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## why does the pencil appear to bend in water

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In given Figure shows a pencil partially immersed in a cup of water. In this case, we're looking at how light waves travel through the air, through water and through glass. This is an important consideration for spearfishing from the surface because it will make the target fish appear to be in a different place, and the fisher must aim lower to catch the fish. • The pencils in the different liquids appear to bend different degrees. Pull it out of the water. Is the pencil actually bent? One of the best liberal arts colleges in Pennsylvania, located in suburban Philadelphia. Drawing rays, explain why the pencil appears more sharply bent when viewed at an angle?
(2) As a result, the immersed part of the pencil does not appear straight with respect to the part outside the water, but appears to be raised. And that makes the pencil look like it bends slightly where it enters the water. Solution Answer: A Justification: Pencils do not bend in water. Explanation of Solution The phenomenon that makes the pencil appear to be bent is called refraction. Incident rays from pencil travel from water to air, and they bend away from normal. When light traveling in one transparent medium,(air) encounters a boundary with a second transparent medium (water), a portion of the light is reflected and a portion is transmitted into the second medium. Refraction is a physics concept that refers to how a wave travels through a medium. The depth that the water appears to be when viewed from above is known as the apparent depth . Why does the pencil appear to be bent? You are describing the physical property called refraction. This happens when light leaves one medium and enters a new one, causing the light to change speed. The refracted rays appear to Why? Due to this, a phenomenon called refraction occurs which causes the light rays to bend and cause the illusion that the pencil is … The pencil is then slowly moved across the middle of the glass from a centered position to an off-center position. Question(s): Why does it do that? Imagine a car driving along a road and suddenly its left wheels hit sticky tar, it would turn left as its right wheels are moving quicker than the left until the right wheels too hit the tar and match speeds resulting in the car now moving in a straight line. Find 25+ flavors, Delivered Directly to Your Door. When viewed from a side, apparent Objects do appear larger (or equivalently nearer) underwater when wearing a mask or goggles. See the image below for confirmation of this fact. Powerful Prototyping and Developer Handoff Axure RP 9 is the most powerful way to plan, prototype, and hand off to developers, all without code. The Broken Pencil A common classroom demonstration involves placing a pencil (or similar object) in an upright position in a round glass of water. 2 See answers Brainly User Brainly User hey buddy! What we found out! This phenomenon occurs due to the difference of refractive indices, because there is a difference in the index of air and in water, so therefore due to that matter the stick appears to bend and that a law of refraction 1. To an observer image of the immersed part of the pencil forms at a different position and pencil appears to be bent. The speed of light in water is less than that in air. Add this limited-time … The only time light rays don't bend as they leave water is when they are perpendicular to the sur- face. Rays of light starting from A suffer refraction at the water surface XY, bend away from normal and appear to come from A'. A bulb is placed 6 m below the water surface in a pool.What should be the area of a carboard, Which has to be placed over the water surface above bulb such that no light rays comes out of the water surface[take refractive index of water as 4/3] Bent Pencil Apparatus transparent container of water, for example a glass beaker, pencil Action The students observe the pencil through side of the container. In thi situation light travels slower in water than it does in air and this difference causes the light to bend. Refraction in a water surface Looking at a straight object, such as a pencil in the figure here, which is placed at a slant, partially in the water, the object appears to bend at the water’s surface. 2. This causes the pencil to appear higher and the water to appear shallower than it really is. Because of refraction parts of the pencil in water appear to rise up. The densities of the liquids are different so Why does the pencil appear broken in the cup of water? the indices of refraction of water and air are different, so light "bends" when it moves from one medium to the other... this causes the pencil to appear … Download a free trial and see why professionals choose Axure RP. Light rays coming from parts of the pencil under water are bent away from the normal as they emerge into the air above. In this lab, we're investigating some of the wave properties of light by examining why a pencil looks different in water. this is because light comes from a direction above the actual position of the pencil immersed in water. Question options: Electromagnetic/Light waves compress Question options: Electromagnetic/Light waves compress If you put a pencil in a cup of water, it looks as if it is broken and larger in the water. If the pencil was viewed from the side of the glass, you wouldn't notice any change apart from slight magnification. Explain with the help of a diagram why pencil partly immersed in water appears to be bent at the water surface. Fill the glass or jar half way with water. of water. The rays enter the eye (or camera) at angles closer to the horizontal, thus the parts of the pencil under water appear closer to the surface than they actually are, so the pencil appears bent. That’s why spearfishing is so hard to master. Place a pencil in a glass filled with water and what do you notice? This makes the beam bend one way or the other based on Rays of light from the tip (B) of the pencil bend away from the normal as they go from water to air. The change in speed makes it appear that the pencil is bent. Therefore, the pencil ABC appears as A'BC, bent at the water When a pencil is obliquely dipped in water , the ray of light travelling from pencil to observer travel from denser medium to rarer medium & hence bend away from normal . Hence, a ABC is a pencil immersed partially in water. During this propagation, they bend away from the normal on refraction. a pencil partly immersed in water appears to be displaced at the interface of air and water due to the phenomenon of refraction. Why does a spoon appear bent in water? Why does the pencil appear to bend where it enters the water? However, looking at this glass from the outside, the pencil appears to be bent. Phenomena: A ruler looks bent in water at different angles. Wait! From far away, the fish you are eyeing always looks nearer to the surface of the water than it actually is. Some of these are: why do fish look larger in the water, what causes a spoon to appear bent in a glass of water, why does light travel indefinitely in an optical … Suppose the water surface had no waves, the pencil you placed in there would appear in its original form. They should note that the pencil appears to bend at the air-water Why is a pencil partially immersed in a cup of water appearing to be bent. Fruit-infused, Flavored Water With no Sugar, No Diet Sweeteners, No Calories. Bending Pencil Materials you will need: • Pencil • Water • Clear Glass or Jar Can light play tricks on your eyes? As you sight at the portion of the pencil that is submerged in the water, light travels from water to air (or from water to glass to air). MEDIUM View Answer A fish is a little away below the surface of a table. Can you tell by the direction of the bend whether the air or water has the higher refractive index? Consider a pencil AB, such that CB portion of the pencil is immersed in water as shown in figure. Why does the pencil appear bent? When stick is immersed partly in water then light coming from stick refracts at the surface of water due to change in density of medium. Why does the pencil appear to be bent? Why is this? Place the pencil in the water. The interface between the water and your mask obeys Snell's law which can be written, in the small angle approximation, as  n_1\theta_1=n_2\theta_2. The part of stick above water is in same medium as your eyes are in, so light travels along Why Does a Pencil Look Bent in Water? How much does light Why does the pencil look bend? As the the water (or other liquid), it changes speed.


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# CriCri Composite

Discussion in 'Composites' started by ULF, Jun 16, 2019.

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1. Jun 24, 2019

### ULF

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Hi Bob,
Searching for carbon fibers I came across bidirectional non- crimp fabric https://shop1.r-g.de/en/art/192150127-EBA, with unidirectional layers +45°/-45, there I saw, some +45°/-45°, and others -45°/+45°.
Ihave never used such bidirectional stuff, I used plain weave, cut 45°.
Do you know about non-crimp fabric? Which one is better,+45°/-45°, or the other way? Do I have to cover it with thin fabric?

John

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2. Jun 24, 2019

### BoKu

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First off, a reminder of my usual warning: I am not an engineer. I do a bit of my own engineering at a very basic level, and I have figured out a regime of composite construction that works well in my remote workshop. But for real engineering, ask Billski or someone like him.

The fabric at your link is what I have heard called a "knitted double-bias." We use fabrics like this in fiberglass, but not in carbon. I think the "non-crimp" feature is because it is not woven, so the fibers stay straighter and have better stiffness. However, I suspect the overall strength and stiffness is probably only modestly greater than conventional woven fabrics. But the weight and price look reasonable, so it is probably worth a try.

Unless you are stressing the material well into its capacity, I think that the order of orientation is probably immaterial; I don't think it matters at all whether it is +/-45 or -/+45.

The 6oz cloth we use a lot of is the 48" 3K Carbon Biaxial 2x2 Twill on this page. The neat thing about it is that it has tiny polyester interstitial threads along the 0 axis so that it does not distort as easily as when you cut +/-45 swaths out of conventional 0/90 cloth. We pre-cut entire 7.5m wing skin swaths, set them aside, and unroll them into the mold when needed. Using this material has reduced the wing skin layup time by more than half compared with cutting individual bias swaths from conventional fabric. It has also made it practical to make each new wing skin set with a new, untrained crew as a training exercise.

Whether you cover the carbon with some other cloth depends a lot on your methodology. Some shops use a thin veil cloth between the carbon and the mold surface. Some shops use a randon-strand veil mat. What we do is use a vinylester in-mold primer (like gelcoat, except filled with microballoons instead of talc so it is lighter and softer), roll on a tie coat of microballoon-thickened epoxy, then put down the first ply of carbon. The in-mold primer gives a good exterior surface on the outside and a good epoxy wetting surface on the inside to eliminate pinholes. The tie coat fills the weave of the carbon to reduce print-through. It's not perfect, but perfectly good for parts with a long in-mold dwell time that get polyurethane paint after assembly.

Thanks, Bob K.

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3. Jun 24, 2019

### Vigilant1

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A slight diversion: We've discussed the minimum exterior skin strength/toughness needed to withstand the real world. If I've got this right, it sounded like Bob had found that 6 oz CF worked well if Divinycell or other substantial core was behind it. Billski has found that a more substantial skin (17+ oz FG, per Rutan el al?) is typical and needed when using massive 2.2 pcf XPS cores.
What would be the likely problem/failure mode in the real world if 6oz CF is laid onto a solid core wing of typical 2.2 pcf XPS? Divinycell has greater compressive strength than XPS, but the skin (thick or thin) is so much stiffer than either it would seem to be taking the point load of the ball-point pen or errant child regardless of which foam is backing it.
I'm not suggesting 6oz CF will work in this application, I'm just trying to understand why it wouldn't.

4. Jun 24, 2019

### Marc Zeitlin

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I've used this material, or at least something extremely similar (knitted +/-45 carbon) on a large aircraft project last year. There is no "other way" - +45/-45 is just -45/+45 turned upside down. If you care which way the first layer goes, just turn it over. In practice, you don't give a crap.

You put a layer of Fiberglass over the carbon if you care about what's going to touch the carbon from a conductivity/corrosion standpoint, or if you want to be able to have a sanding surface that doesn't ever impinge on the carbon.

My \$0.02.

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5. Jun 24, 2019

### BoKu

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I think you'll find that if you flip over a piece of +/-45 over, what you see is still +/-45. If for whatever reason you actually need -/+45, you have to turn the cloth 90 degrees. It's like how if you take a tractor prop and put it on the crankshaft backwards, it's still a tractor prop--just not a very good one.

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6. Jun 25, 2019

### Marc Zeitlin

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Sheesh. You're right - thanks for the correction . But in any case, whether the +45 or -45 is on the top, for normal wing skin type layups, it doesn't much matter.

It MAY matter if you're trying to make a balanced layup with a lot of plies and you can't turn the cloth 90 degrees...

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7. Jun 25, 2019

### wsimpso1

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You are aware that PU foam when heated releases isocynate and related gases, polyol, and other nasty chemicals. This is a high risk process for you and anybody downwind of your operation. Please do some research on the materials and effects in exposure to temperatures reached in hot wire cutting.

For everyone else, we do not hear about people hotwiring polyurethane foam for very good reasons - it will poison you. Just don't.

Billski

8. Jun 25, 2019

### wsimpso1

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Listen to Bob. He has way more experience than I in graphite, plus several of his designs are flying. That counts for a lot...

Technically, if you took all of your BIAX cloth off one roll and used it for both sides of both skins on the wings or on both facings on a massive core, you will have non-symmetric laminate that will have a tendency to twist when temperature changes or is curved under wing bending moments. Thing is that the asymmetry is tiny and so is the twist. Here is why. A BIAX cloth is one layer at +45 and the next at -45, with these two plies 0.0045" apart. Then a 1/4" away you do it again. The twist effects go with the square of the separation, so the bending stiffness compared to the twisting effect is (0.250/0.0045)^2 = 3086. You will have to work really hard to even measure the twist effect while you will see the tips bending under flight loads.

Billski

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9. Jun 25, 2019

### wsimpso1

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I am not saying I have found 3 plies of 7 oz UNI to be adequate, the entire world of Rutan canard ships and their many derivatives have found it to be adequate.

The issue with blue foam at 2 pcf is soft and low strength compared to 6 pcf or 8 pcf Divinycel. Impact damage appears to be the big issue, and the deflections that produce the forces that break the foam from the fiber-resin layer are related to EI of the skins. If 3 UNI in glass is adequate on blue foam, then anything else with that much EI should also be OK.

I was using thickness per ply of 6 oz graphite fiber cloth at 0.0075" and E on the bias of 8.4E6 psi (vacuum bagged). One ply has much lower bending stiffness, two plies is about the same as the glass. That was my basis for same tolerance for impact.

Put one ply of anything on Divinycel foam, and it is way sturdier than that same ply on blue foam. Use one ply 6 oz CF biax cloth on blue foam, and I would expect it to be much more sensitive to impact than 3 UNI (glass).

That being said, 2 plies of GF 6 oz cloth is still much lighter than 3 UNI 7oz glass.

Billski

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10. Jun 25, 2019

### BBerson

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I would try a 30" bow saw (for logs) instead of a hot wire. Turn the blade 90° in the bow.
I haven't tried it so I don't know if it would work.

11. Jun 25, 2019

### Marc Zeitlin

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Just to clarify, while there are areas of Rutan derivative canards (top of wings, fuselage forward of front seatback, etc.) where there are 3 plies of UNI glass, the bottom of the wings and the majority of the fuselage are 2 crossing plies. So from a damage tolerance standpoint, 2 plies is reasonably acceptable. The only place where it isn't (from a damage tolerance standpoint) is on top of the strake where folks kneel to get into the rear seat - there, damage DOES occur on occasion. 3 plies (one spanwise) is use on the top of the wing for buckling resistance. So depending upon the usage, likelihood of finger/pointy object poking, buckling resistance, etc., some places need 2 plies, and some need 3.

And yeah - DON'T hotwire Polyurethane foam. Some folks did that once at Scaled and I blew a gasket - didn't happen again.

12. Jun 25, 2019

### ULF

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Hi there,
Short pieces of 2ft, or under give excellent results hotwired. Very accurate cuts.
John

13. Jun 25, 2019

### wsimpso1

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The Master's Degreed Material Scientist and career long Polymers Engineer in this house (she is my wife) says you are exposing yourself to isocyanates, polyol, and other toxic stuff when you hot wire cut PU or otherwise heat to melting. Please stop...

Billski

14. Jun 25, 2019

### PiperCruisin

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Burning/melting a polyurethane foam will yield hydrogen cyanide, benzene, carbon monoxide, and other fun stuff. So, yes, I concur, hotwire cutting the stuff is simply out.

fyi. Polyurethane comes about when you mix a polyol (organic longish-chain molecule with some OH groups) and an isocyanate. The isocyanate (there are many types) is what you can become sensitized to...so use proper PPE. Of course polyurethanes are very versatile. They can be the foam you are sitting on to high performance paint.

15. Jun 25, 2019

### ULF

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haha, well once every 5 years, on a windy day, Ill probably survive without prtmanent damage! I always wondered what that smell is.
Many thanks to the master!

16. Jun 25, 2019

### BJC

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Unbridled optimism and dismissiveness are among the first symptoms.

BJC

17. Jun 25, 2019

### ULF

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Hi Billski, Bob mentioned you are an engineer, and I wanted to ask for your opinion.
On aircraft, it is not uncommon, to oversize parts sometimes, for practical use, groundhandling and so on.
I did some preliminary calculation about the size of the upper carbon spar capstripe, for 9g.
The aircraft weighs 150kg without wings, that comes up to about 17KN.
With 600MPa published compressive strength of CFK, 17000/600 ~ 28mm² (.043 sq in) of carbon, at the root. Pretty small, but its only a 7ft wing.
Now if I double it, so its an area of 56mm² (.087 sq in). Still not a weight factor.
Now I looked at the original size spar, made of aluminum 2024 T4, with a cap stripe dimension near the root, of 190 mm² (.29 sq in), that is almost seven times that of my carbon cap stripe calculation.
Taken, that carbons compressive strenght is some what double to 2024 T4 (it isn`t really), looks like the original spar could be good for almost 30g, not 9. ?

What would you say, that one should aim for on the capstripe dimension at the root of a cricri composite wing ?

John

18. Jun 25, 2019

### wsimpso1

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The same stuff is in solvent borne polyurethane paints, and you must use supplied air to be resonably safe with that - no filter masks.

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19. Jun 25, 2019

### wsimpso1

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Some folks will use analytical methods to fully size main structures, then add in another ply for grandma (Scaled is reputed to do this). For the most part, you will not find spars to be grossly oversized - the next smaller increment either failed analytical checks of some sort or failed in test and was bumped to the next available thickness. Most oversize I have seen in parts is the result of either shop expedient or min gage restrictions. I would not assume that the original was randomly oversized - Columban did good work in the original.

Basic sizing of composite spars (like you have done for your caps) is just the starting point on composite flanged beam designs. I wrote on it here and elsewhere:

https://www.homebuiltairplanes.com/...ite-plates-beams-and-bigger-structures.29030/

First problem with basic sizing is usually that folks forget that the centroid of the cap is where that force has to be calculated and they just use wing depth. Start with wing thickness, subtract two skin thicknesses, subtract two glue line thicknesses, subtract two spar outer wrap thicknesses, subtract one cap thickness, divide all that by two - now you have about the right arm for sizing symmetric spar caps. Then the stress at the top and bottom fibers of the caps is still higher than that sizing's assumption. And if you go with asymmetric cap sizes, more cap beefing is needed.

Then the spar designed with basic sizing for caps (compression load / compressive stength) and shear web (shear load / shear strength) will almost always indicate failure at lower than intended loads in the web next to the flanges. You gotta beef up something to get to 2.0 times in the analysis, and usually the lightest way to get to strength is beefing both the caps and the web. I advocate surveying the design space for the optimal mix at each of several stations - Left pin, every few inches in the overlap, right pin inboard, right pin outboard, then every few inches as you go towards the tip until you get to MIN Gage on caps and web. Doing this with FOS of 2.0 might find you still with very favorable weights on the main spar.

PM me if you want to share data...

Billski

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20. Jun 25, 2019

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# How would I transport an atmosphere?

An abandoned outpost from a long extinct race is discovered. It can easily be repaired and would make an ideal location for a colony, however it has been breached and any usable air has been long lost to space. The outpost has its own power systems that we are capable of turning back on and using, however there is no method on the station to create a breathable environment for humans. It's quite the journey and a colony ship would need to be in transit for more than a decade to reach its destination.

If a colony ship was dispatched to this location, what would it need to bring to generate enough Earth-like air for the population to breathe and use?

My first idea was solid ammonia and liquid oxygen… Burn the ammonia and let it release its nitrogen along with water. Use electrolysis to recover some of the oxygen back as required (or leave it as usable water). The issues here:

• Way too high of a oxygen to nitrogen ratio… as nice as ammonia is for this, the nitrogen content is relatively low and is created at a 3:1 ratio with oxygen.
• Need to carry too much liquid oxygen. It would be nice if one chemical compound can be released instead of two separate ones… also not a fan of storing high pressure gas on a long trek colony ship
• Too much hydrogen… there are a few uses for it (water/fuel), however the ratio of nitrogen to hydrogen is too low and an excess of hydrogen is created in trying to get Earth-like conditions.

Assume any chemical compound can be reasonably manufactured. Liquid gasses require high pressure and have some degree of danger in space transport… solid compounds preferred, but not an end all.

What would you take on your colony ship to create an Earth-like atmosphere in as little amount as storage space possible and as safely as possible?

It is preferable to bring the material on the ship as opposed to trying to harvest it. The tech level on the propulsion systems is limited and the speed to get there depends on some gravitational slingshotting to get it up to speed already, so there isn't the opportunity to stop to harvest anything and then re accelerate.

• @Nick2253 See this question – DoubleDouble Feb 18 '15 at 22:14
• I figured CO2 could be generated by burning a small amount of the plant matter thats part of the farming operation on the colony ship. Or bring a small dead tree to burn upon arrival. – Twelfth Feb 18 '15 at 22:15
• @DoubleDouble A pure oxygen environment is regularly used on Earth for medical purposes. And without something to oxidize, a pure oxygen atmosphere won't be flammable. – Nick2253 Feb 18 '15 at 22:17
• What's wrong with frozen air? – Oldcat Feb 18 '15 at 22:33
• No one did put Spaceballs referrence here? Am I really the first one to do it? Ok, here you go: youtube.com/watch?v=VptOUWC-Itc – Pavel Janicek Feb 19 '15 at 10:02

## 6 Answers

Per your question, it sounds like you preferred something that was not a pure oxygen atmosphere. In that case, nitrous oxide ($N_2O$) might be an a good match. At 1 atmos. pressure it becomes liquid at -88.5 °C solid at -90.9 °C easily maintained at a 10 year trip in space.

On arrival, $N_2O$ could be easily decomposed into $O_2$ and $N_2$.

I am not aware of any stable compound that has a ratio closer to the N/O ratio of earth's air. The atmos. pressure in your station would only be 60% of "normal" but that might be acceptable.

Ammonium nitrate melts at 169 C, and leaves no residue when heated:

$$\text{NH}_4\text{NO}_3 → \text{N}_2\text{O} + 2\text{H}_2\text{O}$$

Like Gary says $\text{N}_2\text{O}$ can be decomposed into nitrogen and oxygen.

But for the reality check, hydrogen, nitrogen and oxygen are very common elements. The colony ship is bound to find some compounds they could extract it from, although it takes longer.

• Good pt. re: availability. You would also be very likely to want water and food. CHON -- it's what we are made of. I have to assume this question is intended as a startup supply to tide you over before you start mining comets and asteroids. – Gary Walker Feb 19 '15 at 19:17
• +1, I gave the check to Gary's answer, but your expnansion here is very useful. Thankyou. – Twelfth Feb 20 '15 at 17:57
• I definitely like the advantages that this has over N2O, not only that it's a solid at room temperature (much easier to load into your space ship) but that it also decomposes into N2O and water. – Draco18s Jun 21 '18 at 1:29

People in an enclosed environment don't need anything but oxygen. While it's toxic at high partial pressures, you can account for this by having a lower ambient air pressure than what exists on Earth. For example, an atmosphere with a pressure of .2 bar will have enough oxygen for people to breathe, but won't be toxic. Issues with flammability are also tackled by reducing partial pressure, since it is what determines reaction rates in a gas. Reducing partial pressure will also reduce the structural loads on your habitat, making it easier to patch holes and reducing the rate of air loss if there are any small breaches.

If you don't like liquid oxygen, there's another convenient way to transport it: liquid ozone. Ozone has a higher boiling point than oxygen by about 20 degrees, and is also 50% denser than oxygen. It also self-oxidizes, so you can burn it to produce energy, with the byproduct being oxygen.

The downside is that anything that's a strong enough oxidizer that you can burn it to produce another strong oxidizer is a really good oxidizer which means that a tank of liquid ozone laying around needs to be treated really carefully, or it will explode, even without anything else to burn, because ozone will burn itself. But hey, lower volume for storage and free energy if you treat it right.

• Assuming there is some co2 added to the mix, would plant life be able to handle this? And it still seems like a relatively high risk of combustibility, even at the lower pressures...am I wrong there? – Twelfth Feb 18 '15 at 23:14
• Combustibility is mostly a matter of partial pressure rather than fractional concentration. – Mark Feb 19 '15 at 0:21
• I like the answer, but I'm going with the check mark beside the answer that came close to an earth environment. – Twelfth Feb 20 '15 at 17:57
• WRT flammability in a pure oxygen atmosphere, remember Apollo 1. – jamesqf Nov 9 '16 at 17:02
• Ozone is unstable. It will try to revert back to dioxygen, and will produce lots of energy doing so. Hot ozone is more unstable, but even liquid or solid ozone won't last 10 years. Also if it starts to heat up, it will explode. And the very hot, high-pressure newly created oxygen will be happy violently oxidizing everything it touches, be it by corroding it or setting it on fire. – Eth Jun 18 '18 at 15:01

There are several sustances that can use to generate your atmosphere.
Sice they are quite large, right now I'll post only one of them and when I have more time I'll post the other ones.

You will note that this answer is quite large. That is because I also explain how to produce the sustance, and how to produce the sustances which produce it (maybe that information could be useful for someone who is willing to manufacturate the compound instead of bring it from Earth). If you aren't looking for that, only read up to production title (without include it), In order words, only read the first quarter/fifth of the post.

Also note, that the most important part of my answer is under the tile usage.

Finally, if someone have any question, ask in comment. I'll do my best to address the problem.
Good luck reading!

# Nitrous Oxide $\text{N}_2\text{O}$ (Or ammonium nitrate $\text{NH}_4\text{NO}_3$)

$\begin{array}{|c|c} \hline \text{Molar Mass} & 44.013 \text{g/mol} & \\ \text{Density} & 1.977 \text{g/L (gas)} \\ \text{Melting Point} & −90.86 \text{°C} \\ \text{Boiling point} & −88.48 \text{°C} \\ \text{Price} & \text{¿?} \\ \hline \end{array}$

Or ammonium nitrate, see on industrial production.

Introduction: Commonly know as laughing gas or nitrous. At room temperature, it is a colorless non-flammable gas, with a slight metallic scent and taste, and faint, sweet odour. At elevated temperatures, nitrous oxide is a powerful oxidizer similar to molecular oxygen.

Uses: Futhermore build an atmosphere, you can use nitrous oxide in several ways.

• Recreational (due to euphoric effects upon inhaling).
• Anaesthetic and pain reducing on surgery and dentistry. (At 50% concentration can be administred by non-professional people)
• Rocket propellants (used as an oxidizer). Can be use also as a monopropellant rocket with a catalyst.
• Motor racing to increase the power output of engines.

Advantages:

• It's non-toxic. (Long-exposure and abuse can produce vitamin B12 defficiency...)
• It's stable at room temperature -making easy to store and carry on flight-.
• It's possible to decomposed readily to from breathing air.
• It has high density and low storage pressure (when maintained at low tempertarue). That enable it to be highly competitive with stored high-pressure gas systems.

## Storage:

This can be storaged in 20-25 bars tanks at -20 °C or in high pressure tanks at 45-60 bars.

## Usage:

In the presence of a heated catalyst, $\text{N}_2\text{O}$ will decompose exothermically into $\text{N}_2$ and $\text{O}_2$, at a temperature of approximately 577 °C (this even produce energy!).

$$2\text{N}_2\text{O} \Rightarrow 2\text{N}_2 + \text{O}_2 + 82 \text{ kJ/mol}$$ $\begin{array}{|c|c|c|c|c} \hline \text{Examples} & \text{N}_2\text{O} & \text{N}_2 & \text{O}_2 & \text{Energy}\\ \hline \text{by Mass} & 1,000\text{g} & 636.48\text{g} \text{ }(63.64\text{%}) & 363.51\text{g} \text{ }(36.35\text{%}) & 1,863.08\text{ kJ}\\ \text{by Volume} & 1,000\text{cm}^3 & 1,006.33\text{cm} \text{ }(100.63\text{%}) & 502.92\text{cm}^3 \text{ }(50.29\text{%}) & 3,683.32\text{ kJ}\\ \hline \end{array}$

With nitrous oxide you can build and atmosphere of:

$\begin{array}{|cc|r|cc|r|} \hline \text{Chemical} & \text{gr/mol} & \text{Percentage} & \text{Mol Fractal} & \text{Mol} & \text{Partial Pressure} \\ \hline \text{N}_2 & 28.0134 & 63.65\text{%} & 2.272 & 0.666 & 67.550 \text{ kPa} \\ \text{O}_2 & 31.9988 & 36.35\text{%} & 1.135 & 0.333 & 33.775 \text{ kPa} \\ \hline \text{Total} & 60.0122 & 100.00\text{%} & 3.408 & 1.000 & 101.325* \text{ kPa} \\ \hline \end{array}$
101.325 kPa = 1 atm

Okey... that is... fine.

• A bit high oxygen value: Humans need around 21 kPa of oxygen to "work" properly, you will have 33.775 kPa, your people wouldn't suffer hyperoxia. Hyperoxia is produced when oxygen is above 50 kPa it becomes toxic. You will survive, but I think (complete and subjetive personal opinion) your crew will need a few days to adapt their lungs after stop coughing. Also, (again, my opinion) they will suffer some slightly diseases or deficiencies after some years.
• You can improve that reducing the overall pressure from $101.325\text{ kPa} \rightarrow 63\text{ kPa}$. So you will have $42\text{ kPa N}_2$ and $21\text{ kPa O}_2$. Good!
Also, reduce the overall pressure has an advantage: if there is a damage in the ship/station's hull, air will scape slowler from it, giving more time from crew to fix it.

Also, because of the large heat release, the catalytic action rapidly becomes secondary, as thermal autodecomposition becomes dominant (that means you don't need to store much catalyst).
Its catalyst can be any of this (I think): (Note that I don't know about catalysts and their usages)

• Rhodium: ($72.66/g). Density 12.41 g/cm3. • Cobalt: ($ 81.50/kg). Density 8.90 g/cm3.

• Platinum: ($28.56/g). Density 21.45 g/cm3. Strong. • Palladium: ($ 31.85/g). Density 12.023 g/cm3

• Cobalt: ($81.50/kg). Density 8.90 g/cm3. • Copper: ($ 7.20/kg). Density 8.96 g/cm3. Weak.

• Cerium: ($5.51/kg). Density 6.770 g/cm3. Weak. • Iron: ($ 71/t for ore). Density 7.874 g/cm3. Weak.

# Production:

Sadly, the only way to produce all this usances use the same compounds that they produce or with other very difficult to get, that means a circular reference. So you must bring it from Earth.

I am kind a fan of a low presure pure oxygen environment which already mentioned.

So you may use solidified oxygen. In the space there is no need to active cooling. You may keep it with simple insulation and a reflective wrapping from radiant heat sources (like sun) and from the rest of your spaceship.

In fact in the failed Apollo 13 misson there is a malfunctioning "heater" that keeps oxygen from solidification.

Or

You may use hydrogen peroxide H2O2. It's a liquid under the room temprature.

You may even transport it in the solid form (it's melting point is very close to water-ice).

For safety reasons it is usually mixed with water.

It's also used as a propellant in the rockets and the jetpacks.

With the silver catalyst, H2O2 very rapidly breaks up (bursts) to water vapour and a oxygen.

Personal Jetpacks used this reaction for a propulsion.


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## A Tricki issue

This is a mathematical post rather than a Tricki post, but the title comes from the fact that the issue I want to discuss has arisen because I have made a statement in a Tricki article and can’t quite justify it to my satisfaction.

The article in question is about a useful piece of advice, which is that if one is having trouble proving a theorem, it can help a lot to try to prove rigorously that one’s approach cannot work. Sometimes one realizes as a result that it can work, but sometimes this exercise helps one to discover a feature of the problem that one has been wrongly overlooking. I have a few examples written up, but one of them is giving me second thoughts.

The example in question is this. Let $A$ be a subset of $\mathbb{Z}_p$ for some prime $p$. Define a gap in $A$ of length $m$ to be a set $\{x,x+1,\dots,x+m-1\}$ of integers mod $p$ that contains no elements of $A$. A nice (and well-known) question one can ask is the following. Let $A$ be of size at least $\delta p$. How small can one make the largest gap if one dilates $A$? That is, if we write $\gamma(U)$ for the size of the largest gap in a set $U$, what is the largest possible value of $\min_t \gamma(tA)$ over all sets $A$ of size at least $\delta p$? (Here $tA$ stands for the set $\{ta:a\in A\}$.)

If you take a random set $A$ of size $\delta p$, then with high probability the largest gap in any dilate of $A$ will have size at most $C\log p$, where $C$ depends on $\delta$ only. After a bit of experimenting, it becomes natural to conjecture that this is the best possible bound. But is it?

This looks like a combinatorial question until one observes the following. Let $p$ be a prime of the form $4m+3$ and let $A$ be the set of all non-zero quadratic residues mod $p$. Then every dilate of $A$ is either the set of all non-zero quadratic residues or the set of all non-zero quadratic non-residues. Suppose it is the case that all of $1,2,\dots,m$ are quadratic residues mod $p$. Then, since $-1$ is a quadratic non-residue, all of $-1,-2,\dots,-m$ are quadratic non-residues. Therefore, every dilate of $A$ has a gap of length $m$. So the conjecture above will be false if there are infinitely many primes $p$ congruent to $3$ mod $4$ such that the first $m$ numbers are quadratic residues for some $m$ that tends to infinity faster than $\log p$.

Suddenly we have a number-theoretic question to solve, and it turns out to be a well-known unsolved problem. Nobody has even ruled out that there is some $\alpha>0$ such that for infinitely many primes $p$ all the numbers from $1$ to $\lfloor p^\alpha\rfloor$ are quadratic residues mod $p$. Even with the help of the Generalized Riemann Hypothesis this has been improved to $C(\log p)^2$ rather than $C\log p$.

I want to claim that this example demonstrates that the original problem is “not really” a combinatorial problem. But to do that, I have to argue against someone whose reaction to it is, “Ah, this shows that an apparently number-theoretic problem is in fact a combinatorial one.” This is hard, because there are some very nice examples of number-theoretic statements that have been hugely generalized and then solved combinatorially. So what is it about this problem that makes it appear that purely combinatorial arguments are doomed to fail?

One relevant factor is that many combinatorial tools apply to large systems of sets. Initially, it looked as though the set of dilates of $A$ was a large set system, but the example of quadratic residues shows that it can in fact be very small. This fact does seem to lie behind the general feeling (or at least, I think it is a general feeling) that the problem cannot be solved combinatorially, but I would like a more general answer to the following question. Suppose you are trying to prove a theorem of the form “For every $X$, $P(X)$.” Then suppose you come across a particular example of an $X$ for which you cannot prove $P(X)$. When is it clear that you will not be able to tackle the general theorem without first having a thorough understanding of why it is true for $X$?

Clearly the answer is not “always”. For example, $X$ might just be an example that is sufficiently complicated not to have any special features. It may then be slightly easier (for notational reasons, say) to think about $X$, but thinking about $X$ isn’t interestingly different from thinking about the general case in such situations. One relevant feature of the example I have just discussed is that the appropriate tools for tackling the question about quadratic residues seem to be number-theoretic, which makes the special case different from the general case. Another sign of that is that one can easily imagine a solution to the question about quadratic residues not helping all that much to solve the general problem. But why isn’t that an argument for aiming at the general problem and ignoring the particular one?

Another feeling I have about this question, but can’t do proper justice to, relates to the discussion in the previous post. If the conjecture is true, then you can’t find a prime such that the first $m$ integers are quadratic residues for a very large $m$. However, we can create something like a “probably inconsistent model” in which such a prime does exist, and then for any combinatorial argument about a general set $A$, we can interpret it as referring to our fictitious $A$ that consists of quadratic residues modulo our strange prime $p$. And in some sense our combinatorial arguments “never realize” that the model is inconsistent. Are there any reverse mathematicians out there who can make sense of this? Is it conceivable that there is some weak axiom system that is sufficient for combinatorial arguments, in which one might possibly not be able to prove that such a prime cannot exist?

### 15 Responses to “A Tricki issue”

1. Emmanuel Kowalski Says:

It is known that there are infinitely many primes for which the smallest quadratic non-residue is at least c log p, for some positive c, and probably this can be ensured with the condition that p is 3 modulo 4. (This goes back to Chowla and Turan, and can be guessed by considering the values of the Legendre symbol (x/p) for small x, and checking that, essentially, they behave like independent Bernoulli random variables for x up to a small constant times log p). Moreover, on GRH, Montgomery has improved this to c(log p)( log log p), so the natural probabilistic guess is (probably) not true for this problem. (And this tends to confirm that the problem is not purely combinatorial, if we believe that the Riemann Hypothesis is not combinatorial).

2. gowers Says:

Thanks for that — I rather suspected that my knowledge about this problem was incomplete. So perhaps my question should be slightly modified: why does it seem hopeless to find a combinatorial proof even of the weaker assertion that there is a dilate such that the gap has size at most $p^{o(1)}$?

3. Emmanuel Kowalski Says:

Well, from the point of view of classical analytic number theory, the $p^{o(1)}$ bound for the least quadratic non-residue (which is known in that area as a conjecture of Vinogradov) is considered out of reach at the moment because all “standard” attack methods depend on results which more or less directly involve properties of the L-functions which are considered extremely deep. (If not the Riemann Hypothesis, it can be the Lindelöf Hypothesis, or estimates for very short exponential sums).

Actually, that doesn’t mean there couldn’t be a clever combinatorial re-formulation that would lead to a proof of this conjecture. For instance, typically, the standard approaches lead to much more precise results than a bound just for the least quad. non-residue: they lead to equidistribution of residues and non-residues in very small sets; conversely, assuming conjectures on the size of this least quad. non-residue does not imply much about L-functions (as far as I know). So maybe it’s not “hopeless” to go around the L-functions, but at least this special case shows that the problem lies quite deep.

(Maybe an analogy could be drawn with the proof of RH for varieties over finite fields: Grothendieck’s “programme” for that was highly geometric, and maybe it was felt at some point that it was hopeless to look for an arithmetic proof, but in the end Deligne’s work did involve a lot a arithmetic — and a lot of geometry, but it didn’t involve proving first the intermediate results Grothendieck had in mind).

4. Terence Tao Says:

I know that when the sum-product phenomenon was discovered, and Jean Bourgain used it to show that thin sets of ${\Bbb F}_p$ with multiplicative structure (e.g. multiplicative subgroups) expanded rapidly under addition to fill the whole group, there was a bit of excitement as to whether one could show the reverse direction (i.e. whether thin sets of ${\Bbb F}_p$ with additive structure, such as an interval, expanded rapidly under multiplication to fill the whole multiplicative group), since this would in particular say something about the least quadratic non-residue (in particular, the analogue of Bourgain’s results would give a bound of $p^{o(1)}$). But it turns out that the techniques are not symmetric (Bourgain crucially uses at a key point that multiplication distributes over addition, whereas addition does not distribute over multiplication), and besides, the ultimate ingredients used in the sum-product analysis are too “elementary”, and do not seem to point out any obvious inconsistency (or at least improbability) in an alternate universe in which there was a large p with a big interval of quadratic residues.

It seems to me that any proof of the least quadratic nonresidue problem must, at some point, introduce a new (or at least underused) fact about quadratic residues that was not fully exploited in the past, as it seems that having a large least quadratic nonresidue is perfectly compatible with the “standard” facts we know about residues, though there is no real formalisation of this that I know of. (For instance, my theorem with Ben that the primes contained arbitrarily long arithmetic progressions used extremely little number theory, but it still needed to know something moderately non-trivial about the primes, namely that (roughly speaking) they were a dense subset of a pseudorandom set of almost primes – a fact which had been implicitly used a couple times in the past, e.g. in papers of Ramare and of Green, but perhaps not as systematically as it should have been.)

5. Emmanuel Kowalski Says:

Actually, I just read (I didn’t know it before…) that Graham and Ringrose have proved unconditionally that the least quad. non-residue is at least c(log p)(log log log p) for infinitely many primes, so the log p probabilistic estimate is indeed false.

6. Timothy Chow Says:

For the general question—when it is clear that you need to understand a specific X before you can prove it for all X—I am tempted to say that the answer is, “Never.” There are certainly cases where it *seems* that you need to understand the particular case before you can hope to prove the general case, and the quadratic nonresidue problem may be an example. But is this *clearly* true? I’m skeptical. For comparison, note that many people felt that the implication “every elliptic curve is modular => Fermat’s Last Theorem” showed that it was hopeless to prove that every elliptic curve was modular until we understood enough number theory to prove Fermat’s Last Theorem. Strong intuitions can be wrong and different people can have different intuitions about these kinds of things.

The one exception to my “Never” would be a situation where you can easily deduce the general case from the special case.

7. Terence Tao Says:

Certainly generalisation to a more abstract setting that “forgets” some of the original structure can be a powerful technique – though in order for such an argument to not be easily transferable back to the original concrete setting, the argument must somehow exploit a freedom in the abstract setting that is not easily available in the concrete special case. For example, one could imagine working on the gap problem by some sort of iterative procedure that starts with an arbitrary set and performs some operation to increase its min-max gap until it is extremised, at which point some useful structural information on that set is obtained. This is not an approach one would think of if one insisted only on working on the set of quadratic nonresidues. (Though it may end up that the quadratic nonresidues are already somehow the “worst” case, being so self-similar; one may even hope to formalise this with some sort of equivalence between the gap problem and the least quadratic nonresidue problem, though I don’t see how to achieve this. Curiously, Ben Green has recently had some thoughts along these lines, specifically that quadratic residues may somehow be the only obstruction to improvements to the large sieve inequality.)

It is of course quite difficult, as Timothy points out, to definitively rule out the utility of tackling a general case first before looking at the special case, because there may be ways to exploit the generality that one simply has not thought of yet. The one exception I can see is if the general case has an important “near-counterexample” that forces any proof of that general case to be very delicate, whilst the special case, being somehow “very far” from this counterexample, looks more amenable to cruder and simpler techniques. [Of course, if the general case has a genuine counterexample, then it becomes very easy to decide whether to spend any time trying to prove the general case. 🙂 ]

8. gowers Says:

Two small remarks. One is that it may well be that the conjecture suggested by the random case is also something like $C\log p\log\log p$: although you can’t do better than $C\log p$ if you want the gaps to be small with high probability, there is quite a bit of independence around, so it is conceivable that probabilistic methods could show that the $C\log p\log\log p$ bound occurs with positive (but small) probability. I don’t know how the GRH proof goes for the quadratic-residue problem, but my guess is that it is philosophically similar and does in fact show (i) that quadratic residues mod $p$ behave like random sets as $p$ varies and (ii) that for that very reason you expect $C\log p\log\log p$ from time to time.

As for the general case, I suspect that, as has been suggested, it may be hard to give a convincing reason for its being impossible to prove the theorem by purely combinatorial means. However, for any given purely combinatorial attack, it may be possible to use the quadratic-residues example to show that it does not work.

I am trying, but so far failing, to think of a problem where it definitely does happen that a single example kills off all hope of solving the general problem by general means. A first question then might be: if you can’t prove it by general means, then how can you prove it? A possible answer to that is that you first do some sort of classification and then deal with each class separately. So there might be examples in group theory, say, where it is hopeless to prove a theorem without using CSFG. But that would still leave the question of whether for any given problem there are clear signs that one has to use a classification. There are examples in group theory of classification-free proofs of theorems that had been thought to need classification, so perhaps the answer to this question is no.

In the Tricki article, I wanted to find circumstances where it is probably better to concentrate on the special case, so I’d settle for a reasonably convincing and generalizable argument that that is the case here, even if the argument didn’t establish it beyond all possible doubt, and even if one could think of exceptions to the general rule.

9. Emmanuel Kowalski Says:

Here’s an extreme example of interaction between particular and general case, but which may be useful: if there existed a general algorithmic way to decide whether any given diophantine equation has an integral solution, we know from the existence of a universal equation (a particular case!) that we would get a contradiction (i.e., this is the argument solving the Hilbert 10th Problem).

In the opposite direction, but still in this area, Robinson had shown that the general problem would be unsolvable, provided a single particular “exponential” relation could be encoded in polynomial form, and this had apparently been taken to mean that her approach was doomed to fail…(see the famous review MR0133227 (24 #A3061)).

10. Emmanuel Kowalski Says:

P.S. I should have written “Davis, Putnam and Robinson”, not simply “Robinson”.

11. D. Eppstein Says:

By the way, this may seem like a purely theoretical question, but the problem of how big the gaps can be is very closely related to the analysis of a very practical data structure in computer science, hash tables with linear probing. If one views the “gaps” as parts of the table that are filled in by hash table entries, and the dilation mod p as being a hash function, the gap length tells you what the longest access time can be.

See, e.g., [A. Pagh, R. Pagh, and M. Ruzic. Linear probing with constant independence. In Proc. 39th STOC, pages 318–327, 2007], which shows that hash functions with limited independence achieve good hashing behavior. Here “limited independence” is still rather more independence than you would get by dilating by a single randomly-chosen t, though.

12. bengreen Says:

Dear Tim and others,

A couple of points. Firstly, there is something on lower bounds for this gap problem in the book of Konyagin and Shparlinski, but I forget exactly what they prove.

Secondly, the p^{o(1)} conjecture, even when delta = 1/2, implies the Kakeya conjecture! I wrote some notes on this a long time ago, and here they are:

http://www.dpmms.cam.ac.uk/~bjg23/restriction_kakeya_phenomena/notes10.ps

The connection is basically due to Bourgain from about 1994.

Tim’s “probably inconsistent model” is very close to the idea of doing analytic number theory in which one admits the possibility of a zero of some L-function (or, worse, a Siegel zero). Occasionally one can work through arguments keeping track of the contribution of the hypothetical zero and it can even help (cf. Heath-Brown’s paper proving twin primes on the assumption there is a very strong type of Siegel zero)

Best
Ben

13. Scott Carnahan Says:

Minor issue: I think you may want p to be 3 mod 4 instead of 1 mod 4 in order to make this work.

[Thanks — I’ve changed it, and also changed Emmanuel Kowalski’s first comment.]

14. xli Says:

If we examine the random model in the case of quadratic non-residues, then the expected upper bound should be logp loglogp, the same as the lower bound proven by Montgomery conditional on GRH. The model goes something like the following. The values of the quadratic character is completely multiplicative, and hence determined by its values at the primes. If we assume that the character can be modeled by a random variable which takes the values 1 and -1 both with probability 1/2, then the upper bound x should satisfy 2^pi(x) is the same size as p, where pi(x) is the number of primes up to x. Assume pi(x) = x/log x, and solve for x. (Everything depends on x and p being “large”.)

15. Problems related to Littlewood’s conjecture « Gowers’s Weblog Says:

[…] be to start by finding a counterexample to the Littlewood conjecture … (This is an issue that I have discussed before on this blog. Sometimes combinatorial generalizations seem to be no easier to tackle than the […]


text:
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QUESTION

# A random sampling of a company's monthly operating expenses for n = 36 months produced a sample mean of $5938 and a standard deviation of$766. Find a 90% upper confidence bound for the company's mean

A random sampling of a company's monthly operating expenses for n = 36 months produced a sample mean of $5938 and a standard deviation of$766. Find a 90% upper confidence bound for the company's mean monthly expenses. (Round your answer to two decimal places.)

I have tried doing: 5938 + 1.645 * (766 / sqrt(36)) = 6148.01 and 5938 - 1.645 * (766 / sqrt(36)) = 5727.99 but these are incorrect. I don't know what else to try.

The professor never taught us how to do upper confidence bounds so I have no idea how else I could do this question.


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On Tuesday I commented here on the ASA (in their words) “Position on p-values:  context, process, and purpose.” A number of readers replied, some of them positive, some mistakenly thinking I don’t think statistical inferences are needed, and some claiming I overinterpreted the ASA’s statement. I’ll respond in the current post, and will devote most of it to what I believe are the proper alternatives.

First, though, in order to address the question, “What did the ASA really mean?”, I think it may be helpful discuss why the ASA suddenly came out with a statement. What we know is that the ASA statement itself opens with George Cobb’s wonderfully succinct complaint about the vicious cycle we are caught in: “We teach [significance testing] because it’s what we do; we do it because it’s what we teach.” The ASA then cites deep concerns in the literature, with quotes such as “[Significance testing] is science’s dirtiest secret” with “numerous deep flaws.”

As the ASA also points out, none of this is new. However, there is renewed attention, in a more urgent tone than in the past. The ASA notes that part of this is due to the “replicability crisis,” sparked by the Ionnidis paper., which among other things led the ASA to set up a last-minute (one might even say “emergency”) session at JSM 2014. Another impetus was a ban on p-values by a psychology journal (though I’d add that the journal’s statement led some to wonder whether their editorial staff was entirely clear on the issue).

But I also speculate that the ASA’s sudden action came in part because of a deep concern that the world is passing Statistics by, with our field increasingly being seen as irrelevant. I’ve commented on this as being sadly true, and of course many of you will recall then-ASA president Marie Davidian’s plaintive column title, “Aren’t WE Data Science?” I suspect that one of the major motivations for the ASA’s taking a position on p-values was to dispel the notion that statistics is out of data and irrelevant.

In that light, I stand by the title of my blog post on the matter. Granted, I might have used language like “ASA Says [Mostly] No to P-values,” but I believe it was basically a No. Like any statement that comes out of a committee, the ASA phrasing adopts the least common (read most status quo) denominator and takes pains not to sound too extreme, but to me the main message is clear. For example, I believe my MovieLens data example captures the essence of what the ASA said.

What is especially telling is that the ASA gave no examples in which p-values might be profitably used. Their Principle 1 is no more than a mathematical definition, not a recommendation. This is tied to the issue I brought up, that the null hypothesis is nearly always false a priori, which I will return to later in this blog.

Well, then, what should be done instead? Contrary to the impression I seem to have given some readers, I certainly do not advocate using visualization techniques and the like instead of statistical inference. I agree with the classical (though sadly underemphasized) alternative to testing, which is to form confidence intervals.

Let’s go back to my MovieLens example:

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.4725821  0.0482655  71.947  < 2e-16 ***
age         0.0033891  0.0011860   2.858  0.00436 **
gender      0.0002862  0.0318670   0.009  0.99284
...

The age factor is found to have a “highly significant ” impact on movie rating, but in fact the point estimate, 0.0033891, shows that the impact is negligible; a 10-year difference in age corresponds to about 0.03 point in mean rating, minuscule in view of the fact that ratings range from 1 to 5.

A confidence interval for the true beta coefficient, in this case, (0.0011,0.0057), shows that. Tragically, a big mistake made by many who teach statistics is to check whether such an interval contains 0 — which entirely defeats the purpose of the CI. The proper use of this interval is to note that the interval is near 0, in fact even at its upper bound.

So slavish use of p-values would have led to an inappropriate conclusion here (the ASA’s point), and moreover, once we have that CI, the p-value is useless (my point).

My point was also that we knew a priori that H0: β1 = 0 is false. The true coefficient is not 0.0000000000… ad infinitum. In other words, the fundamental problem — “statistics’ dirtiest secret” — is that the hypothesis test is asking the wrong question.  (I know that some of you have your favorite settings in which you think the null hypothesis really can be true, but I would claim that closer inspection would reveal that that is not the case.) So, not only is the test providing no additional value, once we have a point estimate and standard error, it is often worse than useless, i.e. harmful.

Problems like the above can occur with large samples (though there were only 949 users in the version of the movie data that I used). The opposite can occur with small samples. Let’s use the current U.S. election season. Say a staffer for Candidate X commissions a small poll, and the result is that a CI for p, the population proportion planning to vote for X, is (0.48,0.65). Yes, this contains 0.50, but I submit that the staffer would be remiss in telling X, “There is no significant difference in support between you and your opponent.” The interval is far more informative, in this case more optimistic, than that,

One of the most dramatic examples of how harmful testing can be is a Wharton study in which the authors took real data and added noise variables. In the ensuing regression analysis, lo and behold, the fake predictors were found to be “significant.”

Those of us who teach statistics (which I do, in a computer science context, as well as a member of a stat department long ago) have a responsibility to NOT let our students and our consulting clients follow their natural desire for simple, easy, pat answers, in this case p-values. Interpreting a confidence interval takes some thought, unlike p-values, which automatically make our decisions for us.

All this is fine for straightforward situations such as estimation of a single mean or a single regression coefficent. Unfortunately, though, developing alternatives to testing in more advanced settings can be challenging, even in R.  I always praise R as being “Statistically Correct, written by statisticians for statisticians,” but regrettably, those statisticians are the ones George Cobb complained about, and R is far too testing-oriented.

Consider for instance assessing univariate goodness of fit for some parametric model. Note that I use the word assessing rather than testing, especially important in that the basic R function for the Kolmogorov-Smirnov procedure, ks.test(), is just what its name implies, a test. The R Core Team could easily remedy that, by including an option to return a plottable confidence band for the true CDF. And once again, the proper use of such a band would NOT be to check whether the fitted parametric CDF falls entirely within the band; the model might be quite adequate even if the fitted CDF strays outside the band somewhat in some regions.

Another example is that of log-linear models. This methodology is so fundamentally test-oriented that it may not be clear to some what might be done instead. But really, it’s the same principle: Estimate the parameters and obtain their standard errors. If for example you think a  model with two-way interactions might suffice, estimate the three-way interactions; if they are small relative to the lower-order ones, you might stop at two. (Putting aside here the issue of after-the-fact inference, a problem in any case.)

But even that is not quite straightforward in R (I’ve never used SAS, etc. but they are likely the same).  The loglin() function, for instance,  doesn’t even report the point estimates unless one pro-actively requests them — and even if requested, no standard errors are available. If one wants the latter, one must use glm() with the “Poisson trick.”

In the log-linear situation, one might just informally look at standard errors, but if one wants formal CIs on many parameters, one must use simultaneous inference techniques, which brings me to my next topic.

The vast majority of those techniques are, once again, test-oriented.  One of the few, the classic Scheffe’ method, is presented in unnecessarily restrictive form in textbooks (linear model, normality of Y, homoscedasticity, F-statistic and so on).  But with suitable centering and scaling, quadratic forms of asymptotically normally distributed vectors have an asymptotic chi-squared distribution, which can be used to get approximate simultaneous confidence intervals.  R should add a function to do this on vcov() output.

In short, there is a lot that could be done, in our teaching, practice and software. Maybe the ASA statement will inspire some in that direction.


text:
noisy importance sampling

Posted in Statistics with tags , , , on February 14, 2022 by xi'an

A recent short arXival by Fernando Llorente, Luca Martino, Jesse Read, and David Delgado–Gómez in which they analyse settings where (only) a noisy version of the target density is available. Not necessarily in an unbiased fashion although the paper is somewhat unclear as to which integral is targeted in (6), since the integrand is not the original target p(x). The following development is about finding the optimal importance function, which differs from the usual due to the random nature of the approximation, but it does not seem to reconnect with the true target p(x), except when the noisy realisation is unbiased… To me this is a major issue in simulation methodology in that getting away from the unbiasedness constraint opens (rather obviously) a much wider choice of techniques.

what if what???

Posted in Books, Statistics with tags , , , , , on October 7, 2019 by xi'an

[Here is a section of the Wikipedia page on Monte Carlo methods which makes little sense to me. What if it was not part of this page?!]

Monte Carlo simulation versus “what if” scenarios

There are ways of using probabilities that are definitely not Monte Carlo simulations – for example, deterministic modeling using single-point estimates. Each uncertain variable within a model is assigned a “best guess” estimate. Scenarios (such as best, worst, or most likely case) for each input variable are chosen and the results recorded.[55]

By contrast, Monte Carlo simulations sample from a probability distribution for each variable to produce hundreds or thousands of possible outcomes. The results are analyzed to get probabilities of different outcomes occurring.[56] For example, a comparison of a spreadsheet cost construction model run using traditional “what if” scenarios, and then running the comparison again with Monte Carlo simulation and triangular probability distributions shows that the Monte Carlo analysis has a narrower range than the “what if” analysis. This is because the “what if” analysis gives equal weight to all scenarios (see quantifying uncertainty in corporate finance), while the Monte Carlo method hardly samples in the very low probability regions. The samples in such regions are called “rare events”.

a new rule for adaptive importance sampling

Posted in Books, Statistics with tags , , , , , , , , , on March 5, 2019 by xi'an

Art Owen and Yi Zhou have arXived a short paper on the combination of importance sampling estimators. Which connects somehow with the talk about multiple estimators I gave at ESM last year in Helsinki. And our earlier AMIS combination. The paper however makes two important assumptions to reach optimal weighting, which is inversely proportional to the variance:

1. the estimators are uncorrelated if dependent;
2. the variance of the k-th estimator is of order a (negative) power of k.

The later is puzzling when considering a series of estimators, in that k appears to act as a sample size (as in AMIS), the power is usually unknown but also there is no reason for the power to be the same for all estimators. The authors propose to use ½ as the default, both because this is the standard Monte Carlo rate and because the loss in variance is then minimal, being 12% larger.

As an aside, Art Owen also wrote an invited discussion “the unreasonable effectiveness of Monte Carlo” of ” Probabilistic Integration: A Role in Statistical Computation?” by François-Xavier Briol, Chris  Oates, Mark Girolami (Warwick), Michael Osborne and Deni Sejdinovic, to appear in Statistical Science, discussion that contains a wealth of smart and enlightening remarks. Like the analogy between pseudo-random number generators [which work unreasonably well!] vs true random numbers and Bayesian numerical integration versus non-random functions. Or the role of advanced bootstrapping when assessing the variability of Monte Carlo estimates (citing a paper of his from 1992). Also pointing out at an intriguing MCMC paper by  Michael Lavine and Jim Hodges to appear in The American Statistician.

Bayesian intelligence in Warwick

Posted in pictures, Statistics, Travel, University life, Wines with tags , , , , , , , , , , , , on February 18, 2019 by xi'an

This is an announcement for an exciting CRiSM Day in Warwick on 20 March 2019: with speakers

10:00-11:00 Xiao-Li Meng (Harvard): “Artificial Bayesian Monte Carlo Integration: A Practical Resolution to the Bayesian (Normalizing Constant) Paradox”

11:00-12:00 Julien Stoehr (Dauphine): “Gibbs sampling and ABC”

14:00-15:00 Arthur Ulysse Jacot-Guillarmod (École Polytechnique Fedérale de Lausanne): “Neural Tangent Kernel: Convergence and Generalization of Deep Neural Networks”

15:00-16:00 Antonietta Mira (Università della Svizzera italiana e Università degli studi dell’Insubria): “Bayesian identifications of the data intrinsic dimensions”

[whose abstracts are on the workshop webpage] and free attendance. The title for the workshop mentions Bayesian Intelligence: this obviously includes human intelligence and not just AI!

more and more control variates

Posted in Statistics with tags , , , , , , , on October 5, 2018 by xi'an

A few months ago, François Portier and Johan Segers arXived a paper on a question that has always puzzled me, namely how to add control variates to a Monte Carlo estimator and when to stop if needed! The paper is called Monte Carlo integration with a growing number of control variates. It is related to the earlier Oates, Girolami and Chopin (2017) which I remember discussing with Chris when he was in Warwick. The puzzling issue of control variates is [for me] that, while the optimal weight always decreases the variance of the resulting estimate, in practical terms, implementing the method may increase the actual variance. Glynn and Szechtman at MCqMC 2000 identify six different ways of creating the estimate, depending on how the covariance matrix, denoted P(hh’), is estimated. With only one version integrating constant functions and control variates exactly. Which actually happens to be also a solution to a empirical likelihood maximisation under the empirical constraints imposed by the control variates. Another interesting feature is that, when the number m of control variates grows with the number n of simulations the asymptotic variance goes to zero, meaning that the control variate estimator converges at a faster speed.

Creating an infinite sequence of control variates sounds unachievable in a realistic situation. Legendre polynomials are used in the paper, but is there a generic and cheap way to getting these. And … control variate selection, anyone?!


text:
Classical Mechanics

# Momentum Warmup

If a car and a bus are moving with the same momentum, which has greater kinetic energy?

Assumptions and Details

• The car has a much smaller mass than the bus.

A person is falling freely under the gravity. He has a heavy briefcase in his hand. There is swimming pool just beside his vertical line of fall. If he wants to land in the swimming pool, what should he do?

Rain drops fall on the ground with a speed and a momentum. They hit different objects and create sound. Suppose, 100 drops are hitting a horizontal surface of unit area per second. The speed of rain drops 15 m/s in vertically downward direction. The mass of each drop is 0.05 grams. If the rain drops come to rest immediately after hitting the surface then, find the force experienced by the horizontal surface?

When two objects collide, they apply forces to each other called impulse forces. Such forces act for a very small amount of time, yet produce large changes in the momentum of the colliding objects.

Let's consider a case of collision between a ball and a bat. The ball of mass $m$ is approaches the batsman with a speed of $v$ m/s and the bat hits the ball so that it recoils with the same speed, along the same line, but in the opposite direction.

What will be the total impulse of the bat, upon the ball?

When a ball is dropped from a height, after hitting the ground, the ball rebounds. The rebounded height is less than the original height of the drop. What can be said about the collision between the ball and the ground?

×


text:
## Sum of Sines Models

### About Sum of Sines Models

The sum of sines model fits periodic functions, and is given by

`$y=\sum _{i=1}^{n}{a}_{i}\mathrm{sin}\left({b}_{i}x+{c}_{i}\right)$`

where a is the amplitude, b is the frequency, and c is the phase constant for each sine wave term. n is the number of terms in the series and 1 ≤ n ≤ 8. This equation is closely related to the Fourier series described in Fourier Series. The main difference is that the sum of sines equation includes the phase constant, and does not include a constant (intercept) term.

### Fit Sum of Sine Models Interactively

1. Open the Curve Fitter app by entering `curveFitter` at the MATLAB® command line. Alternatively, on the Apps tab, in the Math, Statistics and Optimization group, click Curve Fitter.

2. In the Curve Fitter app, select curve data. On the Curve Fitter tab, in the Data section, click Select Data. In the Select Fitting Data dialog box, select X data and Y data, or just Y data against an index.

3. Click the arrow in the Fit Type section to open the gallery, and click Sum of Sine in the Regression Models group.

You can specify the following options in Fit Options pane:

• Specify the number of terms as a positive integer in the range [1 8]. Look in the Results pane to see the model terms, values of the coefficients, and goodness-of-fit statistics.

• Optionally, in the Advanced Options section, specify coefficient starting values and constraint bounds, or change algorithm settings. The app calculates optimized start points for Sum of Sine fits, based on the data set. You can override the start points and specify your own values in the Fit Options pane.

The Sum of Sine fit has a lower bound constraint on bi of `0`. The default lower bounds for most fits in the Fit Type section are `-Inf`.

For more information on the settings, see Specify Fit Options and Optimized Starting Points.

### Selecting a Sum of Sine Fit at the Command Line

Specify the model type `sin` followed by the number of terms, e.g., `'sin1'` to `'sin8'`.

For example, to load some periodic data and fit a six-term sum of sine model:

```load enso; f = fit( month, pressure, 'sin6') plot(f,month,pressure)```

If you want to modify fit options such as coefficient starting values and constraint bounds appropriate for your data, or change algorithm settings, see the table of additional properties with `NonlinearLeastSquares` on the `fitoptions` reference page.


text:
# When were polynomial equations first factored?

The question pretty much says it all, though I have a specific example in mind. In the mid-1500s while working on his Ars Magna Cardano asked Tartaglia to solve the cubic $$x^3=9x+10$$. Using Tartaglia's formula results in having to take the square root of a negative number, which was undefined at the time; neither Tartaglia nor Cardano could solve the equation.

However, with a little work the above equation can be factored into $$(x+2)(x^2-2x-5)$$, and the three roots readily found using nothing more complicated than the quadratic formula which was known at the time. Clearly neither Tartaglia nor Cardano did this, as they had no solution to the original cubic.

All that aside, when was it realized that such higher-order equations could be factored?

• Well, for polynomials of degree > 4, the answer is "never" :-) – Carl Witthoft Dec 17 '18 at 13:23
• @CarlWitthoft: It is possible to factor many polynomials of high degree, we just can't do it for all such polynomials using radicals. – Rory Daulton Dec 18 '18 at 1:06

I found no trace of factoring polynomials being done before Descartes, in his La Géométrie, written in 1637. In it, he wrote:

It is evident from the above that the sum of an equation having several roots [that is, the polynomial itself] is always divisible by a binomial consisting of the unknown quantity diminished by the false roots. In this way, the degree of an equation can be lowered.

This is how Descartes stated what we call today the factor theorem.

• And the existence of roots from Descartes' Rule of Signs, which I cited in my dissertation. – Andrew Lazarus Jan 18 '19 at 20:53


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# Resolution (mass spectrometry)

Resolution (mass spectrometry)

In mass spectrometry, resolution measures of the ability to distinguish two peaks of slightly different mass-to-charge ratios ΔM, in a mass spectrum.

## Resolution and Resolving Power

There are two different definitions of resolution and resolving power in mass spectrometry.

### IUPAC definition

The IUPAC definition for resolution in mass spectrometry is

$R = \cfrac{M}{\Delta M}$ = resolution
ΔM = resolving power

where a larger resolution indicates a better separation of peaks.[1] This definition is used in a number of mass spectrometry texts.[2][3][4][5][6][7][8][9][10] This use is also implied by the term "high-resolution mass spectrometry."[11]

A high value for resolution corresponding to good separation of peaks is similar to the convention used with chromatography separations,[12] although it is important to note that the definitions are not the same.[13] High resolution indicating better peak separation is also used in ion mobility spectrometry[14]

IUPAC defines Δm as the resolving power.[15]

### Resolving power definition

Some mass spectrometrists use the definition that is similar to definitions used in some other fields of physics and chemistry. In this case, resolving power is defined as:

$R = \cfrac{M}{\Delta M}$ = resolving power.

The minimum peak separation ΔM which allows to distinguish two ion species is then called:

ΔM = resolution.

Resolution and resolving power, when defined in this way, are consistent with IUPAC recommendations for microscopy, optical spectroscopy.[16][17] and ion microscopy (SIMS) [18] but not gas chromatography.[19] This definition also appears in some mass spectrometry texts.[20][21][22]

## Measuring peak separation

There are several ways to define the minimum peak separation ΔM in mass spectrometry, therefore it is important to report the method used to determine mass resolution when reporting its value. The two most widely used are the peak width definition and the valley definition.[1]

### Peak width definition

In the peak width definition, the value of Δm is the width of the peak measured at a specified fraction of the peak height, for example 0.5%, 5%, 10% or 50%. The latter is called the full width at half maximum (FWHM).

### Valley definition

The valley definition defines ΔM as the closest spacing of two peaks of equal intensity with the valley (lowest value of signal) between them less than a specified fraction of the peak height. Typical values are 10% or 50%. The value obtained from a 5% peak width is roughly equivalent to a 10% valley.[1]

## References

1. ^ a b c IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version:  (2006–) "resolution in mass spectroscopy".
2. ^ Biemann, Klaus (1962). Mass Spectrometry: Organic Chemical Applications. New York: McGraw-Hill. p. 13. ISBN 0070052352.
3. ^ Tureček, František; McLafferty, Fred W. (1993). Interpretation of mass spectra. Sausalito, Calif: University Science Books. ISBN 0-935702-25-3.
4. ^ Watson, J. S. (1997). Introduction to mass spectrometry. Philadelphia: Lippincott-Raven. ISBN 0-397-51688-6.
5. ^ Ashcroft, Alison E. (1997). Ionization methods in organic mass spectrometry. Cambridge, Eng: Royal Society of Chemistry. ISBN 0-85404-570-8.
6. ^ JURGEN H. GROSS; Jnrgen H. Gross (2004). Mass Spectrometry: A Textbook. Berlin: Springer-Verlag. ISBN 3-540-40739-1.
7. ^ Todd, John F. J.; March, Raymond E. (2005). Quadrupole ion trap mass spectrometry. New York: Wiley-Interscience. ISBN 0-471-48888-7.
8. ^ Siuzdak, Gary (2006). The Expanding Role of Mass Spectrometry in Biotechnology, Second Edition. MCC Press. ISBN 0-9742451-2-7.
9. ^ Stroobant, Vincent; Hoffmann, Edmond de (2007). Mass spectrometry: principles and applications. London: J. Wiley. ISBN 0-470-03310-X.
10. ^ Ingvar Eidhammer (2007). Computational methods for mass spectrometry proteomics. Chichester: John Wiley & Sons. ISBN 0-470-51297-0.
11. ^ VanLear GE, McLafferty FW (1969). "Biochemical aspects of high-resolution mass spectrometry". Annu. Rev. Biochem. 38: 289–322. doi:10.1146/annurev.bi.38.070169.001445. PMID 4896241.
12. ^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version:  (2006–) "resolution in gas chromatography]]".
13. ^ Blumberg LM, Kle MS (November 2001). "Metrics of separation in chromatography". J Chromatogr A 933 (1–2): 1–11. doi:10.1016/S0021-9673(01)01256-0. PMID 11758739.
14. ^ Karpas, Zeev; Eiceman, Gary Alan (2005). Ion mobility spectrometry. Boca Raton: CRC Press. ISBN 0-8493-2247-2.
15. ^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version:  (2006–) "resolving power in mass spectrometry".
16. ^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version:  (2006–) "resolution in optical spectroscopy".
17. ^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version:  (2006–) "resolving power in optical spectroscopy".
18. ^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version:  (2006–) "resolving power in ion microscopy".
19. ^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version:  (2006–) "resolution in gas chromatography".
20. ^ David O. Sparkman (2006). Mass Spectrometry Desk Reference. Pittsburgh: Global View Pub. ISBN 0-9660813-9-0.
21. ^ Sparkman, O. David (2007). Introduction to mass spectrometry: instrumentation, applications and strategies for data interpretation. Chichester: John Wiley & Sons. ISBN 0-470-51634-8.
22. ^ Dass, Chhabil (2007). Fundamentals of contemporary mass spectrometry. Chichester: John Wiley & Sons. ISBN 0-471-68229-2.

Wikimedia Foundation. 2010.

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text:
Math Help - Initial Value Problem

1. Initial Value Problem

I have a number of equations where I have to solve the initial value problem. I don't really understand what my notes say and the examples they give seem to basic to the ones I have to do.

Could someone please help understand how to go about doing the following?

arcsint = x^4(1-t^2)^1/2 dx/dt, x(0) = 0

2. Originally Posted by JQ2009
I have a number of equations where I have to solve the initial value problem. I don't really understand what my notes say and the examples they give seem to basic to the ones I have to do.

Could someone please help understand how to go about doing the following?

arcsint = x^4(1-t^2)^1/2 dx/dt, x(0) = 0
Your DE is separable so

$
x^4\,dx = \frac{\sin^{-1} t}{\sqrt{1-t^2}}dt
$

Integrate both sides and then use your IC to determine the constant of integration.


text:
# Patterns for eigenvalues of Vandermonde matrix

Let $A$ be a Vandermonde type matrix

$A = \begin{bmatrix} 1 & 1 & \dots & 1 \\ x_1 & x_2 &\dots & x_n \\ \dots& \dots & \dots& \dots\\ x_1^{n-1} &x_2^{n-1} &\dots & x_n^{n-1} \end{bmatrix}$

When I was testing some such real matrices with positive entries for their eigenvalues I have noticed that eigenvalues are also real and positive.

• How such property (if true in general case) can be explained ?

Moreover in examples which I tested (when it was assumed for $i<j$ that $x_i <x_j$) the greatest eigenvalue was always close to the greatest value in this matrix and the smallest one always less than $1$.

• How to explain also these facts ?

Examples of $3 \times 3$ matrices generated from natural numbers

$A = \begin{bmatrix} 1 & 1 & 1 \\ 2 & 3 & 5 \\ 4 & 9 & 25 \end{bmatrix}$

Eigenvalues: $\{ 27.09 , \ 0.12 , \ 1.79 \}$

$A = \begin{bmatrix} 1 & 1 & 1 \\ 3 & 7 & 11 \\ 9 & 49 & 121 \end{bmatrix}$

Eigenvalues: $\{ 125.63, \ 0.37 , \ 3.03 \}$

Of course if two eigenvalues in the examples above are rather small then the third must be rather great from the trace of the matrix, but why these two must be small ?

• If the answer for general case is hard to find let dimension of a matrix be specific i.e. $3 \times 3$ and entries only natural.
• Is this helpful? - mathoverflow.net/questions/155845/… Mar 8, 2017 at 15:07
• @ancientmathematician Yes, it provides some knowledge although not for everything I mentioned here. It seems from mathoverflow that the problem is rather difficult, I'm afraid that explanation there is rather hard for me. What does it mean "strictly totally positive" for the problem( this meaning of minors) I was thinking that the problem is easier.. Mar 8, 2017 at 15:14
• it means all the minors are positive. If one wants to understand the eigenvalues of the Vandermonde matrix, then I think you'd need to grapple with this stuff. (I may be wrong). Mar 8, 2017 at 15:20
• @ancientmathematician so the answer lies somewhere in 35-pages paper, hmm ... it can be rather a long way to understanding .. Mar 8, 2017 at 15:22
• @ancientmathematician That's true :) Step by step makes a mile.. Mar 8, 2017 at 15:32

As such Vandermonde matrix $$A$$ has positive entries, then by Perron-Frobenius theorem it has positive eigenvalue $$\lambda$$ with positive eigenvector $$v$$ (with positive entries). This eigenvalue is the largest by modulus among other eigenvalues.
If you look on the last coordinate of the identity $$Av = \lambda v$$ you will see, that $$\lambda v_n = x_1^{n-1} v_1 + \dots + x_n^{n-1} v_n > x_n^{n-1} v_n$$ because $$v_i$$ are positive. Cancelling $$v_n$$ you get that $$\lambda$$ is greater than $$x_n^{n-1}$$.
For bounding the largest eigenvalue from above you can use Gershgorin circles theorem and its strengthening, which gives in the case $$x_1=1$$, $$x_2 = 7$$, $$x_3 = 11$$ that the largest eigenvalue is not greater than 133.