text:
# NIntegrate 2D highly oscillatory function

I am trying to integrate a function, and the error I get is greater than the result.

So I need to calculate

NIntegrate[osc[n,L,u,x]*f1[u, x]/f2[u, x], {u, 0, 1}, {x, 1, inf}]

where

f1[u_,x_] := u(1-u) ( 1-Cos[x]-x Sin[x] )

f2[u_,x_] := (1-u)^2 x^2 + 2u(1-u) x Sin[x] + 2( 1-Cos[x] ) u^2

and

osc[n_,L_,u_,x_]:= Exp[-I u x(n+L)+2I n ArcTan[(u(1-Cos[x]))/(u Sin[x]+x(1-u))]]

Here, n and L are parameters: n is an integer (1, 2, 3, …), and L is real > 0

The best I got was by using Method -> "LevinRule":

For example, by using

nn=1; LL = 1;

I get

int[n_,L_,u_,x_] := osc[n,L,u,x]*f1[u, x]/f2[u, x];
inf = Infinity;
NIntegrate[ int[nn, LL, u, x], {u, 0, 1}, {x, 1, inf}, Method ->
{"LevinRule", "Kernel" -> osc[nn, LL, u, x]}]

(* -0.0741726 + 0.0851585 I *)

with an error of 0.18027814

Note that the error is greater than the result.

• Welcome to Mathematica.SE! I suggest that: 1) You take the introductory Tour now! 2) When you see good questions and answers, vote them up by clicking the gray triangles, because the credibility of the system is based on the reputation gained by users sharing their knowledge. Also, please remember to accept the answer, if any, that solves your problem, by clicking the checkmark sign! 3) As you receive help, try to give it too, by answering questions in your area of expertise. – bbgodfrey Feb 20 '15 at 20:49
• The option WorkingPrecision might help. – Jinxed Feb 20 '15 at 22:25
• What are int[] and inf? – Dr. belisarius Feb 20 '15 at 22:39
• Sorry, int[n,L,u,x]=osc[n,L,u,x]*f1[u, x]/f2[u, x], and inf= infinity – oaroldan Feb 21 '15 at 14:48
• @Jinxed Thanks Jinxed: increasing WorkingPrecision just give me more digit numbers but doesn’t change the results while spending more time. – oaroldan Feb 21 '15 at 19:23

It seems to me that infinitely many, semi periodic singularities makes for a very difficult integral. If you could find a substitution that would tame all the singularities at once, you could probably succeed in truly taming this integral.

I agree with the OP that each singularity is integrable, but one has also to wonder whether the sum of the infinitely many singularities exists. My feeling is that it does. Even though the magnitude of the integrand looks like it falls off as $1/x$, the oscillatory nature of the function does not rule out convergence, and the numerical calculations I get stay around 0.08(6) + 0.08(1) I, not too far from the OP's result.

My thought was that since the singularities occur as $u \rightarrow 1^-$, $x \rightarrow 2 \pi n^-$, $n$ an integer, we might divide the infinite strip {u, 0, 1}, {x, 0, Infinity} into infinitely many rectangles {u, 0, 1}, {x, 2 Pi n, 2 Pi + 2 Pi n} and sum the integrals. (One has to subtract the integral over {u, 0, 1}, {x, 0, 1} to get the OP's integral, of course.) A singularity occurs only in one corner of each rectangle. This suggests that using Duffy's coordinates might be useful. The substitution transforms a singularity at a corner to one along an edge. If the singularity is of the right type, it can be quite effective. In this case, it helps, but the singularity persists to be a nuisance. However, the IMT singularity handler helps, at least to the extent that with a higher working precision, the integral can be computed with the desired precision.

Summing the integrals over the rectangles can be done with NSum. This means approximating the sum using approximate terms. I'm not real sure what the compounding of the error implies. Given that all terms were computed with a relative precision of MachinePrecision/2 digits and the real and imaginary parts of the terms all have the same sign (except the imaginary part of the first term), the sum should be nearly as good as NSum usually calculates, I suppose.

Another strategy I used was to isolate the singularity in the corner. Away from the corner NIntegrate can accurately compute the integral with the "LevinRule" method. Luckily the singularity can be predicted to lie in the rectangle {u, 1 - 1/Max[n, 1]/4, 1}, {x, 3 Pi/2 + 2 Pi n, 2 Pi + 2 Pi n}. We just have to figure out how to integrate this singularity.

For some reason Method -> {"DuffyCoordinates", "Corners" -> {{1, 1}}} was insufficient to compute the integrals with the desired precision. But the NIntegrate tutorial linked above includes code for manually performing the substitution. We can then call NIntegrate specifying the singularity handler "SingularityHandler" -> "IMT", which finally seems to conquer the problem.

So finally, here is the code that implements the above strategy. The function DuffyCoordinates is given in this section of the NItegrate tutorial. I would include it, but I'm not sure about copyright restrictions. Warning: It takes a long time, about a minute or so per rectangle; hence the results are memoized. Print statements are included for the impatient. The singular integral always gives a singularity warning NIntegrate::slwcon, so I supressed it with Quiet.

ClearAll[intpart7, f1, f2, osc, int];

(* OP's definitions *)
f1[u_, x_] := u (1 - u) (1 - Cos[x] - x Sin[x])
f2[u_, x_] := (1 - u)^2 x^2 + 2 u (1 - u) x Sin[x] + 2 (1 - Cos[x]) u^2
osc[n_, L_, u_, x_] :=
Exp[-I u x (n + L) + 2 I n ArcTan[(u (1 - Cos[x]))/(u Sin[x] + x (1 - u))]]
nn = 1; LL = 1;
int[u_, x_] := osc[nn, LL, u, x]*f1[u, x]/f2[u, x]

(* integrates over the n-th rectangle *)
intpart7[n_Integer] := Block[{x, u},
Print[n];
intpart7[n] = (* memoize for reuse *)
NIntegrate[            (* nonsingular integral *)
Boole[0 <= u <= 1 - 0.25/Max[n, 1] || 2 Pi n <= x <= 2 Pi n + 3 Pi/2]*int[u, x],
{u, 0, 1}, {x, 2 Pi n, 2 Pi + 2 Pi n},
MaxRecursion -> 25,
Method -> {"LevinRule", "Kernel" -> osc[nn, LL, u, x]}}] +
Quiet[Print["."];
NIntegrate @@ Join[  (* singular part *)
DuffyCoordinates[
int[u, x], {u, 1 - 1/Max[n, 1]/4, 1},
{x, 3 Pi/2 + 2 Pi n, 2 Pi + 2 Pi n}, {1, 1}],
{PrecisionGoal -> 8, WorkingPrecision -> 20,
MaxRecursion -> 25,
Method -> {"GlobalAdaptive", "SingularityHandler" -> "IMT"}}
], NIntegrate::slwcon]
];

Grand result

intzero = NIntegrate[int[u, x], {u, 0, 1}, {x, 0, 1}]
(sum = NSum[intpart7[n], {n, 0, Infinity}]) // AbsoluteTiming
sum - intzero
(*
-0.0751088 + 0.0187646 I                (* integral 0 < x < 1 *)
{1891.108293, -0.161177 + 0.0992955 I}  (* {time, sum} *)

-0.0860682 + 0.0805309 I                (* final integral *)
*)

No error estimate warnings!

Remark: The computed terms may be inspected with ? intpart7 or DownValues[intpart7]. The indices n computed are

Cases[DownValues[intpart7], HoldPattern[intpart7[n_Integer]] :> n, Infinity]
(*
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
19, 20, 21, 22, 23, 24, 25, 26, 30, 93305, 93306, 93307}
*)

Accuracy of results

We can see that using more terms in the estimation by NSum tends to increase the magnitude of the result, but by less and less. I let this run, while I was doing other stuff. It probably takes a total of two hours, including the time above. NSum does not seem to estimate the tail of the sum very accurately. It's possible that it diverges very slowly, but the OP seems to think it converges. I have not thought very much about it.

seq = Table[NSum[intpart7[n], {n, 0, Infinity}, NSumTerms -> n] - intzero, {n, 15, 100, 10}]
(*
{-0.0860682 + 0.0805309 I,
-0.0861501 + 0.0805336 I,
-0.0860646 + 0.0805342 I,
-0.0861001 + 0.0805403 I,
-0.0861384 + 0.080535 I,
-0.0861612 + 0.0805338 I,
-0.0861782 + 0.080533 I,
-0.0861947 + 0.080534 I,
-0.0862018 + 0.0805343 I}
*)

ListLinePlot[Abs@Differences@seq, DataRange -> {15, 95}]

Visualization of the Duffy transformation

The plots below show the real and imaginary parts of the integrand in the singular corner of the domain for n = 10 before and after the substitution of Duffy's coordinates. One can see that the singularity is moved to the edge as required by the "IMT" singularity handler.

Block[{n = 10, style, fn},
{style[Re], style[Im]} =
Take["DefaultPlotStyle" /. (Method /. ChartingResolvePlotTheme["Default", Plot3D]), 2];
fn["int"] = {##} &;
fn["Duffy"] = {First@DuffyCoordinates[##], {u, 0, 1/n}, {x, 1 - 2/n, 1}} &;
GraphicsGrid@
Table[Plot3D[part[#1], #2, #3, PlotRange -> 5, MaxRecursion -> 3,
PlotPoints -> 35, PlotStyle -> style[part],
AxesLabel -> Automatic, PlotLabel -> part[f]] & @@
fn[f] @@ {int[u, x], {u, 1 - 1/Max[n, 1]/4, 1}, {x, 3 Pi/2 + 2 Pi n, 2 Pi + 2 Pi n}, {1, 1}},
{part, {Re, Im}}, {f, {"int", "Duffy"}}
]
]

• Michael Thanks a lot! Your help and commitment really motives me to also help other people. I have tested your program and it works perfectly. I will use it to compute a set of integrals that have the same behavior. When finished I'd like to let you now (if you don't bother). – oaroldan Feb 27 '15 at 3:12
• @OmarAlbertoRoldanGarcia You're welcome. Best of luck with the rest of your project. Feel free to let me know how it goes. – Michael E2 Feb 27 '15 at 4:26

If you are happy with the findings so far, and just need help on dynamically creating the exclusions, you might want to proceed with the following index-generation code:

DeleteDuplicates@{#1,#2,Sequence@@Range[Ceiling[#2,2\[Pi]],Floor[#3,2\[Pi]],2\[Pi]],#3}&

which can be called with the variable and its lower and upper bounds like so, e.g.:

DeleteDuplicates@{#1,#2,Sequence@@Range[Ceiling[#2,2\[Pi]],Floor[#3,2\[Pi]],2\[Pi]],#3}&[x,1,10]

So, your NIntegrate becomes:

NIntegrate[int[nn, LL, u, x], {u, 0, 1},
DeleteDuplicates@{#1,#2,
Sequence@@Range[Ceiling[#2,2\[Pi]],Floor[#3,2\[Pi]],2\[Pi]],#3}&[x,xmin,xmax],
Method->{"LevinRule", "Kernel"->osc[nn,LL,u,x]}]
`
• Thanks Jinxed, your tips have been very useful! I took me a little long to answer because I'm on a trip. The errors have indeed diminished. – oaroldan Feb 24 '15 at 22:52
• @OmarAlbertoRoldanGarcia: You are very welcome! – Jinxed Feb 25 '15 at 14:51


text:
# 12 HL The Ratio Test

Complete the following questions before our next class.

Use the ratio test to determine the convergence or divergence of the following series.

1. $\sum_{n=1}^\infty \frac{n^3 3^{n+1}}{4^n}$
2. $\sum_{n=1}^\infty \frac{n^{10}}{10^n}$
3. $\sum_{n=1}^\infty \frac{n^n}{n!}$
Click for a hint to question 3.
Consider the definition of $$e$$.

From the Haese book, complete Exercise K.3 question 1.

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format: plain


text:
# Homework Help: Finding the Temp of the Sun given only 3 variables.

1. Oct 31, 2009

### MarcZero

1. The problem statement, all variables and given/known data

This is for my Optics class:

Given the radius of our planet, its temperature, and the distance to the Sun, estimate the temperature of the Sun (hint: use the Stephan-Boltzmann law).

2. Relevant equations

Radius of Earth $$\widetilde{=}$$ 6,378.1 km
Temperature of the Earth $$\widetilde{=}$$ 14.4 C or 287.55 K
Distance from Sun to Earth $$\widetilde{=}$$ 150,000,000 km

j* = A$$\epsilon\sigma$$T4

(I don't know why it is showing the greek symbols as exponents. I can't seem to get them all to be in the normal line. I apologize about this.)

3. The attempt at a solution

I can use what is given and estimate the surface are of the Earth (AEarth) as:

4$$\pi$$r2 = 5.11 X 108 km2

I am also assuming that the Sun is a perfect blackbody radiation source so that $$\epsilon$$ = 1.

So I am guessing that to find TSun I need to find the Irradiance (j*) of the Sun somehow using the radius and temperature of the earth. This question is from the professor and is not in our Optics book anywhere that I can see (Fowles - Introduction to Modern Optics). I am struggling to figure out how to relate the Earth's info to the Sun's Irradiance. I can't seem to find a formula to plug things into. I can imagine that somehow the Energy of the sun falls off as the distance increases and we could probably find somehow that this is related to the temperature of the Earth, but I am at a loss to do so. I know the units of J* are [W/s/m2], but I can't see how I'm supposed to relate these to what I have to work with.

Any insight or a point in the right direction would be most appreciated. Thank you for your time.

2. Nov 1, 2009

### Redbelly98

Staff Emeritus
I think the key, at least to get started, is to realize that the "equivalent area" that receives radiation from the sun is πr2, while the radiating area of the Earth is its entire surface area.

(I would have thought we'll need the diameter of the sun too, but I haven't work this one out completely yet.)

Oh, here's a copy-and-pastable σ for you, on the house


text:
# Basics and Notation¶

• Regression is the single most important tool at the econometrician's disposal
• Regression analysis is concerned with the description and evaluation of the relationship between a variable typically called the dependent variable, and one or more other variables, typically called the independent or explanatory variables.
• Alternative vocabulary:
y x
dependent variable independent variables
regressand regressors
effect variable causal variables
explained variable explanatory variables
features

### The regression "model" and terminology¶

Suppose you want to describe the relationship between two variables, y and X. You might model that relationship as a straight line:

$$y=a+\beta x$$

But that line can't exactly fit all the points of data, so you need to account for the discrepancies, and so we add a "error term" or "disturbance" denoted by $u$

$$y=a+\beta x+u$$

Now, we want to estimate $a$ and $\beta$ to best "fit" the data. Imagine you pick/estimate $\hat{a}$ and $\hat{\beta}$ to fit the data. If you apply that to all the X data points like $\hat{a} + \hat{\beta}x$ , then you get a predicted value for $y$:

$$\hat{y} = \hat{a} + \hat{\beta}x$$

We call $\hat{y}$ the "fitted values" or the "predicted values" of y. And the difference between each actual $y$ and the predicted $\hat{y}$ is called the residual or error:

$$y-\hat{y} = \hat{u} = \text{"residual" aka "error"}$$

You can see all this visually in this graph, where I've plotted some X and y variable, and decided to see how well a line with $\hat{\alpha}=8$ and $\hat{\beta}=1.5$ fit it:

In [92]:
import seaborn as sns
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.lines as lines

.query('carat < 2.5')               # censor/remove outliers
.assign(lprice = np.log(diamonds['price']),
lcarat = np.log(diamonds['carat']),
const = 1)
.sample(25,random_state = 20)
)

fig, ax = plt.subplots()
sns.scatterplot(x='lcarat',y='lprice',data=diamonds2,ax=ax).set(ylabel="",xlabel="")
plt.axline((0, 8), slope=1.5, color="red", linestyle=(0, (5, 5)))
plt.text(-.5,6.8,'$\hat{y}=8+1.5x$',color='red',size=12)
plt.axvline(0,.51,.8,color='red')
plt.title("Suppose $\hat{a}=8$ and $\hat{B}=1.5$ ", color='red',size=16)
plt.text(-1.1,8.7,'$\hat{u}=8.67-8=0.67$',color='red',size=12,bbox=dict(facecolor='none', edgecolor='red'))
plt.show()

So after you pick an $\hat{a}$ and $\hat{\beta}$, you compute the vertical distance between the fitted line and every point.

The goal of estimation (any estimation, including regression) is to make these errors as "small" as possible. Regression is aka'ed as "Ordinary Least Squares" which as it sounds, takes those errors, squares them, adds those up, and minimizes the sum

$$\min \sum(y-\hat{y})^2$$

and we can substitute $\hat{y}$ in to that and get:

$$\min \sum(y-\hat{a} + \hat{\beta}x)$$

By solving this problem, you're in essence "plotting" a bunch of these lines (with corresponding slopes and intercepts), and each time, you get the error for each datapoint ($\hat{u}$), and choosing the line that fits the data the best.

So, to combine this with the "Modeling process" page, regression follows the same steps as any estimation:

1. Select a model. In a regression, the model is a line (if you only have 1 X variable) or a hyperplane (if you have many X variables).
2. Select a loss function. In a regression, it's the sum of squared errors.
3. Minimize the loss. You can solve the minimum analytically (take the derivative, ...) or numerically (gradient descent). But good news: The python packages we use handle this for you.

Here, the actual regression line is different than what I plotted above, and your eyes probably agree that the blue line fits it quite well!

In [113]:
fig, ax = plt.subplots()
sns.regplot(x='lcarat',y='lprice',data=diamonds2,ax=ax,ci=None,
line_kws={'color':'blue'}).set(ylabel="",xlabel="")
plt.axline((0, 8), slope=1.5, color="red", linestyle=(0, (5, 5)))
plt.text(-.5,6.8,'Not the best fit: $\hat{y}=8+1.5x$',color='red',size=12,bbox=dict(facecolor='none', edgecolor='red'))
plt.text(-1.2,9,'The best fit: $\hat{y}=8.47+1.79$',color='blue',size=12,bbox=dict(facecolor='none', edgecolor='blue'))

plt.title("Suppose $\hat{a}=8$ and $\hat{B}=1.5$ ", color='blue',size=16)
# plt.text(-1.1,8.7,'$\hat{u}=8.67-8=0.67$',color='red',size=12,bbox=dict(facecolor='none', edgecolor='red'))


text:
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text:
I recently stumbled upon a fascinating talk by the physicist Nima Arkani-Hamed on the  The Morality of Fundamental Physics. (“Moral” here is in the sense of “morally correct”, as opposed to understanding the impact of science on society. Perhaps “beauty” would have been a better term.)

In this talk, Arkani-Hamed describes the quest for finding scientific theories in much the same terms as solving an optimization problem, where the solution is easy-to-verify (or “inevitable”, in his words) once you see it, but the problem is that you might get stuck in a local optimum:

The classical picture of the world is the top of a local mountain in the space of ideas. And you go up to the top and it looks amazing up there and absolutely incredible. And you learn that there is a taller mountain out there. Find it, Mount Quantum…. they’re not smoothly connected … you’ve got to make a jump to go from classical to  quantum … This also tells you why we have such major challenges in trying to extend our understanding of physics. We don’t have these knobs, and little wheels, and twiddles that we can turn. We have to learn how to make these jumps. And it is a tall order. And that’s why things are difficult

But what actually caught my attention in this talk is his description that part of what enabled progress beyond Newtonian mechanics was a different, dual, way to look at classical physics. That is, instead of the Newtonian picture of an evolution of particles according to clockwork rules,  we think that

The particle takes every path it could between A and B, every possible one. And then imagine that it just sort of sniffs them all out; and looks around; and says, I’m going to take the one that makes the following quantity as small as possible.

I know almost no physics and a limited amount of math, but this seems to me to be an instance of moving to an external, as opposed to internal definition, in the sense described by Tao. (Please correct me if I’m wrong!) As Arkani-Hamed describes, a hugely important paper of Emmy Noether showed how this viewpoint immediately implies the conservation laws and shows that this second viewpoint, in his words, is

simple, and deep, and will always be the right way of thinking about these things.

Since determinism is not “hardwired” into this second viewpoint, it is much easier to generalize it to incorporate quantum mechanics.

This talk got me thinking about whether we can find an “external” definition for computation.  That is, our usual notion of computation via Turing Machines or circuits involves a “clockwork” like view of an evolving state via composition of some basic steps. Perhaps one of the reasons we can’t make progress on lower bounds is that we don’t have a more “global” or “external” definition that would somehow capture the property that a function F is “easy” without giving an explicit way to compute it. Alas, there is a good reason that we lack such a definition. The natural proofs barrier tell us that any property that is efficiently computable from the truth table and contains all the “easy” functions (which are an exponentially small fraction of all functions) must contain many many other functions (in fact more than 99.99% of all functions) . It is sometimes suggested that the way to bypass this barrier is to avoid the “largeness” condition, as for example even a property that contains all functions except a single function G would be useful to prove a lower bound for G if we can prove that it contains all easy functions. However, I think that to obtain a true understanding of computation, and not just a lower bound for a single function, we will need to find completely new types of nonconstructive arguments.

By Boaz Barak and Omer Reingold

President Trump had just signed a new executive order revising the prior ban on visitors from seven (now six) muslim-majority countries. It is fundamentally the same, imposing a blanket 90-day ban on entry of people from six countries, with the conditions for lifting the ban depending on the cooperation of these countries’ governments.

One good analysis of the original order called it  “malevolence tempered by incompetence”, and indeed the fact that it was incompetently drafted is the main reason why the original ban did not survive court challenges. The new version has obviously been crafted with more input from competent people but it does not change anything about the points we  wrote before.

Every country has a duty to protect its citizens and we have never advocated “open borders”. Indeed, as many people who visited or immigrated to the U.S. know, the visa process is already very arduous, and involves extensive vetting. A blanket policy does not make the U.S. safer. In fact, as opposed to individual vetting,  it actually removes an element of unpredictability for any group that is planning to carry out a terror attack in the U.S. Moreover this policy (whose first, hastily drafted version was crafted without much input from the intelligence community), is not the result of a careful balancing of the risks and benefits but rather an attempt to fulfill an obviously unconstitutional  campaign promise for a “muslim ban” while tailoring it to try to pass it through the courts.

This ban hurts the U.S. and science. Much of progress in science during the 20th century can be attributed to the U.S.’s role in becoming a central hub for scientists, welcoming scientists even from countries that it was in conflict with (including 1930’s Germany and cold-war Soviet Union, and more recently, Iran). This has benefited all the world, but in particular the U.S., which during the 20th century became the world leader in science and technology as a result. Science is not a zero-sum game, and collaborations and interactions are better for all of us. We continue to strenuously object to this ban, and call on all scientists to do the same.

By Boaz Barak and Omer Reingold

Update (1/28): If you are an academic that opposes this action, please consider signing the following open letter.

Today leaked drafts of planned executive actions showed that president Trump apparently intends to issue an order suspending (and possibly permanently banning) entry  to the U.S. of citizens of seven countries:  Iran, Iraq, Libya, Somalia, Sudan, Syria, and Yemen. As Scott Aaronson points out, one consequence of this is that students from these countries would not be able to study or do research in U.S. universities.

The U.S. has mostly known to separate its treatment of foreign governments from its treatment of their citizens, whether it is Cubans, Russians, or other cases. Based on the past records, the danger of terrorism by lawful visitors to the U.S. from the seven countries above is slim to none. But over the years, visitors and immigrants from these countries did contribute immensely to the U.S. society, economy, and the scientific world at large.

We personally have collaborated and built on the scientific works of colleagues from these countries. In particular, both of us are originally from Israel, but have collaborated with scientists from Iran who knew that the issues between the two governments should not stop scientific cooperation.

This new proposed policy is not just misguided, but also directly contradicts the interests of the U.S., and the advancement of science. We call on all our fellow scientists to express their strong  disagreement with it, and their solidarity and gratitude for the contributions of visiting and immigrant scientists, without which the U.S., and the state of human knowledge, would not have been the same.

Update: I made a bit of a mess in my original description of the technical details, which was based on my faulty memory from Laci’s talk a year ago at Harvard. See Laci’s and Harald’s posts for more technical information.

Laci Babai has posted an update on his graph isomorphism algorithm. While preparing a Bourbaki talk on the work Harald Helfgott found an error in the original running time analysis of one of the subroutines. Laci fixed the error but with running time that is quantitatively weaker than originally stated, namely  $\exp(\exp(\sqrt{\log n}))$  time (hiding poly log log factors). Harald has verified that the modified version is correct.

This is a large quantitative difference, but I think makes very little actual difference for the (great) significance of this paper. It is tempting to judge theoretical computer science papers by the “headline result”, and hence be more impressed with an algorithm that improves $2^n$  time to $n^3$ than, say, $n^3$ to $n^{2.9}$. However, this is almost always the wrong metric to use.

Improving quantitative parameters such as running time or approximation factor is very useful as intermediate challenge problems that force us to create new ideas, but ultimately the important contribution of a theoretical work is the ideas it introduces and not the actual numbers. In the context of algorithmic result, for me the best way to understand what a bound such as $\exp(\exp(\sqrt{\log n}))$ says about the inherent complexity of the problem is whether you meet it “on the way up” or “on the way down”.

Often, if you have a hardness result that (for example based on the exponential time hypothesis) shows that some problem (e.g., shortest codeword) cannot be solved faster than  $\exp(\exp(\sqrt{\log n}))$  then you could expect that eventually this hardness would improve further and the true complexity of the problem is $\exp(n^c)$ for some $c>0$ (maybe even $c=1$). That is, when you meet such a bound “on your way up”, it sometimes makes sense to treat $\exp(\sqrt{\log n})$ as a function that is “almost polynomial”.

On the other hand, if you meet this bound “on your way down” in an algorithmic result, such as in this case, or in cases where for example you improve an $n^2$ algorithm to an $n\exp(\sqrt{\log n})$ then one expects further improvements and so in that context it sometimes makes sense to treat $\exp(\sqrt{\log n})$ as “almost polylogarithmic”.

Of course it could be that for some problems this kind of bound is actually their inherent complexity and not simply the temporary state of our knowledge. Understanding whether and how “unnatural” complexity bounds can arise for natural problems is a very interesting problem in its own right.

I’m happy to invite (in the name of the Stanford theory group) applications for the inaugural Motwani Postdoctoral Fellowship in Theoretical Computer Science, made possible by a gift from the Motwani-Jadeja foundation. Please see application instructions . Please apply by Jan 6, 2017 for full consideration.

Economists generally agree that free trade agreements between countries such as the U.S. and Mexico or China that have complimentary strengths result in a net benefit to both sides. But this doesn’t mean that every individual citizen benefits. There are definitely winners and losers, and as we have seen in this election, the losers are anything but “happy campers”.

NAFTA’s effect on U.S. employment  has probably been something between a modest gain to a loss of several hundred thousand U.S. jobs. The effect of trade with China has probably been greater, resulting in a loss of perhaps a million or more jobs. But both of these effects  are likely to be much smaller than the result of the U.S.’s completely unregulated trade with a different country, one that has no labor protections and whose workers work very long hours for very low wages.

I am talking about the “Nation of AI”. According to the Bureau  of Labor and Statistics,  in the U.S. there are 3.85 million drivers (of trucks, buses, taxis, etc..), 3.5 million cashiers, 2.6 million customer service representatives, and many other people working in jobs that could be automated in the near future. It is sometimes said that “routine” jobs are the ones most at risk, but perhaps a better term for this is quantifiable jobs. If your job consists of performing well-defined tasks that have a clear criteria of success (like “getting from point A to point B”) then it is at risk of first being “Uberized” (or “M-Turk’ed”)  and then automated. After all, optimizing well defined objectives is what computers do best.

Of course, like other trade deals and technological advances in the past, it could well be that the the eventual net effect of artificial intelligence on human employment is zero or even positive. But it will undoubtedly involve shifting of jobs, and, especially if it happens on a short time scale, many people whose jobs are eliminated would be unable to acquire the skills for obtaining the jobs that are created.

Understanding how to deal with this (arguably more realistic) type of “AI risk” is a grand challenge on the interface of Economics and Computer Science, as well as many other areas. Like other questions of incentives, privacy, fairness, and others, I believe theoretical computer science can and should play some role in addressing this challenge.

As also posted by Michael Mitzenmacher, we have several postdoc positions at Harvard,  please apply by December 1st.

In particular in 2017-2018, Harvard’s center for mathematical sciences and applications will be hosting a special year on combinatorics and complexity,  organized by  Noga Alon, me, Jacob Fox, Madhu Sudan, Salil Vadhan, and Leslie Valiant. I am quite excited about the workshops and events we have planned, so it should be a great time to be in the area.

The two sister sum-of-squares seminars at Cambridge and Princeton have  been maintaining a fairly extensive set of online lecture notes (with links to videos of Cambridge lectures added as we go along). While these notes are still a work in progress, I am already quite happy  with how they turned out (but would be grateful for any feedback to help make them more accessible).

As I mentioned before, if you want to see the live version, David Steurer and I are going to teach a Sum of Squares Winter Course in UC San Diego in January 4-7, 2017. Should be fun!!

Finally, please send in your suggestions for papers to invite for Theory Fest presentations by December 12, 2016. I’ve been having some issues with the dedicated email I setup for this, so if you sent in a suggestion and didn’t get a response, please send me a copy at my personal email as well.


text:
> **Puzzle 16**. What's going on here? What's the pattern you see, and why is it working this way?

I'm not sure if this is the answer you want John...

I want to expand on Alex Kreitzberg's observation. He is touching on an _alternate definition_ of a Galois pair \$$f \dashv g\$$:

$$f \text{ and } g \text{ are monotone functions and } f(g(b)) \leq b \text{ and } a \leq g(f(a))$$

This is equivalent to the definition Fong, Spivak and you yourself use.

Moreover, if a monotone function has a left (or right) Galois adjoint it is unique.


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Found in: Page 471

College Physics (Urone)

Book edition 1st Edition
Author(s) Paul Peter Urone
Pages 1272 pages
ISBN 9781938168000

Question: (a) The density of water at 0ºC is very nearly (it is actually ), whereas the density of ice at 0ºC is . Calculate the pressure necessary to keep ice from expanding when it freezes, neglecting the effect such a large pressure would have on the freezing temperature. (This problem gives you only an indication of how large the forces associated with freezing water might be.) (b) What are the implications of this result for biological cells that are frozen?

(a) The pressure necessary to keep ice from expanding is 1.9×108 Pa.

(b) Pressure has a keen role to play here so that cells and tissues do not expand and no icing in cells should be there

See the step by step solution

Step 1: Introduction

We calculate the volume change when water is converted to ice and hence by using the equation relating pressure to change in volume, we calculate pressure.

Step 2: Given parameters and formula linking pressure and change in volume

The bulk modulus of water =$\mathbf{2}\mathbf{.}\mathbf{1}\mathbf{×}{\mathbf{10}}^{\mathbf{9}}\text{Pa}$

Let us take the mass of water to be 1000 kg.

The volume of 1000 kg of water will be 1 m3

The volume of 1000 kg of ice will be 1000/917 m3

$Changeinvolume\left(\frac{\Delta V}{V}\right)=\left(\frac{1000{m}^{3}}{917{m}^{3}}-1\right)\phantom{\rule{0ex}{0ex}}=\left(\frac{83}{917}\right)\phantom{\rule{0ex}{0ex}}$

Change in the volume can be expressed as,

$\mathrm{\Delta V}=\frac{\mathrm{FV}}{\mathrm{BA}}$………………(1)

Here,is the bulk modulus of water,is pressure, andis volume change.

Step 3: Calculate pressure

$P\mathbf{=}\left(\frac{\mathbf{\Delta V}}{V}\right)×B\phantom{\rule{0ex}{0ex}}=\left(\frac{83}{917}\right)×\left(2.1×{10}^{9}Pa\right)\phantom{\rule{0ex}{0ex}}=1.9×{10}^{8}Pa\phantom{\rule{0ex}{0ex}}$

Step 3: Implications of the result for frozen biological cells

Biological cells are frozen which are different parts of human organs & tissues are frozen so that the part which was performing some hazard to the body, its functioning can be stopped and hazard minimized. Pressure has a keen role to play here so that cells and tissues do not expand and no icing in cells should be there.

Therefore, the pressure required is $1.9×{10}^{8}Pa$


text:
EXPONENTS OF CARTESIAN PRODUCTS OF TWO DIGRAPHS OF SPECIAL ORDERS

Title & Authors
EXPONENTS OF CARTESIAN PRODUCTS OF TWO DIGRAPHS OF SPECIAL ORDERS
Kim, Byeong-Moon; Rho, Yoo-Mi;

Abstract
In this paper, we find the maximum exponent of D $\small{{\times}}$ E, the cartesian product of two digraphs D and E on n, n + 2 vertices, respectively for an even integer $\small{n\geq4}$. We also characterize the extrema cases.
Keywords
exponents;digraphs;Cartesian products;Wielandt graphs;Frobenius numbers;
Language
English
Cited by
References
1.
A. Brauer, On a problem of partitions, Amer. J. Math. 64 (1942), 299-312.

2.
B. M. Kim, B. C. Song, and W. Hwang, Wielandt type theorem for Cartesian product of digraphs, Linear Algebra Appl. 429 (2008), no. 4, 841-848.

3.
R. Lamprey and B. Barnes, Primitivity of products of digraphs, Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 637-644, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979.

4.
B. L. Liu, B. D. McKay, N. C. Wormald, and K. Zhang, The exponent set of symmetric primitive (0; 1) matrices with zero trace, Linear Algebra Appl. 133 (1990), 121-131.

5.
J. B. Roberts, Note on linear forms, Proc. Amer. Math. Soc. 7 (1956), 465-469.

6.
J. Y. Shao, The exponent set of symmetric primitive matrices, Sci. Sinica Ser. A 9 (1986), 931-939.

7.
H. Wielandt, Unzerlegbare, nicht negative Matrizen, Math. Z. 52 (1950), 642-648.


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# the directed graph for the relation ⎡⎣⎢111101111⎤⎦⎥ is

We need to observe whether the relation is relation reflexive (there is a loop at each vertex), antisymmetric (every edge that Here's an illustrative example to get you going. If E consists of unordered pairs, G is an undirected graph. Are there 300,000 items in the average American household, and 10,000 items in the average European household? In Exercises $21-23$ determine whether the relation with the directed graph shown is an equivalence relation. (e) {extra credit â 3 points} Give the Boolean matrix for this relation. Open Equivalence Relations. 5 poir Let A = {2,3,4,5,6,7,8} and define a relation R on A as follows: for all ye A, * Ry=3(2x - y). (A, R), A = {1, 5, 6, 8, 10} and R denotes the relationA, R), A = {1, 5, 6, 8, 10} and R denotes the relation If E consists of ordered pairs, G is a directed graph. The connectivity relation Ëâconsists of the pairs ( 3,?) find all maximal and minimal elements of this partially ordered set. In the edge (a, b), a is the initial vertex and b is the final vertex. What the text defines (ignoring k=0) is usually called R + Defn: Two vertices in a graph â¦ Deï¬nition 6.1.1. draw the directed graph of the reflexive closure of the relations with the directed graph shown Stack Exchange Network Stack Exchange network consists of 176 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to â¦ Show transcribed image text. Pay for 5 months, gift an ENTIRE YEAR to someone special! To learn more, see our tips on writing great answers. Connectivity In Graphs Let Ëbe a relation on a set Ë. The exact position, length, or orientation of the edges in a graph illustration typically do not have meaning. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. 0 & \color{red}{1} & 0 \\ I'm trying to find a close-up lens for a beginner camera. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. We discuss questions here...not just doing assignment for you. Definition. However, not all relations have functions associated with them. Interactive tool for creating directed graphs using d3.js. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Draw the directed graph for the following relation and determine whether it is a partially ordered relation. 17.1. W h a t a re re la t i o n s? Is the relation represented by the directed graph above, Reflexive? Find the directed graph of the smallest relation that is both reflexive and symmetric that contains each of the relations with directed graphs shown in Exercises 5â7. After eliminating the common sub-expressions, re-write the basic block. A binar y relation from to is a subset of ." This preview shows page 3 - 5 out of 7 pages.. 3. A directed graph is a graph with directions. A directed relation graph showing typical relations of the species. Learning Relation Ties with a Force-Directed Graph in Distant Supervised Relation Extraction. The graphical representationshows different types of data in the form of bar graphs, frequency tables, line graphs, circle graphs, line plots, etc. Welcome to this site! When can a null check throw a NullReferenceException. Netgear R6080 AC1000 Router throttling internet speeds to 100Mbps. (b) This p.o. For instance, a relation is re exive if and only if there is a loop at every vertex of the directed graph, so that every ordered pair of the form (x;x) occurs in the relation. \color{blue}{1} & 0 & 0 \\ A relation is symmetric if â¦ V is a set whose elements are called vertices, nodes, or points;; A is a set of ordered pairs of vertices, called arrows, directed edges (sometimes simply edges with the corresponding set named E instead of A), directed arcs, or directed lines. What causes that "organic fade to black" effect in classic video games? MathJax reference. Is there a specific name for a directed graph that is composed of only loops? DIRECTED GRAPHS, UNDIRECTED GRAPHS, WEIGHTED GRAPHS 745 15 Relationships as a Weighted Graph Figure 17.3: A weighted graph. Can you add some edges to make each of them as an equivalence relation? E can be a set of ordered pairs or unordered pairs. Draw the Hasse diagram of the partial order (, be the first 12 positive integers. This figure shows a simple directed graph with three nodes and two edges. (a) There are two minimal elements: 2, 3. Assume you work as analyst at a firm whose total cost curve is given by the following equation: TC = 0.1Q 3 â 2Q 2 + 60Q + 200. W h a t a re re la t i o n s? (b) Does this p.o. directed graph representing the inverse relation. This type of graph of a relation r is called a directed graph or digraph. Use MathJax to format equations. E is a set of the edges (arcs) of the graph. [Chapter 8.6 Review] a. Hereâs an example. Using the function above and the information about fixed costs, we can create the following cost table: Solved Expert Answer to Find the directed graph of the smallest relation that is both reflexive and symmetric that contains each of the relations with directed graphs â¦ (C) Is the relation antisymmetric? \end{pmatrix}.$$. Procedurally, given the rates of the elementary reactions, the DRG can be constructed in linear time proportional to the number of reactions by evaluating the contribution of each elementary reaction, Ï i â¦ Making statements based on opinion; back them up with references or personal experience. Topics. In this case: Exercise 1.2.1. For the matrix representation, we write 1 in cell (i,j) whenever (i,j) \in R, and 0 otherwise. A graph is an ordered pair G = (V, E) where V is a set of the vertices (nodes) of the graph. We define the notion of vertex degree only in undirected graphs. In order to obtain matrix C_{ij}, you only fill the missing entries of matrix B_{ij} with 1 or 0 according as whether there is a relation between the elements in the horizontal and vertical components or not. FIGURE 5 The Directed Graph of the Relation R . Edges in an undirected graph are ordered pairs. Notice that since 1 r 2 and 2 r 1, we draw a single edge between 1 and 2 with arrows in both directions. These notions are quite similar or even identical, only the languages are diï¬erent. We take the relation$$R=\{\color{red}{(1,2)},\color{blue}{(2,1)},\color{green}{(3,3)}\}$$on the set A=\{1,2,3\}. Draw a directed acyclic graph and identify local common sub-expressions. [[[]]] < == 13. Problem 11 Easy Difficulty. Directed Graph. Figure 2 depicts a directed graph with set of vertices V= {V1, V2, V3}. Updated December 28, 2020. When should one recommend rejection of a manuscript versus major revisions? E is a set of the edges (arcs) of the graph. Let R be a relation on set A = \{1, 2, 3, 4\} defined by$$R = \{(1, 1), (2, 2), (3, 3), (4, 4), (4, 3), (4, 1), (3, 2), (3, 1)\}.$$Find the matrix and directed graph of relation R. (a) Find all minimal and maximal elements. Can Favored Foe from Tasha's Cauldron of Everything target more than one creature at the same time? Problem 11 Easy Difficulty. Is it better for me to study chemistry or physics? It consists of set âVâ of vertices and with the edges âEâ. What have you tried? Discussion. In the directed graph representation, we have the vertex set A and draw an edge i \rightarrow j whenever (i,j) \in R, and no edge otherwise. If E consists of unordered pairs, G is an undirected graph. De nition 3. Why or why not? In formal terms, a directed graph is an ordered pair G = (V, A) where. A graph with directed edges is called a directed graph or digraph. Full list of "special cases" during Bitcoin Script execution (p2sh, p2wsh, etc.)? (a) How many binary relations are there on. ð Give the gift of Numerade. The history of graph theory states it was introduced by the famous Swiss mathematician named Leonhard Euler, to solve many mathematical problems by constructing graphs based on given data or a set of points. Draw the directed graph for the following relation and determine whether it is, 3. 1 2 3 0 FIGURE 6.2.1 The actual location of the vertices is immaterial.$$ Directed graphs apply well to model relationships which are directional and not reciprocal in nature. rev 2021.1.5.38258, 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. Suppose f: R !R is de ned by f(x) = bx=2c. (b)Is the relation symmetric? ð â¦ The directed graph for a relation on the set $= {a,b,c} is shown: (a) Is the relation reflexive? â Beijing Institute of Technology â 0 â share . Directed graphs have edges with direction. $$. Why does nslookup -type=mx YAHOO.COMYAHOO.COMOO.COM return a valid mail exchanger?$$B_{ij}=\begin{pmatrix} Re la t i o n s Relations, properties, operations, and applic ations. Directed Graphs, Boolean Matrices,and Relations The notions of directed graphs, relations, and Boolean matrices are fundamental in computer science and discrete mathematics. The result is Figure 6.2.1. In this example: First and foremost, please note the corrections given by my colleagues. It only takes a minute to sign up. The edges indicate a one-way relationship, in that each edge can only be traversed in a single direction. 0 / Ë 1 3 2o Paths in Directed Graphs Representing relations by directed graphs helps in the construction of transitive closures. 3. \end{bmatrix}. Relation Graphs Trees A Picture of a Binary Relation Types of Graphs Properties of Graphs Directed Graphs A Picture of a Binary Relation Take some binary relation R on A. R ËA A = f(a 1;a 2)jaRb is true g A Graph G = (V;E) is: V is the set of nodes (Vertices) of the graph. Binar y relation Let and be sets. \begin{bmatrix} Equivalence relation. 1 & 0 & 1 & 1 Solved Expert Answer to Determine Whether the relation with the directed graph shown is a partial order \end{pmatrix}$$. Define. relation graph examples, Example. Answer: No, this directed graph does not represent a partial order. Once this is clear to you, you can ask a precise question (notice that your post does not contain a question, just an "order" to perform a task). Remarks: The word directed simply tells you that edge (i,j) is not the same as edge (j,i) while for an undirected graph, edge (i,j) is the same as edge (j,i). What is it that you don't understand? If not, why not? Determine whether the relation with the directed graphs shown below is an equivalence relation. A graph is an ordered pair (V, E) where V is a set and E is a binary relation on V (E â V × V).Elements of E are called edges.We are concerned here with directed graphs (digraphs) that have a loop at every vertex (i.e., (a, a) â E for each a â V).Such digraphs are called reflexive.In this case E â V × V corresponds to a reflexive (and symmetric) binary relation on V. Expert Answer . In some cases the language of graph theory is preferable because of its visual nature. denote the set of divisors of 50, and let “, relation. 1 & 1 & 1 & 0 \\ Directed graphs and representing relations as dir ected graphs. When Q = 0, total cost TC equals 200 and this is the total fixed cost. 1 & 0 & 0 & 0 \\ So sometimes in directed graphs we can just draw an undirected edge, which means that we can both direct the edges between this pair of vertices. 0 & 0 & \color{green}{1} \\ Directed Graphs. For instance, Twitter is a directed graph. Section 5. E can be a set of ordered pairs or unordered pairs. Relation ties, defined as the correlation and mutual exclusion between different relations, are critical for distant supervised relation extraction. Why or why not? (5 points) Draw the directed graph of the reflexive closure of the relations with the directed graph shown below. Course Hero is not sponsored or endorsed by any college or university. Determine whether theâ¦ Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. such that there is a path from 3to ?in Ë.$$C_{ij}=\begin{pmatrix} A graph represents data as a network.Two major components in a graph are â¦ For each relation: a. Relations are represented using ordered pairs, matrix and digraphs: ... A digraph is known was directed graph. Is the graph of$X = {0,1,2,3,4}$with the$<$relation directed or indirected? Question: Determine whether the relation with the directed graph shown is a partial order. A directed graph or digraph is a graph in which edges have orientations.. Thanks for contributing an answer to Mathematics Stack Exchange! Each node is drawn, perhaps with a dot, with itâs name. is the usual “less than or equal to” relation for real numbers. The directed graph representing a relation can be used to determine whether the relation has various properties. A relation is symmetric if â¦ 4& - & - & - & - What is the structure of a directed graph with vertex set A which has a relation R. Name for the type of relation similar to the edge set of a regular directed graph? Directed Graphs and Properties of Relations. This preview shows page 3 - 5 out of 7 pages. 2. set contain the least element and the greatest ele-. 04/21/2020 â by Yuming Shang, et al. Asking for help, clarification, or responding to other answers. A graph is an ordered pair G = (V, E) where V is a set of the vertices (nodes) of the graph. 2& - & - & - & - \\ Notice the previous example illustrates that any function has a relation that is associated with it. Drawing a backward arrow in a flow chart using TikZ, Improve running speed for DeleteDuplicates. set does not have the least element, the greatest element. 0 & 1 & 0 & 0 \\ 1 Add file 10 pa Westfield University assigns housing based on age. Hence, we can eliminate because S1 = S4. For instance, a relation is re exive if and only if there is a loop at every vertex of the directed graph, so that every ordered pair of the form (x;x) occurs in the relation. In Section 7.1, we used directed graphs, or digraphs, to represent relations on finite sets. site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. Directed Multigraph or Directed Simple Graph? Why hasn't JPE formally retracted Emily Oster's article "Hepatitis B and the Case of the Missing Women" (2005)? 1& - & - & - & - \\ Find the directed graph of the smallest relation that is both reflexive and symmetric that contains each of the relations with directed graphs shown in Exercises 5â7. How to install deepin system monitor in Ubuntu? Three properties of relations were introduced in Preview Activity $$\PageIndex{1}$$ and will be repeated in the following descriptions of how these properties can be visualized on a directed graph. Directed acyclic graph and adjacency matrix. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share â¦ Using the function above and the information about fixed costs, we can create the following cost table: (d) Is the relation transitive? Why aren't "fuel polishing" systems removing water & ice from fuel in aircraft, like in cruising yachts? Solution- Directed Acyclic Graph for the given basic block is- In this code fragment, 4 x I is a common sub-expression. 12. A directed graph G D.V;E/consists of a nonempty set of nodes Vand a set of directed edges E. Each edge eof Eis speciï¬ed by an ordered pair of vertices u;v2V. By the graph it can be seen that we have the following elements in our relation, (a, a) , (b, b) , (c, c) view the full answer. If E consists of ordered pairs, G is a directed graph. What is the fastest way to add a poly frame to a window hole? #" # " # 4. In an ideal example, a social network is a graph of connections between people. Set of edges in the above graph can be written as V= {(V1, V2), (V2, V3), (V1, V3)}. In directed graphs, there are two degrees for each vertex. A vertex hereby would be a person and an edge the relationship between vertices. Posted 6 years ago In the following definitions are used: A relation on a set A is defined to be For each of the relations in the referenced exercise, determine whether the relation is irreflexive, asymmetric, intransitive, or none of these. Why or why not? For every node vi 2 V,thedegree d(vi)ofvi is the sum of the weights of the edges adjacent to vi: d(vi)= Xm j=1 wij. b. The Indegree of a vertex is the number of edges ending at v. Answer. Why or why not? By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Relations. University of Maryland, University College, Nanyang Technological University • MH 1812, Nanyang Technological University • MAS 111, University of Maryland, University College • CMSC 150, Nanyang Technological University • SPMS MAS 111. 3& - & - & - & - \\ This video shows how to draw the directed graph for a relation on a set. No Related Subtopics. But this relation is transitive; hence it equals Rt. The directed graph representing a relation can be used to determine whether the relation has various properties. A matrix of the form$C_{ij}$below whose entries are either 0 or 1 is called the adjacency matrix of the directed graph of relation$R$. Transitive? & 1 & 2 & 3 & 4 \\ \hline A graph is a network of vertices and edges. Draw the directed graph for the following relation and determine whether, denote the set of divisors of 18, and let “, Draw the Hasse diagram of the partial order (. What is the matrix and directed graph corresponding to the relation$\{(1, 1), (2, 2), (3, 3), (4, 4), (4, 3), (4, 1), (3, 2), (3, 1)\}$? For example, because$(1,1),(2,2),(4,1)\in R$, we filled their corresponding entries with 1, and since$(2,1),(3,4)\notin R$, we filled the corresponding entries with 0. Supermarket selling seasonal items below cost? Can I repeatedly Awaken something in order to give it a variety of languages? Cases '' during Bitcoin Script execution ( p2sh, p2wsh, etc. ) you. The$ < $relation directed or indirected R! R is called a directed acyclic graph for a can! 3 2o Paths in directed graphs representing relations as dir ected graphs have done so far example! Is an undirected graph be a set, gift an ENTIRE YEAR to special... It would be a set of ordered pairs, G is an undirected graph and not reciprocal in.! For DeleteDuplicates relation with the edges in the graph that is associated with them different relations, properties operations! Even identical, only the languages are diï¬erent a partial order is drawn perhaps. The usual “ less than or equal to ” relation for the directed graph for the relation ⎡⎣⎢111101111⎤⎦⎥ is numbers Script execution ( p2sh, p2wsh etc. Or even identical, only the languages are diï¬erent 0, total cost equals... On writing great answers an ENTIRE YEAR to someone special the average the directed graph for the relation ⎡⎣⎢111101111⎤⎦⎥ is?. Effect in classic video games personal experience ENTIRE YEAR to someone special a backward arrow in single. Hasse diagram of the graph that is associated with them graph ( f ), with name! A common sub-expression is the total fixed cost R! R is de ned by f x. Agree to our terms of service, privacy policy and cookie policy points... Â Beijing Institute of Technology â 0 â share well to model which... Classic video games relation on a set of the pairs ( 3,? do have! Text uses the wrong definition of this partially ordered relation figure 5 the directed graph or digraph a...: two vertices in a graph â¦ 17.1 and minimal elements of the reflexive closure of the edges.... Be a set of the graph that link the vertices have a direction itâs.... 1 add file 10 pa Westfield university assigns housing based on age directed or indirected the (... Directed acyclic graph and identify local common sub-expressions would be a set of pairs! ( ignoring k=0 ) is usually called R + Defn: two vertices in a single direction any! Example to get you going foremost, please note the corrections given by my colleagues figure!: the text uses the wrong definition of this partially ordered relation does not have meaning the correlation and exclusion. '' effect in classic video games figure 2 depicts a directed graph with set vertices!, re-write the basic block Improve running speed for DeleteDuplicates can eliminate because S1 =.... Household, and let “, relation is called a directed graph is! 5 out of 7 pages element and the greatest element one creature at the same time months gift. Of languages a direction vertex is the total fixed cost and an edge relationship... Helps in the average European household has various properties in a flow using...: determine whether the relation with the directed graph of the edges ( arcs of! “ less than or equal to ” relation for real numbers variety of languages real numbers example, social. What you have done so far to subscribe to this RSS feed, copy and this. A social network is a graph in Distant Supervised relation Extraction, is... On opinion ; back them up with references or personal experience f ( x ) =.. Just doing assignment for you and mutual exclusion between different relations, are critical for Distant Supervised relation.! Re re la t i o n s of set âVâ of and! This RSS feed, copy and paste this URL into Your RSS reader and is... Chemistry or physics RSS feed, copy and paste this URL into Your RSS reader the! Text defines ( ignoring k=0 ) is usually called R + Defn: two vertices in a graph 17.1... Awaken something in order to Give it a variety of languages graph that link the is. All relations have functions associated with it the directed graph for the relation ⎡⎣⎢111101111⎤⎦⎥ is with a dot, with itâs name and mutual exclusion between relations! As an equivalence relation drawing a backward arrow in a graph illustration typically do not have meaning great answers rejection... Unordered pairs, G is a graph of the relation has various properties connections people... 2O Paths in directed graphs, there are two the directed graph for the relation ⎡⎣⎢111101111⎤⎦⎥ is elements of this quantity relation can used! Whether theâ¦ a directed graph representing a relation can be used to determine whether is. Backward arrow in a graph in which the edges âEâ graph or digraph is a graph the. Have the least element and the greatest ele- graph of the Missing ''! The average American household, and applic ations a specific name for a relation on a set of the with. Note the corrections given by my colleagues relationship between vertices which edges have..! The$ < $relation directed or indirected speed for DeleteDuplicates each edge can only be traversed in a direction! Edges to make each of them as an equivalence relation graph illustration typically do have... The basic block is- in this example: first and foremost, please the! Should one recommend rejection of a relation that is composed of only loops licensed cc. ; user contributions licensed under cc by-sa directed acyclic graph for the given block... Repeatedly Awaken something in order to Give it a variety of languages minimal:! This directed graph is a question and answer site for people studying math at any level professionals!, length, or responding to other answers can i repeatedly Awaken something order... Into Your RSS reader apply well to model relationships which are directional and not reciprocal in nature elements. ( a ) find all minimal and maximal elements acyclic graph for a relation is transitive ; it! 0, the directed graph for the relation ⎡⎣⎢111101111⎤⎦⎥ is, 2, 3 } ( p2sh, p2wsh, etc. ) the edges a! Hero is not sponsored or endorsed by any college or university ideal example a... / Ë 1 3 2o Paths in directed graphs and properties of.. To make each of them as an equivalence relation i is a subset of. a flow chart TikZ! After eliminating the common sub-expressions, re-write the basic block is- in code. ( p2sh, p2wsh, etc. ) maximal elements -type=mx YAHOO.COMYAHOO.COMOO.COM return a valid mail exchanger formal terms a. Q = 0, total cost TC equals$ 200 and this is the number of ending. Pair G = ( V, a social network is a set of the relations the! One recommend rejection of a relation on a set Westfield university assigns housing based on opinion ; the directed graph for the relation ⎡⎣⎢111101111⎤⎦⎥ is them with...

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text:
Deals Of The Week - hours only!Up to 80% off on all courses and bundles.-Close
Visualize your data – categorical variables
Check yourself 2

## Instruction

Now we've arrived at the second and last stage of mosaic plot construction. This stage consists of dividing each bar according to the second variable. The scheme of this stage is presented below:

The second variable, consumption_cat, dictates how each bar should be divided. Let's add it to our plot.

Look at the highlighted rows in the frequency table. They represent the "low" level of the wealth index. We see that the wealth_index_cat value is constant; the consumption_cat value is what differs between rows. You can also see what percentage each consumption category makes up among countries with a "low" values in wealth_index_cat. The sum of these three percentages is 100%.

We see that low-consumption countries constitute 56.67% among countries with a "low" value in wealth_index_cat. On the chart, this number is encoded as the height of the lightest (and bottom-most) rectangle in the first bar. The legend indicates that this is consumption_cat percentage data.

Once we perform the same steps on the other alcohol consumption categories, we have the final version of our chart. Remember, the goal is to visualize relationships among categorical variables. But how can we understand that relationship here – or even know if there is one? We can't add a trend line or calculate the correlation coefficient; those are only for numerical variables.

What we can do is look at how the rectangles are arranged and assess the trend (or lack of trend) in the data. You can also use a chi-square test to determine if the rectangles are arranged randomly. We won't go into the details of the chi-square test; you can find them here or in any statistics handbook. For our data, the chi-square test suggests that there is a relationship between the variables.


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# Image Retrieval With Sparse Features

This project was inspired from Mahalanobis metric learning which estimates a low rank linear projection of source data, which results in desired distances among the items in a given database.

Instead of generating a dense low rank projection, we aimed at generating a projection to a larger (sub) space, with the constraint of having sparse projections. Such a representation has a much more potential than a low rank projection because it represents a union of subspaces instead of a single subspace.

However the optimization problem is essentially non-linear, hence we opted to use fully connected neural nets with stochastic gradient descent for learning. The resulting algorithm is also extendible to have multiple layers to have a deep network.

The additional benefit of this approach is that it can also be used to learn a low rank Mahalanobis metric projection without a low rank condition during the optimization, hence have a very efficient and parallalizable optimization scheme.

This work was also my first project for which I used python and numpy extensively. I also used the scikit-learn and theano libraries for implementing the (one or multi-layer) neaural network and the stochastic gradient descent algorithm for training it.

# CNN News Footage Dataset

One of the obstacles we have encountered during this project was not having a large enough database to train a neural net for image retrieval. Hence we created our own!

This dataset was generated from images taken from publicly available CNN news footage videos, and labeled according to their similarity. It is the largest database for image retrieval as of 2016. The database is available upon request from the website of Patrick Pérez upon request.

# Relevant Publications and Patents

• Bilen, C.; Zepeda, J.; Pérez, P., ”The CNN news footage datasets: Enabling supervision in image retrieval”, EUSIPCO 2016
• Bilen, C.; Zepeda, J.; Pérez, P., ”Supervised learning of low-rank transforms for image retrieval”, ICIP 2016
• Bilen, Cagdas; Zapeda, Joaquin; Perez, Patrick, Method and Apparatus for image search using sparsifying analysis and synthesis operators", Patent filed (2015)
• Bilen, Cagdas; Zapeda, Joaquin; Perez, Patrick, Method and Apparatus for image search using sparsifying analysis operators", Patent filed (2015)


text:
# equation out of margin?

I am using

    $$equation$$

in order to write formulas

but for example with

$$G(\boldsymbol{x_1}, \overline{z} -dz/2; \boldsymbol{x_2}, \overline{z} +dz/2) = {\displaystyle\int } \frac{d^2 \boldsymbol Q}{{2 \pi}} [\widetilde{C_0}(\boldsymbol{Q}) e^{-i Q^2 dz/2k}] e^{i \boldsymbol{Q} \cdot (\boldsymbol{x_2}-\boldsymbol{x_1})} \times \notag\\ {\displaystyle\int } \frac{d^2 \boldsymbol q}{{2 \pi}} [\widetilde{I_0}(\boldsymbol{q}) e^{-i q^2 dz/8k}] e^{i \boldsymbol{q} \cdot [\boldsymbol{x_2}+\boldsymbol{x_1}]/2} e^{-i \boldsymbol{Q}\cdot \boldsymbol{q}(\overline z /k)}$$

I obtain

How can I correct it?

• It is just too wide. Use e.g. multline and add some line breaks.
– user238301
Mar 27 '21 at 17:21

You can't break lines with \\ inside equation; you need a specific alignment environment, in this case split.

The terms at which it's possible to split are very long, so I suggest to set the left-hand side on a line by itself and indenting the right-hand side split into two lines.

The code:

\documentclass{article}
\usepackage{amsmath,bm}

\newcommand{\vect}[1]{\boldsymbol{#1}}

\begin{document}

$$\begin{split} & G(\vect{x}_1, \bar{z} -dz/2; \vect{x}_2, \bar{z} +dz/2) \\ &\qquad = \int \frac{d^2 \vect Q}{2 \pi} [\tilde{C}_0(\vect{Q}) e^{-i Q^2 dz/2k}] e^{i \vect{Q} \cdot (\vect{x}_2-\vect{x}_1)} \\ &\qquad \times \int \frac{d^2 \vect{q}}{2 \pi} [\tilde{I}_0(\vect{q}) e^{-i q^2 dz/8k}] e^{i \vect{q} \cdot [\vect{x}_2+\vect{x}_1]/2} e^{-i \vect{Q}\cdot \vect{q}(\bar{z}/k)} \end{split}$$

\end{document}

I have also other suggestions:

1. \boldsymbol{x_1} should be \boldsymbol{x}_1
2. It's better to have a more semantic command, so I defined \vect to use \boldsymbol
3. {\displaystyle\int} is wrong: just use \int.
4. Instead of \overline it's better to use \bar over single symbols.
5. Similarly for \widetilde, which need not cover the subscript, and can well be \tilde; don't worry: your readers will see it.

As I said, the equation is just too long. You can either use multline or align with \MoveEqLeft from mathtools. I am also a bit unsure of the notation, the d in dz is not meant to be a differential, is it? In any case, it is very helpful if the reader knows what is meant, and clearly mark differentials. Likewise, you want to have the imaginary i and the Euler e distinguishable from indices and variables. Given that I do not know the context one guess for the equation could be:

\documentclass{article}
\usepackage{mathtools}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
\begin{document}
\begin{align}
\MoveEqLeft
G(\boldsymbol{x}_1, \overline{z} -dz/2; \boldsymbol{x_2}, \overline{z} +dz/2)
= \int\!  \frac{\diff^2 \boldsymbol Q}{{2 \pi}} \,
\bigl[\widetilde{C_0}(\boldsymbol{Q})\, \mathrm{e}^{-\mathrm{i} Q^2 dz/2k}\bigr]\,
\mathrm{e}^{\mathrm{i} \boldsymbol{Q} \cdot
(\boldsymbol{x}_2-\boldsymbol{x}_1)}   \notag\\
&{}\times\int   \!\frac{\diff^2 \boldsymbol q}{{2 \pi}} \,
\bigl[\widetilde{I_0}(\boldsymbol{q})\,
\mathrm{e}^{-\mathrm{i} q^2 dz/8k}\bigr]\,
\mathrm{e}^{\mathrm{i} \boldsymbol{q} \cdot
[\boldsymbol{x}_2+\boldsymbol{x}_1]/2}\,
\mathrm{e}^{-\mathrm{i} \boldsymbol{Q}\cdot \boldsymbol{q}(\overline z /k)}
\;.
\end{align}
\end{document}


text: References of "Rigo, Michel"      in Complete repository Arts & humanities   Archaeology   Art & art history   Classical & oriental studies   History   Languages & linguistics   Literature   Performing arts   Philosophy & ethics   Religion & theology   Multidisciplinary, general & others Business & economic sciences   Accounting & auditing   Production, distribution & supply chain management   Finance   General management & organizational theory   Human resources management   Management information systems   Marketing   Strategy & innovation   Quantitative methods in economics & management   General economics & history of economic thought   International economics   Macroeconomics & monetary economics   Microeconomics   Economic systems & public economics   Social economics   Special economic topics (health, labor, transportation…)   Multidisciplinary, general & others Engineering, computing & technology   Aerospace & aeronautics engineering   Architecture   Chemical engineering   Civil engineering   Computer science   Electrical & electronics engineering   Energy   Geological, petroleum & mining engineering   Materials science & engineering   Mechanical engineering   Multidisciplinary, general & others Human health sciences   Alternative medicine   Anesthesia & intensive care   Cardiovascular & respiratory systems   Dentistry & oral medicine   Dermatology   Endocrinology, metabolism & nutrition   Forensic medicine   Gastroenterology & hepatology   General & internal medicine   Geriatrics   Hematology   Immunology & infectious disease   Laboratory medicine & medical technology   Neurology   Oncology   Ophthalmology   Orthopedics, rehabilitation & sports medicine   Otolaryngology   Pediatrics   Pharmacy, pharmacology & toxicology   Psychiatry   Public health, health care sciences & services   Radiology, nuclear medicine & imaging   Reproductive medicine (gynecology, andrology, obstetrics)   Rheumatology   Surgery   Urology & nephrology   Multidisciplinary, general & others Law, criminology & political science   Civil law   Criminal law & procedure   Criminology   Economic & commercial law   European & international law   Judicial law   Metalaw, Roman law, history of law & comparative law   Political science, public administration & international relations   Public law   Social law   Tax law   Multidisciplinary, general & others Life sciences   Agriculture & agronomy   Anatomy (cytology, histology, embryology...) & physiology   Animal production & animal husbandry   Aquatic sciences & oceanology   Biochemistry, biophysics & molecular biology   Biotechnology   Entomology & pest control   Environmental sciences & ecology   Food science   Genetics & genetic processes   Microbiology   Phytobiology (plant sciences, forestry, mycology...)   Veterinary medicine & animal health   Zoology   Multidisciplinary, general & others Physical, chemical, mathematical & earth Sciences   Chemistry   Earth sciences & physical geography   Mathematics   Physics   Space science, astronomy & astrophysics   Multidisciplinary, general & others Social & behavioral sciences, psychology   Animal psychology, ethology & psychobiology   Anthropology   Communication & mass media   Education & instruction   Human geography & demography   Library & information sciences   Neurosciences & behavior   Regional & inter-regional studies   Social work & social policy   Sociology & social sciences   Social, industrial & organizational psychology   Theoretical & cognitive psychology   Treatment & clinical psychology   Multidisciplinary, general & others     Showing results 61 to 80 of 84     1 2 3 4 5     Pirates informatiques et mathématique modulaireRigo, Michel Learning material (2007)De nos jours, l'envoi de messages secrets requiert la manipulation de nombres ayant plus de cent chiffres décimaux. Nous illustrons une technique cryptographique standard (le RSA) dont la sécurité réside ... [more ▼]De nos jours, l'envoi de messages secrets requiert la manipulation de nombres ayant plus de cent chiffres décimaux. Nous illustrons une technique cryptographique standard (le RSA) dont la sécurité réside dans le fait qu'il est "rapide", sur le plus banal des ordinateurs personnels, de calculer le produit de deux "grands" nombres (cela se compte au pire en secondes), alors que le temps nécessaire pour effectuer l'opération inverse de factorisation prend, dans l'état actuel des connaissances mathématiques, énormément plus de temps (que l'on pourrait estimer en milliards d'années même pour un super-calculateur !). [less ▲]Detailed reference viewed: 55 (6 ULg) On the cost and complexity of the successor functionBerthé, Valérie; Frougny, Christiane; Rigo, Michel et alin Arnoux, P.; Bédaride, N. (Eds.) Proceedings of WORDS 2007 (2007)For a given numeration system, the successor function maps the representation of an integer n onto the representation of its successor n+1. In a general setting, the successor function maps the n-th word ... [more ▼]For a given numeration system, the successor function maps the representation of an integer n onto the representation of its successor n+1. In a general setting, the successor function maps the n-th word of a genealogically ordered language L onto the (n+1)-th word of L. We show that, if the ratio of the number of elements of length n + 1 over the number of elements of length n of the language has a limit b> 1, then the amortized cost of the successor function is equal to b/(b − 1). From this, we deduce the value of the amortized cost for several classes of numeration systems (integer base systems, canonical numeration systems associated with a Parry number, abstract numeration systems built on a rational language, and rational base numeration systems). [less ▲]Detailed reference viewed: 12 (1 ULg) Syntactictal and automatic properties of sets of polynomials over finite fieldsRigo, Michel Conference (2006, October)Detailed reference viewed: 5 (1 ULg) Abstract numeration systems : a survey.Rigo, Michel Conference (2006, August)Detailed reference viewed: 6 (1 ULg) Structural properties of bounded languages with respect to multiplication by a constantCharlier, Emilie ; Rigo, Michel in Actes des Journées Montoises d'Informatique Théorique (2006)We consider the preservation of recognizability of a set of integers after multiplication by a constant for numeration systems built over a bounded language. As a corollary we show that any nonnegative ... [more ▼]We consider the preservation of recognizability of a set of integers after multiplication by a constant for numeration systems built over a bounded language. As a corollary we show that any nonnegative integer can be written as a sum of binomial coefficients with some prescribed properties. [less ▲]Detailed reference viewed: 20 (1 ULg) A note on syndeticity, recognizable sets and Cobham's theoremRigo, Michel ; Waxweiler, Laurent in Bulletin of the European Association for Theoretical Computer Science (2006), 88In this note, we give an alternative proof of the following result. Let p,q>=2 be two multiplicatively independent integers. If an infinite set of integers is both p- and q-recognizable, then it is ... [more ▼]In this note, we give an alternative proof of the following result. Let p,q>=2 be two multiplicatively independent integers. If an infinite set of integers is both p- and q-recognizable, then it is syndetic. Notice that this result is needed in the classical proof of the celebrated Cobham’s theorem. Therefore the aim of this paper is to complete [13] and [1] to obtain an accessible proof of Cobham’s theorem. [less ▲]Detailed reference viewed: 21 (2 ULg) Automates et systèmes de numérationRigo, Michel in Bulletin de la Société Royale des Sciences de Liège (2005), 73Ce survol introductif est basé sur une mini-conférence réalisée à la Société Royale des Sciences de Liège en avril 2004 et sur un exposé réalisé à l’IUFM de Reims en juin 2003 (Integrating Technologies ... [more ▼]Ce survol introductif est basé sur une mini-conférence réalisée à la Société Royale des Sciences de Liège en avril 2004 et sur un exposé réalisé à l’IUFM de Reims en juin 2003 (Integrating Technologies into Mathematics Education). Nous y présentons divers systèmes de numération du point de vue de la théorie des langages formels. On s’attache dès lors à mettre en lumière les liens éventuels entre propriétés arithmétiques des nombres et propriétés syntaxiques de leurs représentations. La première partie de ce texte introduit en particulier la notion d’automate et quelques unes de ses applications. [less ▲]Detailed reference viewed: 16 (2 ULg) Abstract beta-expansion and ultimately periodic representationsRigo, Michel ; Steiner, Wolfgangin Journal de Théorie des Nombres de Bordeaux (2005), 17For abstract numeration systems built on exponential regular languages (including those coming from substitutions), we show that the set of real numbers having an ultimately periodic representation is ... [more ▼]For abstract numeration systems built on exponential regular languages (including those coming from substitutions), we show that the set of real numbers having an ultimately periodic representation is $\mathbb{Q}(\beta)$ if the dominating eigenvalue $\beta>1$ of the automaton accepting the language is a Pisot number. Moreover, if $\beta$ is neither a Pisot nor a Salem number, then there exist points in $\mathbb{Q}(\beta)$ which do not have any ultimately periodic representation. [less ▲]Detailed reference viewed: 11 (2 ULg) Abstract numeration systems and tilingsBerthé; Rigo, Michel in Lecture Notes in Computer Science (2005), 3618An abstract numeration system is a triple S = (L, Sigma, <) where (Z, <) is a totally ordered alphabet and L a regular language over Z; the associated numeration is defined as follows: by enumerating the ... [more ▼]An abstract numeration system is a triple S = (L, Sigma, <) where (Z, <) is a totally ordered alphabet and L a regular language over Z; the associated numeration is defined as follows: by enumerating the words of the regular language L over Z with respect to the induced genealogical ordering, one obtains a one-to-one correspondence between N and L. Furthermore, when the language L is assumed to be exponential, real numbers can also be expanded. The aim of the present paper is to associate with S a self-replicating multiple tiling of athe space, under the following assumption: the adjacency matrix of the trimmed minimal automaton recognizing L is primitive with a dominant eigenvalue being a Pisot unit. This construction generalizes the classical constructions performed for Rauzy fractals associated with Pisot substitutions [16], and for central tiles associated with a Pisot beta-numeration [23]. [less ▲]Detailed reference viewed: 29 (3 ULg) Decidability questions related to abstract numeration systemsHonkala, Juha; Rigo, Michel in Discrete Mathematics (2004), 285(1-3), 329-333We show that some decidability questions concerning recognizable sets of integers for abstract numeration systems are equivalent to classical problems related to HD0L systems. It turns out that these ... [more ▼]We show that some decidability questions concerning recognizable sets of integers for abstract numeration systems are equivalent to classical problems related to HD0L systems. It turns out that these problems are decidable when the sets of representations of the integers are slender regular languages. (C) 2004 Elsevier B.V. All rights reserved. [less ▲]Detailed reference viewed: 13 (0 ULg) Real numbers having ultimately periodic representations in abstract numeration systemsLecomte, Pierre ; Rigo, Michel in Information and Computation (2004), 192(1), 57-83Using a genealogically ordered infinite regular language, we know how to represent an interval of R. Numbers having an ultimately periodic representation play a special role in classical numeration ... [more ▼]Using a genealogically ordered infinite regular language, we know how to represent an interval of R. Numbers having an ultimately periodic representation play a special role in classical numeration systems. The aim of this paper is to characterize the numbers having an ultimately periodic representation in generalized systems built on a regular language. The syntactical properties of these words are also investigated. Finally, we show the equivalence of the classical theta-expansions with our generalized representations in some special case related to a Pisot number theta. (C) 2004 Elsevier Inc. All rights reserved. [less ▲]Detailed reference viewed: 10 (2 ULg) Characterizing Simpler recognizable sets of integersRigo, Michel in Studia Logica (2004), 76For a given numeration system U, a set X of integers is said to be U-star-free if the language of the normalized U-representations of the elements in X is star-free. Adapting a result of McNaughton and ... [more ▼]For a given numeration system U, a set X of integers is said to be U-star-free if the language of the normalized U-representations of the elements in X is star-free. Adapting a result of McNaughton and Papert, we give a first-order logical characterization of these sets for various numeration systems including integer base systems and the Fibonacci system. For k-ary systems, the problem of the base dependence of this property is also studied. Finally, the case of k-adic systems is developed. [less ▲]Detailed reference viewed: 7 (0 ULg) The commutative closure of a binary slip-language is context-free: a new proofRigo, Michel in Discrete Applied Mathematics (2003), 131(3), 665-672Using original arguments about sets of integers satisfying some first-order formula of the Presburger arithmetic , we give a new proof that the commutative closure of a slip-language over a two ... [more ▼]Using original arguments about sets of integers satisfying some first-order formula of the Presburger arithmetic , we give a new proof that the commutative closure of a slip-language over a two letters alphabet is context-free. (C) 2003 Elsevier B.V. All rights reserved. [less ▲]Detailed reference viewed: 14 (4 ULg) Additive functions with respect to numeration systems on regular languagesGrabner, Peter J.; Rigo, Michel in Monatshefte fur Mathematik (2003), 139(3), 205-219Asymptotic formulae for the summatory function of additive arithmetic functions related to numeration systems given by regular languages are derived.Detailed reference viewed: 8 (2 ULg) Construction of regular languages and recognizability of polynomialsRigo, Michel in Discrete Mathematics (2002), 254(1-3), 485-496A generalization of numeration systems in which NI is recognizable by finite automata can be obtained by describing a lexicographically ordered infinite regular language. We show that if P is an element ... [more ▼]A generalization of numeration systems in which NI is recognizable by finite automata can be obtained by describing a lexicographically ordered infinite regular language. We show that if P is an element of Q[x] is a polynomial such that P(N) subset of N then there exists a numeration system in which the set of representations of P(N) is regular. The main issue is to construct a regular language with a complexity function equals to P(n + 1) - P(n) for n large enough. (C) 2002 Elsevier Science B.V. All rights reserved. [less ▲]Detailed reference viewed: 12 (5 ULg) More on generalized automatic sequencesRigo, Michel ; Maes, Arnaudin Journal of Automata, Languages and Combinatorics (2002), 7We give some generalizations of $k$-automatic sequences replacing the $k$-ary system by an abstract numeration system on a regular language. We study some of the closure properties of these sequences and ... [more ▼]We give some generalizations of $k$-automatic sequences replacing the $k$-ary system by an abstract numeration system on a regular language. We study some of the closure properties of these sequences and the possible extension to the multidimensional case or to infinite alphabets. The equivalence of these sequences and morphic predicates is given and the relationship to recognizability is also investigated. [less ▲]Detailed reference viewed: 11 (3 ULg) On the representation of real numbers using regular languagesLecomte, Pierre ; Rigo, Michel in Theory of Computing Systems (2002), 35(1, JAN-FEB), 13-38Using a lexicographically ordered regular language, we show how to represent an interval of R. We determine exactly the possible representations of any element in this interval and study the function ... [more ▼]Using a lexicographically ordered regular language, we show how to represent an interval of R. We determine exactly the possible representations of any element in this interval and study the function which maps a representation onto its numerical value. We make explicit the relationship between the convergence of finite words to an infinite word and the convergence of the corresponding approximations to a real number. [less ▲]Detailed reference viewed: 42 (10 ULg) Characterizing simpler recognizable sets of integersRigo, Michel in Lecture Notes in Computer Science (2002), 2420For the k-ary numeration system, we characterize the sets of integers such that the corresponding representations make up a star-free regular language. This result can be transposed to some linear ... [more ▼]For the k-ary numeration system, we characterize the sets of integers such that the corresponding representations make up a star-free regular language. This result can be transposed to some linear numeration systems built upon a Pisot number like the Fibonacci system and also to k-adic numeration systems. Moreover we study the problem of the base dependence of this property and obtain results which are related to Cobham's Theorem. [less ▲]Detailed reference viewed: 10 (5 ULg) Abstract numeration systems on a regular languages and recognizabilityRigo, Michel Doctoral thesis (2001)Detailed reference viewed: 27 (3 ULg) Numerations systems on a regular languageLecomte, Pierre ; Rigo, Michel in Theory of Computing Systems (2001), 34Generalizations of positional number systems in which N is recognizable by finite automata are obtained by describing an arbitrary infinite regular language according to the lexicographic ordering. For ... [more ▼]Generalizations of positional number systems in which N is recognizable by finite automata are obtained by describing an arbitrary infinite regular language according to the lexicographic ordering. For these systems of numeration, we show that ultimately periodic sets are recognizable. We also study translation and multiplication by constants as well as the order-dependence of the recognizability. [less ▲]Detailed reference viewed: 26 (2 ULg)     1 2 3 4 5


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# Augmented matrices and systems of linear equations

You can think of an augmented matrix as being a way to organize the important parts of a system of linear equations. These “important parts” would be the coefficients (numbers in front of the variables) and the constants (numbers not associated with variables).

## Writing the augmented matrix for a system

Let’s look at two examples and write out the augmented matrix for each, so we can better understand the process. The key is to keep it so each column represents a single variable and each row represents a single equation. The augment (the part after the line) represents the constants.

### Example

Write the augmented matrix for the system of equations:
$$\begin{array}{l}3x_1 + 5x_2 – x_3 = 10\\ x_1 + 4x_2 + x_3 = 7\\ 9x_1 + 2x_3 = 1\\ \end{array}$$

### Solution

There are three variables, and so we will need a column for each. Be careful – notice that the last equation doesn’t have an $$x_2$$. That will be represented with a 0.

### Example

Write the augmented matrix for the system of equations:

$$\begin{array}{l} x_1 – 2x_2 + 8x_3 + x_4 + x_5 = 2\\ 3x_1 – x_2 + x_3 + 2x_4 + 2x_5 = -3\\ \end{array}$$

### Solution

Even though this is not the type of system we are used to seeing in our usual algebra classes, we can still write an augmented matrix to represent it. The augmented matrix for this system would be:

## Common Questions

### Does the order that I write the rows in matter?

No. In algebra, when you were solving a system like $$3x + y = 5$$ and $$2x + 4y = 7$$, it didn’t matter if you wrote one equation first or second. The solution to the problem didn’t change. The same is true when you have more than two equations. Since each row represents an equation, the order that you write the rows in doesn’t matter.

### What is this used for?

Putting a system of equations in this form will allow us to use a new idea called row operations to find its solution (if one exists), describe the solution set (when there are infinitely many solutions), and more. Row operations can help us organize a way to do this regardless of how many variables or how many equations we are given. This will be studied in later articles.