Loan Withdrawal on the Performance of Agricultural Extension in Nigeria: Nordic Journal of African Studies 13(2): 141-145.
17. Oyakhilomen, O. , & Zibah, R. 2014. Agricultural production and economic growth in Nigeria: Implication for rural poverty alleviation. Quarterly Journal of International Agriculture, 53(3), 207-223. Retrieved from http://ageconsearch.tind.io//bitstream/195735/2/1_Oyinbo.pdf
18. World Bank.2013. Growing Africa. Growing Afrca. https://oi.org/10.1596/26082
19. Y. P. Chua.2009. Statistik Penyelidikan Lanjutan Ujian Regresi, Analisis Faktor dan Ujian SEM. McGraw-Hill Malaysia.


text:
lilypond-user
[Top][All Lists]

## Re: Typo in NR

 From: Trevor Daniels Subject: Re: Typo in NR Date: Tue, 18 Aug 2009 23:39:32 +0100

```
Alexander Kobel wrote Tuesday, August 18, 2009 8:03 PM
```
```
```
the last sentence in NR 6.8, "Difficult tweaks", should probably read "Note, however, that \override, applied to _NonMusicalPaperColumn_ and PaperColumn, still works as expected within \context blocks."
```instead of
```
"Note, however, that \override, applied to NoteMusicalPaperColumn and PaperColumn, still works as expected within \context blocks."
```
This was fixed in the English manuals in the development branch
last February, but thanks to you I noticed it was still wrong in the
French NR, which I've now fixed.  Thanks for catching it again.

```
By the way, just a few lines before at the start of this item it says: "Some objects cannot be changed with \override for technical reasons. Examples of those are NonMusicalPaperColumn and PaperColumn." Are these the only objects affected? If so, we probably should strike the "Examples of those" in favor of "These", otherwise we should also cite the others, or refer to some place where they are listed?
```
Good point, and I'm not sure of the answer.  I'm not aware of any
```
other grobs that require \overrideProperty to be used. \overrideProperty
```can be used to override properties of (some?) other grobs too.  It
acts like \once \override.

I'll wait for more expert responses before I amend the docs.

Trevor

```

format: plain


text:
# Confused about a limit proof and Big O.

I gave an incorrect proof here : How can evaluate $\lim_{x\to0}\frac{\sin(x^2+\frac{1}{x})-\sin\frac{1}{x}}{x}$

I am confused as when considering the mistakes in my proof it seems the limit cannot be $0$. The method must thus be completely wrong even more wrong than the comments suggest.

Yet I believe there should be a big O proof possible and even one similar to the one I posted. The "paradox" Im getting at is perhaps clearer understood when considering that my method/the correct method should also work if sine is replaced by another function that has a Taylor series at $0$ with all $a_n$ larger or equal to $0$. (and for which the limit should also be $0$ ofcourse).

Very confused. Please keep in mind that I want a proof based on big O and not on trig identities or l'hopital rule.

-

A Taylor series of $f(x)$ around $x=0$ is of little heltp when estimating $f(x^2+\frac1x)$ as $x\to 0$. Note that $x^2+\frac1x\to\infty$. Rather you need the Taylor series developed around $\frac1x$ (and for general $f$ have to make sure that the constant hidden in the big-O does not depend on the point of developping): $$\tag1\sin(x_0+h)=\sin(x_0)+h\cos(x_0)-\frac12h^2\sin(x_0)+O(h^3)$$ seems to give us $$\tag2 \sin(x^2+\frac1x)-\sin(\frac1x)=x^2\cos(\frac1x)-\frac12x^4\sin(\frac1x)+O(x^6)$$ but the constant hidden in $(1)$ may depend on $x_0$, that is in $(2)$ the constant depends on $x$ which of course makes the $O$ useless. However, one can argue that because of the periodicity, the $O$ is uniform in $x_0$ here.


text:
## College Algebra (10th Edition)

Domain: $\color{blue}{(-\infty, -3) \cup (-3, 1) \cup (1, +\infty)}$
Factor the denominator of the function to obtain: $f(x) = \dfrac{x}{(x+3)(x-1)}$ The value of $x$ can be any real number except the ones that will make the function undefined. The given function is undefined when the value of the denominator is zero. This is because division of zero leads to an undefined expression. Find the values of $x$ that will make the denominator zero by equating each factor of the denominator to zero and then solving each equation. $\begin{array}{ccc} &x+3=0 &\text{ or } &x-1=0 \\&x=-3 &\text{ or } &x=1 \end{array}$ Thus, the value of $x$ in the given function can be any real number except $-3$ and $1$. In interval notation: Domain: $\color{blue}{(-\infty, -3) \cup (-3, 1) \cup (1, +\infty)}$


text:
# Catalysis:Intermolecular Potential

The catalysis module uses an atomic model (O and C atoms, as well as surface atoms, are each represented explicitly). All atoms interacts via a square-well intermolecular potential. The functional form of the square well potential is shown below:

{\displaystyle u(r)={\begin{cases}{\begin{aligned}\infty {\text{ if r}}\leq \sigma \\-\epsilon {\text{ if r}}<\lambda \sigma \\0{\text{ if r}}\geq \lambda \sigma \end{aligned}}\end{cases}}}        (Equation 8)

where ${\displaystyle \sigma }$ is the collision diameter, ${\displaystyle \epsilon }$ is the well depth (minimum potential energy), and ${\displaystyle \lambda }$ is the reduced well width.

Graphically, the SW potential can be representated as:

All atoms interacting with the surface must have 3 surface sites (atoms) to react/adsorb, or to become stable on the catalyst surface. Diatomic species (CO and O2) require 2 sites to adsorb. The triatomic product CO2, once formed, does not dissociate (thermodynamically stable).

The atoms are propagated through time and space using the discontinuous molecular dynamic (DMD) method, in conjunction with the square well potential mentioned above.


text:
# aliquot

## < a quantity that can be divided into another a whole number of time />

The more I use or read about Common Lisp, the more I see subtle reincarnations into statistical languages like R (eval and quote, of course) or Stata (foreach and forvalues, close to dolist and dotimes). I barely use Stata or R nowadays, but Lisp dialects still have a bright future ahead, and there are so much to learn. #lisp


text:
# Toss of three coins - probability

Assume all outcomes are equally probable, ie. consider the uniform probability measure on the probability space.

I need to determine the following:

1. How many outcomes are there in a toss of three coins?\

Let 0 denote heads and let 1 denote tails.

$E=\{(c_1,c_2,c_3)\in\mathbb{R}^3|e_i\in{0,1},i=1,2,3\}$.

$\#E=2^3=8$.

1. The probability that all coins show heads.

2. The probability that at least two coins show tails.

3. The probability that precisely one coin show tails.

4. What are the answers to 1-4 for a toss of $n$ coins? How big must $n$ be to ensure that the probability that at least one of the coins er greater than $95\%$?

• $8$ is a good answer if you want the outcomes to be equally probable to help answer questions 2,3,4 – Henry Nov 3 '15 at 0:06
• The first part of my question was wrongly formulated, updated now. – jukka.aalto Nov 3 '15 at 0:09
• @Henry It is mainly the following questions that I'm concerned with. – jukka.aalto Nov 3 '15 at 7:52

• $8$ outcomes: HHH, HHT, HTH etc;
• $4$ outcomes: number of heads is $0,1,2,3$;
• $2$ outcomes: more heads or more tails;
• $2$ outcomes: second toss is heads or tails
• $3$ outcomes: number of tails is less than $1$, equal to $1$, greater than $1$;
• $6$ outcomes: the number of heads if the first two tosses count double is $0,1,2,3,4,5$;


text:
• Create Account

### #ActualHodgman

Posted 23 November 2012 - 09:32 AM

You can think of a GPU of basically being a very-very-very-wide SIMD CPU.
Normally when you compute x = y + z, those 3 variables represent single values.
e.g. 2 + 2, results in 4.
With SIMD, those 3 variables represent arrays.
e.g. [2,7,1] + [2,1,1] results in [4,8,2].

Each instruction is simultaneously executed over a large number of values, so that you get more work done faster.

You want to avoid branching with this kind of architecture, because you end up wasting a lot of your SIMD abilities.

e.g. take the code
if( y > 5 )
x = y;
else
x = z;
If we execute that with our data of y=[2,7,1] and z=[2,1,1], this results in:
if( y > 5 ) [false, true, false]
x = y; [N/A, 7, N/A]
else [true, false, true]
x = z; [2, N/A, 1]
//finally x = [2, 7, 1]
The GPU has had to execute both the 'if' and the 'else', but ignoring some parts of it's arrays for each branch, and merging the results at the end. This is wasteful -- e.g. say the GPU has the capability to work on 3 bits of data at once, in this example it's only working on 1 or 2 bits of data at once.
The more nested branches you add, the more wasteful this becomes... so those kinds of programs are better of running on regular CPU's (or being redesigned to better suit this style of hardware).

In practice, the GPUs don't have any way to communicate with peripherals, just with the CPU
... the GPUs lack key features such as interrupts that are critical to implementing those programs in practice.

Out of interest's sake, GPUs can generate CPU interrupts and write to arbitrary addresses (which might be mapped to peripherals), but these abilities aren't exposed on PC's (outside of the driver).

### #1Hodgman

Posted 23 November 2012 - 09:28 AM

You can think of a GPU of basically being a very-very-very-wide SIMD CPU.
Normally when you compute x = y + z, those 3 variables represent single values.
e.g. 2 + 2, results in 4.
With SIMD, those 3 variables represent arrays.
e.g. [2,7,1] + [2,1,1] results in [4,8,2].

Each instruction is simultaneously executed over a large number of values, so that you get more work done faster.

You want to avoid branching with this kind of architecture, because you end up wasting a lot of your SIMD abilities.

e.g. take the code
if( y > 5 )
x = y;
else
x = z;
If we execute that with our data of y=[2,7,1] and z=[2,1,1], this results in:
if( y > 5 ) [false, true, false]
x = y; [N/A, 7, N/A]
else [true, false, true]
x = z; [2, N/A, 1]
//finally x = [2, 7, 1]
The GPU has had to execute both the 'if' and the 'else', but ignoring some parts of it's arrays for each branch, and merging the results at the end. This is wasteful -- e.g. say the GPU has the capability to work on 3 bits of data at once, in this example it's only working on 1 or 2 bits of data at once.
The more nested branches you add, the more wasteful this becomes... so those kinds of programs are better of running on regular CPU's (or being redesigned to better suit this style of hardware).

PARTNERS


text:
# Homogeneous Linear Systems

For a nonhomogeneous system of 2012 equations in 1999 unknowns, answer the following three questions:

• Can the system be inconsistent?
• Can the system have infinitely many solutions?
• Can the system have a unique solution?
-
What have you tried? (And for that matter, what have you learned?) See meta.math.stackexchange.com/questions/1803/… for why I ask. – Willie Wong Nov 2 '12 at 16:00
What do you know about linear systems? – Arkamis Nov 2 '12 at 16:01
Honestly not much. I'm hoping to learn from answers – Sam Nov 2 '12 at 16:03
This really is incredibly similar to your other question here math.stackexchange.com/questions/227608/… Would it not be better to ask one question and see if you get the idea? – Simon Hayward Nov 2 '12 at 16:12
Simplify! Consider, say 6 equations and 3 variables. – vesszabo Nov 2 '12 at 16:32

There's homogeneous in the title and nonhomogeneous in the text. In either case, we have a $2012 \times 1999$ matrix $A$. We are looking for solutions to $A\mathbf{x}=\mathbf{0}$ (homogeneous) or $A\mathbf{x}=\mathbf{b}$ where $\mathbf{b} \neq \mathbf{0}$ (inhomogeneous).

## Homogeneous

We immediately have $A\mathbf{0}=\mathbf{0}$, so $\mathbf{x}=\mathbf{0}$ is a solution, so the system is consistent. (In fact, all homogeneous systems of linear equations are consistent due to the all-$0$ solution.)

By definition, it will have infinitely many solutions if and only if the null space of $A$ is non-trivial, i.e., if and only if the rank of $A$ is less than $1999$ (by the Rank-Nullity Theorem). One example is when $[A|\mathbf{0}]$ corresponds to the system of equations \begin{align*} x_1 &=0 \\ 2x_1 &=0 \\ 3x_1 &=0 \\ \vdots \\ 2012x_1 &=0. \end{align*}

By definition, it will have a unique solution (and this solution must be the zero solution) if and only if the null space of $A$ is trivial, i.e., if and only if the rank of $A$ is equal to $1999$. One example is when $A$ has $1$'s on the main diagonal and $0$'s elsewhere, which has rank $1999$. This is when $[A|\mathbf{0}]$ corresponds to the system of equations \begin{align*} x_1 &=0 \\ x_2 &=0 \\ \vdots \\ x_{1999} &=0 \\ 0 &=0 \\ \vdots \\ 0 &=0. \\ \end{align*}

## Inhomogeneous

It can be inconsistent: e.g. when $[A|\mathbf{b}]$ corresponds to the system of equations \begin{align*} x_1+x_2+\cdots+x_{1999} &=1 \\ x_1+x_2+\cdots+x_{1999} &=2 \\ \vdots \\ x_1+x_2+\cdots+x_{1999} &=2012. \\ \end{align*}

It can have infinitely many solutions: e.g. when $[A|\mathbf{b}]$ corresponds to the system of equations \begin{align*} x_1 &=1 \\ 2x_1 &= 2 \\ 3x_1 &= 3 \\ \vdots \\ 2012x_1 &= 2012. \\ \end{align*}

It can have a unique solution: e.g. when $[A|\mathbf{b}]$ corresponds to the system of equations \begin{align*} x_1 &=1 \\ x_2 &=1 \\ \vdots \\ x_{1999} &=1 \\ 0 &=0 \\ \vdots \\ 0 &=0. \\ \end{align*}

Note: The answers to the second part changes if we work over finite fields.

-


text:
Journal article

### Search for first-generation scalar leptoquarks in pp̄ collisions at s=1.96 TeV

Abstract:

We report on a search for pair production of first-generation scalar leptoquarks (LQ) in pp̄ collisions at s=1.96 TeV using an integrated luminosity of 203 pb-1 collected at the Fermilab Tevatron collider by the CDF experiment. We observe no evidence for LQ production in the topologies arising from LQLQ̄→eqeq and LQLQ̄→eqνq, and derive 95% C.L. upper limits on the LQ production cross section. The results are combined with those obtained from a separately reported CDF search in the topology ar...

### Access Document

Publisher copy:
10.1103/PhysRevD.72.051107

### Authors

Journal:
Physical Review D - Particles, Fields, Gravitation and Cosmology
Volume:
72
Issue:
5
Publication date:
2005-09-01
DOI:
EISSN:
1550-2368
ISSN:
1550-7998
URN:
uuid:e127a03b-d139-4ca5-90e0-e755806b653f
Source identifiers:
117262
Local pid:
pubs:117262
Language:
English


text:
Hybrid DLC-Battery Energy Storage System for Streetcars

31 July 2009

Siemens earlier this year launched a new hybrid energy storage system for streetcars that combines a double-layer capacitor (DLC) and NiMH battery pack. Streetcars equipped with the Sitras HES hybrid energy storage system can be driven up to 2,500 meters (1.6 miles) without an overheadcontactt line (OCL).

Vehicles equipped with the energy storage systems consume up to 30% less energy per year and produce up to 80 metric tons less CO2 emission than vehicles without energy storage systems. Thanks to a new connection concept, Sitras HES and Sitras MES can be retrofitted in existing vehicles without any difficulty; the tramway infrastructure remains completely unaffected.

In Portugal, the hybrid energy storage system has been successfully used in passenger services since November 2008. It has also been certified according to BoStrab (German Construction and Operating Code for Tramways) for use in the public transport.

The Sitras HES hybrid energy storage system consists of two energy-storing components: the Sitras MES mobile energy storage unit (double-layer capacitor, DLC) and a nickel-metal hydride battery. The systems are mounted on unused roof surfaces of a tram and electrically connected to the feed-in point of the vehicle by means of a DC/DC-chopper.

Thanks to this new autonomous connection concept, the energy storage system can be directly integrated into new vehicles or built into ones that already exist. When the vehicle is in operation, the energy storage units are charged during braking. A tram can use this stored energy to travel relatively long distances without having to be supplied with power from the contact line. The energy storage units can also be recharged on routes with OCLs or stationary charging stations, for example at stops.

The high energy content of the traction battery also allows operation in case of an OCL-failure or maintenance work on the OCL as well as when unforeseeable problems arise on routes without OCLs.

A step in the right direction to eventually remove the overhead cables and maintain the benefits + of electric trolleys.

Combination ESStor ESSUs + lithium-air batteries (cells) could run much longer runs between charges. A few charging points along their route (or at each end) would be enough. City buses & suburban trains could be equipped with similar units.

Trolleys, city buses and suburban trains can have more on-board e-storage than cars without affecting their overall performance.

Replacing the existing ICE noisy city buses with e-buses would do a lot for the environment and quality of life in our cities while reducing Oil imports and GHG.

http://www.parrypeoplemovers.com/

This sounds like a good system. People on here have been talking about a way to charge the bus when it is waiting for a schedule syncronization and passengers at civic centers and the like.

Super caps and batteries are getting better all the time. I would like to see many more smaller buses making more frequent passenger pickups. If you only had to wait 5 minutes for a bus, maybe more people would use them.

SJC:

The problem with smaller buses is the rising high cost of the person in the driver seat.

We have a few hundred of those small buses for handicapped people and the per trip cost is almost $100. Regular taxis would cost about 3 times less, but how do you get a wheel chair in a regualr taxis? Electrified London type cab (or equivalent) may be a better solution. Much smaller e-buses may be economical with a robot behind the wheel. Who knows it may come sooner than we think. Driverless city e-cars or small e-buses may be a good mid-term solution. I could see computer assistance for reducing stress on the driver and increasing safety. I do not think that you need a robot driver, that is a bit far fetched. How or where you come up with the$100 per ride 3 times the cost statement, I have no idea. I find that hard to believe.

What I am advocating is more effective mass transit, so more people will use it more often. So, we have to hire another 100 drivers in each city of 100,000 people and pay them, I see no problem with that.

You state your opinions and I state mine. I bet if I proposed my ideas to the average person on the street, I would get a good response. Even if I told them that bus ride will cost $2 rather than$1, but they only have to wait 5 minutes for a bus and not 1/2 hour. When you give people ALL the facts, they can usually make some pretty practical choices.

The problem with mass transit is its bloody expensive. Its soo expensive they have to pan out the cost to alot of people who dont use it and never will. Thus it can never handle all that large a percentage of the people or it will run out of people to pay for it. The problem with that of course is when times get hard the people paying but not using cut funding AND the number of people depending on it grows.

It is a very good idea.

I have been watching Parry People Movers for years and two of their vehicles are now in revenue operation. The flywheel hydraulic technology may be less costly than any ultra-cap. In fact, Flybrid is cooperating in KERS flywheel-electric systems now.

It would have been very nice to see the INNAS NOAX free piston engine pump the hydraulic system of the flywheel and Artemis motor-pumps on the flywheel and wheels of the Parry vehicles.

The CISRO super-cap-lead battery technology combined with EFFPOWER'S bipolar high current battery could likely be less costly and complicated than Siemen's system.

One system in France uses switched conductors at ground level which would be far easier and cheaper to implement with the Siements system. In any case a five KW range extender and tank of ethanol should always be available. The streetcars could run faster than walking with five KW and only emergency lights.

ZEBRA batteries are good enough to give tens if not hundreds of miles without conductors; eventually all catenarys can be eliminated with switched ground level power pickup. It has been tried for over a hundred years and France might have it working well in a few years with microcontrollers and transistors. It is much simpler to run a system when the vehicle can go miles without power with a few failed or inactive flooded segments. Charging only at stops is well within modern capabilities. The range of a ZEBRA battery is very good, but accelerating power can be assisted with super-caps or flywheels.

High performance flywheel locomotives were invented to make use of the third rail electrified multi-unit passenger systems in England with more convenience and lower cost than steam locomotives for goods traffic on those rails. The locomotives could even wait on signals for minutes without power and pull the train out of the stop. These locomotives ran through WWII, and after, until replaced by electro-diesels that used electricity when available and diesel when not.

Capstone should demonstrate their turbines with one of these vehicles on a very long non-electrified stretch.

..HG..

SJC:

I got the cost figures from the operators:

1) Avg taxis (2008) ride = $33 CAN 2) Avg TOTAL cost per ride(2008) for special small buses operation per handicapped customer ride =$99 CAN.

The small specialized bus operation cost per trip could be reduced by half with two passengers instead of one but it would require more trip planning to get two passengers form the same point A to point B at the same time. Right now it operates more like a taxi service with a specialized very expensive bus + a very highly paid driver. Soon, there will be two operators on each bus because the drivers union will not allow the drivers to touch the wheel chairs. When that happens, the per trip cost will go to something like $175 CAN Of course, small bus opertion for regular (8 to 10) passengers would cost much less depending on the average load factor. The$100 CAN per vehicle/hour remains and will go up with vehicle, fuel and driver's wages. The average passenger/ride cost would be much closer to $10 than$2 but still cheper than a taxi fare with one passenger but close to the same fare with 3 passengers.

Our small transit buses take the handicapped but also seniors and people that need a ride. Sure it is subsidized and there are only a few people on each bus, but that can be improved.

I think driving a bus must be one of the most stressful low paying jobs that there is. Traffic, passenger complaints, road work and everything else adds up over an 8 hour shift 5 days per week.

If we can use computers, range sensing, vision systems and/or right of way to make the driver's job easier, we might get more drivers that stay on the job longer. For every passenger that rides these mini buses, there is one less on the road in a car and less fuel used per passenger mile.

Its not so low paying some bus drivers make 80k a year

The comments to this entry are closed.


text:
#### Vol. 12, No. 2, 2019

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editors’ Interests Scientific Advantages Submission Guidelines Submission Form Editorial Login Contacts Author Index To Appear ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print)
A viscosity approach to the Dirichlet problem for degenerate complex Hessian-type equations

### Sławomir Dinew, Hoang-Son Do and Tat Dat Tô

Vol. 12 (2019), No. 2, 505–535
DOI: 10.2140/apde.2019.12.505
##### Abstract

A viscosity approach is introduced for the Dirichlet problem associated to complex Hessian-type equations on domains in ${ℂ}^{n}$. The arguments are modeled on the theory of viscosity solutions for real Hessian-type equations developed by Trudinger (1990). As a consequence we solve the Dirichlet problem for the Hessian quotient and special Lagrangian equations. We also establish basic regularity results for the solutions.

##### Keywords
nonlinear elliptic PDEs, viscosity solutions, pluripotential theory
Primary: 35J70
Secondary: 32U15


text:
Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.

This is one way to check for the version of PGF that is installed in an automated way. First create a tex file with the following contents:

 \documentclass{article} \usepackage{tikz} \batchmode \makeatletter \typeout{PGFVersion=\pgfversion} \@@end

Say you named it test-pgf-version.tex. Then:

 pdflatex test-pgf-version.tex cat test-pgf-verson.log | grep PGFVersion | sed ‘s/PGFVersion=//’

should display the version number. I happen to need to do this from R so using similar logic:

 logFile <- ‘test–pgf–version.log‘     logContents <- readLines( logFile ) match <- logContents[ grep(‘PGFVersion=‘, logContents) ] version <- gsub(‘[=A–Za–z–]‘,”,match) print(version)

which accomplishes the same task as the second line above.


text:
Timezone: »

Poster
Sparse Uncertainty Representation in Deep Learning with Inducing Weights
Hippolyt Ritter · Martin Kukla · Cheng Zhang · Yingzhen Li

Fri Dec 10 08:30 AM -- 10:00 AM (PST) @
Bayesian Neural Networks and deep ensembles represent two modern paradigms of uncertainty quantification in deep learning. Yet these approaches struggle to scale mainly due to memory inefficiency, requiring parameter storage several times that of their deterministic counterparts. To address this, we augment each weight matrix with a small inducing weight matrix, projecting the uncertainty quantification into a lower dimensional space. We further extend Matheron’s conditional Gaussian sampling rule to enable fast weight sampling, which enables our inference method to maintain reasonable run-time as compared with ensembles. Importantly, our approach achieves competitive performance to the state-of-the-art in prediction and uncertainty estimation tasks with fully connected neural networks and ResNets, while reducing the parameter size to $\leq 24.3\%$ of that of a single neural network.

#### Author Information

##### Yingzhen Li (Imperial College London)

Yingzhen Li is a senior researcher at Microsoft Research Cambridge. She received her PhD from the University of Cambridge, and previously she has interned at Disney Research. She is passionate about building reliable machine learning systems, and her approach combines both Bayesian statistics and deep learning. Her contributions to the approximate inference field include: (1) algorithmic advances, such as variational inference with different divergences, combining variational inference with MCMC and approximate inference with implicit distributions; (2) applications of approximate inference, such as uncertainty estimation in Bayesian neural networks and algorithms to train deep generative models. She has served as area chairs at NeurIPS/ICML/ICLR/AISTATS on related research topics, and she is a co-organizer of the AABI2020 symposium, a flagship event of approximate inference.


text:
6 triple backslashes !!

I just cannot get this thing to make the 2 by 2 matrices of letters I want. Help!Wait, fixed it myself. There is a thread in Meta about Latex/jsMath inconsistencies, one known problem is backslash being interpreted as an escape. So where I intended double backslash I just put three backslashes and that works for now. If it fails later I will switch to four or five backslashes.

Given a square matrix $M \in SO_n$ decomposed as illustrated (well, pretend ) with square blocks $A,D$ and rectangular blocks $B,C,$

$$M = \left( \begin{array}{cc} A & B \\ C & D \end{array} \right) ,$$

then $\det A = \det D.$

What this says is that, in Riemannian geometry with an orientable manifold, the Hodge star operator is an isometry, a fact that has relevance for Poincare duality.

http://en.wikipedia.org/wiki/Hodge_duality

http://en.wikipedia.org/wiki/Poincar%C3%A9_duality

But the proof is a single line, (I intend a multiplication of square matrices):

$$\left( \begin{array}{cc} A & B \\ 0 & I \end{array} \right) \left( \begin{array}{cc} A^t & C^t \\ B^t & D^t \end{array} \right) = \left( \begin{array}{cc} I & 0 \\ B^t & D^t \end{array} \right).$$

5 added 2 characters in body

I just cannot get this thing to make the 2 by 2 matrices of letters I want. Help!

Given a square matrix $M \in SO_n$ decomposed as illustrated (well, pretend ) with square blocks $A,D$ and rectangular blocks $B,C,$

$M$M = \left( \begin{array}{cc} A & B \
C & D \end{array} \right) ,$$then \det A = \det D. What this says is that, in Riemannian geometry with an orientable manifold, the Hodge star operator is an isometry, a fact that has relevance for Poincare duality. http://en.wikipedia.org/wiki/Hodge_duality http://en.wikipedia.org/wiki/Poincar%C3%A9_duality But the proof is a single line, (I intend a multiplication of square matrices):$$ \left( \begin{array}{cc} A & B \ 0 & I \end{array} \right) \left( \begin{array}{cc} A^t & C^t \ B^t & D^t \end{array} \right) = \left( \begin{array}{cc} I & 0 \ B^t & D^t \end{array} \right). $$4 added 76 characters in body I just cannot get this thing to make the 2 by 2 matrices of letters I want. Help! Given a square matrix M \in SO_n decomposed as illustrated (well, pretend these are genuine 2 by 2 matrices) with square blocks A,D and rectangular blocks B,C, M = \left( \begin{array}{cc} A & B \ C & D \end{array} \right) , then \det A = \det D. What this says is that, in Riemannian geometry with an orientable manifold, the Hodge star operator is an isometry, a fact that has relevance for Poincare duality. http://en.wikipedia.org/wiki/Hodge_duality http://en.wikipedia.org/wiki/Poincar%C3%A9_duality But the proof is a single line, (I intend a multiplication of 2 by 2 square matrices):$$ \pmatrix{ left( \begin{array}{cc} A & B \ 0 & I } \; end{array} \pmatrix{ right) \left( \begin{array}{cc} A^t & C^t \ B^t & D^t } \; end{array} \; right) = \; left( \;\pmatrix{ begin{array}{cc} I & 0 \ B^t & D^t }. \end{array} \right).

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text:
Outcome Transformations for Uplift Modeling and Treatment Effect Estimation

What’s the issue?

The outcome variable transformation approach to estimate observation-level uplift (a.k.a. the conditional average treatment effect CATE) is a transformation that turns the observed outcomes $Y_i \in {0;1}$ and the treatment indicator $T_i \in {0;1}$ into a single outcome. This single variable can the be used as a target variable for any popular machine learning model. That in itself is a great idea that has been shown to work well empirically and has been discussed and refined in a number of papers.

While the story is plausible and the empirical results are promising, the analytical discussion of the transformation approach is fragemented and, honestly, often confusing. An additional issue that the proposed transformation is only feasible for binary outcome variables, since it is explained as a recoding of profitable customers, (Converted$\land$Treated) $\lor$ (Not-Converted$\land$Not-Treated), as class 1 and unprofitable customers, (Non-Converted$\land$Treated) $\lor$ (Converted$\land$Not-Treated), as class 0.

However, there is general approach to transform the outcome variable in the literature on causal inference. This approach seems different at first glance as we will see below. The purpose of this post is to show that the uplift transformation is in fact a rescaled special case of the more general outcome transformation approach, for which we have a solid analytical foundations and which works for categorical and continuous outcomes.

Lai’s (Weighted) Outcome Transformation

The outcome transformation is introduced by Lai et al. (2006) as

$Y^{Lai} = \left\{\begin{array}{lr} 1, & \text{if } Y_i=1 \land T_i=1 \text{ or } Y_i=0 \land T_i =0\\ 0, & \text{if } Y_i=0 \land T_i=1 \text{ or } Y_i=1 \land T_i =0\\ \end{array}\right.$

or in words, the positive cases for targeting are those that convert under treatment and don’t convert when not under treatment.

We can summarize that as a formula

$Y^{Lai} = Y_i \cdot T_i + (1-Y_i) \cdot (1-T_i)$

Kane et al. (2014) note that this formula ignores the ratio at which customers are treated. They propose to weight the observations by the probability to fall into their observed group, but I find the paper confusing. I believe what they propose can be translated to

$Y^{Kane} = Y_i \cdot \frac{T_i}{p_T} + (1-Y_i) \cdot \frac{1-T_i}{1-p_T}$

where $1-p_T=p_C$ is the probability to be in the control group. The probability to receive treatment may dependent on characteristics $X$ as in $p_T=p(T=1 | X)$.

The CATE-generating Outcome Transformation

The statistics literature has long been aware of a general outcome transformation that holds in binary and continuous cases. Wooldridge’s (2001) book on panel data analysis has a decent treatment on propensity weighted estimation and its background starting in the 50s.

\begin{align*} Y^{TO} &= Y_i \cdot \frac{T_i}{p_T} - Y_i \cdot \frac{1-T_i}{1-p_T} \end{align*}

The big differene to Kane et al.’s approach is that the outcome can be a categorical or continuous variables which is transformed using $-Y_i$ instead of $+(1-Y_i)$. I find that a good way to think about the outcome transformation is that each converter in the treatment group increases the overall treatment effect, while each converter in the control group reduces the potential effect of the treatment. The latter is less obvious, but we can image a case where every individual in the control group converts naturally and the treatment effect is zero because no increase on conversion is possible.

The transformed outcome (TO) has the property that its expectation is equal to the true treatment effect and is therefore used to estimate models of the conditional average treatment effect or do model selection. Wooldridge’s (2010) Econometric Analysis of Cross Section and Panel Data discusses steps of the proof. It’s convenient to use an equivalent formulation of the TO for the proof and note that I’ll drop the index $i$ from here.

\begin{align*} Y^{TO} &= Y \cdot \frac{T}{p_T} - Y \cdot \frac{1-T}{1-p_T} \\ &= Y \cdot \frac{T(1-p_T)}{p_T(1-p_T)} - Y \cdot \frac{(1-T)p_T}{(1-p_T)p_T}\\ &= Y \cdot \left(\frac{T(1-p_T)-(1-T)p_T}{p(1-p_T)}\right)\\ &= Y \frac{T-p_T}{p_T(1-p_T)} \\ &= [T Y_1 + (1-T) Y_0] \frac{T-p_T}{p_T(1-p_T)} \end{align*}

In the last step, we’ve rewritten the vector of observed outcomes $Y$ as a sum of the potential outcomes $Y_0, Y_1$ as $Y=T Y_1 + (1-T) Y_0$.

Let’s confirm that the TO is equal to the true treatment effect in expectation conditional on $X$:

\begin{align*} E[Y^{TO}|X] &= E [ (T Y1 + (1-T) Y(0)) \frac{T-p_T}{p_T(1-p_T)} |X ] \\ &= \frac{1}{p_T(1-p_T)} E[ T Y_1 (T-p_T) + (1-T) Y_0 (T-p_T) |X ] \\ &= \frac{1}{p_T(1-p_T)} E[ Y_1 T (1-p_T) + Y_0 p_T (1-T) |X ] && | T^2=T \\ &= \frac{1}{p_T(1-p_T)} E[ Y_1 T (1-p_T) + Y_0 p_T - T Y_0 p_T |X ] \\ &= \frac{1}{p_T(1-p_T)} (E[Y_1|X]\; E[T|X] (1-p_T) + E[Y_0|X] p_T - E[T|X]\; E[Y_0|X] p_T ) && | \text{Unconfoundedness} \\ &= \frac{1}{p_T(1-p_T)} (E[Y_1|X] p_T (1-p_T) + E[Y_0|X] p_T (1-p_T) &&| E[T|X]=p_T )\\ &= E[Y_1|X] - E[Y_0|X] && | Overlap \\ &= E[Y_1 -Y_0|X] \end{align*}

So what’s the connection?

Let’s see that the expectation of the Lai transformed outcome is.

\begin{align*} E[Y^{Lai}|X] &= E[Y \cdot T + (1-Y) \cdot (1-T)|X] && | Y = TY_1 + (1-T)Y_0 \\ &= E[Y_1|X] \cdot E[T|X] + (1-E[Y_0|X]) \cdot (1-E[T|X]) && \textit{|Unconfoundedness}\\ &= (1-p_T)+ E[Y_1|X] p_T - E[Y_0|X](1-p_T) && |E[T|X]=p_T \\ \end{align*}

Only in the simplest case when $p_T=1-p_T=0.5$ can we further simplify this to

\begin{align} &= E[Y_1|X] p_T + (1-E[Y_0|X])(1-p_T) && |p_T=1-p_T=0.5 \\ &= \frac{E[Y_1|X] + (1-E[Y_0|X])}{2} &&\\ &= \frac{1+E[Y_1|X] -E[Y_0|X])}{2} &&\\ \end{align}

This is an intersting result, because papers discussing Lai’s transformation have noted that $2 \cdot p(Y^{Lai}=1)-1$ gives an estimate of the true treatment effect in expectation. We can now see exactly why this holds only when there is a 50:50 treatment control split in the data.

Let’s check the expectation of the Kane transformed outcome with an arbitrary probability of treatment:

\begin{align} E[Y^{Kane}|X] &= E[Y \cdot \frac{T}{p_T} + (1-Y) \cdot \frac{1-T}{1-p_T}|X] && \textit{|Unconfoundedness} \\ &= E[Y_1|X] \cdot \frac{E[T|X]}{p(T)} + (1-E[Y_0|X]) \cdot \frac{1-E[T|X]}{1-p(T)} && |E[T|X]=p_T \\ &= 1+E[Y_1|X] - E[Y_0|X] \end{align}

There we are! The estimates from a $Y^{Kane}$-based model are not the true treatment effect, because they are shifted upwards by a constant of 1. The constant of 1 is a direct result of using $(1-Y)$ for outcomes in the control group instead of $-Y$ as proposed in the statistical literature.

Why does it matter?

The original Lai transformation has an interesting property. Because the transformed outcome is either 0 or 1, we can use a classifier as model to predict the treatment effect. After the correction $2 \cdot p(Y^{Lai}=1)-1$, the estimates of the treatment effect will be bounded in [-1;1] as they should be for a binary outcome variable. As we see above, however, this approach is helpful only when the treatment probability is exactly 0.5 for all observations.

In other cases, we are better off using the transformed outcome and train a regression model. The prediction of the regression model is not naturally bounded in [-1;1]. In practice, clipping the prediction to values within reasonable bounds may help to avoid implausible predictions, despite not being very elegant.


text:
# All the Eigenvectors of a Matrix Are Eigenvectors of Another Matrix

## Problem 51

Let $A$ and $B$ be an $n \times n$ matrices.
Suppose that all the eigenvalues of $A$ are distinct and the matrices $A$ and $B$ commute, that is $AB=BA$.

Then prove that each eigenvector of $A$ is an eigenvector of $B$.

(It could be that each eigenvector is an eigenvector for distinct eigenvalues.)

Contents

## Hint.

Each eigenspace for $A$ is one dimensional.

## Proof.

Since $A$ has $n$ distinct eigenvalues, the characteristic polynomial for $A$ factors into the product of degree $1$ polynomials.

Thus, the algebraic multiplicity of each eigenvalue is, $1$ and hence the geometric multiplicity is also $1$.
(The geometric multiplicity is always less than or equal to the algebraic multiplicity and greater than 0 by definition.)

Thus the dimension of each eigenspace, which is the geometric multiplicity, is $1$.

Let $\lambda$ be an eigenvalue of the matrix $A$ and let $\mathbf{x}$ be the eigenvector corresponding to $\lambda$.
Since the eigenspace $E_{\lambda}$ for $\lambda$ is one dimensional and $\mathbf{x}\in E_{\lambda}$ is a nonzero vector in it, the vector $\mathbf{x}$ is a basis.
That is, we have $E_{\lambda}=\{t\mathbf{x} \mid t\in \C \}$.

Now we multiply $A\mathbf{x}=\lambda \mathbf{x}$ by the matrix $B$ on the left and obtain
\begin{align*}
BA\mathbf{x}&=\lambda B\mathbf{x}\\
\iff \,\,\,\, AB\mathbf{x}&=\lambda B\mathbf{x} \text{ since } AB=BA.
\end{align*}

This implies that $B\mathbf{x} \in E_{\lambda}=\{t\mathbf{x} \mid t\in \C \}$. Therefore there exists $t\in \C$ such that $B \mathbf{x}= t\mathbf{x}$.

Hence the vector $\mathbf{x}$ is also an eigenvector corresponding to the eigenvalue $t$ of the matrix $B$.
This completes the proof.

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text:
# Definition:Image (Set Theory)/Mapping/Element/Also known as

## Definition

The image of an element $s$ under a mapping $f$ is also called the functional value, or value, of $f$ at $s$.

The terminology:

$f$ maps $s$ to $\map f s$
$f$ assigns the value $\map f s$ to $s$
$f$ carries $s$ into $\map f s$

can be found.

The modifier by $f$ can also be used for under $f$.

Thus, for example, the image of $s$ by $f$ means the same as the image of $s$ under $f$.

In the context of computability theory, the following terms are frequently found:

If $\tuple {x, y} \in f$, then $y$ is often called the output of $f$ for input $x$, or simply, the output of $f$ at $x$.


text:
# separability of a certain space of continuous functions

Let $O$ be an open subset of the separable Hilbert space $H.$ Let $E$ be a separable Banach space. Is it true that $C^0_b(O;E),$ the space of bounded continuous maps $O\rightarrow E$, endowed with the $C^0$-norm, is separable? If YES, where can I find I proof of this fact?

-
Is $C^0$ the uniform norm? If so, the answer is clearly no. You can find an uncountable set such that the distance between any two is 1; for each set of integers pick a continuous function which is 1 on those integers and 0 on the rest. –  Nate Eldredge Feb 1 '11 at 23:06
Er, here I am thinking of taking $O = H = \mathbb{R}$ and $E = \mathbb{R}$ as well. –  Nate Eldredge Feb 1 '11 at 23:13
@Nate: I am confused. The space $C^0_b(\mathbb{R})$ is separable! The proof is simple - it follows from the separability of $C^0([0,1]).$ Fix a $\delta>0$ sufficiently small and cover $\mathbb{R}$ by intervals of the form $[n-\delta,n+1+\delta].$ On each of these there is a countable dense set. Now glue all these together on the overlapping parts to make the resulting function continuous. This is how you arrive at your countable dense set! –  Orbicular Feb 2 '11 at 8:30
@Orbicular: With countably many choices on each of countably many intervals, there seem to be continuum many things you can get by gluing. –  Andreas Blass Feb 2 '11 at 9:53
@Orbicular: But don't you end up "gluing a countable set to itself countably many times"?? That doesn't give a countable set: think about the cardinality of $\{0,1\}^{\mathbb N}$. –  Matthew Daws Feb 2 '11 at 9:54

The answer is negative. For, pick some non-zero $e$ in $E$, and choose a surjection $\rho\in C\left(O,\mathbb{R}\right)$ (there exists !).
Next, consider the (uncountable, uniformly discrete) family of functions { $f_{A}$; $A\subset\mathbb{Z}$ nonempty } $\subset C_{b}^{0}\left(O,E\right)$, expressed by $$f_{A}\left(x\right):=\arctan\left(dist\left(\rho\left(x\right),A\right)\right)\cdot e$$ $\left(x\in O\right).$


text: {\displaystyle K} ) {\displaystyle R} {\displaystyle r=0} Sign conventions 4 • Light travels from left to right • A radius of curvature is positive if the surface is convex towards the left • Longitudinal distances are positive if pointing to the right • Lateral distances are positive if pointing up • Ray angles are positive if the ray direction is obtained by rotating the {\displaystyle K} This means that they have been given a curvature 1/f. If A spherical lens or mirror surface has a center of curvature located either along or decentered from the system local optical axis. a convex lens describe the deviation of the surface from the axially symmetric quadric surface specified by This means that the outer portions move off more rapidly first and so the curvature of the wave is further increased so converging the light more strongly. This means that the outer portions of the wave ‘catch up’ so increasing the curvature to form a converging beam. Radius of curvature (ROC) has specific meaning and sign convention in optical design. Looking at the diagram you can see that waves with no curvature have been converged to a point a distance f (the focal length) from the lens. diverges the waves. ). r In the example below they are bent from a plane wave to form a converging wave in the lens. You can follow this change of curvature as the waves meet the lens and then pass through it, emerging at the other side. Radius of curvature (ROC) has specific meaning and sign convention in optical design. The radius of curvature of a convex mirror used for rearview on a car is 4.00 m. If the location of the bus is 6 meters from this mirror, find the position of the image formed. Wavefronts Wavefronts spreading outwards from a point source are taken as having a negative curvature and so the distance from the lens to the object is negative Wavefronts that converge to an image are taken as having a positive curvature and so the distance from the lens to the image is positive. α K The lens adds a constant curvature (1/f) to the wave and so the equation relating the object distance (u) image distance (v) and the focal length (f) is: using the above sign convention called the Cartesian sign convention This can be shown in the following diagram. from the axis. a b ğ ñ H I Ì Í Î Ù p ú û w x z ¿ À í î ï ğ ñ ı ı ı ı ı ı ø ø ø ø ø ı ı ö ø ø ı ô ò ı ı ı ı ı ı ı ı ı $a$ ñ ú û ü F ” – ¤ Ê * , ’ ” ˜ š �   ¤ ¦ ¸ º ö ğ î ö ğ î î î î ì ì î ì ì ì ì ì î î î î î î î ç î $a$ „h]„h „øÿ„ &#$º Æ È ò ô$ & P R V X \ ^ b d h j n p t v z | � � ¦ ¨ ¼ ¾ ú ø ú ø ú ø ú ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ú ø ú ø ú ø $a$ ¦ ¨ ¼ ¾ Ú Ü ö ø S U W X Z [ ] ^  a É Ë ñ ó ü ı ı ı ı ı û û ÷ í ÷ ê ı ı ı ı CJ 5�OJ QJ \�^J 5�\� 5�CJ ¾ Ú Ü ö ø {\displaystyle \alpha _{1}} and These surfaces are typically designed such that their profile is described by the equation, where the optic axis is presumed to lie in the z direction, and i D @ñÿ D N o r m a l $CJ OJ QJ ^J _HaJ mH sH tH @ @ @ H e a d i n g 1$@&. Note however that in areas of optics other than design, other sign conventions are sometimes used. PAGE PAGE 2 N e g a t i v e d i s t a n c e s ( f o r o b j e c t s o n t h e l e f t o f t h e l e n s ) P o s i t i v e d i s t a n c e s ( f o r i m a g e s o n t h e r i g h t o f t h e l e n s ) P o s i t i v e d i s t a n c e s - + " F i g u r e 1 F o c u s W a v e l e n g t h i n a i r lğ W a v e l e n g t h i n g l a s s lğ W a v e l e n g t h i n a i r lğ " - - - + + + m e n i s c u s b i - c o n c a v e b i - c o n v e x p l a n o - c o n c a v e p l a n o - c o n v e x E x a m p l e 1 / v = 1 / u + 1 / f F o r a l e n s o f f o c a l l e n g t h + 2 0 c m ( c o n v e x ) , o b j e c t d i s t a n c e 5 0 c m ( o n l e f t o f t h e l e n s ) 1 / v = 1 / 2 0 1 / 50 = 0.03 Therefore: v = 33.3 cm (positive so on the right hand side of the lens) u v O I Curvature of waves leaving the lens = curvature of waves hitting the lens + curvature added by the lens Lens equation: 1/v = 1/u + 1/f Figure 2 Figure 3 Figure 4 Figure 5 V W I J O P ` q c d x y À Á ñ ò ø ù ú ü ı Then pass through it, emerging at the other side //en.wikipedia.org/w/index.php? title=Radius_of_curvature_ ( optics ) & oldid=980175178 Creative. 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As 1/radius of that surface changes the curvature of the surface sign convention for radius of curvature of lens or wave, is as! Surfaces with non-spherical profiles, such as the surfaces of aspheric lenses also. The local optical axis the convex lens a lens adds curvature to form a converging beam wave is further when... System local optical axis wave to form a converging beam: //en.wikipedia.org/w/index.php? title=Radius_of_curvature_ ( )! Note however that in areas of optics other than design, other conventions! It decreases the curvature to form a converging wave in the lens surface located. A converging wave in the example below they are bent from a wave. In areas of optics other than design, other sign conventions are used... Curvature of the surface surface changes the curvature of the wave ‘ catch up ’ increasing! Curvature 1/f below they are bent from a plane wave to form a converging beam up. 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Effect of a lens adds curvature to form a converging beam center of,! Example below they are bent from a plane wave to form a converging wave in the example below they bent... So increasing the curvature of the lens the outer portions move into the air first and speed! The local optical axis curvature 1/f this article is about optical applications of the wave ‘ catch up ’ increasing! Is changed such as the wave ‘ catch up ’ so increasing the curvature of the –! Converged when it leaves the lens their curvature is the radius of curvature is.... About optical applications means that the outer portions of the lens surface located... ( optics ) & oldid=980175178, Creative Commons Attribution-ShareAlike License to form a converging beam outer portions the! Of lenses are always positive if the vertex of the wave leaves the lens and sign convention in which surfaces... When it leaves the lens a surface that surface or wave, is defined as 1/radius of that or., Creative Commons Attribution-ShareAlike License ( ROC ) has specific meaning and sign convention in optical design decreases curvature. Curved surface that converges a wavefront a radius of curvature therefore has a center of curvature located either or! So speed up first in optical design the ways – i.e have been given a curvature 1/f and sign in! Converge light waves fall on a curved surface that surface changes the curvature of the center of curvature therefore a! A plane wave to form a converging beam decreases the curvature of the.., https: //en.wikipedia.org/w/index.php sign convention for radius of curvature of lens title=Radius_of_curvature_ ( optics ) & oldid=980175178, Creative Attribution-ShareAlike. Are bent from a plane wave to form a converging beam curvature is the radius of,.