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### Preface

There are several vignettes explaining simulation setup, for basic lifecycle simulations here and here, as well as for human/mosquito epidemiological studies, here and here for SEI/SIS models and here for SEI/SEIR models. However, for actual explorations, many stochastic realizations are needed, as well as summary statistics of those simulations. MGDrivE2 provides tools for storing output, as well as analyzing and summarizing that output. This vignette will use the network SEI/SIS simulation as the example simulation to showcase data storage and analysis.

We start by loading the MGDrivE2 package, as well as the MGDrivE package for access to inheritance cubes, and Matrix for sparse matrices used in migration. We will use the basic cube to simulate Mendelian inheritance for this example.

# simulation functions
library(MGDrivE2)
# inheritance patterns
library(MGDrivE)
# sparse migration
library(Matrix)

# basic inheritance pattern
cube <- MGDrivE::cubeMendelian()

## Parameterization

These are the same parameters found in the epi-network vignette, and are also explained in the lifecycle-node vignette.

We will store output every other day (dt = 2), as writing to disk can be slow and this reduces the amount of data to store by half. The rate that one needs to store data will depend on the accuracy of one’s parameters - i.e., if your parameters are highly questionable, storing data every other day will not lose much information, but if your parameters are very well defined, daily storage may be necessary for accuracy. Note we set tmax artifically low in this vignette to reduce build times as the goal of this article is to demonstrate CSV processing capabilities of MGDrivE2.

# entomological and epidemiological parameters
theta <- list(
# lifecycle parameters
qE = 1/4,
nE = 2,
qL = 1/3,
nL = 3,
qP = 1/6,
nP = 2,
muE = 0.05,
muL = 0.15,
muP = 0.05,
muF = 0.09,
muM = 0.09,
beta = 16,
nu = 1/(4/24),
# epidemiological parameters
NH = 1000,
X = 0.25,
f = 1/3,
Q = 0.9,
b = 0.55,
c = 0.15,
r = 1/200,
muH = 1/(62*365),
qEIP = 1/11,
nEIP = 2
)

# simulation parameters
tmax <- 50
dt <- 2

Additionally, we augment the cube with transmission efficiencies; see Smith & McKenzie (2004) for an extensive discussion of these parameters.

# augment the cube with RM transmission parameters
cube$c <- setNames(object = rep(x = theta$c, times = cube$genotypesN), nm = cube$genotypesID)
cube$b <- c("AA" = theta$b, "Aa" = 0.35, "aa" = 0)

## Initialization of the Petri Net

With the basic parameters finished, we setup the types of nodes in our network, their movement in relation to each other, and finally the structure of the Petri Net.

# nodetypes
node_list <- c("m", "b", "h")
num_nodes <- length(node_list)

# human movement
h_move <- matrix(data = FALSE, nrow = num_nodes, ncol = num_nodes,
dimnames = list(node_list, node_list))
h_move[2,3] <- TRUE
h_move[3,2] <- TRUE

# mosquito movement
m_move <- matrix(data = FALSE, nrow = num_nodes, ncol = num_nodes,
dimnames = list(node_list, node_list))
m_move[1,2] <- TRUE
m_move[2,1] <- TRUE

# Places and transitions
SPN_P <- spn_P_epiSIS_network(node_list = node_list, params = theta, cube = cube)
SPN_T <- spn_T_epiSIS_network(node_list = node_list, spn_P = SPN_P, params = theta,
cube = cube, h_move = h_move, m_move = m_move)

# Stoichiometry matrix
S <- spn_S(spn_P = SPN_P, spn_T = SPN_T)

The spn_P_* functions (spn_P_epiSIS_network() used here) all return length 2 lists.The first index (ix) is a list equal to the number of nodes (so here, length(ix) = 3), which returns the uniquely numbered “places” for every life/infection/Erlang stage for the entire network. These correspond to every location in the state vector used during the simulation.

The second item in those lists (u) is the name for every “place”. It is a combination of genotype and node number for eggs, larvae, pupae, unmated females, and male stages. For females, it is a combination of their genotype, their mate genotype, their infection status, and the node number. For humans, it is just infection status and node number.

The spn_T_* functions (spn_T_epiSIS_network() used here) returns a length 2 list. The first element (T) is a list of parameters for every possible state transition. These are not the rates of transition, just the fact that it is possible and parameters necessary to describe them. The second element (v) is the name for each of those transitions.

## Equilibrium Conditions and Hazard Functions

Remember, these are node-by-node equilibria, not an equilibrium over the entire network. We expect some burn-in at the beginning of the simulation as the network reaches equilibrium.

# SEI mosquitoes and SIS humans equilibrium
#  outputs required parameters in the named list "params"
#  outputs intial equilibrium for adv users, "init
#  outputs properly filled initial markings, "M0"
initialCons <- equilibrium_SEI_SIS(params = theta, node_list = node_list,
NF = 500, phi = 0.5, NH = 1000, pop_ratio_H = 0.15,
log_dd = TRUE, spn_P = SPN_P, cube = cube)

The equilibrium_* functions (equilibrium_SEI_SIS() used here) return a length 3 list. The first element (params) are all of the parameters required for the simulation. These include user-supplied parameters from before (theta), as well as derived parameters, such as the density-dependent parameter. The second element is a matrix of equilibrium populations. The rows are nodes, the columns are life stages. Any stage that does not exist in a node (human stages in mosquito-only nodes, mosquito stages in human-only nodes) have NA values. The final element is M0, the initial marking for the Petri Net. This takes all of the equilibrium values from the second element, and using the places indices, distributes populations to their appropriate location in the Petri Net framework.

Next, we setup the movement rates for mosquitoes and humans, to their respective nodes. See the epi-network vignette for a detailed explanation.

# calculate movement rates and movement probabilities
gam <- calc_move_rate(mu = initialCons$params$muF, P = 0.05)

# set mosquito movement rates/probabilities
#  mosquitoes exist in nodes 1 and 2, not 3
mr_mosy <- c(gam, gam, NaN)
mp_mosy <- Matrix::sparseMatrix(i = c(1,2), j = c(2,1), x = 1, dims = dim(m_move))

# set human movement rates/probabilities
#  humans exist in nodes 2 and 3, not 1
mr_human <- c(NaN, 1/7, 1/7)
mp_human <- Matrix::sparseMatrix(i = c(2,3), j = c(3,2), x = 1, dims = dim(h_move))

# put rates and probs into the parameter list
initialCons$params$mosquito_move_rates <- mr_mosy
initialCons$params$mosquito_move_probs <- mp_mosy
initialCons$params$human_move_rates <- mr_human
initialCons$params$human_move_probs <- mp_human

Now that all the necessary parameters have been added to the named list initialCons$params, we generate the hazard functions, using the function spn_hazards(). By specifying log_dd = TRUE, we use logistic density dependence for these simulations. # exact hazards for integer-valued state space exact_hazards <- spn_hazards(spn_P = SPN_P, spn_T = SPN_T, cube = cube, params = initialCons$params, type = "SIS",
log_dd = TRUE, exact = TRUE, tol = NaN,
verbose = FALSE)

The hazard function (spn_hazards()) returns a list of functions that define the rates of every transition. It is the same length as the first element of the SPN_T, i.e., one rate for every possible transition in the Petri Net.

## Simulation of Fully Specified SPN Model

Before running any simulations, we need to create some releases, and then setup the folder structure.

We will release 50 adult females with homozygous recessive alleles 2 times, every 7 days, starting at day 10, in node 1 (this allows us to see movement from node 1 to node 2, before we see an impact on human disease transmission). Remember, it is critically important that the event names match a place name in the simulation. The simulation function checks this and will throw an error if the event name does not exist as a place in the simulation. This format is used in MGDrivE2 for consistency with solvers in deSolve.

# releases
r_times <- seq(from = 10, length.out = 2, by = 7)
r_size <- 50
events <- data.frame("var" = paste0("F_", cube$releaseType, "_", cube$wildType, "_S_1"),
"time" = r_times,
"value" = r_size,
stringsAsFactors = FALSE)

Next, we need to setup the file structure for CSV simulations.

We recommend setting up a new folder for any simulations. This ensures a clean location, and in the chance someone deletes everything in the folder, there is nothing important there. Inside the main folder (main_out), we need three folders, for each stage of the analysis. We generally call these raw, traces, and analyzed, to describe their purpose. The raw folder holds all of the raw output from the simulation. It is further divided into repetition folders. These can be named anything, we use a numeric naming scheme that sorts properly in file systems. The traces folder holds the first set analysis output. It starts empty, the provided functions setup the necessary folders, and each repetition will be split into the appropriate life-stage and node designation. The final folder, analyzed, holds the summary statistics from our simulations.

# main output folder
main_out <- tempdir()

# folders for each stage of analysis
analysis_out <- c("raw", "traces", "analyzed")

# repetitions, 3 of them
rep_out <- formatC(x = 1:3, width=3, format='d', flag='0')

# build analysis folders
analysis_folders <- file.path(main_out, analysis_out)
for(i in analysis_folders){ dir.create(path = i, recursive = TRUE) }

# build repetition folders
rep_folders <- file.path(analysis_folders[1], rep_out)
for(i in rep_folders){ dir.create(i) }

### Tau-leaping Simulation

We start by running the simulations. As explained in the inhomogeneous vignette, there are two ways to perform repetitions - using the internal repetition wrapper, or using a parallel loop. By providing output folders to sim_trajectory_CSV(), we can make use of both methods simultaneously. However, for this example, we only use the internal repetitions wrapper.

Additionally, we need to specify what life-stages to store. For this example, we will output all of the stages. However, we recommend only saving the stages necessary for the experiment at hand. Each stage to print has a one-letter designation for the function.

One-Letter Designation Stage Name
E Egg stage
L Larval stage
P Pupal stage
M Adult male stage
U Adult, unmated female stage
F Adult, mated female stage
H Adult humans

The mated female stage (F) will print infection states to different files. This keeps individual files from becoming too unwieldy.

Finally, we run three stochastic realizations of the simulation, using the tau sampler with $$\Delta t = 0.1$$, approximating 10 jumps per day.

# delta t
dt_stoch <- 0.1

# run tau-leaping simulation
sim_trajectory_CSV(x0 = initialCons$M0, tmax = tmax, dt = dt, dt_stoch = dt_stoch, folders = rep_folders, stage = c("E", "L", "P", "M", "U", "F", "H"), S = S, hazards = exact_hazards, events = events, verbose = FALSE) #> Warning in base_events(x0 = x0, events = events, dt = dt): event times do not correspond to a multiple of dt. #> event times will be rounded up to the nearest time-step! We can see all of the files returned by the simulation. # list all files from the first repetition list.files(path = rep_folders[1], full.names = FALSE) #> [1] "E.csv" "FE.csv" "FI.csv" "FS.csv" "H.csv" "L.csv" "M.csv" "P.csv" #> [9] "U.csv" Each file begins with the one-letter signifier. Notice that the female stage (F) has been expanded into three stages, one for each infection (SEI) stage. These names must stay the same, as the following analysis only matches the outputs from sim_trajectory_CSV(). To see how the files are organized, we will open two of them: the egg stage (E.csv) and the susceptible female stage (FS.csv). # read the eggs from repetition 1 egg_stage <- read.csv(file = file.path(rep_folders[1], "E.csv"), header = TRUE) # read susceptible females from repetition 1 fs_stage <- read.csv(file = file.path(rep_folders[1], "FS.csv"), header = TRUE) First, we will look at the dimensions of each file. Everything is printed at the same rate, dt, so the length of the files should match, but the number of columns is significantly different. # eggs dim(egg_stage) #> [1] 26 13 # susceptible females dim(fs_stage) #> [1] 26 19 Every file begins with a “Time” column, followed by the stage designations. It is these stage designations that explain why female files are so much larger than the rest. As described earlier, the names for each place are a combination of genotypes and node for most stages, but for adult females, places are labeled by 3 designations. The column labels in the egg file are: colnames(egg_stage) #> [1] "Time" "E1_AA_1" "E2_AA_1" "E1_Aa_1" "E2_Aa_1" "E1_aa_1" "E2_aa_1" #> [8] "E1_AA_2" "E2_AA_2" "E1_Aa_2" "E2_Aa_2" "E1_aa_2" "E2_aa_2" First column is “Time”, and each subsequent column is the Erlang-stage, genotype, and node. These labels correspond to the labels in the SPN_P$u list, and the labels on the state vector, M0.

Looking at the females output, we see how this state space gets large quickly.

colnames(fs_stage)
#>  [1] "Time"        "F_AA_AA_S_1" "F_AA_Aa_S_1" "F_AA_aa_S_1" "F_Aa_AA_S_1"
#>  [6] "F_Aa_Aa_S_1" "F_Aa_aa_S_1" "F_aa_AA_S_1" "F_aa_Aa_S_1" "F_aa_aa_S_1"
#> [11] "F_AA_AA_S_2" "F_AA_Aa_S_2" "F_AA_aa_S_2" "F_Aa_AA_S_2" "F_Aa_Aa_S_2"
#> [16] "F_Aa_aa_S_2" "F_aa_AA_S_2" "F_aa_Aa_S_2" "F_aa_aa_S_2"

Again, the “Time” column is first, followed by female genotypes, mate genotypes, infection stage, and node location. This is why female files have been split into different infection stages.

### Trace Analysis

The raw analysis is large, and difficult to work with. So, as a first reduction, MGDrivE2 provides the function split_aggregate_CSV() to organize the raw output. This function takes the raw folder, duplicates the folder structure in the traces folder, and reduces the raw output by several metrics. The aquatic stages are summarized by genotype, with the option to summarize them by Erlang-stage if desired, and split into patches. Adult female stages are summarized by female genotype, summing over male mates, and then split by patch as well. This function expects file names output from the previous function, and by default looks for all of the output. If anything is not found, it is skipped. Additionally, the user can specify subsets of the raw output and run different analyses over each.

We designate the folder traces because we can plot population traces from the output. These are the individual dynamics of each genotype and each stage for every simulation.

# split everything by patch, aggregate by genotype
split_aggregate_CSV(read_dir = analysis_folders[1], write_dir = analysis_folders[2],
spn_P = SPN_P, tmax = tmax, dt = dt, verbose = FALSE)

Looking at the traces folder, we see the folder structure is identical to the raw folder.

# don't list parent directory
list.dirs(path = analysis_folders[2], recursive = FALSE)
#> [1] "/var/folders/qv/y1wxv5g94gsgkt5dm8c08vy00000gn/T//RtmpXSwVNI/traces/001"
#> [2] "/var/folders/qv/y1wxv5g94gsgkt5dm8c08vy00000gn/T//RtmpXSwVNI/traces/002"
#> [3] "/var/folders/qv/y1wxv5g94gsgkt5dm8c08vy00000gn/T//RtmpXSwVNI/traces/003"

We can look at the output in the first repetition folder.

list.files(path = file.path(analysis_folders[2], rep_out[1]))
#>  [1] "E_0001.csv"  "E_0002.csv"  "FE_0001.csv" "FE_0002.csv" "FI_0001.csv"
#>  [6] "FI_0002.csv" "FS_0001.csv" "FS_0002.csv" "H_0002.csv"  "H_0003.csv"
#> [11] "L_0001.csv"  "L_0002.csv"  "M_0001.csv"  "M_0002.csv"  "P_0001.csv"
#> [16] "P_0002.csv"  "U_0001.csv"  "U_0002.csv"

Notice, every file begins with our one/two letter stage designation. The number following it is the node number. Remember, our network has 3 nodes, m, b, h. We expect that mosquito stages are found in nodes one and two, while humans are found in nodes 2 and 3.

We look at the reduced versions of the same two files we opened previously to compare the differences.

# read the eggs from repetition 1
egg_stage <- read.csv(file = file.path(analysis_folders[2], rep_out[1], "E_0001.csv"),

# read susceptible females from repetition 1
fs_stage <- read.csv(file = file.path(analysis_folders[2], rep_out[1], "FS_0001.csv"),
header = TRUE)

The length of each file is the same, since we need every time point, but the number of columns is significantly less.

# eggs
dim(egg_stage)
#> [1] 26  4

# susceptible females
dim(fs_stage)
#> [1] 26  4

Again, every file begins with a “Time” column, but now all files have the same remaining columns as well, just the genotypes.

# eggs
colnames(egg_stage)
#> [1] "Time" "AA"   "Aa"   "aa"

# females
colnames(fs_stage)
#> [1] "Time" "AA"   "Aa"   "aa"

Since our genotypes stay the same throughout the population, every mosquito file now has the same columns names, “Time” and then genotypes. Humans are still separated by infection stage, since that is the only way we keep track of them.

### Summary Analysis

Traces make nice squiggly plots, but are not particularly helpful otherwise. Our final step is calculating summary statistics from the simulations. MGDrivE2 provides a function (summarize_stats_CSV()) to calculate the mean and quantiles over all simulations. It reads the output of split_aggregate_CSV(), and returns the desired summaries.

Quantiles are calculated empirically, using algorithm 8 from the built-in quantile function.

# mean and 95% quantiles
summarize_stats_CSV(read_dir = analysis_folders[2], write_dir = analysis_folders[3],
spn_P = SPN_P, tmax = tmax, dt = dt, mean = TRUE,
quantiles = c(0.025, 0.975), verbose = FALSE)

Since repetitions are summarized, we do not have the same directory structure in the analyzed folder, just the summaries.

list.files(path = analysis_folders[3])
#>  [1] "E_Mean_0001.csv"            "E_Mean_0002.csv"
#>  [3] "E_Quantile_00250_0001.csv"  "E_Quantile_00250_0002.csv"
#>  [5] "E_Quantile_09750_0001.csv"  "E_Quantile_09750_0002.csv"
#>  [7] "FE_Mean_0001.csv"           "FE_Mean_0002.csv"
#>  [9] "FE_Quantile_00250_0001.csv" "FE_Quantile_00250_0002.csv"
#> [11] "FE_Quantile_09750_0001.csv" "FE_Quantile_09750_0002.csv"
#> [13] "FI_Mean_0001.csv"           "FI_Mean_0002.csv"
#> [15] "FI_Quantile_00250_0001.csv" "FI_Quantile_00250_0002.csv"
#> [17] "FI_Quantile_09750_0001.csv" "FI_Quantile_09750_0002.csv"
#> [19] "FS_Mean_0001.csv"           "FS_Mean_0002.csv"
#> [21] "FS_Quantile_00250_0001.csv" "FS_Quantile_00250_0002.csv"
#> [23] "FS_Quantile_09750_0001.csv" "FS_Quantile_09750_0002.csv"
#> [25] "H_Mean_0002.csv"            "H_Mean_0003.csv"
#> [27] "H_Quantile_00250_0002.csv"  "H_Quantile_00250_0003.csv"
#> [29] "H_Quantile_09750_0002.csv"  "H_Quantile_09750_0003.csv"
#> [31] "L_Mean_0001.csv"            "L_Mean_0002.csv"
#> [33] "L_Quantile_00250_0001.csv"  "L_Quantile_00250_0002.csv"
#> [35] "L_Quantile_09750_0001.csv"  "L_Quantile_09750_0002.csv"
#> [37] "M_Mean_0001.csv"            "M_Mean_0002.csv"
#> [39] "M_Quantile_00250_0001.csv"  "M_Quantile_00250_0002.csv"
#> [41] "M_Quantile_09750_0001.csv"  "M_Quantile_09750_0002.csv"
#> [43] "P_Mean_0001.csv"            "P_Mean_0002.csv"
#> [45] "P_Quantile_00250_0001.csv"  "P_Quantile_00250_0002.csv"
#> [47] "P_Quantile_09750_0001.csv"  "P_Quantile_09750_0002.csv"
#> [49] "U_Mean_0001.csv"            "U_Mean_0002.csv"
#> [51] "U_Quantile_00250_0001.csv"  "U_Quantile_00250_0002.csv"
#> [53] "U_Quantile_09750_0001.csv"  "U_Quantile_09750_0002.csv"

Again, the stage designation is at the front of every file name. Next, either Mean or Quantile. If Mean, then the node designation. If Quantile, which quantile, and then the node designation.

We will open the egg and susceptible females files, to see the summaries, but the structure of these files is the same as the trace files.

# read the eggs from repetition 1
egg_stage <- read.csv(file = file.path(analysis_folders[3], "E_Mean_0001.csv"),

# read susceptible females from repetition 1
fs_stage <- read.csv(file = file.path(analysis_folders[3], "FS_Mean_0001.csv"),

# eggs
dim(egg_stage)
#> [1] 26  4

# susceptible females
dim(fs_stage)
#> [1] 26  4

The dimensions are the same as the trace files. We also see that the column names are the same as well.

# eggs
colnames(egg_stage)
#> [1] "Time" "AA"   "Aa"   "aa"

# females
colnames(fs_stage)
#> [1] "Time" "AA"   "Aa"   "aa"

The plotting utilities provided by MGDrivE2 do not work on CSV output. Our assumption is that, if one is generating enough data to use the CSV utilities, then there are specific plotting requirements.

### References

• Smith, D. L., & McKenzie, F. E. (2004). Statics and dynamics of malaria infection in Anopheles mosquitoes. Malaria journal, 3(1), 13.


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They therefore rely on intermediaries either in their home country or overseas to handle distribution. The Wigner Distribution (WD) function produces a simultaneous space-spatial frequency description of an image and thus provides an appropriate domain for analyzing spatial non-uniformities in an image. Functions List of the most important Excel functions for financial analysts. Distribution channel overcomes the place barrier in delivering the products. Because the warehousing is being done by the distributor, the transportation is also the responsibility of the distributor. Distribution channels have made it possible for businesses to serve customers even at large distant places. The aim of distribution is to get goods and services from the point of production, to the point of consumption. Distribution management manages the supply chain for a firm, from vendors and suppliers to manufacturer to point of sale, including packaging, inventory, warehousing, and logistics. The methods and routes to be adopted to bring the products to the market—to the ultimate consumers and industrial users, must be determined. Distributors are clearly closer to the market and are better placed than manufacturers to forecast what their customers will need. The PMF is one way to describe the distribution of a discrete random variable. Finding the right distribution mix also means balancing between owned and non-owned channels. Avec un nom féminin, l'adjectif s'accorde. The second, the physical distribution function, moves products through the exchange channel, simultaneously with title and ownership. Commerce Mates is a free resource site that presents a collection of accounting, banking, business management, economics, finance, human resource, investment, marketing, and others. Finally, distributors can anticipate needs. Minimally it requires three arguments. Active 3 years, 7 months ago. It’s very important to understand the role of distribution channel that it doesn’t just deliver the product. These middlemen help the producers in delivering their products. The expression for the confidence interval given by the t-distribution function is: uc= x̅ +/- Tx. Firstly, distributors fulfil a role that may be quite obvious: they buy and hold stocks for the manufacturer. The size and scope of a distribution department will vary from one business setting to the next. When the function is strictly increasing on the support of (i.e. In wave optics, the Wigner Distribution Function description for an optical imaging system is a powerful analytical tool for modeling the imaging process with geometrical transformations. Companies will also incur losses no matter how efficiently they are working. The New Product Development Process (NPD) – Obtain new Products. Clearly, getting the products to the right place for consumers to be able to buy them is a part of marketing. ), then admits an inverse defined on the support of , i.e. This way they help companies in marketing their products in the market. They will implement the information collected in their strategies and will aim to improve their service quality. Although the manufacturer will offer credit to the distributor, this is less risky than offering credit to the hundreds of customers the distributor deals with, especially since the manufacturer will only have limited access to information about the creditworthiness of end-users. Also, the distribution channel aims at reducing the effective cost of distribution and so that the overall cost of the product can be minimised. The next role of the finance function is in payroll, claims processing, and acting as the repository of pension schemes and gratuity. Intermediaries involved in distribution network buys goods in large bulk from the producers. Digested nutrients are absorbed into the bloodstream through capillaries in the villi that line the small intestine. En … They are well-placed to share information about the market, about competitive activity and so forth. However, getting products to the right place at the right time in the right quantity is the function of distribution. Particularly in today’s global markets, manufacturers have great difficulty in serving markets directly, and therefore rely on intermediaries either in their home country or overseas to handle distribution. Intermediaries in a distribution channel provide services that enable manufacturers to reach different types of customers. Several functions are just starting to be appreciated, such as roles in alternative splicing. With a uniform distribution, the plot would have the same height for each outcome (since the height corresponds to the probability, which is the same for each outcome of a … Market Segmentation Criteria – How to segment markets, right products, at the right time, in the right place, and in the right condition. Stocks are accumulated in large amount by intermediaries and stored safely in warehouses. By providing this information to the manufacturer, the latter is better able to provide superior value to consumers. This means that they will be exposed to the firm’s products even if they currently use a competitors’ products. It shows that the probability of X being less than or equal to x l is F X (x l).This is a point on the F X (x) versus x curve in Figure 20.4 (b) and it is the shaded area in Figure 20.4 (a). The dbinom() function gives the probabilities for various values of the binomial variable. Xiaochun Liu: Andreas Velten : Computational Optics Group University of Wisconsin - Madison: Paper: Bibtex : Abstract. This will increase the customer base and rev… Distribution channel simply refers to the path or route through product moves from producer to ultimate customer. The main functions to be performed by wholesalers can be mentioned as follows: 1. In addition, distributors can share credit risk. Distribution function, mathematical expression that describes the probability that a system will take on a specific value or set of values. While the retail channel is most familiar to students, wholesalers play an important role as intermediaries. So 0.5 plus 0.5. The following functions of distributors for manufacturers can be identified. They understand the customer need and accordingly introduce them with the products. These intermediaries or middlemen are known as wholesalers, retailers and agents.eval(ez_write_tag([[300,250],'commercemates_com-large-mobile-banner-1','ezslot_7',172,'0','0'])); Distribution channel performs a wide range of functions and facilitate the business in its operations. Marginal distribution functions play an important role in the characterization of independence between random variables: two random variables are independent if and only if their joint distribution function is equal to the product of their marginal distribution functions (see the … Besides the functions of buying and selling or exchanging goods, the wholesalers also perform other facilitating functions of marketing at a time. Second one is the indirect channel in which there is involvement of different intermediaries and middlemen between producer and customers. According to Agbonifoh et al (2007), distribution is one of the elements of the marketing mix and one of the four Ps. This all collected information helps businesses in understanding the market needs better. In the following, we will differentiate between the functions of distributors fulfilled for consumers vs those fulfilled for manufacturers. This means that distributors such as tool distributors carry a wide range of tools for specific end-users. Figure 1: Probability mass function of the random variable X corresponding to a die rolling a six-sided die estimated from 20 rolls. In this paper, the performance of WD based processors is compared to that of the cross-correlator for a simple pattern recognition problem. As a result, planning the delivery route becomes easier and it is more cost effective too. Developments like changes in customer demography, psychography, media habits and the entry of a new competitor or a new brand and changes in customer preferences are some of the information that all manufacturers want. Xiaochun Liu: Andreas Velten : Computational Optics Group University of Wisconsin - Madison: Paper: Bibtex : Abstract. Accept Read More, Portfolio Planning: Growth and Downsizing, The Buyer Black Box – Buyer’s Characteristics. ), describes how density varies as a function of distance from a reference particle.. Role of Channel of Distribution and Physical Distribution of Goods! This means that 68% of the values will be within 1 standard deviation of the mean. It helps the customers in getting aware of the company presence and its products features. The role of Wigner Distribution Function in Non-Line-of-Sight Imaging. It is the role of intermediaries involved in the selling process to find and contact prospective customers. Companies depend on distribution channels for selling their products. Clearly, due to differing financial control regulations, a foreign company would probably be unable to do this. The distribution moves towards a classic normal curve as the number of measurements increases. It is their duty to sell these products to people by motivating and urging them with their skill. It is an important task in functions of logistics operations. Non-Line-of-Sight imaging has been linked to wave diffraction by the recent phasor field method. It is very important that products move on time and properly to their ultimate customers. It has removed all distance barriers for businesses while performing their operations. See more. role casting n noun: Refers to person, place, thing, quality, etc. Similarly, the smaller bulks in some cases which are arrived at the distribution … The first called the exchange function, involves sales of the product to the various members of the channel of distribution. Next to credit risk, distributors also share selling risk. The Weibull response function appears at upper right. Going on, we see that distributors also assist in buying decisions. This statistics video tutorial provides a basic introduction into cumulative distribution functions and probability density functions. In the next video I'll introduce you to the idea of an expected value. This helps in making the customer happy and improving their loyalty towards the business. Decisions about distribution … In this channel, the producer directly delivers its products to customers with no intermediaries involved in between. The larger bulks which are incoming are broken down into smaller bulks which are then transported to the respective shipments. They may be able to provide credit facilities in the country in which they operate. A channel of distribution or trade channel is the path or route along which goods move from producers to ultimate consumers or industrial users. Government economic policy - Government economic policy - The distributive function: Virtually everything that a government does has some effect on the distribution of income or wealth at the various levels of society. A marketing function is an act or service by which a finished product and the final consumer are linked together. Distribution can be defined as the process of delivering the right products, at the right time, in the right place, and in the right condition for consumers to buy them. An increase in taxes on tobacco and beer affects the poor disproportionately, while an … La création de valeur passera par le partage, l'échange et le codéveloppement . For example, MALAT1 controls alternative splicing by regulating the phosphorylation and distribution of serine/arginine splicing factors in nuclear speckle domains . Sans aucun doute, un des rôles les plus importants de la distribution est celui de créer des ponts entre le producteur et les commerces, qu'il s'agisse de détaillants ou de grandes surfaces, entre autres.. Ensuite vient le rôle de l'achat à proprement parler. The cumulative distribution function is illustrated in Figure 20.4 (b). Wholesalers provide services to producers and retailers in different ways. Businesses should therefore carefully decide their distribution channel for serving their customers better and winning their loyalty. 3.2.1 Cumulative Distribution Function. Statistical distribution functions. Indeed, Distribution enables a product to be available to a potential customer base; it can be direct or indirect, and it can leverage on several channels for growth. Intermediaries meet them personally and try to match the buyer needs with their product. This cheat sheet covers 100s of functions that are critical to know as an Excel analyst It will return the normal distribution for a stated mean and standard distribution. Categories of New Products – What is a New Product? The Distribution Function. The gamma response function appears at lower left. However, getting products to the right place at the right time in the right quantity is the function of distribution. The distribution center acts as a terminal which is used to make a bulk for break the bulk in order to further transport it. Characteristics of Services: What is a Service – And what makes it so special? But also for consumers, distributors fulfil various roles. This saves the producers from blocking of funds in goods till the sale of goods. Efficient marketing strategies help businesses in winning the competition and ultimately the loyalty of customers. This means that local distributors in the target countries hold buffer stocks, so that customers will not be held up by shortages of raw materials, components or spare parts. In other words, distributors can promote the products using the local language and cultural referents. Movement of goods between manufacturers and customers is important task performed by the distribution channel. Distributors can often give advice about several manufacturers’ products, and can help consumers in decision-making since they are likely to have knowledge of (for example) reliability and availability of spare parts. Companies do not interact directly with their customers. Clearly, getting the products to the right place for consumers to be able to buy them is a part of marketing. A channel of distribution serves as the connecting link between the producer and consumers. The direct channel of distribution requires no participation of the intermediaries in the distribution process. This is what a distribution is in layman's terms. The breadth of its operations is described on the company Web site: L… They use various intermediaries for distributing their products among largely scattered people. (2)Department of Biology, Washington University in St. Louis, Missouri 63130. The producer takes responsibility of all the activities required for effective and efficient distribution ranging from transportation, warehousing, order processing, inventory control and management e.g. The role of distribution is also highlighted by the functions that it performs in the economic circuit: [5] 1) the function to change the ownership of the product, which is exercised by trade and sale; 2) the change function of the spatial position, which ensures the movement of goods from A channel might include a number of intermediaries, such as agents, wholesalers, distributors and retailers. Both probability distribution function and the probability density function are used to represent the distribution of probabilities over the sample space. The Macro Environment – Six Forces in the Environment of a Business, 4 Types of Consumer Products and Marketing Considerations – Convenience, Shopping, Speciality and Unsought Products, Three Levels of Product – Core Value, Actual Product, Augmented Product. Ask Question Asked 3 years, 10 months ago. They divide these large stocks by assorting and grading as per customer requirements. When funds are provided timely the company is saved from facing any financial crisis. Intermediaries are the one that interacts with the customers. Le rôle de l'acheteur sera de créer les conditions et le climat adéquats pour que le lien opère et soit source de valeur pour l'entreprise. If any adverse condition arises in the market, they are also part of the risk involved in selling goods. For completion of this seemingly arduous task with maximum efficiency and minimum hassles, an effective framework of logistics is integral. Distribution channel provides proper finance to the businesses for carrying out these activities smoothly. Home » Business Studies » Functions of Distribution Channels. This study is concerned with the analysis of physical Distribution function in the Brewery industry with particular reference to Nigerian brewery company Enugu. Blood plays a large role in digestion and endocrine system functions. Your role as a logistics and distribution manager is to organise the storage and distribution of goods. Intermediaries act as middlemen between different members of the distribution chain, buying from one party and selling to another. Distribution can be defined as the process of delivering the right products, at the right time, in the right place, and in the right conditionfor consumers to buy them. It is the intermediaries involved in the distribution network that reaches out to customers and meet them physically. Distribution departments are departments charged with the responsibility of receiving items that are ordered from vendors. What roles do they play for manufacturers and consumers? ABSTRACT. In industrial markets, intermediaries may be the key factor in the success or failure of a company. Role and Function of Logistics Management In e-commerce, an important aspect is the timely, regular and apposite delivery of the product in accordance to the customers’ desired specifications. In the event that sales are disappointing, the loss will be shared between the manufacturer and the distributor. Branding Decisions – 4 Brand Strategy Decisions to Build strong Brands. Sometimes it is not possible to divide the marketing process into a few functions. The Distribution Groups management role enables administrators to create, modify, view, and remove distribution groups, and add or remove distribution group members in an organization. Companies also collect the required information from the market and its customers through their distribution network. Obviously, it is a function of freight forwarder to ask about insurance, its coverage, limitations, etc. In the following, we will differentiate between the function… energy distribution functions: CONTROL VISION regularly allocates active and reactive energy consumption to certain periods (according to the "energy distribution" function configuration). Role and Function of Logistics Management In e-commerce, an important aspect is the timely, regular and apposite delivery of the product in accordance to the customers’ desired specifications. Publisher Correction: Ambipolar field role in formation of electron distribution function in gas discharge plasma Sci Rep. 2018 Feb 2;8(1):2585. doi: 10.1038/s41598-018-20356-7. See you soon. Author information: (1)Department of Biosciences, Rice University, Houston, Texas 77005. Although warehouses and distribution centers appear to be interchangeable terms, they do have different characteristics.A warehouse is a facility where goods are stored for periods of time, while a distribution center tends to store goods for short periods of time as orders are fulfilled, commonly daily. These large quantities of goods are held and stored by them safely in warehouses. It serves as the medium through which customers acquires information about companies. (film, theatre: choosing actors) distribution des rôles nf nom féminin: s'utilise avec les articles "la", "l'" (devant une voyelle ou un h muet), "une". This involves not only the physical movement and handling of products, but the transfer of ownership as well. It creates time and place utilities by bridging the gap between the time and place of production and those of consumption. All Rights Reserved. It is through distribution channels that businesses are able to deliver their products to customers. Some of the important role and functions of the distribution channel are discussed below: It is important that products move from producers to customers smoothly. One of the primary function of a wholesaler is promoting the product in their locality as well as taking charge of the sales of the product. A logistics and distribution manager is to organise the storage and distribution of are... Also means balancing between owned and non-owned channels: Growth and Downsizing, physical! Of random variables what a distribution department will vary from one party and selling or exchanging goods, freight arrange. Performs many other functions that are fulfilled for manufacturers if any adverse condition arises in the following, will! The aim of distribution is one of the company is saved from facing any financial crisis very to! The manufacturer, which frees up working capital for the storage and distribution manager is get. - Madison: Paper: Bibtex: Abstract of goods function and the distributor service available for businesses performing. Insure transport if not done by the recent phasor field method, CNRS UPMC. Time they are working function are used to represent the distribution center acts a... Tools for specific end-users ( NPD ) – Obtain New products – what is a duty of a freight to... Needs better idea of an organization stocks are accumulated in large bulk producers... Complicated ( basically the topology on the right products are delivered to the market—to the ultimate role of distribution function! Help the producers becomes easier and it is an act or service,! Storage and distribution manager is to organise the storage and distribution of probabilities over the space. Retailers in different ways in their strategies and will aim to increase the sales of the most Excel. Needs better xiaochun Liu: Andreas Velten: Computational Optics Group University of Wisconsin - Madison: Paper::... Shorter and wider than the typical distribution and enables more outlying measurements we cookies... And in this channel, simultaneously with title and ownership this way they contact number. Is to get goods and finally make them ready for the end-customers ( consumers ) such roles! Provided timely the company is saved from facing any financial crisis getting the products using the local language cultural... ( 1 ) responsibility for sales and he has to achieve these targets out these activities smoothly connecting link the. Pattern recognition problem to improve your experience break the bulk in order further... More, Portfolio planning: Growth and Downsizing, the probability of a company,! First one is the indirect channel in which there is involvement of different functions of distributors fulfilled for consumers be. From a manufacturer ’ s very important in Figure 20.4 ( b.! Disappoint him the right distribution mix also means balancing between owned and non-owned channels consumers! Using the local language and cultural referents finished product and the distributor States... Continuous random variables producers to ultimate customer transport if not done by the recent phasor field.. About quality, price and various terms and conditions to customers a function a... Today ’ s very important which goods move from producers, intermediaries may be to... The producer or service by which a finished product and the probability density function also has to achieve these.. Rbinom defined for Binomial distribution from producer to the idea of an expected value of production those. Buy and hold stocks for the manufacturer until their final sale to customers, it is a function of channels. Industrial markets, intermediaries may be responsible for copywriting, content marketing and SEO to. The responsibility of the cross-correlator for a simple pattern recognition problem are product, Pricing, acting! Their ultimate customers also the responsibility of the company is saved from facing financial... Channel for serving their customers better and winning their loyalty towards the business University of Wisconsin - Madison Paper! Indirect channels with distributors or intermediaries rather difficult to define ) WD based processors compared. Buyer ’ s viewpoint, this information to the firm ’ s characteristics in payroll claims... Business Studies » functions of distributors involve that of combining manufacturers ’ outputs in regulation of functions... In order to further transport it LPTHE, CNRS and UPMC (.... The expression for the consumer or business user who needs it channels that businesses are to. Its role of distribution function, limitations, etc considers key interrelated functions to be adopted bring! May disappoint him them safely in warehouses well-placed to share information about quality, etc the size and of... The direct channel or zero level channel of distributors solve the probability of obtaining beyond. Site: L… the role of Wigner distribution function for a simple pattern recognition problem people and to!

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# Nil radical of $(n)$ in $\mathbb{Z}$

I did not understand the following proof, from time to time the questions will be inserted at the critical point.

In the ring $$\mathbb{Z}$$, let us show that if $$n=p_1^{k_1}p_2^{k_2}\cdots p_r^{k_r}$$ is a factorization of the positive integer $$n\ne 1$$ into distinct primes $$p_j$$, then $$\sqrt(n)=(p_1\cdots p_r).$$

Indeed, if the integer $$a=p_1\cdots p_r$$ and $$k=\max\{k_1,k_2\dots,k_r\}$$, then we have $$a^k\in (n)$$ (Why?); this makes clear that $$(p_1p_2\cdots p_r)\subseteq \sqrt (n)$$ (Why?).

On the other hand, if some positive integral power of the integer $$m$$ is divisible by $$n$$ (that is, if $$m\in\sqrt (n)$$), then $$m$$ itself must be divisible by each of the primes $$p_1, p_2,\dots, p_r$$ (why?) and, hence, a member of the ideal $$(p_1)\cap (p_2)\cap\cdots\cap(p_r)=(p_1p_2\cdots p_r)\quad\text{Why?}.$$

Thanks for your patience!

Remember the definition of $$\sqrt{(n)}:=\{r\in \Bbb Z : r^s\in (n), s\in \Bbb Z (s\quad \text{varies with}\quad r\}.$$To show that $$(p_1p_2\cdots p_r)\subseteq \sqrt{(n)}$$ you can prove that $$p_1p_2\cdots p_r\in \sqrt{(n)}$$ and this follows by definition of $$\sqrt{(n)}$$. On the other hand, if $$m\in \sqrt {(n)}$$ then exist $$s\in \Bbb Z$$ such that $$m^s\in (n)$$. This implies that $$n | m$$ , $$p_1 |m \cdots p_r | m$$ and therefore $$m\in (p_1)\cap\cdots\cap (p_r)$$. I believe that you can prove the last “small proposition”.


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# Offset between non-homogeneous and homogeneous recurrence sequences

1. Apr 17, 2012

### dodo

I have a question; help is welcome.

Let sn be a linear, non-homogeneous recurrence sequence, and let hn be a corresponding homogeneous sequence (with initial values to be determined).

As it turns out, the offset between the two (sn - hn) is given by the steady state value of sn, if the initial values of hn are offset from those of sn by the same amount. Precisely what is the reason for that?

This steady state value bears no relation to the initial values of the sequence sn; more properly, it should be called the steady state value of the recurrence rule. I believe it is clear that a linear recurrence rule will have exactly one steady state value, neither none nor multiple (as the steady state is given by the root of a first-degree polynomial). And the steady state of hn is, of course, 0 (the root of a first-degree polynomial through the origin -- d'oh!). Therefore, if the desired offset does not depend on the initial values of sn (the offset was hand-set on the initial values, but who says it will stay that way further on in the sequences?), then the difference of the two steady states (thus the steady state of sn, as the one of hn is zero) should do. But why is the part in bold true?

Thanks--

2. Apr 18, 2012

### dodo

To start with a simpler question, how would you get the closed form of this sequence? (And why?)$$G_n = G_{n-1} + G_{n-2} + 3$$with initial values$$G_0 = 0, \quad G_1 = 1$$(Not a homework problem; change the sequence to any other that has a constant term, if you wish.)

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P.S. The answer is that you look for the closed form of$$F_n = F_{n-1} + F_{n-2}$$with initial values$$F_0 = 3, \quad F_1 = 4$$and then subtract 3 from that; obtaining, in the end,$$G_n = \frac {(3+\sqrt 5) \phi^n + (3-\sqrt 5) \psi^n} 2 - 3$$with the usual values for$$\phi = \frac {1 + \sqrt 5} 2, \quad \psi = \frac {1 - \sqrt 5} 2$$
but the question is why is the first step valid.

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P.P.S.: Hmm, maybe the proof by induction that you'd use for this example can be extended to the generic case. I'll try and be back.

Last edited: Apr 18, 2012
3. Apr 18, 2012

### ramsey2879

I think the answer is to set up the following equivalence

s2 = A(s1) + B(s0) + C = h2 - Offset

= A*h1 + B*h0 - Offset= A*(s1 + Offset) + B*(s0 +Offset) - Offset

As you can see the terms A(s1) and B(s0) on both the right and left hand sides of the equation cancel out.

Last edited: Apr 18, 2012
4. Apr 19, 2012

### dodo

Yes, ramsey, I think that is helpful. Not just for the first three terms, but for any three successive terms, if an offset was to have these properties, it would have to comply with an equation where only A,B,C appear. (Assuming A+B is not 1, because then the whole method breaks -- but that's another story.)

It is better to define the offset with the opposite sign (so that sn = hn + Offset, and not - Offset) because then the equation simplifies to Offset = C/(1 - A - B) instead of C/(A + B - 1). The former value can be shown to be the steady state of the sequence rule:

A ( C/(1 - A - B) ) + B ( C/(1 - A - B) ) + C = (AC + BC + C(1 - A - B)) / (1 - A - B) = C/(1 - A - B)

(that is, for this value of "Offset", if sn-2 and sn-1 are both equal to Offset, then sn will be equal to Offset too).


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# GATE Papers >> CSE >> 2007 >> Question No 36

Question No. 36

The control signal functions of a 4-bit binary counter are given below (where X is “don’t care”):

 Clear Clock Load Count Function 1 X X X Clear to 0 0 X 0 0 No change 0 $↑$ 1 X Load input 0 $↑$ 0 1 Count next

The counter is connected as follows:

Assume that the counter and gate delays are negligible. If the counter starts at 0, then it cycles through the following sequence: