text:
1. Trigonometric Integrals

How can I solve this problem:

$\int \frac{dx}{\sqrt{3}\sin(x)+\cos(x)}$

Using the fact that $\cos({\frac{\pi}{6}})=\frac{\sqrt{3}}{2}$ ?

Thank you!

How can I solve this problem:

$\int \frac{dx}{\sqrt{3}\sin(x)+\cos(x)}$

Using the fact that $\cos({\frac{\pi}{6}})=\frac{\sqrt{3}}{2}$ ?

Thank you!
http://www.wolframalpha.com/input/?i...Cos[x]%29]

Click Show Steps

3. Well, thank you, but I already know this site and checked for the integral there - the solution did not involve what I need - I have to use that trigonometric equation somehow.

4. $
\int \frac{dx}{\sqrt{3}\sin x+\cos x} = \int \frac{dx}{\frac{\cos\frac{\pi}{6}}{\sin\frac{\pi}{ 6}}\sin x+\cos x} = \sin\frac{\pi}{6}\int \frac{dx}{\cos\frac{\pi}{6}\sin x + \sin\frac{\pi}{6}\cos x} =
$

$
= \frac{1}{2}\int\frac{dx}{\sin(x+\frac{\pi}{6})}
$

I suppose you can continue from here

5. Originally Posted by Unbeatable0
$
\int \frac{dx}{\sqrt{3}\sin x+\cos x} = \int \frac{dx}{\frac{\cos\frac{\pi}{6}}{\sin\frac{\pi}{ 6}}\sin x+\cos x} = \sin\frac{\pi}{6}\int \frac{dx}{\cos\frac{\pi}{6}\sin x + \sin\frac{\pi}{6}\cos x} =
$

$
= \frac{1}{2}\int\frac{dx}{\sin(x+\frac{\pi}{6})}
$

I suppose you can continue from here
Wow, beautiful, thank you very much !!!


text:
## Algebra 2 (1st Edition)

Published by McDougal Littell

# Chapter 10 Counting Methods and Probability - 10.2 Use Combinations and the Binomial Theorem - Guided Practice for Examples 1, 2, and 3 - Page 691: 5

#### Answer

$5400$

#### Work Step by Step

We can choose $3$ out of the $10$ tragedies and $2$ of the $10$ histories, thus the number of possible sets of plays: $_{10}C_3\cdot_{10}C_2=\frac{10!}{7!3!}\frac{10!}{8!2!}=\frac{10\cdot9\cdot8\cdot7!}{7!3!}\frac{10\cdot9\cdot8!}{8!2!}=120\cdot45=5400$

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.

format: plain


text:
# What is Chirped Pulse Amplification, and why is it important enough to warrant a Nobel Prize?

The 2018 Nobel Prize in Physics was awarded recently, with half going to Arthur Ashkin for his work on optical tweezers and half going to Gérard Mourou and Donna Strickland for developing a technique called "Chirped Pulse Amplification".

In general, optical tweezers are relatively well known, but Chirped Pulse Amplification is less well understood on a broader physics or optics context. While normally the Wikipedia page is a reasonable place to turn to, in this case it's pretty technical and flat, and not particularly informative. So:

• What is Chirped Pulse Amplification? What is the core of the method that really makes it tick?
• What pre-existing problems did its introduction solve?
• What technologies does it enable, and what research fields have become possible because of it?

# The problem

Lasers do all sorts of cool things in research and in applications, and there are many good reasons for it, including their coherence, frequency stability, and controllability, but for some applications, the thing that really matters is raw power.

As a simple example, it had long been understood that if the intensity of light gets high enough, then the assumption of linearity that underpins much of classical optics would break down, and nonlinear optical phenomena like second-harmonic generation would become available, getting light to do all sorts of interesting things. Using incoherent light sources, the required intensities are prohibitively high, but once the laser was invented, it took only one year until the first demonstration of second-harmonic generation, and a few short years after that until third-harmonic generation, a third-order nonlinear process that requires even higher intensities.

Put another way, power matters, and the more intensity you have available, the wider a range of nonlinear optical phenomena will be open for exploration. Because of this, a large fraction of laser science has been focused on increasing the available intensities, generally using pulsed lasers to achieve this and with notable milestones being Q-switching and mode-locking.

However, if you try to push onward with a bigger laser amplifier and more and more power, you are basically destined sooner or later to hit a brick wall, rather brusquely, in the form of catastrophic self-focusing. This is a consequence of yet another nonlinear effect, the Kerr effect, happening inside the laser medium itself. At face value, the Kerr effect looks harmless enough: basically, it says that if the intensity is high enough, the refractive index of the material will rise slightly, in proportion to the intensity: $$n(I) = n_0 + n_2\: I.$$ So, what's the big deal? In short, if you have a laser beam propagating through such a medium, then

• the intensity of the light will be higher in the center, which means that the refractive index will be higher in the center.
• In other words, the material's optical properties will look like those of a convex lens, and it will tend to focus the beam.
• This will tend to make the beam sharper, which will increase the intensity at the center, which will raise the refractive index at the center even higher...
• ... which will then focus the beam even more tightly, leading to higher and higher intensities.

This makes up a positive feedback loop, and if the initial intensity is high enough, the medium is long enough, and there isn't enough initial diffraction to counteract it, then it will spiral out of control and cause catastrophic laser-induced damage in the very medium that you're trying to use to amplify that laser beam. (Moreover, it is quite common, particularly in air, that the laser will diffract on the damaged spot and then re-self-focus a bit further down the line, a phenomenon known as laser filamentation. If you get things just right wrong, this can propagate a failure in the gain medium up to the destruction of an entire beamline.)

Image source

This sounds like a funky mechanism, but it was a huge roadblock for a very long time. If you plot the highest laser intensity available at different times since the invention of the laser, it climbs quickly up during the sixties, and then it hits a wall and stays put for some ten to fifteen years:

Image source

This represents the barrier of Kerr-lens self-focusing, and at the time the only way to overcome it was to build a laser which was physically bigger, to dilute the intensity over more gain medium to try to prevent the problem. Until, that is, Chirped Pulse Amplification came around to solve the problem.

# The solution

At its core, Chirped Pulse Amplification (CPA) works by diluting the light, so that it can be amplified to a larger total power without reaching a dangerous intensity, but it does this stretching in time, i.e. longitudinally along the laser pulse.

The basic sequence consists of four steps:

1. First of all, you start with a short laser pulse that you want to amplify

2. You then stretch it in time, by introducing chirp into the signal: that is, you use some sort of dispersive element, like a prism or a diffraction grating, which decomposes the pulse into all of its constituent colors and sends the longer wavelengths first and the shorter wavelengths last. This will naturally reduce the intensity of the pulse.

(Why "chirp"? because the upward (or downward) sweep of frequencies over the pulse is precisely what gives bird chirps their characteristic sound.)

3. You then pass this lower-intensity pulse through your laser amplifier, which is safe because the instantaneous intensity is below the self-focusing damage threshold of your gain medium.

4. Finally, you pass your pulse through a reversed set of gratings which will undo the relative delay between the longer- and shorter-wavelengths of your pulse, putting them all together into a single pulse of the same shape and length as your original pulse...

... but at the much higher amplified power, and at intensities which would be impossible to achieve safely using direct amplification of the pulse.

The core feature that makes the method tick is the fact that, when done correctly, the stretching of the pulse will completely conserve the coherence between the different frequency components, which means that it is fully reversible and when you add a cancelling chirp the pulse will go back to its initial shape.

Furthermore, the method relies on the fact that stimulated emission will completely duplicate, in a coherent way, the photons that it is amplifying, which means that the photons that are introduced by the amplification will have the same frequency and phase characteristics as the initial pulse, which means that when you remove the chirp from the amplified pulse the added-in photons will also compress into a tight envelope.

# Applications

Like I said at the beginning, CPA is particularly useful in places where raw laser power, and particularly concentrated laser power, is of paramount importance. Here are some examples:

• In the same way that lasers gave us nonlinear optics, CPA has been integral in the development of high-order harmonic generation which has pushed past the second- or third-order harmonics to happily produce tens or hundreds of harmonics. (The current record goes all the way to harmonic 5,000.)

This isn't only 'more', it's qualitatively different: it pushes nonlinear optics to regimes where the usual perturbative expansion completely breaks down, and where it needs to be replaced with a completely new set of tools, which revolve around the so-called three-step model, and which involve a nice and quite particular new interface between classical and quantum mechanics, where trajectories do (sort of) exist but over complex-valued time and space, due to the presence of quantum tunnelling.

• It has also helped push the study of light-matter interaction past that same perturbative limit, giving us the tools to extract electrons from molecules and control them in very precise ways, thereby allowing for the creation of tools like e.g. laser-driven electron diffraction, which can be used to image the shapes of molecules as they undergo bending and other vibrations.

• CPA also underpins several breakthrough measurements which have been touched on previously on this site, including the observation of the time-dependent waveform of a light pulse, itself done using high-order harmonic radiation; the observation of charge oscillations when atoms are placed in excited states, again using HHG; or performing electron holography from an atomic target using electrons pulled from that same atom.

• Of course, all the cool laser-driven QED stuff at the top of that second diagram: if your laser is strong enough that, if you release an electron into the focus, the kinetic energy of its oscillations will exceed $$m_e c^2$$, then you can start to have things like laser-driven pair creation, and all sorts of fun stuff. Some of it is already on the table, some of it is in achievable plans for the future, and all of it is made possible by CPA.

• CPA is also extremely useful in delivering sharply controlled bursts of power to materials. This is extremely useful in laser micromachining, for example, where it is routinely used in e.g. using short laser pulses to etch waveguides into dielectrics, which are then extremely useful for chip-based quantum computation and quantum information processing.

• Similarly, the ability to deliver sharply controlled bursts of power is extremely useful in laser microsurgery, and there are several types of eye surgery that exclusively use CPA pulses to provide sharp 'kicks' of power which perform cleaner incisions.

• On a much larger scale, when you really turn up the power to the maximum, CPA is a vital component of laser wakefield acceleration, which uses the ionized pocket left behind by an intense laser pulse as it travels through a gas to accelerate electrons to energies that would otherwise require an extremely large particle accelerator, but which are now available using a much more modest table-top laser system.

• Comments are not for extended discussion; this conversation has been moved to chat. – David Z Oct 5 '18 at 3:06
• I'm struggling to understand step2 here. "You use some sort of dispersive element, like a prism or a diffraction grating." Don't both of these devices (prism & diffraction grating) split up the lights wave-vectors and send off the waves into different directions? Do you want this? I'm not sure your pictures make that clear. If you don't want that, then a simple glass cube would do the job instead of a prism. (Since then you get the dispersive slowdown effect without the angle changing the directions of the frequencies) – Steven Sagona Mar 10 '20 at 23:21
• @Steven The devices are called pulse stretchers and compressors, if you want to look them up. You use a combination of prisms and diffraction gratings, to send the different frequency components into different directions, and then recombine them into a single beam using a second prism or grating. The key is to do that in such a way that the different components pass through different path lengths, so they arrive at the recombination element at different times. (And, conversely, this can only happen if you spread them out in space.) – Emilio Pisanty Mar 11 '20 at 9:28
• @EmilioPisanty For the record, the pulse stretchers in commercial CPA-based systems can be as simple as a long block of glass. The one I’ve used disperses the ~300-nm-bandwidth seed pulse from 6 fs to around 20 ps with nothing more than that. – Gilbert Oct 10 '20 at 2:45

As an addendum to @EmilioPisanty’s excellent review, I’d just like to mention one more application of CPA lasers, which may be overlooked from a theorist’s perspective:

Ultrafast Spectroscopy

Sometimes lower-order nonlinear processes like second-harmonic generation are enough; you just need them done efficiently for practical purposes.

Lasers based on chirped-pulse amplifiers, which have pulses with energy ~5 mJ and duration ~30 fs are the backbone of all but the simplest ultrafast pump-probe spectroscopic techniques. This is because such pulses allow for efficient low-order nonlinearities such as sum-frequency generation, optical parametric amplification, etc., which have greatly expanded the parameter space of these types of experiments. People are clever, and when given more than enough laser power to work with, they come up with increasingly more complex tricks to ferret out the truth.

Without launching into a review of thousands of ultrafast spectroscopic research results, let it suffice to say that humanity knows a lot more about how the world operates on picosecond and sub-picosecond time scales (in fields as disparate as condensed matter physics and quantum biology) thanks to CPA-based lasers.


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main-content

## Inhaltsverzeichnis

### Hyperbolic Geometry and the Moduli Space of Real Binary Sextics

Abstract
The moduli space of real 6-tuples in ℂP 1 is modeled on a quotient of hyperbolic 3-space by a nonarithmetic lattice in IsomH 3. This is partly an expository note; the first part of it is an introduction to orbifolds and hyperbolic reflection groups.
Daniel Allcock, James A. Carlson, Domingo Toledo

### Gauss’ Hypergeometric Function

Abstract
We give a basic introduction to the properties of Gauss’ hypergeometric functions, with an emphasis on the determination of the monodromy group of the Gaussian hypergeometric equation.
Frits Beukers

### Moduli of K3 Surfaces and Complex Ball Quotients

Abstract
These notes are based on a series of talks given by the authors at the CIMPA Summer School on Algebraic Geometry and Hypergeometric Functions held in Istanbul in Summer of 2005. They provide an introduction to recent work on the complex ball uniformization of the moduli spaces of del Pezzo surfaces, K3 surfaces and algebraic curves of lower genus. We discuss the relationship of these constructions with the Deligne-Mostow theory of periods of hypergeometric differential forms. For convenience to a non-expert reader we include an introduction to the theory of periods of integrals on algebraic varieties with emphasis on abelian varieties and K3 surfaces.
Igor V. Dolgachev, Shigeyuki Kondō

### Macbeaths infinite series of Hurwitz groups

Abstract
In the present paper we will construct an infinite series of so-called Hurwitz groups. One possible way to describe Hurwitz groups is to define them as finite homomorphic images of the Fuchsian triangle group with the signature (2, 3, 7). A reason why Hurwitz groups are interesting lies in the fact, that precisely these groups occur as the automorphism groups of compact Riemann surfaces of genus g > 1, which attain the upper bound 84(g − 1) for the order of the automorphism group. For a long time the only known Hurwitz group was the special linear group PSL2($$\mathbb{F}_7$$), with 168 elements, discovered by F. Klein in 1879, which is the automorphism group of the famous Kleinian quartic. In 1967 Macbeath found an infinite series of Hurwitz groups using group theoretic methods. In this paper we will give an alternative arithmetic construction of this series.
Amir Džambić

### Relative Proportionality on Picard and Hilbert Modular Surfaces

Abstract
We introduce “orbital categories”. The background objects are compactified quotient varieties of bounded symmetric domains $$\mathbb{B}$$ by lattice subgroups of the complex automorphism group of $$\mathbb{B}$$. Additionally, we endow some subvarieties of a given compact complex normal variety V with a natural weight > 1, imitating ramifications. They define an “orbital cycle” Z. The pairs V = (V,Z) are orbital varieties. These objects — also understood as an explicit approach to stacks — allow to introduce “orbital invariants” in a functorial manner. Typical are the orbital categories of Hilbert and Picard modular spaces. From the finite orbital data (e.g. the orbital Apollonius cycle on ℙ2) we read off “orbital Heegner series” as orbital invariants with the help of “orbital intersection theory”. We demonstrate for Hilbert and Picard surface F how their Fourier coefficients can be used to count Shimura curves of given norm on F. On recently discovered orbital projective planes the Shimura curves are joined with well-known classical elliptic modular forms.
Rolf-Peter Holzapfel

### Hypergeometric Functions and Carlitz Differential Equations over Function Fields

Abstract
The paper is a survey of recent results in analysis of additive functions over function fields motivated by applications to various classes of special functions including Thakur’s hypergeometric function. We consider basic notions and results of calculus, analytic theory of differential equations with Carlitz derivatives (including a counterpart of regular singularity), umbral calculus, holonomic modules over the Weyl-Carlitz ring.
Anatoly N. Kochubei

### The Moduli Space of 5 Points on ℙ1 and K3 Surfaces

Abstract
We show that the moduli space of 5 ordered points on ℙ1 is isomorphic to an arithmetic quotient of a complex ball by using the theory of periods of K3 surfaces. We also discuss a relation between our uniformization and the one given by Shimura [S], Terada [Te], Deligne-Mostow [DM].
Shigeyuki Kondō

### Uniformization by Lauricella Functions — An Overview of the Theory of Deligne-Mostow

Abstract
This is a survey of the Deligne-Mostow theory of Lauricella functions, or what almost amounts to the same, of the period map for cyclic coverings of the Riemann sphere.
Eduard Looijenga

Abstract
We survey our construction of invariant functions on the real 3-dimensional hyperbolic space ℍ3 for the Whitehead-link-complement group WGL 2(ℤ[i]) and for a few groups commensurable with W. We make use of theta functions on the bounded symmetric domain $$\mathbb{D}$$ of type I 2,2 and an embedding i : ℍ3$$\mathbb{D}$$. The quotient spaces of ℍ3 by these groups are realized by these invariant functions. We review classical results on the λ-function, the j-function and theta constants on the upper half space; our construction is based on them.
Keiji Matsumoto

### On the Construction of Class Fields by Picard Modular Forms

Abstract
The goal of this article is to construct modular functions living on the complex ball of dimension two such that the values in special points — similar to the elliptic modular function — generate class fields. For this purpose we are well prepared by the papers [5] and [6]. The first one classifies the moduli space of abelian 3-folds with a multiplication by ℚ(i) of type (2, 1) as projective surface. Via Jacobians we connect this Shimura surface with the moduli space of a family of curves of Shimura equation type. Thus we are able to continue the construction of the inverse period map of the family by theta constants given in [6]. Knowing the action of the modular group we reach a modular function j by modular forms with respect to the congruence subgroup of level (1 + i) of the full Picard modular group of Gauß numbers. If τ is the period of a (Jacobian of a) curve with complex multiplication the corresponding moduli field is generated over the rational numbers by j(τ). Hence the values in CM points of this function generate abelian extensions of the associated reflex field.
Thorsten Riedel

### Algebraic Values of Schwarz Triangle Functions

Abstract
We consider Schwarz maps for triangles whose angles are rather general rational multiples of π. Under which conditions can they have algebraic values at algebraic arguments? The answer is based mainly on considerations of complex multiplication of certain Prym varieties in Jacobians of hypergeometric curves. The paper can serve as an introduction to transcendence techniques for hypergeometric functions, but contains also new results and examples.
Hironori Shiga, Jürgen Wolfart

### GKZ Hypergeometric Structures

Abstract
This text is based on lectures by the author in the Summer School Algebraic Geometry and Hypergeometric Functions in Istanbul in June 2005. It gives a review of some of the basic aspects of the theory of hypergeometric structures of Gelfand, Kapranov and Zelevinsky, including Differential Equations, Integrals and Series, with emphasis on the latter. The Secondary Fan is constructed and subsequently used to describe the ‘geography’ of the domains of convergence of the Γ-series. A solution to certain Resonance Problems is presented and applied in the context of Mirror Symmetry. Many examples and some exercises are given throughout the paper.
Jan Stienstra

### Orbifolds and Their Uniformization

Abstract
This is an introduction to complex orbifolds with an emphasis on orbifolds in dimension 2 and covering relations between them.
A. Muhammed Uludağ

### From the Power Function to the Hypergeometric Function

Abstract
The hypergeometric function is a slight generalization of the power fucntion. We will see this by the Schwarz map of the hypergeometric equation focussing on the behavior of this map when the local exponent-differences are purely imaginary
Masaaki Yoshida

### Problem Session

Abstract
This article contains the open problems discussed during the problem session of the CIMPA summer school “Arithmetic and Geometry Around Hypergeometric Functions” held at Galatasaray University, İstanbul, 2005.
Celal Cem Sarıoğlu
Weitere Informationen


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## SugarRainbow Group Title without using a calculator find sin beta and cos beta if tan beta=3/4 and sin beta= >0. one year ago one year ago

1. satellite73 Group Title

think of a 3 - 4 - 5 right triangle, take the appropriate ratios

2. SugarRainbow Group Title

how did you get 3-4-5?

3. satellite73 Group Title

|dw:1347251372575:dw|

4. SugarRainbow Group Title

oh wait isn't tangent like the adjacent angles or something?

5. satellite73 Group Title

you have tangent it $$\frac{3}{4}$$ draw the above triangle, you get the hypotenuse is 5 either by pythagoras or by memory

6. satellite73 Group Title

no, it is not adjacent angles. if you are using right triangle trig, tangent is the length of the "opposite" side over the "adjacent" side

7. SugarRainbow Group Title

okay umm so whats sin?


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# Let f(x)=x^2+2x-15. Determine the vaules of x for which f(x)=-12?

Feb 7, 2018

$x = \left\{- 3 , 1\right\}$

#### Explanation:

Setting $f \left(x\right) = - 12$ gives us:

$- 12 = {x}^{2} + 2 x - 15$

To solve quadratic equations, you need to set the equation equal to zero. By adding 12 to both sides, we get:

$0 = {x}^{2} + 2 x - 3$

From here, we can factor the quadratic to $0 = \left(x + 3\right) \left(x - 1\right)$

Using the Zero Product Property, we can solve the equation by setting each factor equal to zero and solving for x.

$x + 3 = 0 \to x = - 3$

$x - 1 = 0 \to x = 1$

The two solutions are -3 and 1

Feb 7, 2018

x=-3 and x=1 .

#### Explanation:

Put f(x)=-12

$- 12 = {x}^{2} + 2 x - 15$
${x}^{2} + 2 x - 15 + 12 = 0$
${x}^{2} + 2 x - 3 = 0$

Time to factorize now
${x}^{2} + 3 x - x - 3 = 0$
$x \left(x + 3\right) + \left(- 1\right) \left(x + 3\right) = 0$

take x+3 common
$\left(x + 3\right) \left(x - 1\right) = 0$

x=-3 and x=1 .

Feb 7, 2018

$1$ or $- 3$

#### Explanation:

Since $f \left(x\right) = - 12$, then ${x}^{2} + 2 x - 15 = - 12$. Solve this by factoring:

${x}^{2} + 2 x - 3 = 0$

$\left(x - 1\right) \cdot \left(x + 3\right) = 0$

$x - 1 = 0$

$x + 3 = 0$

$x = 1 , - 3$


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A closed-loop system is shown in the figure. The system parameter α is not known. The condition for asymptotic stability of the closed loop system is

Free Practice With Testbook Mock Tests

Options:

1. α < -0.5

2. -0.5 < α < 0.5

3. 0 < α < 0.5

4. α > 0.5

Correct Answer: Option 4 (Solution Below)

This question was previously asked in

GATE IN 2017 Official Paper

Solution:

We know that characteristic equation

1 + G(s) H(s) = 0

$$1 + \frac{{s + \alpha }}{{{s^3} + 2\alpha {s^2} + \alpha s + 1}} = 0$$

s3 + 2αs2 + αs + 1 + s + α = 0

s3 + 2αs2 + (1 + α)s + (1 + α) = 0

Now apply R.H. criteria

$$\begin{array}{*{20}{c}} {{s^3}}&1&{1 + \alpha }\\ {{s^2}}&{2\alpha }&{1 + \alpha }\\ {{s^1}}&{\frac{{2\alpha \left( {1 + \alpha } \right) - \left( {1 + \alpha } \right)}}{{2\alpha }}}&0\\ {{s^0}}&{\alpha + 1}&0 \end{array}$$

For sufficient condition

$$\frac{{2\alpha \left( {1 + \alpha } \right) - \left( {1 + \alpha } \right)}}{{2\alpha }} > 0$$

2α > 1

∴ α > 0.5


text:
# Hadoop reducer error: “Shuffle Error: Exceeded the abort failure limit; bailing-out”

I've got a hadoop 0.20 map/reduce job that used to run just fine. In the last few days, it's getting stuck in the reduce phase at 16.66%, and I'm seeing the following error when I look at the reduce task in the jobtracker;

Shuffle Error: Exceeded the abort failure limit; bailing-out.

Can anyone tell me what that means, and maybe point me in the right direction so I can figure out how to fix this?

-
Is there anymore information in the reduce task logs for the failed attempts? –  Chris White Jan 20 '13 at 16:23
There doesn't seem to be a separate log for the reduce task - it's all in 'syslog' files in /var/hadoopdata/mapreduce/userlogs/job_[whatever] But, looking in there, I can see some errors where it's trying to contact the other node using a host name which doesn't resolve. I'm not sure why that would result in the error I'm getting, but it's wrong so I'll try fixing it and seeing if that helps. –  digitalronin Jan 20 '13 at 16:32

This error corresponds to the maximum number of times a reducer tries to fetch a map output before it reports it and maps to the property mapreduce.reduce.shuffle.maxfetchfailures.


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How do you factor by grouping x^3+4x^2+8x+32?

May 29, 2018

$\left({x}^{2} + 8\right) \left(x + 4\right)$

Explanation:

Factor each half of the expression individually first:

$\implies {x}^{2} \textcolor{b l u e}{\left(x + 4\right)} + 8 \textcolor{b l u e}{\left(x + 4\right)}$

Hence $\left(x + 4\right)$ is common in each term, factor this out:

$\implies \textcolor{b l u e}{\left(x + 4\right)} \left[{x}^{2} + 8\right]$

color(red)(= (x^2 + 8 ) ( x + 4)

In terms of real numbers, this is factorised

May 29, 2018

$\left(x + 4\right) \left({x}^{2} + 8\right)$

Explanation:

$= \textcolor{red}{{x}^{2}} \left(x + 4\right) \textcolor{red}{+ 8} \left(x + 4\right)$

$\text{take out the "color(blue)"common factor } \left(x + 4\right)$

$= \left(x + 4\right) \left(\textcolor{red}{{x}^{2} + 8}\right)$

${x}^{3} + 4 {x}^{2} + 8 x + 32 = \left(x + 4\right) \left({x}^{2} + 8\right)$


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# Ideas on using API.AI to report information from BigQuery

If you’ve been paying attention to what I’ve written about lately, you know that I love automation and I love data!

I know that automation is a double edged sword, but I’ve always said that my goal was to automate myself out of a job.

If you’re the one building the tech to do it, that’s great. If you’re the one that gets phased out, that’s not good for anyone. But this post isn’t about the machine uprising. This post is about something fun!!

When we first started using BigQuery at work, one of the ideas I tossed out was the idea that in the future, we may sit in a meeting and be able to ask a device a question and get an immediate response. I didn’t think much about it afterwards then I got a Google Home for Christmas, SQUEE!!!

This little device was amazing. It just “knew” how to respond to so many different versions of the same question. How did it know? After doing some research online, I found that you can build your own conversation actions and that got my creative juices flowing. For example, could you build a way to query a database to get a result?

I found a service called API.AI that lets you build tools to do things exactly like this. So let’s play this out. In an easy example, let’s pretend that we’ve asked the question

“How much money did we make last year?”

Right off the bat, we know three things:

1. We need to sum something
2. What we need to sum is money
3. The time that is needed is last year

That’s a pretty simple SQL that could be written in BigQuery. Let’s take a look at that:

SELECT
SUM(revenue) revenueTotal
FROM
[dataset.table]
WHERE

In this toy example, we didn’t infer if the person asking the question meant Gross or Net Revenue. But with API.AI, we might set up some logic to respond to the person and ask them if they meant gross or net revenue. Then, depending on their answer, we could query Google BigQuery with different queries. You could also change the logic to update the WHERE clause to whatever is applicable to the question.

There are so many other questions that you might want to ask your assistant. For example:

1. What’s our most popular selling product?
• With a reply from the assistant: By units sold or revenue?
2. How did our Adwords marketing campaign do last week?
• With a reply from the assistant: Would you like new customer count or revenue?

## Next Steps

On the surface this might seem pretty rudimentary so you may naturally try to figure out what’s next? Ideally you would want your application to not only report but to also runs simulations, make a prediction or tell you to probabilities for what your next campaign or product might be. This seems simple and straightforward but you can probably guess that something like this might be much more complex.

Until then we just have to keep taking baby steps and innovating as much as possible!


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# The change in the value of g at a height h above the surface of the earth is the same as at a depth d below the surface of earth. When both d an

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in Physics
The change in the value of g at a height h above the surface of the earth is the same as at a depth d below the surface of earth. When both d and h are much smaller than the radius of earth, then which one of the following is correct?
A. x=h
B. x=2h
C. x=h/2
D. x=h^(2)

## 1 Answer

0 votes
by (67.4k points)
Correct Answer - B


text:
# Dixon's Theorem to probabilistically bound largest factor of N

I have recently decided to read up on the current integer factorization algorithms. When looking into some of the algorithms, I came across the following statement:

Say that p is the smallest prime factor of n. By Dixon's Theorem, the probability that the largest factor of n is less than $(p)^\epsilon$ is roughly $\epsilon ^ {- \epsilon}$.

But when I hear Dixon's Theorem, I think of this. I currently don't see how Dixon's Theorem yields such a probabilistic bound. Can you help me with that?

And the disclaimer - I am not currently in any class, so this was not assigned to me, it is not a homework problem, and I will in no way be receiving any sort of academic credit for this work.

-
Congrats on the 10K :-) –  Asaf Karagila Oct 14 '11 at 8:31
@Asaf: Thank you! –  mixedmath Oct 14 '11 at 13:38

Wrong Dixon! Please look at this link to get a start. Unfortunately, the relevant original paper (Mathematics of Computation 36, 1981, pp 255-260) is behind a pay wall unless you are at a subscribing university.

-
–  t.b. Jul 25 '11 at 11:46
@Theo Excellent. I knew I was going no where fast. Thank you for that. –  mixedmath Jul 25 '11 at 15:32


text:
# zbMATH — the first resource for mathematics

## Sussmann, Hector J.

Compute Distance To:
 Author ID: sussmann.hector-j Published as: Sussman, H. J.; Sussman, Hector J.; Sussman, Héctor J.; Sussmann, H.; Sussmann, H. J.; Sussmann, Hector J.; Sussmann, Héctor; Sussmann, Héctor J.
 Documents Indexed: 122 Publications since 1970, including 7 Books
all top 5

#### Co-Authors

 81 single-authored 7 Sontag, Eduardo D. 4 Liu, Wensheng 3 Jurdjevic, Velimir 3 Kokotovic, Petar V. 2 Brockett, Roger Ware 2 Ferreyra, Guillermo S. 2 Hazewinkel, Michiel 2 Levitt, Norman 2 Marcus, Steven I. 2 Nerurkar, Mahesh G. 2 Willems, Jan Camiel 2 Yang, Yudi 1 Baillieul, John B. 1 Bumby, Richard T. 1 Chitour, Yacine 1 Fliess, Michel 1 Gardner, Robert Brown 1 Grasse, Kevin A. 1 Hermes, Henry 1 Kawski, Matthias 1 Lafferriere, Gerardo 1 Łojasiewicz, Stanisław jun. 1 Millman, Richard S. 1 Mordukhovich, Boris S. 1 Piccoli, Benedetto 1 Puel, Jean-Pierre 1 Rampazzo, Franco 1 Saberi, Ali 1 Sastry, Shankar S. 1 Schättler, Heinz Michael 1 Vasconcelos, Wolmer V. 1 Zahler, Raphael S.
all top 5

#### Serials

 11 SIAM Journal on Control and Optimization 7 Journal of Differential Equations 5 Systems & Control Letters 3 IEEE Transactions on Automatic Control 3 Bulletin of the American Mathematical Society 2 Czechoslovak Mathematical Journal 2 Mathematical Systems Theory 2 Proceedings of the American Mathematical Society 2 Synthese 2 Transactions of the American Mathematical Society 2 SIAM Journal on Control 2 The IMA Volumes in Mathematics and its Applications 1 Communications on Pure and Applied Mathematics 1 IEEE Transactions on Information Theory 1 Israel Journal of Mathematics 1 Journal of the Franklin Institute 1 Stochastics 1 ZAMP. Zeitschrift für angewandte Mathematik und Physik 1 The Annals of Probability 1 Canadian Journal of Mathematics 1 Journal of Differential Geometry 1 Journal of Optimization Theory and Applications 1 Journal of Pure and Applied Algebra 1 Memoirs of the American Mathematical Society 1 Boletim da Sociedade Portuguesa de Matemática 1 Revista Matemática Iberoamericana 1 Complex Systems 1 Set-Valued Analysis 1 Proceedings of Symposia in Pure Mathematics 1 Progress in Mathematics 1 Pure and Applied Mathematics, Marcel Dekker 1 SMF Journée Annuelle 1 Journal of Modern Dynamics
all top 5

#### Fields

 74 Systems theory; control (93-XX) 47 Calculus of variations and optimal control; optimization (49-XX) 21 Global analysis, analysis on manifolds (58-XX) 15 Manifolds and cell complexes (57-XX) 13 Ordinary differential equations (34-XX) 11 Probability theory and stochastic processes (60-XX) 9 Nonassociative rings and algebras (17-XX) 8 Dynamical systems and ergodic theory (37-XX) 7 General and overarching topics; collections (00-XX) 7 Several complex variables and analytic spaces (32-XX) 6 Differential geometry (53-XX) 6 Mechanics of particles and systems (70-XX) 5 Operator theory (47-XX) 5 Statistics (62-XX) 3 History and biography (01-XX) 3 Real functions (26-XX) 3 Biology and other natural sciences (92-XX) 2 Topological groups, Lie groups (22-XX) 2 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 2 Information and communication theory, circuits (94-XX) 1 Commutative algebra (13-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Associative rings and algebras (16-XX) 1 Partial differential equations (35-XX) 1 Approximations and expansions (41-XX) 1 Functional analysis (46-XX) 1 Algebraic topology (55-XX) 1 Computer science (68-XX) 1 Operations research, mathematical programming (90-XX)

#### Citations contained in zbMATH Open

107 Publications have been cited 2,323 times in 1,702 Documents Cited by Year
Orbits of families of vector fields and integrability of distributions. Zbl 0274.58002
Sussmann, Hector J.
1973
Controllability of nonlinear systems. Zbl 0242.49040
Sussmann, Hector J.; Judjevic, Velimir
1972
A general result on the stabilization of linear systems using bounded controls. Zbl 0811.93046
Sussmann, Héctor J.; Sontag, Eduardo D.; Yang, Yudi
1994
On the gap between deterministic and stochastic ordinary differential equations. Zbl 0391.60056
Sussmann, Hector J.
1978
Control systems on Lie groups. Zbl 0237.93027
Jurdjevic, Velimir; Sussmann, Hector J.
1972
A general theorem on local controllability. Zbl 0629.93012
Sussmann, H. J.
1987
The peaking phenomenon and the global stabilization of nonlinear systems. Zbl 0749.93070
Sussmann, H. J.; Kokotovic, P. V.
1991
Global stabilization of partially linear composite systems. Zbl 0719.93071
Saberi, A.; Kokotovic, P. V.; Sussmann, H. J.
1990
Subanalytic sets and feedback control. Zbl 0407.93010
Sussmann, Hector J.
1979
A positive real condition for global stabilization of nonlinear systems. Zbl 0684.93066
Kokotovic, P. V.; Sussmann, H. J.
1989
Lie brackets and local controllability: A sufficient condition for scalar-input systems. Zbl 0523.49026
Sussmann, Hector J.
1983
Existence and uniqueness of minimal realizations of nonlinear systems. Zbl 0354.93017
Sussman, Hector J.
1977
Remarks on the pole-shifting problem over rings. Zbl 0455.15009
Bumby, R.; Sontag, E. D.; Sussmann, H. J.; Vasconcelos, W.
1981
Shortest paths for sub-Riemannian metrics on rank-two distributions. Zbl 0843.53038
Liu, Wensheng; Sussman, Héctor J.
1995
Lie brackets, real analyticity and geometric control. Zbl 0545.93002
Sussmann, H. J.
1983
Some properties of vector field systems that are not altered by small perturbations. Zbl 0346.49036
Sussmann, Hector J.
1976
Single-input observability of continuous-time systems. Zbl 0422.93019
Sussmann, Hector J.
1979
Further comments on the stabilizability of the angular velocity of a rigid body. Zbl 0675.93064
Sontag, Eduardo D.; Sussmann, Héctor J.
1989
A sufficient condition for local controllability. Zbl 0391.93004
Sussmann, Hector J.
1978
Regular synthesis and sufficiency conditions for optimality. Zbl 0961.93014
Piccoli, Benedetto; Sussmann, Héctor J.
2000
The structure of time-optimal trajectories for single-input systems in the plane: The general real analytic case. Zbl 0664.93034
Sussmann, H. J.
1987
An interpretation of stochastic differential equations as ordinary differential equations which depend on the sample point. Zbl 0367.60060
Sussmann, Hector J.
1977
A bang-bang theorem with bounds on the number of switchings. Zbl 0427.49008
Sussmann, Hector J.
1979
A generalization of the closed subgroup theorem to quotients of arbitrary manifolds. Zbl 0342.58004
Sussmann, Hector J.
1975
A nonsmooth hybrid maximum principle. Zbl 0967.49016
Sussmann, Héctor J.
1999
Global stabilization of linear discrete-time systems with bounded feedback. Zbl 0901.93053
Yang, Yudi; Sontag, Eduardo D.; Sussmann, Hector J.
1997
Commutators of flow maps of nonsmooth vector fields. Zbl 1137.37010
Rampazzo, Franco; Sussmann, Héctor J.
2007
An introduction to the coordinate-free maximum principle. Zbl 0925.93135
Sussmann, Hector J.
1998
Real-analytic desingularization and subanalytic sets: An elementary approach. Zbl 0696.32005
Sussmann, H. J.
1990
Orbits of families of vector fields and integrability of systems with singularities. Zbl 0255.58001
Sussmann, Hector J.
1973
Regular synthesis for time-optimal control of single-input real analytic systems in the plane. Zbl 0701.93035
Sussmann, H. J.
1987
On controllability by means of two vector fields. Zbl 0313.93006
Levitt, Norman; Sussmann, Hector J.
1975
The ”bang-bang” problem for certain control systems in GL(n,R). Zbl 0242.49041
Sussmann, Hector J.
1972
Noncommutative power series and formal Lie-algebraic techniques in nonlinear control theory. Zbl 0919.93035
Kawski, Matthias; Sussmann, Héctor J.
1997
A differential geometric approach to motion planning. Zbl 0788.70017
Lafferriere, Gerardo; Sussmann, Héctor J.
1992
Global controllability by nice controls. Zbl 0703.93014
Grasse, Kevin A.; Sussmann, H. J.
1990
An extension of a theorem of Nagano on transitive Lie algebras. Zbl 0301.58003
Sussmann, Hector J.
1974
Geometry and optimal control. Zbl 1067.49500
Sussmann, H. J.
1998
A cornucopia of four-dimensional abnormal sub-Riemannian minimizers. Zbl 0862.53033
Sussmann, Héctor J.
1996
A product expansion for the Chen series. Zbl 0619.93021
Sussmann, H. J.
1986
Minimal realizations and canonical forms for bilinear systems. Zbl 0335.93021
Sussmann, Hector J.
1976
Regular synthesis for time-optimal control of single-input real analytic systems in the plane. Zbl 0696.93026
Sussmann, H. J.
1987
Nonexistence of finite-dimensional filters for conditional statistics of the cubic sensor problem. Zbl 0539.93077
Hazewinkel, M.; Marcus, S. I.; Sussmann, H. J.
1983
Bounds on the number of switchings for trajectories of piecewise analytic vector fields. Zbl 0488.34033
Sussmann, Hector J.
1982
Symmetries and integrals of motion in optimal control. Zbl 0891.49011
Sussmann, Hector J.
1995
Limitations on the stabilizability of globally minimum phase systems. Zbl 0708.93070
Sussmann, H. J.
1990
Backpropagation can give rise to spurious local minima even for networks without hidden layers. Zbl 0732.68091
Sontag, Eduardo D.; Sussmann, Héctor J.
1989
Catastrophe theory as applied to the social and biological sciences: A critique. Zbl 0389.92002
Sussmann, Hector J.; Zahler, Raphael S.
1978
300 years of optimal control: From the brachystochrone to the maximum principle. Zbl 1014.49001
Sussmann, Hector J.; Willems, Jan C.
2001
Line-integral estimates and motion planning using the continuation method. Zbl 0946.70003
Chitour, Yacine; Sussmann, Héctor J.
1998
Continuous dependence with respect to the input of trajectories of control-affine systems. Zbl 0919.34003
Liu, Wensheng; Sussmann, Héctor J.
1999
New differential geometric methods in nonholonomic path finding. Zbl 0777.93014
Sussmann, Héctor J.
1992
Some examples of reachable sets and optimal cost functions that fail to be subanalytic. Zbl 0569.49029
Lojasiewicz, Stanisław jun.; Sussmann, H. J.
1985
Semigroup representations, bilinear approximation of input-output maps, and generalized inputs. Zbl 0353.93025
Sussmann, Hector J.
1976
New theories of set-valued differentials and new versions of the maximum principle of optimal control theory. Zbl 0998.93022
Sussmann, Héctor J.
2001
Abnormal sub-Riemannian minimizers. Zbl 0823.49026
Liu, Wensheng; Sussmann, Héctor J.
1994
Optimal control theory and piecewise analyticity of the distance function for some real-analytic sub-Riemannian metrics. Zbl 0772.49019
Sussmann, H. J.
1992
Small-time local controllability and continuity of the optimal time function for linear systems. Zbl 0596.93012
Sussmann, H. J.
1987
A weak regularity theorem for real analytic optimal control problems. Zbl 0638.49018
Sussmann, H. J.
1986
On quotients of manifolds: A generalization of the closed subgroup theorem. Zbl 0301.58002
Sussmann, Hector J.
1974
Limits of the Wong-Zakai type with a modified drift term. Zbl 0733.60082
Sussmann, Héctor J.
1991
Envelopes, conjugate points, and optimal bang-bang extremals. Zbl 0619.49017
Sussmann, H. J.
1986
A Lie-Volterra expansion for nonlinear systems. Zbl 0538.93032
Sussmann, H. J.
1984
Tangent bundles of homogeneous spaces are lomogeneous spaces. Zbl 0256.53039
Brockett, R. W.; Sussmann, H. J.
1972
Control systems on Lie groups. Zbl 0227.93011
Jurdjevic, Velimir; Sussmann, Héctor
1972
Construction of ergodic cocycles that are fundamental solutions to linear systems of a special form. Zbl 1135.37006
Nerurkar, Mahesh G.; Sussmann, Héctor J.
2007
Construction of minimal cocycles arising from specific differential equations. Zbl 0894.93004
Nerurkar, Mahesh G.; Sussmann, Héctor J.
1997
Some recent results on the maximum principle of optimal control theory. Zbl 0883.49020
Sussmann, Hector J.
1997
A strong version of the Lojasiewicz maximum principle. Zbl 0816.49018
Sussmann, H. J.
1994
Synthesis, presynthesis, sufficient conditions for optimality and subanalytic sets. Zbl 0712.49015
Sussmann, H. J.
1990
Product expansions of exponential Lie series and the discretization of stochastic differential equations. Zbl 0651.60065
Sussmann, H. J.
1988
Nonexistence of finite dimensional filters for conditional statistics of the cubic sensor problem. Zbl 0538.93061
Hazewinkel, M.; Marcus, S. I.; Sussmann, H. J.
1984
Minimal realizations of nonlinear systems. Zbl 0291.93007
Sussmann, Hector J.
1973
Smooth distributions are globally finitely spanned. Zbl 1206.58002
Sussmann, Hector J.
2008
Complete controllability of continuous-time recurrent neural networks. Zbl 0901.93005
Sontag, Eduardo; Sussmann, Héctor J.
1997
General classes of control-Lyapunov functions. Zbl 0854.93131
Sontag, Eduardo D.; Sussmann, Héctor J.
1996
Time-optimal control in the plane. Zbl 0516.93025
Sussmann, H. J.
1982
Analytic stratifications and control theory. Zbl 0499.93023
Sussmann, Hector J.
1980
Generic single-input observability of continuous-time polynomial systems. Zbl 0428.93030
Sussmann, Hector J.
1979
Catastrophe theory. Zbl 0309.58008
Sussmann, Hector J.
1975
The control problem $$\dot x$$ = (A(1 - $$u$$) + B$$u$$)$$x$$: A comment on an article by J. Kucera. Zbl 0251.49014
Sussmann, Hector J.
1972
High-order open mapping theorems. Zbl 1161.58301
Sussmann, Héctor J.
2003
Needle variations and almost lower semicontinuous differential inclusions. Zbl 1013.49021
Sussmann, Héctor J.
2002
Multidifferential calculus: Chain rule, open mapping and transversal intersection theorems. Zbl 0981.49013
Sussmann, Héctor J.
1998
Lie bracket extensions and averaging: The single-bracket case. Zbl 0786.70009
Sussmann, Héctor J.; Liu, Wensheng
1992
Recent developments in the regularity theory of optimal trajectories. Zbl 0649.49003
Sussmann, H. J.
1987
On the regularity of optimal controls. Zbl 0636.49009
Schättler, Heinz M.; Sussmann, Héctor J.
1987
Differential geometric control theory. Proceedings of the Conference held at Michigan Technological University, June 28 - July 2, 1982. Zbl 0503.00014
Brockett, Roger W. (ed.); Millman, Richard S. (ed.); Sussmann, Hector J. (ed.)
1983
Approximate finite-dimensional filters for some nonlinear problems. Zbl 0494.93048
Sussmann, Hector J.
1982
On the number of directions needed to achieve controllability. Zbl 0268.93006
Sussmann, Héctor J.
1974
The control problem $$\dot x$$ = A($$u$$)$$x$$. Zbl 0251.49015
Sussmann, Hector J.
1972
Set separation, approximating multicones, and the Lipschitz maximum principle. Zbl 1354.49046
Sussmann, H. J.
2007
The brachistochrone problem and modern control theory. Zbl 1047.93001
Sussmann, Héctor J.; Willems, Jan C.
2002
Some optimal control applications of real-analytic stratifications and desingularization. Zbl 1050.49500
Sussmann, Héctor J.
1998
Nonlinear controllability and optimal control. Based on lectures of a workshop on finite dimensional controllability and optimal control held at Rutgers University, New Brunswick, NJ (USA) on May 18 to 22, 1987. Zbl 0699.00040
Sussmann, H. J. (ed.)
1990
Some recent results on pathwise nonlinear filtering. Zbl 0538.93063
Sussmann, H. J.
1984
The generalized pursuit problem where the pursuer uses band-bang controls. Zbl 0374.90094
Levitt, N.; Sussmann, H.
1977
A Pontryagin maximum principle for systems of flows. Zbl 1201.49024
Sussmann, Héctor J.
2008
Generalized differentials, variational generators, and the maximum principle with state constraints. Zbl 1156.49003
Sussmann, H. J.
2008
Recent results on optimal curves. Zbl 1020.49020
Sussmann, Héctor J.
2000
Smooth distributions are globally finitely spanned. Zbl 1206.58002
Sussmann, Hector J.
2008
A Pontryagin maximum principle for systems of flows. Zbl 1201.49024
Sussmann, Héctor J.
2008
Generalized differentials, variational generators, and the maximum principle with state constraints. Zbl 1156.49003
Sussmann, H. J.
2008
Commutators of flow maps of nonsmooth vector fields. Zbl 1137.37010
Rampazzo, Franco; Sussmann, Héctor J.
2007
Construction of ergodic cocycles that are fundamental solutions to linear systems of a special form. Zbl 1135.37006
Nerurkar, Mahesh G.; Sussmann, Héctor J.
2007
Set separation, approximating multicones, and the Lipschitz maximum principle. Zbl 1354.49046
Sussmann, H. J.
2007
High-order open mapping theorems. Zbl 1161.58301
Sussmann, Héctor J.
2003
Needle variations and almost lower semicontinuous differential inclusions. Zbl 1013.49021
Sussmann, Héctor J.
2002
The brachistochrone problem and modern control theory. Zbl 1047.93001
Sussmann, Héctor J.; Willems, Jan C.
2002
300 years of optimal control: From the brachystochrone to the maximum principle. Zbl 1014.49001
Sussmann, Hector J.; Willems, Jan C.
2001
New theories of set-valued differentials and new versions of the maximum principle of optimal control theory. Zbl 0998.93022
Sussmann, Héctor J.
2001
Regular synthesis and sufficiency conditions for optimality. Zbl 0961.93014
Piccoli, Benedetto; Sussmann, Héctor J.
2000
Recent results on optimal curves. Zbl 1020.49020
Sussmann, Héctor J.
2000
A nonsmooth hybrid maximum principle. Zbl 0967.49016
Sussmann, Héctor J.
1999
Continuous dependence with respect to the input of trajectories of control-affine systems. Zbl 0919.34003
Liu, Wensheng; Sussmann, Héctor J.
1999
An introduction to the coordinate-free maximum principle. Zbl 0925.93135
Sussmann, Hector J.
1998
Geometry and optimal control. Zbl 1067.49500
Sussmann, H. J.
1998
Line-integral estimates and motion planning using the continuation method. Zbl 0946.70003
Chitour, Yacine; Sussmann, Héctor J.
1998
Multidifferential calculus: Chain rule, open mapping and transversal intersection theorems. Zbl 0981.49013
Sussmann, Héctor J.
1998
Some optimal control applications of real-analytic stratifications and desingularization. Zbl 1050.49500
Sussmann, Héctor J.
1998
Essays on mathematical robots. Proceedings of a workshop, Minneapolis, MN, USA, January 25-29, 1993. Zbl 0903.00092
Baillieul, John (ed.); Sastry, Shankar S. (ed.); Sussmann, Hector J. (ed.)
1998
Global stabilization of linear discrete-time systems with bounded feedback. Zbl 0901.93053
Yang, Yudi; Sontag, Eduardo D.; Sussmann, Hector J.
1997
Noncommutative power series and formal Lie-algebraic techniques in nonlinear control theory. Zbl 0919.93035
Kawski, Matthias; Sussmann, Héctor J.
1997
Construction of minimal cocycles arising from specific differential equations. Zbl 0894.93004
Nerurkar, Mahesh G.; Sussmann, Héctor J.
1997
Some recent results on the maximum principle of optimal control theory. Zbl 0883.49020
Sussmann, Hector J.
1997
Complete controllability of continuous-time recurrent neural networks. Zbl 0901.93005
Sontag, Eduardo; Sussmann, Héctor J.
1997
A cornucopia of four-dimensional abnormal sub-Riemannian minimizers. Zbl 0862.53033
Sussmann, Héctor J.
1996
General classes of control-Lyapunov functions. Zbl 0854.93131
Sontag, Eduardo D.; Sussmann, Héctor J.
1996
Shortest paths for sub-Riemannian metrics on rank-two distributions. Zbl 0843.53038
Liu, Wensheng; Sussman, Héctor J.
1995
Symmetries and integrals of motion in optimal control. Zbl 0891.49011
Sussmann, Hector J.
1995
A general result on the stabilization of linear systems using bounded controls. Zbl 0811.93046
Sussmann, Héctor J.; Sontag, Eduardo D.; Yang, Yudi
1994
Abnormal sub-Riemannian minimizers. Zbl 0823.49026
Liu, Wensheng; Sussmann, Héctor J.
1994
A strong version of the Lojasiewicz maximum principle. Zbl 0816.49018
Sussmann, H. J.
1994
A differential geometric approach to motion planning. Zbl 0788.70017
Lafferriere, Gerardo; Sussmann, Héctor J.
1992
New differential geometric methods in nonholonomic path finding. Zbl 0777.93014
Sussmann, Héctor J.
1992
Optimal control theory and piecewise analyticity of the distance function for some real-analytic sub-Riemannian metrics. Zbl 0772.49019
Sussmann, H. J.
1992
Lie bracket extensions and averaging: The single-bracket case. Zbl 0786.70009
Sussmann, Héctor J.; Liu, Wensheng
1992
The peaking phenomenon and the global stabilization of nonlinear systems. Zbl 0749.93070
Sussmann, H. J.; Kokotovic, P. V.
1991
Limits of the Wong-Zakai type with a modified drift term. Zbl 0733.60082
Sussmann, Héctor J.
1991
The leading term of a control variation. Zbl 0796.49040
Sussmann, H. J.
1991
Global stabilization of partially linear composite systems. Zbl 0719.93071
Saberi, A.; Kokotovic, P. V.; Sussmann, H. J.
1990
Real-analytic desingularization and subanalytic sets: An elementary approach. Zbl 0696.32005
Sussmann, H. J.
1990
Global controllability by nice controls. Zbl 0703.93014
Grasse, Kevin A.; Sussmann, H. J.
1990
Limitations on the stabilizability of globally minimum phase systems. Zbl 0708.93070
Sussmann, H. J.
1990
Synthesis, presynthesis, sufficient conditions for optimality and subanalytic sets. Zbl 0712.49015
Sussmann, H. J.
1990
Nonlinear controllability and optimal control. Based on lectures of a workshop on finite dimensional controllability and optimal control held at Rutgers University, New Brunswick, NJ (USA) on May 18 to 22, 1987. Zbl 0699.00040
Sussmann, H. J. (ed.)
1990
A positive real condition for global stabilization of nonlinear systems. Zbl 0684.93066
Kokotovic, P. V.; Sussmann, H. J.
1989
Further comments on the stabilizability of the angular velocity of a rigid body. Zbl 0675.93064
Sontag, Eduardo D.; Sussmann, Héctor J.
1989
Backpropagation can give rise to spurious local minima even for networks without hidden layers. Zbl 0732.68091
Sontag, Eduardo D.; Sussmann, Héctor J.
1989
Product expansions of exponential Lie series and the discretization of stochastic differential equations. Zbl 0651.60065
Sussmann, H. J.
1988
A general theorem on local controllability. Zbl 0629.93012
Sussmann, H. J.
1987
The structure of time-optimal trajectories for single-input systems in the plane: The general real analytic case. Zbl 0664.93034
Sussmann, H. J.
1987
Regular synthesis for time-optimal control of single-input real analytic systems in the plane. Zbl 0701.93035
Sussmann, H. J.
1987
Regular synthesis for time-optimal control of single-input real analytic systems in the plane. Zbl 0696.93026
Sussmann, H. J.
1987
Small-time local controllability and continuity of the optimal time function for linear systems. Zbl 0596.93012
Sussmann, H. J.
1987
Recent developments in the regularity theory of optimal trajectories. Zbl 0649.49003
Sussmann, H. J.
1987
On the regularity of optimal controls. Zbl 0636.49009
Schättler, Heinz M.; Sussmann, Héctor J.
1987
A product expansion for the Chen series. Zbl 0619.93021
Sussmann, H. J.
1986
A weak regularity theorem for real analytic optimal control problems. Zbl 0638.49018
Sussmann, H. J.
1986
Envelopes, conjugate points, and optimal bang-bang extremals. Zbl 0619.49017
Sussmann, H. J.
1986
Some examples of reachable sets and optimal cost functions that fail to be subanalytic. Zbl 0569.49029
Lojasiewicz, Stanisław jun.; Sussmann, H. J.
1985
Lie brackets and real analyticity in control theory. Zbl 0583.93009
Sussmann, Héctor J.
1985
Continuous versions of the conditional statistics of nonlinear filtering. Zbl 0561.60046
Sussmann, H. J.
1985
A Lie-Volterra expansion for nonlinear systems. Zbl 0538.93032
Sussmann, H. J.
1984
Nonexistence of finite dimensional filters for conditional statistics of the cubic sensor problem. Zbl 0538.93061
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1984
Some recent results on pathwise nonlinear filtering. Zbl 0538.93063
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1984
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1983
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Sussmann, H. J.
1983
Nonexistence of finite-dimensional filters for conditional statistics of the cubic sensor problem. Zbl 0539.93077
Hazewinkel, M.; Marcus, S. I.; Sussmann, H. J.
1983
Differential geometric control theory. Proceedings of the Conference held at Michigan Technological University, June 28 - July 2, 1982. Zbl 0503.00014
Brockett, Roger W. (ed.); Millman, Richard S. (ed.); Sussmann, Hector J. (ed.)
1983
Bounds on the number of switchings for trajectories of piecewise analytic vector fields. Zbl 0488.34033
Sussmann, Hector J.
1982
Time-optimal control in the plane. Zbl 0516.93025
Sussmann, H. J.
1982
Approximate finite-dimensional filters for some nonlinear problems. Zbl 0494.93048
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1982
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1981
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Sussmann, Hector J.
1980
Les semi-groupes sous-analytiques et la régularité des commandes en boucle fermée. Zbl 0459.93036
Sussmann, Hector J.
1980
Subanalytic sets and feedback control. Zbl 0407.93010
Sussmann, Hector J.
1979
Single-input observability of continuous-time systems. Zbl 0422.93019
Sussmann, Hector J.
1979
A bang-bang theorem with bounds on the number of switchings. Zbl 0427.49008
Sussmann, Hector J.
1979
Generic single-input observability of continuous-time polynomial systems. Zbl 0428.93030
Sussmann, Hector J.
1979
On the gap between deterministic and stochastic ordinary differential equations. Zbl 0391.60056
Sussmann, Hector J.
1978
A sufficient condition for local controllability. Zbl 0391.93004
Sussmann, Hector J.
1978
Catastrophe theory as applied to the social and biological sciences: A critique. Zbl 0389.92002
Sussmann, Hector J.; Zahler, Raphael S.
1978
On some self-immunization mechanisms of applied mathematics: The case of catastrophe theory. Zbl 0411.00017
Sussmann, H. J.
1978
Existence and uniqueness of minimal realizations of nonlinear systems. Zbl 0354.93017
Sussman, Hector J.
1977
An interpretation of stochastic differential equations as ordinary differential equations which depend on the sample point. Zbl 0367.60060
Sussmann, Hector J.
1977
The generalized pursuit problem where the pursuer uses band-bang controls. Zbl 0374.90094
Levitt, N.; Sussmann, H.
1977
Some properties of vector field systems that are not altered by small perturbations. Zbl 0346.49036
Sussmann, Hector J.
1976
Minimal realizations and canonical forms for bilinear systems. Zbl 0335.93021
Sussmann, Hector J.
1976
Semigroup representations, bilinear approximation of input-output maps, and generalized inputs. Zbl 0353.93025
Sussmann, Hector J.
1976
A generalization of the closed subgroup theorem to quotients of arbitrary manifolds. Zbl 0342.58004
Sussmann, Hector J.
1975
On controllability by means of two vector fields. Zbl 0313.93006
Levitt, Norman; Sussmann, Hector J.
1975
Catastrophe theory. Zbl 0309.58008
Sussmann, Hector J.
1975
An extension of a theorem of Nagano on transitive Lie algebras. Zbl 0301.58003
Sussmann, Hector J.
1974
On quotients of manifolds: A generalization of the closed subgroup theorem. Zbl 0301.58002
Sussmann, Hector J.
1974
On the number of directions needed to achieve controllability. Zbl 0268.93006
Sussmann, Héctor J.
1974
Orbits of families of vector fields and integrability of distributions. Zbl 0274.58002
Sussmann, Hector J.
1973
Orbits of families of vector fields and integrability of systems with singularities. Zbl 0255.58001
Sussmann, Hector J.
1973
Minimal realizations of nonlinear systems. Zbl 0291.93007
Sussmann, Hector J.
1973
Generalized spectral theory and second order ordinary differential operators. Zbl 0222.47013
Sussmann, Hector J.
1973
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text:
# Combining Hardware Acceleration with Low-Precision and Quantization Techniques for more Energy-Efficient, Higher-Performance Computing

EE376A (Winter 2019)

By Juan Camacho

#### Introduction:

The landscape of high-performance parallel computing has been rapidly evolving over these past few years.  With the slowdown of Moore’s Law and the end of Dennard scaling over 1 decade ago, there have not been significant improvements in general-purpose processor speeds and power performances. However, at the same time, we have been seeing massive industry adoptions of emerging applications in “big data” analytics, machine and deep learning,real-time graphics, genomics, among others. Due to a huge interest in deploying these applications in huge datacenters and power-constrained edge devices, both industry and academia practitioners have been investing substantial efforts into developing more specialized. heterogeneous computing platforms for better energy-efficiency compute gains. This, of course, includes the deployment of reconfigurable architectures for hardware acceleration, which has received renewed interest from the computer architecture community to get around the power limitations of CPUs while extracting a lot more parallelism from the applications. In this post, we look at how certain techniques such as low-precision computing and quantization (which are forms of compressing information) can help improve the performance of some data-bound kernels in reconfigurable architectures such as FPGAs. More specifically, we look at the implementation of ML and computer vision kernels such as Stochastic Gradient Descent (for spam detection) and Optical Flow in a research DSL (Domain-Specific Language) for hardware accelerators called Spatial, and how they can be efficiently mapped to hardware constructs.

#### Background:

Although FPGAs (Field Programmable Gate Arrays) has been around for decades, they have recently become quite attractive candidates for implementing hardware accelerators. FPGAs come with flexible logic blocks (in the form of Look-up Tables) that can be reconfigured to implement whatever computation is specified. The picture below shows the internal structure of an FPGA:

Despite this gate-level flexibility, one of the main obstacles to massive FPGA deployment is in its programmability: to reconfigure an FPGA, one needs to use RTL verification and synthesis tools in which the developer writes code in a Hardware-Description Language (e.g. Verilog). Such low-level HDLs are hard to use without proper knowledge of digital hardware design (e.g. one must account for timing of signals, manually handle data movement, etc.), and the whole synthesis flow is usually incredibly slow (a modestly-complex design can take multiple hours before its corresponding FPGA bitstream is generated).  There has been a lot of work into developing higher-level tools that make programming FPGAs easier, one of which is Spatial.

Spatial is a Scala-based DSL that has been developed at Stanford for programming accelerators. The language gives high-level constructs  that improve programmer productivity such as nested-loop pipelining, parameterization of parallelization factors and a simple interface for specifying memory transfers between off-chip and on-chip storage. More importantly, the language allows the programmer to implement parts of the application that require hardware acceleration at a level of abstraction higher than other HDL tools while the compiler performs additional optimizations and handles low-level hardware issues such as register timing and taking into account physical constraints. For those interested, more information can be found here: https://spatial-lang.org/

#### Motivation and Description of Applications:

As already alluded to above, implementing these kinds of applications in reconfigurable hardware can lead to huge boosts in performance per watt when compared to traditional CPUs. For starters, all of the extra hardware overhead required for issuing and executing instructions in CPUs is simply not needed in a reconfigurable setting because in the latter form, specialized hardware logic can be instantiated to perform the desired computation. Also, more complicated parallel patterns and more fine-grain pipelining can be better exploited in hardware, and memory layout can be customized per application for better performance. But more importantly, at least from a information theoretic point of view, the ideas of low-precision computation and quantization can play more significant roles in this context. We show this by going over a hardware-based implementation of SGD for spam detection and optical flow.

Stochastic Gradient Descent is an iterative method for finding a global or local optimal point in a differentiable objective function by using one training example to calculate the current gradient. More formally, for a loss function $J$ and learning rate $\alpha$, the update equation for every sample $z$ is:

$W = W - \alpha \nabla J(W, b, x^{(z)}, y^{(z)})$

In many machine learning applications such as spam detection, SGD with logistic regression as the cost function are commonly used. Here, for our implementation of the SGD update equation over 5 training epochs, we used an input dataset of 5000 emails, with each email being represented by a 1024-dimensional vector. Although the Spatial implementation of SGD plus logistic regression is mostly standard, there are 2 main aspects that differentiate it from a less energy-efficient implementation: our choice of a customizable datatype for low-precision arithmetic (in the form of fixed-point), and quantization of the logistic regression function.

For many ML-based  and other probabilistic-based kernels, there’s been published research indicating that compressing the bit representation of the weights does not usually affect the overall accuracy of the results. Thus, hardware accelerators can take advantage of it because reducing the bit width of ALU engines does reduce the overall cost of computation from both a runtime and power perspective. In our case, we switched to fixed-point computation, which is a numerical format for representing real numbers. Fixed-point representation is similar to Floating-point representation, except that it covers a less range of fractional numbers, so it’s not as accurate. But it’s much cheaper to implement in hardware, and incurs in much less overall latency.

There is also the concern in how to implement logistic regression in hardware, which can be an expensive logic block (since it requires division and exponentiation). Recall that the logistic regression equation is the following:

However, by looking at the logistic regression plot, one key insight comes up: just the middle region of the graph needs to stored. We can quantize this middle region by precomputing a finite number (e.g. around 2000) of values in this range (between $-4 \leq x \leq 4$) and store them in an on-chip LUT (Look-up Table). Therefore, the actual logistic regression block would look like this:

$sig(x) = \begin{cases} 0 &\quad x < -4 \\ 1 &\quad x > 4 \\ SigLUT(x) &\quad -4 \leq x \leq4 \end{cases}$

Although this will not be as accurate as the actual logistic regression equation, this is much more hardware-friendly.

For our second application, we implemented Optical Flow, which is an object detection-like algorithm that finds motion patterns of objects between consecutive frames (e.g. people in a surveillance video). Below is a diagram of the whole image processing pipeline for optical flow, where each individual block is simply a convolution engine and the whole pipeline is implemented in a streaming fashion (i.e each engine is enabled every time there is new data coming in).

Just like with SGD, we took advantage of low-precision computing here by leveraging fixed-point arithmetic instead of floating-point arithmetic where it was applicable without losing too much accuracy. In fact, all of the compute engines in optical flow with the exception of the last stage (“Compute Flow”) could make use of fixed-point ALUs, and we were able to get significant performance gains in runtime while getting output values close enough to the original ones.

The choice of these applications was inspired by a HLS benchmark suite for software-programmable FPGAs published last year in the FPGA’ 18 conference [2].

#### Results:

For our results, we’re mostly concentrating on overall runtime for each applications. We’ve tested both SGD and optical flow on a Xilinx FPGA running at a 100 Mhz clock frequency. Here are, for example, the runtime values for spam detection on actual hardware (one for a CPU implementation, another one for a High-level Synthesis implementation reported in [2], and one for a Spatial implementation):

The Spatial implementation achieves a ~92% test accuracy, which is close to what’s expected for this dataset. We also see that Spatial achieves the lowest runtime for training using SGD plus logistic regression. This is desirable since reducing overall latency reduces energy footprint.

Now, here are the results for optical flow:

The HLS implementation uses Floating-point arithmetic for all of its computation, but we use Fixed-point arithmetic for most of the stages. The above graph shows how switching to a more compressed datatype representation for decimal numbers can help in reducing the overall latency of an accelerator.

For future work, we plan to measure performance per watt numbers (using a multimeter on the FPGA device) for each application.

#### Conclusion:

Compression and quantization techniques inspired by ideas from information theory have been incredibly useful in the design of more energy-efficient hardware accelerators, and it has quite recently become a commonly studied topic in the field of reconfigurable computing. There’s still a lot of other work in academia and industry that’s being done on this space, including other kinds of applications such as binarized neural networks (where the weights are represented by just 1 bit each). For me, it’s interesting to see how information theory concepts are being applied to make better heterogeneous computing systems.

#### Outreach Activity:

My outreach activity was actually not related to the project described above. For my outreach activity, I had a list of simple English sentences that were intentionally jumbled up (e.g swapped letters, missing letters, letters replaced by other similarly-looking symbols). I’ve had the kids try to read them and see if they could instead understand it despite the intentional misspellings. I believed this would be a fun way of showing how humans can still process and understand information correctly even if that information was not presented in its original form, which is something that has been pointed out in various other studies about the human brain. It’s also a fun way to teach how this is analogous to correcting information that was transmitted through noisy channels.

Here is the list of sentences I brought to the event:

• Ths sentence looks mostly fine, but there is one type. Spot it!
• Thiz sentnce haz intntional mizpelingz, but yu can stil read ti.
• Yuo cna porbalby raed tihs esaliy desptie teh msispeillgns.
• Alth0ugh this sentence l00ks like it doesn’t h@ve any mispellings, there’s 0ne pr0blem with it. Wh@t’s the c0mm0n err0r th@t you see here?

The outreach event was overall a fun experience, and the kids were really into the activities.

#### References:

[1] Koeplinger et al. Spatial: a language and compiler for application accelerators, Proceedings of the 39th ACM SIGPLAN Conference on Programming Language Design and Implementation, June 18-22, 2018, Philadelphia, PA, USA

[2] Zhou et al. Rosetta: A Realistic High-Level Synthesis Benchmark Suite for Software Programmable FPGAs, Proceedings of the 2018 ACM/SIGDA International Symposium on Field-Programmable Gate Arrays, February 25-27, 2018, Monterey, CALIFORNIA, USA

[3] De Sa et al. 2018. High-accuracy Low-Precision Training. Arxiv, 1803.03383.

[4] John L. Hennessy , David A. Patterson, Computer Architecture, Sixth Edition: A Quantitative Approach, Morgan Kaufmann Publishers Inc., San Francisco, CA, 2017