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# Distance sampling simulation from Buckland et al. (2015)

Description and R code to produce simulation described in Sections 3.5.2 and 11.1.1 in Buckland et al. (2015); demonstrating the use of the package DSsim to evaluate subjective vs random placement of transects.

Laura Marshall http://distancesampling.org (CREEM, Univ of St Andrews)https://creem.st-andrews.ac.uk
2020-04-10

This case study shows you how to use the R package DSsim (Marshall, 2019) to compare the performance of different survey designs. We will replicate the analyses of Sections 3.5.2 and 11.1.1 of (Buckland, Rexstad, Marques, & Oedekoven, 2015).

# Getting started

Ensure you have administrator privileges on your computer and install the necessary R packages.

needed.packages <- c("DSsim", "mrds", "shapefiles", "splancs")
myrepo <- "http://cran.rstudio.com"
install.packages(needed.packages, repos = myrepo)

## Directory structure for files in this project

In addition to the R packages, there are additional files required by this analysis. All necessary material are included in a Zip archive file entitled DSsim_study.zip. When that archive is uncompressed, the directory structure will be created as described.

Examine the other files and folders in the “DSsim_study” folder. There are three files starting with the name “Region” and ending with .dbf, .shp and .shx, these files make up the shapefile for the survey region. The density.surface.robj file is the density surface for the survey region. The Survey Transects folder contains a folder for each of the designs you are asked to look at, these contain the transect shapefiles. The Results folder contains the results from 999 replications as this can take a good few hours to run. Select the directory that contains these files.

## Alternative analysis using Distance for Windows

Some parts of this exercise can also be performed using Distance for Windows (Thomas et al., 2010). That software cannot recreate the subjective survey design (see below), but can assess the performance of other survey designs described here. The Zip archive DSsimExercise.zip can be downloaded to your computer and opened with Distance for Windows (version 7.3 and above).

# Creating a new simulation

## Creating a region object

Read the Region shapefile into R.

library(shapefiles)
region.shapefile <- read.shapefile("DSsim_study/Region")

Next create the region object using this shapefile. As there are no strata in this example, you just need to provide a name for your survey region and specify the units (metres, m).

region <- make.region(region.name = "Survey Region", units = "m", shapefile = region.shapefile)

View the resulting object:

plot(region, plot.units = "km")

## Creating a density object

Now create a density object within this region. For this study, a density surface has already been created, but you can experiment with the options in the next code chunk to define one yourself.

You can create other density surfaces by creating a density object based on a uniform density grid over the area, and adding some hot spots (or low spots).

density <- make.density(region = region, x.space = 1000, y.space = 1000,
constant = 4e-08)
density <- add.hotspot(density, centre = c(-2500, 2224000), sigma = 10000,
amplitude = 1e-08)
density <- add.hotspot(density, centre = c(0, 2184000), sigma = 18000,
amplitude = -5e-09)

Load the density surface and view the data that comprise the surface.

kable(density.surface[[1]][1:5, ])
x y density
17071.09 2152699 1e-07
18071.09 2152699 1e-07
15071.09 2153699 1e-07
16071.09 2153699 1e-07
17071.09 2153699 1e-07

The density surface is a data set of x and y locations and corresponding densities at each point.

To create the density object, you need to provide the density surface, the region object for which it was created, and the grid spacing that was used. I used a grid spacing of 1000m in both the x and y directions to create this density surface.

pop.density <- make.density(region = region, density.surface = density.surface,
x.space = 1000, y.space = 1000)

View the density surface.

plot(pop.density, plot.units = "km")
plot(region, add = TRUE)

### Population size

Fix the population size at 1500 individuals. Using make.population.description set the first argument to the desired population abundance and set the second argument such that exactly this number of individuals is generated in each realisation (fixed.N = TRUE).

pop.description <- make.population.description(region.obj = region, density.obj = pop.density,
N = 1500, fixed.N = TRUE)

### True detection function

We select a half-normal detection function with a $$\sigma$$ (scale.param) of 500m and a truncation distance of 1000m.

detect <- make.detectability(key.function = "hn", scale.param = 500, truncation = 1000)

## Creating the survey design object

We first consider the subjective design (Section 11.1.1 of (Buckland et al., 2015)). The subjective design uses existing paths together with additional transects chosen to achieve a more even coverage of the survey region.

NOTE: you will need to edit the path argument to describe where the files are on your machine. Look back to the path you found when you were setting up and add “/Survey Transects/Subjective Design”.

new.directory <- paste(getwd(), "DSsim_study/Survey_Transects/Subjective_Design",
sep = "/")
subjective.design <- make.design(transect.type = "Line", design.details = c("user specified"),
region = region, plus.sampling = FALSE, path = new.directory)
subjective.design@filenames

[1] "New_Des1"

Previous code not only constructs the subjective design, but prints out the names of the shapefiles that will be used in the simulation. This is a check that the working directory has been set properly. The command subjective.design@filenames is a check to make sure the correct folder has been identified. This will tell you the names of the shapefiles the simulation will use.

## Creating the analyses object

Describe the analyses to carry out on the simulated data. Here we propose both half-normal and hazard-rate models (neither with covariates) and choose between them based on the AIC values.

ddf.analyses <- make.ddf.analysis.list(dsmodel = list(~cds(key = "hn",
formula = ~1), ~cds(key = "hr", formula = ~1)), method = "ds", criteria = "AIC")

## Creating the simulation object

We can finally put it all together and have a look at some example populations, transects and survey data. Set the number of repetitions (reps) to be fairly low initially to avoid long run times. For the subjective design, specify that the same set of transects is to be used in each replicate, single.transect.set = TRUE.

my.simulation.subjective <- make.simulation(reps = 10, single.transect.set = TRUE,
region.obj = region, design.obj = subjective.design, population.description.obj = pop.description,
detectability.obj = detect, ddf.analyses.list = ddf.analyses)

Before running the simulation, you can check to see that it is doing what you want. These commands will allow you to investigate the simulation properties.

# set the display window up for 4 plots
par(mfrow = c(2, 2))
# generate and plot and example population
pop <- generate.population(my.simulation.subjective)
plot(region)
par(cex = 0.3)
plot(pop)
# generate (or rather load from file) the transects
transects <- generate.transects(my.simulation.subjective)
plot(region)
plot(transects, col = 4, lwd = 2)
# simulate the survey process of detection
eg.survey <- create.survey.results(my.simulation.subjective)
plot(eg.survey)
par(cex = 1)
# have a look at the distance data from the simulated survey
dist.data <- get.distance.data(eg.survey)
hist(dist.data$distance, xlab = "Distance (m)", main = "Distance Data")  par(mfrow = c(1, 1)) If the previous plots lead you to believe you have properly parameterised your simulation, it is time to run it. Be patient, as it will take a few minutes to complete, even though there are only 10 replicates.  my.simulation.subjective.run <- run(my.simulation.subjective) Information about effort, estimated abundance, estimated density and detection probability $$(P_a)$$ is available in the resulting object. Results from each replicate for abundance estimation can be examined, such as  kable(t(my.simulation.subjective.run@results$individuals$N[, , 5]), digits = 3, caption = "Estimate of abundance (with measures of precision) for the fifth replicate of the simulation performed above.") Table 1: Estimate of abundance (with measures of precision) for the fifth replicate of the simulation performed above. Estimate se cv lcl ucl df 1855.342 586.617 0.316 957.703 3594.32 14.174 or average abundance estimates across replicates  kable(t(my.simulation.subjective.run@results$individuals$N[, , 11]), digits = 3, caption = "Average estimate of abundance (with measures of precision) across all replicates of the simulation performed above.") Table 2: Average estimate of abundance (with measures of precision) across all replicates of the simulation performed above. Estimate se cv lcl ucl df 1456.045 631.259 0.49 629.711 3641.815 14.177 There is also a summary() function for simulation results objects that provide more exhaustive results. # On to randomised designs You will need to create 2 new simulations each with a new design object, one for the parallel design and one for the zigzag design. The other objects (region, density, population description etc.) should not be changed. Here is how to do it for the parallel design (path directed to folder containing shape files associated with a parallel transect survey design). Note: We now wish different transects to be used on each repetition (single.transect.set = FALSE).  parallel.directory <- paste(getwd(), "DSsim_study/Survey_Transects/Parallel_Design", sep = "/") parallel.design <- make.design(transect.type = "Line", design.details = c("Parallel", "Systematic"), region.obj = region, design.axis = 45, spacing = 12000, plus.sampling = FALSE, path = parallel.directory) my.simulation.parallel <- make.simulation(reps = 10, single.transect.set = FALSE, region.obj = region, design.obj = parallel.design, population.description.obj = pop.description, detectability.obj = detect, ddf.analyses.list = ddf.analyses) The code does not complete the process of analysing and summarising the simulation of systematic parallel line transects. The user can complete this investigation. ## Contrast bias and precision from subjective and randomised surveys More than 10 replicates are needed to adequately address the question of bias. Results of a more extensive simulation study are stored as workspace objects. The following code loads these results.  load("DSsim_study/Results/simulation.subjective.robj") load("DSsim_study/Results/simulation.parallel.robj") load("DSsim_study/Results/simulation.zigzag.robj") We can extract components from these R objects to approximate the findings presented in Table 2.1, page 29 of (Buckland et al., 2015).  extract.metrics <- function(simobj) { nreps <- simobj@reps data <- simobj@results$individuals$summary effort.avg <- data[, , nreps + 1][3]/1000 detects.avg <- data[, , nreps + 1][4] abund <- simobj@results$individuals\$N
abund.avg <- abund[, , nreps + 1][1]
true.N <- simobj@population.description@N
est.bias <- (abund.avg - true.N)/true.N
est.bias.pct <- est.bias * 100
mean.stderr <- abund[, , nreps + 1][2]
stddev <- abund[, , nreps + 2][1]
result.vector <- unname(c(effort.avg, detects.avg, abund.avg, est.bias.pct,
mean.stderr, stddev))
return(result.vector)
}
subjective <- extract.metrics(simulation.subjective)
parallel <- extract.metrics(simulation.parallel)
zigzag <- extract.metrics(simulation.zigzag)

result.table <- data.frame(subjective, parallel, zigzag, row.names = c("Mean effort(km)",
"Mean sample size", "Mean abundance estimate", "Estimated percent bias",
"Mean SE of abundance estimate", "SD of abundance estimates"))
kable(result.table, digits = 1, caption = "Simulation results summary for each design.  Estimates based upon 999 simulations with true abundance of 1500 objects.")
Table 3: Simulation results summary for each design. Estimates based upon 999 simulations with true abundance of 1500 objects.
subjective parallel zigzag
Mean effort(km) 336.8 473.5 694.6
Mean sample size 85.0 146.5 215.2
Mean abundance estimate 1214.5 1484.5 1480.2
Estimated percent bias -19.0 -1.0 -1.3
Mean SE of abundance estimate 283.4 216.2 173.6
SD of abundance estimates 225.0 193.9 154.9

Buckland, S. T., Rexstad, E. A., Marques, T. A., & Oedekoven, C. S. (2015). Distance sampling: Methods and applications. Retrieved from https://www.springer.com/gp/book/9783319192185

Marshall, L. (2019). DSsim: Distance sampling simulations. Retrieved from https://CRAN.R-project.org/package=DSsim

Thomas, L., Buckland, S. T., Rexstad, E. A., Laake, J. L., Strindberg, S., Hedley, S. L., … Burnham, K. P. (2010). Distance software: Design and analysis of distance sampling surveys for estimating population size. Journal of Applied Ecology, 47(1), 5–14. Retrieved from https://doi.org/10.1111/j.1365-2664.2009.01737.x


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# p-value of multivariate response in partykit::ctree

I wonder if anyone can help me to understand the two questions regarding partykit::ctree:

1. what's the difference between "quadratic" and "maximum" in ctree_control(teststat = c("quadratic", "maximum"))? When should I use one vs the other?

2. For multivariate response, what's the meaning of p-value at each splitting node, separately for regression and classification? Here is one example for classification:

library(partykit)
airq <- subset(airquality, !is.na(Ozone))
airct2 <- ctree(Ozone + Temp ~ ., data = airq)
airct2
plot(airct2)

$$$$

As for the question "quadratic" vs. "maximum"`: In the case with two numeric response variables, the former will typically be more powerful if the mean of both response variables differs for each split found in the tree. In contrast, if there are many splits where only one or the other mean of the responses changes but the other mean remains constant, the maximum test may have somewhat better power. Of course, in practice it is typically unkown (before fitting the tree) which situation you are in.

format: plain


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# Stability region of explicit midpoint method

Consider the explicit midpoint method, i.e $$y_{n+1}-y_{n-1} = 2hf(y_n).$$

I'm asked to apply this method to the linear test equation, $$f(y_n) = \lambda y_n,$$ in order to find the method's stability region.

So I have $$y_{n+1}-y_{n-1} = 2h\lambda y_n.$$

In the problems I've done before I wouldn't have the $$y_{n-1}$$ term so I could just easily write $$y_{n+1} = R(z)y_n.$$ How would I solve this?

## 1 Answer

You have written down the midpoint rule as a two-step method, a member of the family of multi-step methods. For these methods, one can show that a multi-step method $$\alpha_{k} y_{n+k} + \alpha_{k-1} y_{n+k-1} + \ldots + \alpha_{0} y_{n} = h\left( \beta_{k} f_{n+k} + \ldots + \beta_{0} f_{n}\right)$$ is called stable if the polynomial $$\rho(z) = \alpha_{k} z^{k} + \alpha_{k-1} z^{k-1} + \ldots + \alpha_{0}$$ has only roots lying on or within the unit circle and that roots lying on the unit circle are simple roots.

So it is up to you to demonstrate for which stepsize $$h$$ the roots of the polynomial $$\rho(z) = z^{2} - 2h z - z$$ satisfy the above criterion.

The other approach would be as you hint through the test equation. And then, you should indeed re-write the method as a single-step method:$$y_{n+1} = y_{n} + h\lambda y_{n+\frac{1}{2}} = y_{n} + h\lambda \left(y_{n} + \frac{1}{2}h \lambda y_{n}\right) = y_n \left(1+h\lambda + \frac{1}{2}\left(h\lambda\right)^{2}\right)$$ and this gives you your $$R(z)$$ function as $$R(z) = 1+h\lambda + \frac{1}{2}\left(h\lambda\right)^{2}$$ so your stability criterion in $$(x,y)$$ form says: $$|1+x+\frac{x^2-y^2}{2}| \leq 1 \ \text{and}\ |y+xy| \leq 1$$

Solving this gives you a graph like below.


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Browse Questions

# Choose the correct answer in the slope of the normal to the curve $y = 2x^2 + 3 sin\: x\; at\;$ $x = 0$ is

$(A)\; 3 \quad (B)\;\frac{1}{3} \quad (C)\; -3 \quad (D)\; -\frac{1}{3}$

Can you answer this question?

Toolbox:
• If a tangent line to the curve $y=f(x)$ makes an angle $\theta$ with $x$-axis in the +ve direction,then $\large\frac{dy}{dx}$=slope of the tangent=$\tan \theta$
• Slope of the normal is $\large\frac{-1}{\Large\frac{dy}{dx}}$
Step 1:
Equation of the curve is $y=2x^2+3\sin x$
Differentiating w.r.t $x$ we get,
$\large\frac{dy}{dx}$$=4x+3\cos x Hence the slope of the tangent at x=0 \large\frac{dy}{dx}_{(x=0)}$$=4(0)+3\cos 0$
But $\cos 0=1$
$\qquad\quad\;\;=3\times 1=3$
Step 2:
Hence the slope of the tangent at $x=0$ is $3$
Therefore the slope of the normal is $\large\frac{-1}{3}$
The correct option is $D$
answered Jul 12, 2013


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# $H_\infty$ control design

In this tutorial, we will design a controller for a DC-servo (electrical motor).

The servo takes a torque command as input and we are interested in controlling the angle of the output shaft. An approximate model for a DC servo, valid for low frequencies, is

$$$\tau = k ( J \ddot{\phi} + c\dot{\phi} )$$$

where $\tau$ is the torque and $\phi$ is the angle. The parameter $J$ denotes the inertia of the motor and load, $c$ is a viscous friction parameter and $k$ is a torque constant (gain).

This is a simple SISO example with a pole in the origin. Poles on the stability boundary are problematic for several numerical routines, and this tutorial demonstrates a common trick applied in these situations.

We start by defining the process model. We will use the parameter values $J=1, c=0.12, k=11.2$ which yields the transfer function

$$$G(s) = \dfrac{k}{s(Js + c)} = \dfrac{11.2}{s^2 + 0.12s + 0}$$$
using JuliaSimControls, Plots
Gtrue = tf([11.2], [1, 0.12, 0])

When designing a controller using $H_\infty$ synthesis, we formally specify an optimization problem where the cost function is the $H_\infty$ norm of a suitably chosen transfer function, and the optimization variable is the controller. In this example we will optimize

$$$\operatorname{minimize}_K \begin{Vmatrix} W_S S \\ W_U KS \end{Vmatrix}_\infty$$$

where $K$ is the controller and $S$ is the sensitivity function $(1+GK)^{-1}$. The transfer functions $W_S$ and $W_U$ are weight functions that emphasize different frequency ranges, for example, if $W_S(i\omega)$ is large for a particular frequency $\omega$, then $S$ is forced to be small at $\omega$ in order for the $H_\infty$ norm to be small.

# Sensitivity weight function
WS = makeweight(1e5, 0.05, 0.5) |> tf

# Output sensitivity weight function. Increase this value to penalize controller effort more
WU = ss(1.0)

# Complementary sensitivity weight function
WT = [] # We do not put any weight on T in this example

To solve this problem, we partition the system $P$ in a two-input, two output configuration such that $\operatorname{lft}_{l}(P,K)$ forms the system we want to minimize the $H_\infty$ of. To help with this partitioning, we have the function hinfpartition:

P = hinfpartition(Gtrue, WS, WU, WT)

Before solving, we may check if the synthesis problem is feasible

hinfassumptions(P, verbose=true)
false

The problem is not feasible, in this case due to the integrator pole on the stability margin. We thus modify the plant description by moving the integrator pole in the origin slightly ($\vareplsilon$) into the stable region

ε = 1e-4
G = tf([11.2], [1, 0.12]) * tf([1], [1, ε])

If we now perform the assumption check again, it passes

P = hinfpartition(G, WS, WU, WT)
hinfassumptions(P)
true

## Synthesize the H-infinity optimal controller

With the problem properly defined, we may call hinfsyn_lmi to solve it. The result is a controller K and a performance index $\gamma$:

K, γ = hinfsyn_lmi(P, γrel=1.05, ftype=BigFloat)
γ
3.1460364963313867

The achieved performance level is indicated by $\gamma$, this number is the norm we are optimizing and it should be as low as possible.

For verification purposes, we may extract some transfer functions defining common sensitivity functions:

Pcl, S, KS, T = hinfsignals(P, G, K)

We may also verify that the closed-loop system has $H_\infty$ norm $\gamma$

Pcl == lft(P, K)
isapprox(hinfnorm2(Pcl)[1], γ, rtol=1e-5)
true

## Plot the specifications

The resulting sensitivity functions can be plotted together with the inverse weighting functions multiplied by $\gamma$ to get a feeling for where the constraints are active.

specificationplot([S, KS, T], [WS, WU, WT], γ)

In this case, the noise amplification constraint is active between about $10^0 - 10^{1.5}$ rad/s and the sensitivity function is pushed down for lower frequencies. In this case, the complimentary sensitivity function $T = (1+GK)^{-1}GK$ has desireable properties without us specifying and penalizing this function in the optimization problem. If we would like to push this function down at certain frequencies, we may specify a non-empty weight $W_T$ as well.


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# Math Help - vectors

1. ## vectors

if you have 3 vectors: them being (x,1,3)
(0,1,2)
(x,1,x)

how do you find the values of x to which you cant form a base of R^3.

doesn't even make sense at least i dont think it does.

2. The determinate cannot be zero.
$\left| {\begin{array}{ccc} x & 1 & 3 \\ 0 & 1 & 2 \\ x & 1 & x \\
\end{array}} \right| = x^2 - 3x$

What values of x gives a determinate of zero?


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## August 19, 2017

### Simplicial Sets vs. Simplicial Complexes

#### Posted by John Baez

I’m looking for a reference. Homotopy theorists love simplicial sets; certain other topologists love simplicial complexes; they are related in various ways, and I’m interested in one such relation.

Let me explain…

There’s a category $SSet$ of simplicial sets and a category $SCpx$ of simplicial complexes.
There is, I believe, a functor

$F: SCpx \to SSet$

that takes the simplices in a simplicial complex, which have unordered vertices, and creates a simplicial set in which the vertices of each simplex are given all possible orderings.

There is a geometric realization functor

$| \cdot | : SSet \to Top$

There’s also a geometric realization functor for simplicial complexes. I’d better give it another name… let’s say

$[ \cdot ] : SCpx \to Top$

Here’s what I want a reference for, assuming it’s true: there’s a natural transformation $\alpha$ from the composite

$SCpx \stackrel{F}{\longrightarrow} SSet \stackrel{|\cdot|}{\longrightarrow} Top$

to

$SCpx \stackrel{[\cdot]}{\longrightarrow} Top$

and for any simplicial complex $X$

$\alpha_X : |F(X)| \to [X]$

is a homotopy equivalence. (I don’t think it’s a homeomorphism; a bunch of simplices need to be squashed down.)

Since this field is fraught with confusion, I’d better say exactly what I mean by some things. By a simplicial set I mean a functor $X : \Delta^{op} \to Set$, and a morphism between simplicial sets to be a natural transformation between them. That seems pretty uncontroversial. By a simplicial complex I mean what Wikipedia calls an abstract simplicial complex, to distinguish them from simplicial complexes that are made of concrete and used as lawn ornaments… or something like that. Namely, I mean a set $X$ equipped with a family $U_X$ of non-empty finite subsets that is downward closed and contains all singletons. By a map of simplicial complexes from $(X,U_X)$ to $(Y,U_Y)$ I mean a map $f: X \to Y$ such that the image of any set in $U_X$ is a set in $U_Y$.

On MathOverflow the topologist Allen Hatcher wrote:

In some areas simplicial sets are far more natural and useful than simplicial complexes, in others the reverse is true. If one drew a Venn diagram of the people using one or the other structure, the intersection might be very small.

Perhaps this cultural divide is reflected in the fact that the usual definition of an abstract simplicial complex involves a set that contains the set of vertices, whose extra members are entirely irrelevant when it comes to defining morphisms! It’s like defining a vector space to consist of two sets, one of which is a vector space in the usual sense and the other of which doesn’t show up in the definition of linear map. Someone categorically included would instead take the approach described on the nLab.

Ultimately you get equivalent categories either way. And as Alex Hoffnung and I explain, the resulting category of simplicial complexes is a quasitopos of concrete sheaves:

A quasitopos has many, but not quite all, of the good properties of a topos. Simplicial sets are a topos of presheaves, so from a category-theoretic viewpoint they’re more tractable than simplicial complexes.

Posted at August 19, 2017 7:48 AM UTC

TrackBack URL for this Entry:   https://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/2978

### Re: Simplicial Sets vs. Simplicial Complexes

The claim is not true. While there is always a map $|F(X)| \to [X]$ that forgets the ordering of vertices, it is almost never a homotopy equivalence. The first counterexample is when $X$ is the 1-simplex! In this case $[X]$ is an interval and $|F(X)|$ is a circle. (The two orderings of the 1-simplex produce two different 1-simplices that combine to form the circle.)

The functor $F$ is much-studied. If $X$ is the $n$-simplex with vertices $0,1,\ldots,n$ then $|F(X)|$ is called the “complex of injective words”. It is homotopy equivalent to a wedge of $n$-spheres, and the number of spheres in the wedge is equal to the number of derangements of the set $\{0,1,...,n\}$. (A derangement is a permutation with no fixed points.)

The complex of injective words is an important tool in proofs of homological stability for the symmetric group and for various types of configuration spaces. That’s why I’m interested in it. It is also of interest in combinatorial topology, though I don’t know why, and there’s apparently also a connection to knot theory. I’ll try to provide references later.

A web search for “complex of injective words” should prove helpful!

Brief moment of pickiness: It is most common to define $F(X)$ as a semi-simplicial set. In this case it is easy to make your definition of $F(X)$ precise: an $n$-simplex of $F(X)$ is an $n$-simplex of $X$, together with an ordering of its vertices, and the $i$-th face of such a simplex is obtained by erasing the $i$-th vertex. If you wanted to do it all simplicially then you would have to talk about degenerate sinplices and so on. Anyhow, the good news is that you could rephrase everything you wrote to be semi-simplicial, and that in the end $|F(X)|$ doesn’t change!

Posted by: Richard Hepworth on August 19, 2017 10:26 AM | Permalink | Reply to this

### Re: Simplicial Sets vs. Simplicial Complexes

In the interest of sending John back to his project with something more hopeful, there should also be a “Simplicial Realization” functor $/\cdot/ : SCpx \to SSet$, with $/X/_n$ all the sequences of vertices in $X$ that happen to be contained in some abstract-simplex of $X$ (ignoring which one, in case there are many). The SCpx 1-cell $\{ \{1\},\{2\},\{1,2\}\}$ has, in $/X/_1$ both $(1,2)$ and $(2,1)$ but also $(1,1)$ and $(2,2)$ and in $/X/_2$ any triangles you like, such as $(1,1,2)$ and $(1,2,1)$ and …

Posted by: Jesse C. McKeown on August 19, 2017 3:19 PM | Permalink | Reply to this

### Re: Simplicial Sets vs. Simplicial Complexes

Jesse wrote:

In the interest of sending John back to his project with something more hopeful, there should also be a “Simplicial Realization” functor $/\cdot/ : SCpx \to SSet$, with $/X/_n$ all the sequences of vertices in $X$ that happen to be contained in some abstract-simplex of $X$ (ignoring which one, in case there are many).

Yes, that’s what I really should have used instead of my hastily described $F$. And if you don’t mind, I’d prefer to call your functor $G: SCpx \to SSet$, since it’s an alternative to my $F: SCpx \to SSet$, and we already have two funky bracket-shaped functors.

And though I said

(I don’t think it’s a homeomorphism; a bunch of simplices need to be squashed down.)

I now believe that when we use this $G$, all the simplices that need to be squashed down, like those triangles you mention, are degenerate already and thus automatically squashed down by $|\cdot| : SSet \to Top$.

So in fact I now believe that the composite

$SCpx \stackrel{G}{\longrightarrow} SSet \stackrel{|\cdot|}{\longrightarrow} Top$

is naturally isomorphic to

$SCpx \stackrel{[\cdot]}{\longrightarrow} Top$

Someone must have figured this stuff out already. Does anyone know a reference?

Posted by: John Baez on August 19, 2017 3:47 PM | Permalink | Reply to this

### Re: Simplicial Sets vs. Simplicial Complexes

It is not true that $|G(X)|$ is naturally isomorphic to $[X]$.

For example, if we take $X=\{\{1\},\{2\},\{1,2\}\}$ as in Jesse’s example, then $G(X)$ contains simplices of the form $(1,2,1,2,1,\ldots)$ that are non-degenerate (they do not have consecutive repeated vertices) and have dimension as large as you like. This means that $|G(X)|$ is infinite-dimensional, unlike $[X]$ which is 1-dimensional.

However, it sounds plausible that $|G(X)|\to [X]$ is a homotopy equivalence. For example, in $|G(X)|$ the 2-simplex $(1,2,1)$ appears as a globular 2-cell with boundary the edges $(1,2)$ and $(2,1)$, so the problematic circle formed by $(1,2)$ and $(2,1)$ can now be contracted.

Posted by: Richard Hepworth on August 19, 2017 4:12 PM | Permalink | Reply to this

### Re: Simplicial Sets vs. Simplicial Complexes

The usual way to get a simplicial set from an abstract simplicial complex is via the “nerve” functor $N\colon \mathrm{SCpx}\to\mathrm{SSet}$, which sends $(X,U_X)$ to the usual nerve of the poset $U_X$. In this case there is a very natural homeomorphism $|N(X)|\approx [X]$, which is easy to write down. Of course, there is no bijective correspondence between simplices of $X$ and non-degenerate simplices of $N(X)$.

Posted by: Charles Rezk on August 19, 2017 3:57 PM | Permalink | Reply to this

### Re: Simplicial Sets vs. Simplicial Complexes

This is very nice, but alas it’s probably not the relation between simplicial sets and simplicial complexes that I need to think about. I think I need the functor called $G$ above, which takes a simplicial complex $X$ and turns it into a simplicial set for which an $n$-simplex is an ordered $(n+1)$-tuple of vertices of a single simplex of $X$. I’m hoping that someone has shown $|G(X)|$ is naturally homotopy equivalent to $[X]$.

I’m trying to relate two pieces of existing work, and I think bringing in the nerve would distract from the way some people are using simplicial complexes where others are using simplicial sets: it seems the simplicial complex people are using $X$ where the simplicial set people would use $G(X)$.

If nobody has ever thought about this functor $G$, I’ll either have to figure it out or wriggle around it with some expository trick.

It’s kinda neat how this thread has unearthed 3 fairly interesting functors from $SCpx$ to $SSet$.

Posted by: John Baez on August 20, 2017 1:45 AM | Permalink | Reply to this

### Re: Simplicial Sets vs. Simplicial Complexes

The classical quick reference for some of this is in the little early survey article by Curtis: Simplicial homotopy theory, Advances in Math. 6 (1971), 107 – 209. He attributes to Barratt a result that the geometric realisation of a simplicial set can be triangulated, and thus is the geometric realisation (in the simplicial complex sense) of a simplicial complex. The proof is sketched in Curtis and involves the double barycentric subdivision of the simplicial set. (The discussion is in the last section of the paper.)

Posted by: Tim Porter on August 20, 2017 6:52 AM | Permalink | Reply to this

### Re: Simplicial Sets vs. Simplicial Complexes

Thanks!

My own puzzle does not involve barycentric subdivisions, but Curtis’ paper (free online from the Evil Empire) is still relevant.

In Example 1.3, Curtis describes a way to get a simplicial set from a simplicial complex. This should be approximately functor I’m calling $G : SCpx \to SSet$ in my conversation with Richard and Jesse above. However, Curtis works with “geometrical” simplicial complexes rather than the “abstract” simplicial complexes I’d prefer to think about. So, I should be careful.

In Section 1.29, Curtis describes the usual geometric realization of a simplicial set $X$, which he calls $R(X)$ rather than the now-standard $|X|$. He then claims that if we take a geometrical simplicial complex $X$, and let $K$ be the simplicial set we get from it by the procedure in Example 1.3, then $R(K)$ is homeomorphic to $X$.

This seems to contradict Richard’s claim (above) that that $R(G(X))$ is not homeomorphic to $X$. Perhaps the distinction between “geometrical” and “abstract” simplicial complexes is a bigger deal than I imagined. Or conceivably Curtis is just wrong, since he doesn’t actually prove his claim. Or maybe Richard is wrong… or more likely, I’m just mired in confusion.

Nonetheless, I feel I’m making progress! So thanks!

Posted by: John Baez on August 20, 2017 7:28 AM | Permalink | Reply to this

### Re: Simplicial Sets vs. Simplicial Complexes

My understanding is that to get from a simplicial complex $X$ to a simplicial set $T(X)$ (I should denote it by something different) in a sensible way you choose a total order on the set of vertices then then the $n$-simplices $T(X)$ will be the $n+1$-tuples $x_0\leq x_1\leq \ldots \leq x_n$. Richard’s $(1,2,1,2,1,2)$ is thus disallowed. (This construction is Curtis’ 1.3 I think.) This is NOT a functor from $SCpx$ to $SSet$ as it requires a choice, but study what happens when one changes the relative position in the total order of two vertices. The study of that situation makes me think that the end results will have homeomorphic realisations and also that there is a specifiable concordance’ in some sense between them, i.e., a simplicial complex structure or triangulation of $|X|\times I$ with one ordering on the top and the other on the bottom of the cylinder. (I used something like this in my papers on Dave Yetter’s TQFT.) In other words, the lack of well-definition of $T$ is controlled.

I will not explore the question the behaviour of $T$ with respect to simplicial maps, but that does need exploring.

Posted by: Tim Porter on August 20, 2017 11:09 AM | Permalink | Reply to this

### Re: Simplicial Sets vs. Simplicial Complexes

Tim wrote approximately:

My understanding is that to get from a simplicial complex $X$ to a simplicial set $K$ in a sensible way you choose a total order on the set of vertices; then the $n$-simplices of $T(X)$ will be the $n+1$-tuples $x_0\leq x_1\leq \ldots \leq x_n$. Richard’s $(1,2,1,2,1,2)$ is thus disallowed. (This construction is Curtis’ 1.3 I think.) This is NOT a functor from $SCpx$ to $SSet$ as it requires a choice.

Brilliant! Yes, this is the construction in Curtis’ Example 1.3, and he starts by randomly choosing a total order on the vertices of $X$, destroying functoriality but eliminating the extra simplices Richard found.

So, the ‘paradox’ that was bothering me seems to be resolved. We seem to be getting this: any simplicial complex $X$ we are finding some simplicial set $K$ whose geometric realization $|K|$ is homeomorphic to the geometrical realization $[X]$.

This procedure is not functorial, but $K$ has the same set of vertices as $X$, and I believe there’s a canonical (meaning: no further choices are required) bijection between the $n$-simplices of $X$ and the nondegenerate $n$-simplices of $K$.

On the other hand, Charles has pointed out a way to construct a functorial way to construct a simplicial set $N(X)$ from $X$ for which $|N(X)|$ is homeomorphic to $[X]$, but which has more vertices, and in general more $n$-simplices.

Ordinarily I’d prefer the functorial construction, but for what I’m doing — trying to connect up to other people’s work — the non-functorial construction may be best.

Posted by: John Baez on August 20, 2017 11:51 AM | Permalink | Reply to this

### Re: Simplicial Sets vs. Simplicial Complexes

If I am not wrong, Charles’ construction gives the barycentric subdivision of the one I mentioned but does not involve a choice.

(Some of this was discussed in a paper by Abels and Holz: Higher generation by subgroups, J. Alg, 160, (1993), 311– 341. It also discusses some applications of these ideas in geometric and combinatorial group theory. These involve calculation of higher syzygies from presentations of subgroups. Great Fun!)

Posted by: Tim Porter on August 20, 2017 4:37 PM | Permalink | Reply to this

### Re: Simplicial Sets vs. Simplicial Complexes

Tim wrote:

If I am not wrong, Charles’ construction gives the barycentric subdivision of the one I mentioned but does not involve a choice.

That sounds right to me.

Posted by: John Baez on August 21, 2017 1:40 AM | Permalink | Reply to this

### Re: Simplicial Sets vs. Simplicial Complexes

Since I said it is easy to describe the homeomorphism $|N(X)|\xrightarrow{\sim} [X]$, I should probably describe it.

For a simplicial complex $(X,U_X)$, the space $|N(X)|$ is a quotient of the space whose points are data:

$(S_0\subseteq \cdots \subseteq S_k, \; t_0,\dots, t_k)$

where $S_i\in U_X$ and $t_i\in [0,1]$ with $\sum t_i=1$. (The identifications are the “obvious” ones: (i) if $t_i=0$ then delete $S_i$ and $t_i$, or (ii) if $S_i=S_{i+1}$ then delete $S_{i+1}$ and replace $t_i,t_{i+1}$ with $t_i+t_{i+1}$. Thus, every point of $|N(X)|$ has a unique “canonical form”, in which all $S_i\neq S_{i+1}$ and all $t_i\neq0$.)

Per nLab, $[X]$ is the space of probability measures on the set $X$ of vertices which are supported on some element of $U_X$.

The function $|N(X)|\to [X]$ sends a point $(\{S_i\}, \{t_i\})$ to the measure $\mu$ defined by

$\mu(A) = \sum_{j=0}^k t_j \frac{|A\cap S_j|}{|S_j|}.$

Conversely, given $\mu\in [X]$ we can recover $(\{S_i\}, \{t_i\})$ as follows. Let

$M_\mu = \{ \mu(\{x\})\; |\; x\in X,\; \mu(\{x\})\gt 0\},$

and list the elements of $M_\mu$ as $1\geq m_0\gt \cdots \gt m_k\gt 0$, with $m_{k+1}=0$. Set

$S_j = \{ x\in X\; | \; \mu(\{x\})\geq m_j \}$

and

$t_j = |S_j|(m_j - m_{j+1}).$

Then it’s easy to see that $(\{S_i\}, \{t_i\}) \mapsto \mu$, and not hard to see that this is the only point of $|N(X)|$ which is sent to $\mu$.

Posted by: Charles Rezk on August 20, 2017 5:58 PM | Permalink | Reply to this

### Re: Simplicial Sets vs. Simplicial Complexes

Nice! It’s neat how a spoonful of probability theory helps the medicine go down. Has anyone used this connection to do anything interesting?

The formula for $\mu(A)$ reminds me of the formula for conditional probability, and now one of the pictures in the Wikipedia article on conditional probability reminds me of something about simplicial complexes.

Is there a nice probabilistic interpretation of your map $|N(X)| \to [X]$, or am I just imagining things?

Posted by: John Baez on August 21, 2017 1:52 AM | Permalink | Reply to this

### Re: Simplicial Sets vs. Simplicial Complexes

Posted by: Allen Knutson on August 20, 2017 11:43 AM | Permalink | Reply to this

### Re: Simplicial Sets vs. Simplicial Complexes

These concrete simplicial complexes are just as ugly as I’d imagined!

Posted by: John Baez on August 21, 2017 1:38 AM | Permalink | Reply to this

### Re: Simplicial Sets vs. Simplicial Complexes

Where is this by the way?

Posted by: Todd Trimble on August 21, 2017 4:56 AM | Permalink | Reply to this

### Re: Simplicial Sets vs. Simplicial Complexes

The Juno Beach Centre in Normandy, apparently. Image appears here.

Posted by: David Roberts on August 21, 2017 7:05 AM | Permalink | Reply to this

### Re: Simplicial Sets vs. Simplicial Complexes

Over on G+, Allen Knutson and I are having a parallel conversation that might be interesting. I could use some help!

I wrote:

I just explained the category of abstract simplicial complexes on Wikipedia, where they start the article with a truly wretched — but apparently widespread — definition of these objects.

He wrote:

Oh no. Are you to blame for all the “non-empty” on that page?

There should be a correspondence between abstract simplicial complexes on (say) $\{1,...,n\}$ and monomial ideals in $k[x_1...x_n]$. Two such monomial ideals are $\langle 1 \rangle$, and $\langle x_1,...,x_n \rangle$.

I wrote:

I didn’t change the definition of abstract simplicial complex on Wikipedia, which is one of the standard ones. I just pointed out that it gives an equivalent category to another of the standard ones. I think you’re advocating a third nonequivalent one, which is good for Stanley–Reisner rings.

He wrote:

Here’s a nice construction that’s only reasonable if you allow the set $V$ of vertices to be larger than the set of 1-element faces. (I call the extra ones phantom vertices.)

Given a simplicial complex $\Delta$ (considered as a collection of subsets of the vertex set, closed under going down), define a new one $\Delta^c$ where $F$ is a face of $\Delta^c$ iff $V \ F$ is a face of Delta. IIRC this is the basis of Alexander duality.

People are used to the idea that Delta might have “cone vertices”, that are in every face. Regardless of how you choose $V$, if $\Delta$ has a cone vertex, then $\Delta^c$ will have a phantom vertex. I see no reason to say “you’re not allowed to take Alexander dual unless you ditch all the cone vertices first”; it’s like Moore forbidding the empty set and having to always check, before taking an intersection, whether it’s nonempty.

I wrote:

I like your thing. I wonder if it deserves to be called an ‘augmented’ simplicial complex, since it reminds me slightly of an augmented simplicial set — a contravariant functor from the category of finite ordinals to Set, where we drop the usual eye-popping but ultimately reasonable condition that the ordinals be nonempty. An augmented simplicial set has (-1)-simplices as well as simplices of dimensions 0, 1, 2, ….

I’m not sure if your idea is connected to augmented simplicial sets or not! Over on the $n$-Café we’ve been discussing functors (and a non-functorial process) relating simplicial complexes and simplicial sets. If these extend gracefully to functors relating your things and augmented simplicial sets, then you’re really talking about augmented simplicial complexes.

It’s then a separate argument whether the augmented things are the real things.

… and I’m less interested in that other argument, than in understanding Allen’s concept of simplicial complex and its relation, if any, to augmented simplicial sets.

Posted by: John Baez on August 21, 2017 8:35 AM | Permalink | Reply to this

### Re: Simplicial Sets vs. Simplicial Complexes

Maybe this is part of what’s going on. Let a ‘simplicial complex’ be the kind of simplicial complex I like. Then there’s another concept, possibly better in some ways, which I’ll call an ‘augmented’ simplicial complex.

Definition 1. A simplicial complex is set $V$ of vertices together with a collection $\Delta$ of nonempty finite subsets of $V$ such that:

• $\Delta$ contains all singletons.

• If $F \in \Delta$ and $E \subseteq F$ is nonempty then $E \in \Delta$.

For $n \ge 1$, the $n$-element subsets in $\Delta$ are called $(n-1)$-simplices.

Definition 2. Let an augmented simplicial complex be a set $S$ together with a collection $\Delta$ of finite subsets of $S$ such that:

• If $F \in \Delta$ and $E \subseteq F$ then $E \in \Delta$.

The elements $v \in S$ such that $\{v\} \in \Delta$ are called vertices, and the set of vertices is called $V \subseteq S$. For $n \ge 0$, the $n$-element subsets in $\Delta$ are called $(n-1)$-simplices. The elements of $S - V$ are called phantoms.

There’s a category $Inj$ consisting of nonempty finite sets and injections. A simplicial complex determines a presheaf

$F : Inj^{op} \to Set$

where $F(n)$ is the set of $(n-1)$-simplices for $n \ge 1$. However, not every presheaf on $Inj$ arises this way.

Puzzle 1. Which presheaves on $Inj$ arise (up to isomorphism) from simplicial sets?

There’s also a category $Inj_+$ consisting of finite sets and injections. An augmented simplicial complex determines a presheaf

$F: Inj_+^{op} \to Set$

where $F(n)$ is the set of $(n-1)$-simplices for $n \ge 0$. However, not every presheaf on $Inj_+$ arises this way.

Puzzle 2. Which presheaves on $Inj_+$ arise (up to isomorphism) from augmented simplicial sets?

I feel I’m muddling some things here: for example, the phantoms versus the $(-1)$-simplices. But I wanted to get this idea out in public, since I may not get around to thinking about it myself.

Posted by: John Baez on August 22, 2017 5:55 AM | Permalink | Reply to this

### Re: Simplicial Sets vs. Simplicial Complexes

My understanding is that the $-1$-simplices are the connected components of the augmented simplicial set (which is not the same thing as being the connected components of the un-augmented simplicial set). This is because the connected components functor is left adjoint to the discrete inclusion of sets into augmented simplicial sets, and is therefore calculated by evaluating at the terminal object of the site. Since $[-1]$ is the terminal object, the $-1$-simplices have to be the connected components. I’m not sure this is a good way to think about things – it’s more like “formal” connected components (since the $-1$-simplices are not necessarily the limit of all the other ones).

Do your phantom vertices relate to connected components in this way? I don’t immediately see a relation, but I haven’t sat down with it.

Posted by: David Jaz Myers on August 23, 2017 5:45 PM | Permalink | Reply to this

### Re: Simplicial Sets vs. Simplicial Complexes

Yes, I think of the $(-1)$-simplices as something like ‘formal connected components’, with the property that any number of actual connected components can belong to each formal connected component. I think they’re pretty different from the phantoms.

For those a bit intimidated by the adjunction you mentioned, a nice lowbrow way to understand all this is to note that there’s a face map

$\partial_0 : X_1 \to X_0$

assigning a $(-1)$-simplex to each vertex of our augemented simplicial set, and the relations on face maps imply that if two vertices are connected by an edge, they must be assigned the same $(-1)$-simplex. So, two vertices lying in the same connected component must belong to the same ‘formal connected component’, but not conversely.

Todd Trimble puts all this to good use in his note on bar constructions. I’ll quote a bit.

B.1. Acyclic structures

We follow the algebraists' convention, taking the simplicial category
to mean the category of finite ordinals (including the empty ordinal)
and order-preserving maps. It is well-known that \Delta is initial
amongst strict monoidal categories equipped with a monoid, which in
\Delta is 1. This induces a monad 1 + - on \Delta, called the
translation monad. By composition, this in turn induces a pullback
comonad P on simplicial sets; by the Kan construction, it has a left

To explain this terminology, we recall the topologists' convention,
where \Delta is restricted to the full subcategory \Delta_+ of
non-empty ordinals.  If S-set denotes the category of simplicial
sets under the algebraists' convention, and S_{+}-set that under the
topologists' convention, then by restriction we get a functor

S-set --> S_{+}-set,

which has a left adjoint. The left adjoint augments a S_{+} set X by
its set of path components \pi_{0}(X). Starting with X, we can apply
the left augmentation, followed by the cone monad, followed by
restriction S-set --> S_{+} set: this gives a monad which, passing
to geometric realization, is the mapping cone of X --> \pi_{0}(X).
category of algebras over the monad is equivalent to the category
of coalgebras over the comonad. This category of algebras/coalgebras
could be called the "acyclic topos": the algebras X are acyclic as
simplicial sets.  More to the point, the algebra structure CX --> X
witnesses this acyclicity by providing a representative basepoint
for each path component of X, together with a well-behaved simplicial
homotopy which contracts each component down to its basepoint.

Definition: An acyclic structure is an algebra over C (or coalgebra
over P). An S-acyclic structure is one augmented over a set S.

A morphism between acyclic structures is just a C-algebra map; a
morphism between S-acyclic structures is one whose component at S is
the identity.

It doesn't matter whether the monad C is taken under the algebraists'
or topologists' convention: the category of C-algebras in S-set is
equivalent to the category of C-algebras in S_{+}-set, since given a
C-algebra in S-set,

-->
... X_{1} --> X_{0} --> X_{-1},

it follows from acyclicity that this portion of the simplicial
structure is a split coequalizer, so that the augmentation map is
the usual augmentation to its set of path components.

In the sequel, it will be useful to regard an acyclic structure,
as a functor X: \Delta^{op} --> Set, as a right coalgebra XT --> X
over the translation comonad T: \Delta^{op} --> \Delta^{op}.
`
Posted by: John Baez on August 24, 2017 2:40 AM | Permalink | Reply to this

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text:
## Singh, Bhagat

Compute Distance To:
 Author ID: singh.bhagat Published as: Singh, Bhagat; Singh, B.
 Documents Indexed: 155 Publications since 1945, including 1 Book 1 Contribution as Editor · 3 Further Contributions Co-Authors: 6 Co-Authors with 38 Joint Publications 100 Co-Co-Authors
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### Co-Authors

 69 single-authored 28 Dahiya, Rajbir S. 7 Pande, Hemant D. 6 Kusano, Takaŝi 3 Jaiswal, Anuradha 3 Sharma, Prem Lata 2 Chouhan, M. S. 2 Eisenbud, David 2 Jha, Navnit 2 Vogel, Wolfgang 2 Yadava, S. R. 1 Arora, J. L. 1 Azaiez, Jalel 1 Bailar, B. A. 1 Bailey, Leslie 1 Bhagat, Prakash 1 Brock, Dwight B. 1 Chariar, V. R. 1 Corby, C. 1 Cumberland, William G. 1 Dagum, Estelle Bee 1 Fienberg, Stephen Elliot 1 Finkner, A. L. 1 Freeman, Daniel H. jun. 1 Gopal, Venu 1 Gupta, U. S. 1 Hanson, M. H. 1 Hartley, Hermann Otto 1 Horwitz, D. G. 1 Iseki, Kiyoshi 1 Jain, Akansha 1 Jain, Shishir 1 Joorel, Jagdish Prasad Singh 1 Joshi, Devi Datt 1 Kaushik, Surendra K. 1 Khan, R. U. 1 Kish, Leslie 1 Kmenta, Jan 1 Lanphier, C. M. 1 Madow, William G. 1 Murari, K. 1 Murthy, M. N. 1 Namboodiri, N. Krishnan 1 Naps, Thomas L. 1 Naveen, K. F. 1 Nisselson, Harold 1 Pandey, Rajiv Kumar 1 Platek, Richard A. 1 Qureshi, Noshad Ahmed 1 Rangachariar, V. 1 Rao, Jonnagadda Nalini Kanth 1 Royall, Richard M. 1 Saini, Rajesh Kumar 1 Shah, B. V. 1 Sharma, Prem Hari 1 Shrivastava, Sonal 1 Smith, T. M. Fred 1 Sukavanam, Nagarajan 1 Tomar, Sushil Kumar 1 Tremblay, V. 1 Tyagi, Dinesh Kumar 1 Verma, Med Ram 1 Vuagnat, P. 1 Yen, Shee-Mang
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### Serials

 9 Hiroshima Mathematical Journal 8 Indian Journal of Pure & Applied Mathematics 8 Journal of Mathematical Analysis and Applications 8 Tamkang Journal of Mathematics 6 Acta Ciencia Indica. Mathematics 5 Mathematica Japonica 4 Bulletin of the Calcutta Mathematical Society 4 Funkcialaj Ekvacioj. Serio Internacia 4 International Journal of Mathematics and Mathematical Sciences 4 Sankhyā. Series B. Methodological 4 SIAM Journal on Mathematical Analysis 3 Journal of Mathematical and Physical Sciences 3 Journal of Sound and Vibration 3 Archivum Mathematicum 3 Czechoslovak Mathematical Journal 3 The Journal of the Indian Academy of Mathematics 3 Journal of the Indian Society of Agricultural Statistics 2 The Mathematics Education 2 Bulletin Mathématique de la Société des Sciences Mathématiques de la République Socialiste de Roumanie. Nouvelle Série 2 Bulletin of the Technical University of Istanbul 2 The Journal of the Indian Mathematical Society. New Series 2 Journal of M.A.C.T. 2 Bulletin of the Allahabad Mathematical Society 2 Atti della Accademia Nazionale dei Lincei. Serie Ottava. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali 2 International Journal of Applied Mechanics and Engineering 2 Bulletin Société Mathématique de Grèce. Nouvelle Série 2 Teubner-Texte zur Mathematik 1 AIAA Journal 1 Bulletin of the Australian Mathematical Society 1 International Journal for Numerical and Analytical Methods in Geomechanics 1 Journal of Technical Physics 1 Rocky Mountain Journal of Mathematics 1 Ukrainian Mathematical Journal 1 Acta Mexicana de Ciencia y Tecnologia 1 Analele Universității din Timișoara. Științe Matematice 1 Biometrical Journal 1 Calcutta Statistical Association. Bulletin 1 Canadian Journal of Mathematics 1 Chinese Journal of Mathematics 1 Gaṇita 1 Jnanabha 1 Kyungpook Mathematical Journal 1 The Mathematics Student 1 Mitteilungen. Vereinigung Schweizerischer Versicherungsmathematiker (VSV) 1 Portugaliae Mathematica 1 Proceedings of the National Academy of Sciences, India. Section A 1 Progress of Mathematics 1 Quarterly of Applied Mathematics 1 Revue de la Faculté des Sciences de l’Université d’Istanbul. Série A 1 Studia Scientiarum Mathematicarum Hungarica 1 Yokohama Mathematical Journal 1 Applied Mathematics and Mechanics. (English Edition) 1 Computational Mathematics and Modeling 1 Fukuoka University Science Reports 1 Aequationes Mathematicae 1 Annales de la Société Scientifique de Bruxelles. Série I 1 Communications in Statistics. Theory and Methods 1 SIAM Journal on Applied Mathematics 1 Physics of Fluids 1 Differential Equations and Dynamical Systems 1 Ultra Scientist of Physical Sciences 1 Advances in Difference Equations 1 International Journal of Applied Mathematics and Mechanics 1 Asian Journal of Civil Engineering (Building and Housing) 1 Acta Ciencia Indica 1 Proceedings of the Indian Academy of Sciences 1 International Mathematical Forum 1 Commentationes Mathematicae 1 ProbStat Forum 1 Vijnana Parishad Anusandhan Patrika 1 Nonlinear Analysis. Theory, Methods & Applications 1 Journal of the Indian Statistical Association
all top 5

### Fields

 66 Ordinary differential equations (34-XX) 15 Special functions (33-XX) 14 General topology (54-XX) 13 Statistics (62-XX) 10 Mechanics of deformable solids (74-XX) 8 Integral transforms, operational calculus (44-XX) 8 Fluid mechanics (76-XX) 7 Differential geometry (53-XX) 6 Partial differential equations (35-XX) 5 Operator theory (47-XX) 5 Numerical analysis (65-XX) 4 Harmonic analysis on Euclidean spaces (42-XX) 4 Integral equations (45-XX) 3 General and overarching topics; collections (00-XX) 2 Algebraic geometry (14-XX) 2 Real functions (26-XX) 2 Difference and functional equations (39-XX) 1 Commutative algebra (13-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Abstract harmonic analysis (43-XX) 1 Probability theory and stochastic processes (60-XX) 1 Computer science (68-XX) 1 Classical thermodynamics, heat transfer (80-XX) 1 Operations research, mathematical programming (90-XX) 1 Biology and other natural sciences (92-XX)

### Citations contained in zbMATH Open

60 Publications have been cited 213 times in 96 Documents Cited by Year
A Lyapunov inequality and non-oscillation theorem for a second order non- linear differential-difference equation. Zbl 0239.34033
Dahiya, R. S.; Singh, Bhagat
1973
Forced oscillations in general ordinary differential equations. Zbl 0339.34039
Singh, Bhagat
1975
A necessary and sufficient condition for the oscillation of an even order nonlinear delay differential equation. Zbl 0273.34047
Singh, Bhagat
1973
On oscillatory behavior of even order delay equations. Zbl 0225.35017
Dahiya, R. S.; Singh, B.
1973
Asymptotic nature of nonoscillatory solutions of $$n$$-th order retarded differential equations. Zbl 0314.34082
Singh, Bhagat
1975
Positive solutions of functional differential equations with singular nonlinear terms. Zbl 0558.34057
Kusano, Takaŝi; Singh, Bhagat
1984
A correction to ”forced oscillations in general ordinary differential equations with deviating arguments”. Zbl 0409.34070
Singh, Bhagat
1979
Asymptotically vanishing oscillatory trajectories in second order retarded equations. Zbl 0321.34058
Singh, Bhagat
1976
On the oscillation of a Volterra integral equation. Zbl 0847.45003
Singh, Bhagat
1995
Asymptotic behavior of oscillatory solutions of a differential equation with deviating arguments. Zbl 0472.34044
Singh, Bhagat; Kusano, Takasi
1981
On oscillation of second order retarded equations. Zbl 0281.34066
Singh, Bhagat; Dahiya, R. S.
1974
Cone 2-metric space and fixed point theorem of contractive mappings. Zbl 1304.54095
Singh, B.; Jain, Shishir; Bhagat, Prakash
2012
Nonoscillations of fourth order retarded equations. Zbl 0294.34051
Singh, Bhagat
1975
Weak-compatibility and fixed point theorems in fuzzy metric space. Zbl 1211.54066
Singh, B.; Jain, S.
2005
Necessary and sufficient condition for maintaining oscillations and nonoscillations in general functional equations and their asymptotic properties. Zbl 0397.34038
Singh, Bhagat
1979
On asymptotic limits of nonoscillations in functional equations with retarded arguments. Zbl 0447.34067
Singh, Bhagat; Kusano, Takasi
1980
Oscillation and nonoscillation of even-order nonlinear delay-differential equations. Zbl 0273.34046
Singh, Bhagat
1973
Necessary and sufficient conditions for eventually vanishing oscillatory solutions of functional equations with small delays. Zbl 0389.34048
Singh, Bhagat
1978
Transverse vibrations of an elliptic plate with variable thickness. Zbl 0585.73103
Singh, B.; Tyagi, D. K.
1985
Slowly oscillating and nonoscillating trajectories in second order retarded sublinear equations. Zbl 0429.34063
Singh, Bhagat
1980
Axisymmetric vibration of a circular plate with exponential thickness variation. Zbl 1232.74035
Singh, B.; Saxena, V.
1996
Forced nonoscillations in second order functional equations. Zbl 0411.34042
Singh, Bhagat
1977
Necessary and sufficient condition for eventual decay of oscillations in general functional equations with delays. Zbl 0449.34051
Singh, Bhagat
1980
Nonoscillation of third order retarded equations. Zbl 0266.34039
Singh, Bhagat; Dahiya, R. S.
1974
Forced oscillations in general ordinary differential equations with deviating arguments. Zbl 0329.34057
Singh, Bhagat
1976
On CATANOVA method for analysis of two-way classified nominal data. Zbl 0873.62068
Singh, B.
1996
On the analysis of variance method for nominal data. Zbl 0790.62025
Singh, B.
1993
Stability of miscible displacements of shear thinning fluids in a Hele-Shaw cell. Zbl 1185.76037
Azaiez, J.; Singh, B.
2002
A family of compact finite difference formulations for three-space dimensional nonlinear Poisson’s equations in Cartesian coordinates. Zbl 1387.35165
Jha, Navnit; Gopal, Venu; Singh, Bhagat
2018
Dispersion relations in a generalized monoclinic thermoelastic solid half-space. Zbl 1208.74066
Singh, B.
2006
Vanishing nonoscillations of Lienard type retarded equations. Zbl 0362.34053
Singh, Bhagat
1977
Damped trajectories and slow oscillation in forced Lienard type functional equations. Zbl 0439.34060
Singh, Bhagat
1980
Certain results on nonoscillation and asymptotic nature of delay equations. Zbl 0302.34088
Dahiya, R. S.; Singh, Bhagat
1975
Impact of delays on oscillation in general functional equations. Zbl 0317.34059
Singh, Bhagat
1975
On certain integrals involving $$\tilde\omega_{\mu,\nu} (x)$$. Zbl 0144.36902
Singh, B.
1965
On quick oscillations in functional equations with deviating arguments. Zbl 0556.34070
Singh, Bhagat
1984
Minimal existence of nonoscillatory solutions in functional differential equations with deviating arguments. Zbl 0558.34056
Singh, Bhagat
1984
CATANOVA for analysis of nominal data from repeated measures designs. Zbl 1188.62181
Singh, B.
2004
General functional differential equations and their asymptotic oscillatory behavior. Zbl 0361.34062
Singh, Bhagat
1976
Comparative oscillation of non-homogeneous even order delay differential equations. Zbl 0227.35015
Dahiya, R. S.; Singh, B.
1973
On existence of nonoscillatory solutions in forced nonlinear differential equations. Zbl 0587.34051
Singh, Bhagat
1985
Necessary and sufficient conditions for finally vanishing oscillatory solutions in second order delay equations. Zbl 0702.34061
Singh, Bhagat
1989
Existence of slow oscillations in functional equations. Zbl 0704.34077
Singh, Bhagat; Dahiya, R. S.
1990
Asymptotic nonoscillation under large amplitudes of oscillating coefficients in second order functional equations. Zbl 0574.34046
Singh, Bhagat
1985
Nonlinear oscillations in disconjugate forced functional equations with deviating arguments. Zbl 0507.34058
Singh, Bhagat
1983
On oscillation and asymptotic nonoscillation of functional retarded equations. Zbl 0395.34066
Singh, B.
1975
On slow oscillation and nonoscillation in retarded equations. Zbl 0412.34018
Singh, Bhagat
1979
On Hankel transformation involving hypergeometric functions. Zbl 0252.44004
Dahiya, R. S.; Singh, Bhagat
1973
On some results involving generalized hypergeometric polynomials. Zbl 0257.33009
Dahiya, R. S.; Singh, Bhagat
1974
Integral transforms of Bessel product functions and their integrals. Zbl 0264.44002
Dahiya, R. S.; Singh, Bhagat
1973
Comparative oscillation of non-homogeneous even order delay differential equations. Zbl 0276.34065
Dahiya, R. S.; Singh, Bhagat
1973
On the oscillation of a second order delay equation. Zbl 0289.34105
Dahiya, R. S.; Singh, Bhagat
1974
Comparative study of asymptotic nonoscillation and quick oscillation of second order linear differential equations. Zbl 0298.34033
Singh, Bhagat
1974
Damp nonoscillation in third order retarded equations. Zbl 0317.34058
Singh, Bhagat
1975
The special infinitesimal projective transformation in an $$n$$-dimensional special Kawaguchi space. Zbl 0325.53028
Pande, H. D.; Singh, B.
1975
On the nonoscillation of integro-differential difference equation of first order. Zbl 0343.45007
Singh, Bhagat
1975
On the oscillation of an elliptic equation of fourth order. Zbl 0857.35011
Singh, Bhagat
1996
Moments of progressively type-II right censored order statistics from additive Weibull distribution. Zbl 1422.62170
Singh, B.; Khan, R. U.
2019
A comparison of variance component estimators under unbalanced situations. Zbl 0711.62060
Singh, B.
1989
On the nonoscillation of Lienard type retarded equations. Zbl 0354.34075
Singh, Bhagat; Dahiya, R. S.
1976
Moments of progressively type-II right censored order statistics from additive Weibull distribution. Zbl 1422.62170
Singh, B.; Khan, R. U.
2019
A family of compact finite difference formulations for three-space dimensional nonlinear Poisson’s equations in Cartesian coordinates. Zbl 1387.35165
Jha, Navnit; Gopal, Venu; Singh, Bhagat
2018
Cone 2-metric space and fixed point theorem of contractive mappings. Zbl 1304.54095
Singh, B.; Jain, Shishir; Bhagat, Prakash
2012
Dispersion relations in a generalized monoclinic thermoelastic solid half-space. Zbl 1208.74066
Singh, B.
2006
Weak-compatibility and fixed point theorems in fuzzy metric space. Zbl 1211.54066
Singh, B.; Jain, S.
2005
CATANOVA for analysis of nominal data from repeated measures designs. Zbl 1188.62181
Singh, B.
2004
Stability of miscible displacements of shear thinning fluids in a Hele-Shaw cell. Zbl 1185.76037
Azaiez, J.; Singh, B.
2002
Axisymmetric vibration of a circular plate with exponential thickness variation. Zbl 1232.74035
Singh, B.; Saxena, V.
1996
On CATANOVA method for analysis of two-way classified nominal data. Zbl 0873.62068
Singh, B.
1996
On the oscillation of an elliptic equation of fourth order. Zbl 0857.35011
Singh, Bhagat
1996
On the oscillation of a Volterra integral equation. Zbl 0847.45003
Singh, Bhagat
1995
On the analysis of variance method for nominal data. Zbl 0790.62025
Singh, B.
1993
Existence of slow oscillations in functional equations. Zbl 0704.34077
Singh, Bhagat; Dahiya, R. S.
1990
Necessary and sufficient conditions for finally vanishing oscillatory solutions in second order delay equations. Zbl 0702.34061
Singh, Bhagat
1989
A comparison of variance component estimators under unbalanced situations. Zbl 0711.62060
Singh, B.
1989
Transverse vibrations of an elliptic plate with variable thickness. Zbl 0585.73103
Singh, B.; Tyagi, D. K.
1985
On existence of nonoscillatory solutions in forced nonlinear differential equations. Zbl 0587.34051
Singh, Bhagat
1985
Asymptotic nonoscillation under large amplitudes of oscillating coefficients in second order functional equations. Zbl 0574.34046
Singh, Bhagat
1985
Positive solutions of functional differential equations with singular nonlinear terms. Zbl 0558.34057
Kusano, Takaŝi; Singh, Bhagat
1984
On quick oscillations in functional equations with deviating arguments. Zbl 0556.34070
Singh, Bhagat
1984
Minimal existence of nonoscillatory solutions in functional differential equations with deviating arguments. Zbl 0558.34056
Singh, Bhagat
1984
Nonlinear oscillations in disconjugate forced functional equations with deviating arguments. Zbl 0507.34058
Singh, Bhagat
1983
Asymptotic behavior of oscillatory solutions of a differential equation with deviating arguments. Zbl 0472.34044
Singh, Bhagat; Kusano, Takasi
1981
On asymptotic limits of nonoscillations in functional equations with retarded arguments. Zbl 0447.34067
Singh, Bhagat; Kusano, Takasi
1980
Slowly oscillating and nonoscillating trajectories in second order retarded sublinear equations. Zbl 0429.34063
Singh, Bhagat
1980
Necessary and sufficient condition for eventual decay of oscillations in general functional equations with delays. Zbl 0449.34051
Singh, Bhagat
1980
Damped trajectories and slow oscillation in forced Lienard type functional equations. Zbl 0439.34060
Singh, Bhagat
1980
A correction to ”forced oscillations in general ordinary differential equations with deviating arguments”. Zbl 0409.34070
Singh, Bhagat
1979
Necessary and sufficient condition for maintaining oscillations and nonoscillations in general functional equations and their asymptotic properties. Zbl 0397.34038
Singh, Bhagat
1979
On slow oscillation and nonoscillation in retarded equations. Zbl 0412.34018
Singh, Bhagat
1979
Necessary and sufficient conditions for eventually vanishing oscillatory solutions of functional equations with small delays. Zbl 0389.34048
Singh, Bhagat
1978
Forced nonoscillations in second order functional equations. Zbl 0411.34042
Singh, Bhagat
1977
Vanishing nonoscillations of Lienard type retarded equations. Zbl 0362.34053
Singh, Bhagat
1977
Asymptotically vanishing oscillatory trajectories in second order retarded equations. Zbl 0321.34058
Singh, Bhagat
1976
Forced oscillations in general ordinary differential equations with deviating arguments. Zbl 0329.34057
Singh, Bhagat
1976
General functional differential equations and their asymptotic oscillatory behavior. Zbl 0361.34062
Singh, Bhagat
1976
On the nonoscillation of Lienard type retarded equations. Zbl 0354.34075
Singh, Bhagat; Dahiya, R. S.
1976
Forced oscillations in general ordinary differential equations. Zbl 0339.34039
Singh, Bhagat
1975
Asymptotic nature of nonoscillatory solutions of $$n$$-th order retarded differential equations. Zbl 0314.34082
Singh, Bhagat
1975
Nonoscillations of fourth order retarded equations. Zbl 0294.34051
Singh, Bhagat
1975
Certain results on nonoscillation and asymptotic nature of delay equations. Zbl 0302.34088
Dahiya, R. S.; Singh, Bhagat
1975
Impact of delays on oscillation in general functional equations. Zbl 0317.34059
Singh, Bhagat
1975
On oscillation and asymptotic nonoscillation of functional retarded equations. Zbl 0395.34066
Singh, B.
1975
Damp nonoscillation in third order retarded equations. Zbl 0317.34058
Singh, Bhagat
1975
The special infinitesimal projective transformation in an $$n$$-dimensional special Kawaguchi space. Zbl 0325.53028
Pande, H. D.; Singh, B.
1975
On the nonoscillation of integro-differential difference equation of first order. Zbl 0343.45007
Singh, Bhagat
1975
On oscillation of second order retarded equations. Zbl 0281.34066
Singh, Bhagat; Dahiya, R. S.
1974
Nonoscillation of third order retarded equations. Zbl 0266.34039
Singh, Bhagat; Dahiya, R. S.
1974
On some results involving generalized hypergeometric polynomials. Zbl 0257.33009
Dahiya, R. S.; Singh, Bhagat
1974
On the oscillation of a second order delay equation. Zbl 0289.34105
Dahiya, R. S.; Singh, Bhagat
1974
Comparative study of asymptotic nonoscillation and quick oscillation of second order linear differential equations. Zbl 0298.34033
Singh, Bhagat
1974
A Lyapunov inequality and non-oscillation theorem for a second order non- linear differential-difference equation. Zbl 0239.34033
Dahiya, R. S.; Singh, Bhagat
1973
A necessary and sufficient condition for the oscillation of an even order nonlinear delay differential equation. Zbl 0273.34047
Singh, Bhagat
1973
On oscillatory behavior of even order delay equations. Zbl 0225.35017
Dahiya, R. S.; Singh, B.
1973
Oscillation and nonoscillation of even-order nonlinear delay-differential equations. Zbl 0273.34046
Singh, Bhagat
1973
Comparative oscillation of non-homogeneous even order delay differential equations. Zbl 0227.35015
Dahiya, R. S.; Singh, B.
1973
On Hankel transformation involving hypergeometric functions. Zbl 0252.44004
Dahiya, R. S.; Singh, Bhagat
1973
Integral transforms of Bessel product functions and their integrals. Zbl 0264.44002
Dahiya, R. S.; Singh, Bhagat
1973
Comparative oscillation of non-homogeneous even order delay differential equations. Zbl 0276.34065
Dahiya, R. S.; Singh, Bhagat
1973
On certain integrals involving $$\tilde\omega_{\mu,\nu} (x)$$. Zbl 0144.36902
Singh, B.
1965
all top 5

### Cited by 80 Authors

 11 Agarwal, Ravi P. 11 Singh, Bhagat 10 Grace, Said Rezk 10 Özbekler, Abdullah 8 Dahiya, Rajbir S. 7 Kusano, Takaŝi 6 Graef, John R. 5 Onose, Hiroshi 4 Panigrahi, Saroj 4 Parhi, Narahari 4 Spikes, Paul W. 3 Cahlon, Baruch 3 Çakmak, Devrim 3 Cecchi, Mariella 3 Grammatikopoulos, Myron K. 3 Lalli, Bikkar S. 3 Li, Wan-Tong 3 Marini, Mauro 3 Schmidt, Darrell 3 Tiryaki, Aydin 3 Yeh, Cheh-Chih 3 Zhang, Qiming 2 Aftabizadeh, A. R. 2 Ahmad, Sufia Zulfa 2 Chen, Lu-San 2 El-Beltagy, Mohamed A. 2 He, Xiaofei 2 Ismail, Fudziah Bt. 2 Jleli, Mohamed Boussairi 2 Naito, Manabu 2 O’Regan, Donal 2 Samet, Bessem 2 Senu, Norazak 2 Staikos, Vasilios A. 2 Ünal, Mehmet 2 Villari, Gabriele 2 Wiener, Joseph 2 Zafer, Ağacık 1 Bainov, Drumi Dimitrov 1 Brychkov, Yury A. 1 Cetin, Erbil 1 Chen, Mingpo 1 Cheng, Sui Sun 1 Domoshnitsky, Alexander I. 1 Egwurube, Michael 1 Foster, Kent 1 Hasanbulli, Mustafa 1 Hill, Mary 1 Jha, Navnit 1 Jiang, Jianchu 1 Karakostas, George L. 1 Kirane, Mokhtar 1 Koplatadze, R. G. 1 Kumar, Neelesh 1 Liu, Yingfan 1 Marichev, Oleg Igorevich 1 Meng, Fanwei 1 Mishev, Dimitar P. 1 Mytropol’skyĭ, Yuriĭ Oleksiĭovych 1 Mursaleen, Mohammad 1 Oguz, Arzu Denk 1 Ozair, Muhammad 1 Pourhadi, Ehsan 1 Prudnikov, Anatoliĭ Platonovich 1 Rogovchenko, Yuri V. 1 Rusnák, Marián 1 Sficas, Yiannis G. 1 Shevelo, Vladimir Nikolajevič 1 Smith, Wanda W. 1 Šoltés, Vincent 1 Stavroulakis, Ioannis P. 1 Sztaba, Urszula 1 Tang, Xianhua 1 Wang, Xiaoping 1 Wu, Yumei 1 Wyrwinska, Aleksandra 1 Yu, Cheng-Shu 1 Zhang, Meirong 1 Zhong, Chengkui 1 Zhou, Yong
all top 5

### Cited in 41 Serials

 26 Journal of Mathematical Analysis and Applications 5 Annali di Matematica Pura ed Applicata. Serie Quarta 4 Computers & Mathematics with Applications 4 Czechoslovak Mathematical Journal 3 Bulletin of the Australian Mathematical Society 3 Ukrainian Mathematical Journal 3 Applied Mathematics and Computation 3 Journal of Computational and Applied Mathematics 3 Journal of Differential Equations 3 Mathematica Slovaca 3 Proceedings of the American Mathematical Society 3 Applied Mathematics Letters 3 Abstract and Applied Analysis 2 Journal of Inequalities and Applications 2 Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM 1 Applicable Analysis 1 Mathematical Methods in the Applied Sciences 1 Periodica Mathematica Hungarica 1 Journal of Soviet Mathematics 1 Mathematische Nachrichten 1 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 1 Proceedings of the Edinburgh Mathematical Society. Series II 1 Rendiconti del Circolo Matemàtico di Palermo. Serie II 1 Tohoku Mathematical Journal. Second Series 1 Applied Mathematics and Mechanics. (English Edition) 1 Acta Mathematicae Applicatae Sinica. English Series 1 Journal of Integral Equations and Applications 1 Aequationes Mathematicae 1 Turkish Journal of Mathematics 1 Mathematical Problems in Engineering 1 Differential Equations and Dynamical Systems 1 Discrete Dynamics in Nature and Society 1 Analysis (München) 1 Journal of the Australian Mathematical Society 1 Journal of Applied Mathematics 1 Communications on Pure and Applied Analysis 1 Advances in Difference Equations 1 Proceedings of the Japan Academy 1 Journal of Mathematical Inequalities 1 Journal of Mathematics 1 Nonlinear Analysis. Theory, Methods & Applications
all top 5

### Cited in 12 Fields

 83 Ordinary differential equations (34-XX) 9 Integral equations (45-XX) 5 Real functions (26-XX) 5 Numerical analysis (65-XX) 4 Partial differential equations (35-XX) 3 Difference and functional equations (39-XX) 3 Operator theory (47-XX) 2 Special functions (33-XX) 2 Integral transforms, operational calculus (44-XX) 2 Mechanics of particles and systems (70-XX) 2 Systems theory; control (93-XX) 1 Dynamical systems and ergodic theory (37-XX)