text:
## A multiobjective and fixed elements based modification of the evolutionary structural optimization method.(English)Zbl 1120.74708

Summary: In 1993, Y.M. Xie and G.P. Steven introduced an approach called evolutionary structural optimization (ESO). ESO is based on the simple idea that the optimal structure (maximum stiffness, minimum weight) can be produced by gradually removing the ineffectively used material from the design domain. The design domain is constructed by the FE method, and furthermore, external loads and support conditions are applied to the element model. Considering the engineering aspects, ESO seems to have some attractive features: the ESO method is very simple to program via the FEA packages and requires a relatively small amount of FEA time. On the other hand, different constraints cannot be added into the problem. In the ESO optimization the results supposedly approach truss-like, fully stressed topologies, which have the maximum stiffness with respect to the volume. Generally, these types of structures correspond to least-weight trusses.
Although ESO is not capable of handling general stress or displacement constraints, the design problems are often such that these constraints do not need to be included in the topology optimization, especially if the design optimization task is divided into two stages. In the first stage, only the overall geometry is outlined, and for that reason, the actual constraints do not have to be activated. In the second stage, the sizing optimization is performed. It can be concluded that ESO is well suited to solve the first stage optimization problems.
In some design problems it may occur that the structure cannot attain the fully stressed state because of geometrical constraints. It follows that the topology having the maximum stiffness with respect to the volume does not necessarily produce the least-weight structure when the stress constraints are applied in the second stage optimization. The geometrical constraints may force some structural components to be subject to a understressed state, i.e. to carry some “waste material”. As a consequence, the aim of this paper was to study whether ESO can be modified so that some geometrical constraints can be taken into account already in the first stage topology optimization. The modification was based on the assumption that if the stress level of otherwise understressed structural components can be increased during the $$compliance-volume$$ product minimization, a lighter topology may be obtained. This new approach, the multiobjective and fixed elements based modification of the evolutionary structural optimization (MESO) utilizes a new optimization objective in which the overall stiffness of the structure and the loading of some parts of the structure are increased simultaneously. The gradient vector of the MESO objective function was determined by the FE method. Some of the partial derivatives involved were first presolved and then approximated. This approach was justified by large savings in the analysis time. Yet, MESO cannot take the general constraints into account.
To study the performance of the MESO optimization, two numerical examples were evaluated. The main purpose of these examples was to study whether MESO can produce structures lighter than the ESO results for problems having both stress and geometrical constraints.
These example were based on the two-stage optimization approach. In both example the MESO truss turned out to be lighter than the corresponding ESO truss. However, the ESO truss was the stiffest one also having a smaller overall stress value. In the question of shallow structures the deflection criteria may be predominant, and as a consequence, the ESO optimization may yield lighter structures than MESO.

### MSC:

 74P10 Optimization of other properties in solid mechanics 74P15 Topological methods for optimization problems in solid mechanics 65K10 Numerical optimization and variational techniques

CONMIN; ESOFRAME
Full Text:

### References:

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This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
−(x − 3) + 4 < 5 − 2x
Question:

−(x − 3) + 4 < 5 − 2x

Solution:

$-(x-3)+4<5-2 x$

$\Rightarrow-x+3+4<5-2 x$

$\Rightarrow-x+7<5-2 x$

$\Rightarrow-x+2 x<5-7 \quad[$ Transposing $-2 \mathrm{x}$ to LHS and 7 to RHS $]$

$\Rightarrow x<-2$

Hence, the solution set of the given inequation is $(-\infty,-2)$.

format: plain


text:
# Probability distribution for the position of a biased random walker on the positive integers

I initialize a biased one-dimensional random walk on the positive integers at the origin, $x = 0$, which also serves as a reflecting boundary blocking steps onto the negative integers. Let's say that a $+1$ step for the walker has probability $p$, and an $-1$ step (away from the origin) has probability $q$, where $(p+q) \leq 1$ allowing the walker to stay in place with probability $r = 1 - (p+q)$.

If $p < q$, what is the probability distribution for the walker on the positive integers? What are the implications if $p = q$ provided that a 1D random walk shouldn't be transient?

I've always appealed to simulations when problems like this arise, however, the formulation of this system seems simple enough that I'd imagine an analytic solution should exist (beyond the generalization that we should have exponential increasing hitting times as we linearly increase the distance to the origin)? Is there a formulation of, say, Gambler's ruin that deals with this problem? We can of course eliminate the reflecting boundary by making $p$ the probability of stepping away from the origin, $q$ the probability of stepping towards the origin, etc.

Some additional questions that come to mind: how long would it take a randomly placed walker to achieve this distribution? If we place a walker at the origin, how fair or unfair would it be to assume that the hitting time at some site $x_{t}$ would be proportional to its occupancy probability from the distribution? And, in the case that this is unfair, what does the probability distribution look like for initial hitting times on the positive integers conditioned on the walker being initialized at the origin?

Clarification - A walker can occupy the origin, though here, we have a $-1$ step probability of $q = 0$, the same $+1$ step probability $p$, and a probability of remaining at the origin for a step of $r = (1 - p)$.

-

For your stable distribution try $$P(X=n) = \left(\frac{p}{q}\right)^n \left(1-\frac{p}{q}\right)$$ which satisfies $P(X=n)= p P(X=n-1) +(1-p-q) P(X=n) + q P(X=n+1)$ and sums to $1$.
If $p=q$ you are in a similar situation to a standard random walk, in that the probability of a return to the origin is $1$, but there is no stable distribution as any initial distribution widens over time.
@PlacidLake If the reflecting barrier was not there and $p \lt q$ then the walk would drift negative, eventually permanently. The reflecting barrier does not have much of a substantial effect for $p=q$ beyound an absolute value type of impact. – Henry Dec 13 '12 at 17:57
You can calculate the probability of each position using $P(X_{k+1}=n)= p P(X_k=n-1) +(1-p-q) P(X_k=n) + q P(X_k=n+1)$ and $P(X_{k+1}=0)= (1-q) P(X_k=0) + q P(X_k=1)$ starting for example at $P(X_0=0)=1$ and $P(X_0=n)=0$ for $n\gt 0$ and see what happens as $k$ increases. – Henry Dec 14 '12 at 7:13


text:
# Estimating training values in machine learning

I've started to learn machine learning using the book 'Machine Learning' by Tom Mitchell. The first chapter introduces the reader to machine learning and to the first problem: learning to play checkers. It starts by choosing a target function $V: Board \rightarrow \mathbb{R}$, which maps a particular board setting to a score (how good the current board state is). A score of $100$ means the game was won, $-100$ it was lost, $0$ draw. It then chooses a representation for the board: features.

$$V = w_0 + x_1*w_1 + x_2*w_2 + ...$$

where $w_i$ are weights and $x_i$ are features (for example the number of pieces you currently have on the board).

Features can be determined from the board configuration itself, but the weights have to be adjusted by the training examples. Each training example is a pair of the form $\langle b, V_{train}(b)\rangle$, where $b$ is the board state described by its features and $V_{train}(b)$ is the training value for $b$. Determining $V_{train}(b)$ for boards that are won or lost are easy, but we also need training examples for intermediate board states.

This is where I start being confused. It states a good approximation for intermediate boards is $V_{train}(b) \leftarrow V(successor(b))$, where $successor(b)$ denotes the next board state following $b$ for which it is again the program's turn to move (i.e., the board state following the program's move and the opponent's response). How am I supposed to know what's $successor(b)$? Which move am I supposed to make and how do I know what will be the opponent's move?

• Sounds a lot like an awful way to start teaching machine learning--confusing the reader with an unrelated problem. Jul 15 '14 at 20:44

## 1 Answer

successor(b) is calculated by our current approximation of V. Our next move would be calculated by the move which maximizes V and the opponent's move would be which minimizes the function V.


text:
# algebra equations

• March 13th 2008, 12:41 PM
miramontes
algebra equations
Hi

I need help on these problems, these are practice problems for my final because the questions will be similar to these but I need help with these.
I am lost.

Any help would be greatly appreciated (Speechless)(Thinking)(Crying)

1) find average speed if you drive 100 miles half the trip at 60mph nd half
30mph?

and then
Driving 30 minutes @ 45mph and 30 minutes @15mph

2) of a collection of mango fruits the king took 1/6 the queen 1/5 of the
raineder the three chief princes 1/4,1/3 and 1/2 of the successive remainders
and the youngest child took the reamaining 3 mangoes o you who are clever in
miscellaneous problems on fractions give out the measure of the collection of
mangoes???

3) I NEED TO SOLVE THIS PROBLEM BACKWARD BUT I DONT KNOW
i began the day feeling very rich as i was walking to school i met a friend who
had loaned me money and thought this would be a great opportunity to pay her
back so i gave her $15 theni remembered i nedd to buy some school supplies and that took one third of the money in my pocket then i met another friend and she asked me for a loan so i gave her three fourths of the money in my pocket now i had only$10.00 in my pocket how much money didi i start with??

ANY HELP WOULD BE GREATLY APPRECIATED, THANKS IN ADVANCE (Nerd)
• March 13th 2008, 08:03 PM
SengNee
Quote:

Originally Posted by miramontes
1) find average speed if you drive 100 miles half the trip at 60mph nd half
30mph?

and then
Driving 30 minutes @ 45mph and 30 minutes @15mph

$v_1=60$
$\frac{s_1}{t_1}=60$
$\frac{50}{t_1}=60$
$t_1=\frac{5}{6}$

$v_2=30$
$\frac{s_2}{t_2}=30$
$\frac{50}{t_2}=30$
$t_2=\frac{5}{3}$

$t=\frac{5}{6}+\frac{5}{3}=2.5$

$v=\frac{s}{t}=\frac{100}{2.5}=40$mph

and then
Driving 30 minutes @ 45mph and 30 minutes @15mph
this means???

Quote:

Originally Posted by miramontes
2) of a collection of mango fruits the king took 1/6 the queen 1/5 of the
raineder the three chief princes 1/4,1/3 and 1/2 of the successive remainders
and the youngest child took the reamaining 3 mangoes o you who are clever in
miscellaneous problems on fractions give out the measure of the collection of
mangoes???

Let, number of mangoes is $x$.

1st,
King takes $\frac{1}{6}$
Remains $\frac{5}{6}$

2nd,
Queen takes $\frac{1}{5}$
Remains $\frac{4}{5}$

3rd,
The 3 chief princes take $\frac{1}{4}$, $\frac{1}{3}$ and $\frac{1}{2}$ of the successive remainders
Remain,
$\frac{1}{2}(\frac{2}{3})(\frac{3}{4})(\frac{4}{5}) (\frac{5}{6}x)$

Last,
The youngest child took the reamaining 3 mangoes
Therefore,
$\frac{1}{2}(\frac{2}{3})(\frac{3}{4})(\frac{4}{5}) (\frac{5}{6}x)=3$
$\frac{1}{6}x=3$
$x=18$

Quote:

Originally Posted by miramontes
3) I NEED TO SOLVE THIS PROBLEM BACKWARD BUT I DONT KNOW
i began the day feeling very rich as i was walking to school i met a friend who
had loaned me money and thought this would be a great opportunity to pay her
back so i gave her $15 theni remembered i nedd to buy some school supplies and that took one third of the money in my pocket then i met another friend and she asked me for a loan so i gave her three fourths of the money in my pocket now i had only$10.00 in my pocket how much money didi i start with??

Let, your money is $x$.

1st,
You remain $x-15$

2nd,
You used $\frac{1}{3}$
therefore, you remain $\frac{2}{3}$
$\frac{2}{3}(x-15)$

Last,
You used $\frac{3}{4}$
Therefore, you remain $\frac{1}{4}$
$\frac{1}{4}[\frac{2}{3}(x-15)]$

$\frac{1}{4}[\frac{2}{3}(x-15)]=10$
$x-15=60$
$x=75$
• March 13th 2008, 09:05 PM
miramontes
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
thank you so much for your help, I really appreciate it(Rock)(Cool)(Nerd)

and then
Driving 30 minutes @ 45mph and 30 minutes @15mph
this means???

this means apply driving 100 miles to the question above.
• March 14th 2008, 04:32 PM
SengNee
Quote:

Originally Posted by miramontes
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
thank you so much for your help, I really appreciate it(Rock)(Cool)(Nerd)

and then
Driving 30 minutes @ 45mph and 30 minutes @15mph
this means???

this means apply driving 100 miles to the question above.

You mean,
1) find average speed if you drive 100 miles 30 minutes @ 45mph and 30 minutes @15mph?

If your mean is like that, I can't solve......

Because,

If driving with speed 45mph for 0.5 hour,
the distance travelled is 0.5(45)miles.

If driving with speed 15mph for 0.5 hour,
the distance travelled is 0.5(15)miles.

Therefore,

The total distance travelled is 30miles.
I am not finish the 100miles distance.


text:
Individualised therapies are increasingly important for drug development. One reason for individualised therapies is concentration related toxicity and high between-patient variability of exposure levels. In such a case each patient can be monitored for the concentration levels and the drug dose will be adjusted when the individual concentration level exceeds a certain threshold. In this example we demonstrate how Simulx is used to run clinical trial simulations with an adaptive dosing. We will simulate a PK/PD model with a decision rule to adjust the dose based on monitored concentration levels. In addition, we will include a decision rule for the efficacy as well to ensure that desired efficacy level will be maintained.

# The PK model

Tne PK model is a one compartment model for oral administration with first order absorption and elimination.

Parameters $$ka$$, $$V$$ and $$k$$ are log-normally distributed

This PK model is implemented in a text file cts1a.txt:

[LONGITUDINAL]
input = {ka, V, k, a1}
EQUATION:
Cc = pkmodel(ka, V, k)
DEFINITION:
y1 = {distribution=lognormal, prediction=Cc, sd=a1}

[INDIVIDUAL]
input={ka_pop, omega_ka, V_pop, omega_V, k_pop, omega_k}
DEFINITION:
ka = {distribution=lognormal, prediction=ka_pop, sd=omega_ka}
V  = {distribution=lognormal, prediction=V_pop,  sd=omega_V}
k  = {distribution=lognormal, prediction=k_pop,  sd=omega_k}

# Simulation of PK data

We can use simulx for simulating a clinical trial with 20 patients to whom a dose of 20mg is administrated orally every 12 hours starting at time 0.

We then want to compute for these 20 patients:

• the predicted concentration $$Cc$$ every hour between times 0h and 440h
• the observed concentration $$y_1$$ every 24 hours between time 4h and 440h

adm <- list(time=seq(0,to=440,by=12), amount=20)

ppk <- c(ka_pop=0.4,   V_pop=10,    k_pop=0.05,
omega_ka=0.3, omega_V=0.5, omega_k=0.1,
a1=0.05)

g   <- list(size=20, level='individual')

Cc <- list(name='Cc', time=seq(0,to=440,by=1))
y1  <- list(name='y1', time=seq(4,to=440,by=24))

s   <- list(seed=1234)

res1 <- simulx(model     = "model/cts1a.txt",
parameter = ppk,
group     = g,
output    = list(Cc, y1),
settings  = s)

(we use seed = 1234 for reproducibility reason. Different results will be obtained with different values of the seed.)

We can now plot the predicted and observed concentrations:

library("gridExtra")
library(ggplot2)
theme_set(theme_bw())
plot1 <- ggplot(data=res1$Cc, aes(x=time, y=Cc, colour=id)) + geom_line(size=0.5) + xlab("time (hour)") + ylab("Cc (mg/l)") + theme(legend.position="none") plot2 <- ggplot(data=res1$y1, aes(x=time, y=y1, colour=id)) +
geom_line(size=0.5) + geom_point(size=2) +
xlab("time (hour)") + ylab("measured conc. (mg/l)") +
theme(legend.position="none")
grid.arrange(plot1, plot2, ncol=2)

# Introducing a single decision rule

Assume now that, for safety reason, the concentration should remain below a given threshold of 5 mg/l.

The exposure, i.e. the true’’ concentration $$Cc$$, is above this threshold for several patients. Since $$Cc$$ cannot be measured in practice, we can then decide to reduce the dose for those patients which measured concentration is above 5 mg/l.

We thus define an adaptive individual design as follows:

• Visits are scheduled every 24 hours, starting at time 124h.
• If, at a given visit, the observed concentration $$y_1$$ of a patient is greater than 5, then the next doses administrated to this patient will be reduced, by multiplying the current amount by a factor 0.75.

This decision rule is implemented as a R list:

r1 <- list(name='y1',
time=seq(124,to=440,by=24),
condition="y1>5",
factor=0.75)

R function titration is a variant of simulx which now takes this rule into account.

Remark: function titration.R is not part of the mlxR package. This is just an example of what can be done: this function can be freely used/modified by anyone.

source("titration.R")
res2 <- titration(model     = "model/cts1a.txt",
parameter = ppk,
output    = list(Cc, y1),
rule      = r1,
group     = g,
settings  = s)

Outputs of titration and simulx have similar structure. We can therefore display the predicted and observed concentrations obtained with titration as we did previously with the output of simulx:

plot3 <- ggplot(data=res2$Cc, aes(x=time, y=Cc, colour=id)) + geom_line(size=0.5) + xlab("time (hour)") + ylab("Cc (mg/l)") + theme(legend.position="none") + geom_hline(yintercept=5) plot4 <- ggplot(data=res2$y1, aes(x=time, y=y1, colour=id))     +
geom_line(size=0.5) + geom_point(size=2) +
xlab("time (hour)") + ylab("measured conc. (mg/l)")           +
theme(legend.position="none") + geom_hline(yintercept=5)
grid.arrange(plot3, plot4, ncol=2)

What was expected is now clearly observed: high concentrations tend to reduce after the first visit at time 124.

# Combining several decision rules

We can introduce several decision rules. Imagine for instance that efficacy requires a concentration above 3 mg/l. We can then decide, for example, to multiply the dose by a factor 1.5 when the observed concentration is below this value at a given visit.

We can then implement this new decision rule as a new list:

r2 <- list(name='y1',
time=seq(124,to=440,by=24),
condition="y1<3",
factor=1.5)

We will now use titration combining the two decisions rules r1 and r2:

res3 <- titration(model     = "model/cts1a.txt",
parameter = ppk,
output    = list(Cc, y1),
rule      = list(r1,r2),
group     = g,
settings  = s)

plot5 <- ggplot(data=res3$Cc, aes(x=time, y=Cc, colour=id)) + geom_line(size=0.5) + xlab("time (hour)") + ylab("Cc (mg/l)") + theme(legend.position="none") + geom_hline(yintercept=c(3,5)) plot6 <- ggplot(data=res3$y1, aes(x=time, y=y1, colour=id))     +
geom_line(size=0.5) + geom_point(size=2) +
xlab("time (hour)") + ylab("measured conc. (mg/l)")   +
theme(legend.position="none") + geom_hline(yintercept=c(3,5))
grid.arrange(plot5, plot6, ncol=2)

# Extension to PKPD data

Assume now that we have PK and PD data. In this example, the PD model is a Emax model.

The joint PKPD model is implemented in a new file cts1b.txt:

[LONGITUDINAL]
input = {ka, V, k, Emax, EC50, a1, a2}
EQUATION:
Cc = pkmodel(ka, V, k)
E  = Emax*Cc/(EC50+Cc)

DEFINITION:
y1 = {distribution=lognormal, prediction=Cc, sd=a1}
y2 = {distribution=normal, prediction=E,  sd=a2}

[INDIVIDUAL]
input={
ka_pop, omega_ka, V_pop, omega_V, k_pop, omega_k,
Emax_pop, omega_Emax, EC50_pop, omega_EC50
}
DEFINITION:
ka   = {distribution=lognormal, prediction=ka_pop,   sd=omega_ka}
V    = {distribution=lognormal, prediction=V_pop,    sd=omega_V}
k    = {distribution=lognormal, prediction=k_pop,    sd=omega_k}
Emax = {distribution=lognormal, prediction=Emax_pop, sd=omega_Emax}
EC50 = {distribution=lognormal, prediction=EC50_pop, sd=omega_EC50}

We need to provide the values of the population PD parameters for simulating PKPD data from this model.

ppd <- c(Emax_pop=100,   EC50_pop=3,
omega_Emax=0.1, omega_EC50=0.2, a2=5)

E <-  list(name='E',  time=seq(0,to=440,by=1))
y2 <- list(name='y2', time=seq(16,to=440,by=36))

res4 <- simulx(model     = "model/cts1b.txt",
parameter = c(ppk,ppd),
settings  = s)
pl1=ggplot(data=res4$Cc, aes(x=time, y=Cc, colour=id))+geom_line(size=0.5)+ xlab("time (hour)") + ylab("Cc (mg/l)") + theme(legend.position="none") pl2=ggplot(data=res4$E, aes(x=time, y=E, colour=id))+geom_line(size=0.5)  +
pl3=ggplot(data=res4$y1, aes(x=time, y=y1, colour=id))+geom_line(size=0.5)+ geom_point(size=2) + xlab("time (hour)") + ylab("measured conc (mg/l)") + theme(legend.position="none") pl4=ggplot(data=res4$y2, aes(x=time, y=y2, colour=id))+geom_line(size=0.5)+
grid.arrange(pl1, pl2, pl3, pl4, ncol=2)


text:
# sigmaplot

Plot singular values of frequency response with additional plot customization options

## Syntax

``h = sigmaplot(sys)``
``h = sigmaplot(sys1,sys2,...,sysN)``
``h = sigmaplot(sys1,LineSpec1,...,sysN,LineSpecN)``
``h = sigmaplot(___,w)``
``h = sigmaplot(___,type)``
``h = sigmaplot(AX,___)``
``h = sigmaplot(___,plotoptions)``

## Description

`sigmaplot` lets you plot the singular values (SV) of frequency response of a dynamic system model with a broader range of plot customization options than `sigma`. You can use `sigmaplot` to obtain the plot handle and use it to customize the plot, such as modify the axes labels, limits and units. You can also use `sigmaplot` to draw an SV plot on an existing set of axes represented by an axes handle. To customize an existing SV plot using the plot handle:

1. Obtain the plot handle

2. Use `getoptions` to obtain the option set

3. Update the plot using `setoptions` to modify the required options

For more information, see Customizing Response Plots from the Command Line. To create SV plots with default options or to extract the frequency response data, use `sigma`.

example

````h = sigmaplot(sys)` plots the singular values (SV) of the frequency response of the dynamic system model `sys` and returns the plot handle `h` to the plot. You can use this handle `h` to customize the plot with the `getoptions` and `setoptions` commands.```

example

````h = sigmaplot(sys1,sys2,...,sysN)` plots the SV of multiple dynamic systems `sys1,sys2,…,sysN` on the same plot. All systems must have the same number of inputs and outputs to use this syntax.```

example

````h = sigmaplot(sys1,LineSpec1,...,sysN,LineSpecN)` sets the line style, marker type, and color for the SV plot of each system. All systems must have the same number of inputs and outputs to use this syntax.```

example

````h = sigmaplot(___,w)` plots singular values for frequencies specified by the frequencies in `w`.If `w` is a cell array of the form `{wmin,wmax}`, then `sigmaplot` plots the singular values at frequencies ranging between `wmin` and `wmax`.If `w` is a vector of frequencies, then `sigmaplot` plots the singular values at each specified frequency.You can use `w` with any of the input-argument combinations in previous syntaxes.See `logspace` to generate logarithmically spaced frequency vectors.```

example

````h = sigmaplot(___,type)` plots the modified singular value responses based on the `type` argument. Specify `type` as:`1` to plot the SV of the frequency response H-1, where H is the frequency response of `sys`.`2` to plot the SV of the frequency response I+H.`3` to plot the SV of the frequency response I+H-1.You can only use the `type` argument for square systems, that is, systems that have the same number of inputs and outputs.```
````h = sigmaplot(AX,___)` plots the singular values on the `Axes` object in the current figure with the handle `AX`.```

example

````h = sigmaplot(___,plotoptions)` plots the singular values with the options set specified in `plotoptions`. You can use these options to customize the SV plot appearance using the command line. Settings you specify in `plotoptions` overrides the preference settings in the MATLAB® session in which you run `sigmaplot`. Therefore, this syntax is useful when you want to write a script to generate multiple plots that look the same regardless of the local preferences.```

## Examples

collapse all

For this example, use the plot handle to change the frequency units to Hz and turn on the grid.

Generate a random state-space model with 5 states and create the sigma plot with plot handle `h`.

```rng("default") sys = rss(5); h = sigmaplot(sys);```

Change the units to Hz and turn on the grid. To do so, edit properties of the plot handle, `h` using `setoptions`.

`setoptions(h,'FreqUnits','Hz','Grid','on');`

The sigma plot automatically updates when you call `setoptions`.

Alternatively, you can also use the `sigmaoptions` command to specify the required plot options. First, create an options set based on the toolbox preferences.

`p = sigmaoptions('cstprefs');`

Change properties of the options set by setting the frequency units to Hz and enable the grid.

```p.FreqUnits = 'Hz'; p.Grid = 'on'; sigmaplot(sys,p);```

You can use the same option set to create multiple sigma plots with the same customization. Depending on your own toolbox preferences, the plot you obtain might look different from this plot. Only the properties that you set explicitly, in this example `Grid` and `FreqUnits`, override the toolbox preferences.

For this example, create a sigma plot that uses 15-point red text for the title. This plot should look the same, regardless of the preferences of the MATLAB session in which it is generated.

First, create a default options set using `sigmaoptions`.

`plotoptions = sigmaoptions;`

Next, change the required properties of the options set `plotoptions`.

```plotoptions.Title.FontSize = 15; plotoptions.Title.Color = [1 0 0]; plotoptions.FreqUnits = 'Hz'; plotoptions.Grid = 'on';```

Now, create a sigma plot using the options set `plotoptions`.

`h = sigmaplot(tf(1,[1,1]),plotoptions);`

Because `plotoptions` begins with a fixed set of options, the plot result is independent of the toolbox preferences of the MATLAB session.

For this example, create a sigma plot of the following continuous-time SISO dynamic system. Then, turn the grid on, rename the plot and change the frequency scale.

`$sys\left(s\right)=\frac{{s}^{2}+0.1s+7.5}{{s}^{4}+0.12{s}^{3}+9{s}^{2}}.$`

Create the transfer function `sys`.

`sys = tf([1 0.1 7.5],[1 0.12 9 0 0]);`

Next, create the options set using `sigmaoptions` and change the required plot properties.

```plotoptions = sigmaoptions; plotoptions.Grid = 'on'; plotoptions.FreqScale = 'linear'; plotoptions.Title.String = 'Singular Value Plot of Transfer Function';```

Now, create the sigma plot with the custom option set `plotoptions`.

`h = sigmaplot(sys,plotoptions);`

`sigmaplot` automatically selects the plot range based on the system dynamics.

For this example, consider a MIMO state-space model with 3 inputs, 3 outputs and 3 states. Create a sigma plot with linear frequency scale, frequency units in Hz and turn the grid on.

Create the MIMO state-space model `sys_mimo`.

```J = [8 -3 -3; -3 8 -3; -3 -3 8]; F = 0.2*eye(3); A = -J\F; B = inv(J); C = eye(3); D = 0; sys_mimo = ss(A,B,C,D); size(sys_mimo)```
```State-space model with 3 outputs, 3 inputs, and 3 states. ```

Create a sigma plot with plot handle `h` and use `getoptions` for a list of the options available.

`h = sigmaplot(sys_mimo);`

`p = getoptions(h)`
```p = FreqUnits: 'rad/s' FreqScale: 'log' MagUnits: 'dB' MagScale: 'linear' IOGrouping: 'none' InputLabels: [1x1 struct] OutputLabels: [1x1 struct] InputVisible: {0x1 cell} OutputVisible: {0x1 cell} Title: [1x1 struct] XLabel: [1x1 struct] YLabel: [1x1 struct] TickLabel: [1x1 struct] Grid: 'off' GridColor: [0.1500 0.1500 0.1500] XLim: {[1.0000e-03 1]} YLim: {[-25 15]} XLimMode: {'auto'} YLimMode: {'auto'} ```

Use `setoptions` to update the plot with the requires customization.

`setoptions(h,'FreqScale','linear','FreqUnits','Hz','Grid','on');`

The sigma plot automatically updates when you call `setoptions`.

For this example, compare the SV for the frequencies of a parametric model, identified from input/output data, to a non-parametric model identified using the same data. Identify parametric and non-parametric models based on the data.

Load the data and create the parametric and non-parametric models using `tfest` and `spa`, respectively.

```load iddata2 z2; w = linspace(0,10*pi,128); sys_np = spa(z2,[],w); sys_p = tfest(z2,2);```

`spa` and `tfest` require System Identification Toolbox™ software. The model `sys_np` is a non-parametric identified model while, `sys_p` is a parametric identified model.

Create an options set to turn the grid on. Then, create a sigma plot that includes both systems using this options set.

```plotoptions = sigmaoptions; plotoptions.Grid = 'on'; h = sigmaplot(sys_p,'b--',sys_np,'r--',w,plotoptions); legend('Parametric Model','Non-Parametric model');```

Consider the following two-input, two-output dynamic system.

`$H\left(s\right)=\left[\begin{array}{cc}0& \frac{3s}{{s}^{2}+s+10}\\ \frac{s+1}{s+5}& \frac{2}{s+6}\end{array}\right].$`

Plot the singular value responses of H(s) and I + H(s). Set appropriate titles using the plot option set.

```H = [0, tf([3 0],[1 1 10]) ; tf([1 1],[1 5]), tf(2,[1 6])]; opts1 = sigmaoptions; opts1.Grid = 'on'; opts1.Title.String = 'Singular Value Plot of H(s)'; h1 = sigmaplot(H,opts1);```

Use input 2 to plot the modified SV of type, I + H(s).

```opts2 = sigmaoptions; opts2.Grid = 'on'; opts2.Title.String = 'Singular Value Plot of I+H(s)'; h2 = sigmaplot(H,[],2,opts2);```

## Input Arguments

collapse all

Dynamic system, specified as a SISO or MIMO dynamic system model or array of dynamic system models. Dynamic systems that you can use include:

• Continuous-time or discrete-time numeric LTI models, such as `tf`, `zpk`, or `ss` models.

• Sparse state-space models, such as `sparss` or `mechss` models. Frequency grid `w` must be specified for sparse models.

• Generalized or uncertain LTI models such as `genss` or `uss` (Robust Control Toolbox) models. (Using uncertain models requires Robust Control Toolbox™ software.)

• For tunable control design blocks, the function evaluates the model at its current value to plot the SV.

• For uncertain control design blocks, the function plots the SV at the nominal value and random samples of the model.

• Frequency-response data models such as `frd` models. For such models, the function plots the SV at frequencies defined in the model.

• Identified LTI models, such as `idtf` (System Identification Toolbox), `idss` (System Identification Toolbox), or `idproc` (System Identification Toolbox) models. For such models, the function can also plot confidence intervals and return standard deviations of the frequency response. (Using identified models requires System Identification Toolbox™ software.)

Line style, marker, and color, specified as a character vector or string containing symbols. The symbols can appear in any order. You do not need to specify all three characteristics (line style, marker, and color). For example, if you omit the line style and specify the marker, then the plot shows only the marker and no line.

Example: `'--or'` is a red dashed line with circle markers

Line StyleDescription
`-`Solid line
`--`Dashed line
`:`Dotted line
`-.`Dash-dot line
MarkerDescription
`'o'`Circle
`'+'`Plus sign
`'*'`Asterisk
`'.'`Point
`'x'`Cross
`'_'`Horizontal line
`'|'`Vertical line
`'s'`Square
`'d'`Diamond
`'^'`Upward-pointing triangle
`'v'`Downward-pointing triangle
`'>'`Right-pointing triangle
`'<'`Left-pointing triangle
`'p'`Pentagram
`'h'`Hexagram
ColorDescription

`y`

yellow

`m`

magenta

`c`

cyan

`r`

red

`g`

green

`b`

blue

`w`

white

`k`

black

Frequencies at which to compute and plot SV of the frequency response, specified as the cell array `{wmin,wmax}` or as a vector of frequency values.

• If `w` is a cell array of the form `{wmin,wmax}`, then the function plots the SV at frequencies ranging between `wmin` and `wmax`.

• If `w` is a vector of frequencies, then the function plots the SV at each specified frequency. For example, use `logspace` to generate a row vector with logarithmically spaced frequency values.

Specify frequencies in units of rad/`TimeUnit`, where `TimeUnit` is the `TimeUnit` property of the model.

Option to plot modified singular values, specified as one of the following options:

• `1` to plot the SV of the frequency response H-1, where H is the frequency response of `sys`.

• `2` to plot the SV of the frequency response I+H.

• `3` to plot the SV of the frequency response I+H-1.

You can only use the `type` argument for square systems, that is, systems that have the same number of inputs and outputs.

Target axes, specified as an `Axes` object. If you do not specify the axes and if the current axes are Cartesian axes, then `sigmaplot` plots on the current axes.

Sigma plot options set, specified as a `SigmaPlotOptions` object. You can use this option set to customize the SV plot appearance. Use `sigmaoptions` to create the option set. Settings you specify in `plotoptions` overrides the preference settings in the MATLAB session in which you run `sigmaplot`. Therefore, `plotoptions` is useful when you want to write a script to generate multiple plots that look the same regardless of the local preferences.

For the list of available options, see `sigmaoptions`.

## Output Arguments

collapse all

Plot handle, returned as a `handle` object. Use the handle `h` to get and set the properties of the SV plot using `getoptions` and `setoptions`. For the list of available options, see the Properties and Values Reference section in Customizing Response Plots from the Command Line.

## Version History

Introduced before R2006a


text:
# Increase height of fancyhdr header

How would one increase the height of the \fancyhead{} so that the text in the main body, doesn't collide with the title?

I have the following MWE:

\documentclass[a4paper,10pt,oneside]{article}
\usepackage{fancyhdr}
\usepackage[english]{babel} % Required to compile in Windows
\usepackage[letterspace=150]{microtype}
\usepackage{fontspec}

\pagestyle{fancy}
\setmainfont {Adobe Garamond Pro} % Main document font

}

\begin{document}
\section*{Foo}
\end{document}

However, the section foo starts right in the header, therefore I want to add some vspace in between header and main body. I tried \vspace, \bigskip, and all other sorts of vertical space commands, but none worked.

Updated MWE:

\documentclass[a4paper,10pt,oneside]{article}
\usepackage{marvosym} % Allows the use of symbols
\usepackage[english]{babel} % Required to compile in Windows
\usepackage[letterspace=150]{microtype}
\usepackage{fontspec}
\usepackage{fancyhdr}

\pagestyle{fancy}
\setmainfont {Adobe Garamond Pro} % Main document font

}

\fancyfoot[C]{
\lsstyle
\scshape{
\renewcommand{\\}{\ {\large\textperiodcentered}\ }
Foo Street \\Bar\\Foocountry
}\\
{\Large\Letter} foo@bar.com \ {\Large\Telefon} (+001) 000-0000000
}

\begin{document}
\date{} % hide date
\section*{Foo Bar}
\end{document}

-

You log file says (in the warning)

Package Fancyhdr Warning: \headheight is too small (12.0pt):
Make it at least 103.50645pt.
We now make it that large for the rest of the document.
This may cause the page layout to be inconsistent, however.

Hence a safe head height will be around 110pt. To change this, you can use

\setlength{\headheight}{110pt}

or

\usepackage[headheight=110pt,showframe]{geometry}

# Code:

\documentclass[a4paper,10pt,oneside]{article}
\usepackage{fancyhdr}
\usepackage[english]{babel} % Required to compile in Windows
\usepackage[letterspace=150]{microtype}
\usepackage{fontspec}

\pagestyle{fancy}
\setmainfont {Adobe Garamond Pro} % Main document font

\addfontfeature{LetterSpace=20.0}\fontsize{30}{30}\selectfont\scshape Foo Bar Camp Names\\[5mm] %% why this 5mm vertical space??
}

\begin{document}
\section*{Foo}
\end{document}

As a side note, I did not understand the reason for 5mm vertical space you left in the header. I think you tried to correct the spacing. If so, better way is to use geometry and adjust head height. Without this 5mm the head height of 65pt is sufficient.

# Update:

\documentclass[a4paper,10pt,oneside]{article}
\usepackage{marvosym} % Allows the use of symbols
\usepackage[english]{babel} % Required to compile in Windows
\usepackage[letterspace=150]{microtype}
\usepackage{fontspec}
\usepackage{fancyhdr}

\pagestyle{fancy}
\setmainfont {Adobe Garamond Pro} % Main document font

}

\fancyfoot[C]{
\lsstyle
\scshape{
\renewcommand{\\}{\ {\large\textperiodcentered}\ }
Foo Street \\Bar\\Foocountry
}\\
{\Large\Letter} foo@bar.com \ {\Large\Telefon} (+001) 000-0000000
}

\begin{document}
\date{} % hide date
\section*{Foo Bar}
\end{document}

The gap between the header and the main body is decided by the length headsep. This can be controlled by the option (to geometry) as headsep=1cm (change the length 1cm as needed):

\usepackage[top=1.5in,bottom=1in,right=1in,left=1in,headheight=65pt,headsep=1cm]{geometry}

-
Thanks! The 5mm was a remainder from my actual code, where I had a second headline. My LaTeX parser didn't give me the warning you got... weird. –  cherrung Dec 2 '12 at 11:50
@cherrung -- when you say your latex parser didn't give the warning, do you mean on your screen, or did you look in the log? the log contains a lot more information than what's shipped out to the screen, and sometimes it's really important. –  barbara beeton Dec 2 '12 at 13:27
@cherrung: See the updated answer. You can adjust the headsep. Give suitable values as you need. Also play with other values like the margin. –  Harish Kumar Dec 3 '12 at 0:54


text:
# Solving a convex resource allocation problem

I want to solve the following optimization problem

$$\begin{array}{ll} \text{maximize} & \displaystyle\sum_{i=1}^n \log_{2}(1+a_i x_i)\\ \text{subject to} & \displaystyle\sum_{i=1}^n (c_i - x_i) a_i = b\\ & 0 \leq x_i \leq c_i\end{array}$$

where $a_i \geq a_2 \geq \dots \geq a_n$, $b \in \mathbb R^+$ and $c_1 = c_2 = \dots = c_n$ are given.

I tried to solve this problem using Lagrange multipliers

$$\mathcal{L}(x,\lambda,\mu,\nu)=\sum_{i=1}^n\log_{2}(1+a_ix_i) + \lambda \left( \sum_{i=1}^n(c_i-x_i)a_i - b \right) - \mu_i (x_i-c_i) + \nu_ix_i$$

We need to have $\frac{d\mathcal{L}}{dx_i}=0$; therefore

$$\sum_{i=1}^n \frac{a_i}{\ln(2)\times(1+a_ix_i)}-\lambda a_i-\mu_{i}+\nu_{i}=0$$

In particular, my problem always has

$$\sum_{i=1}^{n-1} a_i c_i < b < \sum_{i=1}^n a_i c_i$$

How can I obtain the optimal values of $x_i$

• One "trick" would be to discard the inequality constraints and add $$\gamma \displaystyle\sum_{i=1}^n \left( \log (x_i) + \log (c_i -x_i) \right)$$ to the objective function, where $\gamma > 0$. In other words, "hard" constraints are replaced by "soft" constraints. – Rodrigo de Azevedo Mar 28 '18 at 14:15
• @RodrigodeAzevedo Thank you for your response.Does this mean that I need to solve the following optimization problem$\gamma \displaystyle\sum_{i=1}^n \left( \log (x_i) + \log (c_i -x_i) \right)$constrained by $\sum_{i}^{n}(c_i-x_i)a_i=b$. What abot $\gamma$? How can I deal with it? – user841315 Mar 28 '18 at 17:13
• No, you solve $$\begin{array}{ll} \text{maximize} & \displaystyle\sum_{i=1}^n \log_{2}(1+a_i x_i) + \gamma \displaystyle\sum_{i=1}^n \left( \log (x_i) + \log (c_i -x_i) \right)\\ \text{subject to} & \displaystyle\sum_{i=1}^n (c_i - x_i) a_i = b\end{array}$$ As you vary parameter $\gamma > 0$, you weigh more the original objective or the soft constraints. The optimal solution should not be the same as the optimal solution of the original problem, but it should be much easier to find. – Rodrigo de Azevedo Mar 28 '18 at 17:31
• @RodrigodeAzevedo Thank you for the clarification. It seems the optimal values of $x_i$ can be obtained only numerically. I thought an analytical solution exists for this problem. – user841315 Mar 28 '18 at 18:26

I think your best approach is duality. First, note that $\log_2$ and $\ln$ differ by a multiplicative constant $\ln(2)$. Thus, we can minimize with $\ln$ and obtain the same optimal value, and I will use $\ln$ instead of $\log_2$ in my derivations. Second, I will treat it as a minimization problem, and therefore write $-\ln$ instead of $\ln$.

Giving a Lagrange multiplier only to the equality constraint, you obtain the Lagrangian \begin{aligned} L(x, \lambda) &= -\sum_{i=1}^n \ln(1+a_i x_i) + \lambda \left[ \sum_{i=1}^n (c_i - x_i) a_i - b \right] \\ &= \sum_{i=1}^n \left[-\ln(1+a_i x_i) - \lambda a_i x_i \right] + \lambda \Bigl[\underbrace{-b + \sum_{i=1}^n a_i c_i}_{C} \Bigr] \end{aligned} Since the Lagrangian and the constraints $0 \leq x_i \leq c_i$ are both separable in $x_i$, the dual objective is: $$q(\lambda) = \sum_{i=1}^n \min_{x_i} \biggl\{ -\ln(1+a_i x_i) - \lambda a_i x_i :~ 0 \leq x_i \leq c_i \biggr\} + \lambda C$$ Note, that each of the elements in the sum above is a minimization of a convex function of the form $-\ln(1+ax) -bx$ over an interval, and it is quite easy to find a closed form solution (if you need help, please say so). From this solution you will also obtain a formula from recovering the optimal $x_i$ from the optimal $\lambda$.

Now, your dual problem is the one-dimensional optimization problem $\max_{\lambda} q(\lambda)$, which can be easily solved by a numerical method, such as the Golden Section search. After finding the optimal $\lambda^*$ you can recover $x^*$.

P.S. If you really want a really reliable method in terms of approximating the optimum in terms of the objective value, you can compute an approximate $x_i$ after every iteration of the the Golden Section search, and stop when the primal-dual gap is below some threshold.

# Update

Since there are many questions about computing $q(\lambda)$, here are the details. First, I will denote by $g_i(x_i, \lambda)$ the objective inside the minimum. That is, we solve $$\min_{x_i} \biggl\{g_i(x_i, \lambda) = -\ln(1+a_i x_i) - \lambda a_i x_i :~ 0 \leq x_i \leq c_i \biggr\}$$ The minimum is either a stationary point inside the interior of the interval $I = [0, c_i]$ or one of its endpoints. First, note that if $\lambda \geq 0$ then $g_i$ is a decreasing function, and in this case there are no stationary points. Thus, for any stationary point we must have $\lambda < 0$.

To find a stationary point, we solve: $$\frac{d g_i}{d x_i} = -\frac{a_i}{1+a_i x_i} - \lambda a_i = 0.$$ Since $a_i > 0$ we can divide both sides by $a_i$, and obtain the equation $$\frac{1}{1+a_i x_i} = -\lambda$$ Since $\lambda \neq 0$, we take the inverse on both sides, multiply by $a_i$, and obtain $$1+a_i x_i = -\frac{1}{\lambda},$$ which is equivalent to $$a_i x_i = -1-\frac{1}{\lambda},$$ or $$x_i = -\frac{1+\lambda}{a_i \lambda}.$$ Using the fact that $\lambda < 0$, we conclude that the point above is inside the interior of $I$ if and only if $$-1 < \lambda < -\frac{1}{1 + c_i a_i},$$ in that case the expression for $x_i$ above is the minimizer, and the minimum is $$g_i\left(-\frac{1+\lambda}{a_i \lambda}, \lambda\right) = \ln(-\lambda) + \lambda + 1$$ Otherwise, either the left endpoint $x_i=0$ or the right endpoint $x_i=c_i$ is the minimizer. To summarize, $$q_i(\lambda) = \min_{x_i} \biggl\{g_i(x_i, \lambda) :~ 0 \leq x_i \leq c_i \biggr\} = \begin{cases} \ln(-\lambda) + \lambda + 1 & -1 < \lambda < -\frac{1}{1 + c_i a_i} \\ \min(0, -\ln(1+a_i c_i)-\lambda a_i c_i) & \text{otherwise} \end{cases}$$ The dual objective is $$q(\lambda) = \sum_{i=1}^n q_i(\lambda) + \lambda C$$ It is possible to simplify the expressions above, but it is not required to do so in order to write code which evaluates $q(\lambda)$ in order to solve it numerically.

• Thank you for your efforts. Do you mean that I need to minimize the function $-\ln(1+ax)-bx$ to obtain $x$ as a function of $a$ and $b$ then substitute this $x$ in the given Lagrangian $q(\lambda)$ and finally numerically solve $\text{max}_{\lambda}q(\lambda)$? – user841315 Mar 28 '18 at 20:48
• Exactly. You need to find a closed form expression for the minimizer $x$ and the resulting minimum as a function of $a,b$ and the endpoints of the interval. In that way you will have a formula for $q$ and you can use it to numerically maximize $q$. – Alex Shtof Mar 29 '18 at 7:08
• I have computed the minimizer $x$ by taking the derivative of $\tilde{f}(x)=-\ln(1+ax)-bx$ then $\tilde{f}'(x)=0$ so we have $x=\frac{-a-b}{ab}$. I'm confused about the value of $x$ because it always returns a negative number and the $\ln$-function is undefined then – user841315 Mar 29 '18 at 10:05
• First, the sign of $-b$ is not always negative in our case So it depends on whether $\lambda$ is positive or negative. Second and much more important, when minimizing over an interval you cannot just use the stationary point obtained from the "derivative=zero" equation. You need to take the endpoints of the interval into account too. – Alex Shtof Mar 29 '18 at 13:15
• The strong duality theorem ensures that a pair feasible solutions, one for the primal and the other for the dual, are optimal if and only if the objectives of both problems at these points are equal. You can certify optimality by taking the results you obtained numerically and show that the difference between the primal and the dual objectives is less than some very small threshold, say $10^{-6}$. – Alex Shtof Apr 3 '18 at 5:41


text:
# Hopfian group

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
This is a variation of finite group|Find other variations of finite group |

This property makes sense for infinite groups. For finite groups, it is always true

## Definition

A group is termed Hopfian if it satisfies the following equivalent conditions:

## Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
subgroup-closed group property No Hopfianness is not subgroup-closed It is possible to have a Hopfian group $G$ and a subgroup $H \le G$ such that $H$ is not Hopfian.
quotient-closed group property No Hopfianness is not quotient-closed It is possible to have a Hopfian group $G$ and a normal subgroup $H$ of $G$ such that the quotient group $G/H$ is not Hopfian.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
finite group order is finite Finitely generated Hopfian group, Finitely generated profinite group, Finitely generated residually finite group, Finitely presented conjugacy-separable group, Group satisfying ascending chain condition on normal subgroups, Noetherian group|FULL LIST, MORE INFO
finitely generated free group free group on finite generating set finitely generated and free implies Hopfian Finitely generated Hopfian group, Finitely generated residually finite group, Finitely presented conjugacy-separable group, Finitely presented residually finite group|FULL LIST, MORE INFO
finitely generated residually finite group finitely generated and residually finite finitely generated and residually finite implies Hopfian Finitely generated Hopfian group|FULL LIST, MORE INFO
group satisfying ascending chain condition on normal subgroups there is no infinite strictly ascending chain of normal subgroups ascending chain condition on normal subgroups implies Hopfian |FULL LIST, MORE INFO
Noetherian group (also called slender group) every subgroup is finitely generated; equivalently, no infinite strictly ascending chain of subgroups Noetherian implies Hopfian (proof is ascending chain condition on normal subgroups) Finitely generated Hopfian group, Group satisfying ascending chain condition on normal subgroups, Group satisfying ascending chain condition on subnormal subgroups|FULL LIST, MORE INFO
finitely generated Hopfian group finitely generated and Hopfian (by definition) Hopfian not implies finitely generated |FULL LIST, MORE INFO
simple group has no proper nontrivial normal subgroups simple implies Hopfian Group in which every endomorphism is trivial or injective, Group satisfying ascending chain condition on normal subgroups, Group satisfying ascending chain condition on subnormal subgroups|FULL LIST, MORE INFO
group in which every endomorphism is trivial or an automorphism every endomorphism is either trivial or is an automorphism every endomorphism is trivial or an automorphism implies Hopfian Group in which every endomorphism is trivial or injective|FULL LIST, MORE INFO
Group in which every endomorphism is trivial or injective every endomorphism is either trivial or an injective endomorphism every endomorphism is trivial or injective implies Hopfian (finite counterexamples) |FULL LIST, MORE INFO
finitely generated abelian group finitely generated abelian implies Hopfian (also via Noetherian) any finite non-abelian group Finitely generated Hopfian group, Finitely generated conjugacy-separable group, Finitely generated residually finite group, Finitely presented conjugacy-separable group, Finitely presented residually finite group, Noetherian group|FULL LIST, MORE INFO
finitely generated nilpotent group finitely generated and nilpotent implies Hopfian (via Noetherian) (via Noetherian, also any finite non-nilpotent counterexample) Noetherian group|FULL LIST, MORE INFO


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Papers:

 Tame covers of curves: p-ranks and fundamental groups PhD thesis. The p-rank of ramified covers of curves Comp. Math. 126(2001), no. 3, 295--322. The p-rank of curves and covers of curves Courbes semi-stables et groupe fondamental en géométrie algébrique (Luminy, 1998), 267--277, Progr. Math., 187, Birkhäuser, Basel, 2000. Reduction of covers and Hurwitz spaces.  With Stefan Wewers. Crelle J. 574 (2004), 1--49. Rigidity, Reduction, and Ramification. With Rachel Pries. Math. Ann 326 (2003), 803--824. Stable reduction of modular curves. With Stefan Wewers. math.AG/0210363. Modular curves and abelian varieties, Progress in Math. 224 (2004), 1--22, Birkhäuser. Reduction of the Hurwitz space of metacyclic covers. Duke Math. J. 121 (2004), 75--111. Ordinary elliptic curves of high rank over $\bar F_p(x)$ with constant j-invariant. With Claus Diem and Jasper Scholten. Manuscripta Math. 114 (2004), 487--501 Alternating groups as monodromy groups in positive characteristic. With Stefan Wewers. Pacific J. Math. 222 (2005), 185-?? Asymptotically good towers and differential equations. With Peter Beelen. Compositio Math. 141 (2005), 1405--1424. The local lifting problem for dihedral groups. With Stefan Wewers. Duke Math J. 134 (2006), 412--452. Rational functions with given monodromy on generic curves. With Stefan Wewers. Problems from the workshop on "automorphisms of curves" (Leiden 2004). Rend. Sem. Math. Univ. Padova. 113 (2005), 129--177. Invariants of ternary quadratic forms. Asterique 312, 2007. Pseudo-elliptic bundles, deformation data, and the reduction of Galois covers Habilitation thesis. Indigenous bundles with nilpotent p-curvature. With Stefan Wewers. Internat. Math. Res. Notices (2006), 1-37. Teichmueller curves, triangle groups, and Lyapunov exponents. With Martin Moeller. Annals of Math. 172 (2010). The accessory parameter problem in positive characteristic. Preprint Construction of covers in positive characteristic via degeneration. Proc. A.M.S. 137 (2009), 3169--3176. Differential equations associated with nonarithmetic Fuchsian groups With Martin Moeller Journal of the London Math. Soc. 81 (2010), 65--90. Deformation data, Belyi maps, and the local lifting problem, with Stefan Wewers and Leonardo Zapponi. Trans. AMS. 362 (2009), 6645--6659. Some 4-point Hurwitz numbers in positive characteristic, with Brian Osserman. Trans. AMS.303, 2011, 6685–6711. Covers of the affine line in positive characteristic with prescribed ramification In WIN, Fields Institute Communications 60, 193-200, 2011. Syndrom decoding for Hermite codes with a Sugiyama algorithm, with Sabine Kampf Advances of math Communications 6, 2012, 419-442. Existence of covers with fixed ramification in positive characteristic, with Leonardo Zapponi IJNT 9(2013), 1475-1489, arXiv:1206.1262 Computing L-functions and semistable reduction of superelliptic curves, with Stefan Wewers Glasgow math J. 59,  77-108, 2017. arXiv:1211.4459 Bad reduction of genus-3 curves with complex multiplication, with Jenny Cooley, Kristin Lauter, Elisa Lorenzo Garcia, Michelle Manes, Rachel Newton, Ekin Ozman. in: Women in numbers - Europe, research directions in number theory, AWM series Vol. 2, Bertin  et al (eds), 109-151, Springer,  2015. arXiv:1407.3589 Zeta functions of a class of Artin-Schreier curves with many automorphisms, with Wei Ho, Beth Malmskog, Renate Scheidler, Padma Srinivasan, and Christelle Vincent. in: Directions in Number Theory. Proceedings of the 2014 WIN Workshop. AWM series Vol. 3, Eischen et al (eds), 87-124, Springer, 2016. arXiv:1410.7031 The functional equation for L-functions of hyperelliptic curves, with Michel Börner and Stefan Wewers. Experimental Math. 16 (2017), 396-411. arXiv:1504.00508 Picard curves with small conductor, with Michel Börner and Stefan Wewers. In: Algorithmic and experimental methods in algebra, geometry, and number theory, 97-122, Springer, 2017. arXiv:1701.01986 Dynamical Belyi maps, with Jackie Anderson, Özlem Ejder, Neslihan Girgin, Valentijn Karemaker, and Michelle Manes. In: Women in Numbers Europe II, Contributions to arithmetic geometry and number theory, AWM Vol. 11, Bouw et al (eds), 57-82, Springer, 2018. arXiv:1703.08563. Improved power decoding of interleaved one-point Hermitian codes, with Sven Puchinger and Johan Rosenkilde. Designs, Codes and Cryprography 87 (2019), 589–607. arXiv: 1801.07006. Dynamical Belyi maps and arboreal Galois groups, with Özlem Ejder and Valentijn Karemaker. arXiv:1811.10086. To appear in Manuscripta Math. Conductor and discriminant of Picard curves, with Angelos Koutsianas, Jeroen Sijsling, and Stefan Wewers. arXiv:1902.09624. To appear in Journal of the London Math. Soc. Reduction type of genus-3 curves in a special stratum of their moduli space, with Nirvana Coppola, Pinar Kilicer, Sabrina Kunzweiler, Elisa Lorenzo Garcia, and Anna Somoza. arXiv:2003.07633. To appear in Women in Numbers Europe III.


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# vector question in multivariable calculus

• Nov 30th 2006, 01:14 AM
Jenny20
vector question in multivariable calculus
Question
If two forces Fsub1 and Fsub2 each of magnitude 10N act on an object from perpendicular directions, then the effect is the same as that of a single force of 10N acting in a direction 45 degrees to each of the forces Fsub1 and Fsub2.

The answer is false.

Could you explain to me why the answer is false? If possible , could you please show me the working step? Thank you very much.
• Nov 30th 2006, 03:55 AM
topsquark
Quote:

Originally Posted by Jenny20
Question
If two forces Fsub1 and Fsub2 each of magnitude 10N act on an object from perpendicular directions, then the effect is the same as that of a single force of 10N acting in a direction 45 degrees to each of the forces Fsub1 and Fsub2.

The answer is false.

Could you explain to me why the answer is false? If possible , could you please show me the working step? Thank you very much.

Add the two forces vectorally.

Let $\vec{F}_1 = 10 \hat{i}$ N and $\vec{F}_2 = 10 \hat{j}$ N. Then
$\vec{F}_{tot} = \vec{F}_1 + \vec{F}_2$

$\vec{F}_{tot} = 10\hat{i} + 10\hat{j}$ N

which has a magnitude of $\left | \vec{F}_{tot} \right | = \sqrt{10^2 + 10^2} = 10\sqrt{2}$ N, not 10 N.

Note: The angle mentioned in the problem is correct. I leave that to you to show.

-Dan


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# Communications of the ACM

### 121 - 130 of 2,536 for bentley

#### Personal knowledge questions for fallback authentication: security questions in the era of Facebook

Security questions (or challenge questions) are commonly used to authenticate users who have lost their passwords. We examined the password retrieval mechanisms for a number of personal banking websites, and found that many of them rely in part on security questions with serious usability and security weaknesses. We discuss patterns in the security questions we observed. We argue that today's personal security questions owe their strength to the hardness of an information-retrieval problem. However, as personal information becomes ubiquitously available online, the hardness of this problem, and security provided by such questions, will likely diminish over time. We supplement our survey of bank security questions with a small user study that supplies some context for how such questions are used in practice.

https://dl.acm.org/ft_gateway.cfm?id=1408667&dwn=1

#### BSkyTree: scalable skyline computation using a balanced pivot selection

Skyline queries have gained a lot of attention for multi-criteria analysis in large-scale datasets. While existing skyline algorithms have focused mostly on exploiting data dominance to achieve efficiency, we propose that data incomparability should be treated as another key factor in optimizing skyline computation. Specifically, to optimize both factors, we first identify common modules shared by existing non-index skyline algorithms, and then analyze them to develop a cost model to guide a balanced pivot point selection. Based on the cost model, we lastly implement our balanced pivot selection in two algorithms, BSkyTree-S and BSkyTree-P, treating both dominance and incomparability as key factors. Our experimental results demonstrate that proposed algorithms outperform state-of-the-art skyline algorithms up to two orders of magnitude.

https://dl.acm.org/ft_gateway.cfm?id=1739067&dwn=1

#### An experimental investigation of set intersection algorithms for text searching

The intersection of large ordered sets is a common problem in the context of the evaluation of boolean queries to a search engine. In this article, we propose several improved algorithms for computing the intersection of sorted arrays, and in particular for searching sorted arrays in the intersection context. We perform an experimental comparison with the algorithms from the previous studies from Demaine, López-Ortiz, and Munro [ALENEX 2001] and from Baeza-Yates and Salinger [SPIRE 2005]; in addition, we implement and test the intersection algorithm from Barbay and Kenyon [SODA 2002] and its randomized variant [SAGA 2003]. We consider both the random data set from Baeza-Yates and Salinger, the Google queries used by Demaine et al., a corpus provided by Google, and a larger corpus from the TREC Terabyte 2006 efficiency query stream, along with its own query log. We measure the performance both in terms of the number of comparisons and searches performed, and in terms of the CPU time on two different architectures. Our results confirm or improve the results from both previous studies in their respective context (comparison model on real data, and CPU measures on random data) and extend them to new contexts. In particular, we show that value-based search algorithms perform well in posting lists in terms of the number of comparisons performed.

https://dl.acm.org/ft_gateway.cfm?id=1564507&dwn=1

#### Adding an interactive display to a public basketball hoop can motivate players and foster community

Interactive displays that aim to engage people through play have been successfully deployed in urban environments. However, there has been little work bringing interactive displays into existing public game spaces like outdoor basketball courts. To explore this, we designed an interactive display for a public half-court basketball hoop. We studied the impact of 3 different display modes over a 10-week period through interviews with players, spectators, and passers-by. Our findings suggest 3 dimensions for the design space of such interactive displays: balancing noticeability across different user groups, support for different play action, and support for connecting user groups. We also present 6 design tactics along these dimensions to help designers create engaging interactive displays for public game spaces. using it to facilitate engaging playful experiences.

https://dl.acm.org/ft_gateway.cfm?id=2493478&dwn=1

#### Social audio features for advanced music retrieval interfaces

The size of personal music collections has constantly increased over the past years. As a result, the traditional metadata based lists to browse these collections have reached their limits. Interfaces that are based on music similarity offer an alternative and thus are increasingly gaining attention. Music similarity is typically either derived from audio-features (objective approach) or from user driven information sources, such as collaborative filtering or social tags (subjective approach). Studies show that the latter techniques outperform audio-based approaches when it comes to describe the perceived music similarity. However, subjective approaches typically only define pairwise relations as opposed to the global notion of similarity given by audio-feature spaces. Many of the proposed interfaces for similarity based music access inherently depend on this global notion and are thus not applicable to user driven music similarity measures. The first contribution of this paper is a high dimensional music space that is based on user driven similarity measures. It combines the advantages of audio-feature spaces (global view) with the advantages of subjective sources that better reflect the users' perception. The proposed space compactly represents similarity and therefore is well suited for offline use, such as in mobile applications. To demonstrate the practical applicability, the second contribution is a comprehensive mobile music player that incorporates several smart interfaces to access the user's music collection. Based on this application, we finally present a large-scale user study that underlines the benefits of the introduced interfaces and shows their great user acceptance.

https://dl.acm.org/ft_gateway.cfm?id=1874007&dwn=1

#### Finding Probabilistic k-Skyline Sets on Uncertain Data

Skyline is a set of points that are not dominated by any other point. Given uncertain objects, probabilistic skyline has been studied which computes objects with high probability of being skyline. While useful for selecting individual objects, it is not sufficient for scenarios where we wish to compute a subset of skyline objects, i.e., a skyline set. In this paper, we generalize the notion of probabilistic skyline to probabilistic k-skyline sets (Pk-SkylineSets) which computes k-object sets with high probability of being skyline set. We present an efficient algorithm for computing probabilistic k-skyline sets. It uses two heuristic pruning strategies and a novel data structure based on the classic layered range tree to compute the skyline set probability for each instance set with a worst-case time bound. The experimental results on the real NBA dataset and the synthetic datasets show that Pk-SkylineSets is interesting and useful, and our algorithms are efficient and scalable.

https://dl.acm.org/ft_gateway.cfm?id=2806452&dwn=1

#### Sampling Big Trajectory Data

The increasing prevalence of sensors and mobile devices has led to an explosive increase of the scale of spatio-temporal data in the form of trajectories. A trajectory aggregate query, as a fundamental functionality for measuring trajectory data, aims to retrieve the statistics of trajectories passing a user-specified spatio-temporal region. A large-scale spatio-temporal database with big disk-resident data takes very long time to produce exact answers to such queries. Hence, approximate query processing with a guaranteed error bound is a promising solution in many scenarios with stringent response-time requirements. In this paper, we study the problem of approximate query processing for trajectory aggregate queries. We show that it boils down to the distinct value estimation problem, which has been proven to be very hard with powerful negative results given that no index is built. By utilizing the well-established spatio-temporal index and introducing an inverted index to trajectory data, we are able to design random index sampling (RIS) algorithm to estimate the answers with a guaranteed error bound. To further improve system scalability, we extend RIS algorithm to concurrent random index sampling (CRIS) algorithm to process a number of trajectory aggregate queries arriving concurrently with overlapping spatio-temporal query regions. To demonstrate the efficacy and efficiency of our sampling and estimation methods, we applied them in a real large-scale user trajectory database collected from a cellular service provider in China. Our extensive evaluation results indicate that both RIS and CRIS outperform exhaustive search for single and concurrent trajectory aggregate queries by two orders of magnitude in terms of the query processing time, while preserving a relative error ratio lower than 10\%, with only 1% search cost of the exhaustive search method.

https://dl.acm.org/ft_gateway.cfm?id=2806422&dwn=1

#### CHI '07: Proceedings of the SIGCHI Conference on Human Factors in Computing Systems

Welcome to the CHI 2007 proceedings. We believe the technical papers and notes herein present some of the best current work in the diverse and dynamic field of human-computer interaction (HCI).

CHI is the leading HCI conference. Creating the technical program requires a huge investment of time and effort from members of the research community. 840 submissions were processed (571 papers, 269 notes), requiring over 3000 reviews. We thank all the reviewers for the dedication with which they undertook this task. We are particularly indebted to the papers and notes program committee members, also known as the Associate Chairs (ACs). Balancing areas of expertise, ACs were selected from the field's leading researchers. The AC role included recruiting all reviewers, moderating and supervising the review process to ensure a high-quality set of reviews was obtained, initiating and organizing author rebuttal and reviewer discussions and approving final submissions. The estimated time expenditure to serve as an AC was 11 days of full-time work; many committee members spent more time than that. Papers ACs came to San Jose in December 2006 from around the world for two intense days of review, debates, and deliberation; Notes ACs who could not attend the parallel notes meeting in San Jose engaged in a virtual conference. The committee was extremely serious and careful in making CHI paper and note decisions, with many submissions receiving multiple discussions, before and during the program committee meetings. No review process can guarantee perfect decisions, but we are confident that every possible effort was made to ensure fair process and high quality decision-making. This year's program committee certainly has our respect and gratitude, and deserves the sincere appreciation of the entire HCI community. We would also like to thank the ACs and their organizations for underwriting the travel expenses for meeting.

CHI is both a journal-quality archival forum and a community-building conference. To encourage quality in the written presentation of accepted work, all of the 142 full paper and 40 note acceptances were provisional. As a result, authors actively responded and incorporated feedback from the reviews into the final versions of the papers that appear here.

Twenty-eight accepted papers and four accepted notes (5% of submissions) deemed to make an especially noteworthy contribution to human-computer interaction research were nominated by the program committee for Best Paper and Best Note Awards; these nominated papers and notes are identified in the Final Program. At the conference, up to six of these will be announced as winners of a CHI Best Paper Award (1% of submissions), and one note will be selected as an exemplary note. While all papers accepted into the CHI technical papers program have passed a rigorous examination of their quality, the Best Paper and Best Notes Awards signal and reward particularly outstanding contributions in each year.

https://dl.acm.org/ft_gateway.cfm?id=1240624&dwn=1

#### CHI '08: Proceedings of the SIGCHI Conference on Human Factors in Computing Systems

https://dl.acm.org/ft_gateway.cfm?id=1357054&dwn=1

#### Exact L∞ nearest neighbor search in high dimensions

We present an algorithm for solving the nearest neighbor problem with respect to $L_{\infty}$-distance. It requires no preprocessing and storage only for the point set $P$. Its average runtime assuming that the set $P$ of $n$ points is drawn randomly from the unit cube $[0,1]^{d}$ under uniform distribution is essentially $\Theta (nd/ln\; n)$ thereby improving the brute-force method by a factor of $\Theta (1/ln\; n)$. Several generalizations of the method are also presented, in particular to other “well-behaved” probability distributions and to the important problem of finding the $k$ nearest neighbors to a query point.

https://dl.acm.org/ft_gateway.cfm?id=378655&dwn=1


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# Use Gauss' Law to Derive the Expression for the Electric Field (Vece)Due to a Straight Uniformly Charged Infinite Line of Charge Density λ C/M. - CBSE (Science) Class 12 - Physics

#### Question

Use Gauss' law to derive the expression for the electric field (vecE)due to a straight uniformly charged infinite line of charge density λ C/m.

#### Solution

Field due to an infinitely long straight uniformly charged wire

Consider a thin, infinitely long straight line charge of linear charge density λ.

Let P be the point at a distance a from the line. To find the electric field at point P, draw a cylindrical surface of radius ‘a’ and length l.

If E is the magnitude of electric field at point P, then electric flux through the Gaussian surface,

Φ = E × Area of the curved surface of a cylinder of radius r and length l

As electric lines of force are parallel to end faces (circular caps) of the cylinder, there is no component of the field along the normal to the end faces.

Φ = E × 2πal … (i)

According to Gauss's Theorem,

phi = q/epsilon_0

∵ q = lambdal

:. phi = (lambdal)/epsilon_0  ...(ii)

From equations (i) and (ii), we get:

E xx 2pial = (lambdal)/epsilon_0

Is there an error in this question or solution?

#### Video TutorialsVIEW ALL [4]

Solution Use Gauss' Law to Derive the Expression for the Electric Field (Vece)Due to a Straight Uniformly Charged Infinite Line of Charge Density λ C/M. Concept: Applications of Gauss’s Law.
S


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# Permutations/Combinations - Could someone please check my work?

1. Nov 24, 2005

### rocketboy

Hey everyone,

I have an assignment on Permutations and Combinations and was wondering if some of you could go over my work and ensure that I have this aced. I really need to get my math mark up for University admissions so I need to ace this assignment.

Thanks a lot for your help!
-Jonathan

1) In how many distinguishable ways can the letters of the word MISSISSIPPI be arranged?

Number of distinguishable arrangements = 11!/(4!4!2!) = 34,650

2) Find the number of ways of arranging all the letters AAABBBCCDD in a row so that no two A's are side-by-side.

Number of arrangements = 7!/(3!3!2!2!) = 35

what I did: I paired the letter A up with another letter, by viewing them as 1 letter no 2 A's would ever be side by side. This makes 7 letters, and then I divided out the repeats.

3) How many distinguishable four letter arrangements can be made from teh letters of the word MISSISSIPPI?

Number of four letter arrangements = (ways with no repeats) + (ways with all 4 letters repeated) + (ways with 2 doubles) + (ways with a triple and a single)

= (4c4)x4 + (2c1) + (3c2)x(4!/2!2!) + (2c1)(1c1)x(4!/3!)
= 24 + 2 + 18 + 8
= 52

*****If it is unclear what I have done hear please ask me*****

Those are the first 3 questions out of 5. I want to make sure I'm on the right track before I finish the next 2, as they are significantly more difficult.

2. Nov 24, 2005

### rocketboy

I don't need to know the exact answers, just if i'm correct or not.

Thanks.

Last edited: Nov 24, 2005
3. Nov 26, 2005

### IB

You got #1 right. It is indeed (11!)/(4!*4!*2!) = 34,650.
--------------------------------------------------------

For #2:

Step one: calculate all possible ways taking into account repetitions: 10!/(3!*3!*2!*2!)

Step two: consider AAA as one letter and thus calculate in how many varinats they are to gether (8!/(3!*2!*2!)

Step three: consider AA as one letter and calculate in how many cases they are together: (9!/(3!2!2!)

Step four: substract step 3 and step 2 from step one
-----------------------------------------------------

For #3 I'm still thinking. The only solution I can see right now is to count it manually. But meh, that isn't a good approach to combinatorics. Allow me a few more minutes to think of a better solution. Will post again soon.

Last edited: Nov 26, 2005
4. Nov 26, 2005

### IB

Okay, for #3, there can be many cases (possibilities-which letters we choose, not their arrangements in the 4 letter word):

(a) All four letters are different (MISP) 4*3*2*1=24

(b) If two letters repeat (2I, 2P or 2S) AND there is M

There are 6 possibilities (M2IP/S; M2PI/S; M2SP/I).
Therefore: 6*([4*3*2*1]/2)=72

(c) If 2 and 2 letters repeat then there are 3 possibilities (2I2P; 2I2S; 2P2S)
Therefore: 3*([4*3*2*1]/4)=18

(d) If 3 letters repeat there are 3 possibilities (3SM; 3SP; 3SI)
Therefore 4*3=12

Therefore we obtain: 24+72+18+12=126

I hope you got the method.

5. Nov 28, 2005

### rocketboy

Is this correct? Can anybody verify please?

6. Nov 28, 2005

### rocketboy

Ok, that is what I was doing in my first post, so far so good.:tongue2:

Ooooh...kk i forgot this. THANKS IB!

EDIT: I just worked this part out myself, and got a different answer for this case.

There are 3 sets of letters to choose doubles from, and we want 1 of them to be chosen. Then we want 2 singles from the 3 sets of singles that are left to be chosen from. (ex, if 2 I's are chosen, we need 2 singles, fro meither S, P or M.) Then we have to multiply by 4! (for different positions) and divide by 2! (the repeats).

Thus we get (3c1)*(3c2)*(4!/2!) = 108 ways.

Hmmm....I don't think this is correct. I had the same idea, and now that I think of it I think we are both wrong.

for each of the triples that can occur, there has to be another letter present. This means that if there are 3 I's, then the fourth can be an S, M or P. And, they can be in four different spots. I's and S's are the only letters that can give tripples. For each of them, there is a fourth letter needed, one of 3 letters possible. Then, for each of these arrangements there are 4! ways of arranging the letters. Thus I think the answer for this case is (2c1)*(3c1)*4!

Where (2c1) means "two choose one" (combinations).

So there should be 144 in this case....wierd, that can't be.

Can anyone help?

EDIT: I forgot to divide that answer by 3! to get rid of the repeats. This means that there are 24 ways for this case.

IB, I think you also forgot to include the arrangements where there are only repeats, such as (IIII) or (SSSS). Thanks for your help though, I appreciate it. Without it I would never have seen my errors.

Well, I redid it, and I get a total of 176 ways.

Who is right? or are we both wrong?

Last edited: Nov 28, 2005


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Home OALib Journal OALib PrePrints Submit Ranking News My Lib FAQ About Us Follow Us+
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 Alfredo Donno Mathematics , 2013, Abstract: Inspired by the definition of generalized wreath product of permutation groups, we define the generalized wreath product of graphs, containing the classical Cartesian and wreath product of graphs as particular cases. We prove that the generalized wreath product of Cayley graphs of finite groups is the Cayley graph of the generalized wreath product of the corresponding groups.
 Mathematics , 2015, Abstract: A new class of subgroups is obtained for symmetric groups using wreath products and graphical representation of the elements of wreath products in the set of signed Brauer diagrams.
 Vahagn H. Mikaelian International Journal of Mathematics and Mathematical Sciences , 2002, DOI: 10.1155/s0161171202012528 Abstract: We outline results on varieties of groups generated by Cartesian and direct wreath products of abelian groups and pose two problems related to our recent results in that direction. A few related topics are also considered.
 Mathematics , 2011, Abstract: The wreath product of two permutation groups G < Sym(Gamma) and H < Sym(Delta) can be considered as a permutation group acting on the set Pi of functions from Delta to Gamma. This action, usually called the product action, of a wreath product plays a very important role in the theory of permutation groups, as several classes of primitive or quasiprimitive groups can be described as subgroups of such wreath products. In addition, subgroups of wreath products in product action arise as automorphism groups of graph products and codes. In this paper we consider subgroups X of full wreath products Sym(Gamma) wr Sym(Delta) in product action. Our main result is that, in a suitable conjugate of X, the subgroup of Sym(Gamma) induced by a stabilizer of a coordinate delta in Delta only depends on the orbit of delta under the induced action of X on Delta. Hence, if the action of X on Delta is transitive, then X can be embedded into a much smaller wreath product. Further, if this X-action is intransitive, then X can be embedded into a direct product of such wreath products where the factors of the direct product correspond to the X-orbits in Delta. We offer an application of the main theorems to error-correcting codes in Hamming graphs.
 Vahagn H. Mikaelian Mathematics , 2015, DOI: 10.13140/2.1.2300.9607 Abstract: We present a survey of our recent research on varieties, generated by wreath products of groups. In particular, the full classification of all cases, when the (cartesian or direct) wreath product of any abelian groups $A$ and $B$ generates the product variety $\var{A} \var{B}$, is given. The analog of this is given for sets of abelian groups. We also present new, unpublished research on cases when the similar question is considered for non-abelian, finite groups.
 Anton Evseev Mathematics , 2012, Abstract: The Alperin--McKay conjecture relates irreducible characters of a block of an arbitrary finite group to those of its $p$-local subgroups. A refinement of this conjecture was stated by the author in a previous paper. We prove that this refinement holds for all blocks of symmetric groups. Along the way we identify a "canonical" isometry between the principal block of $S_{pw}$ and that of $S_p\wr S_w$. We also prove a general theorem on expressing virtual characters of wreath products in terms of certain induced characters. Much of the paper generalises character-theoretic results on blocks of symmetric groups with abelian defect and related wreath products to the case of arbitrary defect.
 F. K. Indukaev Mathematics , 2008, Abstract: The RP-property of Fel'shtyn and Troitsky is proved for wreath products of finitely generated Abelian groups with the group of integers. Such wreath products become the first known example of finitely generated RP-groups being not almost polycyclic.
 Mathematics , 2015, Abstract: We introduce a notion of partition wreath product of a finite group by a partition quantum group, a construction motivated on the one hand by classical wreath products and on the other hand by the free wreath product of J. Bichon. We identify the resulting quantum group in several cases, establish some of its properties and show that when the finite group in question is abelian, the partition wreath product is itself a partition quantum group. This allows us to compute its representation theory, using earlier results of the first named author.
 Sean Cleary Mathematics , 2005, Abstract: Wreath products such as Z wr Z are not finitely-presentable yet can occur as subgroups of finitely presented groups. Here we compute the distortion of Z wr Z as a subgroup of Thompson's group F and as a subgroup of Baumslag's metabelian group G. We find that Z wr Z is undistorted in F but is at least exponentially distorted in G.
 Mathematics , 2008, DOI: 10.4171/GGD/81 Abstract: Let G be a group which has for all n a finite number r_n(G) of irreducible complex linear representations of dimension n. Let $\zeta(G,s) = \sum_{n=1}^{\infty} r_n(G) n^{-s}$ be its representation zeta function. First, in case G is a permutational wreath product of H with a permutation group Q acting on a finite set X, we establish a formula for $\zeta(G,s)$ in terms of the zeta functions of H and of subgroups of Q, and of the Moebius function associated with the lattice of partitions of X in orbits under subgroups of Q. Then, we consider groups W(Q,k) which are k-fold iterated wreath products of Q, and several related infinite groups W(Q), including the profinite group, a locally finite group, and several finitely generated groups, which are all isomorphic to a wreath product of themselves with Q. Under convenient hypotheses (in particular Q should be perfect), we show that r_n(W(Q)) is finite for all n, and we establish that the Dirichlet series $\zeta(W(Q),s)$ has a finite and positive abscissa of convergence s_0. Moreover, the function $\zeta(W(Q),s)$ satisfies a remarkable functional equation involving $\zeta(W(Q),es)$ for e=1,...,|X|. As a consequence of this, we exhibit some properties of the function, in particular that $\zeta(W(Q),s)$ has a singularity at s_0, a finite value at s_0, and a Puiseux expansion around s_0. We finally report some numerical computations for Q the simple groups of order 60 and 168.
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18.3. Command Line Introduction¶

Sometimes we will be directing you to use a command window or terminal to compile and run C# programs. [1]

Reasons to use the command line:

• The command line precedes the graphical user interface (GUI) used in modern operating systems and provides a simpler interface for input and output that is very flexible and powerful for knowledgeable users.
• Input comes from the keyboard as typed characters (no mouse processing). Commands are only processed once you press Enter.
• Output goes to the monitor as textual information (no window processing).
• In C# these input/output mechanisms are called Console processing.
• Input from and output to files is done in a very similar way, simplifying learning.
• Many software development organizations use command line processing to automate creating, compiling (“building”), and running or executing software programs.
• Command line “scripts” can be created to automate routine tasks.
• Command line scripts are similar to C# and other computer programs.
• Serious software developers should be familiar with the command line.

The most direct way to access the command line (often called a command shell):

• On Windows, to have easy access to Mono tools, press the Windows key (lower left or lower right on the keyboard) and type Mono, then select the Mono Command Prompt and press Enter.

Alternately, if you do not need Mono tools for sure, the general way to get a command window is to press the Windows key and R (lower or upper case) together, then type cmd and press the Enter or Return key – this brings up the basic command processing program, cmd.

• On a Mac just open a Terminal window – this is fine for the Mono SDK commands.

Mac OSX, Linux and other Unix variants work basically the same once you get to a terminal, so we will only distinguish Windows and Mac OS-X.

18.3.2. Common Commands¶

The command shell waits for you to type in a command (a short name that the shell recognizes) followed by 0 or more parameters separated by spaces (and Enter). Note that if a parameter contains spaces you must surround the parameter value with matching single or double quotes – you’ll see an example later.

We are going to mention some of the simplest uses of basic commands. More advanced documentation would include more options.

Some commands are common between the Windows and Mac shells:

dir (Windows) or ls (Mac)
to list all the files a in the current directory or a named directory.
cd

stands for Change Directory – you can use this command to change the current working directory to a different one.

You can use this command to change to directories where your C# program source files are located, if different from the initial directory.

On Windows, suppose you created a directory C:\COMP170\hello; to change to that, type cd C:\COMP170\hello and press Enter – the shell prompt will change to show this new directory location and programs like mcs and mono will be able to access files there, directly by name. If the Comp170 directory was you current directory, it would be shorter to use relative paths and just cd hello. Remember if you want a different Windows drive, you must first use a drive change command.

On a Mac absolute or relative paths work with cd. There is no issue with drives.

If you included a space in one or more of the directory names, for example C:\COMP 170\hello (a space between COMP and 170) you should enclose the path in quotes like: cd "C:\\COMP 170\\hello"

Mac Note: if you type just cd and press Enter you will change back to your home directory. There is also a shorthand name for your home directory in command paths: tilde (~), often shifted backquote on the keyboard. Sorry, no such thing with Windows.

mkdir

stands for make directory – you can use mkdir to create a new empty directory in the current directory.

For example, on a Mac with current directory /Users/YourLogin, type mkdir hello and press Enter – this will create a new directory /Users/yourLogin/hello if it did not exist before; you can now create a C# source file in that directory and enter cd hello in the command shell.

An optional Windows abbreviation is md.

rmdir

removes an empty directory that you give as parameter, e.g.,

rmdir hello

With Mono installed (and for Windows, with a Mono command window), the programs associated with Mono can be used:

mcs
compiles one or more listed C# source files without using Xamarin Studio.
csharp
is the interactive C# statement testing program.

Other useful commands, with different names for Windows and Mac, are listed next by generic function, with general Windows syntax first and Mac second, and then often examples in the same order:

Display the contents of a text file in the command window. The Unix/Mac name origin: a more complicated use of cat is to concatenate files.

type textFileName
cat textFileName
type my_program.cs
cat my_program.cs

Make a copy of a file. Caution: If the second file already exists, you wipe out the original contents!

copy originalFile copyName
cp originalFile copyName
copy prog.cs prog_bak.cs
cp prog.cs prog_bak.cs

Erase or remove a file:

erase fileToKill
rm fileToKill
erase poorAttempt.cs
rm poorAttempt.cs

Another Windows equivalent is del (short for delete).

We have explained the simplest use of many of the commands above. Many modifiers are possible.

Help on a command:

help commandName;
commandName –help

Note the double dash above: This sometimes works for concise help on a Mac while you can generally get an immensely detailed help overload on a Mac from

man commandName

18.3.3. Scripts¶

This is not a subject of this course, but commands can be combined into script files.

Scripting languages are in fact whole new specialized programming languages, that include many of the types of programming statements found in C#.

18.3.4. Copy and Paste¶

Copying or pasting with a Mac is is the same with a terminal as in other editing: Use the same Apple Command key with C or P, and you can select with the mouse.

In Windows it is more complicated when using a command window: You can paste into the current command line by right clicking on the Command Window Title bar, and select edit and then paste.

By default a Windows command window is not sensitive to the mouse. You can change so that it is sensitive for select and copy: Right click in the title bar, select defaults, and make sure the check boxes under edit options are all checked. (The last two are explained in the next section.) Click OK. Then you can select with mouse and press Enter for the selection to be remembered in the copy buffer.

18.3.5. Command Line Shortcuts¶

Both Mac and Windows (with the right options selected, like the Windows check boxes in the last section), allow you to reduce typing:

You can bring back a previous command for the history of commands that are automatically remembered: Use the up and down arrows. This makes it very easy to run the same command again, or to make slight edits.

Both Windows and OS-X can see what files are in any directory being referred to. If you just start to type a file or folder name and then press the Tab key, both Windows and OS-X will do file completion to complete the name if there is no ambiguity. If there is ambiguity, they work differently:

• Windows will cycle through all the options as you keep pressing Tab.
• On the first tab OS-X will do nothing but give a sound if there is ambiguity, but the second tab will list all the options. Then you need to type enough more to disambiguate the meaning.
 [1] Thanks to Dr. Robert Yacobellis for elaborations to this section.


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# Optimal command pipelining that maximizes total value in a given time duration

Consider I have two types of commands A and B which can be pipelined. Each of these types have a certain time duration and a value associated with it. For example,

A = <|"time"->180, "value"->5|>;
B = <|"time"->60, "value"->1|>;

Minimum timing allowed of A and B is as follows

tAAsame = 180; (* Cmd_A (same) to Cmd_A (same) *)
tAAdiff = 90; (* Cmd_A (diff) to Cmd_A (diff) *)
tBBsame = 60; (* Cmd_B (same) to Cmd_B (same) *)
tBBdiff = 10; (* Cmd_B (diff) to Cmd_B (diff) *)
tABsame = 180; (* Cmd_A (same) to Cmd_B (same) *)
tABdiff = 10; (* Cmd_A (diff) to Cmd_B (diff) *)

Now, consider I have commands: A0, A1, B0, B1. I would like to fit these commands in time 1000s which maximizes the total value of the commands in that time duration.

Here is an example of how these commands can be spaced. This is not optimal.

I looked into KnapSackSolve function but couldn't clearly think of how to input command to command timing constraint into it. Is there a way to input such information? if not what other ways are there to solve this command pipelining problem.