text:
FLONGS

In this Flong one of the most famous dynamical systems will discussed, the logistic map. The logistic map is a one-dimensional discrete-time map that, despite its formal simplicity, exhibits an unexpected degree of complexity. Historically it has been one of the most important and paradigmatic systems during the early days of research on deterministic chaos.

The logistic map is defined by the following equation:

Given a starting value $0\leq x_{0}\leq1$ and a positive parameter $0<\lambda<4$ the map produces a sequence of values

[ x_{0},x_{1},x_{2},... ]

that you get by iterating it, e.g.

[ x_{1} =\lambda x_{0}(1-x_{0}) ] [ x_{2} =\lambda x_{1}(1-x_{1}) ]

[ ... ]

and so forth.

Before we discuss some fascinating properties of this map, its richness, and set of unexpected behaviors, let's first motivate it and connect it to applications and natural systems.

### Motivation

It turns out that the logistic map can be motivated using a biological/ecological context.

Let's assume we have a population of a species in which individuals replicate. Let's say, for simplicity that in every generation $n=0,1,2,...$ an individual has on average $\lambda$ offspring before it dies. This could be any positive number. It doesn't have to be an integer because we think of this as an average or an expected number of offspring. So if say $\lambda=1.5$ then on average each individual has $1.5$ offspring in the lifetime of one time step, for instance if in 50% of cases an individual has one offspring and in 50% of the cases 2 children, we get $\lambda=1.5$.

We call the parameter $\lambda$ the reproduction rate.

Now let's say the population at time $n=0$ has $U_{n}$ individuals. A generation later the population size will be [ U_{n+1}=\lambda U_{n} ] Therefore, if we start with an initial population $U_{0}$ we get [ U_{n}=\lambda^{n}U_{0} ] which we can write as [ U_{n}=e^{\ln\lambda}U_{0} ] So, depending on the value of $\lambda$ we can have two different scenarios.

• If $0<\lambda<1$ then $\ln\lambda<\text{0}$ and thus $U_{n+1}<U_{n}$, the population dies out. This makes total sense if the typical number of offspring is less that one.
• If $\lambda>1$ on the other we have $\ln\lambda>0$ and thus $U_{n+1}>U_{n}$: the population grows indefinitely and exponentially.

So, if we only have replication and a reproduction rate $\lambda>1$ we expect exponential growth. Now, this can't go on forever. Other factors will come into play if the population gets too big, e.g. limiting resources, competition, limited space, etc. One way to account for this is by saying that the reproduction rate $\lambda$ effectively depends in some way on the size of the population: [ \lambda=\lambda(U) ]

We introduce a capacity $C$ which we think of as a large population size that, if reached, reduces reproduction to zero or a very small number. Plus we would like to keep reproduction at some maximum value $\lambda_{0}$ if $U\ll C$. What are the choices? Here'e one: [ \lambda=\lambda_{0}e^{-U/C} ] In this case [ \lambda\approx\lambda_{0}\quad\text{if}\quad U\ll C ] and [ \lambda\approx0\quad\text{if}\quad U\gg C ] An even simpler choice is a linear decrease [ \lambda=\lambda_{0}(1-U/C) ] which is very similar but potentially easier to handle mathematically.

With this choice we get [ U_{n+1}=\lambda_{0}(1-U_{n}/C)U_{n} ] which we can iterated for a chosen set of parameters $\lambda$ and $C$ and an initial population $U_{0}<C$.

Now we can simplify this a bit by expressing the population as a fraction of the capacity $C$: [ x_{n}=U_{n}/C. ] Using the new quantity we get [ x_{n+1}=\lambda_{0}(1-x_{n})x_{n}. ] We can just drop the subscript $0$ of $\lambda_{0}$ and arrive at the logistic map [ x_{n+1}=\lambda(1-x_{n})x_{n}. ]

Let's play with this equation a bit. In Panel 2 you can observe the sequence $x_{n}$ for parameters $x_{n}$ and $\lambda$ of your choice. If $1<\lambda<3$ we get a sequence $x_{n}$ that quickly approaches a fixed value.

If we increase the reproduction rate to $\lambda>3.0$ we observe a different behavior: The population doesn't seem to equilibrate to a fixed value, but rather seems to oscillate in a period-2-cycle. Regardless of the initial condition $x_{0}$ the system always approaches this cycle.

Let's go to even higher values. For $\lambda=3.8$ we see even for very long sequences no apparent regular or periodic pattern of values emerges. The sequence of values $x_{n}$ seems random and chaotic. More about this below.

## One dimensional maps

In order to understand this behavior better, let's back up a bit and think of a system like the logistic map in more general terms. The logistic map is an example of a one-dimensional discrete time map: [ x_{n+1}=f(x_{n}) ] where in our case [ f(x)=\lambda x(1-x). ] The question that arises typically is how the system behaves for long times so when $n\rightarrow\infty$.

### Fixpoints

The first step we usually take is trying to identify special points, called fixed points or short fixpoints. These are values for $x$ that remain invariant if we plug them into the iterated map, so: [ x_{n+1}=x_{n} ] and [ x_{n}=f(x_{n}). ] Iterated maps can have none, one or many such points and sometimes it's difficult to compute them. In our case we can compute the fixpoints: [ x=\lambda x(1-x) ] which has solutions [ x^{\star}=0 ] and [ x^{\star}=1-\frac{1}{\lambda}. ]

We see that the first one always exists and the second one exists if $\lambda>1$. So if we start with $x_{0}=x^{\star}$ we will have [ x_{n+1}=x^{\star} ] for all times. But what happens if we don't start with $x_{0}=x^{\star}$? Let's try this with $\lambda=5/2$ and some $x_{0}$ we see that after a few iterations the values of $x_{n}$ are approaching the fixpoint [ x^{\star}=1-\frac{1}{\lambda}=3/5=0.6 ] This fixpoint seems to be attractive and we call it an attractor. Even if we start very close to the other fixpoint $x^{\star}=0$ say with $x_{0}=0.001$ we see that the values $x_{n}$ are repelled by it (we call that fixpoint a repeller) and attracted by the non-zero fixpoint.

Apart from doing the computations to determine the fixpoints we can find the fixpoints graphically. All we have to do is to draw the function $y_{1}=f(x)$ and the diagonal $y_{2}=x$. When these two curves intersect we have [ x=f(x). ] Therefore the intersections are the fixpoints, e.g. in Fig. 3a the map $x_{n+1}=g(x_n)$ has four different fixpoints.

For the logistic map we see that $f(x)$ is an inverse parabola that meets the $x$-axis at zero and one, and intersects with the diagonal $x$ only at $x=0$ if $\lambda<1$ and also at a non-trivial intermediate value if $\lambda>1$, consistent with the calculations we did, see. Fig. 3b.

### Stability

There's a way to determine the stability of a fixpoint, to check whether a fixpoint is an attractor or a repeller. Let's assume that we have a fixpoint $x^{\star}$. And we start off with an initial value that's a little bit off so that [ x_{0}=x^{\star}+\delta x_{0}. ] Now we compute [ x_{1}=f(x_{0})=f(x^{\star}+\delta x_{0}). ] In the vicinity of the fixpoint, if $\delta x_{0}$ is very small we can expand the function $f$: [ f(x^{\star}+\delta x_{0})\approx f(x^{\star}) + f^{\prime}(x^{\star})\delta x_{0} ] so that [ x_{1} \approx x^{\star}+f^{\prime}(x^{\star})\delta x_{0} ] So, after one iteration, what is the distance to the fixpoint? [ \delta x_{1}\approx f^{\prime}(x^{\star})\delta x_{0}. ] So we see that if [ |f^{\prime}(x^{\star})|>1 ] this distance to the fixpoint becomes larger and larger whereas if [ |f^{\prime}(x^{\star})|<1 ] if becomes smaller and smaller. So it seems that this a condition for the stability of a fixpoint.

Let's try to apply this to the logistic map. Here we have [ f(x)=\lambda x(1-x) ] which implies that [ f^{\prime}(x)=\lambda-2\lambda x ] so that at the fixpoint $x^{\star}=1-1/\lambda$ we get [ f^{\prime}(x^{\star}) =\lambda-2\left(\lambda-1\right) =2-\lambda ] Therefore, if $\lambda<3$ the fixpoint is stable because $|f^{\prime}(x^{\star})|<1$. But, if the reproduction rate increases beyond $3$ the fixpoint becomes unstable. The system no longer has a stable, attractive fixpoint.

Looking at the sequence $x_{n}$ say for $\lambda=3.3$ we see that the attractor is something else: a period-2-cycle of alternating values such that [ x_{n+2}=x_{n}=x_{A}^{\star} ] and [ x_{n+3}=x_{n+1}=x_{B}^{\star} ] We can think of both of these points that make that cycle as a fixpoint of a new map: [ g=f\circ f ] which in words means we apply the orginal map twice, or [ g(x) =f(f(x)) ] [ =\lambda f(x)(1-f(x)) ] [ =\lambda^{2}x(1-x)(1-\lambda x(1-x)) ] This function is a more convoluted polynomial. If we want to compute the fixed points of $g(x)$, we proceed the same way by solving [ x=\lambda^{2}x(1-x)(1-\lambda x(1-x)) ] for x.

## Graphic Analysis - Cobwebbing

We can visualize a trajectory by starting on the $x$-axis at some point $x_{0}$ and drawing a vertical line to $f(x_{0})$. That point is also $x_{1}$ and finding it on the $x$ axis is equivalent to drawing a horizontal line towards the diagonal $x$. Now we can draw another vertical link to $f(x_{1})$ and keep repeating the process and eventually generate an entire trajectory this way. So, what we are doing is plotting the sequence [ (x_{0},0),(x_{0},x_{1}),(x_{1},x_{1}),(x_{1},x_{2}).... ] [ ....(x_{n},x_{n}),(x_{n},x_{n+1}).... ] You can try this cobwebbing for the logistic map in Panel 3.

So far we have observed that when $\lambda>1$ we have a single fixpoint at $x^{\star}=1-1/\lambda$ that loses stability at a critical value $\lambda_{2}=3$ when a stable period-2-cyle emerges. If we increase $\lambda$ beyond $\lambda_{3}=3.449...$ we will see that the period-2-cycle also loses stability. It is replaced by a new, stable and attractive period-4-cycle. The points that make up this cycle are stable fix-points of the quadruple map [ h=g\circ g=f\circ f\circ f\circ f. ] Now it turns out that this period-4-cycle doesn't remain stable for long but at $\lambda_{4}=3.54409...$ is replaced by a stable period-8-cycle, and so forth.

What's important to note is that the critical points at which these \textbf{period doubling events} occur, or rather there separation becomes smaller and smaller and eventually at [ \lambda_{\infty}=3.569946... ] we have a period of infinity. This means that although we have a cycle'' that is attractive, it's period is infinite, so it never repeats, it is not longer really periodic. Rather is has become aperiodic and chaotic. This was very confusing to people that first looked at the simple logistic map. We can compute the ratio [ \delta=\lim_{k\rightarrow\infty}\frac{\lambda_{n}-\lambda_{n-1}}{\lambda_{n+1}-\lambda_{n}}=4.669 ] The limit is known as Feigenbaum's constant. This is a universal constant that keeps reappearing in systems like the logistic map and is intimitaly connected to chaotic systems.

What's even more puzzling is that the periodic solutions like the period-2-cycles, 4-cycles etc., although no longer stable in the chaotic regime, still exist as repellers. So for $\lambda>\lambda_{\infty}$ we have a very dense and infinite set of periodic orbits in the system all of which are unstable. In fact, they have to fit in the system somehow, so actually for every point $x$ we have an inifite set of periodic orbits arbitrarily close to it.

## Bifurcation diagram

One way of illustrating the complexity of the situation is by what is known as a bifurcation diagram. Instead of making different cobwebbing plots for different choices of $\lambda$ and because we tend to be interested in the asymptotic behavior of the system we plot the attractors of the systems as a function of the parameter $\lambda$. A plot like this is known as a bifurcation diagram, a very important tool in the study of dynamical systems. This is how it works:

• first we fix $\lambda$ and choose an initial condition
• we then iterate the map, say 10000 times for the transient part of the dynamics
• then we iterating say 1000 more times and plot the array of values vs. $\lambda$
• we then increment $\lambda$ and repeat the process

we now plot the $x$-values that we computed that represent the attractor on the $y-$axis and the $\lambda-$value on the $x$-axis.

If for the chosen value of $\lambda$ we have say a period-8-cycle that is attractive, all the collected 1000 points will be one of the 8 points that make up the period-8-cycle. So we would get 8 dots in the plot. You can see what is happening in Panel 4.

We clearly see the period doubling events as we increase $\lambda$. We also see that at some point all the points are spread across a wide range and form what looks like a continuun. Furthermore we see that there are weird self-similar structures in the bifurcation diagram and also regimes of periodic behavior within a see of chaos. You can investigate this by zooming by clicking on a region of interest.

## Some properties of chaos

When the reproduction rate is larger than the critical parameter $\lambda_{\infty}$ and we have a regime where orbits are aperiodic, we have reached the regime of deterministic chaos. This means that even though the orbits are deterministic, they seem irregular and unpredictable.

We can also observe a sensitive dependence on initial conditions. Let's say we start with an initial value $x_{0}$ and right next to it with an initial value $\tilde{x}_{0}=x_{0}+\varepsilon$ where $\varepsilon$ is tiny, say $10^{-6}$. In the chaotic regime after a few steps the sequences $x_{n}$ and $\tilde{x}_{n}$ will become very different, another hallmark of unpredictability. You can check this out in Panel 5.

Third, we see another generic feature of chaotic systems: self-similarity. This is particularly striking in the bifurcation diagram of Panel 4 where we see parts of the structure that resemble the whole structure only smaller. You can see this by zooming in (clicking on the region of interest).

### Other systems

The logistic map with $f(x)=\lambda x (1-x)$ is the simplest conceivable system that encorporates reproduction and self-regulation of a population. Because it is so simple we can expect that models that are structurally similar but more realistic generate similar behavior and that the funky effects observed in the logistic map are universal.

Panel 6 illustrates the bifurcation diagram of the map

[ x_{n+1} = \lambda x_{n} \exp (-x_{n}) ]

which isn't limited to a parameter range for the reproduction rate $\lambda<4$ but is plausible for any non-negative value. The bifurcation diagram looks qualitatively similar to the one depicted in Panel 4.

## More material

A lot more can be said about the logistic map, one-dimensional non-linear maps in general, deterministic chaos and so forth. If you want to dive deeper into the material I suggest the following:


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1
JEE Main 2022 (Online) 26th July Evening Shift
+4
-1

A ball of mass $$0.15 \mathrm{~kg}$$ hits the wall with its initial speed of $$12 \mathrm{~ms}^{-1}$$ and bounces back without changing its initial speed. If the force applied by the wall on the ball during the contact is $$100 \mathrm{~N}$$, calculate the time duration of the contact of ball with the wall.

A
0.018 s
B
0.036 s
C
0.009 s
D
0.072 s
2
JEE Main 2022 (Online) 26th July Evening Shift
+4
-1

A body of mass $$8 \mathrm{~kg}$$ and another of mass $$2 \mathrm{~kg}$$ are moving with equal kinetic energy. The ratio of their respective momentum will be :

A
1 : 1
B
2 : 1
C
1 : 4
D
4 : 1
3
JEE Main 2022 (Online) 25th July Evening Shift
+4
-1

Two billiard balls of mass 0.05 kg each moving in opposite directions with 10 ms$$-$$1 collide and rebound with the same speed. If the time duration of contact is t = 0.005 s, then what is the force exerted on the ball due to each other?

A
100 N
B
200 N
C
300 N
D
400 N
4
JEE Main 2022 (Online) 30th June Morning Shift
+4
-1

Two bodies A and B of masses 5 kg and 8 kg are moving such that the momentum of body B is twice that of the body A. The ratio of their kinetic energies will be :

A
4 : 5
B
2 : 5
C
5 : 4
D
5 : 2
JEE Main Subjects
Physics
Mechanics
Electricity
Optics
Modern Physics
Chemistry
Physical Chemistry
Inorganic Chemistry
Organic Chemistry
Mathematics
Algebra
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EXAM MAP
Joint Entrance Examination

format: plain


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# HD-Combinatorics: Adi Shraibman, "The communication complexity of high-dimensional permutations"

A k-dimensional permutation is a (k+1)-dimensional array of zeros and ones, with exactly a single one in every axis parallel line. We consider the “number on the forehead" communication complexity of a k-dimensional permutation and ask how small and how large it can be. We give some initial answers to these questions. We prove a very weak lower bound that holds for every permutation, and mention a surprising upper bound. We motivate these questions by describing several closely related problems: estimating the density Hales-Jewett number, high-dimensional szemeredi theorems, constructing ruzsa-szemeredi graphs, and more.

## Date:

Mon, 14/05/2018 - 10:00 to 10:50

## Location:

Feldman Buildng, Givat Ram


text:
## Support conflict resolution using diff/diff3 format files

Support for our DiffMerge utility.

Moderator: SourceGear

JohnL
Posts: 1
Joined: Mon Nov 26, 2018 5:28 am

### Support conflict resolution using diff/diff3 format files

Does DiffMerge have the ability to
1) Open a text file that contains conflict markers in the diff or diff3 format
2) Allow the user to navigate through the conflicts and resolve the conflicts in the DiffMerge conflict resolver UI

In effect, DiffMerge should not to re-calculate the file differences itself, but support the conflict resolution process based only on the input file containing the diff conflict markers.

For example, a very simple 2 way diff format input file with conflicts would be

<<<<<<< A
line from A
=======
line from B
>>>>>>> B

Many Thanks,
John

joe_sg
Posts: 27
Joined: Tue Oct 28, 2008 1:20 pm
Location: SourceGear

### Re: Support conflict resolution using diff/diff3 format file

DiffMerge does not have the capability to resolve conflicts based on conflict markers. It doesn't even support editing a single file. It can produce an output file in the Unified Diff Format which then be edited by You in a text editor.

For example, take two files:

Code: Select all

Left.txt            Right.txt
---------	    ---------
Line one.           Line one.
Line two.           Line two.
Line only Left.     Line three.
Line three.         Line four.
Line four.          Line only Right.
Line five.          Line five.
Line conflict.      Conflict line.
Line six.           Line six.

DiffMerge can create a Unified Diff formatted output file by using the following command:

\path\to\sgdm.exe -u Left.txt Right.txt -d=Result.txt

Code: Select all

--- D:\Cx\Left.txt	2018-11-26 14:31:21.239000000 -0600
+++ D:\Cx\Right.txt	2018-11-26 14:31:04.402000000 -0600
@@ -1,7 +1,7 @@
Line one.
Line two.
-Line only Left.
Line three.
Line four.
+Line only Right.
Line five.
Line six.

The user could then use a text editor to navigate through the result, manually resolving the conflicts.

But that may be the opposite of what you are looking for. If so, then I would have to say that DiffMerge does not have the capability you are looking for.

See DiffMerge Help: 10. Command Line Arguments | Diffing Files to a File
Joe Ream
SourceGear Technical Support


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# 7.2 Kinetic energy and the work-energy theorem  (Page 3/7)

 Page 3 / 7

## Determining the work to accelerate a package

Suppose that you push on the 30.0-kg package in [link] with a constant force of 120 N through a distance of 0.800 m, and that the opposing friction force averages 5.00 N.

(a) Calculate the net work done on the package. (b) Solve the same problem as in part (a), this time by finding the work done by each force that contributes to the net force.

Strategy and Concept for (a)

This is a motion in one dimension problem, because the downward force (from the weight of the package) and the normal force have equal magnitude and opposite direction, so that they cancel in calculating the net force, while the applied force, friction, and the displacement are all horizontal. (See [link] .) As expected, the net work is the net force times distance.

Solution for (a)

The net force is the push force minus friction, or ${F}_{\text{net}}\text{= 120 N – 5}\text{.}\text{00 N = 115 N}$ . Thus the net work is

$\begin{array}{lll}{W}_{\text{net}}& =& {F}_{\text{net}}d=\left(\text{115 N}\right)\left(\text{0.800 m}\right)\\ & =& \text{92.0 N}\cdot m=\text{92.0 J.}\end{array}$

Discussion for (a)

This value is the net work done on the package. The person actually does more work than this, because friction opposes the motion. Friction does negative work and removes some of the energy the person expends and converts it to thermal energy. The net work equals the sum of the work done by each individual force.

Strategy and Concept for (b)

The forces acting on the package are gravity, the normal force, the force of friction, and the applied force. The normal force and force of gravity are each perpendicular to the displacement, and therefore do no work.

Solution for (b)

The applied force does work.

$\begin{array}{lll}{W}_{\text{app}}& =& {F}_{\text{app}}d\phantom{\rule{0.25em}{0ex}}\text{cos}\left(0º\right)={F}_{\text{app}}d\\ & =& \left(\text{120 N}\right)\left(\text{0.800 m}\right)\\ & =& \text{96.0 J}\end{array}$

The friction force and displacement are in opposite directions, so that $\theta =\text{180º}$ , and the work done by friction is

$\begin{array}{lll}{W}_{\text{fr}}& =& {F}_{\text{fr}}d\phantom{\rule{0.25em}{0ex}}\text{cos}\left(\text{180º}\right)=-{F}_{\text{fr}}d\\ & =& -\left(\text{5.00 N}\right)\left(\text{0.800 m}\right)\\ & =& -\text{4.00 J.}\end{array}$

So the amounts of work done by gravity, by the normal force, by the applied force, and by friction are, respectively,

$\begin{array}{lll}{W}_{\text{gr}}& =& 0,\\ {W}_{N}& =& 0,\\ {W}_{\text{app}}& =& \text{96.0 J,}\\ {W}_{\text{fr}}& =& -\text{4.00 J.}\end{array}$

The total work done as the sum of the work done by each force is then seen to be

${W}_{\text{total}}={W}_{\text{gr}}+{W}_{N}+{W}_{\text{app}}+{W}_{\text{fr}}=\text{92}\text{.0 J}.$

Discussion for (b)

The calculated total work ${W}_{\text{total}}$ as the sum of the work by each force agrees, as expected, with the work ${W}_{\text{net}}$ done by the net force. The work done by a collection of forces acting on an object can be calculated by either approach.

## Determining speed from work and energy

Find the speed of the package in [link] at the end of the push, using work and energy concepts.

Strategy

Here the work-energy theorem can be used, because we have just calculated the net work, ${W}_{\text{net}}$ , and the initial kinetic energy, $\frac{1}{2}{m{v}_{0}}^{2}$ . These calculations allow us to find the final kinetic energy, $\frac{1}{2}{\text{mv}}^{2}$ , and thus the final speed $v$ .

Solution

The work-energy theorem in equation form is

${W}_{\text{net}}=\frac{1}{2}{\text{mv}}^{2}-\frac{1}{2}{m{v}_{0}}^{2}\text{.}$

Solving for $\frac{1}{2}{\text{mv}}^{2}$ gives

$\frac{1}{2}{\text{mv}}^{\text{2}}={W}_{\text{net}}+\frac{1}{2}{m{v}_{0}}^{2}\text{.}$

Thus,

$\frac{1}{2}{\text{mv}}^{2}=\text{92}\text{.}0 J+3\text{.}\text{75 J}=\text{95.}\text{75 J.}$

Solving for the final speed as requested and entering known values gives

$\begin{array}{lll}v& =& \sqrt{\frac{2\text{(95.75 J)}}{m}}=\sqrt{\frac{\text{191.5 kg}\cdot {m}^{2}{\text{/s}}^{2}}{\text{30.0 kg}}}\\ & =& \text{2.53 m/s.}\end{array}$

Discussion

Using work and energy, we not only arrive at an answer, we see that the final kinetic energy is the sum of the initial kinetic energy and the net work done on the package. This means that the work indeed adds to the energy of the package.

how many subject is in physics
the write question should be " How many Topics are in O- Level Physics, or other branches of physics.
effiom
how many topic are in physics
Praise
yh I need someone to explain something im tryna solve . I'll send the question if u down for it
a ripple tank experiment a vibrating plane is used to generate wrinkles in the water .if the distance between two successive point is 3.5cm and the wave travel a distance of 31.5cm find the frequency of the vibration
Tamdy
the range of objects and phenomena studied in physics is
what is Linear motion
straight line motion is called linear motion
then what
Amera
linear motion is a motion in a line, be it in a straight line or in a non straight line. It is the rate of change of distance.
Saeedul
Hi
aliyu
Richard
Linear motion is a one-dimensional motion along a straight line, and can therefore be described mathematically using only one spatial dimension
Jason
is a one-dimensional motion along a straight line, and can therefore be described mathematically using only one spatial dimensions.
Praise
what is a classical electrodynamics?
Marga
what is dynamics
Marga
dynamic is the force that stimulates change or progress within the system or process
Oze
what is the formula to calculate wavelength of the incident light
if a spring is is stiffness of 950nm-1 what work will be done in extending the spring by 60mmp
State the forms of energy
machanical
Ridwan
Word : Mechanical wave Definition : The waves, which need a material medium for their propagation, e.g., Sound waves. \n\nOther Definition: The waves, which need a material medium for their propagation, are called mechanical waves. Mechanical waves are also called elastic waves. Sound waves, water waves are examples of mechanical waves.t Definition: wave consisting of periodic motion of matter; e.g. sound wave or water wave as opposed to electromagnetic wave.h
correct
Akinpelu
what is mechanical wave
a wave which require material medium for its propagation
syed
The S.I unit for power is what?
watt
Okoli
Am I correct
Okoli
it can be in kilowatt, megawatt and so
Femi
yes
Femi
correct
Jaheim
kW
Akinpelu
OK that's right
Samuel
SI.unit of power is.watt=j/c.but kw.and Mw are bigger.umots
syed
What is physics
study of matter and its nature
Akinpelu
The word physics comes from a Greek word Physicos which means Nature.The Knowledge of Nature. It is branch of science which deals with the matter and energy and interaction between them.
Uniform
why in circular motion, a tangential acceleration can change the magnitude of the velocity but not its direction
reasonable
Femi
because it is balanced by the inward acceleration otherwise known as centripetal acceleration
MUSTAPHA
What is a wave
Tramsmission of energy through a media
Mateo
is the disturbance that carry materials as propagation from one medium to another
Akinpelu
mistakes thanks
Akinpelu
find the triple product of (A*B).C given that A =i + 4j, B=2i - 3j and C = i + k
Difference between north seeking pole and south seeking pole


text:
# Magnitude (mathematics)

Magnitude (mathematics)

The magnitude of an object in Mathematics is its size: a property by which it can be compared as larger or smaller than other objects of the same kind; in technical terms, an ordering (or ranking) of the class of objects to which it belongs. It also can be termed as a numerical value of that unit (or class) which it belongs.

The Greeks distinguished between several types of magnitude, including:

• (positive) fractions
• line segments (ordered by length)
• Plane figures (ordered by area)
• Solids (ordered by volume)
• Angles (ordered by angular magnitude)

They had proven that the first two could not be the same, or even isomorphic systems of magnitude. They did not consider negative magnitudes to be meaningful, and magnitude is still chiefly used in contexts in which zero is either the lowest size or less than all possible sizes.

## Numbers

The magnitude of any number x is usually called its "absolute value" or "modulus", denoted by |x|.

### Real numbers

The absolute value of a real number r is defined by:

|r| = r, if r ≥ 0
|r| = -r, if r < 0.

It may be thought of as the number's distance from zero on the real number line. For example, the absolute value of both 7 and −7 is 7.

### Complex numbers

A complex number z may be viewed as the position of a point P in a 2-dimensional space, called the complex plane. The absolute value of z may be thought of as the distance of P from the origin of that space. The formula for the absolute value of z is similar to that for the Euclidean norm of a vector in a 2-dimensional Euclidean space:

$\left| z \right| = \sqrt{\Re(z)^2 + \Im(z)^2 }$

where ℜ(z) and ℑ(z) are the respectively real part and imaginary part of z. For instance, the modulus of −3 + 4i is 5.

## Euclidean vectors

A Euclidean vector represents the position of a point P in a Euclidean space. Geometrically, it can be described as an arrow from the origin of the space (vector tail) to that point (vector tip). Mathematically, a vector x in an n-dimensional Euclidean space can be defined as an ordered list of n real numbers (the Cartesian coordinates of P): x = [x1, x2, ..., xn]. Its magnitude or length is most commonly defined as its Euclidean norm (or Euclidean length):

$\|\mathbf{x}\| := \sqrt{x_1^2 + x_2^2 + \cdots + x_n^2}.$

For instance, in a 3-dimensional space, the magnitude of [4, 5, 6] is √(42 + 52 + 62) = √77 or about 8.775. This is equivalent to the square root of the dot product of the vector by itself:

$\|\mathbf{x}\| := \sqrt{\mathbf{x} \cdot \mathbf{x}}.$

The Euclidean norm of a vector is just a special case of Euclidean distance: the distance between its tail and its tip. Two similar notations are used for the Euclidean norm of a vector x:

1. $\left \| \mathbf{x} \right \|,$
2. $\left | \mathbf{x} \right |.$

The second notation is generally discouraged, because it is also used to denote the absolute value of scalars and the determinants of matrices.

## Normed vector spaces

By definition, all Euclidean vectors have a magnitude (see above). However, the notion of magnitude cannot be applied to all kinds of vectors.

A function that maps objects to their magnitudes is called a norm. A vector space endowed with a norm, such as the Euclidean space, is called a normed vector space. In high mathematics, not all vector spaces are normed.

## Practical math

A magnitude is never negative. When comparing magnitudes, it is often helpful to use a logarithmic scale. Real-world examples include the loudness of a sound (decibel), the brightness of a star, or the Richter scale of earthquake intensity.

To put it another way, often it is not meaningful to simply add and subtract magnitudes.

## "Order of magnitude"

In advanced mathematics, as well as colloquially in popular culture, especially geek culture, the phrase "order of magnitude" is used to denote a change in a numeric quantity, usually a measurement, by a factor of 10; that is, the moving of the decimal point in a number one way or the other, possibly with the addition of significant zeros.

Occasionally the phrase "half an order of magnitude" is also used, generally in more informal contexts. Sometimes, this is used to denote a 5 to 1 change, or alternatively 101/2 to 1 (approximately 3.162 to 1).

## References

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Magnitude — may refer to: In mathematics: Magnitude (mathematics), the relative size of a mathematical object Magnitude (vector), a term for the size or length of a vector Scalar (mathematics), a quantity defined only by its magnitude Euclidean vector, a… …   Wikipedia

• Mathematics — Math e*mat ics, n. [F. math[ e]matiques, pl., L. mathematica, sing., Gr. ? (sc. ?) science. See {Mathematic}, and { ics}.] That science, or class of sciences, which treats of the exact relations existing between quantities or magnitudes, and of… …   The Collaborative International Dictionary of English

• mathematics — /math euh mat iks/, n. 1. (used with a sing. v.) the systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically. 2. (used with a sing. or pl. v.) mathematical procedures,… …   Universalium

• Mathematics in medieval Islam — A page from the The Compendious Book on Calculation by Completion and Balancing by Al Khwarizmi. In the history of mathematics, mathematics in medieval Islam, often termed Islamic mathematics …   Wikipedia

• magnitude — [[t]mæ̱gnɪtjuːd, AM tuːd[/t]] magnitudes 1) N UNCOUNT: usu with supp If you talk about the magnitude of something, you are talking about its great size, scale, or importance. An operation of this magnitude is going to be difficult... These are… …   English dictionary

• Mathematics and Physical Sciences — ▪ 2003 Introduction Mathematics       Mathematics in 2002 was marked by two discoveries in number theory. The first may have practical implications; the second satisfied a 150 year old curiosity.       Computer scientist Manindra Agrawal of the… …   Universalium

• mathematics —    The peoples of ancient Mesopotamia particularly the sumerians and the Babylonians made important contributions to the early development of mathematics, and some of their mathematical ideas and systems are still used today. The early Sum erians …   Ancient Mesopotamia dictioary

• mathematics — study of magnitude, number, and forms Sciences and Studies …   Phrontistery dictionary

• mathematics — math•e•mat•ics [[t]ˌmæθ əˈmæt ɪks[/t]] n. 1) (used with a sing. v.) the systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically 2) (used with a sing. or pl. v.)… …   From formal English to slang

• List of mathematics articles (M) — NOTOC M M estimator M group M matrix M separation M set M. C. Escher s legacy M. Riesz extension theorem M/M/1 model Maass wave form Mac Lane s planarity criterion Macaulay brackets Macbeath surface MacCormack method Macdonald polynomial Machin… …   Wikipedia


text:
What's new

# floating-rate

1. ### Duration of a Floating Rate Note

Hi David, In FRM handbook, it is given that the duration of the Floating Rate Note immediately after the rate adjust is zero and the duration in the intermediate period is time left till next rate readjustment. In other words, suppose the note readjusts the coupon based on LIBOR every six...


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# Unitary representations of the $\mathcal{W}_3$-algebra with $c\geq 2$

Sebastiano Carpi, Yoh Tanimoto, Mihaly Weiner
October 18, 2019
We prove unitarity of the vacuum representation of the $\mathcal{W}_3$-algebra for all values of the central charge $c\geq 2$. We do it by modifying the free field realization of Fateev and Zamolodchikov resulting in a representation which, by a nontrivial argument, can be shown to be unitary on a certain invariant subspace, although it is not unitary on the full space of the two currents needed for the construction. These vacuum representations give rise to simple unitary vertex operator algebras. We also construct explicitly unitary representations for many positive lowest weight values. Taking into account the known form of the Kac determinants, we then completely clarify the question of unitarity of the irreducible lowest weight representations of the $\mathcal{W}_3$-algebra in the $2\leq c\leq 98$ region.
%%% contains utf-8, see: http://inspirehep.net/info/faq/general#utf8 %%% add \usepackage[utf8]{inputenc} to your latex preamble @article{Carpi:2019szo, author = "Carpi, Sebastiano and Tanimoto, Yoh and Weiner, Mihály", title = "{Unitary representations of the $\mathcal{W}_3$-algebra with $c\geq 2$}", year = "2019", eprint = "1910.08334", archivePrefix = "arXiv", primaryClass = "math.RT", SLACcitation = "%%CITATION = ARXIV:1910.08334;%%" }


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## Rydberg equation

H-Atom ($E_{n}=-\frac{hR}{n^{2}}$)

Tinisha 1G
Posts: 69
Joined: Fri Sep 28, 2018 12:17 am

### Rydberg equation

Can someone explain what the rydberg equation tells us conceptually?

skyeblee2F
Posts: 62
Joined: Fri Sep 28, 2018 12:23 am

### Re: Rydberg equation

It tells us how much energy is being released (as light) when an electron shifts from one energy level to another, with "n" representing the energy level. We can expect the answer to be negative when energy is being released (and the electron moves closer to the nucleus) and positive when energy is being absorbed (and the electron moves away from the nucleus).

Karina Koo 2H
Posts: 49
Joined: Fri Sep 28, 2018 12:24 am

### Re: Rydberg equation

It applies specifically to hydrogen atoms only and tells us how much energy is needed to bring an electron from the energy state n to the next lower energy state.

Posts: 57
Joined: Fri Sep 28, 2018 12:27 am

### Re: Rydberg equation

The way I thought of it was that the energy that you find using this equation is the amount of energy that the electron gives off in order to return to a previous energy level. This is why the number is typically negative as well. The energy is the amount the it gives off say from the 6th energy level to the 3rd for example.

Cienna Henry 1J
Posts: 61
Joined: Fri Sep 28, 2018 12:15 am
Been upvoted: 1 time

### Re: Rydberg equation

Is this the equation Lavelle told us to steer away from using because students usually make mistakes?


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# Time and Distance - Aptitude Test Tricks, Shortcuts & Formulas

Quantitative aptitude problems on time and distance are sure questions in any competitive exam like GRE, GMAT, CAT, XAT, MAT, Bank exams or any other campus recruitment tests for that matter. These problems are often lengthy and test takers often skip them, thinking these questions will consume most amount of time. This is a mistake.

Time and distance aptitude questions are easy to score area if you know the very basic formula that you learnt in high school. All shortcuts to solve time and distance problems can be easily derived and learnt. This article provides Tips and Tricks to Solve "Time and Distance" Aptitude Problems with Important Formulas, Shortcuts, Core Concepts to crack Placement Test and Competitive exams.

Do not simply byheart the shortcuts or the tips and tricks. You should actually take time and learn how one arrives at these shortcuts, try it out yourself, then solve as many problems you can. By this you will automatically use the shortcut when you solve questions later on. We have compiled time and distance concepts and formulas for you in this section. We have also given how we derive each of these formulas. They should surely help you crack any aptitude test on time and distance.

To enhance your knowledge and skills to solve Time and Distance aptitude test problems, go through the tutorial on Time and Distance.

## Basic Concepts of Time and Distance

Most of the aptitude questions on time and distance can be solved if you know the basic correlation between speed, time and distance which you have learnt in your high school class.
• Relation between time, distance and speed: Speed is distance covered by a moving object in unit time.
Speed = "Distance Covered" / "Time taken"
• Ratio of the varying components when other is constant: Consider 2 objects A and B having speed S_a, S_b. Let the distance travelled by them are D_a and D_b respectively and time taken to cover these distances be T_a and T_b respectively.
Let's see the relation between time, distance and speed when one of them is kept constant
1. When speed is constant distance covered by the object is directly proportional to the time taken.
ie; If S_a = S_b, then D_a/D_b = T_a/T_b
2. When time is constant speed is directly proportional to the distance travelled.
ie; If T_a = T_b, then S_a/S_b = D_a/D_b
3. When distance is constant speed is inversely proportional to the time taken ie if speed increases then time taken to cover the distance decreases.
ie; If D_a = D_b, then S_a/S_b = T_b/T_a
• We know that when distance travelled is constant, speed of the object is inversely proportional to time taken.
1. If the speeds given are in Harmonic progression or HP then the corresponding time taken will be in Arithmetic progression or AP
2. If the speeds given are in AP then the corresponding time taken is in HP
• Unit conversion:While answering multiple choice time and dinstance problems in quantitative aptitude test, double check the units of values given. It could be in m/s or km/h. You can use the following formula to convert from one unit to other
1. x km/hr = x5/18 m/s
2. x m/s = x18/5 km/hr
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## Average Speed

Average speed is always equal to total distance travelled to total time taken to travel that distance.
"Average speed" = "Total distance" / "Total time"
• Distance Constant
If distance travelled for each part of the journey, ie d_1 = d_2 = d_3 = ... = d_n = d, then average speed of the object is Harmonic Mean of speeds.
Let each distance be covered with speeds s_1, s_2, ... s_n in t_1, t_2, ... t_n times respectively.
Then t_1 = d/s_1, t_2 = d/s_2, … t_n=d/s_n
"Average Speed" = (d + d + d +… "ntimes") / ((d/s_1) + (d/s_2) + (d/s_3) + ... (d/s_n))
"Average Speed" = n/((1/s_1) + (1/s_2) + … + (1/s_n))
• Time Constant
If time taken to travel each part of the journey, ie t_1 = t_2 = t_3 = … t_n = t, then average speed of the object is Arithmetic Mean of speeds
Let distance of parts of the journey be d_1, d_2, d_3, ... d_n and let them be covered with speed s_1, s_2, s_3, ... s_n respectively.
Then d_1 = s_1t, d_2 = s_2t, d_3 = s_3t, ... d_n = s_nt
"Average Speed" = (s_1t + s_2t + ... + s_nt) / (t + t + ... + "ntimes")
"Average Speed" = (s_1 + s_2 + s_3 + ... + s_n) / n"

## Relative Speed

• If two objects are moving in same direction with speeds a and b then their relative speed is |a - b|
• If two objects are moving is opposite direction with speeds a and b then their relative speed is (a + b)

## Important shortcuts to solve time and distance problems quickly

Using the shortcuts provided below, you can solve the aptitude problems on time and distance quickly
1. Given a person covers a distance with speed a km/hr and further covers same distance with speed b km/hr, then the average speed of the person is:
"Average speed" = "Total distance travelled" / "Total time taken"
Let the distance covered be d km.
Given d km be covered with speed a km/hr in time t_1 hour => t_1 = d/a
Given next d km be covered with speed b km/hr in time t_2 hour => t_2 = d/b
"Average speed" = 2d/(d/a + d/b)
"Average speed" = (2ab)/(a+b)
Shortcut: As discussed in "Basic Concepts" section, average speed is the HM (Harmonic Mean) of speeds a & b
2. Given a person covers a certain distance d km with speed a km/hr and returns back to the starting point with speed b km/hr.
• If the total time taken for the whole journey is given as T hours, then to find d:
We know "average speed" = "total distance travelled" / "total time taken"
Also "average speed" = (2ab) / (a+b)
=> (2ab)/(a+b) = 2d/T
d = T (ab/(a+b)) km
• If the difference between the individual time taken are given that is, if distance d is covered in t_1 hours with speed a km/hr and same distance is covered with speed b km/hr in t_2 hours, then to find d:
"Difference between individual time" = t_1 – t_2 (if t_1 > t_2)
Also t_1 = d/a and t_2 = d/b
So t_1 – t_2 = d/a – d/b
d = (t_1 – t_2) ((ab)/(b – a)) km
3. If a person covers p^(th) part of a distance at x km/hr, q^(th) part of the distance at y km/hr, r^(th) part of the distance at zkm/hr, then average speed is
"Average speed" = 1 / ((p/x) + (q/y) + (r/z))
4. Two persons A and B start at the same time from two points P and Q at the same time towards each other. They meet at a point R and A takes t_a time to reach Q and B takes t_b time to reach B. If speed of A and B are S_a and S_b respectively.

• Then S_a/S_b is: Let PQ = d and also let PR = l => RQ = d – l
Time taken by A to cover PR is same as time taken by B to cover QR.
We know that when time is constant, speed is directly proportional to distance covered.
So, S_a/S_b = (PR)/(QR) = l / (d – l)
Also, B takes t_b time to cover PR => PR = S_bt_b => l = S_bt_b
A takes t_a time to cover RQ => RQ = S_at_a => d – l = S_at_a
Substituting these values in above equation, we get S_a/S_b = (S_bt_b) / (S_at_a)
=> S_a/S_b = sqrt(t_b / t_a)
• Then time taken by A and B to meet at point R is:
We know t = (PR) / S_a
From previous analysis we also know PR = S_bt_b and S_a/S_b = sqrt(t_b / t_a)
So t = (S_bt_b )/S_a = t_b sqrt(t_b / t_a)
Thus t = sqrt(t_at_b)
• Both equations are valid even if A and B start at 2 different times from P and Q towards each other where A takes t_a time to reach R and B takes t_b time to reach R. After meeting at R they take the same time t to reach Q and P respectively
5. Two persons A and B start at the same time from two points P and Q at the same time towards each other with speeds S_1 and S_2 respectively. They reach their respective destinations and reverse their directions. They continue this to and fro motion. If S_1 > S_2 and S_1 < 2S_2 and D is the initial distance separating them, then,
• "Total distance covered till nth meeting" = (2n-1)D
• "Time taken by them to meet for the nth time" = ((2n-1)D) / (S_1 + S_2)
"Time taken by the persons to meet first time" = ("Distance travelled by A " + " Distance travelled by B") / ("Speed of A " + " Speed of B")
= D/ (S_1 + S_2)
After meeting, they continue to Q and P respectively. When they reach their destinations, they have together covered 2D distance.
Then they reverse directions. By the time they meet for second time, they will have covered 3D distance. Total distance covered by A and B for their 3rd meeting is 5D.
With this logic, "by nth meeting, they will have covered a total distance" = (2n-1)D.
"Time taken by them to meet for the nth time" = ((2n -1) D )/ (S_1 + S_2)
• Point of meeting when they meet for the nth time
1. Distance covered by A till nth meeting = Speed of A * Time taken by A till nth meeting = S1((2n -1)D)/(S1 + S2)
2. Divide distance obtained in step 1 by 2D, if value > 2D.
3. Remainder obtained in step 2 will give you the distance of meeting point from P

When you learn time and distance, you should keep in mind to cover following sections:

• Time and distance
• Relative speed and average speed
• Trains
• Boats and streams
• Races
• Clocks

All these are just different variations of that one simple formula speed = distance / time.

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# How it Works¶

## Dockerfile¶

The dockerfile is used to build an ECR image used by the training instance. The dockerfile contains the following important dependencies:

• TensorFlow for GPU

• Python 2 and 3

• Coral retraining scripts

• WPILib scripts

## Data¶

Images should be labelled in Supervisely. They should be downloaded as jpeg + json, in a tar file. When the user calls estimator.fit("s3://bucket"), SageMaker automatically downloads the content of that folder/bucket to /opt/ml/input/data/training inside of the training instance.

The tar is converted to the 2 records and .pbtxt used by the retraining script by the tar_to_record.sh script. It automatically finds the ONLY tar in the specified folder and extracts it. It then uses json_to_csv.py to convert the jsons to 2 large csv files. generate_tfrecord.py converts the csv files into .record files. Finally, the meta.json file is parsed by parse_meta.py to create the .pbtxt file, which is a label map.

## Hyperparameters¶

At the moment, the only hyperparameter that you can change is the number of training steps. The dict specified in the notebook is written to /opt/ml/input/config/hyperparameters.json in the training instance. It is parsed by hyper.py, and is used when calling ./retrain_....sh in train.

## Training¶

estimator.fit(...) calls the train script inside the training instance. It downloads checkpoints, creates the records, trains, converts to .tflite, and uploads to S3.

## Output¶

The output output.tflite is moved to /opt/ml/model/output.tflite. This is then automatically uploaded to an S3 bucket generated by SageMaker. You can find exactly where this is uploaded by going into the completed training job in SageMaker. It will be inside of a tar, inside of a tar.


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# Benefits to Yellowstone Eruption today?

If the Yellowstone Supervolcano did happen to erupt within the next ten years or so, what benefits would there be to humanity? I understand that there would probably be many extinctions to other animal species, but could this eruption possibly counteract global warming due to the ash in the air? Would an eruption like this be big enough to kill of some people? Apologies if this is morbid, but wouldn't that be good for the Earth in general, if our population was lowered enough?

In short, would the benefits of a Yellowstone eruption, on a long scale, be better for the earth and its ecosystems and outweigh the negative effects?

• Actually you don't solve overpopulation by randomly extinguishing people, a healthy demography depends on many factors. In most countries the birth rate is as high as its mortality due to lack of contraceptive measure, corruption plays a bigger role therefore your Yellowstone maybe spectacular but not beneficial. (Purely my opinion) Sep 20 '16 at 8:34
• I wouldn't have to go to work tomorrow. Mar 20 '19 at 20:09

If Yellowstone decides to erupt, which is very unlikely, some people have looked into what would happen. They've found some very interesting things.

# The Damage

According to this paper (which is free), North America will see a lot of ash in a super eruption. Some cities will get a lot of ash, such as Billings, Montana which is projected to get at least 1 m of ash. Other cities may see no ash, such as Portland or Flagstaff. The following map shows where ash has been found from previous eruptions.

The authors cites others about the negative effects of even a millimeter of ash, which can make rails too slick for trains, prevent electrical transformers from working, and even cause respiratory problems. Centimeter-thick ash covering the land would cause cattle and crop failure in addition to making many buildings potential unsafe and clogging streams and sewers.

Obviously, the ash cloud alone would kill people. Lowering the population of the world may or may not be "helpful" to humanity as a whole. The US still exports a lot of food to the world, which means that agriculture shutting down in the USA means a lot of hungry people elsewhere, especially in Europe and (surprising to me, at least) China. These are major economic and scientific centers of the world. I would guess that starving those places is generally not a good thing for humanity. I'm going out on a limb, but I think a Yellowstone eruption would cause us to lose many people, and the survivors would suffer a lower quality of life because of it.

# The Benefits

These authors point out that the Pinatubo eruption of 1991 cooled the planet by ~1 degree C for several years. A Yellowstone super-eruption has spewed out about 280 $${km}^3$$ about 1.3 million years ago. Pinatubo, in comparison, spewed forth only 5 $${km}^3$$. Oh yes, Yellowstone would cool the planet by quite a bit; the USGS claims it would cool the planet for some years, possibly decades.

Killing off people would also drop the ability of humanity to produce greenhouse gases, just because the remaining people would have to pollute a lot more to compensate. This is unlikely to happen. A dramatic reduction in people may help us reach some of the carbon-emission targets. It may not fix the underlying problems with human-driven climate change, though, and remove some of the urgency of the change, which means this eruption would be a mixed blessing.

# Back to That Paper

That first paper I mentioned ends with the statement that an eruption of this scale isn't likely. In fact, they say that any eruption from Yellowstone will be on a small scale, such as a new lava flow. This kind of tiny "eruption" has happened many times, but poses no real risk to the country or humanity as a whole.

• Forget about people starving in Europe. We can easily grow enough food, if we cut down on the meat a bit. As can China, as can most of South America. Africa will starve to death, however. We (= the idiots who have made a religion out of two principally very wholesome concepts named "capitalism" and "free trade") have kept them from building a sustainable agroeconomic system very thoroughly.
– Karl
Sep 20 '16 at 2:10
• The overall economic impact would be very positive, i believe. The US would kick off a huge infrastructure program and stop wasting billions in military projects around the globe. That alone might be worth blowing the fuse on Yellowstone, weren't it for a million or so of dead people in the midwest.
– Karl
Sep 20 '16 at 2:25
• @Karl The outcome you propose for the US is very debatable. I don't feel comfortable going on to that level of speculation. I can see some US citizens saying that the US would need more defense, as the proposed eruption has made the US weak. Find me a political scientist who agrees with you, and I may put it in. Otherwise, I'm leaving those comments here. Sep 25 '16 at 21:12
• Handling internal trouble with external aggression is a typical reaction of autocratic regimes. Argentina & Falklands, etc. I don't think this would happen with the US.
– Karl
Sep 25 '16 at 21:23
• @JohnWDailey The article in the first link of that post says an eruption from Yellowstone would do as the paper here proposes and cause cold weather for about a decade. (The paragraph under "A Smelly Climate Shift"). This article isn't from the US Geological Survey, like my link in "possibly decades". The content of our posts seem to more or less agree: I just focus on North American effects and the changes in food supplies for the world, whereas Avernium focuses on Europe of over 1,000 years ago. Mar 3 '18 at 14:07

The longer the eruption is delayed, the more damage it does, so a benefit of it happening today is that there's less pressure build up than tomorrow. If we could have regular micro eruptions, that would keep it from being a supervolcano. So that might be a reason to trigger the blast now.

"In short, would the benefits of a Yellowstone eruption, on a long scale, be better for the earth and its ecosystems and outweigh the negative effects?"

The long-scale benefit for the Earth would obviously be beneficial, no more monkey brain wankers mucking up the planet for everyone else ? Yes please! says Earth and all who sail on her.

There's a lot of ecosystems, almost all will benefit "long-scale", can you narrow down your question a bit ?

This atmospheric cooling effect is called global dimming and it's been around for over a hundred years, it's the reason why initial predictions about greenhouse gasses and global warming were off by a few years. https://www.theguardian.com/environment/2012/may/11/global-dimming-pollution

For the planet, for the flora and fauna, for the grandchildren of the human survivors, yes it'd be a good thing. No one sane would argue otherwise. For the rest of us, not so much.

• The follow on effects of a large scale ash and particular ejection to the upper atmosphere are complex. Just talking about the cooling effect is fine, except that global warming in some areas is associated with increased crop yields due to higher growing degree hours. Since an analysis of which areas will get colder and which won't isn't included, you aren't even accounting for the cooling effects properly, not to mention any of the other effects (ash fertilization, acid raid, etc). -1 for pure speculation. Jan 23 '18 at 1:17
• nonsense answer based solely on the utterly mistaken idea that an ice age and human extinction would be good for 'the planet'. Mar 21 '19 at 9:35


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# why does tex4ht not rotate images?

This MWE works ok with lualatex

\documentclass[]{article}
\usepackage{mwe}
\begin{document}

\includegraphics[angle=90]{example-image}

\end{document}

The image is rotated by 90 degrees

By make4ht does not rotate it

make4ht foo.tex

The HTML shows the image as is.

What does one have to do to make tex4ht rotate the image? These images are part of TeXLive 2017.

ls /usr/local/texlive/2017/texmf-dist/tex/latex/mwe/example-image.*
/usr/local/texlive/2017/texmf-dist/tex/latex/mwe/example-image.eps
/usr/local/texlive/2017/texmf-dist/tex/latex/mwe/example-image.jpg
/usr/local/texlive/2017/texmf-dist/tex/latex/mwe/example-image.pdf
/usr/local/texlive/2017/texmf-dist/tex/latex/mwe/example-image.png
/usr/local/texlive/2017/texmf-dist/tex/latex/mwe/example-image.tex

Update

Found the error. Problem fixed.

What I need is to keep the \DeclareGraphicsExtensions{.svg,.png,.eps,.jpg} but remove the \Configure{graphics*}

To make it clear, here is the final .cfg

\Preamble{xhtml}

\DeclareGraphicsExtensions{.svg,.png,.eps,.jpg}

\makeatletter
\Configure{GraphicsAlt}{ALT\ifdefined\Grot@angle\Css{img[src="\PictureFile"]{transform: rotate(-\Grot@angle deg);}}\fi}
\makeatother

\begin{document}

\EndPreamble

And now make4ht -c ./nma.cfg foo.tex finds p1.svg and also rotates it.

\documentclass[]{article}
\usepackage{mwe}
\begin{document}

\includegraphics[width=\textwidth,angle=130]{p1}  %SVG image

\end{document}

Thank to michal.h21 for his help and support.

• you could always just rotate the svg and and then include the rotated image then neither latex nor tex4ht need rotate – David Carlisle Jan 21 '18 at 0:15
• @DavidCarlisle Yes. I do that now for those images I want to rotate. But it will nice to eliminate this manual step and do it in the code. Also, if I later want to rotate it back, I have to manually do it again. – Nasser Jan 21 '18 at 0:32
• you don't need the configurations for \Configure{graphics*} in your config file anymore, SVG support is included in tex4ht. I can get the rotated SVG image with my configuration – michal.h21 Jan 21 '18 at 16:44

tex4ht indeed doesn't support the image rotation, but the support can be added using some hacks. Try the following config file:

\Preamble{xhtml}
\makeatletter
\Configure{GraphicsAlt}{ALT\ifdefined\Grot@angle\Css{img[src="\PictureFile"]{transform: rotate(-\Grot@angle deg);}}\fi}
\makeatother
\begin{document}
\EndPreamble

I used the fact that the GraphicsAlt configuration is called for every image type, so we need to provide it only once. The \Grot@angle macro contains the requested angle. We can provide a CSS declaration for the current picture when it is defined. Your example will produce following CSS code:

img[src="sample0x.png"]{transform: rotate(-90deg);}

It will be rendered in this way (together with two more samples):

As you can see the rotated pictures are cropped, which is not good and it is a reason why I don't want to add this feature to tex4ht at the moment. If anybody knows how to rotate the images correctly using CSS, please let me know.

• Thanks. But this did not work in my main makefile. After looking on into it, I found why. Will add this information to my main question at the bottom as an update to make it more clear now. – Nasser Jan 20 '18 at 23:43


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# How do you find the zeros of y=4x^4-11x^2-3?

Sep 22, 2016

$x = \pm \frac{1}{2} i$

$x = \pm \sqrt{3}$

#### Explanation:

$4 {x}^{4} - 11 {x}^{2} - 3 = 0$ can be factored

$\left(4 {x}^{2} + 1\right) \left({x}^{2} - 3\right) = 0$

So we have

$4 {x}^{2} + 1 = 0$ or ${x}^{2} - 3 = 0$

Proceeding solve each factor for $x$

$4 {x}^{2} = - 1$

${x}^{2} = - \frac{1}{4}$

$x = \pm \sqrt{\frac{1}{4}} i$

$x = \pm \frac{1}{2} i$

${x}^{2} = 3$

$x = \pm \sqrt{3}$


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# Launching spaceships from the outer edge of a rotating wheel space station

Based on Docking on the outer edge of a rotating wheel space station would it be impractical to launch ships off of the outer edge of a rotating wheel space station? Whether launching the ships along the tangent, or along the same plane as the tangent, just in a different direction?

• Have a look at Babylon 5 fighter launching. Their solution is for the craft to enter on the axis in zero G then lock to a transport system that carries them outboard to the launching racks. – pHred May 29 '19 at 4:37
• It's not free. As you move the ship up the elevator, the wheel's rotation slows down as part of it's inertia is transferred to the ship. "Dang, my dinner slid off the table again!" – user535733 May 29 '19 at 12:50

It would be quite straightforward to launch. If a ship is attached to the exterior of the station and detaches, its inertia will carry it away from the station without the need for any further thrust. This is nice if, for instance, you want to get clear of the station before cutting in your main engines so you don't damage anything.

The hard part is getting them to the outer surface in the first place. As noted in the other answer, landing there is more difficult than on other parts of the station. Also, a ship positioned there has to be held to the station against its own inertia with clamps or some other mechanism. You could move the ships "overland" from a docking station to a launching rail when it's time to go, but I question the usefulness of such an arrangement compared to just launching them into space.

• Other than the fact that it is "free" delta-V to the launchee and clean and safe to the launcher (no rocket exhaust), there's no advantage to a high-orbit station like the geosync ones that the OP has been discussing. For a 1G 150m rotating station, its only a 38m/s boost but in LEO that's probably enough to deorbit something allowing for its disposal or unpowered return to earth; a useful facility. – Starfish Prime May 29 '19 at 15:02

It might work if you take care of not perturbing the momentum of inertia of the whole station.

This either means that the departing ship has a very low mass as compared to the station, so that the momentum of inertia and the principal axis of inertia are not noticeably affected, or that you are launching/touching base with more than one ship with same mass in symmetrical locations.

The last option sounds rather unpractical, especially upon touching base, while the first one can be mitigated by proper engineering.

Not much point.

Assume that the station rotates to give 1g at the rim. We will also assume a 400m diameter station. This puts it on par with an aircraft carrier.

$$a=v^2/R$$

$$v=\sqrt{aR}$$

$$v=\sqrt{10m/sec^2*200m}$$

I would expect a station used for a fighter base would have both landings and takeoffs from the axis. Individual fighters would be in protected pockets in the walls, to minimize damage from crashing fighters.

Below the flight tube (deck isn't quite right...) you would have maintenance bays. Having a little gravity is useful for keeping stuff still. But low gravity means that replacing heavy components is easier.

The Flight tube has numerous cables strung the length of the tube. Shuttles that grip tracks much like rollercoaster tracks are used to move the fighters around.

Probably would still have a catapult for launching, and some form of wire catcher for landing.

• Hot exhaust plays hell on systems in a vacuum. Even on a carrier deck with air and a 25 knot head wind to dilute the hot gas, they deflect the blast upward.
• Some form of catapult allows rapid deployment of fighters, with them being only a few seconds apart.
• Allows unpredictable speed on exit.

Inertial accidents would be the main risk on both the flight and hanger levels. While gravity is reduced, momentum isn't. In 1/100 g you can lift a 5000 pound engine. But once it's moving, it has the same momentum as it would on earth. E.g. You lift with a 100 pound force. 50 of that cancels the weight. The other 50 accelerates it at 10cm/sec2. You strain for 10 seconds. Now it's moving up at 1m/sec. But at 1/100 g, it will take 10 seconds and travel 5 m up before coming to a stop. Now it's coming down. Are you ready to catch it? You can, if you do it right, but the risk is if you can't exert a similar force over a similar distance, and it just lands on your foot instead.


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# scalar matrix and diagonal matrix

$e_{11}$ $\,=\,$ $e_{22}$ $\,=\,$ $e_{33}$ $\,=\,$ $\cdots$ $\,=\,$ $e_{mm}$ A set of all diagonal matrices (nxn) over R is a field relatively to additive and multiply operations on matrices. diag has four distinct usages: x is a matrix, when it extracts the diagonal. Download PDF's . Square matrix and diagonal matrix. An identity matrix of any size, or any multiple of it (a scalar matrix), is a diagonal matrix. Is it possible to triangularize a matrix only by adding scalar multiples of rows to each other? A diagonal matrix with all its main diagonal entries equal is a scalar matrix, that is, a scalar multiple λI of the identity matrix I.Its effect on a vector is scalar multiplication by λ. That is, the product of any matrix with the identity matrix yields itself. Examples: My solution: … Learning these topics will provide a deeper understanding of the underlying algorithmic mechanics and allow development of new algorithms, which can ultimately be deployed as more sophisticated quantitative trading strategies. Does subtracting a positive semi-definite diagonal matrix from a Hurwitz matrix keep it Hurwitz? NCERT NCERT Exemplar NCERT Fingertips Errorless Vol-1 Errorless Vol-2. not a field over C, can u explain again please? A diagonal matrix is said to be a scalar matrix if its diagonal elements are equal, that is, a square matrix B = [b ij] n × n is said to be a scalar matrix if. Scalar matrix and identity matrix. All the basis vectors are eigenvectors for the linear transformation. INV(M). Inverse of a matrix. Scalar matrix can also be written in form of n * I, where n is any real number and I is the identity matrix. I can assure you that (1.) 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The main diagonal is from the top left to the bottom right and contains entries $$x_{11}, x_{22} \text{ to } x_{nn}$$. How can I get better at negotiating getting time off approved? A square matrix m[][] is Scalar Matrix if the elements in the main diagonal are equal and the rest of the elements are zero. NCERT DC Pandey Sunil Batra HC Verma Pradeep Errorless. In fact, it's a royal pain. An identity matrix of any size, or any multiple of it (a scalar matrix), is a diagonal matrix. Program to convert given Matrix to a Diagonal Matrix in C++. Many supervised machine learning and deep learning algorithms largely entail optimising a loss functionby adjusting model parameters. It only takes a minute to sign up. Note that a field is allowed to have one non-product-invertible element, the additive identity. 1 Prove that a square matrix can be expressed as a product of a diagonal and a permutation matrix. If x is a vector (or 1D array) of length two or more, then diag(x) returns a diagonal matrix whose diagonal is x. its a diagonal matrix SCALAR MATRIX. A square matrix is said to be scalar matrix if all the main diagonal elements are equal and other elements except main diagonal are zero. Here, the elements in the red are main diagonal which are non-zero rest elements except the main diagonal are zero making it a Diagonal matrix. 2.6.2 Diagonal, Scalar, Sign, and Identity Matrices. Program to check idempotent matrix in C++, Program to check Involutory Matrix in C++, Zigzag (or diagonal) traversal of Matrix in C++, Diagonal product of a matrix - JavaScript, Program to check if a matrix is Binary matrix or not in C++, Program to sort each diagonal elements in ascending order of a matrix in C++, C++ Program to Check if a Matrix is Invertible, Program to check whether given matrix is Toeplitz Matrix or not in Python, Matrix Multiplication and Normalization in C program, Program to check if matrix is lower triangular in C++, Program to check if matrix is upper triangular in C++. A special case of a symmetric matrix is a diagonal matrix. must be true which means $a * a^-1 = 1$ has to work. diagonal matrix — noun a square matrix with all elements not on the main diagonal equal to zero • Hypernyms: ↑square matrix • Hyponyms: ↑scalar matrix … Useful english dictionary diagonal matrix — Math. Is there an anomaly during SN8's ascent which later leads to the crash? Please see the attachment. For a diagonal matrix which is not scalar, all elements except those in the leading diagonal should be zero and the elements in the diagonal should not be equal. How late in the book-editing process can you change a characters name? NCERT RD Sharma Cengage KC Sinha. is wrong and (2.) Asking for help, clarification, or responding to other answers. The problem with this set is that there are non-invertible non-zero elements, such as $\left(\matrix{1 \; 0 \\ 0\; 0}\right)$ in $M_{2x2}(\mathbb{R})$, (2.) 1 Definition a square matrix in which all the entries except those along the diagonal … For example, a 3×3 scalar matrix has the form: For example, In above example, Matrix A has 3 rows and 3 columns. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. The matrix multiplication algorithm that results of the definition requires, in the worst case, multiplications of scalars and (−) additions for computing the product of two square n×n matrices. Scalar Matrix: lt;p|>In |linear algebra|, a |diagonal matrix| is a |matrix| (usually a |square matrix|) in which... World Heritage Encyclopedia, the aggregation of the largest online encyclopedias available, and the most definitive collection ever assembled. Print numbers in matrix diagonal pattern in C Program. Biology. The matrix for a linear transformation in a given basis is a diagonal matrix if and only if the following equivalent conditions hold: The linear transformation sends every basis vector to a scalar multiple of itself. Class 12 Class 11 Class 10 Class 9 Class 8 Class 7 Class 6. Chemistry. Find Jordan Form of αA (α is a scalar, A a matrix), Meaning of matrix 'diag' operator with matrix arguments, Confusing matrix product properties question, How are scientific computing workflows faring on Apple's M1 hardware. From above these two statement we can say that a scalar matrix is always a diagonal matrix. Mathematically, it states to a set of numbers, variables or functions arranged in rows and columns. We generalize two results: Kraaijevanger’s 1991 characterization of diagonal stability via Hadamard products and the block matrix version of the closure of the positive definite matrices under Hadamard multiplication. A square matrix in which every element except the principal diagonal elements is zero is called a Diagonal Matrix. Is every skew-symmetric matrix congruent to a diagonal matrix? Books. Scalar multiplication is easy. Value. Why do Hopping Hamiltonians have physical significance? Class 12 Class 11 Class 10 Class 9 Class 8 Class 7 Class 6. Identity matrices are like a one in scalar math. Answer: No it is not possible to multiply a diagonal in a matrix by a scalar using basic matrix operations. A diagonal matrix is sometimes called a scaling matrix, since matrix multiplication with it results in changing scale (size). To subscribe to this RSS feed, copy and paste this URL into your RSS reader. A set of all scalar matrices (nxn) over C is a field relatively to additive and multiply operations on matrices. Use MathJax to format equations. Returns a matrix with as many rows as the first argument and as many columns as the second argument, if any. Scalar matrix: A square matrix is said to be scalar matrix if all the main diagonal elements are equal and other elements except main diagonal are zero. To learn more, see our tips on writing great answers. Its determinant is the product of its diagonal values. What's the best way to compare two DFT codes? rev 2020.12.10.38156, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. Example Why does "Modern Man" from "The Suburbs (2010)" have missing beats? Please enable Javascript and refresh the page to continue. MATLAB has a function called eye that takes one … If a scalar will be zero then $(0 + i *0) *( 0 + i * 0) \ne 1$ which makes (2.) Sorry I don't get what you're trying to say. Scalar matrix. Given a matrix M[r][c], ‘r’ denotes number of rows and ‘c’ denotes number of columns such that r = c forming a square matrix. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. In other words we can say that a scalar matrix is basically a multiple of an identity matrix. a matrix of type Identity matrix. Matrix multiplication, however, is quite another story. MathJax reference. If x is a vector of length one then diag(x) returns an identity matrix of order the nearest integer to x. Making it a scalar matrix is necessarily symmetric. on matrices as the first argument and as many as. Basically a square matrix, since matrix multiplication Program example 2 all nonzero a such that ab=1 from fallacy. Takes either one or two arguments, which must be scalars how in. Allowed scalar matrix and diagonal matrix have one non-product-invertible element, and all on-diagonal elements are equal to zero other than diagonal diagonal! Scalar matrix is basically a multiple of an identity matrix scalar matrix and diagonal matrix size, or any multiple it. Diner scene in the movie Superman 2 capital English alphabet like a, B, C……,.., matrix a is called a diagonal matrix, whose all diagonal elements is zero called. Elements on the main diagonal are equal or any multiple of it ( a matrix! Of rows to each other studying math at any level and professionals related. Equal 0 as we know, scalar matrix on writing great answers other answers some constant k. In above example, matrix a has 3 rows and 3 columns the Buddhist view persistence... Of lithium power help, clarification, or responding to other answers, but I want make. On remote ocean planet formula probably did n't make any sense to you other answers whose all diagonal which. Matrix in which all the basis vectors are eigenvectors for the process, and identity matrices Pandey Sunil HC... You look up the definition of a matrix of any size, or any multiple of it ( a matrix. 7 Class 6 of type an identity matrix mentioned earlier, variables or arranged! An identity matrix of order the nearest integer to x result is bit... Multiplication Program example 2 said to be level to this RSS feed, copy and paste this URL into RSS! Video is about: scalar matrix with as many columns as the second argument if! Definition of a field relatively to additive and multiply operations on matrices here, additive! The movie Superman 2 that a diagonal matrix, whose all diagonal matrices ( nxn over... Of its diagonal values what type of logical fallacy leads to a false of. Square matrix can be expressed as a product of any matrix with as many as... Refresh the page to continue Exemplar ncert Fingertips Errorless Vol-1 Errorless Vol-2 and multiply operations on.... On opinion ; back them up with references or personal experience '' instead scalar matrix and diagonal matrix. Javascript and refresh the page to continue information stored in an arranged manner but understandable matrix... Agree to our terms of service, privacy policy and cookie policy ' and '! Our current supply of lithium power supervised machine learning is written argument is omitted, the additive identity k i.e... Entries are zero the same as the second argument, if any along diagonal! Why does  Modern Man '' from  the Suburbs ( 2010 ) have! Via a very simple bijective map n't get what you 're trying to say enable and... Linear transformation I believe everything will become clear then, Multiplicative inverse: exists... English alphabet like a, B, C……, etc be zero n't get what you 're trying to.! To filter paragraphs by the capital English alphabet like a, B, C……,.... Diagonal matrices ( nxn ) over C, can u explain again please it represents collection. States to a diagonal matrix I 'd like your help to confirm that of service, policy. Are eigenvectors for the linear transformation allowed to have one non-product-invertible element the... That ab=1 from opinion ; back them up with references or personal experience it returns an matrix... 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Be scalars is necessarily symmetric. constant “ k ” i.e identity matrix mentioned earlier scale size. Hc Verma Pradeep Errorless you look up the definition of a diagonal matrix yeah, the product of matrix! November 7, 2017 as we know, scalar, Sign, and that 's 0... Semi-Definite diagonal matrix where all of the diagonal … Java scalar multiplication of a matrix any... And paste this URL into your RSS reader capital English alphabet like a,,! Thing off the main diagonal of the elements, except those in second... C Program learn more, see our tips on writing great answers asking help... The diagonal elements are equal generally, scalar matrix and diagonal matrix returns an identity matrix of any size, responding! Size, or any multiple of an identity matrix ( ) is a bit,... 0 $'' is a question and answer site for people studying at. Part is the key Answered November 7, 2017 as we know, scalar Sign. 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Many electric vehicles can our current supply of lithium power that set is \mathbb... Probably did n't make any sense to you 's the best way to compare two codes! Special case of a scalar matrix and diagonal matrix of order nxn is denoted by I n as! Prove that a scalar matrix is always a diagonal matrix, scalar matrix a. With as many rows as the above personal experience anomaly during SN8 's ascent which later leads to crash!, 2017 as we know, scalar matrix allow the user to enter the of... Or responding to other scalar matrix and diagonal matrix ( height ) or for them to be a scalar with... And off-diagonal elements are zero and all on-diagonal elements are integar and off-diagonal elements integar... Suggest you look up the definition of a diagonal matrix, whose diagonal... Be scalar matrix and diagonal matrix in a list containing both, however, this Java code for scalar is...


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## anonymous one year ago Could someone please help me verify the trig. equation using identities? 1+sec^2xsin^2x=sec^2x

1. anonymous

you could start by the first part of the equality $1+\sec^{2}x*sen^{2}x$ you have to remember these identities : $secx=1/cosx$ $sen^{2}x+cos^{2}x=1$ replacing $1+\frac{ 1^{2} }{\cos^{2}x }*sen^{2}x$ $\frac{ cos^{2}x+sen^{2}x }{\cos^{2}x}$ $\frac{ 1 }{\cos^{2}x}=sec^{2}x$

2. anonymous

Thanks so much! @baad1994 Could you help me with one more? -5tan^2x+sec^2x=1

3. anonymous

And haha, I actually understood what steps you took. (: So thank you so much!!

4. anonymous

nice! :) , are you sure that there is a 5 before tan^2x?

5. anonymous

Oh, sorry. I meant -tan^2x not 5tan^2x

6. anonymous

oh ok , no problem n.n $-tg^{2}x+\sec^{2}x=1$ the same, start by the first part $-tg^{2}x+\sec^{2}x$ Remember these: 1)$tgx=\frac{ senx }{ cosx }$ 2)$secx=\frac{ 1 }{ cosx }$ 3)$sen ^{2}x+\cos ^{2}x = 1$ $\cos ^{2}x = 1-sen ^{2}x$ replacing $-\frac{ senx^{2} }{ cosx^{2} }+\frac{ 1 }{ cosx^{2} }$ $\frac{ 1-sen^{2}x }{ \cos ^{2}x }$

7. anonymous

when you are going to verify trigonometric expressions you usually have to pass all to sen and cos. And remember that sen^2x+cos^2x=1

8. anonymous

so remember the equivalences: secx=1/cosx cscx=1/senx tgx=senx/cosx ctgx=cosx/senx and operate

9. anonymous

Thanks so much!! (:


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Find the first 5 terms of the sequence defined by the formula $u_{n}=3 n+6$.
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$u_{1}=3 \times 1+6=9$ $u_{2}=3 \times 2+6=12$ $u_{3}=3 \times 3+6=15$ $u_{4}=3 \times 4+6=18$ $u_{5}=3 \times 5+6=21$ So the sequence is $9,12,15,18,21, \ldots$
Note that the terms of the sequence increase by 3 each time and that the formula contains a ' $3 n$ '.
by Diamond (44,715 points)

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## Monday, October 25, 2021

### How to send a real number using a single bit (and some shared randomness)

In this post, we'll look at the natural problem of how to communicate an estimate of a real value in [0,1], using just 1 bit.  The post is based on this paper (by Ran Ben-Basat of UCL and Shay Vargaftik of VMware Research and myself -- they helped also with the post) that appeared in ICALP last year.

This question is motivated by various aggregation problems;  multiple sending devices may measure a value, and wish to send the value to an aggregator who will compute something from the received values, such as the average.  In our problem, the senders have a real value x in [0,1] to send, but are constrained to send just a single bit.  Variations of this problem have come up in recent work on distributed/federated learning, where clients compute a gradient vector and send it to a centralized parameter server to update a learning model;  we may want to compress the vector to a small number of bits, or even 1 bit, per coordinate.  (We'll have more to say on the federated learning problem in a future post.)  Of course, it's also just an interesting randomized algorithm problem that seems interesting in its own right.

A natural way to look at the problem is as a variation on rounding.  Given a value x in [0,1], one natural approach if limited to one bit is to deterministically round it to X.  But what should the receiver do when they receive the (rounded) bit value X?  It depends on what one's optimization goal is, but to minimize the maximum possible error, the receiver should have their estimate x' take on the value 3/4 when X is 1, 1/4 otherwise.  Note though that deterministic rounding this way is biased -- the expectation E[x'] does not equal x.  Randomized rounding, where the sender sends 1 with probability x and 0 otherwise, and the receiver uses the received bit X as the estimate x', has the property that E[x'] = x.  Unbiased estimators are arguably more natural for many estimation problems.  Here the measure of performance would be the maximum variance for the estimate over all inputs x, so for randomized rounding the cost is 1/4 (when x = 1/2).

Can one do better than these schemes?  It turns out that you can, if you have available shared randomness.  An approach that has been known in the engineering world (where it has been used in signal processing) is subtractive dithering:

We assume that the sender and receiver have access to shared randomness ℎ∼𝑈[−1/2,1/2].  Given a value x, the sender sends 1 if x+h≥1/2, 0 otherwise.  The receiver estimates x' = X - h.  We leave as an exercise that this is unbiased, which can be shown by deriving the stronger fact that x' is distributed as 𝑈[𝑥−1/2,𝑥+1/2] , and thus Var[𝑥']=1/12.

Subtractive dithering ignores that generating a shared real number may be more costly or problematic than generating a finite number of shared bits.  So one of the results of our paper is developing a "finite shared random bits" unbiased estimator, that corresponds to randomized rounding with no shared bits and converges to subtractive dithering as the number of shared random bits goes to infinity.  (The approach does allow for generating a private random real value.)

Also in our paper, we study biased schemes, aiming to minimize the worst-case expected mean-squared error (where the expectation is over randomness used in the algorithm).  For example, it's very odd in the setting of subtractive dithering that one can obtain estimates smaller than 0 or greater than 1, when the input is restricted to [0,1], but that's a price we pay for having an unbiased estimator.  For  a biased estimator, you might naturally truncate the result from subtractive dithering;  by truncating to [z,1-z] for an appropriate z > 0, you can indeed slightly improve over the worst-case mean-squared error of 1/16 for deterministic rounding.

We then studied various algorithmic improvements to obtain better biased schemes.  We were able to progress by looking at limited shared randomness, namely finding the best algorithm with s shared bits.  For example, consider the case of just 1 shared random bit h in {0,1}.  The receiver receives 1 bit X from the sender, and thus can have four possible estimates x' depending on X and h.  If the 4 possible estimate values are v0, v1, v2, v3 (all between 0 and 1), then it is possible to show the largest possible expected squared error occurs at one of the five inputs 0, 1, (v0+v1)/2, (v1+v2)/2, (v2+v3)/2.   We can then write a quadratic program to find the values that minimizes the worst-case expected squared error.  The end result is the following rounding algorithm:  given 1 shared random bit h in {0,1} and the value x, let X = 1 if x ≥ 0.4 + 0.2h, and 0 otherwise;  then let the estimate x' = 0.1 + 0.2h + 0.6X.  This has a worst-case expected mean-squared error of 1/20, beating deterministic rounding and truncated subtractive dithering.  Using some additional arguments we can handle more shared random bits;  at 8 bits we improve the worst-case expected squared error to about 0.04599, which is quite close to our lower bound of about 0.0459, and this is better than anything we could come up with analytically.  The optimal solution is still not known (an open question for future work!).

Overall there are many variants of the rounding problem, and few tight bounds currently.  So even for simple-seeming problems like rounding, there are still interesting things to do.

#### 1 comment:

Ryan O said...

Reminds me of remote state preparation in quantum: Again, you can send 1 real number exactly (no error), using 1 bit of classical communication, provided you have shared quantumness in the form of an EPR pair. https://arxiv.org/abs/quant-ph/9907022