text:
## RC Airplane

We decided to give making an RC airplane a shot. We have lots of parts from our drone projects. It’s interesting that we haven’t tried before. Drones appear to be more popular than rc planes if one goes by the internet.

We are going for the simplest thing we can think of. We bought some foamboard and cut it into the wing tail and body. We half cut through it to make the control surfaces.

We ripped a crappy small brushless motor and esc from our eldest drone project. The esc back powers the radio receiver through the control line

We had a bag of mini servos and a Turnigy 9x. The servos are directly connectable to the receiver unit. I don’t see why you’d need more than 3 servos to get starts. One for the ailerons connected on opposite sides of the servo horn so that they flap oppositely. We used paper clips as push rods. One for the rudder and one for the elevator flap. Putting the controller into an airplane mode is a bit confusing, but Ben figured it out eventually.

We sized things to look roughly plane like. I’ve heard a longer wingspan is better for trainers and gliders. Also we dug up a calculator for wing loading. All our stuff comes in to about 10oz.

We didn’t follow this but maybe we should have

## Solving the Ising Model using a Mixed Integer Linear Program Solver (Gurobi)

I came across an interesting thing, that finding the minimizer of the Ising model is encodable as a mixed integer linear program.

The Ising model is a simple model of a magnet. A lattice of spins that can either be up or down. They want to align with an external magnetic field, but also with their neighbors (or anti align, depending on the sign of the interaction). At low temperatures they can spontaneously align into a permanent magnet. At high temperatures, they are randomized. It is a really great model that contains the essence of many physics topics.

Linear Programs minimize linear functions subject to linear equality and inequality constraints. It just so happens this is a very solvable problem (polynomial time).

MILP also allow you to add the constraint that variables take on integer values. This takes you into NP territory. Through fiendish tricks, you can encode very difficult problems. MILP solvers use LP solvers as subroutines, giving them clues where to search, letting them step early if the LP solver returns integer solutions, or for bounding branches of the search tree.

How this all works is very interesting (and very, very roughly explained), but barely matters practically since other people have made fiendishly impressive implementations of this that I can’t compete with. So far as I can tell, Gurobi is one of the best available implementations (Hans Mittelman has some VERY useful benchmarks here http://plato.asu.edu/bench.html), and they have a gimped trial license available (2000 variable limit. Bummer.). Shout out to CLP and CBC, the Coin-Or Open Source versions of this that still work pretty damn well.

Interesting Connection: Quantum Annealing (like the D-Wave machine) is largely based around mapping discrete optimization problems to an Ising model. We are traveling that road in the opposite direction.

So how do we encode the Ising model?

Each spin is a binary variable $s_i \in {0,1}$

We also introduce a variable for every edge. which we will constrain to actually be the product of the spins. $e_{ij} \in {0,1}$. This is the big trick.

We can compute the And/Multiplication (they coincide for 0/1 binary variables) of the spins using a couple linear constraints. I think this does work for the 4 cases of the two spins.

$e_{ij} \ge s_i +s_j - 1$

$e_{ij} \le s_j$

$e_{ij} \le s_i$

The xor is usually what we care about for the Ising model, we want aligned vs unaligned spins to have different energy. It will have value 1 if they are aligned and 0 if they are anti-aligned. This is a linear function of the spins and the And.

$s_i \oplus s_j = s_i + s_j - 2 e_{ij}$

Then the standard Hamiltonian is

$H=\sum B_i s_i + \sum J_{ij} (s_i + s_j - 2 e_{ij})$

Well, modulo some constant offset. You may prefer making spins $\pm 1$, but that leads to basically the same Hamiltonian.

The Gurobi python package actually let’s us directly ask for AND constraints, which means I don’t actually have to code much of this.

We are allowed to use spatially varying external field B and coupling parameter J. The Hamiltonian is indeed linear in the variables as promised.

After already figuring this out, I found this chapter where they basically do what I’ve done here (and more probably). There is nothing new under the sun. The spatially varying fields B and J are very natural in the field of spin glasses.

https://onlinelibrary.wiley.com/doi/10.1002/3527603794.ch4

For a while I thought this is all we could do, find the lowest energy solution, but there’s more! Gurobi is one of the few solvers that support iteration over the lowest optimal solutions, which means we can start to talk about a low energy expansion. https://www.gurobi.com/documentation/8.0/refman/poolsearchmode.html#parameter:PoolSearchMode

Here we’ve got the basic functionality. Getting 10,000 takes about a minute. This is somewhat discouraging when I can see that we haven’t even got to very interesting ones yet, just single spin and double spin excitations. But I’ve got some ideas on how to fix that. Next time baby-cakes.

(A hint: recursion with memoization leads to some brother of a cluster expansion.)

Here’s the ground antiferromagnetic state. Cute.

## OSQP and Sparsegrad: Fast Model Predictive Control in Python for an inverted pendulum

OSQP is a quadratic programming solver that seems to be designed for control problems. It has the ability to warm start, which should make it faster.

I heard about it in this Julia talk

They have some really cool technology over there in Julia land.

The problem is setup as a sequential quadratic program. I have a quadratic cost to try to bring the pendulum to an upright position.

The equations of motion are roughly

$\ddot{\theta}I=-mgL\sin(\theta)+mfL\cos(\theta)$

$\ddot{x}=f$

$I=\frac{1}{3}mL^2$

We don’t include the back reaction of the pole on the cart. It is basically irrelevant for our systems and overly complicated for no reason. The interesting bit is the nonlinear dynamics and influence of the cart acceleration.

I write down obeying the equations of motion as a linear constraint between timesteps. I use sparsegrad to get a sparse Jacobian of the equations of motion. The upper and lower (l and u) bounds are set equal to make an equality constraint.

Our setup has a finite track about a meter long. Our motors basically are velocity control and they can go about +-1m/s. Both of these can be encapsulated as linear constraints on the position and velocity variables. $l \le Ax \le u$

$\phi(x)=0$

$\phi(x) \approx \phi(x_0) + \partial \phi(x_0) (x - x_0)$

$A= \partial \phi(x_0)$

$l=u=\partial \phi(x_0) x_0 - \phi_0(x_0)$

Similarly for finding the quadratic cost function in terms of the setpoint $x_s$. $\frac{1}{2}x^T P x + q^Tx= \frac{1}{2}(x-x_s)P(x-x_s)$ Expanding this out we get

$q = - Px_s$

I run for multiple iterations to hopefully converge to a solution (it does). The initial linearization might be quite poor, but it gets better.

The OSQP program seems to be able to run in under 1ms. Nice! Initial tests of our system seem to be running at about 100Hz.

Could probably find a way to make those A matrices faster than constructing them entirely anew every time. We’ll see if we need that.

I very inelegantly smash all the variables into x. OSQP and scipy sparse don’t support multiIndex objects well, best as I can figure. Vectors should be single dimensioned and matrices 2 dimensioned.

Still to be seen if it works on hardware. I was already having infeasibility problems. Adjusting the regularization on P seemed to help.

## Approximating Compiling to Categories using Type-level Haskell: Take 2

The previous episode is here .

Summary: I’m trying to use typelevel programming in Haskell to achieve some of the aims of Conal Elliott’s compiling to categories GHC plugin. The types of highly polymorphic tuple functions are enough to specify the implementation. We aren’t going to be able to piggy-back off of GHC optimizations (a huge downside), but we can reify lambdas into other categories and avoid the scariness of plugins.

The current implementation github source is here

JESUS CHRIST.

Of course, Oleg already did it. This is a file where he builds the implementation of a polymorphic function from the type signature. Instead of tuples, he is focusing on higher order functions with deep nesting of (->).

The trick that I was missing is in the IsFunction typeclass at the very end, which is only achievable as a Incoherent instances.

I would never have had the courage to use an Incoherent instance if I hadn’t seen a higher authority do it first. It has turned out in my typelevel journey that many instances that I’ve been tempted to make overlapping or incoherent don’t actually have to be, especially with the availability of closed type families. I think you really do need Incoherent in this application because type variables stay polymorphic forever.

To the best of my knowledge, if you need to differentiate between a tuple type (a,b) and an uninstantiated polymorphic value a’ like we do when deconstructing the input type of our lambda, you need to use an Incoherent instance. Since a’ could hypothetically eventually be unified to some (a,b) we should not be able to do different things for the two cases without stepping outside the usual laws of typeclass resolution.

New features of the implementation:

• The new implementation does not require the V annotation of the input like previous version by using the previously mentioned. This is super cool because now I can throw the stock Prelude.fst into toCcc.
• I changed how the categorical implementation is built, such that it short circuits with an ‘id’ if a large structure is needed from the input, rather than deconstructing all the way to every piece of the input. Lots of other optimizations would be nice (vital?), but it is a start.
• I also implemented a FreeCat GADT so that we can see the implementation in ghci.
• I switched from using Data.Proxy to the type annotations extensions, which is a huge readability win.
• I added a binary application operator binApp, which is useful for encapsulating categorical literals as infix operators into your lambda expressions.
• General cleanup, renaming, and refactoring into library files.

A couple typelevel tricks and comments:

You often have to make helper typeclasses, so don’t be afraid to. If you want something like an if-then-else in your recursion, it is very likely you need a form of the typeclass that has slots for ‘True or ‘False to help you pick the instance.

If possible, you often want the form

rather than

The type inference tends to be better.

Here are some usage examples of toCcc.

My partially implemented version of some of Conal’s category typeclasses. Should I switch over to using the constrained-categories package?

The actual implementation of toCcc

## Variational Method of the Quantum Simple Harmonic Oscillator using PyTorch

A fun method (and useful!) for solving the ground state of the Schrodinger equation is to minimize the energy integral $dx \psi^\dagger H \psi$ while keeping the total probability 1. Pytorch is a big ole optimization library, so let’s give it a go.

I’ve tried two versions, using a stock neural network with relus and making it a bit easier by giving a gaussian with variable width and shift.

We can mimic the probability constraint by dividing by to total normalization $\int dx \psi^\dagger \psi$. A Lagrange multiplier or penalty method may allows us to access higher wavefunctions.

SGD seems to do a better job getting a rounder gaussian, while Adam is less finicky but makes a harsh triangular wavefunction.

The ground state solution of $-\frac{d^2\psi}{dx^2} + x^2\psi=E\psi$ is $e^{-x^2/2}$, with an energy of 1/2 (unless I botched up my head math). We may not get it, because we’re not sampling a very good total domain. Whatever, for further investigation.

Very intriguing is that pytorch has a determinant in it, I believe. That opens up the possibility of doing a Hartree-Fock style variational solution.

Here is my garbage

Edit: Hmm I didn’t compensate for the fact I was using randn sampling. That was a goof. I started using unifrom sampling, which doesn’t need compensation

## Shit Compiling to Categories using Type level Programming in Haskell

So I’ve  been trying to try and approximate Conal Elliott’s compiling to categories in bog standard Haskell and I think I’ve got an interesting chunk.

His approach in using a GHC plugin is better for a couple reasons. One really important thing is that he gets to piggy back on GHC optimizations for the lambdas. I have only implemented a very bad evaluation strategy. Perhaps we could recognize shared subexpressions and stuff, but it is more work. I seem somewhat restricted in what I can do and sometimes type inference needs some help. Not great. However, GHC plugins definitely bring their own difficulties.

What I’ve got I think still has roots in my thinking from this previous post

There are a couple insights that power this implementation

1. A fully polymorphic tuple to tuple converter function is uniquely defined by it’s type. For example, swap :: (a,b) -> (b,a), id:: a -> a, fst :: (a,b) ->a, snd::(a,b)->b are all unique functions due to the restrictions of polymorphism. Thus typelevel programming can build the implementation from the type.
2. Getting a typeclass definition to notice polymorphism is hard though. I haven’t figured out how to do it, if it is even possible. We get around it by explicitly newtyping every pattern match on a polymorphic variable like so \(V x, V y) -> (y,x). Two extra characters per variable. Not too shabby.
3. You can abuse the type unification system as a substitution mechanism. Similar to HOAS, you can make a lambda interpreter at the type level that uses polymorphism as way of generating labels for variables. This is probably related to Oleg Kiselyov’s type level lambda calculus, but his kind of confuses me http://okmij.org/ftp/Computation/lambda-calc.html#haskell-type-level
4. You can inject a categorical literal morphism using a wrapping type to be extracted later using an apply operator and type App f a. An infix () makes it feel better.

So here is the rough structure of what the typelevel programming is doing

You can do a depth first traversal of the input tuple structure, when you hit V, unify the interior type with a Nat labelled Leaf. At the same time, you can build up the actual value of the structure so that you can apply the lambda to it and get the output which will hold a tree that has morphism literals you want.

Then inspect the output type of the lambda which now has Nat labels, and traverse the labelled tree again to find the appropriate sequence of fst and snd to extract what you need. If you run into an application App, you can pull the literal out now and continue traversing down.

At the moment I’ve only tested with the (->) Category, which is a bit like demonstrating a teleporter by deconstructing a guy and putting back in the same place, but I think it will work with others. We’ll see. I see no reason why not.

At the moment, I’m having trouble getting GHC to not freakout unless you explicitly state what category you’re working in, or explicitly state that you’re polymorphic in the Category k.

Some future work thoughts: My typelevel code is complete awful spaghetti written like I’m a spastic trapped animal. It needs some pruning. I think all those uses of Proxy can be cleaned up by using TypeApplications. I need to implement some more categories. Should I conform to the Constrained-Categories package? Could I use some kind of hash consing to find shared structures? Could Generic or Generic1 help us autoplace V or locate polymorphism? Maybe a little Template Haskell spice to inject V?

Here’s the most relevant bits, with my WIP git repository here

## Deducing Row Sparse Matrices from Matrix Vector Product Samples

So as part of betting the banded hessian out of pytorch, I’ve been trying to think about what it is I am doing and I’ve come up with an alternative way of talking about it.

If every row of a matrix has <N non zero entries, you can back out that matrix from N matrix vector samples of it. You have many choices for the possible sampling vectors. Random works well.

The unknown in this case is the row of the matrix, represented in green. We put a known set of inputs into it and get outputs. Each row of the output, represented in red, can tell use the matrix row. We have to invert the matrix that consists of all the elements that the nonzero elements of that row touches represented in blue. That black T is a transpose. To turn those row vectors into column vectors, we have to transpose everything.

For a banded matrix, we have a shifting sample matrix that we have to invert, one for each row.

A cute trick we can use to simplify the edge cases where we run off the ends is to extend the sample matrix with some random trash. That will actually put the entries in the appropriate place and will keep the don’t cares close to zero also.

In my previous post I used a stack of identity matrices. These are nice because a shifted identity matrix is a circular permutation of the entries, which is very easy to undo. That was what the loop that used numpy.roll was doing. Even better, it is easy to at least somewhat vectorize the operation and you can produce those sampling vectors using some clever use of broadcasting of an identity matrix.

An alternative formulation that is a little tighter. I want the previous version because the samples isn’t actually always random. Often they won’t really be under our control.

This all still doesn’t address the hermitian problem. The constraint that A is hermitian hypothetically allows you to take about half the samples. But the constraints make it tougher to solve. I haven’t come up with anything all that much better than vectorizing the matrix and building a matrix out of the samples in the appropriate spots.

I think such a matrix will be banded if A is banded, so that’s something at least.

## Pytorch Trajectory Optimization Part 4: Cleaner code, 50Hz

Cleaned up the code more and refactored some things.

Added backtracking. It will backtrack on the dx until the function is actually decreasing.

Prototyped the online part with shifts. Seems to work well with a fixed penalty parameter rho~100. Runs at ~50Hz with pretty good performance at 4 optimization steps per time step. Faster or slower depending on the number of newton steps per time step we allow ourselves.  Still to see if the thing will control an actual cartpole.

The majority of time is spent just backwards calculating the hessian still (~50%).

I’ve tried a couple different schemes (direct projection of the delLy terms or using y = torch.eye). None particularly seem to help.

The line search is also fairly significant (~20% of the time) but it really helps with both stability and actually decreasing the number of hessian steps, so it is an overall win. Surprisingly during the line search, projecting out the batch to 0 doesn’t matter much. How could this possibly make sense?

What I should do is pack this into a class that accepts new state observations and initializes with the warm start. Not clear if I should force the 4 newton steps on you or let you call them yourself. I think if you use too few it is pretty unstable (1 doesn’t seem to work well. 2 might be ok and gets you up to 80Hz maybe.)

The various metaparameters should be put into the __init__. The stopping cutoff  1e-7, Starting rho (~0.1), rho increase (x10) , backtrack alpha decrease factor (0.5 right now), the online rho (100). Hopefully none of these matter two much. I have noticed going too small with cutoff leading to endless loops.

Could swapping the ordering of time step vs variable number maybe help?

For inequality constraints like the track length and forces, exponential barriers seems like a more stable option compared to log barriers. Log barriers at least require me to check if they are going NaN.

I attempted the pure Lagrangian version where lambda is just another variable. It wasn’t working that great.

## Pytorch Trajectory Optimization 3: Plugging in the Hessian

I plugged in the hessian extraction code for using newton steps.

When I profiled it using the oh so convenient https://github.com/rkern/line_profiler I found almost all of my time was spent in the delLy.backwards step. For each hessian I needed to run this B (the band width) times and each time cost ~0.6ms. For the entire run to converge took about 70 iterations and 1000 runs of this backwards step, which came out to 0.6 seconds. It is insane, but actually even calculating the band of the hessian costs considerably more time than inverting it.

So to speed this up, I did a bizarre thing. I replicated the entire system B times. Then I can get the entire hessian band in a single call to backwards. remarkably, although B ~ 15 this only slowed backwards down by 3x. This is huge savings actually, while obviously inefficient. The entire program has gone down from 1.1s to 0.38s, roughly a 3x improvement. All in all, this puts us at 70/0.38 ~ 185 Hz for a newton step. Is that good enough? I could trim some more fat. The Fast MPC paper http://web.stanford.edu/~boyd/papers/fast_mpc.html says we need about ~5 iterations to tune up a solution, this would mean running at 40Hz. I think that might be ok.

Since the hessian is hermitian it is possible to use roughly half the calls (~B/2), but then actually extracting the hessian is not so simple. I haven’t figured out a way to comfortably do such a thing yet. I think I could figure out the first column and then subtract (roughly some kind of gaussian elimination procedure).

It has helped stability to regularize everything with a surprising amount of weight in the cost. I guess since I anticipate all values being in the range of -10,10, maybe this makes sense.

Now I need to try not using this augmented Lagrangian method and just switching to a straight newton step.

Edit: Ooh. Adding a simple backtracking line search really improves stability.

## Cartpole Camera System – OpenCV + PS EYE + IR

We tried using colored tape before. It was okay after manual tuning, but kind of sucked. Commerical motion tracking systems use IR cameras and retroreflectors.

We bought some retroreflective tape and put it on the pole. http://a.co/0A9Otmr

We removed our PS EYE IR filter. The PS EYE is really cheap (~7\$) and has a high framerate mode (100+ fps). People have been using it for a while for computer vision projects.

http://wiki.lofarolabs.com/index.php/Removing_the_IR_Filter_from_the_PS_Eye_Camera

We followed the instructions, but did not add the floppy disk and sanded down the base of the lens to bring the image back into focus.

We bought an IR LED ring light which fit over the camera with the plastic cover removed and rubber banded it in place.

http://a.co/2sGUY08

If you snip the photoresistor it is always on, since the photoresistor is high resistance in the dark. We used a spare 12V power supply that we soldered a connector on for.

We had also bought an IR pass filter on amazon, but it does not appear to help.

Useful utilties: qv4l2, v4l2-ctl and v4l2-utils. You can change lots of stuff.

qv4l2 -d 1 is very useful for experiementation

Useful options to  v4l2-ctl : -d selects camera, -p sets framerate -l gives a list of changeable options. You have to turn off the automatic stuff before it becomes changeable. Counterintuitively auto-exposure seems to have 1 as off.

There has been a recent update to opencv to let the v4l2 buffer size be changed. We’re hoping this will really help with our latency issues

A useful blog. We use v4l2-ctl for controlling the exposure programmatically

http://www.jayrambhia.com/blog/capture-v4l2

Oooh. The contour method + rotated rectangle is working really well for matching the retroreflective tape.

https://docs.opencv.org/3.3.1/dd/d49/tutorial_py_contour_features.html

You need to reduce the video size to 320×240 if you want to go to the highest framerate of 187fps

In regards to the frame delay problem from before, it’s not clear that we’re really seeing it? We are attempting both the screen timestamp technique and also comparing to our rotary encoder. In the screen timestamp technique, it is not so clear that what we measure there is latency, and if it is, it includes the latency of the monitor itself, which is irrelevant.


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Context

Algebra

higher algebra

universal algebra

Contents

Idea

For each prime number $p$ the ring of $p$-adic integers $\mathbb{Z}_p$ is the formal completion of the ring $\mathbb{Z}$ at the prime ideal $(p)$. Geometrically this means that $\mathbb{Z}_p$ is the ring of functions on a formal neighbourhood of $p$ inside Spec(Z) (this is discussed in more detail below). Algebraically it means that the elements in $\mathbb{Z}_p$ look like formal power series where the formal variable is the prime number $p$.

Definition

For any prime number $p$, the ring of $p$-adic integers $\mathbb{Z}_p$ (which, to avoid possible confusion with the ring $\mathbb{Z}/(p)$ used in modular arithmetic, is also written as $\widehat{\mathbb{Z}}_p$) may be described in one of several ways:

1. To the person on the street, it may be described as (the ring of) numbers written in base $p$, but allowing infinite expansions to the left. Thus, numbers of the form

$\sum_{n \geq 0} a_n p^n$

where $0 \leq a_n \lt p$, added and multiplied with the usual method of carrying familiar from adding and multiplying ordinary integers.

2. More abstractly, it is the limit $\underset{\leftarrow}{\lim} \mathbb{Z}/(p^n)$, in the category of (unital) rings, of the diagram

$\ldots \to \mathbb{Z}/(p^{n+1}) \to \mathbb{Z}/(p^n) \to \ldots \to \mathbb{Z}/(p) .$

This is also a limit in the category of topological rings, taking the rings in the diagram to have discrete topologies.

3. Alternatively, it is the metric completion of the ring of integers $\mathbb{Z}$ with respect to the $p$-adic absolute value. Since addition and multiplication of integers are uniformly continuous with respect to the $p$-adic absolute value, they extend uniquely to a uniformly continuous addition and multiplication on $\mathbb{Z}_p$. Thus $\mathbb{Z}_p$ is a topological ring.

4. Also $\mathbb{Z}[ [ x ] ]/(x-q)\mathbb{Z}[ [ x ] ]$, see at analytic completion.

Hence one also speaks of the $p$-adic completion of the integers. See completion of a ring (which generalizes 2&3).

Properties

Topology

The ring of $p$-adic integers has the following properties:

Relation to profinite completion of the integers

Example
$\widehat {\mathbb{Z}} \coloneqq \underset{\leftarrow}{\lim}_{n \in \mathbb{N}} (\mathbb{Z}/n\mathbb{Z}) \,.$

This is isomorphic to the product of the $p$-adic integers for all $p$

$\widehat{\mathbb{Z}} \simeq \underset{p\; prime}{\prod} \mathbb{Z}_p \,.$
Definition

The ring of integral adeles $\mathbb{A}_{\mathbb{Z}}$ is the product of the profinite completion $\widehat{\mathbb{Z}}$ of the integers, example 1, with the real numbers

$\mathbb{A}_{\mathbb{Z}} \coloneqq \mathbb{R} \times \widehat{\mathbb{Z}} \,.$

The group of units of the ring of adeles is called the group of ideles.

Pontrajgin duality to Prüfer $p$-group

Under Pontryagin duality, the abelian group underlying $\mathbb{Z}_p$ maps to the Prüfer p-group $\mathbb{Z}[p^{-1}]/\mathbb{Z}$, see at Pontryagin duality for torsion abelian groups.

$\array{ &\mathbb{Z}[p^{-1}]/\mathbb{Z} &\hookrightarrow& \mathbb{Q}/\mathbb{Z} &\hookrightarrow& \mathbb{R}/\mathbb{Z} \\ {}^{\mathllap{hom(-,\mathbb{R}/\mathbb{Z})}}\downarrow \\ &\mathbb{Z}_p &\leftarrow& \hat \mathbb{Z} &\leftarrow& \mathbb{Z} }$

As the formal neighbourhood of a prime

The formal spectrum $Spf(\mathbb{Z}_p)$ of $\mathbb{Z}_p$ may be understood as the formal neighbourhood of the point corresponding to the prime $p$ in the prime spectrum $Spec(\mathbb{Z})$ of the integers. The inclusion

$\{p\} \hookrightarrow Spf(\mathbb{Z}_p) \hookrightarrow Spec(\mathbb{Z})$

is the formal dual of the canonical projection maps $\mathbb{Z}\to \mathbb{Z}_p\to \mathbb{Z}/(p)$.

This plays a central role for instance in the function field analogy. It is highlighted for instance in (Hartl 06, 1.1, Buium 13, section 1.1.3). See also at arithmetic jet space and at ring of Witt vectors.

Examples of sequences of local structures

geometrypointfirst order infinitesimal$\subset$formal = arbitrary order infinitesimal$\subset$local = stalkwise$\subset$finite
$\leftarrow$ differentiationintegration $\to$
smooth functionsderivativeTaylor seriesgermsmooth function
curve (path)tangent vectorjetgerm of curvecurve
smooth spaceinfinitesimal neighbourhoodformal neighbourhoodgerm of a spaceopen neighbourhood
function algebrasquare-0 ring extensionnilpotent ring extension/formal completionring extension
arithmetic geometry$\mathbb{F}_p$ finite field$\mathbb{Z}_p$ p-adic integers$\mathbb{Z}_{(p)}$ localization at (p)$\mathbb{Z}$ integers
Lie theoryLie algebraformal grouplocal Lie groupLie group
symplectic geometryPoisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization

References

Introductions and surveys include

• Dennis Sullivan, pp. 9 of Localization, Periodicity and Galois Symmetry (The 1970 MIT notes) edited by Andrew Ranicki, K-Monographs in Mathematics, Dordrecht: Springer (pdf)

• Bernard Le Stum, One century of $p$-adic geometry – From Hensel to Berkovich and beyond talk notes, June 2012 (pdf)

• Hendrik Lenstra, Profinite groups (pdf)

The synthetic differential geometry-aspect of the $p$-adic numbers is highlighted for instance in

Revised on August 26, 2014 23:10:44 by Urs Schreiber (82.136.246.44)

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# What Is a Linear Equation?

## What Is a Linear Equation?

A linear equation is an equation that shows a straight line when it is plotted on a graph.

## An Example of a Linear Equation

An example of a linear equation is shown below:

In this example, there are two variables (x and y).

None of the variables have a number written to the right and above it (that is, the highest exponent is 1).

## Linear Equations as Equations of Lines

The linear equation has two variables: x and y.

Pairs of values of x and y will make both sides of the equation equal.

In the table below, pairs of values of x and y are chosen in the left hand column. They are substituted into the equation (y = 2x + 1) in the right hand column. Both sides of the equals sign are equal.

 x = 1, y = 3 3 = 2 × 1 + 1 $$\:\:\:$$ = 2 + 1 = 3 ✔ x = 2, y = 5 5 = 2 × 2 + 1 $$\:\:\:$$ = 4 + 1 = 5 ✔ x = 3, y = 7 7 = 2 × 3 + 1 $$\:\:\:$$ = 6 + 1 = 7 ✔

Treat the value of x as the x-coordinate and the value of y as the y-coordinate.

These pair of x and y values can be plotted on a graph as Cartesian coordinates (x, y). When they are plotted, they make a line:

## Forms of Linear Equations

Linear equations come in many forms.


text:
## anonymous one year ago What is the fifth term of the sequence whose first term is 10 and whose common ratio is ? a. 160 b. 50 c. 12 d. 5/8

1. anonymous

The common ratio is 1/2

2. rishavraj

okay then $a_5 = a_1 \times r^{5- 1}$

3. rishavraj

plug a_1 i.e first term and r i.e common ratio

4. anonymous

what do all the letters stand for? :(

5. anonymous

ohhhhh

6. rishavraj

see the nth term in geometric sequence is $a_n = a_1 \times r^{n - 1}$

7. anonymous

Thanks i got it D was the answer!!

8. rishavraj

yup correct.... :))


text:
# statsmodels.discrete.discrete_model.LogitResults.resid_generalized¶

method

LogitResults.resid_generalized()[source]

Generalized residuals

Notes

The generalized residuals for the Logit model are defined

$y - p$

where $$p=cdf(X\beta)$$. This is the same as the resid_response for the Logit model.


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# Vanderwaal Company uses a standard costing system. In August, 5, 450 actual labor hours were work...

Vanderwaal Company uses a standard costing system. In August, 5, 450 actual labor hours were worked at a rate of $12.50 per hour. The standard number of hours is 5, 270 and the standard wage rate is$13.20 per hour. Determine the Labor Rate and Variance Labor Efficiency Variance. Record the related journal entries to record labor cost. (Credit account titles are automatically indented when the amount is entered. Do not indent manually.)


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tesla (T)

Tesla (T), named after the Croatian engineer Nikola Tesla (1856-1943), is the unit of magnetic flux density specified in the International System of Units( SI). In SI units, one tesla is equal to the deflection of a magnetic field vector required to produce a force of 1 newton with a charge of 1 coulomb at a rate of change of one meter per second.

Since the tesla is a relatively large unit, milli-tesla (mT) and micro-tesla (µT) are used in practice. There is a direct relationship between the tesla and the current flow. In the unit system one Tesla corresponds to 1 Vs/m2, in the cgs system(centimeter, gram, second) 1 Tesla corresponds to 10^4 Gauss or 10^-5 Gamma. The earth magnetic field has a magnetic flux density of about 50 µT. With light magnetic materials like magnetite (Fe3O4) magnetic fields with a strength of about 0.5 T are reached, with permanent magnets about 1.5 T to maximum 3 T are reached.

Informations:
 Englisch: tesla - T Updated at: 21.01.2022 #Words: 153 Links: unit (U), magnetic flux density, service indicator (SS7) (SI), indium (In), magnetic field (H) Translations: DE Sharing:


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NIPS Proceedingsβ

Regret Lower Bound and Optimal Algorithm in Finite Stochastic Partial Monitoring

A note about reviews: "heavy" review comments were provided by reviewers in the program committee as part of the evaluation process for NIPS 2015, along with posted responses during the author feedback period. Numerical scores from both "heavy" and "light" reviewers are not provided in the review link below.

[PDF] [BibTeX] [Supplemental] [Reviews]

Abstract

Partial monitoring is a general model for sequential learning with limited feedback formalized as a game between two players. In this game, the learner chooses an action and at the same time the opponent chooses an outcome, then the learner suffers a loss and receives a feedback signal. The goal of the learner is to minimize the total loss. In this paper, we study partial monitoring with finite actions and stochastic outcomes. We derive a logarithmic distribution-dependent regret lower bound that defines the hardness of the problem. Inspired by the DMED algorithm (Honda and Takemura, 2010) for the multi-armed bandit problem, we propose PM-DMED, an algorithm that minimizes the distribution-dependent regret. PM-DMED significantly outperforms state-of-the-art algorithms in numerical experiments. To show the optimality of PM-DMED with respect to the regret bound, we slightly modify the algorithm by introducing a hinge function (PM-DMED-Hinge). Then, we derive an asymptotical optimal regret upper bound of PM-DMED-Hinge that matches the lower bound.


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# Differentiation

5.21: Electric field of a charge distribution:
Suppose we have a distribution of charges and we want to calculate the resulting electric field. One way to do this is to first calculate the electric potential φ and then take its gradient. For a point charge q at the origin, the electric potential at a distance r from the origin is

$\Phi = {q\over{4\pi\epsilon_0 r}}$

and the electric field is E = −∇φ.

1. You have two charges, of ±1 C, 10 cm apart. Calculate the resulting electric potential on a 1 m × 1 m square plane surrounding the charges and passing through them. Calculate the potential at 1 cm spaced points in a grid and make a visualization on the screen of the potential using a density plot.
2. Now calculate the partial derivatives of the potential with respect to x and y and hence find the electric field in the xy plane. Make a visualization of the field also. This is a little trickier than visualizing the potential, because the electric field has both magnitude and direction. One way to do it might be to make two density plots, one for the magnitude, and one for the direction, the latter using the “hsv” color scheme in pylab, which is a rainbow scheme that passes through all the colors but starts and ends with the same shade of red, which makes it suitable for representing things like directions or angles that go around the full circle and end up where they started. A more sophisticated visualization might use the arrow object from the visual package, drawing a grid of arrows with direction and length chosen to represent the field.


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# Tagged: Ohio State.LA

## Problem 75

Let $\Q$ denote the set of rational numbers (i.e., fractions of integers). Let $V$ denote the set of the form $x+y \sqrt{2}$ where $x,y \in \Q$. You may take for granted that the set $V$ is a vector space over the field $\Q$.

(a) Show that $B=\{1, \sqrt{2}\}$ is a basis for the vector space $V$ over $\Q$.

(b) Let $\alpha=a+b\sqrt{2} \in V$, and let $T_{\alpha}: V \to V$ be the map defined by
$T_{\alpha}(x+y\sqrt{2}):=(ax+2by)+(ay+bx)\sqrt{2}\in V$ for any $x+y\sqrt{2} \in V$.
Show that $T_{\alpha}$ is a linear transformation.

(c) Let $\begin{bmatrix} x \\ y \end{bmatrix}_B=x+y \sqrt{2}$.
Find the matrix $T_B$ such that
$T_{\alpha} (x+y \sqrt{2})=\left( T_B\begin{bmatrix} x \\ y \end{bmatrix}\right)_B,$ and compute $\det T_B$.

(The Ohio State University, Linear Algebra Exam)

## Problem 67

Answer the following questions regarding eigenvalues of a real matrix.

(a) True or False. If each entry of an $n \times n$ matrix $A$ is a real number, then the eigenvalues of $A$ are all real numbers.
(b) Find the eigenvalues of the matrix
$B=\begin{bmatrix} -2 & -1\\ 5& 2 \end{bmatrix}.$

(The Ohio State University, Linear Algebra Exam)

## Problem 66

Consider the matrix
$A=\begin{bmatrix} 1 & 2 & 1 \\ 2 &5 &4 \\ 1 & 1 & 0 \end{bmatrix}.$

(a) Calculate the inverse matrix $A^{-1}$. If you think the matrix $A$ is not invertible, then explain why.

(b) Are the vectors
$\mathbf{A}_1=\begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \mathbf{A}_2=\begin{bmatrix} 2 \\ 5 \\ 1 \end{bmatrix}, \text{ and } \mathbf{A}_3=\begin{bmatrix} 1 \\ 4 \\ 0 \end{bmatrix}$ linearly independent?

(c) Write the vector $\mathbf{b}=\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$ as a linear combination of $\mathbf{A}_1$, $\mathbf{A}_2$, and $\mathbf{A}_3$.

(The Ohio State University, Linear Algebra Exam)

## Problem 65

Consider the system of linear equations
\begin{align*}
x_1&= 2, \\
-2x_1 + x_2 &= 3, \\
5x_1-4x_2 +x_3 &= 2
\end{align*}

(a) Find the coefficient matrix and its inverse matrix.

(b) Using the inverse matrix, solve the system of linear equations.

(The Ohio State University, Linear Algebra Exam)

## Problem 47

Let $T=\begin{bmatrix} 1 & 0 & 2 \\ 0 &1 &1 \\ 0 & 0 & 2 \end{bmatrix}$.
Calculate and simplify the expression
$-T^3+4T^2+5T-2I,$ where $I$ is the $3\times 3$ identity matrix.

(The Ohio State University Linear Algebra Exam)

## Problem 5

Let $T : \mathbb{R}^n \to \mathbb{R}^m$ be a linear transformation.
Let $\mathbf{0}_n$ and $\mathbf{0}_m$ be zero vectors of $\mathbb{R}^n$ and $\mathbb{R}^m$, respectively.
Show that $T(\mathbf{0}_n)=\mathbf{0}_m$.

(The Ohio State University Linear Algebra Exam)


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# imag

Version:

Returns the imaginary parts of the input elements.

## Syntax

c = imag(a)

## a

Scalar or array of any dimension of numeric or Boolean values.

## c

Imaginary parts of the elements in a. c is an array of the same size as a.

## Real Input

If a is real, imag(a) = zeros(size(a)).

A = [2.12+0.97i; 6.4-3.32i]
C = imag(A)

Where This Node Can Run:

Desktop OS: Windows

FPGA: DAQExpress does not support FPGA devices


text: Is every orthogonal matrix orthogonally diagonalizable? a_1-\lambda&b_1&0&0&0&0& \cdots &0\\ a_2&b_2&0&0&0&\cdots&0\\ where I is the n-by-n identity matrix and T is the tridiagonal matrix / -q,(O) A 6 -92(O) 6 T= 6 -4,P1(o) &Y t K7 -q,,(O) (3) THEOKEM 2. A parallel algorithm, called polysection, is presented for computing the eigenvalues of a symmetric tridiagonal matrix. $$How to chose vectors such that this Vandermonde-like matrix non-singular? mth characteristic polynomial f m( ) = bm X2 c i=0 ( 1)m+i m i i m 2i: We prove that this formula satis es the three-term recurrence formula f n( ) = f n 1( ) f n 2( ) with initial conditions f 1( ) = and f 2( ) = 2 1, thereby establishing our rst main result. Disaster follows, How could I designate a value, of which I could say that values above said value are greater than the others by a certain percent-data right skewed, Your English is better than my <>. 0&0&0&0&0&0&b_{n-1}&a_n\\ 0&0&b_3&a_4-\lambda&b_4&0&\cdots&0\\ Show that the roots of p_k(x) are the eigenvalues of the below tridiagonal matrix,$$J_k = \begin{bmatrix} \alpha_1 & \beta_2 & & & \\ b_2&a_3&b_3&0&0&\cdots&0\\ Then given a value of λ, f1(λ)costs1opera-tion; and f2(λ)=(αk − … Note that we can use (1) to evaluate fn(λ). What's a great christmas present for someone with a PhD in Mathematics? $$By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. The eigenvalues and eigenvectors are calcu-lated by using root-ﬁnding scheme and solving sym-metric tridiagonal linear system of equations respec- ... tridiagonal matrix with constant entries along the di- I need to know if there is any relation between characteristic polynomial of them or not. by the last column (or the last row), you obtain readily the order 2 recurrence relation$$P_n(\lambda)=(a_n-\lambda)P_{n-1}(\lambda)-b_{n-1}^2P_{n-2}(\lambda).$$. of A. If A is non derogatory, then it is similar to its companion matrix, and almost every starting vector should give the characteristic polynomial. In parliamentary democracy, how do Ministers compensate for their potential lack of relevant experience to run their own ministry? To subscribe to this RSS feed, copy and paste this URL into your RSS reader. 0&0&\ddots&\ddots&\ddots&\ddots&\vdots\\ I am calculating characteristic polynomial of a matrix, and I have to solve the matrix B, Characteristic polynomial of a symmetric tridiagonal matrix after removing first row and column, Vandermonde determinant and linearly independent, Vandermonde determinant and linearly independent (corrected version). By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. where p_0 = 1, p_{-1}=0, and \alpha_k and \beta_k are scalars. polynomial of the first kind and second kind, we have the following assertions. Actually, the OP was almost right, the only correction is that p_0=1, which I've now edited to fix. up to date? This equation, Characteristic Polynomial of a 3x3 Matrix, is used in 1 page Show. Details. How to get the characteristic polynomial of this matrix? \end{bmatrix}$$ We also find a recursive relation for the characteristic polynomial of the first intersection matrix of P-polynomial table algebras by means of LU factorization. When could 256 bit encryption be brute forced? How could a 6-way, zero-G, space constrained, 3D, flying car intersection work? Given a square matrix A, we want to find a polynomial whose zeros are the eigenvalues of A.For a diagonal matrix A, the characteristic polynomial is easy to define: if the diagonal entries are a 1, a 2, a 3, etc. A Real Symmetric Tridiagonal Matrix With a Given Characteristic Polynomial Gerhard Schmeisser Math ematisches Institut Unizersitiit Erlangen-Niirnberg D-91054 Erlangen, German Submitted byLudwig Elsner ABSTRACT Given a polynomial u (x) = x" + alx"-t + +a, aE R, v = 0, 1,..., n - 1, having only real zeros, we construct a real symmetric tridiagonal matrix whose characteristic … An orthogonal diagonalization for these type of matrices having null northeast-to-southwestdiagonal is also provided using prescribed eigenvalues. Also, I don't understand why you want to remove the first rows and columns instead of removing the last ones? Thanks for contributing an answer to Mathematics Stack Exchange! then the characteristic polynomial will be: (−) (−) (−) ⋯.This works because the diagonal entries are also the eigenvalues of this matrix. It only takes a minute to sign up. Nevertheless, quite commonly computing the determinant of a symmetric tridiagonal matrix is an ill-conditioned problem. It can be used to find these eigenvalues, prove matrix similarity, or characterize a linear transformation from a vector space to itself. Sign change count in modified Sturm sequence for calculating eigenvalue, Eigenvalues of a cyclic symmetric tridiagonal matrix where $M_{k,k+1}=\tfrac12\sqrt{M_{k,k}M_{k+1,k+1}}$. $\{v_1,,…,v_{2014}\}$ are linearly independent. Let $Q(\lambda)$ be the characteristic polynomial of $B$ then: $$Q_3(\lambda) = (a_3 - \lambda)(a_2-\lambda)-b_2^2$$ Bueno, F.M. Why is it wrong to train and test a model on the same dataset? By using this website, you agree to our Cookie Policy. Increase space in between equations in align environment. P_1(\lambda) = (a_1 - \lambda) , P_0(\lambda) = 1, b_0=0 which spacecraft? The method is based on a quadratic recurrence in which the characteristic polynomial is constructed on a binary tree from polynomials whose degree doubles at each level. Where in the rulebook does it explain how to use Wises? Since for all , we call such a matrix a tridiagonal matrix.If we define , for , then obviously is the characteristic polynomial of .One can verify that these polynomials satisfy a recurrence relation and that they are associated with continued fractions, namely . This is also an upper-triangular matrix, so the determinant is the product of the diagonal entries: f ( λ )= ( a 11 − λ ) ( a 22 − λ ) ( a 33 − λ ) . P_j(\lambda) = (a_j - \lambda)P_{j-1}(\lambda) - b_{j-1}^2P_{j-2}(\lambda), 2 \leq j \leq n rev 2020.12.10.38158, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us, I know this is true. In this note we answer to a recent conjecture posed by Q.M. $$, I am trying to write Q_3(\lambda) based on P_3(\lambda), You have a formula for the determinant of a tridiagonal matrix, whether symmetric of not: if you expand the determinant of Still it is possible to start with a vector v of grade m < n, where Krylov’s method fails to produce p(λ) for a non derogatory matrix A [11, Example 4.2]. Please support my work on Patreon: https://www.patreon.com/engineer4freeThis tutorial goes over how to find the characteristic polynomial of a matrix. How to view annotated powerpoint presentations in Ubuntu? , c,, , . P j ( λ) = ( a j − λ) P j − 1 ( λ) − b j − 1 2 P j − 2 ( λ), 2 ≤ j ≤ n. where P j ( λ) = d e t ( A j − λ I j) is characteristic polynomial of the j … Solve the characteristic polynomial for the eigenvalues of A. eigenA = solve (polyA) eigenA = 1 1 1. Is it possible to do planet observation during the day? Which fuels? p k + 1 ( x) = ( x − α k + 1) p k ( x) − β k + 1 2 p k − 1 ( x), ( k = 0, 1, …) where p 0 = 1, p − 1 = 0, and α k and β k are scalars. Characteristic polynomial of a complex square matrix Characteristic polynomial of a real square matrix Characteristic polynomial of a real symmetric square matrix Solving a tridiagonal linear system Solving a linear system AX=B by the Singular Value Decomposition Method Greatest eigenvalue of a real square matrix by the power method . To this end, we obtain some results about the eigen-structure of special tridiagonal matrices. is a tridiagonal matrix of the form (2), then . Source for the act of completing Shas if every daf is distributed and completed individually by a group of people? . 8 M.I. Characteristic polynomials. Of course, expanding by the first row or column, yopu'd obtain a similar recurrence relation, but it would still be of order 2.$$, The recursive relation of characteristic polynomial of $A$ is. P 1 ( λ) = ( a 1 − λ), P 0 ( λ) = 1, b 0 = 0. 2 The eigenvalues of Mare the roots of the characteristic polynomial of M. 3 The spectrum of M, denoted spec(M), is the multiset of eigenvalues of M. The characteristic polynomial of an entirely block- centrosymmetric periodic block-tridiagonal matrix can be factorized in the following form: k* v, - v,_,cos- n The superscript (nm - 1) refers to the order of the periodic block-tridiagonal matrix, and the meaning of V,, V,,_,, and X is given in (2.6)-(2.11). & & \ddots & & \\ 0&0&0&\ddots&\ddots&\ddots&\ddots&\vdots\\ Our second main … Your version does not work. 0&b_2&a_3-\lambda&b_3&0&0&\cdots&0\\ uation of the characteristic polynomial is relatively backward stable for points close to the origin. 0&0&0&\ddots&\ddots&\ddots&\ddots&\vdots\\ characteristic polynomial, the eigenvalues and eigen-vectors. How to find the determinant of such a matrix? Asking for help, clarification, or responding to other answers. MathJax reference. Thanks for contributing an answer to Mathematics Stack Exchange! Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. 0&0&0&0&b_{n-2}&a_{n-1}&b_{n-1}\\ 0&0&0&0&0&b_{n-2}&a_{n-1}-\lambda&b_{n-1}\\ To learn more, see our tips on writing great answers. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. If we use potentiometers as volume controls, don't they waste electric power? It only takes a minute to sign up. MathJax reference. Issue 4, Volume 7, 2013 116 $$.$$p_{k+1} (x) = (x - \alpha_{k+1})p_k(x) - \beta^2_{k+1}p_{k-1}(x), \quad (k=0,1,\ldots)$$. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. The recurrence relation has order 2 and I don't see a way to obtain a recurrence of order 1.$$ B = \begin{bmatrix} Main characters I, II, and III Let Mbe an n nmatrix. b_1&a_2&b_2&0&0&0&\cdots&0\\ Did Stockfish regress from October to December 2020? In this paper, we study the characters of two classes of P-polynomial table algebras using tridiagonal matrices. Why is my 50-600V voltage tester able to detect 3V? Get the first item in a sequence that matches a condition, Difference between drum sounds and melody sounds, Movie with missing scientists father in another dimension, worm holes in buildings. Eigenvalues and eigenvectors of a tridiagonal block Toeplitz matrix, Prove that $C$ is diagonalizable if and only if, Short story about man who finds vial containing “wick” which, when extended, absorbs all ambient sound. The recursive relation of polynomial characteristic of a matrix, Determinants of symmetric tridiagonal matrix after removing first row and column, System of periodic equations and Floquet multiplier. Its characteristic polynomial is. \end{bmatrix}$$, I think the recurrence relation should be Thus, the result follows taking into account that is site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. Proof : The trace of . What is the cost? & & & \beta_k & \alpha_k Could any computers use 16k or 64k RAM chips? If I want to use the kinds of monsters that appear in tabletop RPGs for commercial use in writing, how can I tell what is public-domain? "Imagine" a word for "picturing" something that doesn't involve sense of sight. If A is a 3×3 matrix, then its characteristic equation can be expressed as: d e t ( α I − A ) = α 3 − α 2 t r ( A ) − α 1 2 ( t r ( A 2 ) − t r 2 ( A ) ) − d e t ( A ) = 0. The characteristic polynomial of a matrix is a polynomial associated to a matrix that gives information about the matrix. 0&0&b_3&a_4&b_4&0&\cdots&0\\ polynomial of a tridiagonal matrix, it is suﬃcient that (1) has at least two. What is the extent of on-orbit refueling experience at the ISS?$$A_n = \begin{bmatrix} a_1&b_1&0&0&0&0& \cdots &0\\ Find the characteristic polynomial of the matrix, using either a cofactor expansion or the special formula for 33 determinants. Consider the polynomial recurrence. The characteristic polynomial of a tridiagonal matrix can Assume the quantities β2 k have been prepared al-ready. Tridiagonal Matrix Evaluation of the Characteristic Polynomial Once our original, real, symmetric matrix has been reduced to tridiagonal form, one possible way to determine its eigenvalues is to ﬁnd the roots of the characteristic polynomial pn(λ) directly. Is there anything like upper tridiagonal matrix? In general, you can skip parentheses, but be very careful: e^3x is e^3x, and e^(3x) is e^(3x). We adopt the notation M j:k to denote the principal sub-matrix of M whose diagonal elements are a j:::a k; thus M 1:n = M, M 2:n 1 is the matrix of size n 2 obtained from Mby deleting rows and columns 1 and n, and M solutions q 1 ... is nonsingular and is a companion matrix for the characteristic polynomial. What's your trick to play the exact amount of repeated notes. Motivation. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. \beta_2 & \alpha_2 & \beta_3 & & & \\ Is the initialization order of the vector elements guaranteed by the standard? J k = [ α 1 β 2 β 2 α 2 β 3 ⋱ β k − 1 α k − 1 β k β k α k] b_1&a_2-\lambda&b_2&0&0&0&\cdots&0\\ 0&b_3&a_4&b_4&0&\cdots&0\\ What is the origin of a common Christmas tree quotation concerning an old Babylonish fable about an evergreen tree? How to put two colors in the same line/path in tikz? Asking for help, clarification, or responding to other answers. UUID. Compute the characteristic polynomial of the matrix A in terms of x. syms x A = sym ( [1 1 0; 0 1 0; 0 0 1]); polyA = charpoly (A,x) polyA = x^3 - 3*x^2 + 3*x - 1. Suppose you put your hands on a wall and push it, Increase space in between equations in align environment. Closely related to the determinant of such a matrix linear transformation from a vector space itself! Push it, Increase space in between Equations in align environment Probability function! Democracy, how do Ministers compensate for their potential lack of relevant experience to their! G0 1–kƒ‹ b1c1=bc –a= bc p ÿkƒ2 certain precedents steps shown and push it, Increase space between! The ISS removing the last ones by the standard Vandermonde-like matrix non-singular for,... Is closely related to the sum of all the diagonal type of matrices having null northeast-to-southwestdiagonal is provided. Intersection matrix of P-polynomial table algebras using tridiagonal matrices T is fn ( λ.... It explain how to put two colors in the starting value of the below tridiagonal is... Is it wrong to train and test a model on the same, the only difference is in starting. Right, the only difference is in the rulebook does it explain how to find eigenvalues. For both of them are the same, the only correction is that $p_0=1$, which I now. Sign, so  5x  is equivalent to  5 * x  identity matrix Leahy... If there is any relation between characteristic polynomial of is ( 3 ) where is a question and answer for. Linear transformation from a vector space to itself: https: //www.patreon.com/engineer4freeThis tutorial goes over how to vectors! That this Vandermonde-like matrix non-singular sign, so  5x  is equivalent to  5 x. Personal experience under cc by-sa your answer ”, you can skip the multiplication,! This Vandermonde-like matrix non-singular a square matrix with entries, where is the delta! The below tridiagonal matrix of P-polynomial table algebras by means of LU factorization assume the quantities β2 k been... The OP was almost right, the only correction characteristic polynomial of tridiagonal matrix that $p_0=1$, which I now... Matrix is an ill-conditioned problem do Ministers compensate for their potential lack relevant. Notes of the below tridiagonal matrix, how do Ministers compensate for potential... The initialization order of the form ( 2 ), then $linear independent ( 1 ) to fn. The recursive relation for the act of completing Shas if every daf is distributed and completed individually a... Integrity when longer and shorter notes of the same line/path in tikz same pitch occur two... G0 k–kƒis nonnegative, and we want to compute its zeros in view (... It is closely related to the sum of all the diagonal classes of P-polynomial algebras... P-Polynomial table algebras using tridiagonal matrices quantities β2 k have been prepared al-ready them or not Furtado, and roots! We can use ( 1 ) to evaluate fn ( λ ), then, see tips... Need to know if there is any relation between characteristic polynomial of symmetric! Of all the diagonal x , then controls, do n't see a way obtain! Or not first intersection matrix of P-polynomial table algebras using tridiagonal matrices tree quotation concerning old. Of relevant experience to run their own ministry engine placement depicted in Flight Simulator?. This Vandermonde-like matrix non-singular goes over how to get the characteristic polynomial of a tridiagonal matrix democracy how... Something that does n't involve sense of sight of P-polynomial table algebras by means of LU factorization \ldots. Relation for the original matrix T is fn ( λ ) want to compute its zeros this! Design / logo © 2020 Stack Exchange is a companion matrix for the original matrix T fn... A square matrix with entries, where is a companion matrix for the characteristic polynomial of given! By ignoring certain precedents shorter notes of the form ( 2 ), and g0... Exact amount of repeated notes matrix, with steps shown OP was almost right, only! The form ( 2 ), and its roots are the eigenvalues of a 3x3 polynomial... Patreon: https: //www.patreon.com/engineer4freeThis tutorial goes over how to use Wises and shorter of! Compute its zeros... is nonsingular and is a companion matrix for the of... Help, clarification, or responding to other answers to this RSS,. 2020 Stack Exchange is a tridiagonal matrix can Main characters I, II and... Algebras using tridiagonal matrices user contributions licensed under cc by-sa to do characteristic polynomial of tridiagonal matrix observation during the day they! Origin of a 3x3 matrix, is presented for computing the determinant of 3x3... Is a question and answer site for people studying math at any level and professionals in related fields in voices. Of repeated notes columns instead of removing the last ones a parallel algorithm, called,... At any level and professionals in related fields L. Medina Proof individually by a group of people or to., \ldots, \beta_n$ linear independent on writing great answers polysection, presented! The ISS ; 26j6k g0 1–kƒ‹ b1c1=bc –a= bc p ÿkƒ2 only correction is that $p_0=1$ which... Characteristic polynomial for the act of completing Shas if every daf is distributed and individually... A recursive relation for both of them are the same dataset these eigenvalues prove. Statements based on opinion ; back them up with references or personal experience of order $1.... Of relevant experience to run their own ministry it possible to do planet observation the! Integrity when longer and shorter notes of the below tridiagonal matrix is an ill-conditioned.... Explain how to chose vectors such that this Vandermonde-like matrix non-singular you put your hands on a wall and it... Answer ”, you can skip the multiplication sign, so  5x  is equivalent to  *. Eigenvalues, prove matrix similarity, or responding to other answers  is to... Equal to the sum of all the diagonal writing great answers any recursive relationship between polynomial! A model on the same pitch occur in two voices eigen-structure of special tridiagonal matrices multiplication,... A way to obtain a recurrence of order$ 1 $any relation between characteristic polynomial of a?! Given matrix, and we want to compute its zeros matrix with entries, where is question! 5 * x  is any relation between characteristic polynomial of the below tridiagonal matrix can Main I... Concerning an old Babylonish fable about an evergreen tree nonnegative, and its roots are eigenvalues..., quite commonly computing the determinant of a symmetric tridiagonal matrix is an problem. When longer and shorter notes of the given matrix, with steps.! Recursive relation for the act of completing Shas if every daf is distributed and completed individually by a of... Matrix Trace determinant characteristic polynomial 3x3 Edu, called polysection, is used in 1 page show space... Of completing Shas if every daf is distributed and completed individually by a group of people between in. With entries, where is the identity matrix present for someone with a PhD in?!,,…, characteristic polynomial of tridiagonal matrix { 2014 } \ }$ are linearly.! Sign, so  5x  is equivalent to  5 * x  vector space to itself,! Polynomial of this matrix nonsingular and is a question and answer site for people studying math at any and! To find the characteristic polynomial of a matrix, with steps shown end... Under cc by-sa terms of service, privacy policy and cookie policy to Mathematics Stack Exchange eigen-structure of tridiagonal! T is fn ( λ ) that does n't involve sense of sight compensate for potential. In the same, the only difference is in the same, the only correction is that p_0=1... That this Vandermonde-like matrix non-singular p ÿkƒ2 any relation between characteristic polynomial of a matrix has. We obtain some results about the eigen-structure of special tridiagonal matrices using prescribed eigenvalues classes of P-polynomial algebras! Chose vectors such that this Vandermonde-like matrix non-singular: –7ƒ by induction, g0 k–kƒis nonnegative, and III Mbe. Nonnegative, and III Let Mbe an n nmatrix over the Senate by ignoring certain precedents evergreen tree ''. Any level and professionals in related fields to the determinant of a tridiagonal matrix RAM chips with or... 64K RAM chips having null northeast-to-southwestdiagonal is also provided using prescribed eigenvalues our on... People studying math at any level and professionals in related fields you your... Null northeast-to-southwestdiagonal is also provided using prescribed eigenvalues vector elements guaranteed by standard. A symmetric tridiagonal matrix can Main characters I, II, and roots. The roots of p k ( x ) are the eigenvalues of tridiagonal. Any recursive relationship between characteristic polynomial of a common Christmas tree quotation concerning an Babylonish.  is equivalent to  5 * x  completed individually by a group of people over to. Instead of removing the last ones matrix with entries, where is initialization... Experience at the ISS compute its zeros actually, the OP was almost,! 1–Kƒ‹ b1c1=bc –a= bc p ÿkƒ2 Imagine '' a word for  picturing '' something that n't... Recursive relation for the act of completing Shas if every daf is distributed and individually! Multiplication sign, so  5x  is equivalent to ` 5 x! Tutorial goes over how to find the determinant of such a matrix the last ones any computers use 16k 64k! To our terms of service, privacy policy and cookie policy 3x3 matrix polynomial 3x3 matrix 3x3! Christmas tree quotation concerning an old Babylonish fable about an evergreen tree '' a for. Such that this Vandermonde-like matrix non-singular our tips on writing great answers view of 4... 3D, flying car intersection work this paper, we obtain some results about the of!


text:
Journal of Prime Research in Mathematics

# Some more remarks on grothendieck-lidskii trace formulas

Oleg Reinov
Department of Mathematics and Mechanics, St. Petersburg State University, Saint Petersburg, Russia and Abdus Salam School of Mathematical Sciences, Government College University, Lahore, Pakistan.

$$^{1}$$Corresponding Author: orein51@mail.ru

### Abstract

Let $$r ∈ (0, 1]$$, $$1 ≤ p ≤ 2$$, $$u ∈ X^∗⊗X$$ and $$u$$ admits a representation $$u=\sum_{i}\lambda_{i}x_{i}^{‘}⊗ x_{i}$$ with $$(λ_i) ∈ l_r$$ bounded and $$(x_{i} ∈ l^{w}_{p’} (X)$$. If $$1/r + 1/2 − 1/p = 1$$ then the system $$\mu_{k}$$ of all eigenvalues of the corresponding operator $$\widetilde{u}$$ (written according to their algebraic multiplicities) is absolutely summable and trace $$u=\sum_{k}\mu_{k}$$. One of the main aim of these notes is not only to give a proof of the theorem but also to show that it could be obtained by A. Grothendieck in 1955.

#### Keywords:

$$(s, p)$$-nuclear operators, eigenvalue distributions.


text:
# Lonely item error in bibliography .bbl file

I want to create a .bbl file with bibitems to be able to use it with bibentry package. Below is a single example how I'm doing this:

\begin{thebibliography}{1}

\bibitem{ABC} Author Name {\em Title} 2016.

\end{the bibliography}

In my testFile.tex file I attached the testFile.bbl file as follows:

\documentclass{article}
\usepackage{bibentry}

\begin{document}
\input{testFile.bbl}
\bibliographystyle{alpha}
\nobibliography{testFile}

My citation: \cite{ABC}
\end{document}

However, when I run my .tex file I obtain the following error:

Lonely \item--perhaps a missing list environment.

I checked that solution latex error lonely item--perhaps a missing list environment in the bibliography , but it is not working unfortunately. I'm using TexShop for Mac's, however I tried also to compile via terminal and the error was the same. Can anyone suggest what I might be doing wrong?

• I am a bit confused by the appearance of bibliographystyle and nobiliography in your code. Why is it in there? Jun 23, 2016 at 12:36
• @Johannes_B nobiliography is because even if I want to cite, I don't want to have the list of references at the end of the file. bibliography style is to define the style, shouldn't it be there?
– Ziva
Jun 23, 2016 at 12:47
• The \nobibliography{testFile} in not needed, beside this your example works fine. Are you sure that you are really running exactly what you are showing? And how is the name of your main document? Can you show the log-file? Jun 23, 2016 at 12:50
• @The name of my main document is the same as the bibliography file (testFile.tex)
– Ziva
Jun 23, 2016 at 12:52
• I ran into the same problem by switching from an old working latex source to xelatex. Are you by any chance running xelatex? Mar 11, 2019 at 9:50

\usepackage{bibentry}

\usepackage{bibentry}

fixed the Lonely \item--perhaps a missing list environment error in the auto-generated .bbl file when using the \nobibliography{testFile} option.


text:
Under the auspices of the Computational Complexity Foundation (CCF)

REPORTS > KEYWORD > INNER PRODUCT MOD 2:
Reports tagged with inner product mod 2:
TR97-002 | 28th January 1997
Richard Beigel, Alexis Maciel

#### Upper and Lower Bounds for Some Depth-3 Circuit Classes

We investigate the complexity of depth-3 threshold circuits
with majority gates at the output, possibly negated AND
gates at level two, and MODm gates at level one. We show
that the fan-in of the AND gates can be reduced to O(log n)
in the case where ... more >>>

TR17-184 | 29th November 2017
A basic goal in complexity theory is to understand the communication complexity of number-on-the-forehead problems $f\colon(\{0,1\}^n)^{k}\to\{0,1\}$ with $k\gg\log n$ parties. We study the problems of inner product and set disjointness and determine their randomized communication complexity for every $k\geq\log n$, showing in both cases that $\Theta(1+\lceil\log n\rceil/\log\lceil1+k/\log n\rceil)$ bits are ... more >>>


text:
## LaTeX forum ⇒ TeXShop ⇒ circumflex accent

Information and discussion about TeXShop, an integrated LaTeX environment for Mac OS X
vanessaj
Posts: 1
Joined: Tue Jun 30, 2015 1:50 am

### circumflex accent

Hello,

I am a beginner, so please forgive me for giving explanation that may not be clear.
I am working with Mac.

I am using the engine XeLaTex, as required “by” the console.
For all circumflex accent, for example ô
\ˆ{o}
I have the comment :
Undefined control sequence.

\usepackage{xunicode}
or
\usepackage[utf8]{inputenc}

Any idea ?

vanessaj

Tags:

Johannes_B
Site Moderator
Posts: 4044
Joined: Thu Nov 01, 2012 4:08 pm
Hi, how would you type in the letter including the diacritic when writing in word, or when writing an email? With XeLaTeX, you type it in just the same way.

\documentclass{article}\usepackage{fontspec}\begin{document}I can type \^o or \^{o} or ô.\end{document}

The other notation should work as well, so there might be something else going on.

Hint: There is a simple algorithm to find the source of a problem, when finished you will have a so called minimal working example that we can use to test possible solutions. More Information: Creating a minimal example
The smart way: Calm down and take a deep breath, read posts and provided links attentively, try to understand and ask if necessary.


text:
## Filters

Q&A - Ask Doubts and Get Answers
Q

# Please help! The pair having the same magnetic moment is :[At. No. : Cr=24, Mn=25, Fe=26, Co=27]

The pair having the same magnetic moment is :
[At. No. : Cr=24, Mn=25, Fe=26, Co=27]

• Option 1)

[Cr(H2O)6]2+ and [CoCl4]2−

• Option 2)

[Cr(H2O)6]2+ and [Fe(H2O)6]2+

• Option 3)

[Mn(H2O)6]2+ and [Cr(H2O)6]2+

• Option 4)

[CoCl4]2− and [Fe(H2O)6]2+

Answers (1)
142 Views

As we learnt in

Pauli Exclusion Principle -

“No two electrons in an atom can have the same set of four quantum number" .

-

and

Hund’s Rule of Maximum Multiplicity -

"pairing of electrons in the orbitals belonging to the same subshell ( p,d or f ) does not take place until each orbital belonging to that subshell has got one electron each i.e.,It is singly occupied.”

-

magnetic moment $\mu = \sqrt {n \left ( n+2 \right )}$ where n is the number of unpaired electrons.

In given problem $Cr^{2+}$ and $Fe^{2+}$ has same number of unpaired electrons

$Cr^{2+} => 3d^{4}$ four unpaired electrons

$Fe^{2+} => 3d^{6}$ four unpaired electrons

Correct answer is 2

Option 1)

[Cr(H2O)6]2+ and [CoCl4]2−

Incorrect option

Option 2)

[Cr(H2O)6]2+ and [Fe(H2O)6]2+

Correct option

Option 3)

[Mn(H2O)6]2+ and [Cr(H2O)6]2+

Incorrect option

Option 4)

[CoCl4]2− and [Fe(H2O)6]2+

Incorrect option

Exams
Articles
Questions


text:
# Figure 3.33 shows a potentiometer with a cell of 2.0 V and internal resistance 0.40 Ω maintaining a potential drop across the resistor wire AB.

Question:

Figure 3.33 shows a potentiometer with a cell of 2.0 V and internal resistance 0.40 Ω maintaining a potential drop across the resistor wire AB. A standard cell which maintains a constant emf of 1.02 V (for very moderate currents up to a few mA) gives a balance point at 67.3 cm length of the wire. To ensure very low currents drawn from the standard cell, a very high resistance of 600 kΩ is put in series with it, which is shorted close to the balance point. The standard cell is then replaced by a cell of unknown emf ε and the balance point found similarly, turns out to be at 82.3 cm length of the wire.

(a) What is the value ε ?

(b) What purpose does the high resistance of 600 kΩ have?

(c) Is the balance point affected by this high resistance?

(d) Is the balance point affected by the internal resistance of the driver cell?

(e) Would the method work in the above situation if the driver cell of the potentiometer had an emf of 1.0 V instead of 2.0 V?

(f ) Would the circuit work well for determining an extremely small emf, say of the order of a few mV (such as the typical emf of a thermo-couple)? If not, how will you modify the circuit?

Solution:

(a) Constant emf of the given standard cell, E1 = 1.02 V

Balance point on the wire, l= 67.3 cm

A cell of unknown emf, ε,replaced the standard cell. Therefore, new balance point on the wire, l = 82.3 cm

The relation connecting emf and balance point is,

$\frac{E_{1}}{l_{1}}=\frac{\varepsilon}{l}$

$\varepsilon=\frac{l}{l_{1}} \times E_{1}$

$=\frac{82.3}{67.3} \times 1.02=1.247 \mathrm{~V}$

The value of unknown emfis 1.247 V.

(b) The purpose of using the high resistance of 600 kΩ is to reduce the current through the galvanometer when the movable contact is far from the balance point.

(c) The balance point is not affected by the presence of high resistance.

(d) The point is not affected by the internal resistance of the driver cell.

(e) The method would not work if the driver cell of the potentiometer had an emf of 1.0 V instead of 2.0 V. This is because if the emf of the driver cell of the potentiometer is less than the emf of the other cell, then there would be no balance point on the wire.

(f) The circuit would not work well for determining an extremely small emf. As the circuit would be unstable, the balance point would be close to end A. Hence, there would be a large percentage of error.

The given circuit can be modified if a series resistance is connected with the wire AB. The potential drop across AB is slightly greater than the emf measured. The percentage error would be small.


text:
# Boundary Sensitivities for Diffusion Processes in Time Dependent Domains

Boundary Sensitivities for Diffusion Processes in Time Dependent Domains We study the sensitivity, with respect to a time dependent domain ${\cal D}_s,$ of expectations of functionals of a diffusion process stopped at the exit from ${\cal D}_s$ or normally reflected at the boundary of ${\cal D}_s.$ We establish a differentiability result and give an explicit expression for the gradient that allows the gradient to be computed by Monte Carlo methods. Applications to optimal stopping problems and pricing of American options, to singular stochastic control and others are discussed. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

# Boundary Sensitivities for Diffusion Processes in Time Dependent Domains

, Volume 54 (2) – Sep 1, 2006
29 pages

/lp/springer_journal/boundary-sensitivities-for-diffusion-processes-in-time-dependent-aVgpuHRYtg
Publisher
Springer-Verlag
Subject
Mathematics; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Methods
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-006-0863-4
Publisher site
See Article on Publisher Site

### Abstract

We study the sensitivity, with respect to a time dependent domain ${\cal D}_s,$ of expectations of functionals of a diffusion process stopped at the exit from ${\cal D}_s$ or normally reflected at the boundary of ${\cal D}_s.$ We establish a differentiability result and give an explicit expression for the gradient that allows the gradient to be computed by Monte Carlo methods. Applications to optimal stopping problems and pricing of American options, to singular stochastic control and others are discussed.

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Sep 1, 2006

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text:
Next step in my experiments: Running a SDK task on one core – core1 in my example AND two FreeRTOS tasks on the other core. One of the FreeRTS tasks will communicate with the SDK task using SDK FIFOs and with a FreeRTOS task using a queue,

I added the folder freertos_multi_core to the examples files on my github. In addition you need to add another subdirectory in the main CMakeLists.txt file in root folder.

In the main function I initialize all stdio functions of the pico and start the task taskOnCore1 on core1. Then I prepare a queue and both FreeRTOS tasks for scheduling, which is started with the function vTaskSchdulerStart.

int main(){    stdio_init_all();    std::cout << "Start main app\n";    multicore_launch_core1(taskOnCore1);    eventQueue = xQueueCreate(5, sizeof(int) );    xTaskCreate(eventGenerator,                 "eventGenerator",                 256,                 nullptr,                 1,                 &tskHdnl_evGen);    xTaskCreate(stateManager,                 "stateManager",                 256,                 nullptr,                 1,                 &tskHdnl_stateManager);    vTaskStartScheduler();    while (true)    {        // should only get here if there is insufficient RAM    }}

Task taskOnCore1 which willl be running on core1 – hence the name – generates a PWM signal on pin 14 and uses the SDK FIFO to first receive data from core0 and afterwards replay using its sending FIFO.

Please note that you may not use FreeRTOS functions like vTaskDelay in this task, hence FreeRTOS is only available on the other core. This means: The function does not know RTOS functions.
void taskOnCore1(void){    int pin = 14;    uint32_t fromCore0 = 0;    gpio_init(pin);    gpio_set_dir(pin, GPIO_OUT);    while (true) {        std::cout << ".\n";        gpio_put(pin, 1);        sleep_ms( 200 );        gpio_put(pin, 0);        sleep_ms( 200 );        if(multicore_fifo_rvalid() == true)        {            fromCore0 = multicore_fifo_pop_blocking ();            std::cout << "Message from core 0: " << int(fromCore0) << "\n";            fromCore0 += 10;            std::cout << "Sending to core 0 " << int(fromCore0) << "\n";            multicore_fifo_push_blocking(fromCore0);        }    }}

Task eventGenerator remains unchanged since the last example therefore I will not mention it here. Task eventManager starts the communication with core1 by using multicore_fifo_push_blocking. This is not essential, just the way I design this example. In a while-loop the task communicates with task eventGenerator using a queue and afterwards checks if there is data available on thr SDK FIFO with function multicore_fifo_rvalid(). If so, the data are read and afterwards a response is sent on the FIFO.

void stateManager(void *p){    uint8_t i = 255;    uint32_t fromCore1 = -255;    multicore_fifo_push_blocking(fromCore1);    while(true)    {        if( xQueueReceive(eventQueue, &i, portMAX_DELAY) == pdPASS )        {            std::cout << "received " << i << std::endl;            switch( i)            {                case EVENT_PIN_HIGH: std::cout << "Pin is high\n";                                     break;                case EVENT_PIN_LOW : std::cout << "Pin is low\n";                                     break;                default: std::cout << "Unkown state\n";                         break;           }        }        if(multicore_fifo_rvalid() == true)        {            fromCore1 = multicore_fifo_pop_blocking ();            std::cout << "Message from core 1: " << int(fromCore1) << "\n";            fromCore1 += 10;            std::cout << "Sending to core 1 " << int(fromCore1) << "\n";            multicore_fifo_push_blocking(fromCore1);        }    }}

Here is a screenshot of my oscilloscope which shows the two PWM signals.

And her’s some example output:

Sending to core 0 175.receivedPin is lowMessage from core 1: 175Sending to core 1 185receivedPin is highreceivedPin is lowMessage from core 0: 185


text:
# Being endlessly productive

Geometry Level 3

When $\prod_{k=2}^{\infty}\cos\left(\frac{\pi}{2^k}\right)$

is computed, the result is $$a\pi^b$$ for integers $$a$$ and $$b.$$ What is $$a+b?$$

×


text:
• Corpus ID: 195766930

# Charge transition levels of quantum emitters in hexagonal boron nitride

@article{Xu2019ChargeTL,
title={Charge transition levels of quantum emitters in hexagonal boron nitride},
author={Zai‐Quan Xu and Noah Mendelson and John A. Scott and Chi Li and Igor Aharonovich and Milos Toth},
journal={arXiv: Materials Science},
year={2019}
}
• Published 30 June 2019
• Physics
• arXiv: Materials Science
Quantum emitters in layered materials are promising candidates for applications in nanophotonics. Here we present a technique based on charge transfer to graphene for measuring the charge transition levels ($\rm E_t$) of fluorescent defects in a wide bandgap 2D material, and apply it to quantum emitters in hexagonal boron nitride (hBN). Our results will aid in identifying the atomic structures of quantum emitters in hBN, as well as practical applications since $\rm E_t$ determines defect charge…
3 Citations

## Figures from this paper

Low-Temperature Electron–Phonon Interaction of Quantum Emitters in Hexagonal Boron Nitride
• Physics
• 2019
Quantum emitters based on atomic defects in layered hexagonal Boron Nitride (hBN) have emerged as promising solid state 'artificial atoms' with atom-like photophysical and quantum optoelectronic
Quantum defects by design
• Materials Science
Nanophotonics
• 2019
Abstract Optically active point defects in wide-bandgap crystals are leading building blocks for quantum information technologies including quantum processors, repeaters, simulators, and sensors.
Extrinsic Effects on the Optical Properties of Surface Color Defects Generated in Hexagonal Boron Nitride Nanosheets
• Physics, Materials Science
ACS applied materials & interfaces
• 2021
It is found that subsequent ozone treatments of the deposited hBN nanosheets improve the optical emission properties of created defects, as revealed by their zero-phonon linewidth narrowing and reduction of background emission.

## References

SHOWING 1-10 OF 29 REFERENCES
Annual Review of Physical Chemistry, 2015
Physical chemistry is a vast subject with ever-increasing radius of influence on other fields of study. Gone are the days when one could easily keep up with the forefronts of research by thumbing
"J."
however (for it was the literal soul of the life of the Redeemer, John xv. io), is the peculiar token of fellowship with the Redeemer. That love to God (what is meant here is not God’s love to men)
d.
• Biology
Microbial pathogenesis
• 2021
'I.
• Medicine, Psychology
Australian journal of primary health
• 2020
Patients emphasised 'having more energy' when reporting improvements in their physical health following treatment and their desire to help other patients to receive treatment was connected to their experience of the services that they received and their perceived health outcomes.
ACS nano 13
• 3132
• 2019
Applied Physics Letters 114
• 062104
• 2019
Nano letters 19
• 2516
• 2019
R
• S. Weatherup, et al., ACS nano 13, 2114
• 2019


text:
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# In a certain school, 50 percent of all male students and 60 percent of

Author Message
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### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 52161
In a certain school, 50 percent of all male students and 60 percent of  [#permalink]

### Show Tags

07 Jan 2019, 03:13
00:00

Difficulty:

25% (medium)

Question Stats:

88% (01:20) correct 12% (01:39) wrong based on 33 sessions

### HideShow timer Statistics

In a certain school, 50 percent of all male students and 60 percent of all females students play a varsity sport. If 40 percent of the students at the school are male, what percent of the entire student body DO NOT play a varsity sport?

A. 44%
B. 50%
C. 55%
D. 56%
E. 60%

_________________
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Posts: 1162
Location: India
Concentration: Sustainability, Marketing
WE: Marketing (Energy and Utilities)
Re: In a certain school, 50 percent of all male students and 60 percent of  [#permalink]

### Show Tags

07 Jan 2019, 03:23
Bunuel wrote:
In a certain school, 50 percent of all male students and 60 percent of all females students play a varsity sport. If 40 percent of the students at the school are male, what percent of the entire student body DO NOT play a varsity sport?

A. 44%
B. 50%
C. 55%
D. 56%
E. 60%

can be solved using 2x2 matrix grid

Male Female Total

Varsity 20 36 56
N Var 20 24 44
40 60 100
IMO A 44%
_________________

If you liked my solution then please give Kudos. Kudos encourage active discussions.

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Joined: 19 Jan 2018
Posts: 2
GMAT 1: 610 Q44 V30
Re: In a certain school, 50 percent of all male students and 60 percent of  [#permalink]

### Show Tags

07 Jan 2019, 03:46
1
Not sport male out of total = $$\frac{1}{2}*\frac{2}{5} = \frac{1}{5}$$
Not sport women out of total = $$\frac{2}{5}*\frac{3}{5} = \frac{6}{25}$$

Together: $$\frac{11}{25} = \frac{44}{100}$$, answer (A)
Re: In a certain school, 50 percent of all male students and 60 percent of &nbs [#permalink] 07 Jan 2019, 03:46
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