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Definitions

# Linear filter

A linear filter applies a linear operator to a time-varying input signal. Linear filters are very common in electronics and digital signal processing (see the article on electronic filters), but they can also be found in mechanical engineering and other technologies.

They are often used to eliminate unwanted frequencies from an input signal or to select a desired frequency among many others. There are a wide range of types of filters and filter technologies, of which this article will present an overview.

Regardless of whether they are electronic, electrical, or mechanical, or what frequency ranges or timescales they work on, the mathematical theory of linear filters is universal.

## Classification by transfer function

### Impulse response

Linear filters can be divided into two classes: infinite impulse response (IIR), and finite impulse response (FIR) filters. In general, a filter with a compact frequency response will have an infinite impulse response and a filter with a compact impulse response will have an infinite frequency response.

An FIR filter may be described as a weighted sum of delayed inputs. For such a filter, if the input becomes zero at any time, then the output will eventually become zero as well, as soon as enough time has passed so that all the delayed inputs are zero, too. Therefore the impulse response lasts only a finite time, hence the name finite impulse response. The transfer function of such a filter contains only zeros, and no poles.

In an IIR filter, by contrast, if the input is set to 0, then the output will decay exponentially, but never become precisely zero. Therefore the impulse response extends to infinity, and the filter is said to have an Infinite Impulse Response. The transfer function of such a filter will contain poles as well as zeros.

Until about the 1970s, only analog IIR filters were practical to construct. However, in digital logic, both FIR and IIR filters are straightforward, and both techniques are commonly used in digital filters. Analog FIR filters are still uncommon, though they can be built with analog delay lines.

### Frequency response

Here is an image comparing the most popular IIR filters: Butterworth, Chebyshev, and elliptic filters. The filters in this illustration are all fifth-order low-pass filters. The particular implementation -- analog or digital, passive or active -- makes little difference; the output from any implementation would still match this image.

As is clear from the image, elliptic filters are sharper than the others, but they show ripples on the whole bandwidth.

There are several common kinds of linear filters:

Band-stop and band-pass filters can be constructed by combining low-pass and high-pass filters. A popular form of 2 pole filter is the Sallen-Key type. This is able to provide low-pass, band-pass, and high pass versions. A particular bandform of filter can be obtained by transformation of a prototype filter of that class.

## Mathematics of filter design

LTI system theory describes linear time-invariant (LTI) filters of all types. LTI filters can be completely described by their frequency response and phase response, the specification of which uniquely defines their impulse response, and vice versa. From a mathematical viewpoint, continuous-time IIR LTI filters may be described in terms of linear differential equations, and their impulse responses considered as Green's functions of the equation. Continuous-time LTI filters may also be described in terms of the Laplace transform of their impulse response, which allows all of the characteristics of the filter to be analyzed by considering the pattern of poles and zeros of their Laplace transform in the complex plane. Similarly, discrete-time LTI filters may be analyzed via the Z-transform of their impulse response.

Before the advent of computer filter synthesis tools, graphical tools such as Bode plots and Nyquist plots were extensively used as design tools. Even today, they are invaluable tools to understanding filter behavior. Reference books had extensive plots of frequency response, phase response, group delay, and impulse response for various types of filters, of various orders. They also contained tables of values showing how to implement such filters as RLC ladders - very useful when amplifying elements were expensive compared to passive components. Such a ladder can also be designed to have minimal sensitivity to component variation a property hard to evaluate without computer tools.

Many different analog filter designs have been developed, each trying to optimise some feature of the system response. For practical filters, a custom design is sometimes desirable, that can offer the best tradeoff between different design criteria, which may include component count and cost, as well as filter response characteristics.

These descriptions refer to the mathematical properties of the filter (that is, the frequency and phase response). These can be implemented as analog circuits (for instance, using a Sallen Key filter topology, a type of active filter), or as algorithms in digital signal processing systems.

Digital filters are much more flexible to synthesize and use than analog filters, where the constraints of the design permits their use. Notably, there is no need to consider component tolerances, and very high Q levels may be obtained.

FIR digital filters may be implemented by the direct convolution of the desired impulse response with the input signal. They can easily be designed to give a matched filter for any arbitrary pulse shape.

IIR digital filters are often more difficult to design, due to problems including dynamic range issues, quantization noise and instability. Typically digital IIR filters are designed as a series of digital biquad filters.

All low-pass second-order continuous-time filters have a transfer function given by

$H\left(s\right)=frac\left\{K omega^\left\{2\right\}_\left\{0\right\}\right\}\left\{s^\left\{2\right\}+frac\left\{omega_\left\{0\right\}\right\}\left\{Q\right\}s+omega^\left\{2\right\}_\left\{0\right\}\right\}.$

All band-pass second-order continuous-time have a transfer function given by

$H\left(s\right)=frac\left\{K frac\left\{omega_\left\{0\right\}\right\}\left\{Q\right\}s\right\}\left\{s^\left\{2\right\}+frac\left\{omega_\left\{0\right\}\right\}\left\{Q\right\}s+omega^\left\{2\right\}_\left\{0\right\}\right\}.$

where

• K is the gain (low-pass DC gain, or band-pass mid-band gain) (K is 1 for passive filters)
• Q is the Q factor
• $omega_\left\{0\right\}$ is the center frequency
• $s=sigma+jomega$ is the complex frequency

## External links and references

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Using Perlmutter¶

Perlmutter is not a production resource

Perlmutter is not a production resource and usage is not charged against your allocation of time. While we will attempt to make the system available to users as much as possible, it is subject to unannounced and unexpected outages, reconfigurations, and periods of restricted access. Please visit the timeline page for more information about changes we've made in our recent upgrades.

Current Known Issues¶

Known Issues on Perlmutter

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ssh saul-p1.nersc.gov

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Last Updated: Dec 10th, 2021.

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Generally we recommend that you use the most recent programming environment installed on Perlmutter. However, sometimes it is convenient to have access to previous programming environments to check things like compile options and libraries, etc. You can use module load cpe/YY.XX to load the previous programming environment from year YY and month XX. We will remove cpe modules for environments that no longer work on our system due to changes in underlying dependencies like network libraries.

Load correct cpe-cuda module for gcc and nvcc compatibility

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E.g., If cpe/YY.XX is loaded please use module load cpe-cuda/YY.XX replacing the values of Y and X.

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There are 3 cudatoolkit modules per nvhpc (NVIDIA HPC SDK). This will give users access to a specific CUDA version and its associated libraries for that version of nvhpc. For example, the cudatoolkit/21.9_11.4 module for sdk nvhpc/21.9 will give you access to CUDA v11.4.1 and the CUDA APIs supported in that release.

If you are looking for a specific CUDA version that does not match the cudatoolkit version (like 11.3), please consider trying the CUDA Compatibility Libraries which are availble in all cudatoolkit modules. These are meant to make later CUDA versions backwards compatible with earlier versions. For example, you could load cudatoolkit/21.9_11.4 which should enable you to run a code using CUDA 11.3.

Note

You can only have one cudatoolkit module at a time.

A cudatoolkit module must be loaded to compile any GPU code on the system for any programming environment.

GPU-aware MPI¶

HPE/Cray MPI (CrayMPICH) is a CUDA-aware MPI Implementation, which allows the programmers to use pointers to GPU device memory in MPI buffers. See the CUDA-Aware MPI section for an example code.

You must set MPICH_GPU_SUPPORT_ENABLED to use CUDA-aware MPI

To use CUDA-aware MPI at runtime you must set export MPICH_GPU_SUPPORT_ENABLED=1 in your batch script or interactive node. Without this setting, you may get a segfault.

Building your application with CUDA-aware MPI¶

In addition to linking your application with HPE/Cray MPI wrappers (cc/ CC) you must also link your application with the HPE/Cray GPU Transport Layer (GTL) library which enables the use of CUDA-Aware MPI. The accelerator target needs to be set to nvidia-80 during your compilation step to achieve this; ways to enable:

• module load craype-accel-nvidia80 (one of the cudatoolkit modules must be loaded before loading this module), or,
• set environment variable: export CRAY_ACCEL_TARGET=nvidia80, or,
• pass the compiler flag -target-accel=nvidia80

Failing to do one of the above steps, your application will not be able to use CUDA-Aware MPI/ GPU Direct RDMA and at runtime you will see errors like:

MPIDI_CRAY_init: GPU_SUPPORT_ENABLED is requested, but GTL library is not linked

Known issues with CUDA-aware MPI¶

Bug in Cray MPICH may require GPU binding for jobs with many MPI ranks

Due to an outstanding bug with our vendor, users with many MPI ranks may also require GPU binding. This is because the MPI ranks are incorrectly allocating GPU memory, and too many MPI ranks that allocate this memory will cause the program to segfault (this segfault might happen during execution, or before the first statement is executed, and may happen only when multiple nodes are used). One workaround is to use gpu-binding to evenly spread the allocated memory. Here is an example of using gpu-binding in a 4 node job:

srun --ntasks=32 --ntasks-per-node=8 -G 4 --gpu-bind=single:2 python -m mpi4py.bench helloworld

Even with GPU binding, users may find that the number of MPI ranks they can use within a job are limited. Note that this also impacts CPU-only code that is using CUDA-aware MPI. We expect a fix for this problem soon.

We currently recommend using either PrgEnv-nvidia or PrgEnv-gnu to compile all your applications based on the current A100 support available through each of these environments.

PrgEnv-nvidia¶

Under the program environment nvidia, host compilers that are available for C, C++ and Fortran applications are nvc, nvc++ and nvfortran. You can see the versions of each compiler by running:

C Compiler:

cc --version

nvc 21.9-0 64-bit target on x86-64 Linux -tp zen-64
NVIDIA Compilers and Tools

C++ Compiler:

CC --version

nvc++ 21.9-0 64-bit target on x86-64 Linux -tp zen-64
NVIDIA Compilers and Tools

Fortran Compiler:

ftn --version

nvfortran 21.9-0 64-bit target on x86-64 Linux -tp zen-64
NVIDIA Compilers and Tools

Note

As you can see above the Cray wrappers (cc, CC, ftn) in the PrgEnv-nvidia, points to the relative host compilers nvc, nvc++ and nvfortran. You must use these wrappers to compile your application.

Compiling CUDA code with Cray wrappers¶

The host compilers nvc / nvc++ (accessible through the cc/ CC wrapper) in NVIDIA SDK has CUDA opt-in support. To compile a single source C / C++ code (host & device code in the same source file) with the Cray wrappers you must add the -cuda flag to their compilation step which notifies the nvc/ nvc++ compiler to accept CUDA runtime APIs. Omitting the -cuda flag will result in your application compiling without any of the CUDA API calls, and will generate an executable with undefined behavior. Failing to do so you will also notice warnings like:

nvc-Warning-The -gpu option has no effect unless a language-specific option to enable GPU code generation is used (e.g.: -acc, -mp=gpu, -stdpar, -cuda)

To compile an OpenMP application one must pass the -mp=gpu flag in the compilation step. For additional details on OpenMP support on the system please see the OpenMP section in our Perlmutter readiness page.

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Enabling offloading support with these programming models. You must set the accelerator target to nvidia80 to allow code generation for A100s. Ways to enable:

• module load craype-accel-nvidia80 (one of the cudatoolkit modules must be loaded before loading this module), or,
• set environment variable: export CRAY_ACCEL_TARGET=nvidia80, or,
• pass the compiler flag -target-accel=nvidia80

PrgEnv-gnu¶

Under the program environment gnu host compilers that are available for C, C++ and Fortran applications are gcc, g++ and gfortran. Switching to the PrgEnv-gnu will reload certain modules to match the Cray MPI installations and other libraries.

module load PrgEnv-gnu

Lmod is automatically replacing "nvidia/21.9" with "gcc/11.2.0".

Lmod is automatically replacing "PrgEnv-nvidia/8.2.0" with "PrgEnv-gnu/8.2.0".

Due to MODULEPATH changes, the following have been reloaded:
1) cray-mpich/8.1.11

Similar to the PrgEnv-nvidia, compiler wrappers (cc, CC, ftn) will now point to the corresponding host compilers in PrgEnv-gnu:

C Compiler:

cc --version

gcc (GCC) 11.2.0 20210728 (Cray Inc.)
Copyright (C) 2021 Free Software Foundation, Inc.
This is free software; see the source for copying conditions.  There is NO
warranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

C++ Compiler:

CC --version

g++ (GCC) 11.2.0 20210728 (Cray Inc.)
Copyright (C) 2021 Free Software Foundation, Inc.
This is free software; see the source for copying conditions.  There is NO
warranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

Fortran Compiler:

ftn --version

GNU Fortran (GCC) 11.2.0 20210728 (Cray Inc.)
Copyright (C) 2021 Free Software Foundation, Inc.
This is free software; see the source for copying conditions.  There is NO
warranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

GCC compatibility with the nvcc compiler¶

When using the PrgEnv-gnu environment in conjunction with the cudatoolkit module (i.e., if compiling any application for both host and device side), you must note that not every version of gcc is compatible with every version of nvcc. Current Perlmutter default cudatoolkit version cudatoolkit/21.9_11.4 (note: nvcc version for this release is v11.4.1) supports GCC 11.x (see document outlining supported host compilers for each nvcc installation).

You may use the cpe-cuda module available on the system to automatically downgrade the gcc version to gcc/9.3.0 or manually load the gcc/10.3.0 if you wish to use a different cudatoolkit (e.g., cudatoolkit/21.9_11.0 will only support upto GCC 10.x, and you must downgrade the gcc version after loading the cudatoolkit module).

If using the cpe-cuda module, it must be loaded after loading the PrgEnv-gnu:

    module load PrgEnv-gnu

To run a job on Perlmutter GPU nodes, you must submit the job using a project GPU allocation account name, which ends in _g (e.g., m9999_g). An account name without the trailing _g is for charging CPU jobs on Cori and Phase 2 CPU-only nodes.

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# Upper bound of the norm of a matrix difference using an absolutely converging geometric series and Neumann's theorem

I am having trouble proving the following statement:

Let $$\mathbf{A}\in\mathbb{C}^{n\times n}$$ be a square matrix such that $$\|\mathbf{A}\|<1$$, for some induced norm $$\|.\|$$. Then, $$\|(\mathbf{I}-\mathbf{A})^{-1}-\mathbf{I}\|\leq\dfrac{\|\mathbf{A}\|}{1-\|\mathbf{A}\|}$$.

I have been able to prove that $$\|(\mathbf{I}-\mathbf{A})^{-1}\|\leq\dfrac{1}{1-\|\mathbf{A}\|}$$ by using the absolutely converging geometric series $$(1-z)^{-1} = \sum_{n=0}^{\infty}{z^n}$$, having $$|z|<1$$, together with Neumann's theorem $$(\mathbf{I}-\mathbf{A})^{-1} = \sum_{n=0}^{\infty}{\mathbf{A}^n}$$, but I can't prove the above "extension" statement from the inequalities that emerge after using the triangle inequality: $$\|(\mathbf{I}-\mathbf{A})^{-1}-\mathbf{I}\|\leq\|(\mathbf{I}-\mathbf{A})^{-1}\|+\|\mathbf{I}\|$$.

Can anyone please provide me with a hint/help on how to proceed? Thanks in advance!

Note The formulation to be proven appears in Generalizations of the Condition Number by Predrag S. Stanimirovic.

$$(I-A)^{-1}=I+A+A^{2}+\cdots$$ (the series converging in the norm since $$\|A\|<1$$) and so $$(I-A)^{-1}-I=A+A^{2}+\cdots$$. So $$\|(I-A)^{-1}-I\| \leq \|A\|+\|A\|^{2}+\cdots=\frac {\|A\|} {1-\|A\|}$$.

It is :

$$\mathbf{\|(1-A)^{-1}\| \leq \frac{1}{1-\|A\|}}$$

But, also, as you proved by using the triangle inequality :

\begin{align*} \|(\mathbf{1}-\mathbf{A})^{-1}-\mathbf{1}\| &\leq\|(\mathbf{1}-\mathbf{A})^{-1}\|+\|\mathbf{1}\|\\ & \leq \mathbf{\frac{1}{1-\|A\|}} + \|\mathbf{1}\| \leq \mathbf{\frac{\mathbf{\|A\|}}{1-\|A\|}} \end{align*}

Important note : Since we are talking about matrices, try to be careful not to make things tangled with the number $$1 \in \mathbb R$$ and the $$\mathbf{1} \equiv \mathbf{I}$$ matrix which is essentialy the element $$1$$ but in $$C^{n \times n}$$.

• Thanks, but I think the $\mathbf{1}$ in the denominator of the right hand side should be a scalar $1$ since $\|\mathbf{A}\|$ is a nonnegative scalar. – Im YoungMin Dec 2 '18 at 1:00


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# How to keep mages from taking over the world?

In my world anyone can use magic with the proper training. There are two levels of magic users:

• The first level is apprentice (someone with 4 to 8 years of magic education). They can move people and medium size objects with their mind, sense what a person is feeling, and they have limited control over one of the elements (earth, fire, air, water).
• The second level is master (for this you must have between 20 and 30 years of training). They can move large objects such as buildings and large boulders with their minds, read and control the minds of others, control all four elements and summon large storms or earthquakes.

My question is this: given how powerful the mages are, how can the kings and lords of the land prevent them from uniting together and taking over the world?

The tech level is same as that of the early colonization of America.

• What about the bit inbetween Apprentice and Master? there is a 12 year gap of doing what? I assume they would be somewhat powerful as well Jul 1, 2016 at 13:23
• You say anyone can be a mage, so I read the question as "How do I prevent a group of people from starting a revolution?" The answer is you don't really. Jul 1, 2016 at 13:45
• I suggest reading the Discworld series. Specially a book called Sourcery. There are more reasons why wizards don't rule the world than all your teeth, fingers and toes combined, you can extract a lof of them from these books. Jul 1, 2016 at 14:04
• I think this is analogous to "People can own guns, so why haven't people with guns taken over the world." This is answered by "because the people in power also have guns. Jul 1, 2016 at 14:58
• The unasked question here is why would they WANT to take over the world? Ruling the world is a lot of work, and mages presumably have more interesting things to do, while having sufficient power to keep the ostensible rulers from controlling them. You might similarly ask why so few scientists go into politics. Jul 3, 2016 at 5:42

How do you keep people who've taken over the world from becoming mages?

Since anyone can learn magic young members of the nobility would routinely study magic. Being able to read and control the minds of others is a hell of an advantage for a king or any senior noble.

In this world since anyone can learn but only those with the money and time for training can become really powerful whichever way you spin it the nobles are going to make themselves mages even if nobody takes over.

The mages won't revolt because the richest and most powerful people already running the country will have trained as mages already. If you were king in this world you'd be sending your kids to learn magic and taking your own private classes.

Some mages might be more powerful than the king but at that point they're either invested in the system or are facing hundreds of powerful mages in the nobility who are invested in the system.

According to your answer magic isn't powered by anything external, it comes from themselves so you can't strip the manna from around mages.

If you want it to turn out differently you need to change the rules for who can use magic and how in this world.

• So your answer is that the mages won't take over the world, they already did by default. Why would anyone NOT train to be a mage given the opportunity? Jul 1, 2016 at 15:57
• This would also have an interesting effect on succession: if you're in the line of succession and need to be a master mage to rule, then you can't succeed the previous king until you're 35-40 years old (assuming you start training at age 15). If people have a life expectancy after childhood of 50-60 years, then most "kings" would actually spend most of their life as princes-in-waiting, and agnatic seniority would be popular as a way for another highly-trained mage to take over while the king's children finished their long training periods. Jul 1, 2016 at 18:13
• What would really be a good twist based off this plot would be a peasant that has an exceptional ability to learn the arts of magic, and within 8 years of training has already mastered the art of the second level mages. Then later surpasses what any 'master mage' has ever learned and revolts against the rulers! Jul 1, 2016 at 18:35
• @Timmy, and promptly gets squashed: there's only one of him, and hundreds or thousands of nobles.
– Mark
Jul 1, 2016 at 21:07
• ...and, as in real life, anything that threatens the existing power structure would be controlled by the power structure. They'd all learn magic, then pass laws and use existing religion and social structures to prevent the filthy masses from threatening the existing order. Only those with noble blood and members of the clergy may study magic, on penalty of death and eternal damnation. (This literally happened with written language in many parts of the world, not to mention military technologies and the like.) Jul 2, 2016 at 22:28

I feel like there is a flaw in the logic of your premise.

If, as mentioned in your setting, any one can use magic with the proper training what's to say that your kings are not already mages. I would personally find it hard to believe that they are not trained.

In the pre-industrial world education was considerably more limited and focused primarily on the upper classes. Odds are with the amount of training required your nobles/leadership caste are going to be mages more often than the common folks anyway. Its hard to study magic when you are a subsistence farmer trying to survive.

Section 2 To expand on this from our conversation in the comments...

It is unlikely that kings would become full blown master mages due to what appear to be quite stringent training requirements for becoming a mage.

Most leaders would have some degree of magical training and a few would have the opportunity and inclination to become masters (Some dad's just wont die!).

On the flip side few mages are going to have the world experience and general education that future kings are going to receive.

In this case the ridiculous amount of training to become a master may end up being what keeps mages from taking over...they wouldn't even know what to do with their power.

This could lead to a couple take-over attempts by mages that end up going horribly for the kingdoms and the mages that take over. Over time they realize that maybe it just makes more sense to let the kings rule...

• And there beings the conspiracy Jul 1, 2016 at 13:39
• @James you can take eight years just to reach The Apprentice level. To be a full Master takes 20 to 30 years of training. A king would too busy ruling or leading troops into battle or doing other kingly stuff to study Magic. Jul 1, 2016 at 13:39
• Just to point it out, alot of princes learnt sword fighting so if they were attacked they could defende themself or part of some tradition, learning magic to defend yourself or to show off your prowess would also be a valid reason Jul 1, 2016 at 13:46
• @James that's actually a pretty good point. I suppose it would make sense for Kings to have some training on how to use Magic. But I still think that it is unlikely that they will have time to become full masters. Unless of course there father of lives for a very long time. Also remember that a king needs training in other stuff to like politics and the sword fight and Military strategy. This might slow down their magic training some Jul 1, 2016 at 13:50
• @James a good answer. Jul 1, 2016 at 14:08

Try the Pratchett option

Hand any two wizards a piece of rope and they would instinctively pull in opposite directions.

If they could co-operate they'd take over the world in a couple of days, but that would mean sharing power. They're too busy struggling among themselves to actually care what anyone else is doing.

The quote in full (Sourcery):

The reason that wizards didn’t rule the Disc was quite simple. Hand any two wizards a piece of rope and they would instinctively pull in opposite directions. Something about their genetics or their training left them with an attitude towards mutual co-operation that made an old bull elephant with terminal toothache look like a worker ant.

• Please add a reference to the source of the quote (book and chapter). (I tried searching on Google and there are no matches for this exact phrase) Jul 3, 2016 at 2:49
• @yoniLavi, someone has corrected my misremembered quote, it's from Sourcery. Jul 4, 2016 at 8:15

No time for such mundane matters

A profession that takes 20-30 years to master is going to take a significant amount of upkeep to maintain that level of expertise. Almost all of a master's time would be devoted to meditation and continued study of the magic arts. Taking time away from their study to deal with political matters or military conquest would drastically decrease their power.

For a real life example, look at the budō martial arts, such as kyūdō. Although originating in samurai combat training, some schools of kyūdō also have a spiritual aspect. They teach self-discipline and to involve oneself completely to the art of shooting, very reminiscent of Zen philosophy. Applying this to your world, you could have your mages constantly seek perfection in their art. To sully themselves with political affairs or to use their magic for destructive means would defy the foundations of their teachings.

• It is possible to defy the foundations of your teachings. Though this might cause all other non political mages to come after you. Jul 2, 2016 at 17:30

Create a legend to protect the Royalty's claim to the Throne.

In ages past, when great monster's ruled the realm, Magus The Mighty worked a wonderous enchantment, entrapping the great beasts behind a wall of impenetrable force and burying them deep beneath our feet. He saved us from the creatures who would make us food.

But he did more than that! He searched across the land until he found a young boy of noble and righteous countenance. Then he worked the boy's blood into the spell so that the wall would hold, only as long as the boy or his lineage ruled the land. Magus created our First King and set him on the throne which his line must keep, lest the monsters return to our lands. In doing this, Magus saved us again! This time, from the monsters which we spell-casters might become!

• Not going to lie, I can only see Magnus being a douch in this, as humanity evolves and the kings become more aware of the fact if they die so does everyone else that they will abuse it, the origional might have been righteous but time changes everything. Still a great answer +1 Jul 1, 2016 at 13:36
• Did I forget to mention that the legend could be a complete fabrication? Magnus The Mighty might not be a douch... He might be a myth. :^) Jul 1, 2016 at 14:12
• I keep seeing conspiracys in all these answers now, since he may be a myth the nobles are manipulating the lower classes with fake stories Jul 1, 2016 at 14:20
• Absolutely! A conspiracy is just a management strategy in camouflage. What's wrong with camouflage? Kings are expected to have strategies to keep their thrones. That's not only ethical, its common sense. If a lie will work better than a sword, who's to say that the lie is a bad thing. It's keeping the sword from doing the job. Jul 1, 2016 at 14:24
• That hard logic right there made we see alot more good in conspiracys Jul 1, 2016 at 14:26

Maybe you only can activate magic powers if your heart is pure, and your heart is pure only if you are completely free from ambition for worldly power?

Or of course, maybe magic demands "mana" and mana is a scarce resource, requiring that wizards live in limited areas, close to the sources of it?

Or the warlocks know that they would struggle for power against each others, with ruinous results, so they have a pact among them that they will refrain from it?

... or... perhaps the mages already do have the power, but prefer to pretend not, for reasons of better using/enjoying it - and the power of kings, lords, and other laymen, is actually an illusion?

• No, the kings etc. are powerful. But remember, the mages have the ability to control other's minds … how do you tell whether the king's decisions are really his own, and not the ones of mages messing with his mind? For the mages, it's much more comfortable that way, as even if there's a revolution, the revolutionaries will go after the king/nobility, and then eventually there will be a new leadership whose mind can be manipulated. Jul 3, 2016 at 8:45

Do what happened in Great Britain during the Dark Ages Witch hunt.

Incite fear of magic, claim it is evil, corrupt or devil worship and will bring death to you and all you love.

Have a reward for either capturing or killing someone who practice the cursed arts.

Or

If you want a reason for the mages to not try and take over the world, then make it so they don't care for money/power such that a king or lord has, they are after knowledge and improving their own ability.

Mages may not want to share what they have learnt with other mages for fear of it being used against them.

• Spanish Inquisition! Jul 1, 2016 at 21:33
• A Witch Hunt is only successful if the hunted are relatively powerless. It's not going to be successful against powerful mages. Jul 3, 2016 at 10:49
• @Christian Catch them while they are young and inexperienced,for the more powerful ones more tatic is needed but it isnt impossible, we have questions here on how to capture gods after all :D Jul 3, 2016 at 22:21
• @beppe9000 Didn't expect the "Spanish Inquisition". Jul 4, 2016 at 0:21

Kings and Queens are the descendants of warriors and leaders of warriors who conquered the area. In a world with magic, warriors mean mages.

By the time you build up a large civilization, the combat capabilities of an individual pale before the coordinated combat capabilities of a trained military force. Technology and techniques will be developed for the use of magic in mass-warfare, and governments who arrange for effective use will dominate, and those who do not will be conquered.

At colonization of the americas equivalent stage, I'd expect the equivalent of Britian's empire would be fueled by mass magical education, where they crack the technique required for anyone to become a mage, and maybe ritual mass-combat magic, where even relative apprentices can be leveraged for mass magical effects, and industrial magic.

If a civilization figured that out, while others did not, it could experience something like the UK's industrial revolution (where a small island dominated the world). Others would start to copy it (but, mass education requires mass educators, and the social structure of a state where almost everyone has magic is going to be different from the typical magocracy) so this is going to take decades to catch up).

Meanwhile, at the top, the monarch is likely to have some magical education. But they need not be a top-tier combat or industrial mage: their power comes from the huge mass of the state, not from their individual magic. They are likely to have combat mage bodyguards, much like the monarch is going to have elite guards.

Rulers of monarchies in the age of steam where not expert engineers or top-notch warriors. They usually had some military experience and a good education, but that was to season them: their power did not flow from their personal prowess. The admirals and warriors and engineers served the crown, they did not rule it.

So I'd build the world where most of the world magical education is limited to the nobility, and are secrets passed down in families. The rising power has created a mass magical education system, where even peasants are taught narrow types of magic be more productive, and every soldier (often drafted from the peasants!) is a combat mage capable of helping in mass combat ritual magic and individual skirmishes. All backed by industrial-scale mass production of enchanted goods (even if they are just "modern" strength alloys).

You've only described the strengths of the mages. If you want to change the balance of the game, you should give them some weaknesses also. Here are some ideas for weaknesses:

• Channeling time: The most powerful spells could take a long time to cast, (e.g. days or weeks) leaving the mage defenseless while they channel.
• Cooldown time: After casting a powerful spell, the mage would probably be weakened and unable to do much for several weeks. Study what it's like for someone to recover from surgery.
• Mana cost: As someone else mentioned, you can introduce the concept of mana as a scarce resource which might be hard to find, or maybe it's easy to find but it takes a long time for you to get enough of it.
• Maybe the telepathy backfires on the mages, and normal people can sense the presence of their telepathic mind, making it impossible for a mage to hide from an angry crowd with pitchforks. Mages are persecuted and have to live far from the non-mages, in their own small groups.
• +1. Expanding beyond mana: magic might require reagents. Perhaps common reagents at the basic levels, but increasingly more rare and expensive reagents at the highest levels. Control the reagent, control the magic. Also, how about anti-magic and magic resistance? Perhaps certain materials in this world exert a large sphere of anti-magic around them not allowing any magical effect to take place within that radius. The king's castle, crown, carriage, etc, are all made of this material. Jul 2, 2016 at 21:34

It would seem to me that your description of magic is a little simplistic in the question from this standpoint. It seems almost as if you're saying that strength is based only on time put in, sort of like the way unions might promote a member from apprentice to journeyman with minimal accepted skill level + minimum number of years of experience. But just like there are generalists and experts in carpentry and plumbing, there would be generalists and experts in magic. One could be a general master mage which would mean, they have the experience to safely do advanced level spells, but maybe not all of them based on their discipline of study. Requirement for Mastery of magic might mean 20 years, plus the ability to cast safely and sufficiently strong a set of master level spells in each of fire, water, air, and earth diciplines. However, there may be more than 5 master spells each. In fact the master generalist can safely explore the master spells, but might not bother mastering all the master level fire spells because his interest lies elsewhere.

It could also be that there are hidden/rare/ultimate spells that are kept from the general public and invested in royalty/ruling class that are only know to those of highest government wizard rank. Not having access to, or in fact, knowledge that certain spells exist could be one way of controlling them. Another might be wealth and/or the existence of mana/runestones/wands/objects of power that might be licensed or sold by ruling class. If someone were to say requisition a huge amount of mana potions(or whatever) in a short period of time, it might raise red flags to watch said individuals.

Then there is just human nature. Those that strive for power, inevitably don't want to share it. It would be difficult to gather together a group of power hungry wanna-be-take-over-the-world mages together without them starting to squabble, duel or otherwise fight amongst themselves. Then there could be some magic or artifact that the rulers have that could negate magic(perhaps the castles are all erected on anti-magic fields) or have null magic shields that prevent any spells from entering. This is sort of like the steadings in the Wheel of Time series of books. Magic just doesn't work there. Then it also could be that magicians are by their very nature sort of like the stereotypical "mad scientist" who tend to be more interested in the "science" or "magic" itself rather than application or consequences of said experiments. Finally, it could just be a matter of you got your mages and the government has theirs. They might try to take it down and the government has some teeth to prevent it(not always working, after all just like governments take over each other in real life).

Brian

• Also, it's your story, not mine, however, I would tend to think that a more RPG like descriptions of "level" might be easier to classify mages than just two. There has to be a difference between casting evaporate on a glass of water, doing it to a lake, an ocean, the planet, removing all water from the atmosphere. Of course, there could be this, but you define a master someone who can achieve level 14 spells from the test booklet(or whatever). Still it seems more than two might be useful to you. Jul 1, 2016 at 21:45
• Another thought. It may be that the knowledge needed to do magic is a bit at odds with the knowledge that is needed to manage a kingdom/empire. Just because you can take over a place, doesn't mean you can manage(rule) worth a darn. It's could be akin to the difference between a general and a master swordsman. The swordsman could kill a general perhaps with ease, but may have no ability to lead men, plan a strategy, or feed an army(logistics). Sure men might follow a winning champion, but eventually they will be overcome by strategy. Jul 1, 2016 at 21:51

given how powerful the mages are how can the kings and lords of the land prevent them from uniting together and taking over the world?

Who's to say they haven't already?

Given the kind of society you're describing, I would expect that there would be powerful mages lurking behind the seats of power, pulling the strings. The kings and lords may think they have power, but the real power lies with their trusted "advisors" who manipulate every major decision toward their long term goals.

To prevent mages from taking over, you need to create real obstacles to becoming a mage.

Say for apprentice level, you need to spend a year in a cave, learning how to channel the magical energy. And if you fail you may die.

For master level, you need to spend 10 years in such cave. While other mages (employed by other powers) might try to detect and interfere.

To be a king, to rule, you have to be able to inspire the people. It's very unlikely that a Mage would have the charisma or be a people person given the level of solitary study required. Rather like Steve jobs, techno wiz but no ambition to be a politician. While trump couldn't change a fuse but can inspire people to vote for him. Of course, a Mage may rule the world through a puppet King... Mwahaha :)

• Good point: mages are just like engineers in our universe. Jul 2, 2016 at 3:47
• A person with mind control magic doesn't need natural charisma. Jul 3, 2016 at 10:51

As Ronan and some others said, the Discworld offers some ideas here.

The Discworld MUD has gone through this too and they balanced off magic use quite nicely I think.

One reason that wasn't mentioned is that magic on the disc is

(1) dangerous as it is unreliable and has a mind of it's own which can easily backfire and kill the caster. More powerful magic = higher risk (2) is a little inconvenient to use as it often requires components and ingredients you have to carry around on you.

There have been a lot of good answers, and a number retreading the same ground. I'd just like to add one more.

Consider adding a Bane. Something antithetical to magic.

Like the classic, Cold Iron. Iron forged without getting above a certain temperature (say, it can be heated but not to the point of actual visible glow). Cold Iron sinks and dispels magic with 100% efficiency. So a Master Mage can crush a castle and burn an army with a wave of his hand, but a single warrior wearing cold iron plate and carrying a cold iron sword is immune, possibly pulling and grounding out spells like a lightning rod, protecting those around them as well.

A king or queen keeps at least a small elite guard equipped with the magic bane, a royal crown is lined with small amounts to protect the lords and ladies minds from being controlled, and so on.

Sheers numbers.

Wizardry takes a lot of time in your world to be truly powerful. COnsider this similar to a PhD in our world. Only a small percentage of the world population has a PhD, so it would follow only a small percentage of your worlds population would be master level wizards.

Sure, a master level wizard could kill 1000 men. But could he kill 10,000? 20,000? Eventually, a master wizard could be taken down by enough men if they stepped out of line.

You should watch the X-men movies through. The mutants are essentially mages, and indeed, a group of them, led by Magneto, basically keeps trying to take over the world.

Humans don't stand much of a chance against them, there's only another group of good mutants that can successfully do that - - the X-men. The X-men believe in peace and coexistence with humans. Many humans, however, do not understand the mutants thus fearing them and thus acting hostile towards them. For Magneto this basically justifies his war.

So in your world, why would the mages take over? Have they been given any reason to do such a thing? It doesn't sound like that, I gather that they are actually valued members of the community. And like someone else already mentioned, ruling is hard work, many of the mages can certainly think of something more interesting to do.

There's also that quote that says that often those who crave for power the most are the ones who are the least capable of handling it. Take some power hungry mage, and you can be certain there are other mages who would much rather not see them reign.

So overall I suppose I can summarize by saying that there are luckily more good people than bad people, within mages too.

Mages are like feudal knights. They are experts at what they do and are devoted to their lord/king/queen. This arrangement is mutually beneficial - mages are given the time and resources to devote themselves to their craft, and the rulers gain powerful allies.

The rulers themselves are rarely mages, because the dedication required conflicts with the responsibilities of ruling. However, it is common for the non-inheriting children of important families to study magic, and they become important contributors to the family's strength.

Sufficiently intelligent and studious people of lower rank can rise in status by becoming mages - the usual route for this is to seek patronage from an important family, and then repay them with service. Self-taught prodigies are extremely rare. Mages who betray their patron are seen as treacherous and ungrateful. Eventually, a "commoner" mage can gain their own lands and start their own noble house, so they are rarely interested in bringing the whole feudal system toppling down.

When a mage exercises power they are not generating the energy involved, they are only bending and directing what is already around them. Two mages cannot manipulate the same energy at the same time. Similar to destructive interference.

The more powerful or skilled or experienced a mage, is the wider an area they will be able to effect in their manipulations. Consequently, they do not want another fully trained mage within their territory.

The result is that they don't work together, or even gather together.

Getting trained as an apprentice (the only stage the requires formal education, the remainder is up to self study) requires a teacher willing to set aside time where they relax their grip on the ether while the students practice. Even so, dedicated students will find that they need to travel into the wilder areas to gain the solitude necessary to gain skill. This is part of the reason it takes so long. Additionally, a teaching mage will require some exorbitant fees in order to mitigate the loss of time spent within the ecstasy of total mastery of their demesne.

You specifically note magic education. As it doesn't seem that you have any issues with the nobility being magically gifted as well, perhaps magic instructors are prevented from teaching the common folk or are so rare that the nobility snatches them up as tutors before any old dirt farmer could learn powers that make rebellion a bit more possible.

Perhaps the common folk of your world can get somewhere with self study, but you could use this to limit their magic to things that wouldn't assist a rebellion. For example, farmers use their knowledge of water magic to irrigate their farms completely.

In addition, perhaps the commoners won't rebel if they don't think they can. Have the average citizen believe magic is only for basic support and making lives easier. Perhaps the kingdom has instituted public education to teach just that, at schools or churches. Thus the common folk live believing magic is a convenient tool, and the nobility can rest easy.

If you look at the world we live in, the world is not taken over by those with the most physical prowess, not the smartest (arguably), and not those with the most refined social skills who hold the power.

Power resides where people think it resides. It's a trick

Power is largely held by frail old men with weired quirks and poor behaviour. It's more a matter of intent and being in tune with the intent of the group mind.

The mages in your would would probably - most of them - be instruments of someone else actually making plans and holding power.

Why isn't the most powerful person king of everything?

The same reason why the best fighter in this universe isn't the king of everything.

Being a fighter is hard work. You commit your life to it.

Being a king is also hard work. You commit your life to it. If you need to fight, you get a whole group of top fighters to fight for you. You can't be a good fighter yourself, you're too busy running a kingdom.

The question is moot, because it cannot happen. There is a symbiotic relationship between magic and goodness. Look at the grand magic that Jesus pulled off, water into wine, parting of the seas, etc.. But the best example is of course God. The creator of life itself, there is no greater magic trick than something from nothing.All that was created in goodness.

The goodness is so entwined with the magic that one cannot exist without the other. A simple example of reading another's mind is impossible without the honesty needed to communicate on that level. If the motive was to take over the world, then that would poison the attempted mind communication to an extent that the mind reading [earthquake, bending spoon, interstellar travel etc..] simply would not take place as the energy is bad, like trying to run an electric car on sand, it won't work.

This was the fail-safe code written into magic by God. It must be good or it cannot happen.

If you want to be a grand master of magic, first you must be a grand master of goodness. If you are, you will not want to take over the world.

• to be fair in the the bible some bad people were able to do a some supernatural feets, however they weren't very affect against someone with the Creator on his side, the Witch of endor or the Egyptian prest for example. Jul 4, 2016 at 10:26
• 2 Kings 21:6, for starters, or Acts 8: 9-13. There are lots of people and demons using magic for 'bad' purposes in the bible. It clearly can happen, unless the Bible is a falsification of history...? Jul 13, 2016 at 18:43

A simple way of preventing mages from just overthrowing everything is to make them vulnerable. I propose adding some such problems, they work on their own but in combination they are lethal to mages who get rebellious ideas.

In order of importance

• Make it really obvious to anyone when someone uses magic. I would make it so that even untrained people would sense when a mage cast a spell (say within twice the casting range) and allow other mages to trivially identify exactly which spell and what the targets and other parameters are within 10 times the casting distance.
• Ensure that there is no magic that easily protects from a well placed arrow
• Dissalow affecting things you can not see or in some other way sense
• Disallow or restrict magic enhancing the senses, so that a mage can not sense things more easily (adding senses like read thoughts is ok, but not improving existing ones)

This combination of restrictions makes mages vulnerable to anyone with a bow and better eyesight than them. I would envision that this makes mages valued members of any despotic system of governance while still allowing for non magic users to be at the top, a king can just order them shot on sight in the capital if he is paranoid.

1. Casting time: how long the spell takes to cast
2. Mana: how often you can cast spells
3. "Miscasting": Your head explodes because you swapped a syllable or got distracted.
4. Eldritch Beings: scary creatures that are drawn to magical power
5. Weapons designed to kill magic users, made by other magic users, ironically
6. Range: anything less than the range of a bow and arrow and you have a problem
7. Rarity of Books: books are rare and expensive
8. Misinformation: Some/Many/Most books on magic have "traps" written in...
9. Shady Guilds: anyone whose not a member has a bounty placed on their head.
10. Etc.


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# Mathematica: What is happening?

• Mathematica

## Main Question or Discussion Point

Ok, I have either completely lost it (very likely) or Mathematica is being weird (much less likely). The finction D[f,x] is supposed to return the partial of f with respect to x, $\partial{f}/\partial{x}$.

So I have a function $V_\theta = (A/r^2 + B)\sin\theta$ where A and B are constants. So we should have $\partial{V_\theta}/\partial{\theta} = (A/r^2 + B)\cos\theta$.

But here is what mathematica is returning. What does that even mean? Why does it just say "sin" and not have the argument? Moreover, why does it not say cos ?!

#### Attachments

• 3.4 KB Views: 400


text:
Goto Chapter: Top 1 2 3 4 5 Bib Ind

### 3 Nilpotent Orbits

This chapter contains functions for dealing with the nilpotent orbits of a semisimple Lie algebra K under its adjoint group G. We refer to the book by Collingwood and McGovern, [CM93] (and the references therein) for an account of the theory of nilpotent orbits. A nilpotent orbit has two important attributes: the weighted Dynkin diagram, and an sl_2-triple. The weighted Dynkin diagram is represented by a list of integers in {0,1,2} of length equal to the rank of the Lie algebra. The i-th position in this list correponds to the i-th node of the Dynkin diagram of the root system. The Dynkin diagram of the root system is described by the Cartan matrix of the root system. Now in GAP this Cartan matrix can be somewhat different from the more usual forms. This holds most particularly for type F4, where the enumeration of the simple roots is rather different from the one usually found. So when using the functions in this chapter one should keep this in mind.

Every nilpotent orbit has an sl_2-triple, that is a triple (y,h,x) of elements of the simple Lie algebra with [x,y]=h, [h,x]=2x, [h,y]=-2y. The nilpotent orbit corresponding to this is the orbit of the element x under the action of the adjoint group.

Let P be a parabolic subalgebra of K (i.e., generated by the Cartan subalgebra of K, all positive root vectors, along with the negative simple root vectors corresponding to a given subset of the basis of simple roots), L the corresponding Levi subalgebra (i.e., the reductive part of P), and N the nilradical of P. Let O_L be a nilpotent orbit in L. There exists a unique nilpotent orbit O_K in K such that the intersection of O_K and O_L + N is dense in the latter. In this situation O_K is said to be induced from O_L. Nilpotent orbits in K which are not induced are said to be rigid.

Now consider the variety of all G-orbits in K of a given dimension d. The irreducible components of this variety are called the sheets of K. Every sheet has a unique nilpotent orbit. Moreover this nilpotent orbit is induced from an orbit O_L, and O_L is rigid in L. So the sheets are parametrised by pairs (L,O_L), where L is a Levi subalgebra, and O_L a rigid nilpotent orbit in it. This data can conveniently be given by a sheet diagram: this is the Dynkin diagram of K, were the nodes that do not correspond to simple roots of L have label 2. So, leaving out the nodes with label 2, one obtains the Dynkin diagram of L. The remaining labels in the sheet diagram then correspond to the weighted Dynkin diagram of the nilpotent orbit O_L. Since this orbit is rigid, its weighted Dynki diagram has labels 0 or 1. From that it follows that one can recover L and O_L from the sheet diagram. The rank of a sheet is defined as the dimension of the centre of L, obviously that is equal to the number of 2's in the sheet diagram.

#### 3.1 The functions

##### 3.1-1 NilpotentOrbit
 ‣ NilpotentOrbit( L, wd ) ( operation )

Here L is a simple Lie algebra and wd a weighted Dynkin diagram (i.e., a list containing the weights of the weighted Dynkin diagram, in the same order as the nodes of the Dynkin diagram of the root system of L; that order can be deduced from the Cartan matrix of the same root system). The corresponding nilpotent orbit is returned. It is the responsibility of the user to make sure that the weighted Dynkin diagram corresponds to a nilpotent orbit.

gap> L:= SimpleLieAlgebra("E",6,Rationals);;
gap> o:= NilpotentOrbit( L, [1,2,0,0,0,1] );
<nilpotent orbit in Lie algebra of type E6>

##### 3.1-2 NilpotentOrbits
 ‣ NilpotentOrbits( L ) ( attribute )

Here L is a semisimple Lie algebra. This function returns the list of all nilpotent orbits of L. If L is simple of classical type, then the nilpotent orbits correpond to partitions (of n+1 for type A_n, of 2n+1 for type B_n, of 2n for type C_n and of 2n for type D_n, see [CM93]). If L is of one of these types then the orbits returned by this function have the attribute OrbitPartition set, which returns the corresponding partition.

gap> L:= SimpleLieAlgebra("E",6,Rationals);;
gap> orbs:= NilpotentOrbits(L);;
gap> orbs[10];
<nilpotent orbit in Lie algebra of type E6>
gap> Length(orbs);
20
gap> L:= SimpleLieAlgebra("B",4,Rationals);;
gap> orbs:= NilpotentOrbits(L);;
gap> OrbitPartition( orbs[10] );
[ 5, 2, 2 ]

##### 3.1-3 WeightedDynkinDiagram
 ‣ WeightedDynkinDiagram( o ) ( attribute )

Here o is a nilpotent orbit; this function returns its weighted Dynkin diagram.

##### 3.1-4 WeightedDynkinDiagram
 ‣ WeightedDynkinDiagram( L, x ) ( method )

Here L is a semisimple Lie algebra, and x a nilpotent element. This function returns the weighted Dynkin diagram of the orbit containing x.

gap> L:= SimpleLieAlgebra("B",3,Rationals);;
gap> WeightedDynkinDiagram( L, L.1+L.9 );
[ 2, 0, 0 ]
gap> L:= SimpleLieAlgebra("E",6,Rationals );;
gap> WeightedDynkinDiagram(L, L.1+L.6+L.20+2*L.32 : table:= true );
[ 0, 0, 0, 1, 0, 0 ]
gap> time;
2048
gap> WeightedDynkinDiagram(L, L.1+L.6+L.20+2*L.32 : table:= true );
[ 0, 0, 0, 1, 0, 0 ]
gap> time;
64

##### 3.1-5 AmbientLieAlgebra
 ‣ AmbientLieAlgebra( o ) ( attribute )

Here o is a nilpotent orbit; this function returns the Lie algebra it lives in.

##### 3.1-6 SemiSimpleType
 ‣ SemiSimpleType( o ) ( attribute )

Here o is a nilpotent orbit; this function returns the type of the Lie algebra it lives in.

##### 3.1-7 SL2Triple
 ‣ SL2Triple( o ) ( attribute )

Here o is a nilpotent orbit; this function returns an sl_2-triple (y,h,x) corresponding to o. For the exceptional types the x is as in the paper [Gra08]. For the classical types the x is computed on the fly.

gap> L:= SimpleLieAlgebra("E",6,Rationals);;
gap> orbs:= NilpotentOrbits(L);;
gap> SL2Triple( orbs[10] );
[ (4)*v.51+(3)*v.53+(3)*v.56+v.59, (4)*v.73+(6)*v.74+(8)*v.75+(11)*v.76+(
8)*v.77+(4)*v.78, v.15+v.17+v.20+v.23 ]

##### 3.1-8 RandomSL2Triple
 ‣ RandomSL2Triple( o ) ( operation )

Here o is a nilpotent orbit; this function returns a random sl_2-triple (x,h,y) corresponding to o. This means that every call (potentially) returns a different sl_2-triple.

gap> L:= SimpleLieAlgebra("E",6,Rationals);;
gap> orbs:= NilpotentOrbits(L);;
gap> RandomSL2Triple( orbs[10] );
[ (3)*v.49+(3)*v.50+v.51+(4)*v.59, (4)*v.73+(6)*v.74+(8)*v.75+(11)*v.76+(
8)*v.77+(4)*v.78, v.13+v.14+v.15+v.23 ]
gap> RandomSL2Triple( orbs[10] );
[ (3)*v.50+(3)*v.53+v.54+(4)*v.57, (4)*v.73+(6)*v.74+(8)*v.75+(11)*v.76+(
8)*v.77+(4)*v.78, v.14+v.17+v.18+v.21 ]

 ‣ SL2Grading( L, h ) ( operation )

Here L is a Lie algebra, and h is an element of it, such that there is an sl_2 triple of which it is the Cartan element (the system does not check that). This function returns the grading of L in eigenspaces of h. A list containing three lists is returned: the first list contains bases of the components with degrees 1,2,3,... the second list has bases of the components with degrees -1,-2,-3,..., the last list contains a basis of the zero component.

gap> L:= SimpleLieAlgebra("F",4,Rationals);;
gap> orbs:= NilpotentOrbits(L);;
gap> sl2:= RandomSL2Triple( orbs[6] );
[ v.36+(2)*v.40+(2)*v.42, (3)*v.49+(4)*v.50+(6)*v.51+(8)*v.52, v.12+v.16+v.18
]
[ [ [ v.3, v.5, v.7, v.8, v.9, v.11 ],
[ v.10, v.12, v.13, v.14, v.15, v.16, v.17, v.18, v.20 ],
[ v.19, v.21 ], [ v.22, v.23, v.24 ] ],
[ [ v.27, v.29, v.31, v.32, v.33, v.35 ],
[ v.34, v.36, v.37, v.38, v.39, v.40, v.41, v.42, v.44 ],
[ v.43, v.45 ], [ v.46, v.47, v.48 ] ],
[ v.1, v.2, v.4, v.6, v.25, v.26, v.28, v.30, v.49, v.50, v.51, v.52 ] ]

##### 3.1-10 SL2Triple
 ‣ SL2Triple( L, x ) ( operation )

Here L is a simple Lie algebra, and x is a nilpotent element of it. A list of three elements is returned, forming an sl_2-triple, the last of which is equal to x.

gap> L:= SimpleLieAlgebra("F",4,Rationals);;
gap> SL2Triple( L, L.1+L.20 );
[ v.16+v.25, v.49, v.1+v.20 ]

##### 3.1-11 InducedNilpotentOrbits
 ‣ InducedNilpotentOrbits( L ) ( attribute )

Here L is a simple Lie algebra. This function returns the list of all induced nilpotent orbits of L. An induced orbit is given by a record containing two fields: sheetdiag, which is a diagram describing the Levi subalgebra and the rigid nilpotent orbit in it from which the nilpotent orbit is induced, and norbit, which is the induced nilpotent orbit in L. The sheet diagram is a labeled Dynkin diagram, and the labels are 0, 1 or 2. If we take the Dynkin diagram and erase the nodes which have label 2 then we obtain the Dynkin diagram of the Levi subalgebra. Moreover, the labels 0 and 1 on that diagram give the rigid nilpotent orbit in the Levi subalgebra. From this pair the nilpotent orbit norbit is induced. It may happen that the same nilpotent orbit is induced from more pairs consisting of a Levi subalgebra and a rigid nilpotent orbit in it. In that case the same nilpotent orbit appears more than once in the list, each time with a different sheet diagram attached. This function works for the Lie algebras of exceptional type and for the Lie algebras of type A regardless of the rank. It works for the Lie algebras of the other types up to rank 10.

gap> L:= SimpleLieAlgebra("E",6,Rationals);;
gap> s:= InducedNilpotentOrbits(L);;
gap> s[19];
rec( norbit := <nilpotent orbit in Lie algebra of type E6>,
sheetdiag := [ 2, 0, 0, 1, 0, 2 ] )
gap> WeightedDynkinDiagram( s[19].norbit );
[ 0, 0, 0, 2, 0, 0 ]

##### 3.1-12 RigidNilpotentOrbits
 ‣ RigidNilpotentOrbits( L ) ( attribute )

Here L is a simple Lie algebra. This function returns the list of all rigid nilpotent orbits of L, except the zero orbit (which is always rigid).

gap> L:= SimpleLieAlgebra("E",6,Rationals);;
gap> RigidNilpotentOrbits(L);
[ <nilpotent orbit in Lie algebra of type E6>,
<nilpotent orbit in Lie algebra of type E6>,
<nilpotent orbit in Lie algebra of type E6> ]
gap> List( last, WeightedDynkinDiagram );
[ [ 0, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0 ], [ 1, 0, 0, 1, 0, 1 ] ]

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Brian, you are such an inspiration. I wonder how do you get all these hacks and then publish them for all of us. I have been reading your stuff from quite a time now, but I have a problem. Every time I read something you post I feel overwhelmed but I haven’t been really able to generate any fruitful results on any of my sites. I just don’t know where to start. Imagine I don’t even have an email list.
SEO often involves the concerted effort of multiple departments within an organization, including the design, marketing, and content production teams. While some SEO work entails business analysis (e.g., comparing one’s content with competitors’), a sizeable part depends on the ranking algorithms of various search engines, which may change with time. Nevertheless, a rule of thumb is that websites and webpages with higher-quality content, more external referral links, and more user engagement will rank higher on an SERP.
##### Danny Sullivan was a journalist and analyst who covered the digital and search marketing space from 1996 through 2017. He was also a cofounder of Third Door Media, which publishes Search Engine Land, Marketing Land, MarTech Today and produces the SMX: Search Marketing Expo and MarTech events. He retired from journalism and Third Door Media in June 2017. You can learn more about him on his personal site & blog He can also be found on Facebook and Twitter.

Digital marketing became more sophisticated in the 2000s and the 2010s, when[13][14] the proliferation of devices' capable of accessing digital media led to sudden growth.[15] Statistics produced in 2012 and 2013 showed that digital marketing was still growing.[16][17] With the development of social media in the 2000s, such as LinkedIn, Facebook, YouTube and Twitter, consumers became highly dependent on digital electronics in daily lives. Therefore, they expected a seamless user experience across different channels for searching product's information. The change of customer behavior improved the diversification of marketing technology.[18]
Backlink is a link one website gets from another website. Backlinks make a huge impact on a website’s prominence in search engine results. This is why they are considered very useful for improving a website’s SEO ranking. Search engines calculate rankings using multiple factors to display search results. No one knows for sure how much weight search engines give to backlinks when listing results, however what we do know for certain is that they are very important.

I’ve never been particularly enamoured with nofollow, mainly because it breaks the “do it for humans” rule in a way that other robots standards do not. With other standards (e.g. robots.txt, robots meta tag), the emphasis has been on crawling and indexing; not ranking. And those other standards also strike a balance between what’s good for the publisher and what’s good for the search engine; whereas with nofollow, the effort has been placed on the publisher with most of the benefit enjoyed by the search engine.

PageRank is one of many, many factors used to produce search rankings. Highlighting PageRank in search results doesn’t help the searcher. That’s because Google uses another system to show the most important pages for a particular search you do. It lists them in order of importance for what you searched on. Adding PageRank scores to search results would just confuse people. They’d wonder why pages with lower scores were outranking higher scored pages.

### Denver CO Search Engine Optimization

In regards to link sculpting I think the pro’s of having the “no follow” attribute outweigh the few who might use it to link sculpt. Those crafty enough to link sculpt don’t actually need this attribute but it does make life easier and is a benefit. Without this attribute I would simply change the hierarchy of the internal linking structure of my site and yield the same results I would if the “no follow” attribute didn’t exist.
At the time I was strongly advocating page rank sculting by inclusion of no follow links on “related product” links. It’s interesting to note that my proposed technique would have perhaps worked for a little while then would have lost its effectiveness. Eventualy I reached the point where my efforts delivered diminishing returns which was perhaps unavoidable.

Thanks for the info on nofollow and pagerank. It makes sense that this will always be a moving target less everyone will eventually game the system until it’s worthless but at the same time it’s worth it to know a few tricks. I still have open concerns on how freshness of content factor in, the only time i’m ever annoyed by search results these days is when the only links available (on the first page at least) are articles from 4 years ago.

#### The total number of backlinks can often include many links from the same referring domain or multiple referring domains. It’s common for referring domains to link back to your content if it is relevant, authoritative or useful in some way to their own domain. In an ideal world, that’s how backlinks are accumulated; unique content that other websites want to be associated with.

Bad Idea, many, many websites nowadays use nofollow in every single external link to sustain the value of their PageRank, not only to prevent comments spam, which is in my opinion defeating the original purpose of nofollow. Google should’ve “marketed” nofollow as nospam without going into much details of how that links to PageRank to keep webmasters away from getting into the idea that nofollow means better PageRank
PageRank was developed by Google founders Larry Page and Sergey Brin at Stanford. In fact the name. PageRank is a likely play on Larry Page's name. At the time that Page and Brin met, early search engines typically linked to pages that had the highest keyword density, which meant people could game the system by repeating the same phrase over and over to attract higher search page results. Sometimes web designers would even put hidden text on pages to repeat phrases.

Webmasters and content providers began optimizing websites for search engines in the mid-1990s, as the first search engines were cataloging the early Web. Initially, all webmasters needed only to submit the address of a page, or URL, to the various engines which would send a "spider" to "crawl" that page, extract links to other pages from it, and return information found on the page to be indexed.[5] The process involves a search engine spider downloading a page and storing it on the search engine's own server. A second program, known as an indexer, extracts information about the page, such as the words it contains, where they are located, and any weight for specific words, as well as all links the page contains. All of this information is then placed into a scheduler for crawling at a later date.

Search engine optimization is a key part of online marketing because search is one of the primary ways that users navigate the web. In 2014, over 2.5 trillion searches were conducted worldwide across search engines such as Google, Bing, Yahoo, Baidu, and Yandex. For most websites, traffic that comes from search engines (known as "natural" or "organic" traffic) accounts for a large portion of their total traffic.
Ask for a technical and search audit for your site to learn what they think needs to be done, why, and what the expected outcome should be. You'll probably have to pay for this. You will probably have to give them read-only access to your site on Search Console. (At this stage, don't grant them write access.) Your prospective SEO should be able to give you realistic estimates of improvement, and an estimate of the work involved. If they guarantee you that their changes will give you first place in search results, find someone else.
Make sure your backlinks appear to be natural. Don’t ask webmasters to link back to your pages with a specific anchor text since this can haphazardly result in a pattern that may get noticed by search engines and cause you to get a linking penalty, a la Penguin. Also, don’t do anything shady or unnatural to create backlinks, like asking a site to put a link in the footer of every page on their site.
NOTE: You may be curious what your site’s or your competitor’s PR score is. But Google no longer reveals the PageRank score for websites. It used to display at the top of web browsers right in the Google Toolbar, but no more. And PR data is no longer available to developers through APIs, either. Even though it’s now hidden from public view, however, PageRank remains an important ingredient in Google’s secret ranking algorithms.
I am not worried by this; I do agree with Danny Sullivan (Great comment Danny, best comment I have read in a long time). I will not be changing much on my site re: linking but it is interesting too see that Google took over a year to tell us regarding the change, but was really happy to tell us about rel=”nofollow” in the first place and advised us all to use it.
For most parts the sophistication in this system is simplified here. I still have trouble understanding the difference between letting link flow withing my pages without thinking about a loop. For example, page A, B and C link to each other from all angles therefore the link points should be shared. But in this loop formula, page B does not link to A. It just goes to C and loops. How does this affect navigation bars? As you know they are meant to link stay on top and link to all pages. I’m lost.
There are simple and fast random walk-based distributed algorithms for computing PageRank of nodes in a network.[33] They present a simple algorithm that takes {\displaystyle O(\log n/\epsilon )} rounds with high probability on any graph (directed or undirected), where n is the network size and {\displaystyle \epsilon } is the reset probability ( {\displaystyle 1-\epsilon } is also called as damping factor) used in the PageRank computation. They also present a faster algorithm that takes {\displaystyle O({\sqrt {\log n}}/\epsilon )} rounds in undirected graphs. Both of the above algorithms are scalable, as each node processes and sends only small (polylogarithmic in n, the network size) number of bits per round.
I work on a site that allows users to find what they are looking for by clicking links that take them deeper and deeper into the site hierarchy. Content can be categorised in lots of different ways. After about three steps the difference between the results pages shown is of significance to a user but not to a search engine. I was about to add nofollow to links that took the browser deeper than 3 levels but after this announcement I won’t be…
It’s hard to believe that the Internet is now multiple decades old. Affiliate marketing has been around since the earliest days of online marketing. It’s a great solution for businesses that are risk-averse or don’t have the budget to spend on upfront marketing costs. Use affiliate marketing to build a new revenue stream for your ecommerce or B2B business.

## The third and final stage requires the firm to set a budget and management systems; these must be measurable touchpoints, such as audience reached across all digital platforms. Furthermore, marketers must ensure the budget and management systems are integrating the paid, owned and earned media of the company.[67] The Action and final stage of planning also requires the company to set in place measurable content creation e.g. oral, visual or written online media.[68]

PageRank gets its name from Google cofounder Larry Page. You can read the original ranking system to calculate PageRank here, if you want. Check out the original paper about how Google worked here, while you’re at it. But for dissecting how Google works today, these documents from 1998 and 2000 won’t help you much. Still, they’ve been pored over, analyzed and unfortunately sometimes spouted as the gospel of how Google operates now.

# Our digital agency offers both traditional targeted online display advertising as well as behavioral retargeting. Through an intense discovery process, our team will determine the most optimal marketing mix for your online media plan. We will leverage ad network partnerships for planning the ideal media buys and negotiating the best possible pricing.

In order to be a data driven agency, we foster a culture of inspired marketing entrepreneurs that collaborate, innovate, and are constantly pushing the threshold of marketing intelligence. Our analytics team is well versed in mathematics, business analytics, multi-channel attribution modeling, creating custom analytics reporting dashboards, and performing detailed analysis and reporting for each client.
There are things like solo ads that you can buy or join up with JV partners to help sell whatever it is that you're peddling. But at the outset, you're going to find it difficult to succeed in the internet marketing field. Ensure that you learn the right skills along the way so that you can become an effective online marketer. It doesn't matter how slow you go, as long as you don't stop, you'll reach your goals.
Do you regularly publish helpful, useful articles, videos or other types of media that are popular and well produced? Do you write for actual human beings rather than the search engine itself? Well, you should. Latest research from Searchmetrics on ranking factors indicates that Google is moving further towards longer-form content that understands a visitor’s intention as a whole, instead of using keywords based on popular search queries to create content.

#### Since heading tags typically make text contained in them larger than normal text on the page, this is a visual cue to users that this text is important and could help them understand something about the type of content underneath the heading text. Multiple heading sizes used in order create a hierarchical structure for your content, making it easier for users to navigate through your document.

How does this all relate to disallows in the robots.txt? My ecom site has 12,661 pages disallowed because we got nailed for duplicate content. We sale batteries so revisons to each battery where coming up as duplicate content. Is PageRank being sent (and ignored) to these internal disallowed links as well? One of our category levels has hundreds of links to different series found under models, the majority of these series are disallowed. If PageRank acts the same with disallows as it does with nofollows, are these disallowed links are hurting our
Backlinks take place across the Internet when one website mentions another website and links to it. Also, referred to as “incoming links,” backlinks make their connection through external websites. These links from outside domains point to pages on your own domain. Whenever backlinks occur, it is like receiving a vote for a webpage. The more votes you get from the authoritative sites creates a positive effect on a site’s ranking and search visibility.

#### In order to be a data driven agency, we foster a culture of inspired marketing entrepreneurs that collaborate, innovate, and are constantly pushing the threshold of marketing intelligence. Our analytics team is well versed in mathematics, business analytics, multi-channel attribution modeling, creating custom analytics reporting dashboards, and performing detailed analysis and reporting for each client.

[43] Katja Mayer views PageRank as a social network as it connects differing viewpoints and thoughts in a single place. People go to PageRank for information and are flooded with citations of other authors who also have an opinion on the topic. This creates a social aspect where everything can be discussed and collected to provoke thinking. There is a social relationship that exists between PageRank and the people who use it as it is constantly adapting and changing to the shifts in modern society. Viewing the relationship between PageRank and the individual through sociometry allows for an in-depth look at the connection that results.
I agree that there is no point in trying to over analyze how the PageRank is flowing through your site. Just focus on great content. Link out when it actually helps the reader. This is what Google wants – for you to give good quality content to their users. So if you are doing that, they will reward you in the long run. No need to worry yourself with these types of link strategies.
If you are serious about improving web traffic to your website, we recommend you read Google Webmasters and Webmaster Guidelines. These contain the best practices to help Google (and other search engines) find, crawl, and index your website. After you have read them, you MUST try our Search Engine Optimization Tools to help you with Keyword Research, Link Building, Technical Optimization, Usability, Social Media Strategy and more.
There's a lot to learn when it comes to the internet marketing field in general, and the digital ether of the web is a crowded space filled with one know-it-all after another that wants to sell you the dream. However, what many people fail to do at the start, and something that Sharpe learned along the way, is to actually understand what's going on out there in the digital world and how businesses and e-commerce works in general, before diving in headfirst.
Social Media Marketing - The term 'Digital Marketing' has a number of marketing facets as it supports different channels used in and among these, comes the Social Media. When we use social media channels ( Facebook, Twitter, Pinterest, Instagram, Google+, etc.) to market a product or service, the strategy is called Social Media Marketing. It is a procedure wherein strategies are made and executed to draw in traffic for a website or to gain attention of buyers over the web using different social media platforms.
On a blog the page rank should go to the main article pages. Now it just gets “evaporated” if you use “nofollow” or scattered to all the far flung nooks and crannys which means google will not be able to see the wood for the trees. The vast majority of a site’s overall page rank will now reside in the long tail of useless pages such as commentors profile pages. This can only make it harder for google to serve up the most relevant pages.
Due to the importance of backlinks, there are lots of bad practices followed by website owners to gain backlinks. Some of these bad practices are: purchasing backlinks, link exchange networks, selling backlinks, etc. Most of these practices are not recommended by search engines. They usually deindex and penalize websites suspected of involvement in such practices.
Going into network marketing? Understand that if you're not close to the top of the food chain there, your ability to generate any serious amount of income will be limited. Be wary of the hype and the sales pitches that get you thinking that it's going to work the other way. Simply understand that you're going to have to work hard no matter what you pick to do. Email marketing? Sure. You can do that. But you'll need a massive and very targeted list to make any dent.
A breadcrumb is a row of internal links at the top or bottom of the page that allows visitors to quickly navigate back to a previous section or the root page. Many breadcrumbs have the most general page (usually the root page) as the first, leftmost link and list the more specific sections out to the right. We recommend using breadcrumb structured data markup28 when showing breadcrumbs.
While there are several platforms for doing this, clearly YouTube is the most popular for doing this. However, video marketing is also a great form of both content marketing and SEO on its own. It can help to provide visibility for several different ventures, and if the video is valuable enough in its message and content, it will be shared and liked by droves, pushing up the authority of that video through the roof.
Matt, my biggest complaint with Google and this “page Rank” nofollow nightmare is it seems we need to have a certain type of site to get ranked well or to make your crawler happy, you say you want a quality site, but what my users deem as quality (3000 links to the best academic information on the planet for business development) is actually looked at by Google as a bad thing and I do not get any rank because of it, makes it hard for my site to be found, and people that can really use the information can not find it when you yourself would look at the info and think it was fantastic to find it all in one place.


text:
# How to partition a graph into N groups with M elements nearest?

Hi.

This problem probably already here but I could not find the right words to find it.

I have a list with 1700 points (geographic coordinates) and a need to separate into 17 groups with 100 nearest. I mapped this as a graph where each node is the point and each edge weight is the real distance of points connected by this. Then the problem is how to partition this graph into 17 groups of 100 elements each that minimizes the sum of the inter-partition edges weight.

Someone has a better idea to map this problem?

I tried to use METIS but the best result was using the edge-cut that minimize the sum of the weights of the edges removed (for this case I use the inverse of the distance as weight). I also tried using k-means but it does not guarantee the number of elements in each group.

Thanks in advance!

-
I suspect that a small variation on $k$-means clustering would work. You might want to try asking on stats.stackexchange.com where there may be more people with experience with this type of problem. –  Douglas Zare Aug 21 '12 at 18:40

## 1 Answer

Unfortunately I have just an approach, not a tool for you to use.

I would suggest using a local search algorithm that maintains the size of the sets at 100. You seek a clustering that consists of a partition of the vertices int 17 clusters, each of size 100. The cost of each cluster is the sum over all pairs of vertices in the cluster of the distance between the vertices. The cost of a clustering is the sum of the 17 cluster costs.

Your move in the local search space should be exchanging a vertex in one cluster with a vertex in another cluster. In this way you can compute the change in cost that results from exchanging two vertices.

Depending on how you select a move, this may be very time consuming. If you try this kind of approach, you might want to do it on 10% of the data set just to make sure it's not going to take an absurd amount of time to find a decent solution.

-


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# Development of reflective low concentrated photovoltaic/ thermal system

Algareu, Abdulmaged Omer (2017). Development of reflective low concentrated photovoltaic/ thermal system. University of Birmingham. Ph.D.

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Algareu17PhD.pdf
PDF - Accepted Version

## Abstract

This work aims to investigate the performance of a new design for CPV/T system using 3D flat sided (3D V-trough) concentrators named squared (SAC), Hexagonal (HAC), Octagonal (OAC) and Circular (CAC) inlet and exit Aperture concentrators, with an effective cooling facility that keeps PV temperature within the Normal operating range. Novel mathematical optical models were generated, to the HAC and OAC geometries and validated using OPTISORKS software, to calculate the geometrical concentration ratio ($$GCR$$) and actual concentration ratio ($$ACR$$ by the inlet aperture area
(A$$_i$$$$_n$$) a function of aperture width (W$$_i$$$$_n$$) and number of reflections (n), and material reflectivity ($$p$$) and at any concentrator side angle (Ψ), consequently the optical performance. Results showed that the optimum concentrator side angels for GCR of 2, 4, 6, 8 and 10 are 35°, 30°, 20°, 20° and 15°, respectively for all investigated geometries. Also COMSOL Multiphysics software was used for thermal modelling. Optical, thermal and electrical investigation results highlight that the designed CPV/T system is beneficial enough and feasible to be used in generating electrical and thermal powers for domestic use, as one useful package of energy with high output compared with the flat PV modules which generate only electrical power.

Type of Work: Thesis (Doctorates > Ph.D.)
Award Type: Doctorates > Ph.D.
Supervisor(s):
Supervisor(s)EmailORCID
Licence:
College/Faculty: Colleges (2008 onwards) > College of Engineering & Physical Sciences
School or Department: School of Engineering, Department of Mechanical Engineering
Funders: None/not applicable
Subjects: T Technology > TJ Mechanical engineering and machinery
URI: http://etheses.bham.ac.uk/id/eprint/7674

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text:
# The “beach problem”: does anyone know it? or know how to solve it?

The following problem was given some years ago in the German computer-science contest for pupils ("Bundeswettbewerb Informatik"). It was originally wrapped in a story which I will briefly translate below. My second "translation" already includes my understanding of the problem in terms of pure mathematics.

The fascinating thing for me is the following: Although the problem doesn't mention any numbers, my computer program only found a solution for 17 tents (but none for 18 and higher). I wonder if there is a rigorous proof for this.

Consider a beach of a finite length. During one summer period there are several campers coming to this beach (one after each other). The first camper may put his tent anywhere he likes. When the second camper arrives the beach will be divided into two equal parts and the second one must put his tent into the free half. Generally, if the k'th camper arrives, the beach is divided into k equal parts and there must be only one tent per part.

The question is the following: Given that all campers choose there positions perfectly, how many campers may arrive until there is no way to add another one (besides moving tents)?

My mathematical translation assumes the following:

• Tents are point-like
• The beach has a size of 1 (consider the real numbers from 0 to 1)
• If a camper puts his tent on the position 0.5 he is by definition in the right half ([0.5, 1])
• Therefore we can also think of the beach as the open interval $[0, 1)$

For a given natural number N, is there a way to choose a sequence of (ordered) points $P_1, P_2, \dots, P_N$ on the interval $[0, 1)$ such that for each $k \in \{1,\dots,N\}$ and each $l \in \{1, \dots,k\}$ there exists only one $i \in \{1, \dots, k\}$ such that

$\frac{l-1}{k} \le P_i < \frac{l}{k}$

I would like to know the following:

• Is there a solution for any N or is there a maximum N?
• If there is a maximum N, which one is it?

Investigating this problem I was only able to simplify it a bit. For example: it is enough to consider the "boundaries" as possible points for the tents. Assume you want to find a solution for $N=4$ (which is quite easy :-)). It is enough to consider the following possible positions:

$0, 1/4, 1/3, 1/2, 2/3, 3/4$

Thinking about this also brought up Eulers $\varphi$ function, because the number of such possible positions is $\sum_{k=1}^N \varphi(k)$. So the problem could be related to prime numbers.

Any hints pointing to a solution are appreciated. I also would like to know if you are aware of similar problems or maybe other formulations of the same problem?

-
This shows up every now and then. I've seen it expressed in terms of using flagpoles as opposed to tents. I may have seen it on this forum or another. The first google hit I got is mathforum.org/kb/message.jspa?messageID=391377&tstart=0 A brute force computer check shows that 17 is the maximum. I don't recall having seen an elegant solution. – Jyrki Lahtonen Jul 7 '11 at 12:20
Thanks for the link. At least this is a hint that my program worked correctly. I still wonder how this "17" pops up. – shark.dp Jul 7 '11 at 12:31
This was discussed at math.stackexchange.com/questions/43311/… and good references are given there. One shows that 17 is maximal. – Ross Millikan Jul 7 '11 at 12:38
Thanks a lot, this is what I've been searching for.. just didn't have any name. – shark.dp Jul 7 '11 at 12:40
This has little to do with prime numbers or infinite series, so I replaced those tags with "uniform distribution". There's probably something better that I'm not thinking of. – Gerry Myerson Jul 7 '11 at 12:45


text:
×

# Bug in Brilliant!

Hi Team!

Just found a bug in the site, thought to report it to you. After solving a problem we are asked whether we would like to view the discussion or submit our solution. But if one opens the same page in a new tab and clicks join discussion in the new tab and submit solution in the old then he/she can view the discussion even before submitting the solution and the solution submission gives no error!

Note by Ayush Pateria
4 years, 2 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

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Hey Ayush Please send me the KVPY interview ebook

- 3 years, 10 months ago

- 3 years, 10 months ago

yes true.....observed but ignored thanks for reporting

- 4 years, 1 month ago

Thanks for the report. We'll fix it!

Staff - 4 years, 2 months ago

Ok, should be fixed now. Thanks again for reporting it!

Staff - 4 years, 2 months ago