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Research Article

# Latrine Utilization and Factors Associated among Rural Community in Chire Woreda at Sidama National Regional state, Ethiopia: A Community-based Cross-sectional Study

### Demelash Wachamo Wako* and Yadessa Tegene Woldie

Department of Public Health, Paradise Valley College, West Arsi Zone, Oromia, Ethiopia

*Address for Correspondence: Demelash Wachamo Wako, Department of Public Health, Paradise Valley College, West Arsi Zone, Oromia, Ethiopia, E-mail: demmenew1@gmail.com

Submitted: 27 August 2020; Approved: 09 September 2020; Published: 10 September 2020

Citation this article: Wako DW, Woldie YT. Latrine Utilization and Factors Associated among Rural Community in Chire Woreda at Sidama National Regional state, Ethiopia: A Community-based Cross-sectional Study. American J Epidemiol Public Health. 2020;4(3): 090-00. https://dx.doi.org/10.37871/ajeph.id37

Copyright: © 2020 Wako DW, et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

Keywords: Latrine utilization; Rural communities; Chire district; Sidama regional state

Background: In low income countries like Ethiopia; lack of sanitation is a serious health risk, affecting billions of people, particularly the poor people. The better health and quality of life, sanitation, and hygiene conditions are among the major causes of public health problems. Therefore, this study aimed to assess utilization and associated factor of latrine in rural communities.

Methods: A community-based cross-sectional study was conducted among 793 randomly selected households. The data were collected through an interview by pre-tested questionnaires and observation checklist. Data entry and analysis for descriptive and logistic regression models by SPSS v.20.

Results: The utilization of latrine was 71.5%. It associated with being younger age [AOR 1.83, 95% CI:1.11, 3.02], who had attended secondary & above [AOR 4.17, 95% CI:1.10, 15.94], monthly income more than 1000 Ethiopian Birr (ETB), [AOR 5.17, 95% CI: 3.10, 8.72], family size [AOR 3.19, 95% CI: 1.97, 5.17], latrine inside the compound [AOR 2.47, 95% CI: 1.64, 3.71] as compared with their counterparts were more used latrine.

Conclusions: The utilization of latrine was low and still needs, to improve awareness of the community there is a need of health education programs regarding improved use of latrine, cleanliness of latrine, constructing latrine inside the compound and maintenance of latrine for the proper utilization including children should be considered.

## Introduction

Worldwide lack of sanitation is a serious health risk, affecting billions of people around the world, particularly the poor and disadvantaged people around the world [1]. Lack of sanitation facilities compels people to practice open defecation and this increases the risk of transmission of diseases [2]. The disease burden associated with poor water, sanitation, and hygiene is estimated to account for 4.0% of all deaths and 5.7% of the total disease burden in Disability-Adjusted Life Year (DALYs) in worldwide, principally through diarrheal diseases. About 1.8 million people die every year due to diarrheal diseases, and children under the age of 5 years account for 90% of diarrheal deaths [3,4]. Moreover, 88% of diarrheal diseases are attributed to unsafe water supply, inadequate sanitation, and poor hygiene [5].

In most developing countries, especially in Sub- Saharan Africa (SSA), the basic causes of more than 80% of the diseases are inadequate and unsafe water supply and improper disposal of waste. Ethiopia is among the poorest countries in the world, ranking 170 out of 177 in the UN human development index and is the second-most populous country in Africa (population estimated above 80 million). Yet, Ethiopia’s rural populations are among the least served with a rural water supply and sanitation access at only 24% and 8% respectively [6].

However, the Provision of sanitation facilities initiated in all parts of Ethiopia with interventions of a health extension program and continued investments to increase access to safe water and improved sanitation [7]. The studies conducted in different parts of Ethiopia showed that the latrine utilization level differs from region to region of the country and from district to district within the same region depending on many factors. In the study area, there is no available research conducted on a similar topic. Therefore, this study was designed to assess the latrine utilization level and associated factors of rural community separately in the study area. This study can provide evidence or information regarding the latrine utilization in open defecation and its determinant factors which can be used for public health officials, clinicians, and health planners to reduce the impacts of poor utilization of latrine.

## Materials and Methods

### Study setting and design

This study was conducted in Chire District, at Sidama National Regional state, Ethiopia, Southern Ethiopia, which located about 396 km away to the South of Addis Ababa, the capital city of Ethiopia. By the end of 2018, latrine coverage was 99.8% but utilization not known. It has a climatic condition of Dega and Weyna Dega. The total number of households is estimated to be 41355 within the average household size of 4.9 and of these households, about 97% of them have latrine facilities. The source population was all rural community households. While all selected households with latrine facilities in the rural community of districts were considered as the study population. Among this, all randomly selected adults of the household and who are residents in selected kebele were included in the study. All adults who were critically ill that were not able to respond appropriately for the interviews were excluded from the study.

### Sampling and sampling technique

The sample size (n) was calculated using the following single population proportion formula based on the assumption of 61% proportion (p) from Dembia district [8], 95% C.I (1.96), 5% margin of error (d), 2 design effect and adding 15% contingency. n =

${\left({Z}_{1-\alpha }{2}}\right)}^{2}\frac{P\left[1-P\right]}{{d}^{2}}$

n = (1.96)2(0.61) (0.39) = 365

Therefore, the required sample size was n = (365 x 2) = 10% (730) +73.0 = 803 HHs with latrines included in the study.

A multistage random sampling technique. In the 1st stage, ten out of 27 rural kebeles were selected by Simple Random Sampling (SRS) technique. In the 2nd stage, the sample size was proportionally allocated. In the last stage, only one randomly selected eligible person was interviewed (Figure 1).

### Data collection tools, and procedures

The data were collected through an interview by pre-tested questionnaires and observation checklist. The questionnaires were translated into the “Sidamic language” and validated before the study time was done outside of the study area and necessary modifications were done based on the findings. Data on utilization of latrine, a survey was carried out with direct observation check list the status of the households in selected kebele. Trained data collectors, health workers have collected the data. Principal investigators and supervisors follow the data collection process and check them for consistency and completeness.

### Data analysis

Data entry, cleaning, and analysis were done by SPSS V. 20. Descriptive analysis including frequency distribution and the percentage was made to determine the latrine utilization, to describe socio-economic and demographic, environmental, behavioral, knowledge and attitude related variables. All factors with p-value < 0.25 in the bivariate logistic regression analysis were a candidate to the multivariable model to control confounding effects. The Hosmer-Lemeshow goodness-of-fit statistic was used to assess whether the necessary assumptions for the application of multiple logistic regression are fulfilled. Odds Ratios (OR) with 95% Confidence Intervals (CI) were calculated. Finally, p-value < 0.05 declared a significant association.

### Operational definitions

Satisfactory latrine utilization: is a latrine that provided services at the time of data collection even if the latrine required maintenance. A latrine is utilized when households had functional latrines, no observable faces in the compound, observable feces through the squat hole, and the foot-path to the latrine is uncovered with grasses.

Status of latrine: which needs maintenance or not at the time of data collection on it’s a door (any cover), a leaking roof and sagging walls.

The critical time for handwashing practice – handwashing practices mainly after visiting latrines or cleaning bottoms of children, before preparing food and before feeding children

A Child-friendly feature of latrine facility: means availability of at least one of the following features; small squatting hole, lower seat, and presence of potty.

## Results

### Socio-demographic characteristics

A total of 793 participants were interviewed yielding a response rate of 98.75%. The majority of the studied participants were 482 (60.8%) female by sex and 770(97.1%) were married. About 322 (40.6%) participants were in the age range of 26-35 years, while 281 (35.4%) mothers and 288 (36.3%) fathers had completed primary level education, respectively. More than half, 530 (66.8%) of the respondents were housewives. Nearly 693 (87.4%) of the households had under five years old children in the household (Table 2).

### Knowledge and attitude towards of latrine utilization

The overall knowledge of latrine utilization was 590 (74.4%) had good knowledge. The majority of 715 (90.2%) knew the benefit of using a latrine. The study participants claim to use latrine because of 77.5% privacy, and 69.5% prevent excreta related diseases. Among study participants, only, 335 (42.2%) had a positive attitude towards latrine utilization. The study participants who agreed that the main obstacles to using latrine were beliefs 55.6% were lack of construction tools, 67.2% believe that it is the responsibility of husband and 33.3% government.

### Environmental and behavioral characteristics

The majority of the study participants prefer to locate their latrine 520 (65.6%) were inside the compound. More than half 513 (64.7%) were constructed between 1-3 years ago. While 202 (25.5%) households had feces and urine around the latrine and 89 (11.2%) had still faeces and urine around the home. On another hand, 91.0% had no handwashing facility near the latrine, 91.6% had no water for handwashing. Regarding the behavioral practice, Out of 793 respondents who had latrine facility, 90.0% had simple pit latrine, improved ventilated 2.4% and 7.6% were other types of the latrine (Figure 2). While, out of 229 (28.9%), 32.0% pit, 38.7% floor, and 29.3% whole part of the latrine needs maintenance, respectively. The reasons given by respondents for why under-five children did not use the latrines were: being just a child 258 (37.2%), large squatting hole 226 (32.6%), and the floor was not safe to stand on 209 (30.2%).

### Utilization of latrine

Out of 793 respondents, 567 (71.5%) households had satisfactory utilized latrines with [95% CI: 68.3% - 74.4%] (Figure 3). Out of 212, the study participant’s claim as the main reason for not using a latrine, 59.7% reports latrine is not functional, 22.7% due to far from the house and 9.2% claims using latrine is inconvenient during the rainy season and at night without proper roof and door.

### Associated factors of the utilization of latrine

In bivariable logistic regression analysis the variables which had statistically significant association and the highest latrine utilization was noticed among; younger age had, who attend primary and above education fathers, attend primary and above for the mother, the estimated average monthly income of the household above 24 USD$per a month, and the households who had latrine inside had p-value < 0.001 and higher proportion of the utilization of the latrine. In addition to this, occupation of mother, had under 5 years children in the household and high number of household size can affect latrine use also had p-value < 0.05 and higher proportion of the utilization of the latrine. In the multivariate analysis age of the respondent, the educational status of the mother, average monthly income of the household, household size can affect latrine use and location of the latrine was remained associated with latrine utilization. The younger age had [AOR 1.83, 95% CI: 1.11, 3.02], a significant association with latrine utilization as compared with the 35 years and older age. A mother who had primary education [AOR 3.68, 95% CI: 1.47, 9.17] and secondary and above [AOR 4.17, 95% CI: 1.10, 15.94] as compared with no formal education. The household monthly income more than 66.4$, [AOR 5.17, 95% CI: 3.10, 8.72], Household family size [AOR 3.19, 95% CI: 1.97, 5.17], the households who had latrine inside the compound [AOR 2.47, 95% CI: 1.64, 3.71] as compared with their counterparts (Table 2).

## Discussions

This community-based cross-sectional study revealed that the utilization of latrine among selected households was 71.5% with [95% CI: 68.3% - 74.4%]. Out of 212, the study participants claim as the main reason for not using the latrine, 59.7% reports latrine is not functional, 22.7% due to far from the house and 9.2% claims using latrine is inconvenient during the rainy season and at night without proper roof and door. The reasons given by respondents for why under-five children did not use the latrines were: being just a child 37.2%, large squatting hole 32.6%, and the floor was not safe to stand on 30.2%. This was consistent with study findings done in Nigeria level of latrine utilization `was 69% [9], 73% rural Becho district of central Ethiopia [10]. This similarity may due to the study participants in the Africa had almost had at similar socio-economic status and practice of the latrine utilization at the rural community.

This finding was lower than 88% Democratic Republic of Congo [6], 89.9% rural village of Eastern Nepal [11], 93% Mirab Abaya and Alaba [12], 86.7% Hulet Ejju Enessie district [13]. Higher than study in Bahr Dar Zuria (62%) [14] and 61.2% in rural areas of Denbia district, Northwest Ethiopia [8]. This difference in the utilization of latrine was among selected households from Chire district might be due to socio-economic differences, different interventions among locally officials and interventions of the different projects among different regions in the present day and time of the study [15]. This implies that the health officials, and focal needs work forward to maximize the utilization of the latrine by setting different strategies.

This study shows that the younger age of the respondent had a significant association with the utilization of latrine as compared with older age. This finding is similar to the study conducted in India [16], rural Bangladesh [17], North West Ethiopia [18] and rural communities in the District of Bahir Dar Zuria, Ethiopia [19]. This may explain that uneducated and elder people in a rural area may find it difficult to get some when latrine construction not easy to use by elders due to different reasons like privacy and cultural issues when latrine lacks door and appropriate cover. In addition to this, they are also economically dependent and they cannot afford the construction of the latrine.

This study reveals that the high education level of a mother had a significant association with latrine utilization. Similarly, a study conducted in India [17,18,20], Tigray Hawzien district, Northern Ethiopia [21], and Gedeo Zone, South Ethiopia [22]. This could be attributed to the impact of education on behavior change and the adoption of good latrine hygiene practices at the household level high among educated mothers. Furthermore, the monthly income of the household had also a significant association with latrine utilization. This result agrees with the Gulomekada District, Tigray Region, North Ethiopia [23], and Bangladesh [24]. This may due to Household’s monthly income determines the availability and quality of latrine which are the important predictors of utilization of latrine. Also, this might be due to low-income households had a shortage of money to constrict latrine rather than other important materials and utilities for daily consumption.

Household family size was found to be a significant association with latrine utilization. Similarly, North West Ethiopia [20] and rural coastal Odisha, India [17]. This might be due to the presence of family and cultural practices promote small children to defecate around the house rather than go to open whole latrine use. On the other hand, this may also be related to the economic status of the household. The location of the latrine being inside the compound had also a statistically significant association with latrine utilization. This result is in line with Chencha District, Southern Ethiopia [5], and Bangladesh [26]. This may due to fear to utilize the latrine during at night and raining because it is far from the resident’s house and no roof to prevent raining and it is exposed to an animal attack.

This study result shows, there was a low utilization of latrine. There is a significant number of the household still not using the latrine. The main reason was reported; the latrine is not functional, far from the house, when in the rainy season and at night without proper roof and door. This implies that a lot has to be done on awareness creation about the proper utilization of latrine and support on the maintenance of the poor family latrine that needs attention to improve rural community health. Provide different types of strategies to a rural community with an affordable cost to facilitate children’s toilet training.

There might be a potential for recall and social desirability bias in the utilization of latrine and socioeconomic. In addition to this, the odds ratios of the cross-sectional study did not show the strength of an association.

## Conclusions

This study result shows that the utilization of latrine needs improvements and attention to enhance the proper and adequate utilization of the latrine. It associated with being younger age, maternal education, monthly income, family size, latrine inside the compound as compared with their counterparts were found to be associated factors of the utilization of latrine. Hence, this needs to improve awareness of the community there is a need for health education programs regarding improved use of latrine, cleanliness of latrine, constructing latrine inside the compound and maintenance of latrine for the proper utilization including children should be considered. In addition to this all actors to bridge the apparent gap between knowledge and practice pertinent to upscaling latrine use. Facilitate women education, training on latrine construction skills and capacity building continuously for the community. Not only this they need support on the utilization of the matching resources to tackle the sanitation disparities while utilizing socio-culturally appropriate technological options, suitable for the study community at affordable prices.

### Supplementary Materials

There is no remaining pertinent data and materials, all information is presented in the main manuscript.

### Author Contributions

Conceptualization, DW; methodology, DW & YT; software, validation, and formal analysis, investigation, resources, data curation, writing-original draft preparation, writing-review and editing, visualization, supervision, project administration, DW& YT; funding acquisition, DW & YT. All authors have read and agreed to the published version of the manuscript.

## Acknowledgments

The authors would like to thank the SNNPR health bureau for ethical approval and at Sidama National Regional state and Chire district/woreda health bureau for their cooperation on providing, information and support letter. We would like to provide our gratitude for Paradise Valley College for its cooperation and financial supports during data collection they paid peridium for data collectors and stationery support, which did not affect the finding of this result. The authors are also grateful to all data collectors and study participants for their valuable contributions.

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Spectral order statistics of Gaussian random matrices: large deviations for trapped fermions and associated phase transitions

# Spectral order statistics of Gaussian random matrices: large deviations for trapped fermions and associated phase transitions

## Abstract

We compute the full order statistics of a one-dimensional gas of fermions in a harmonic trap at zero temperature, including its large deviation tails. The problem amounts to computing the probability distribution of the th smallest eigenvalue of a large dimensional Gaussian random matrix. We find that this probability behaves for large as , where is the Dyson index of the ensemble. The rate function , computed explicitly as a function of in terms of the intensive label , has a quadratic behavior modulated by a weak logarithmic singularity at its minimum. This is shown to be related to phase transitions in the associated Coulomb gas problem. The connection with statistics of extreme eigenvalues of random matrices is also elucidated.

\DeclareBoldMathCommand\boldlangle\DeclareBoldMathCommand\boldrangle

Introduction - Recent spectacular progresses in the fabrication of devices for the manipulation of cold atoms [[1], [2]] have given a formidable impulse towards the theoretical understanding of the behavior of quantum many-body systems using tools and techniques borrowed from classical statistical physics.
Perhaps the simplest conceivable setting is a system of fermions confined by optical laser traps into a limited region of space [[2], [3], [4], [5], [6], [7], [8]]. In this experimentally rather common setup, there is a well-known connection between the ground state many-body wavefunction and the statistics of eigenvalues of a certain class of random matrices.
More precisely, in presence of a harmonic potential , the single particle eigenfunctions are given by , where are Hermite polynomials and . For a non-interacting Fermi gas of particles in a 1D harmonic trap (in the same spin state) the spatial ground-state manybody wavefunction is the Slater determinant of the first eigenstates. The explicit evaluation of this determinant yields

 |Ψ0(x)|2=CNe−mωℏ∑Ni=1x2i∏j

where are the positions of the particles on the line and is the normalization constant such that . As firstly noticed in [[9]], (1) can be interpreted as the joint probability density (jpd) of the real eigenvalues of a matrix belonging to the Gaussian Unitary Ensemble (GUE) [[10]]. The general jpd for Gaussian matrices is indeed written as

 Pβ(λ)=1ZN,βe−βN2∑Ni=1λ2i∏j

with the Dyson index of the ensemble ( for GUE) and a normalization constant. The jpd of real eigenvalues (2) can be written as

 Pβ(λ)=e−βE(λ)ZN,β ,E(λ)=N2N∑i=1λ2i−12∑j≠kln|λj−λk| . (3)

Written in this form, (2) can be interpreted as the canonical Boltzmann-Gibbs weight of an associated thermodynamical system: a gas of charged particles confined to a line, and in equilibrium at inverse temperature under competing interactions, the quadratic confining potential and a logarithmic (2D-Coulomb) repulsive interaction. Then, through Random Matrix Theory (RMT), observables of a (quantum) non-interacting 1D Fermi gas at zero temperature are mapped to thermodynamical properties of a (classical) 2D-Coulomb gas confined on the line at temperature . In the Fermi gas picture, the logarithmic repulsion of the Coulomb gas is essentially a manifestation of the exchange interaction between fermions.
In this Letter, we employ this mapping to study the order statistics of trapped fermions, i.e. the distribution of the th leftmost particle of a Fermi gas on the line. In the RMT language, this amounts to computing the full statistics of the th smallest eigenvalue, including its large deviation (LD) tails, of large Gaussian matrices (for all ). Results on the order statistics abound for i.i.d. random variables [[23]], but are much scarcer for correlated variables (see [[24], [25], [26], [27], [28]] for most interesting recent developments). Our approach provides a unified and transparent framework to probe interesting features of these distributions, such as non-Gaussian tails and a non-analytic behavior at the peak, which is a direct consequence of phase transitions of the underlying Coulomb gas system. We find that the spectral order statistics are governed by a rate function that is a function of two variables, which we are able to compute explicitly, with the “intensive” label of the th eigenvalue. Quite remarkably, by taking different limits, this single function is able to recover at once i.) the law of small fluctuations of the th smallest eigenvalue [[29], [30]] in the bulk, ii.) the distribution of the number of eigenvalues lying to the left of a barrier at (), and iii.) known results on left and right LD for the extreme eigenvalues [[32], [33], [34], [35]].
Setting and summary of results - We consider -Gaussian ensembles ( being the Dyson index) of random matrices (see [[10], [31]])) whose eigenvalues have law (2). Using the 2D-Coulomb gas picture, it is well known [[10]] that, for large , the average density of eigenvalues (normalized to unity) of (2) approaches the celebrated Wigner’s semicircle law on the single support , . We now arrange the eigenvalues in increasing order and we denote them by . It is easy to notice that the cumulative distribution of the th smallest eigenvalue is related in a simple way to the tail-cumulative distribution of (the number of eigenvalues smaller than ),

 P[λ(k)k] . (4)

Indeed, the th eigenvalue is smaller than if and only if there are at least eigenvalues to the left of . Using (2), the probability density of is

 P[Nx=cN]=∫RNdλPβ(λ)δ(cN−N∑i=1Θ(x−λi)) . (5)

The full statistics of was computed for large in [[17], [18]], while for a symmetric interval around the origin , the full distribution of (including large deviation tails) and its variance in all regimes of have been computed in [[11]] (see [[13], [14], [15], [16], [19], [20], [21]] for related results). These results are in agreement for with numerical and analytical estimates provided in recent literature on 1D Fermi systems [[22]].
Computing the large behavior of (5) and using (4), we establish the following LD estimates

 P[Nx=cN]≈e−βN2ψ(c,x);P[λ(k)=x]≈e−βN2ψ(k/N,x) , (6)

where stands for logarithmic equivalence. The central result of our paper is the calculation of the large deviation (or rate) function (with and ), given explicitly in Eq. (15) and plotted in Fig. 1. Interestingly, the same function (viewed in turn as a function of or of , the other variable acting as a parameter) governs both the LD behavior of the cumulative distribution of and of the th eigenvalue . We will show below that, as an additional bonus, it also recovers in certain limits the right and left LD laws for the extreme eigenvalues of Gaussian matrices [[32], [33], [34]].

The rate function is convex, it satisfies and has a minimum (zero) on the red line in Fig. 1, identified by the equation

 c⋆(x)=∫x−∞dzρsc(z)=1−θx−sinθx2π, (7)

with for , while (resp. ) if (resp. ). Thus e.g. the distribution of is peaked around which is precisely its mean value for large , while the distribution of is peaked around , where and its generalized inverse are given by (7).
Asymptotics - Expanding the rate function (15) around its minima in the two directions, we get access to the law of typical fluctuations of and . We find

 Missing or unrecognized delimiter for \right (8)

where we have denoted and is a constant independent of and . The fact that is not simply harmonic in and close to its minima but contains also a logarithmic contribution has important consequences on the variance of the two random variables, as it is customary in this type of problems [[11], [17], [18]]. Indeed, inserting this behavior back into (6) [[36]], we find that the extensive variables and in the bulk have Gaussian fluctuations

 P[Nx=cN]≃e−(Nc−Nc⋆(x))22Δ1;Δ1=1βπ2lnρ3⋆N (9) P[yk=y]≃e−(y−√Nx⋆(c))22Δ2;Δ2=1βπ2lnNρ2⋆N, (10)

up to corrections for large , in agreement with earlier results in both cases [[29], [30], [17], [18], [37]]. These estimates are respectively valid for such that , i.e. for “deeply” within the semicircle edges, and , i.e. . However, for larger fluctuations, the Gaussian form is no longer valid, and the distribution has non-Gaussian LD tails described by the rate function (15). Furthermore, the limit (or symmetrically ) of (15) at fixed (corresponding to the largest or smallest eigenvalue [[38], [39]]) is interesting. We find

 limc→1−ψ(c,x)1−c =Ψ+(x),x>√2 (11) limc→1−ψ(c,x) =Ψ−(x),x<√2 , (12)

where ([[34]], r.h.s. of Eq. 13 with ) and ([[33]], Eq. 59) are respectively the right and left LD functions for the largest eigenvalue of Gaussian matrices. The different scalings in for the two branches of the largest eigenvalue ( on the left and on the right) can be recovered using a simple argument (see [[11]]). We find particularly interesting (as already discovered for the number statistics problem in a bounded interval [[11]]) that a rate function obtained with a Coulomb gas calculation is able to recover not only the “Coulomb gas” branch , but also the “energetic” (or instantonic) branch , which was originally calculated using an entirely different approach [[34]] (see also [[40]] for a subsequent elaboration in the context of quantum gravity and string theory). In next section, we sketch the derivation of the rate function (15).

Derivation - In order to evaluate the multiple integral (5) for large , we resort to the “constrained” Coulomb gas method (introduced in [[32], [33]] and subsequently used in many different problems [[41], [42], [43], [44], [45], [46], [47], [48], [20], [21], [49]]), where a multiple integral over an eigenvalue jpd (like (5)) is converted into the partition function of the associated 2D-Coulomb gas with density , in equilibrium at inverse temperature . Skipping details [[36]], the multiple integral (5) can be written as a functional integral over , , where

 E[ρ]=12∫dλρ(λ)λ2−12∬dλdλ′ρ(λ)ρ(λ′)ln|λ−λ′| +A1(∫dλρ(λ)Θ(x−λ)−c)+A2(∫dλρ(λ)−1) (13)

is the action (the integral version of the energy function ) and and are Lagrange multipliers constraining a fraction of eigenvalues to the left of , and enforcing normalization of the density to . The functional integral will then be dominated by the equilibrium density of eigenvalues (Coulomb gas particles) minimizing the action (13).
Using the resolvent method [[36]], we find that the equilibrium (saddle-point) density has the general form

 ρ⋆c,x(λ)=1π√(λ+−λ)(λ0−λ)(λ−λ−)x−λ (14)

for such that the radicand is positive. The edge points (depending parametrically on ) are the three roots of the polynomial , and the function is determined by the solution of the constraint . It turns out that, depending on the values of and , five different phases of the gas of eigenvalues are possible (see Fig. 2). For a generic position of the barrier , whenever we recover the semicircle law (phase ). For (resp. ) and (resp. ) we have a single support solution (phases and , respectively). For all the other values the solution has double support (phases and ) and generalizes the density in [[17], [18]] to the case . One can see that the double-support phases and are separated by the unperturbed phase (the semicircle law) that lies on the curve . This tiny separation in the plane between the double support phases is ultimately responsible for the weak non-analytic behavior of the rate function close to its minimum in both the and directions (see (8)).
Once the saddle-point density is known, we precisely find the law (6) with (where comes from the normalization factor in (2) and needs to be subtracted), where

 ψ(c,x)=∫c⋆(x)cdc′(λ20(c′)−x22+∫xλ0(c′)dzS(z,c′)), (15)

and . The rate function in (15) can be written in closed form in terms of elliptic integrals [[36]]. We have checked that for we recover the rate function for the so called index problem [[17], [18]], and all the results have been checked numerically with excellent agreement [[36]].

Conclusions - In summary, we provided a complete characterization of the spectral order statistics of large dimensional Gaussian random matrices. We showed that the problem is amenable to a Coulomb gas treatment through a simple relation between the distribution of the th smallest eigenvalue and the number of eigenvalues smaller than a threshold . Details on the derivations, numerical checks and the outlook for future research will be provided elsewhere [[36]]. In the future, it will be interesting to study the order statistics for other ensembles and the crowding effects close to a specific eigenvalue in the bulk (see [[50]] for the first eigenvalue of GUE), as well as to investigate finite corrections. In the Fermi gas picture (), our results provide the full statistics of particle number on a semi-infinite line (extending recent results [[11], [17], [18]]) and single-particle fluctuations in the bulk of a system of 1D fermions in a harmonic trap, including their LD tails. We expect the results presented here to apply also to bosonic systems in presence of very strong repulsive interactions between bosons [[51]]. The link between 1D Fermi gases and RMT makes it possible to speculate about the possibility of “simulating” RMT eigenvalues in laboratory by manipulating Fermi gas systems, and the experimental verification of LD results is an intriguing challenge whose realization is very much called for.

Acknowledgments - IPC acknowledges hospitality by the LPTMS (Univ. Paris Sud) where this work was initiated. He also acknowledges multiple discussions with Pierpaolo Vivo and Fabio Cunden as well as the work done on the manuscript.

### References

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20. R. Marino, S. N. Majumdar, G. Schehr, and P. Vivo, J. Phys. A: Math. Theor. 47, 055001 (2014).
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23. H. A. David and H. N. Nagaraja, Order Statistics (Wiley, New Jersey, 2003), 3rd ed.
24. E. Brunet and B. Derrida, Europhys. Lett. 87, 60010 (2009); J. Stat. Phys. 143, 420 (2011).
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27. K. Ramola, S. N. Majumdar, and G. Schehr, Phys. Rev. Lett. 112, 210602 (2014).
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# Need expertise in controlling precision across calculations: specific case - elementary symmetric functions

I've seen several questions here concerning precision issues when performing multiple floating point calculations, but unfortunately got completely confused. I really don't understand what precision means. Actually I don't even understand difference between accuracy and precision in Mathematica. And I badly need this understanding. I really hope somebody can explain all this to me, or point to an answer to a question where it is clearly written.

I thought maybe I would understand better if explained on my own example where I want to achieve precise result and cannot.

So, I need to compute some expressions involving $n$th elementary symmetric function in $2n$ real numbers, for $n$ around 50. The initial numbers are of magnitude $\pm10$ and are rounded to $.001$, and I need the output also up to $.001$. I am using Newton's formulæ: for, say, $x_1$, ..., $x_{100}$ I first compute $p_k=x_1^k+...+x_{100}^k$, $k=1,2,...$, and then recursively find $$e_j=\frac1j(p_1e_{j-1}-p_2e_{j-2}+p_3e_{j-3}-...)$$ starting from $e_0=1$ and going up to $e_{50}$. I seem to lose precision catastrophically, since the final expressions where I have to use the $e_{50}$ must be between 0 and 1 while I sometimes obtain something above $10^{25}$ and sometimes something negative.

So, what is the correct way to calculate $e_{50}$ correctly to $.001$ in this situation?

And it would be really great if the answer would reveal some general principle for dealing with similar calculations, since obviously there is a general issue of this kind which does not involve specifics of my case at all.

• To clarify: all you have are the power sums, and you need to convert them to elementary symmetric polynomials? If you have the arguments themselves, then use SymmetricPolynomial[] directly. (In general, Newton-Girard is not very numerically reliable, so you need very high precision if you don't have exact numbers.) – J. M. will be back soon Apr 7 '17 at 9:21
• @J.M. I tried this, maybe I do something wrong but I got the message Cannot create a polynomial with 100891344545564193334812497256 terms. As for power sums - maybe I lose precision already finding them too, I don't know. I have the $x$es and I need their elementary symmetric functions – მამუკა ჯიბლაძე Apr 7 '17 at 9:27
• Hmm, yes, SymmetricPolynomial[50, RandomReal[1, 100]] fails; I haven't looked at it in a while, but I'm not sure why there's this limitation. Can you try SeriesCoefficient[Product[c t - 1, {c, xlist}], {t, 0, 50}], please? – J. M. will be back soon Apr 7 '17 at 9:30
• If you don't mind a bit of a wait, I'll write one later; I've got something running... – J. M. will be back soon Apr 7 '17 at 10:23
• Various ways of calculating elementary symmetric polynomial can be found in: Finding column with maximum sum of by-element products of column subsets efficiently? – jkuczm Apr 7 '17 at 10:46


text:
# Fixed-point theorems in infinite-dimensional spaces

In mathematics, a number of fixed-point theorems in infinite-dimensional spaces generalise the Brouwer fixed-point theorem. They have applications, for example, to the proof of existence theorems for partial differential equations.

The first result in the field was the Schauder fixed-point theorem, proved in 1930 by Juliusz Schauder (a previous result in a different vein, the Banach fixed-point theorem for contraction mappings in complete metric spaces was proved in 1922). Quite a number of further results followed. One way in which fixed-point theorems of this kind have had a larger influence on mathematics as a whole has been that one approach is to try to carry over methods of algebraic topology, first proved for finite simplicial complexes, to spaces of infinite dimension. For example, the research of Jean Leray who founded sheaf theory came out of efforts to extend Schauder's work.

Schauder fixed-point theorem: Let C be a nonempty closed convex subset of a Banach space V. If f : CC is continuous with a compact image, then f has a fixed point.

Tikhonov (Tychonoff) fixed point theorem: Let V be a locally convex topological vector space. For any nonempty compact convex set X in V, any continuous function f : XX has a fixed point.

Browder fixed point theorem: Let K be a nonempty closed bounded convex set in a uniformly convex Banach space. Then any non-expansive function f : KK has a fixed point. (A function ${\displaystyle f}$ is called non-expansive if ${\displaystyle \|f(x)-f(y)\|\leq \|x-y\|}$ for each ${\displaystyle x}$ and ${\displaystyle y}$.)

Other results include the Markov–Kakutani fixed-point theorem (1936-1938) and the Ryll-Nardzewski fixed-point theorem (1967) for continuous affine self-mappings of compact convex sets, as well as the Earle–Hamilton fixed-point theorem (1968) for holomorphic self-mappings of open domains.

Kakutani's fixed-point theorem: Every correspondence that maps a compact convex subset of a locally convex space into itself with a closed graph and convex nonempty images has a fixed point.


text:
## Chatter [NCA / SHAM]

Hi Nastia!

» So what are the main requirements for the ideal PK metric?

You are asking a nasty question – rattling the foundations of BE by NCA. I like it.

» Not to change and have low variability?

IMHO, it should be selective, i.e., reflect changes of what we are interested in. For the extent of absorption it’s clearly AUC0–x. It’s another question what the x is… As we have seen in the other thread starting there, we get an unbiased estimate of T/R with AUC0–∞ and AUC0–t(common). How early could we stop sampling (x = ?) to get an unbiased estimate is – as you know – on my to-do list. I still hold that for IR products it’s much earlier than regulators think.
For the rate of absorption? Well, cough. Since Cmax is a composite metric, it can’t be selective for ka. We don’t have to re-invent the wheel; reading educates.1–6 Though I know >50% of the authors personally, I’m not biased.
Interesting the abstract6

Results: The outcome of a bioequivalence trial was shown to depend on the measure. Cmax/AUC reflected changes in ka, but not in F. AUC showed dependence on F, but virtually no dependence on ka. For Cmax, a 3- to 4-fold increase in ka and a concomitant 20% decrease in F, as well as corresponding changes in the opposite directions, resulted in bioequivalent outcomes.

But then

Conclusions: It was concluded that use of Cmax/AUC should be discouraged and that defining bioequivalence in terms of rate and extent of absorption has major problems.

(my emphasis)
IMHO, the conclusion contradicts the results. How come? Only because two authors were from the FDA and they didn’t want to change the rules? Why am I not surprised?
A picture tells more than a thousand words. A funny one of the paper:

A Cmax/AUC, B AUC, C Cmax

With $$\small{\frac{k_{a,T}}{k_{a,R}}=5\,\wedge \frac{F_T}{F_R}=0.8}$$ one has a high chance of passing Cmax (given, will fail on AUC). OK but as stated in the abstract there are combinations where ka and F are clearly different though both Cmax and AUC will pass. Bravo, well done! Great metric.
Reminds me on another paper7 by authors of the FDA assessing the performance of AUC0–t and AUC0–∞. Results: T/R-ratios very similar, AUC0–∞ more variable. The FDA’s consequences: Use both.

» Returning to Cmax/AUC - do we need more strict conditions to prove BE on its base keeping in mind its much lower variability (narrow CI for example) - cause it's always within the limits?

Why? The conventional limits are based on the assumption that a of 20% is clinically not relevant. Leaving NTIDs aside, do we narrow the limits (say, for AUC) only because the variability of some drugs is low? Nope. If we would make the limits dependent on the variability, we would end up in reference-scaling chaos.

» (In old (e.g. below 2013) russian protocols I even saw the opposite situation when the confidence limits for Cmax/AUC…

I know. Strange.

» … (as well as Cmax) were choosen to be "75,00-133,00%")

I understand that. Before the European 2001 Note for Guidance, many products got even an approval with 30% (70.00–142.86%).

» Can you please also clarify the distribution of T1/2: it is always presented as Mean±SD, is it correct? (Mean=Arithmetic Mean).

You rub salt into my wounds. Let’s step back. Rate constants have a unit of 1/time. Hence, the correct location parameter is the harmonic mean. Its dispersion parameter is the jackknife standard deviation (in WinNonlin’s terminology: Pseudo SD). For t½ you have two options: Use the same as well (as I do though I’m not sure about the distribution; Γ?) or go with nonparametrics (x̃, quartiles).

» It seemed to me I've read somewhere on the forum that for study planning we should use not the mean T1/2 but the confidence level for it, but I couldn't find this thread now.

I’m too lazy to search as well. This one about extremes?

» Is it correct to use standard approach to calculate CI for untransformed T1/2 as normally distributed data …

Anything is better than the mean (see this presentation, slides 64–66).

» … or should we use other approaches like nonparametric median confidence intervals or bootstrap?

Sounds good though “nonparametric CI” is a little bit strange. I think that I once saw a paper about it, not sure. However, if you don’t have data of a previous study… What you find in the public domain is often x±SD or min/max.

» » E.g., for biphasic methylphenidate the cut-off time (FDA: 3 h fasting, 4 h fed) is based on PD indeed (at that time ~90% of patients show the maximum effect). […]
»
» That's correct but too much individual approach - we couldn't know beforehand all the thin nuances of the PD of the specific drug, sometimes it is even impossible to find in literature any PK data for some drugs while planning new study... So I think the overall approach should be more general.

Early and late partial AUCs are only relevant for multiphasic MR products. Luckily I met only three so far: zolpidem, methylphenidate, amphetamine(s). For the first two there is a PD based justification possible. For the last one – no idea.

» Why not? The static moments of the plain curve 'Concentration-time' (C(t)) are:
»
»

$$M_t=\int t\cdot C(t)dt$$;
» $$M_C=1/2\cdot\int C(t)^2dt$$

» First is connected with MRT (muliplyed by total square of the curve), and the second should reflect some ideal Concentration-unit parameter (the height of the gravity center). Why not to use it like some alternative to Cmax?

Back in the days when I used my own software, I reported all it could calculate (hey, look how clever I am). Arrogant attitude. Confused sponsors and regulators. What I learned: The variability of VRT sucks. Not surprising cause we have $$\small{t^2}$$ and $$\small{C^2}$$ in it.

» » Only nice to print a profile, cut it a out, push a pin through it, and make a weird whizz wheel for kids.
»
» Oh, if we'll proceed further in the power of moments we should be able to calculate moment of inertia - then one can take that curve to the orbital station in order to research Dzhanibekov effect

Didn’t know that one! BTW, the center of gravity can be outside of the profile (scroll down in this post). Aboriginals know that for ages.

» » To get a reliable estimate of AUC one has to cover 95% (!) of AUC0–∞ (note that I’m not taking about BE but hard-core PK). For AUMC is should be 99%. I’m quoting Les Benet. Don’t blame me.
»
» Could you please explain this more detaily? What do you mean in getting a reliable estimate of AUC?

I was talking about PK modeling. IIRC, Les Benet said that at the BioInternational in Munich 1994. Not sure whether it’s mentioned in the book.

1. Endrényi L, Fritsch S, Yan W. Cmax/AUC Is a Clearer Measure Than Cmax for Absorption Rates in Investigations of Bioequivalence. Int J Clin Pharm Ther Toxicol. 1991;29(10):394–9. PMID 1748540.
2. Schall R, Luus HG. Comparison of absorption rates in bioequivalence studies of immediate release drug formulations. Int J Clin Pharmacol Ther. 1992;30(5):153–9. PMID 1592542.
3. Endrényi L, Yan W. Variation of Cmax and Cmax/AUC in investigations of bioequivalence. Int J Clin Pharm Ther Toxicol. 1993;31(4):184–9. PMID 8500920.
4. Lacey LF, Keene ON, Duquesnoy C, Bye A. Evaluation of Different Indirect Measures of Rate of Drug Absorption in Comparative Pharmacokinetic Studies. J Pharm Sci. 1994;83(2):212–5. doi:10.1002/jps.2600830219.
5. Schall R, Luus HG, Steinijans VW, Hauschke D. Choice of Characteristics and Their Bioequivalence Ranges for the Comparison of Absorption Rates of Immediate-Release Drug Formulations. Int J Clin Pharmacol Ther. 1994;32(7):323–8. PMID 7952792.
6. Tozer TN, Bois FY, Hauck WW, Chen M-L, Williams RL. Absorption Rate Vs. Exposure: Which Is More Useful for Bioequivalence Testing? Pharm Res. 1996;13(3):453–56. doi:10.1023/a:1016061013606.
7. Bois FY, Tozer TN, Hauck WW, Chen M-L, Patnaik R, Williams RL. Bioequivalence: Performance of Several Measures of Extent of Absorption. Pharm Res. 1994;11(5):715–22. doi:10.1023/A:1018932430733.

My post № 5,000.

Edit: I explored my data. Drug X, 5–60 mg (linear PK proven), same bioanalytical method (enantioselective stable isotope IS GC/MS, LLOQ dependent on the dose), sampling for 24 h, 3–7 time points for the estimation of λz, extrapolated AUC <10%.

I was wrong for many years. Seems that I have to revise my procedures and go with the median or geometric mean in the future.

Dif-tor heh smusma 🖖
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes


text:
# If $A$ is a semisimple algebra, then every irreducible $A$-module is cyclic over $A$

Let $A$ be a finite-dimensinoal associative algebra with unit $1$ over $K$. Then $A$ is a $A$-right module (by module I always mean right module) over itself by right multiplication (the so called right regular module), denote it by $A^{\circ}$. An $A$-right module $V$ is semisimple if it is the sum of irreducible $A$-submodules, and this is equivalent with the fact that for every $A$-submodule $U \le V$ there exists another $A$-submodule $W$ such that $V = U\oplus W$. An algebra $A$ is called semisimple if $A^{\circ}$ is semisimple as a module. With this we could show that if $A^{\circ}$ is semisimple, then every irreducible $A$-right module is isomorphic to a submodule of $A^{\circ}$.

Now I noticed the following, but see it nowhere mentioned. So I ask myself if my reasoning is correct. If $W$ is an irreducible $A$-module for a semisimple algebra $A$, then I guess $W$ must be cyclic as an $A$-module, i.e. generated by a single element. This follows because an $A$-isomorphism commutes with multiplication by elements from $A$, and every element from $A$ (or $A^{\circ}$) could be written as a multiple of $1 \in A$.

Let $A$ be a semisimple algebra and let $W$ be an irreducible $A$-right module, then $W$ is generated by a single element.

Proof: By the above there exists a submodule $W_0$ of $A^{\circ}$ such that $W_0 \cong W$. Let $\varphi \in \operatorname{Hom}_A(W_0, W)$ be such an isomorphism, then for $x \in W_0$, as also $x \in A$, we have $\varphi(x) = \varphi(x1) = \varphi(1x) = \varphi(1)x$, hence every element $y \in W$ has the form $\varphi(1)\cdot x = y$ where $x \in A$ is choosen such that $\varphi(x) = y$, i.e. we have $\varphi(1)A = W$. $\square$

Is this correct? Or have I overlooked something? And is this of use, or not; as I see it nowhere mentioned?

• You should not call such module $1$-dimensional, as this sounds like they are free (which they need not be). Being generated by a single element is called being cyclic, and any simple module is cyclic, regardless of the algebra. – Tobias Kildetoft Feb 22 '16 at 11:20
• Thanks for your comment. I am not that much into module theory and just used the terminology which is common for vector spaces. Maybe you want to make an answer out of your comment; maybe with some further remarks about cyclic modules, where they are used and so on... – StefanH Feb 22 '16 at 11:25

You're overthinking! ;-)
Let $M$ be an irreducible left module over the (unital) ring $R$. By definition, $M\ne\{0\}$ and the only submodules of $M$ are $\{0\}$ and $M$.
Take $x\in M$, $x\ne0$; then $Rx$ is a nonzero submodule of $M$, so $Rx=M$.


text:
## router interface shows on public, instead of hosted website (ehcp)

I used EHCP for quick setup since im new to this, just dont understand how to setup the ip…
basically i can access my router from public ip, and my website only from my machine…

what could be the error in ip config, do I need to change my router or my config in ehcp ? there are no errors or anything i will dump my config which should be pretty… amateur

EHCP

``````Array gosteriliyor:
Array
(
[dnsip] => 188.79.170.95
[ehcpdir] => /var/www/new/ehcp
[banner] => this is banner.. you may write here something using server settings

[defaulttemplate] => sky
[defaultlanguage] => en
[updatehostsfile] => Yes
[messagetonewuser] => Dns servers for our server:

87.216.1.65

87.216.1.66

(This will be sent to new users)
[backupdir] => /var/backup
[quotaupdateinterval] => 6
[webservertype] => apache2
[webservermode] => nonssl
[mysqlcharset] => DEFAULT CHARACTER SET utf8 COLLATE utf8_turkish_ci
[enablewebstats] => Yes
[versionwarningcounter] => 12
[initialize_domain_files] => Yes
[switchtoapacheonerror] => Yes
[localip] => 127.0.1.1
[lastquotaupdate] => 2018-10-08 09:19:33
[activewebserverip] => 188.79.170.95
[dnsipv6] =>
[updatednsipfromweb] => Yes
[morethanoneserver] =>
[server_id] =>
[disablecustomhttp] => Yes
[disableeditwebservertemplate] =>
[disableeditdnstemplate] =>
[userscansignup] =>
[enablewildcarddomain] =>
[freednsidentifier] =>
[defaultdnsserverips] =>
[defaultwebserverips] =>
[defaultmailserverips] =>
[singleserverip] => 188.79.170.95
)
``````

## Some index files failed to download / public key not available

When I try to use

``````sudo apt-get update
``````

it gives this error:

``````Fetched 5.530 kB in 53s (104 kB/s)
W: GPG error: http://extras.ubuntu.com precise Release: The following signatures couldn't be verified because the public key is not available: NO_PUBKEY 16126D3A3E5C1192
W: Failed to fetch http://dl.google.com/linux/chrome/deb/dists/stable/Release Unable to find expected entry 'main/binary-i386/Packages' in Release file (Wrong sources.list entry or malformed file)
``````

When I try to access to etc/apt folder, it said no such file or directory.

How can I fix this?

## Problem with apt-get update: Some index files failed to download (public key not available)

When I try to use

``````sudo apt-get update
``````

it gives this error:

``````Fetched 5.530 kB in 53s (104 kB/s)
W: GPG error: http://extras.ubuntu.com precise Release: The following signatures couldn't be verified because the public key is not available: NO_PUBKEY 16126D3A3E5C1192
W: Failed to fetch http://dl.google.com/linux/chrome/deb/dists/stable/Release Unable to find expected entry 'main/binary-i386/Packages' in Release file (Wrong sources.list entry or malformed file)
``````

When I try to access to etc/apt folder, it said no such file or directory.

How can I fix this?

## Setting up public key authentication to Linux server from Windows (ppk private key)

I created a public and private key using PuTTYgen and copied the public key to `.ssh/authorized_keys` under my user account.

Then I try to specify the private key when trying to log in, but apparently it doesn’t pick it up and keep asking for the username/password I originally had. I’m using WinSCP to connect and specify the private key in Advanced/Authentication section. Am I missing any steps in WinSCP?

The content of `authorized_keys` looks like as below

``````ssh-rsa AAAAB3NzaC1yc2EAAAABpEVSiiRXi7tOHpkOyFa9w2OLpBep31k9lePCK7RQxsdfs9u11+rdu0XCidRKOY5j4anD1eDaNBj87wqZbsreRe5cFcsakyGUAYXAvqgGApvsep31k9lePCK7RQxlOY5j4anD1eDaNBj8LJO++K3SkUN8E0srRBO8YyMT6Y03/F7+AAAAB3NzaC1yc2Q4h2RLGtr12CDKSBVAnFEc+JucuF4uF0WY4Sh66MSFI63mCQFu9iYNYwWyT6lUo6sks4WypEVSiiRXi7tOHpkOyFa9w2OLpBzAlTA/VSQwdNTFYUI1vquaufZ9ORzTa6dkbBRo/mLVdevYSRMSDw1BUcinYz/ogdxRvw==
``````

I changes the permission to `.ssh` to 700 and `authorized_keys` to 600.

Although I go to authentication section in WinSCP and specify the private key (as shown in the screenshot) it looks like it still need username and password and doesn’t pick it up.

When I use PuTTY and specify the private key, after entering the login username it says

Server refused our key

Here is the log from PuTTY

``````2018-04-28 17:43:05 Connecting to 158.85.98.202 port 22
2018-04-28 17:43:05 We claim version: SSH-2.0-PuTTY_Release_0.70
2018-04-28 17:43:05 Server version: SSH-2.0-OpenSSH_7.4
2018-04-28 17:43:05 Using SSH protocol version 2
2018-04-28 17:43:05 Doing ECDH key exchange with curve Curve25519 and hash  SHA-256
2018-04-28 17:43:05 Server also has ecdsa-sha2-nistp256 host key, but we don't know it
2018-04-28 17:43:05 Host key fingerprint is:
2018-04-28 17:43:05 ssh-ed25519 256         6b:0d:e2:f6:c5:9e:15:84:0c:1b:2c:19:62:cd:5b:ef
2018-04-28 17:43:05 Initialised AES-256 SDCTR client->server encryption
2018-04-28 17:43:05 Initialised HMAC-SHA-256 client->server MAC algorithm
2018-04-28 17:43:05 Initialised AES-256 SDCTR server->client encryption
2018-04-28 17:43:05 Initialised HMAC-SHA-256 server->client MAC algorithm
2018-04-28 17:43:05 Reading key file "C:Users\Desktopprivate_key.ppk"
2018-04-28 17:43:09 Offered public key
2018-04-28 17:43:09 Server refused our key
2018-04-28 17:43:09 Using SSPI from SECUR32.DLL
2018-04-28 17:43:09 Attempting GSSAPI authentication
2018-04-28 17:43:09 GSSAPI authentication request refused
``````

## Assign public IP of /29 block directly to a connected device in pfSense

Ultimately what I want to do is connect a second physically separate gateway, and assign its WAN port one of the public IP addresses given by our ISP.

So I have the following setup currently and is working.

1. Fibre leased line from ISP.
2. Fibre comes to ISP box
3. Ethernet from ISP box plugs into pfSense WAN port
4. pfSense WAN port set as static IP assignment IP: xxx.xxx.xxx.99, GW: xxx.xxx.xxx.98/30
5. Add one of the public IP addresses as a virtual IP address in pfSense IP: xxx.xxx.xxx.105/29
6. Create a new private network and assign it to a spare ethernet port IP: 10.61.1.5/30
7. Connect the second gateway wan port to pfSense and assign the wan a static IP: 10.61.1.6
8. In pfSense setup 1:1 NAT and outbound NAT to connect all traffic xxx.xxx.xxx.105 <- between-> 10.61.1.6
9. Setup firewall rules in pfSense to allow all traffic between WAN xxx.xxx.xxx.105 and LAN 10.61.1.6

While this works and the new device talks over the public IP address, the actual gateway thinks it’s public IP address is 10.61.1.6, not xxx.xxx.xxx.105. This make configuration of VPN serves impossible for me as the device is wrongly thinking its public IP is a private one.

To clarify, which is my understanding, I might be wrong, the ISP gateway is xxx.xxx.xxx.98 on a /30 network and have given us a /29 block of IPs that are routable through xxx.xxx.xxx.98/30. From my testing the above rules out being able to connect a switch between the ISP box and pfSense WAN and just assign devices those public IPs of the /29 block.

Is there any way I can configure the WAN port on the secondary device with the public IP address, connect it to pfSense someway and just get pfSense to route it out to xxx.xxx.xxx.98?

## KDF Salt: How/When is it Okay for it to be public?

Post Structure:

1. Prior source material studied

2. Question & Details

3. Assumed threat model

4. Example

5. Assumptions

6. Clarifications

Prior source material studied

0.1. RFC 2898

0.2. HAC (Handbook of Applied Cryptography)

0.3. Crypto 101

0.4. SE answers (ex: If attacker knows salt and hash, how is salt effective?) and (Use of salt to hash a password)

0.5 2018 DBIR

Question:

How is a KDF using a known salt and a low-entropy password more secure than the so-called “completely broken and not at all secure” primitive hash with salt?

1.1 Details:

Based on the available subject material, the KDF salt is not required to be kept a secret.

Does the recommendation imply a starting assumption in the threat model that the user-provided secret-key/password will always have sufficiently high entropy, or won’t be an instance of weak one? Or does the recommendation only apply to specific instances where the same or different data is encrypted regularly with a different salt/key each time?

The author of Crypto101 writes this about generic Crypto Hash Primitives:

“While this [rainbow tables based on a known single salt] would have been prevented by using a different salt for each user, systems that use a cryptographic hash with a per-user salt are still considered fundamentally broken today; they are just harder to crack, but not at all secure.”

If the threat model always assumes (as it should) that the user-provided secret key could be weak, then how does using KDF with a known salt become more secure than the aforementioned broken pattern of using a Hash Primitive with unknown (high-entropy) salt? (see Clarifications for answer to an obvious counter here)

(Aspects of) Assumed threat model for the question:

2.1 The user creates a low entropy password (TW37VE), which needs to be stored, and used for login.

2.2 The threat actor will use the same kdf (with correctly-guessed iteration) to compute the key from this common low-entropy password.

2.3 Salt needed to add entropy to the user-provided password (edit: a typo originally stated ‘secret key’ instead of password here) — which is assumed to be from low-entropy space — will be assumed here to be known to the threat actor (based on the recommendations in the documents).

2.4 The slow-computation/high-memory cost of KDF is bearable, if only for access to even a single user data: for instance, it could be a case of an insider-threat actor willing to access data for a specific user.

2.5 Internal/Insider threat actors: Based on 2018 DBIR, 28% of all the reported data breaches involved an insider/internal actor. The threat model thus accounts for an internal actor as a potential threat even in the absence of an external one. Thus, the need to secure the salt as well. (See clarifications for answer to the obvious counter)

Example:

Used KDF:

Say for password TW37VE, PBKDF2_SHA256 generates a key with the random salt 0CE@N’s, and 100000 iterations, such that the derived key is:

kdf(TW37VE, 0CE@N’s, 100000) :
pbkdf2-sha256$$100000$$OCE@N’s\$!@#%^&

Threat KDF:
kdf(harvested-weak-password = TW37VE, not-secret-salt = 0CE@N’s, guessed-iterations = 100000):

pbkdf2-sha256$$100000$$OCE@N’s\$!@#%^&

If a legitimate user could validate itself using the known low-entropy password and the retrieved salt, what would be the deterrant in case of an attacker armed with (a) the harvested password (or it’s precomputed dictionary/rainbow hash), and (b) the not-kept-a-secret salt?

Even if it is trivial for the threat actor to scan-and-match the password against the, say 64-bit, salt key space, the resulting potential delay wouldn’t hurt.

4.1. The non-random part of the derived key is irrelevant to the security model strength.

4.2. The attacker could just as well use pbkdf2-sha256 with the known salt, the harvested secret key, and the correctly-guessed iterations.

Clarification

5.1 The drawbacks of using certain primitive-based approaches are understood; and, so are and the advantages of random-salt KDF’s in differentiating similar contents.

5.2 The question focuses on why a primitive-based pattern under the assumed threat model is said to be broken, whereas the KDF pattern which simply yields to similar threat models under similar conditions is considered as an improvement over the former.

5.3 The DBIR reported “28% internal actors” figure includes different types of actors with different actions, which — although likely to be small — in this case is assumed to be a developer from the application-server team gone rogue (or maybe it was always bad?).

5.4 The home-grown roll: there is a way to keep the salt encrypted without the threat actor having the access to the salt-decryption key unless the entire database server is compromised.

Summary:

To reiterate, how does NOT keeping the kdf salt a secret help, when exposing it could have the potential to be harmful, and security-equivalent of an unsecure primitive method?

## “Public relationship”

What does the author of this sentence (from a GRE practice test) mean by “public relationship”? :

The governor has long been obsessed with excising the media from the politician public relationship. That’s been the unifying aim of all her seemingly disconnected ventures since entering public life: a determination to erode, and eventually end, the media’s hold on political communication.

## Public relationship? I don’t understand at all what the author want to say

The governor has long been obsessed with excising the media from the politician public relationship. That’s been the unifying aim of all her seemingly disconnected ventures since entering public life: a determination to erode, and eventually end, the media’s hold on political communication.


text:
'

# Random Question & Answer

Select Category:

### Differentiation Question 10z

10. Differentiate with respect to the given variable.

z)

10. Differentiate with respect to the given variable.

10. Differentiate with respect to the given variable.

z)

Solution:

z)

Remember

The derivative of .

We choose so that Hence the derivative of is:

Derivative of a .

If

then since

Hence,


text:
## University Physics with Modern Physics (14th Edition)

a. The Heisenberg uncertainty principle limits our knowledge. $$\Delta x \Delta p_x \geq \frac{\hbar}{2}$$ $$\Delta x\;m \Delta v_x \geq \frac{\hbar}{2}$$ We are told that $\ \Delta v_x = (0.010)v_x$ Set the product of uncertainties to its minimum value. $$\Delta x\;m \Delta v_x = \frac{\hbar}{2}$$ $$\Delta x\;m (0.010)v_x = \frac{\hbar}{2}$$ $$v_x =\frac{\hbar}{2m (0.010) \Delta x}=19.3m/s$$ b. Repeat, using the mass of a proton, to find 11 mm/s.


text:
Precalculus (10th Edition)

$1.047\text{ in}^2$
Here, $r$ denotes the radius of the sector of the circle. The Area $A$ of sector can be computed using the formula: $A=\frac{\theta}{360}\cdot \pi \cdot r^2$ where $\theta$ is the central angle in in degrees. Hence, here we have $A=\frac{30}{360}\cdot \pi \cdot 2^2=\frac{\pi}{3}\approx1.047\text{ in}^2$