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# Ramsey's theory inequality with $t$-subsets

Let $q_{1},\, q_{2}, \ldots, q_{k},t$ be positive integers, where $q_{1}\geq t, q_{2}\geq t, \ldots, q_{k}\geq t$. Let $m$ be the largest of $q_{1},q_{2}, \ldots, q_{k}$.

Show that $r_{t}(m,m,\ldots,m)\geq r_{t}(q_{1},q_{2}, \ldots,q_{k})$

Conclude that to prove Ramsey's theorem, it is enough to prove it in the case that $q_{1}=q_{2}= \ldots =q_{k}.$

If I understand correctly, the LHS means:

Assume we have $n$ objects and $k$ colors and we color subsets of $t$ elements. then if $n=r_{t}(m,m, \ldots, m)$ means that there must be $m$ such subsets of the same color. Am I right? Any help is apreciated.

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Here's one way to look at it:

Let $r_1:=r_t(m,m,\cdots m)$ and $r_2=r_t(q_1,q_2,\cdots q_k)$.

If we $k$-color the $t$-order subsets of $\{1,2,\cdots r_1\}$ then by the definition of $r_1$ we are guaranteed that a $m$-order subset of $\{1,2,\cdots r_1\}$ exists such that its $t$-order subsets will all be of the same color (say color $i$).

Clearly, then since $m\ge q_i$ any $q_i$ elements in that $m$-subset constitute a $q_i$-order subset of $\{1,2,\cdots r_1\}$ such that its $t$-order subsets are all of the same color. In this way we see that for any coloring of the first $r_1$ numbers, there is such a subset for some $i$. Define this property as property $R$. Hence $r_1$ satisfies property $R$. By definition $r_2$ is the least such positive number so that property $R$ holds, and hence we have $r_2\le r_1$.

Note that we do not use the fact that $m=max\{q_1,\cdots q_k\}$; we only require the weaker condition that $m\ge q_i\forall i$.

Now by the well ordering principle if one could show that there is a non-empty subset of $\mathbb{N}$ satisfying property $R$, then it will have a least element. So it is required to prove only the existence of $r_1$ to establish the existence of $r_2$. Further, once we have established that numbers of the kind $r_1$ (call them diagonal ramsey numbers) exist, the proof shows that any $r_t(q_1,\cdots q_k)$ exists. So the existence of all diagonal numbers, i.e. the case when all arguments are equal, is the only thing that requires a proof.

For another interpretation of the statement of Ramsey's theorem through hypergraphs and a proof for the diagonal case see my blog post here.

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# Lesson 14

More Arithmetic with Complex Numbers

## 14.1: Which One Doesn’t Belong: Complex Expressions (5 minutes)

### Warm-up

This warm-up prompts students to compare four expressions. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.

### Launch

Arrange students in groups of 2–4. Display the expressions for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning why a particular item does not belong, and together find at least one reason each item doesn't belong.

### Student Facing

Which one doesn’t belong?

A. $$i^2$$

B. $$(1 + i) + (1 - i)$$

C. $$(1 + i)^2$$

D. $$(1 + i)(1 - i)$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given are correct. During the discussion, ask students to explain the meaning of any terminology they use, such as real part or imaginary part. Also, press students on unsubstantiated claims.

## 14.2: Powers of $i$ (15 minutes)

### Optional activity

This activity is optional because it goes beyond the depth of understanding required to address the standard.

In this activity, students use repeated reasoning to find patterns in the powers of $$i$$. Students find that powers of $$i$$ correspond to a repeating sequence $$1, i, \text-1, \text-i$$.

Monitor for students who:

• Break the expressions into repeated factors of $$i^2$$
• Discover and use the fact that $$i^4=1$$ to reduce the exponent to the remainder upon division by 4
• Use the exponent rule $$x^a \boldcdot x^b = x^{a+b}$$ to calculate
• Use the exponent rule $$(x^a)^b=x^{ab}$$ to calculate

### Launch

Arrange students in groups of 2. Encourage students to check with their partner about each power of $$i$$ before going on to the next power. Follow with a whole-class discussion.

Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to help students improve their written explanations for finding the values of $$i^{100}$$ and $$i^{38}$$. Give students time to meet with 2–3 partners to share and receive feedback on their responses. Display feedback prompts that will help students strengthen their ideas and clarify their language. For example, “What is the value of $$i^4$$?”, “Which exponent rule did you use?”, “How did you use the exponent rule to find the value of $$i^{100}$$?” Invite students to go back and revise or refine their written responses based on the feedback from peers. This will help students explain their strategy for calculating powers of $$i$$.
Design Principle(s): Optimize output (for justification); Cultivate conversation

### Student Facing

1. Write each power of $$i$$ in the form $$a+bi$$, where $$a$$ and $$b$$ are real numbers. If $$a$$ or $$b$$ is zero, you can ignore that part of the number. For example, $$0+3i$$ can simply be expressed as $$3i$$.

$$i^0$$

$$i^1$$

$$i^2$$

$$i^3$$

$$i^4$$

$$i^5$$

$$i^6$$

$$i^7$$

$$i^8$$

2. What is $$i^{100}$$? Explain your reasoning.
3. What is $$i^{38}$$? Explain your reasoning.

### Student Response

For access, consult one of our IM Certified Partners.

### Student Facing

#### Are you ready for more?

1. Write each power of $$1+i$$ in the form $$a+bi$$, where $$a$$ and $$b$$ are real numbers. If $$a$$ or $$b$$ is zero, you can ignore that part of the number. For example, $$0+3i$$ can simply be expressed as $$3i$$.

1. $$(1+i)^0$$
2. $$(1+i)^1$$
3. $$(1+i)^2$$
4. $$(1+i)^3$$
5. $$(1+i)^4$$
6. $$(1+i)^5$$
7. $$(1+i)^6$$
8. $$(1+i)^7$$
9. $$(1+i)^8$$
2. Compare and contrast the powers of $$1+i$$ with the powers of $$i$$. What is the same? What is different?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

Select previously identified students to share the patterns they found in this order:

• Break the expressions into repeated factors of $$i^2$$
• Discover and use the fact that $$i^4=1$$ to reduce the exponent to the remainder upon division by 4
• Use the exponent rule $$x^a \boldcdot x^b = x^{a+b}$$ to calculate
• Use the exponent rule $$(x^a)^b=x^{ab}$$ to calculate

Ask students to compare how well these strategies work when the exponent is very large, like with the last two problems. Students might find that combining exponent rules with the fact that $$i^2=\text-1$$ and $$i^4=1$$ makes it more efficient to compute powers of $$i$$ with very large exponents. Close by asking students to compute $$i$$ to a very large power, like the calendar year. For example, $$i^{2018}=i^{2000}\boldcdot i^{16} \boldcdot i^2 = (i^4)^{500} \boldcdot (i^4)^4 \boldcdot i^2 = 1 \boldcdot 1 \boldcdot \text-1 = \text-1$$.

Action and Expression: Internalize Executive Functions. Provide students with a two-column table to record responses and identify a pattern. Use the left column to list each given power of $$i$$ and the right column to list expressions in the form $$a+bi$$ for students to identify patterns.
Supports accessibility for: Organization; Attention

## 14.3: Add 'Em Up (or Subtract or Multiply) (15 minutes)

### Optional activity

This activity is optional because it is an opportunity for extra practice that not all classes may need.

The structure of a row game gives students an opportunity to construct viable arguments and critique the reasoning of others (MP3). In a row game, pairs of students do different problems that have the same answer. If there are discrepancies in their answers, students must communicate with each other to resolve those discrepancies.

Look for groups who disagree but then work to reach agreement to share during the synthesis. Also look for any groups who initially agree, but upon deeper inspection, were both incorrect.

### Launch

Arrange students in groups of 2, assigning one student as partner A and the other as partner B. Explain to students that there will be two columns of problems and that they only do the problems in their column. Students are to complete the problems, and then compare answers with their partner. If they do not get the same answer, they should work together to find the error.

Speaking: MLR8 Discussion Supports. To help students explain their strategy for simplifying the expression, provide sentence frames such as: “First, I _____ because _____. Then, I…”, “I tried _____ and what happened was…”, “How did you get…?”, and “How do you know…?”
Design Principle(s): Optimize output (for explanation)
Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite pairs to agree on 4 of the 6 rows to complete. Chunking this task into more manageable parts may also support students who benefit from additional processing time.
Supports accessibility for: Organization; Attention; Social-emotional skills

### Student Facing

For each row, your partner and you will each rewrite an expression so it has the form $$a+bi$$, where $$a$$ and $$b$$ are real numbers. You and your partner should get the same answer. If you disagree, work to reach agreement.

partner A partner B
$$(7 + 9i) + (3 - 4i)$$ $$5i(1 - 2i)$$
$$2i(3 + 4i)$$ $$(1 + 2i) - (9 - 4i)$$
$$(4 - 3i)(4 + 3i)$$ $$(5 + i) + (20 - i)$$
$$(2i)^4$$ $$(3 + i \sqrt{7})(3 - i \sqrt{7})$$
$$(1 + i \sqrt{5}) - (\text- 7 - i \sqrt{5})$$ $$(\text- 2i)(\text- \sqrt{5} + 4i)$$
$$\left( \frac12 i \right) \left( \frac13 i \right) \left( \frac34 i \right)$$ $$\left( \frac12 i \right)^3$$

### Student Response

For access, consult one of our IM Certified Partners.

### Anticipated Misconceptions

In the last problem, students may stop with the answer $$\frac18 i^3$$. Remind these students of their work in the previous activity.

### Activity Synthesis

Select previously identified groups to share how they resolved disagreement, or how they figured out they were both incorrect. To involve more students in the conversation, consider asking:

• “Who can restate $$\underline{\hspace{.5in}}$$’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to $$\underline{\hspace{.5in}}$$’s strategy?”
• “Do you agree or disagree? Why?”

## Lesson Synthesis

### Lesson Synthesis

In this lesson, students practiced arithmetic with complex numbers. Here are some questions for discussion:

• “How is arithmetic with complex numbers the same as with real numbers? How is it different?” (The arithmetic is the same as with real numbers because it still involves multiplying, exponentiating, adding, and subtracting. It is different because we didn’t divide complex numbers, and also there is an extra consideration that $$i^2=\text-1$$.)
• “What are some things about complex numbers or radicals you didn’t understand very well at first, but now you feel you understand much better?” (I didn’t understand the difference between the $$\sqrt{}$$ symbol and its connection to the solutions of a quadratic equation like $$x^2=5$$. I now know from graphs that $$x^2=5$$ has two solutions, a positive and a negative. The positive solution is written as $$\sqrt{5}$$ and the negative solution is written as $$\text-\sqrt{5}$$.)

## 14.4: Cool-down - Operate on Complex Numbers (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.

## Student Lesson Summary

### Student Facing

Suppose we want to write the product $$(3+5i)(7-2i)$$ in the form $$a+bi$$, where $$a$$ and $$b$$ are real numbers. For example, we might want to compare our solution with a partner’s, and having answers in the same form makes that easier. Using the distributive property,

\displaystyle \begin{align} (3+5i)(7-2i) &= 21 - 6i +35i - 10i^2 \\ &= 21 + 29i - 10(\text-1) \\ &= 21+ 29i +10 \\ &= 31 +29i \end{align}

Keeping track of the negative signs is especially important since it is easy to mix up the fact that $$i^2=\text-1$$ with the fact that $$\text-i^2=\text-(\text-1) = 1$$.

Next, suppose we want to write the difference $$(\text-6+3i) - (2-4i)$$ as a single complex number in the form $$a+bi$$. Distributing the negative and combining like terms, we get:

\displaystyle \begin{align} (\text-6+3i) - (2-4i) &= \text-6 +3i - 2 - (\text-4i) \\ &= \text-8 +3i +4i \\ &= \text-8 +7i \end{align}

Again, it is important to be precise with negative signs. It is a common mistake to just subtract $$4i$$ rather than subtracting $$\text-4i$$.

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# How do you find the derivative of xe^y=y-1?

Feb 19, 2015

Use implicit differentiation. On the left hand side use the product rule.

We are differentiating with respect to x

${e}^{y} + x {e}^{y} \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{dy}}{\mathrm{dx}}$

get everything with $\frac{\mathrm{dy}}{\mathrm{dx}}$ on one side of the equation

$x {e}^{y} \frac{\mathrm{dy}}{\mathrm{dx}} - \frac{\mathrm{dy}}{\mathrm{dx}} = - {e}^{y}$

Factor left hand side

$\frac{\mathrm{dy}}{\mathrm{dx}} \left(x {e}^{y} - 1\right) = - {e}^{y}$

now divide both sides by $x {e}^{y} - 1$

$\frac{\mathrm{dy}}{\mathrm{dx}} = - {e}^{y} / \left(x {e}^{y} - 1\right)$


text: Publications of Michaël Gillon     Results 181-200 of 364.   1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19   TRAPPIST monitoring of comets C/2012 S1 (Ison) and C/2013 R1 (Lovejoy)Opitom, Cyrielle ; Jehin, Emmanuel ; Manfroid, Jean et alConference (2014, June)Detailed reference viewed: 38 (5 ULiège) A window on exoplanet dynamical histories: Rossiter-McLaughlin observations of WASP-13b and WASP-32bBrothwell, R.D.; Watson, C.A.; Hébrard, G. et alin Monthly Notices of the Royal Astronomical Society (2014), 440(4), 3392-3401We present Rossiter-McLaughlin observations of WASP-13b and WASP-32b and determine the sky-projected angle between the normal of the planetary orbit and the stellar rotation axis (lambda). WASP-13b and ... [more ▼]We present Rossiter-McLaughlin observations of WASP-13b and WASP-32b and determine the sky-projected angle between the normal of the planetary orbit and the stellar rotation axis (lambda). WASP-13b and WASP-32b both have prograde orbits and are consistent with alignment with measured sky-projected angles of lambda =8°^{+13}_{-12} and lambda =-2°^{+17}_{-19}, respectively. Both WASP-13 and WASP-32 have Teff < 6250 K, and therefore, these systems support the general trend that aligned planetary systems are preferentially found orbiting cool host stars. A Lomb-Scargle periodogram analysis was carried out on archival SuperWASP data for both systems. A statistically significant stellar rotation period detection (above 99.9 per cent confidence) was identified for the WASP-32 system with Prot = 11.6 ± 1.0 days. This rotation period is in agreement with the predicted stellar rotation period calculated from the stellar radius, R*, and vsin i if a stellar inclination of i* = 90° is assumed. With the determined rotation period, the true 3D angle between the stellar rotation axis and the planetary orbit, psi, was found to be psi = 11° ± 14°. We conclude with a discussion on the alignment of systems around cool host stars with Teff < 6150 K by calculating the tidal dissipation time-scale. We find that systems with short tidal dissipation time-scales are preferentially aligned and systems with long tidal dissipation time-scales have a broad range of obliquities. [less ▲]Detailed reference viewed: 25 (1 ULiège) Transiting hot Jupiters from WASP-South, Euler and TRAPPIST: WASP-95b to WASP-101bHellier, Coel; Anderson, D. R.; Collier Cameron, A. et alin Monthly Notices of the Royal Astronomical Society (2014)We report the discovery of the transiting exoplanets WASP-95b, WASP-96b, WASP-97b, WASP-98b, WASP-99b, WASP-100b and WASP-101b. All are hot Jupiters with orbital periods in the range 2.1-5.7 d, masses of ... [more ▼]We report the discovery of the transiting exoplanets WASP-95b, WASP-96b, WASP-97b, WASP-98b, WASP-99b, WASP-100b and WASP-101b. All are hot Jupiters with orbital periods in the range 2.1-5.7 d, masses of 0.5-2.8 MJup and radii of 1.1-1.4 RJup. The orbits of all the planets are compatible with zero eccentricity. WASP-99b produces the shallowest transit yet found by WASP-South, at 0.4 per cent. The host stars are of spectral type F2-G8. Five have metallicities of [Fe/H] from -0.03 to +0.23, while WASP-98 has a metallicity of -0.60, exceptionally low for a star with a transiting exoplanet. Five of the host stars are brighter than V = 10.8, which significantly extends the number of bright transiting systems available for follow-up studies. WASP-95 shows a possible rotational modulation at a period of 20.7 d. We discuss the completeness of WASP survey techniques by comparing to the HATnet project. 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Based on our analysis of the Rossiter-McLaughlin effect, we find sky-projected spin-orbit angles of β[SUB]p[/SUB] = -52° ± 6° and β[SUB]s[/SUB] = 3° ± 7° for the primary and secondary stars (B2.5V + B2.5V, P = 6.9 days). We combine this information with several measurements of changing projected stellar rotation speeds (vsin i [SUB]sstarf[/SUB]) over the last 30 yr, leading to a model in which the primary star's obliquity is ≈65°, and its spin axis precesses around the total angular momentum vector with a period of about 140 yr. The geometry of the secondary star is less clear, although a significant obliquity is also implicated by the observed time variations in the vsin i [SUB]sstarf[/SUB]. By integrating the secular tidal evolution equations backward in time, we find that the system could have evolved from a state of even stronger misalignment similar to DI Herculis, a younger but otherwise comparable binary. Based on observations made with ESOs 2.2 m Telescopes at the La Silla Paranal Observatory under programme ID 084.C-1008 and under MPIA guaranteed time. [less ▲]Detailed reference viewed: 48 (1 ULiège) Disentangling planetary orbits from stellar activity in radial-velocity surveysHaywood, R. D.; Cameron, A. Collier; Queloz, D. et alin International Journal of Astrobiology (2014), 13The majority of extra-solar planets have been discovered (or confirmed after follow-up) through radial-velocity (RV) surveys. Using ground-based spectrographs such as High Accuracy Radial Velocity ... [more ▼]The majority of extra-solar planets have been discovered (or confirmed after follow-up) through radial-velocity (RV) surveys. Using ground-based spectrographs such as High Accuracy Radial Velocity Planetary Search (HARPS) and HARPS-North, it is now possible to detect planets that are only a few times the mass of the Earth. However, the presence of dark spots on the stellar surface produces RV signals that are very similar in amplitude to those caused by orbiting low-mass planets. Disentangling these signals has thus become the biggest challenge in the detection of Earth-mass planets using RV surveys. To do so, we use the star's lightcurve to model the RV variations produced by spots. Here we present this method and show the results of its application to CoRoT-7. [less ▲]Detailed reference viewed: 41 (2 ULiège) TRAPPIST monitoring of comet C/2012 F6 (Lemmon)Opitom, Cyrielle ; Jehin, Emmanuel ; Manfroid, Jean et alPoster (2014, April)C/2012 F6 (Lemmon) is a long period comet discovered by the Mount Lemmon Survey on 2012 March 23 at 5 AU from the sun. C/2012 F6 (Lemmon) reached perihelion on March 23, 2013 at 0.73 AU from the sun. In ... [more ▼]C/2012 F6 (Lemmon) is a long period comet discovered by the Mount Lemmon Survey on 2012 March 23 at 5 AU from the sun. C/2012 F6 (Lemmon) reached perihelion on March 23, 2013 at 0.73 AU from the sun. In December 2012 the comet was unexpectedly bright, allowing us to make an extensive monitoring during several months with both broadband and narrowband filters to follow the evolution of the comet chemical composition. The monitoring was made with TRAPPIST robotic telescope installed at La Silla observatory [1]. TRAPPIST is a 60-cm telescope dedicated to the study of exoplanets and small bodies in the solar system. The telescope is equipped with a 2Kx2K FLI Proline CCD camera very sensitive in the blue and the red. A set of narrowband cometary filters designed by the NASA for the Hale-Bopp Observing Campaign [2] is permanently mounted on the telescope along with classic Johnson-Cousins B, V, Rc, and Ic filters. We observed the comet from December 11, 2012 to March 4, 2013 (pre-perihelion) and from April 29, 2013 to June 11, 2013 (post-perihelion). At least 2 or 3 observing runs per week were programmed during this period. We collected 1358 images on 52 nights. In January and February the comet visibility allowed us to make several long runs and to detect the comet rotational variability. From the comet images in narrowband filters we studied the gaseous coma chemical composition and activity by deriving OH, NH, CN, C2 and C3 production rates using a classical Haser model [3]. The production and properties of the dust component were studied through the observation of C/2012 F6 (Lemmon) with narrowband continuum filters at 344.2 nm (UC), 444.9 nm (BC), 525.7 nm (GC) and 713.0 nm (RC). We used A(θ)fρ [4] parameter as a proxy for the dust production. 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Measurements of the transmission spectra of these planets offer the opportunity to resolve this degeneracy by directly constraining the scale heights and corresponding mean molecular weights of their atmospheres. We present Hubble Space Telescope near-infrared spectroscopy of two transits of the newly discovered transiting super-Earth HD 97658b. We use the Wide Field Camera 3's scanning mode to measure the wavelength-dependent transit depth in thirty individual bandpasses. Our averaged differential transmission spectrum has a median 1 sigma uncertainty of 19 ppm in individual bins, making this the most precise observation of an exoplanetary transmission spectrum obtained with WFC3 to date. Our data are inconsistent with a cloud-free solar metallicity atmosphere at the 17 sigma level. They are a good match for flat models corresponding to either a metal-rich atmosphere or a solar metallicity atmosphere with a cloud or haze layer located at pressures of a mbar or higher. [less ▲]Detailed reference viewed: 23 (0 ULiège) Transiting planets from WASP-South, Euler and TRAPPIST: WASP-68 b, WASP-73 b and WASP-88 b, three hot Jupiters transiting evolved solar-type starsDelrez, Laetitia ; Van Grootel, Valérie ; Anderson, D. R. et alin Astronomy and Astrophysics (2014)Using the WASP transit survey, we report the discovery of three new hot Jupiters, WASP-68 b, WASP-73 b and WASP-88 b. The planet WASP-68 bhas a mass of 0.95 ± 0.03 MJup, a radius of 1.24-0.06+0.10 RJup ... [more ▼]Using the WASP transit survey, we report the discovery of three new hot Jupiters, WASP-68 b, WASP-73 b and WASP-88 b. The planet WASP-68 bhas a mass of 0.95 ± 0.03 MJup, a radius of 1.24-0.06+0.10 RJup, and orbits a V = 10.7 G0-type star (1.24 ± 0.03 M&sun; 1.69-0.06+0.11 R&sun;, Teff = 5911 ± 60 K) with a period of 5.084298 ± 0.000015 days. Its size is typical of hot Jupiters with similar masses. The planet WASP-73 bis significantly more massive (1.88-0.06+0.07 MJup) and slightly larger (1.16-0.08+0.12 RJup) than Jupiter. It orbits a V = 10.5 F9-type star (1.34-0.04+0.05 M&sun;, 2.07-0.08+0.19 R&sun;, Teff = 6036 ± 120 K) every 4.08722 ± 0.00022 days. Despite its high irradiation (~2.3 × 109 erg s-1 cm-2), WASP-73 b has a high mean density (1.20-0.30+0.26 rhoJup) that suggests an enrichment of the planet in heavy elements. The planet WASP-88 bis a 0.56 ± 0.08 MJuphot Jupiter orbiting a V = 11.4 F6-type star (1.45 ± 0.05 M&sun;, 2.08-0.06+0.12 R&sun;, Teff = 6431 ± 130 K) with a period of 4.954000 ± 0.000019 days. With a radius of 1.70-0.07+0.13 RJup, it joins the handful of planets with super-inflated radii. The ranges of ages we determine through stellar evolution modeling are 4.5-7.0 Gyr for WASP-68, 2.8-5.7 Gyr for WASP-73 and 1.8-4.3 Gyr for WASP-88. The star WASP-73 appears to be significantly evolved, close to or already in the subgiant phase. The stars WASP-68 and WASP-88 are less evolved, although in an advanced stage of core H-burning. [less ▲]Detailed reference viewed: 53 (8 ULiège) WASP-20b and WASP-28b: a hot Saturn and a hot Jupiter in near-aligned orbits around solar-type starsAnderson, D. R.; Collier Cameron, A.; Hellier, C. et alE-print/Working paper (2014)We report the discovery of the planets WASP-20b and WASP-28b along with measurements of their sky-projected orbital obliquities. WASP-20b is an inflated, Saturn-mass planet (0.31 $M_{\rm Jup}$; 1.46 $R ... [more ▼]We report the discovery of the planets WASP-20b and WASP-28b along with measurements of their sky-projected orbital obliquities. WASP-20b is an inflated, Saturn-mass planet (0.31$M_{\rm Jup}$; 1.46$R_{\rm Jup}) in a 4.9-day, near-aligned (\lambda = 8.1 \pm 3.6^\circ$) orbit around CD-24 102 ($V$=10.7; F9). WASP-28b is an inflated, Jupiter-mass planet (0.91$M_{\rm Jup}$; 1.21$R_{\rm Jup}) in a 3.4-day, near-aligned (\lambda = 8 \pm 18^\circ$) orbit around a$V$=12, F8 star. As intermediate-mass planets in short orbits around aged, cool stars ($7^{+2}_{-1}$Gyr for WASP-20 and$5^{+3}_{-2}$Gyr for WASP-28; both with$T_{\rm eff}\$ < 6250 K), their orbital alignment is consistent with the hypothesis that close-in giant planets are scattered into eccentric orbits with random alignments, which are then circularised and aligned with their stars' spins via tidal dissipation. [less ▲]Detailed reference viewed: 25 (0 ULiège) High-frequency A-type pulsators discovered using SuperWASPHoldsworth, Daniel L.; Smalley, B.; Gillon, Michaël et alin Monthly Notices of the Royal Astronomical Society (2014)We present the results of a survey using the WASP archive to search for high-frequency pulsations in F-, A- and B-type stars. Over 1.5 million targets have been searched for pulsations with amplitudes ... [more ▼]We present the results of a survey using the WASP archive to search for high-frequency pulsations in F-, A- and B-type stars. Over 1.5 million targets have been searched for pulsations with amplitudes greater than 0.5 millimagnitude. We identify over 350 stars which pulsate with periods less than 30 min. Spectroscopic follow-up of selected targets has enabled us to confirm 10 new rapidly oscillating Ap stars, 13 pulsating Am stars and the fastest known δ Scuti star. We also observe stars which show pulsations in both the high-frequency domain and the low-frequency δ Scuti range. This work shows the power of the WASP photometric survey to find variable stars with amplitudes well below the nominal photometric precision per observation. [less ▲]Detailed reference viewed: 29 (0 ULiège) WASP-103 b: A new planet at the edge of tidal disruptionGillon, Michaël ; Anderson, D. R.; Collier-Cameron, A. et alin Astronomy and Astrophysics (2014)We report the discovery of WASP-103b, a new ultra-short-period planet (P=22.2 hr) transiting a 12.1 V-magnitude F8-type main-sequence star (1.22+-0.04 Msun, 1.44-0.03+0.05 Rsun, Teff = 6110+-160 K). WASP ... [more ▼]We report the discovery of WASP-103b, a new ultra-short-period planet (P=22.2 hr) transiting a 12.1 V-magnitude F8-type main-sequence star (1.22+-0.04 Msun, 1.44-0.03+0.05 Rsun, Teff = 6110+-160 K). WASP-103b is significantly more massive (1.49+-0.09 Mjup) and larger (1.53-0.07+0.05 Rjup) than Jupiter. Its large size and extreme irradiation (around 9 10^9 erg/s/cm^2) make it an exquisite target for a thorough atmospheric characterization with existing facilities. Furthermore, its orbital distance is less than 20% larger than its Roche radius, meaning that it might be significantly distorted by tides and might experience mass loss through Roche-lobe overflow. It thus represents a new key object for understanding the last stage of the tidal evolution of hot Jupiters. [less ▲]Detailed reference viewed: 24 (4 ULiège) Rossiter-McLaughlin Observations of 55 Cnc eLopez-Morales, Mercedes; Triaud, Amaury H. M. J.; Rodler, Florian et alin Astrophysical Journal. Letters (2014), sous presseWe present Rossiter-McLaughlin observations of the transiting super-Earth 55 Cnc e collected during six transit events between January 2012 and November 2013 with HARPS and HARPS-N. We detect no radial ... [more ▼]We present Rossiter-McLaughlin observations of the transiting super-Earth 55 Cnc e collected during six transit events between January 2012 and November 2013 with HARPS and HARPS-N. We detect no radial-velocity signal above 35 cm/s (3-sigma) and confine the stellar v sin i to 0.2 +/- 0.5 km/s. The star appears to be a very slow rotator, producing a very low amplitude Rossiter-McLaughlin effect. Given such a low amplitude, the Rossiter-McLaughlin effect of 55 Cnc e is undetected in our data, and any spin-orbit angle of the system remains possible. We also performed Doppler tomography and reach a similar conclusion. Our results offer a glimpse of the capacity of future instrumentation to study low amplitude Rossiter-McLaughlin effects produced by super-Earths. [less ▲]Detailed reference viewed: 40 (1 ULiège) Eclipsing Am binary systems in the SuperWASP surveySmalley, B.; Southworth, J.; Pintado, O. I. et alin Astronomy and Astrophysics (2014), 564The results of a search for eclipsing Am star binaries using photometry from the SuperWASP survey are presented. The light curves of 1742 Am stars fainter than V = 8.0 were analysed for the presence of ... [more ▼]The results of a search for eclipsing Am star binaries using photometry from the SuperWASP survey are presented. The light curves of 1742 Am stars fainter than V = 8.0 were analysed for the presence of eclipses. A total of 70 stars were found to exhibit eclipses, with 66 having sufficient observations to enable orbital periods to be determined and 28 of which are newly identified eclipsing systems. Also presented are spectroscopic orbits for 5 of the systems. The number of systems and the period distribution is found to be consistent with that identified in previous radial velocity surveys of "classical" Am stars. [less ▲]Detailed reference viewed: 18 (0 ULiège) Six newly-discovered hot Jupiters transiting F/G stars: WASP-87b, WASP-108b, WASP-109b, WASP-110b, WASP-111b \amp WASP-112bAnderson, D. R.; Brown, D. J. A.; Collier Cameron, A. et alE-print/Working paper (2014)Detailed reference viewed: 30 (4 ULiège) A Photometric Study of the Hot Exoplanet WASP-19bLendl, M.; Gillon, Michaël ; Queloz, D. et alin Astronomy and Astrophysics (2013), 552Context. The sample of hot Jupiters that have been studied in great detail is still growing. In particular, when the planet transits its host star, it is possible to measure the planetary radius and the ... [more ▼]Context. The sample of hot Jupiters that have been studied in great detail is still growing. In particular, when the planet transits its host star, it is possible to measure the planetary radius and the planet mass (with radial velocity data). For the study of planetary atmospheres, it is essential to obtain transit and occultation measurements at multiple wavelengths. Aims: We aim to characterize the transiting hot Jupiter WASP-19b by deriving accurate and precise planetary parameters from a dedicated observing campaign of transits and occultations. Methods: We have obtained a total of 14 transit lightcurves in the r'-Gunn, I-Cousins, z'-Gunn, and I + z' filters and 10 occultation lightcurves in z'-Gunn using EulerCam on the Euler-Swiss telescope and TRAPPIST. We also obtained one lightcurve through the narrow-band NB1190 filter of HAWK-I on the VLT measuring an occultation at 1.19 μm. We performed a global MCMC analysis of all new data, together with some archive data in order to refine the planetary parameters and to measure the occultation depths in z'-band and at 1.19 μm. Results: We measure a planetary radius of Rp = 1.376 ± 0.046 RJ, a planetary mass of Mp = 1.165 ± 0.068 MJ, and find a very low eccentricity of e = 0.0077-0.0032+0.0068, compatible with a circular orbit. We have detected the z'-band occultation at 3σ significance and measure it to be δFocc,z' = 352 ± 116 ppm, more than a factor of 2 smaller than previously published. The occultation at 1.19 μm is only marginally constrained at δFocc,NB1190 = 1711-726+745 ppm. Conclusions: We show that the detection of occultations in the visible range is within reach, even for 1 m class telescopes if a considerable number of individual events are observed. Our results suggest an oxygen-dominated atmosphere of WASP-19b, making the planet an interesting test case for oxygen-rich planets without temperature inversion. [less ▲]Detailed reference viewed: 52 (3 ULiège) HD 97658 and its super-Earth: Spitzer transit analysis and seismic modeling of the host starVan Grootel, Valérie ; Gillon, Michaël ; Valencia, Diana et alConference (2013, December)Detailed reference viewed: 18 (0 ULiège) Comet C/2012 S1 (Ison)Opitom, Cyrielle ; Jehin, Emmanuel ; Manfroid, Jean et alin Central Bureau Electronic Telegrams (2013), 3693CBET 3693 available at Central Bureau for Astronomical Telegrams.Detailed reference viewed: 23 (7 ULiège) Comet C/2012 S1 (Ison)Crovisier, J.; Colom, P.; Biver, N. et alin Central Bureau Electronic Telegrams (2013), 3711CBET 3711 available at Central Bureau for Astronomical Telegrams.Detailed reference viewed: 187 (6 ULiège) WASP-71b: a bloated hot Jupiter in an 2.9-day, prograde orbit around an evolved F8 starSmith, A. M. S.; Anderson, D. R.; Bouchy, F. et alin Astronomy and Astrophysics (2013), 552We report the discovery by the WASP transit survey of a highly-irradiated, massive (2.242 +/- 0.080 MJup) planet which transits a bright (V = 10.6), evolved F8 star every 2.9 days. The planet, WASP-71b ... [more ▼]We report the discovery by the WASP transit survey of a highly-irradiated, massive (2.242 +/- 0.080 MJup) planet which transits a bright (V = 10.6), evolved F8 star every 2.9 days. The planet, WASP-71b, is larger than Jupiter (1.46 +/- 0.13 RJup), but less dense (0.71 +/- 0.16 {\rho}Jup). We also report spectroscopic observations made during transit with the CORALIE spectrograph, which allow us to make a highly-significant detection of the Rossiter-McLaughlin effect. We determine the sky-projected angle between the stellar-spin and planetary-orbit axes to be {\lambda} = 20.1 +/- 9.7 degrees, i.e. the system is 'aligned', according to the widely-used alignment criteria that systems are regarded as misaligned only when {\lambda} is measured to be greater than 10 degrees with 3-{\sigma} confidence. WASP-71, with an effective temperature of 6059 +/- 98 K, therefore fits the previously observed pattern that only stars hotter than 6250 K are host to planets in misaligned orbits. We emphasise, however, that {\lambda} is merely the sky-projected obliquity angle; we are unable to determine whether the stellar-spin and planetary-orbit axes are misaligned along the line-of-sight. With a mass of 1.56 +/- 0.07 Msun, WASP-71 was previously hotter than 6250 K, and therefore might have been significantly misaligned in the past. If so, the planetary orbit has been realigned, presumably through tidal interactions with the cooling star's growing convective zone. [less ▲]Detailed reference viewed: 38 (6 ULiège)


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# A body is projected from the earth at angle $30^{\circ}$ with the horizontal with some initial velocity. If its range is 20 m, the maximum height reached by it is : (in meters)

$\begin {array} {1 1} (a)\;5 \sqrt 3 & \quad (b)\;\frac{5}{\sqrt 3} \\ (c)\;\frac{10}{\sqrt 3} & \quad (d)\;10 \sqrt 3 \end {array}$
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## 1 Answer

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$(b)\;\frac{5}{\sqrt 3}$
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# A cylindrical pillar is 50 cm in diameter and 3.5 m in height. Find the cost of painting the curved surface of the pillar at the rate of Rs 12.50 per m^2.

Q : 5    A cylindrical pillar is  $\small 50\hspace{1mm}cm$ in diameter and $3.5\ m$ in height. Find the cost of painting the curved surface of the pillar at the rate of $\small Rs \hspace{1mm} 12.50$  per $\small m^2$

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Given,

Radius of the cylindrical pillar, r = $\frac{50}{2}\ cm= 25\ cm= 0.25\ m$

Height of the cylinder, h  = $3.5\hspace{1mm}m$

We know,

Curved surface area of a cylinder = $2\pi r h$

$\therefore$ Curved surface area of the pillar = $2\times\frac{22}{7}\times 0.25 \times3.5$

$= 1\times22\times 0.25\ m^2$

$= 5.5\ m^2$

Now,

Cost of painting $\small 1\ m^2$ of the pillar =$\small Rs \hspace{1mm} 12.50$

$\therefore$ Cost of painting the curved surface area of the pillar = $\small Rs.\ (12.50\times5.5)$

$\small = Rs.\ 68.75$

Therefore, the cost of painting curved surface area of the pillar is $\small Rs.\ 68.75$

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# What is the unitary operator realizing a given CPTP operator

Complete Positive Trace Preserving Map (CPTP) operator is the most general operation that can be performed on a quantum system. This post mentioned that a CPTP operator is nothing but a unitary operator on the system after adding few ancilla bits. I would like to know how to realize a given CPTP operator.

Choi's theorem states that any CPTP operator $$\Phi(\cdot) : C^*_{n\times n} \rightarrow C^*_{m \times m}$$ can be expressed as $$\Phi(\rho) = \sum_{j=1}^r F_j^\dagger \rho F_j$$, for some $$n \times m$$ matrices $$F_j$$ such that $$\sum_j F_j F_j^\dagger = I_n$$.

Using this fact, can we come up with unitary operation corresponding to the given CPTP operator $$\Phi$$?

An isometry $$U:S\rightarrow S\otimes E$$, where $$S$$ is your system and $$E$$ is an environment, such that
$$\mathrm{Tr}_E(U\rho U^\dagger)=\Phi(\rho)$$ is called a Stinespring dilation of $$\Phi$$. An easy way to construct a Stinespring dilation from the Kraus operators is to consider $$\mathcal{H}_E=\mathrm{span}\{|j\rangle\}_{j=1}^r$$ and
$$U=\sum_j F_j^S\otimes|j\rangle^E$$
$$\mathrm{Tr}_E(U\rho U^\dagger)= \mathrm{Tr}_E\left(\sum_jF_j\rho F_k^\dagger \otimes |j\rangle\langle k|\right)= \sum_jF_j\rho F_j^\dagger=\Phi(\rho)$$


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Industrial Utility Efficiency

# Pneumatics

The industrial internet of things (IIoT) and industry 4.0 have digitally transformed the way facilities operate, enabling processes that are smarter and more efficient than ever before. But digital transformation is still new to some and may seem abstract or ambiguous. In actuality, it’s quite the opposite.

## Machine Builders Optimize Pneumatic Circuit Tubing Volumes

The advent of manifold mounted, plug-in pneumatic valves has been a boon for machine builders. It allows them to mount complete valve packages in a safe and secure location on a machine. Using a D-sub connector, serial interface module, or similar single-point wiring system, all of the electrical control outputs can feed into one location on the manifold, greatly simplifying the wiring. Plumbing issues are reduced, since a single air pressure line can be used to feed a common pressure gallery. The same advantage applies to the common exhaust gallery. No longer would both a plumber and an electrician be required to replace a valve, since any valve can be replaced without disturbing electrical connections or plumbing lines.

## Advanced Automation Lowers Labor Cost and Improves Performance at the Las Palmas, California, Waste Water Treatment Plant

Recently the capacity of the Las Palmas, California, waste water treatment operations were expanded by combining two plants and making one centralized filtration center. The new center expanded the flow capacity from 162,000 Gallons per Day (GPD) combined to 288,000 GPD when the manually controlled reclaimed water operations were updated to a state-of-the-art automated system. Reclaimed water from the plant irrigates local community green spaces. The new automated system ensures lower labor costs, consistent quality, and peak efficiency in the process of reclaiming waste water for irrigation.

## Pneumatic Circuit Analysis for Energy Efficiency

Currently, and for good reason, much attention is being focused on the conservation of energy. Compressed air, like electricity and gas, is an energy resource. It has often been referred to as the third utility. As with all energy sources, our global environment demands that it be conserved and used wisely.

## Improved Aeration Efficiency through Design and Control

With the recent and future increases of the cost of energy, operating a wastewater treatment plant (WWTP) as efficiently as possible has become one of the most important factors that operators and managers are facing today. The implementation of a properly designed aeration control system has been reported by the United Sates Environmental Protection Agency to reduce aeration energy by 25 to 40 percent.

## How Efficient Are Those Pneumatic Valves?

Back when gasoline was 35 cents a gallon, the term “environmental technology” was not well known. Engineers did not often promote the benefits of building low-energy consumption pneumatic valves among their peers. Recycling or conservation of resources was seldom discussed with any seriousness. In reality, the conversation was more likely to have turned to the muscle cars of the day and how much horsepower they would generate.

## Pneumatic and Vacuum Energy Optimization in OEM Machines

Machine builders aiming to improve the energy efficiency of their machines tend to focus on using energy media other than pneumatics (typically electro-mechanical or hydraulic) since pneumatics, as traditionally applied, is viewed by some as inefficient due to factors like leakage and over-pressurization (i.e.: supplying a higher pressure in an actuator to accomplish a task which is endemic in practice). But they shouldn't, with its low cost of ownership, pneumatics when properly used remains a viable and many times preferable energy source for a given application. When generating and using compressed air, it's true that there are many places in the system where energy can be lost, however targeted measures within a comprehensive energy saving concept can prevent these losses and significantly reduce energy consumption at the machine level.

## PET Bottle Blowing Efficiency

The PET industry is in a state of flux right now. A number of new bottle blowing facilities are being brought on-line. They are in the “discovery” phase right now as they realize how challenging the required compressed air systems are to manage – from an energy efficiency standpoint. The average high-volume stretch blow molder (SBM) working with PET usually has 2,000 to 4,000 horsepower of installed air compressors with the related energy costs running between \$1 to \$4 million per year. This typically represents 35-40% of the facilities’ total energy bill.

## Festo Optimizes Compressed Air Consumption at Volkswagen

Many passenger cars on roads in Germany contain efficiency concepts that make a considerable contribution to lowering emissions. Automotive manufacturers such as VW have gone even further than this, by applying efficiency strategies in their own value added chain. Because the benefits of pneumatics in automotive industry production processes have seen pneumatic actuation win over other drive technologies, efficient use of compressed air plays a key role in increasing energy efficiency.

## A View From Canada: Tembec Sawmill and Pneu-Logic Stabilize Pressure

Pneumatic air cylinders play a major role in allowing a modern sawmill to produce at the high-speed production rates required. Stable air pressure is critical to allow the air cylinders to respond in a timely manner and avoid any production delays.


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# Greater Tuna

Welcome to Greater Tuna, a simulation game exploring sustainable management of renewable resources. This page is the instructions; the game itself is available via the link at the bottom of the page.

### Object of the game:

Find the profit-maximizing level of fishing effort for a sustainable harvest of a search fishery.

### How to play:

Each turn you must decide how much effort to devote to fishing. Enter the value using the Effort slider. When you're happy with your level of effort (!), click the "Next Turn" button to see how things turn out.

### Details about the fish:

This particular fishery is described by a biological growth function of the following form:

• Spawn = NatGrow*StartPop*(CarryCap-StartPop)/CarryCap
where Spawn is the number of new fish, StartPop is the population at the beginning of the period, and NatGrow and CarryCap are parameters. The population at the end of the period will be given by:
• EndPop = StartPop + Spawn - Harvest

where Harvest is the number of fish harvested by people (more on that below). This will become the starting population for the next period. (Just to make sure there aren't any ambiguities please note that Spawn, Harvest, EndPop and StartPop may vary with time while NatGrow and CarryCap are parameters.) It will be up to you to determine the numerical values of NatGrow and CarryCap. This should be a matter of observing a few data points and doing a little algebra.

### Details about the industry:

The harvest of fish is given by a function of the form:

• Harvest = Tech*StartPop*(Effort/100)
where Harvest is the number of fish caught, Tech is a parameter (initially unknown), and Effort is the level of effort. (The factor of 100 appears here for computer convenience rather than economics: it allows the slider to read from 0 to 100 instead of from 0 to 1.) Finally, the fishery is also a price-taker in both the fish and labor markets. Its total revenue and total cost are given by:
• Revenue = $1*Harvest • Cost =$2000*(Effort/100)
The goal of the game is to find the profit-maximizing level of effort AND to end up with a sustainable population--that is, one for which StartPop equals EndPop at the optimum level of effort.

### Link to the game itself:

URL: https://wilcoxen.maxwell.insightworks.com/pages/408.html
Peter J Wilcoxen, The Maxwell School, Syracuse University
Revised 03/12/2008


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# Arithmetic mean and control groups

Define arithmetic mean arithmetic mean synonyms, arithmetic mean pronunciation, arithmetic mean translation, english dictionary definition of arithmetic mean n the value obtained by dividing the sum of a set of quantities by. Arithmetic mean with more than two objects what is the average age of the students in the two groups combined (19) trail mix consists of walnuts and raisins. The difference between arithmetic mean and geometric mean this lesson demonstrates the difference between average (or arithmetic mean) and geometric meanthat were introduced in two previous lessons. We produce a self contained account of the relationship between the gaussian arithmetic-geometric mean iteration and the fast computation of elementary functions a particularly pleasant algorithm for $\pi$ is one of the by-products. An arithmetic-logic unit pure risk, also called absolute risk, is a category of threat that is beyond human control and has only one possible outcome if. Among other things, an anova table stores the arithmetic meanfor each sample the example below allows you to control the group mean differences. About arithmetic mean if there are two groups containing n ₁ and n ₂ observations x̄ ₁ and x̄ ₂ are the respective arithmetic means. How do i compare sample means in this experimental-control group group to the sample mean of three control groups arithmetic mean of a sufficiently.

A scientific experiment may be designed with a control group here's what a control group is and how it helps increase the validity of an experiment. Groups: arithmetic mean 936, median 12 in the unordered set of data, 14 points (in our schools this used to mean grade 3 if 20 was total number of points. 114743 uploaded by mani variation the arithmetic mean is found by adding the process is in a state of statistical control a sample is a group of units. Averages: arithmetic mean (average), median, and mode descriptive statistics are measures of a population or data set in other words, you have a group of.

Mean or average or arithmetic mean is one of the representative values of data we can find the mean of observations by dividing the sum of all the observations by the total number of observations. Arithmetic mean is defined as: arithmetic mean is a quotient of sum of the given values and number of the given values arithmetic mean can be computed for both ungrouped data (raw data: data without any statistical treatment) and grouped data (data arranged in tabular form containing different groups.

Start studying statistics test 1 (1-4 why is random assignment used to assign people to treatment groups and control groups in a the mean tends to. The precision of the arithmetic mean, geometric mean and percentiles groups of researchers may also be compared for theoretical reasons, such as to contrast the. The standard deviation for the experimental group was only slightly higher at 7 78hrs than that of the control group’s 7 2hrs, however the mean of the experimental group of 14 63hrs is much higher than the control group’s 12 70hrs having an average labor time of almost 2 hours more is significant.

## Arithmetic mean and control groups

Values represent arithmetic mean (standard deviation) breakdown of healthcare resource use and associated cost of intervention and control groups costs. Two commonly used means are the arithmetic mean and the geometric mean differences in arithmetic & geometric mean // leaf group education adchoices. Arithmetic mean is the most commonly used average or measure of the central tendency applicable only in case of quantitative data it is also simply called the mean.

• Free sample essay on arithmetic mean and control groups.
• Mean values of these variables were obtained for treatment and control groups at each site and a standard statistical test of the difference between these means 7 was conducted this statistical test provides us with an indication of whether any observed differences between the two groups on average should be considered large relative.
• In mathematics and statistics, the arithmetic mean ( /ˌærɪθˈmɛtɪk ˈmiːn/, stress on third syllable of arithmetic), or simply the mean or average when the context is clear, is the sum of a collection of numbers divided by the number of numbers in the collection.
• Weighted means and means as weighted sums the ordinary mean is sometimes called the “arithmetic mean” to distinguish it to compare means of two groups.

An investigation of four properties issues and preliminary results the arithmetic mean is probably the most commonly taught and different age groups. Regional groups sas user groups us timepoint statistic ----- pre-dose n arithmetic mean geometric mean sd %cv except for the geometric mean. The mean calculated in this manner is called weighted arithmetic mean the computation of weighted arithmetic is useful in many situations where different items are of. A knowledge structure for the arithmetic mean: relationships between statistical conceptualizations and and a control group 113 arithmetic mean as a.

Arithmetic mean and control groups
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# Notes on Covariance

$X,Y\,independent \rightarrow cov(X,Y) = 0$

Assume random variables $$X$$ and $$Y$$ are discrete. That is, assume that there is a finite or denumerable sample space which is a set of $$w_i$$ and a set of quantities $$x_i$$ and $$y_i$$ defined.

Definition $$X$$ and $$Y$$ are independent if

$prob((X = x)\,and\,(Y = y)) = prob(X = x)prob(Y = y)$

in which $$x$$ is some $$x_i$$ and $$y$$ is some $$y_j$$.

Then if $$X$$ and $$Y$$ are independent,

$E(XY) = E(X)E(Y)$

Proof:

$E(XY) = \sum _{i,j} \ x_iy_jprob(XY = x_iy_j)$

$= \sum _{i,j} \ x_iy_jprob((X = x_i)\,and\,(Y = y_j))$

$= \sum _{i,j} \ x_iy_jprob(X = x_i)prob(Y = y_j)$

$= \sum _{i} \ x_iprob(X = x_i)\sum _{j} \ y_jprob(Y = y_j) = E(X)E(Y)$

Then if $$X$$ and $$Y$$ are independent,

$cov(X,Y) = E[(X - E(X))(Y - E(Y))]$

$=E[XY - XE(Y) - YE(X)+E(X)E(Y)]$

$=E[XY] - E(X)E(Y) - E(Y)E(X) + E(X)E(Y) = 0$


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10. There are three artists named MIG.

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# zbMATH — the first resource for mathematics

LTL with arithmetic and its applications in reasoning about hierarchical systems. (English) Zbl 1415.68144
Barthe, Gilles (ed.) et al., LPAR-22. 22nd international conference on logic for programming, artificial intelligence and reasoning, Awassa, Ethiopia, November 17–21, 2018. Selected papers. Manchester: EasyChair. EPiC Ser. Comput. 57, 343-362 (2018).
Summary: The computational bottleneck in model-checking applications is the blow-up involved in the translation of systems to their mathematical model. This blow up is especially painful in systems with variables over an infinite domain, and in composite systems described by means of their underlying components. We introduce and study linear temporal logic with arithmetic (LTLA), where formulas include variables that take values in $$\mathbb{Z}$$, and in which linear arithmetic over these values is supported. We develop an automata-theoretic approach for reasoning about LTLA formulas and use it in order to solve, in PSPACE, the satisfiability problem for the existential fragment of LTLA and the model-checking problem for its universal fragment. We show that these results are tight, as a single universally-quantified variable makes the satisfiability problem for LTLA undecidable.
In addition to reasoning about systems with variables over $$\mathbb{Z}$$, we suggest applications of LTLA in reasoning about hierarchical systems, which consist of subsystems that can call each other in a hierarchical manner. We use the values in $$\mathbb{Z}$$ in order to describe the nesting depth of components in the system. A naive model-checking algorithm for hierarchical systems flattens them, which involves an exponential blow up. We suggest a model-checking algorithm that avoids the flattening and avoids a blow up in the number of components.
For the entire collection see [Zbl 1407.68021].

##### MSC:
 68Q60 Specification and verification (program logics, model checking, etc.) 03B44 Temporal logic 68Q45 Formal languages and automata
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# Hole-doped room-temperature superconductivity in H3S1-xZ (Z=C, Si)

@article{Ge2020HoledopedRS,
title={Hole-doped room-temperature superconductivity in H3S1-xZ (Z=C, Si)},
author={Yanfeng Ge and Fan Zhang and Ranga P. Dias and Russell J. Hemley and Yugui Yao},
journal={arXiv: Superconductivity},
year={2020}
}
• Yanfeng Ge, +2 authors Yugui Yao
• Published 25 November 2020
• Physics, Materials Science
• arXiv: Superconductivity
We examine the effects of the partial substitution of S atoms by C and Si atoms on the superconductivity of H$_3$S with the $Im\bar{3}m$ structure at megabar pressure. The low-level substitution can fine-tune the Fermi energy to reach the electronic density-of-states peak maximizing the electron-phonon coupling. This can boost the critical temperature from the original 203 K to 289 K and 283 K, respectively, for H$_3$S$_{0.962}$C$_{0.038}$ at 260 GPa and H$_3$S$_{0.960}$Si$_{0.040}$ at 230 GPa…
17 Citations

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It is found that high-temperature superconductivity is closely associated with H clathrate structures, with large H-derived electronic densities of states at the Fermi level and strong electron-phonon coupling related to the stretching and rocking motions of H atoms within the cages.
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The AAS WorldWide Telescope from Python

# pywwt: AAS WorldWide Telescope from Python/Jupyter

Note: our cloud servers usually start up quickly, but if they were recently updated you may have to wait a few minutes for the backing software images to be rebuilt.

The pywwt package is the official toolkit for visualizing astronomical data in Python using AAS WorldWide Telescope (WWT). WWT is a free, open-source tool for visually exploring humanity’s scientific understanding of the Universe. It includes a sophisticated 4D WebGL rendering engine and a cloud-based web service for sharing and visualizing terabytes of astronomical data. WWT is brought to you by the non-profit American Astronomical Society (AAS), the major organization of professional astronomers in North America, and the .NET Foundation.

With pywwt you can:

• Visualize and explore astronomical data interactively in the Jupyter and JupyterLab environments through an HTML widget
• Do the same in standalone applications with a Qt widget
• Load data from common astronomical data formats (e.g. AstroPy tables) into WWT
• Control a running instance of the WWT Windows application

The full documentation, including installation instructions, can be found at http://pywwt.readthedocs.io/.

## For Developers: Testing

To test your pywwt checkout, use the pytest command.

The pywwt test suite includes a set of image tests that generate imagery using the WWT Qt widget and compare the results to a set of reference images. This component of the test suite can be finicky, even when everything is working properly, because the details of the rendering are dependent upon your operating system and OpenGL implementation. If your setup is yielding visually correct results, but the test suite is not passing for you, you can fix that as described below.

For a bit more context, each “image test” generates a WWT visual and compares it to multiple reference images. If any of those images is sufficiently close to the WWT result, the test passes. So if you’re running the test suite and the comparisons are failing, you need add appropriate new images to the corpus.

For a test like image_layer_equ, the reference images are stored in the subdirectory pywwt/tests/data/refimg_image_layer_equ. The filenames of the reference images within that directory don't matter, and are intentionally uninformative since the same reference image might match a wide variety of rendering platforms.

If you run the test suite with the environment variable $PYWWT_TEST_IMAGE_DIR set to a non-empty value, the WWT visuals generated during the test run will be saved in the named directory. For any images that fail tests, difference images with names resembling image_layer_equ_vs_a.png will also be saved. So to update the image corpus so that the test suite passes for you, run the test suite in this mode, then copy the failing images to the appropriate reference image data directories. Don't forget to git add the new files! And you should also verify that your new images do in fact look “reasonable” compared to what’s expected for the test. You can also run python -m pywwt.tests$imgdir1 $imgdir2 ..., where $imgdirN are paths to directories or Zip files containing images generated during one or more test runs. This will compare those images to the current corpus of reference images, and indicate whether there are images in the reference corpus that could potentially be removed. Note, however, that this is only safe if your collection of \$imgdirN spans all pywwt rendering platforms of interest. If there’s a developer that runs the test suite on MacOS 10.10 and your collection doesn't include those samples, you run the risk of breaking the test suite for that person if you remove the reference files that they need. That being said, it is quite possible for reference images to get out-of-date as the rendering code and test suite evolve. On the third hand, deleting files from the Git repository doesn't actually make it smaller, so removing old reference images only helps a bit with housekeeping.

## Reporting issues

If you run into any issues, please open an issue here.

## Acknowledgments

The AAS WorldWide Telescope (WWT) system, including pywwt, is a .NET Foundation project. Work on WWT and pywwt has been supported by the American Astronomical Society (AAS), the US National Science Foundation (grants 1550701, 1642446, and 2004840), the Gordon and Betty Moore Foundation, and Microsoft.

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# Computing only one byte of a cryptographically secure hash function

Let $H$ be a cryptographically secure hash function, like SHA256. The output of $H$ is $N$ bytes long, with $N$ usually in the order of $32$ for commonly used hash functions.

Now, be $X$ a string, I can compute $Y = H(X)$, which is a sequence $Y_0, \ldots, Y_{N-1}$ of bytes. Now, I was wondering: is there any way to have some $H^{(i)}$ so that $\forall X,\; H^{(i)}(X) = H(X)_i$ (i.e., a function that extracts the $i$-th byte of $H(X)$) but so that computing $H^{(i)}$ is faster than computing $H$, then extracting the $i$-th byte?

In other words, if I want to just compute one byte of $H$, can I avoid carrying out some computations and save some CPU?

(I hope not. If not, is there a proof for this?)

• I'm pretty sure it's a common security assumption (which hasn't failed us so far) that short-cutting the hash is impossible. – SEJPM Oct 26 '16 at 14:36
• It probably depends on how much faster and if you want this property for all $i$. Certainly if $H(X)$ can at best computed in time $T$, there is no way that $H^{(i)}(X)$ could be faster than $T/N$ for all $i$ ... because otherwise you could just use the $H^{(i)}$ functions to compute $H$. However, they maybe certain bytes that can be computed slightly faster. – Guut Boy Oct 26 '16 at 14:57
• Please define what you mean by "cryptographically secure". The answer depends on the precise definition. Do you mean "behaves like a random oracle"? Or do you mean "one-way and collision-resistant"? The question will have two different answers, depending on which you mean. – D.W. Oct 26 '16 at 18:56
• If there was a significant reduction, bitcoin miners would already be using it. – k-l Oct 26 '16 at 23:54

For SHA-256, given its structure, the savings achievable when computing a single byte of the output is small. Argument: the computation consists of 64 rounds in cascade (or several times that for input of at least 56 bytes); full diffusion is achieved after 8 rounds, therefore all but the last 7 rounds must be computed in full (and then a large fraction of the rest, varying with which byte is kept), therefore savings are (much) less than 7/64 on the rounds, and a little extra operation on the output (worth less than 8 32-bit additions). Thanks to tylo for reporting that diffusion is much slower than I stated.

This is largely true for all common hash functions, but we can not generalize to any hash function. It is easy to construct a hash function of cost about proportional to the width needed (but somewhat inefficient, or insecure).

• That's what I wanted to hear. – Matteo Monti Oct 26 '16 at 14:56
• Actually, in the cascade most of the variables are just shuffled around. Only two of them depend on the entire previous round. But that's a minor detail, and even then the saving is ranging from nothing at all (working variable $A$ requires the execution of the full round) to 2 rounds (for variables $C$ and $H$). In the end it doesn't change much, though. – tylo Oct 26 '16 at 15:18
• @tylo: Looking at it more carefully, I now believe that A has no way to influence H before performing 7 rounds; and then I'm not sure that's quite full diffusion, so I added an extra round to err on the safe side. – fgrieu Oct 26 '16 at 19:25
• @fgrieu Yes, it's a little more complex than 0-2 rounds, but looking at "half rounds" (calculation of $E$ but not $A$) and comparing them to full rounds would actually require looking into more details even. About the example with $A$ influencing $H$: you're right, that $A$ itself needs 7 roundsfor that, but from that round you do need $E$, so that the saving isn't the full round. – tylo Oct 27 '16 at 9:50
• @tylo: I think we can safely conclude that the achievable saving is much less than 7/64 and call it a day ! – fgrieu Oct 27 '16 at 9:57

There are definitely secure hash functions for which it is possible to calculate certain bytes fairly quickly.

There are some concatenation constructions (SHA-1(X)||MD5(X)) for which this is blatantly obvious.

While they do exist for password hashing, I am not aware of any requirements for computation time for cryptographic hash functions in general. It should therefore be possible to construct one that can calculate small parts faster than the entire function value.

• The question here is not to build a broken scheme, we assume that $H$ is secure, and we would like to know if it is possible to short cut some computations in order to target some parts of the output. – Biv Oct 26 '16 at 15:14
• Yes, but from the fact that H is cryptographically secure (i.e. collision and preimage resistant) it does not follow that there is no faster way to calculate the entire H(X) or parts of H(X). – Elias Oct 26 '16 at 15:17
• There could for example simply be a more efficient implementation of the entire algorithm for computing H. This does not influence the cryptographic strength. – Elias Oct 26 '16 at 15:19
• I don't know why this answer is down voted. It appears to be correct, and it answers the question. The question was "are we guaranteed that...?" and the answer is "no; here's a counterexample". That's a perfectly valid answer. @Biv, I disagree; a counterexample (what you call "building a broken scheme") is a perfectly acceptable way to demonstrate that a particular statement is false. Note that the concatenation construction is secure, under at least one standard definition of security for hash functions. – D.W. Oct 26 '16 at 19:01
• +1 to counteract the (in this case unjustified) kneejerk reaction that some people automatically downvote answers that suggest MD5(). Perhaps replacing that example with ( SHA3(X) || Argon2(X) ) would have prevented those (unjustified) downvotes. – David Cary Oct 27 '16 at 20:46

As SEJPM said, one of the common security assumption for a hash function is that there is no way to retrieve the hash but to calculate the whole function. In other word there should be no short-cutting.

Let us have a look of how we could accelerate calculation. This could be done either by trying to reduce the numbers of iterations over the message and by tweaking the main function.

Considering the 2 mains constructions schemes we have two cases:

In both case the message has to be completely absorbed in order to retrieve the hash. So no way to cheat on that part, we will need to process the whole message.

Why is this necessary ? Because each bit of the message has an influence on the hash. If it wasn't the case it would pose some serious security threat: the difficulty to forge a hash would be less complex.

However, we know that a hash function is composed of a round function applied multiple times. Can we simplify this process and have a tweaked round function that compute only the desired bits ?

It could be an idea, but you will notice that after few rounds, the bits that you are targeting depends on full states. E.g. for MD5, $19$ rounds (over the $64$) implies a full propagation of a modified bit (source code). Therefore the you need to compute the exact full state for the $64 - 19 = 45$ iterations left of the round function.

Therefore you during the main absorption phase you won't be able to cheat and only after fully absorbed the message, you will have to do $45$ normal rounds before using your $19$ tweaked ones. This surely is not a significant gain of speed.

Also notice that the number of operations (without considering the data provenances) will be relatively similar.

• "one of the common security assumption for a hash function is that there is no way to retrieve the hash but to calculate the whole function" - No, that's not a common assumption. There are two standard assumptions for modeling hash functions: (a) the random oracle model, or (ii) one-way + collision-resistant. I don't think either one necessarily implies that there's no speedup; certainly one can easily construct hash functions where there is a speedup but that satisfy (ii). – D.W. Oct 26 '16 at 18:59