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# Which of the following can be the ratio of sides of a triangle?

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Which of the following can be the ratio of sides of a triangle?  [#permalink]

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28 Feb 2016, 08:26
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Which of the following can be the ratio of sides of a triangle?

I. $$3:\sqrt{4}:5$$
II. $$\sqrt{3}:4:5$$
III. $$3:4:\sqrt{5}$$

A. II only
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III

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Re: Which of the following can be the ratio of sides of a triangle?  [#permalink]

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29 Feb 2016, 12:15
1
For any triangle with sides a,b,c, the following rules for sides MUST be followed:

|a-b| << c << a+b (this has to be true for ALL the sides!)

II and III: both satisfy the conditions
Hence, Ans: D
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Re: Which of the following can be the ratio of sides of a triangle?  [#permalink]

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29 Feb 2016, 21:35
1
Bunuel wrote:
Which of the following can be the ratio of sides of a triangle?

I. $$3:\sqrt{4}:5$$
II. $$\sqrt{3}:4:5$$
III. $$3:4:\sqrt{5}$$

A. II only
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III

The sum of any two sides of a triangle must be greater than the third side.

I. $$3:\sqrt{4}:5$$
The ratio of sides are 3, 2 and 5
But 3 + 2 = 5 which is not acceptable. The sum of any two sides must be greater than the third side.
Hence these cannot be the ratio of the sides of the triangle.

II. $$\sqrt{3}:4:5$$

The ratio of sides are 1.7, 4 and 5.
The sum of any two is greater than the third.
So these can be the ratio of the sides of the triangle.

III. $$3:4:\sqrt{5}$$

The ratio of sides are 3, 4 and 2.2
The sum of any two is greater than the third.
So these can be the ratio of the sides of the triangle.

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Re: Which of the following can be the ratio of sides of a triangle?  [#permalink]

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05 Dec 2017, 00:32
VeritasPrepKarishma wrote:
Bunuel wrote:
Which of the following can be the ratio of sides of a triangle?

I. $$3:\sqrt{4}:5$$
II. $$\sqrt{3}:4:5$$
III. $$3:4:\sqrt{5}$$

A. II only
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III

The sum of any two sides of a triangle must be greater than the third side.

I. $$3:\sqrt{4}:5$$
The ratio of sides are 3, 2 and 5
But 3 + 2 = 5 which is not acceptable. The sum of any two sides must be greater than the third side.
Hence these cannot be the ratio of the sides of the triangle.

II. $$\sqrt{3}:4:5$$

The ratio of sides are 1.7, 4 and 5.
The sum of any two is greater than the third.
So these can be the ratio of the sides of the triangle.

III. $$3:4:\sqrt{5}$$

The ratio of sides are 3, 4 and 2.2
The sum of any two is greater than the third.
So these can be the ratio of the sides of the triangle.

VeritasPrepKarishma, Bunuel,

Is it sufficient to just check for condition - The sum of any two is greater than the third.

or do we also have to check that the difference of any two sides should be less than the third side

Thanks
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Re: Which of the following can be the ratio of sides of a triangle?  [#permalink]

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05 Dec 2017, 01:34
KGump wrote:
VeritasPrepKarishma wrote:
Bunuel wrote:
Which of the following can be the ratio of sides of a triangle?

I. $$3:\sqrt{4}:5$$
II. $$\sqrt{3}:4:5$$
III. $$3:4:\sqrt{5}$$

A. II only
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III

The sum of any two sides of a triangle must be greater than the third side.

I. $$3:\sqrt{4}:5$$
The ratio of sides are 3, 2 and 5
But 3 + 2 = 5 which is not acceptable. The sum of any two sides must be greater than the third side.
Hence these cannot be the ratio of the sides of the triangle.

II. $$\sqrt{3}:4:5$$

The ratio of sides are 1.7, 4 and 5.
The sum of any two is greater than the third.
So these can be the ratio of the sides of the triangle.

III. $$3:4:\sqrt{5}$$

The ratio of sides are 3, 4 and 2.2
The sum of any two is greater than the third.
So these can be the ratio of the sides of the triangle.

VeritasPrepKarishma, Bunuel,

Is it sufficient to just check for condition - The sum of any two is greater than the third.

or do we also have to check that the difference of any two sides should be less than the third side

Thanks

These are just different ways of saying the same thing.

a + b > c
or
a > c - b

Say sides are 2, 3 and 6.
Sum of 2 and 3 is not greater than 6.
Also difference of 6 and 3 is not less than 2.
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Re: Which of the following can be the ratio of sides of a triangle?  [#permalink]

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05 Dec 2017, 01:42
VeritasPrepKarishma wrote:

These are just different ways of saying the same thing.

a + b > c
or
a > c - b

Say sides are 2, 3 and 6.
Sum of 2 and 3 is not greater than 6.
Also difference of 6 and 3 is not less than 2.

Oh.. i see now.. always had this stupid doubt..
Thankyou for clearing this
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Re: Which of the following can be the ratio of sides of a triangle?  [#permalink]

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18 Aug 2019, 17:08
Official Explanation:

Solution: The sum of any two sides of a triangle must be greater than the third side.

4–√=2
. If the ratio of the sides is 3:2:5, the triangle cannot be formed because 3+2 is equal to 5, not greater.
3–√≈1.73
(however, all we need to know is that 3–√
is larger than 1 and smaller than 2). If the ratio of the sides is 1.73:4:5, the sum of any two sides will be greater than the third side. So the triangle can be formed.
5–√≈2.2
(again, we just need to know it is larger than 2 but smaller than 3). If the ratio of the sides is 3:4:2.2, the sum of any two sides will be greater than the third side. So the triangle can be formed.
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Re: Which of the following can be the ratio of sides of a triangle?   [#permalink] 18 Aug 2019, 17:08
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# Raptor Composite Aircraft

### Help Support HomeBuiltAirplanes.com:

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Not open for further replies.

#### Jay Kempf

##### Curmudgeon in Training (CIT)
That’s what polyurethane floor coating is for. That’s how basketball courts stay shiny without losing traction.
Until anything gets spilled on it.

#### pictsidhe

##### Well-Known Member
Can you cite that again, as a diesel flyer I'd like to read that?

M
Tests were on unlubricated ULSD, not Jet-A. But the results should still be relevant.

#### Attachments

• 130.7 KB Views: 8

#### pictsidhe

##### Well-Known Member
The landing gear emergency drop didn't work properly either. Peter thought that was ok as he couldn't think of a scenario where there could be 0V on the battery.
So, gear misbehaves. Belly flop if it has stayed up, or pull the chute if it partially dropped?

#### BoKu

##### Pundit
HBA Supporter
...So, gear misbehaves. Belly flop if it has stayed up, or pull the chute if it partially dropped?
I'd take the belly flop. If it grinds off 1/4" of carbon, there's still 1/4" left.

#### Marc Zeitlin

##### Exalted Grand Poobah
If there is a delimination at this place, there is a 90% probability that there will be delamintions elsewhere...
Where'd that 90% estimate come from? Is there some study showing that if one finds a disbond in a layup (and this is a disbond, not a delamination - let's be clear on terminology) that there's a 90% chance that you'll find another one?

and that the glue fillets are insufficient. Did he do bag load testing?
So in this case, the disbond is probably caused by an extremely poor choice of location for aileron hard stops (at the aileron, rather than in the control circuit upstream of the aileron). Pull hard on the stick, put stress on the wing skin bond joint at the hard stop. So a disbond here is only an indication of a lack of understanding of aircraft control system design, not an indication that the fabricators did a poor job of bonding the plane together.

My $0.02. #### berridos ##### Well-Known Member Thanks for the delamination/disbond clarification. Instinctively I was worrying about wrong surface treatment or contamination pre bonding or insufficient fillets. An issue frequently neglected. A wrong peel ply or some other factor and you have the structure significantly weakened. #### swoodman ##### New Member To be quite fair to Peter while he is showing the positive outlook on all this from his side, I think he's earnestly showing his side of the story. He's unaware of the best or "proper" way to solve a lot of these issues, he's learning as he goes, and while sometimes he acts incredulous at the need for certain tasks he does eventually seem to do them and it seems he tends to come away from those experiences satisfied in the outcome having been worthwhile. His positive outlook lately has been strained by all the reality catching up, the things he's learning, and I think bit by bit he's starting to get some of it just by being faced with the bill of it all. The lack of experience and knowledge really clearly comes through when seen from the Wasabi video, but it makes plain that he's not so much intentionally hiding the bad stuff, because he simply doesn't know what's bad and should be hidden. When he has all these deep complicated explanations for how certain things that should just work all have their certain way of making it work, it's clear he's worked it out in his head that it all makes sense and that specific thing might be slightly odd but it works fine; but I don't think he has taken the moment to step back and reflect on all of those quirks in aggregate and whether they add up to the type of machine someone wants to jump into. He's gotta explain every little thing and justify it. He doesn't see that as an issue cuz he has lived with all this knowledge for years. And one or two quirks is probably fine. But everything here has things that just don't quite work right, and so while it works-ish. it's all clearly half-finished prototype stage solutions. And in retrospect when there's a better design worked out (hopefully) he'll clearly see how far he had to go. But that's the future. But somewhere I think he knows there are big gaps in his personal experience, which is why despite all of his hubris at times, he still knows enough (I want to hope) to continue to defer to people with experience and real critical eyes when it comes to flying the thing, and has to have inside his mind at least a bit of the real concern that the creature may bite. I read every post; have done for four years or however long it has been now, and I was (until 2 weeks ago) one of the people with a deposit in escrow (I cancelled without any argument from Peter BTW). I think these comments are spot on. My opinion, is that the truth is in the middle between some of the more extreme viewpoints. On this forum they tend to be very negative at times but the balanced engineering and technical expertise is what I come back for almost every day. I respect Peter for not doing the stupid thing and trying to fly the plane. I also admire that even though he absolutely knew that he wasn't going to like stuff, he was good for Wasabi to post anyway. I hold out hope that those molds will perhaps at some point lead to a lighter and better version of the Velocity - I think we all want that. #### Rik- ##### Well-Known Member He switched to diesel after getting many YouTube comments about Jet-A being a BAD IDEA. He does sometimes listen, eventually. I cited a study showing how effective his chosen lubricity additive is(n't). What exactly is wrong with Jet A? #### pictsidhe ##### Well-Known Member What exactly is wrong with Jet A? It's for jet engines! Diesel engines evolved with a fuel that is a fair lubricant. Jet A has poor lubricity, so diesel pumps that need a certain level of Iubrication are prone to siezing up. Jet A also does not have a controlled cetane number, another important specification for diesel useage. It would be very useful if someone could research an additive package to 'dieselise' Jet A. May take some dollars, though. #### Rik- ##### Well-Known Member It's for jet engines! Diesel engines evolved with a fuel that is a fair lubricant. Jet A has poor lubricity, so diesel pumps that need a certain level of Iubrication are prone to siezing up. Jet A also does not have a controlled cetane number, another important specification for diesel useage. It would be very useful if someone could research an additive package to 'dieselise' Jet A. May take some dollars, though. Jet A, House Heating Oil, Kerosene and Diesel #1 are all the same exact fluid. In the winter months in NY and NJ, the fuel stations sell Diesel #1 as it does not have the tendency to gel in the extreme winter temperatures. I've purchased over 500,000 gallons of Jet A and used it in my VW Diesel Rabbit as well as my CAT diesel engines (as well as my Turbines) and never had a problem. I've heard the rumors of the fuel pumps would not like it but he engine doesn't seem to care. #### rbarnes ##### Well-Known Member Thanks for the delamination/disbond clarification. Instinctively I was worrying about wrong surface treatment or contamination pre bonding or insufficient fillets. An issue frequently neglected. A wrong peel ply or some other factor and you have the structure significantly weakened. Actual lay up of the carbon fiber is the one place in the project where Peter had someone doing the work that actually knew what they were doing. and no nothing was sand bag tested, drop tested, or any other kind of tested beyond running up and down the runway at 80+kts #### pictsidhe ##### Well-Known Member They are most certainly not the same. Do you really think that oil refiners would use a specific cetane cut and add lubricant additives to kerosene or heating oil? I haven't played with Jet-A, buy I believe that it is pretty close to kerosene. Kerosene is noticeably thinner and less lubricious than diesel. It also smells different, especially when burned. People can spot kero burning diesel cars by the smell. I would mix it with 10% biodiesel, which is very slippy stuff. I've seen several people kill their pumps with pure kero. #### Rik- ##### Well-Known Member Actual lay up of the carbon fiber is the one place in the project where Peter had someone doing the work that actually knew what they were doing. and no nothing was sand bag tested, drop tested, or any other kind of tested beyond running up and down the runway at 80+kts He bounced it pretty good when he slammed it back onto the runway! That's gotta be worth something. After all the wings didn't fall off. #### Rik- ##### Well-Known Member They are most certainly not the same. Do you really think that oil refiners would use a specific cetane cut and add lubricant additives to kerosene or heating oil? I haven't played with Jet-A, buy I believe that it is pretty close to kerosene. Kerosene is noticeably thinner and less lubricious than diesel. It also smells different, especially when burned. People can spot kero burning diesel cars by the smell. I would mix it with 10% biodiesel, which is very slippy stuff. I've seen several people kill their pumps with pure kero. Here is what and why I know this to be true. Every week I was get 1,000 gallons of Fuel delivered to my work for 20+ years. My fuel supplier and my driver were both saying the exact same thing. You ask for Jet A we charge you X, you ask for Diesel #1, we charge you X, you as for house heating oil we charge you X. What's the difference (Mr. Chevron Refinery in Richmond CA) Price.. other than that not a **** thing so if you want to pay$3/gallon order Jet A. Want to pay more Order Diesel #1, want to pay less order House Heating Oil/ aka Kersosene.

So having done this game for 30+ years I can tell you with a straight face, it's exactly the same stuff.

#### TFF

##### Well-Known Member
When I worked for a regional airline, the only car/ truck to survive the Jet A was VW Rabbits and old style Mercedes. The Ford and Chevy trucks died expensive deaths. I think unless he has said anything different, we are talking #2 diesel out of the pump. That’s what is around here. If your stuff survived , great, but I bet you would be surprise how different jet A and house kerosene is. Both kerosene but A is way dryer. The beauty of a diesel is just about anything combusts. The fuel delivery is what takes the beating. Anyway $3 a gal is retail for Jet A here, bulk it’s about$1.20g.

#### narfi

##### Well-Known Member
Log Member
different additives will destroy different seals.
Science and politics promote the use of different additives in the different named fuels.

It is my experience that at times all three fuels delivered to my village originate from the same pump, however if they have a customer requiring the newer diesel (with additives for newer diesel engines) that they have to rinse their tanks before delivering jet fuel as it will destroy the seals in our turbine engines.

For the most part I understand the underlying fuel will burn for most of the 3 main purposes (jet/turbine, diesel engine and heaters) but the wrong one in any given system could destroy the seals and or pumps. Those delivering the fuel know what is on the work order as well as the real life end use for the fuel and should ensure that the fuel they deliver is compatible with its end use as well as communicating the reality of the standards of the actual fuel being delivered.

(I am not directly involved in any of this except to receive the fuel for our village generators, so accept my experience with a grain of salt)

#### BJC

##### Well-Known Member
HBA Supporter
I've purchased over 500,000 gallons of Jet A and used it in my VW Diesel Rabbit
Wow! I drove a Diesel Rabbit about 225,000 miles, but it didn’t take close to 500,000 gallons.

I burned lots of Jet A in industrial boilers back in the early 1980’s at dirt cheap prices. The fuel terminal had take-or-pay contracts for Jet A, the airlines weren’t burning much, more fuel was in transit, so it was cheaper that either natural gas of Number 2 fuel oil. No problems.

BJC

#### Rik-

##### Well-Known Member
Wow! I drove a Diesel Rabbit about 225,000 miles, but it didn’t take close to 500,000 gallons.

I burned lots of Jet A in industrial boilers back in the early 1980’s at dirt cheap prices. The fuel terminal had take-or-pay contracts for Jet A, the airlines weren’t burning much, more fuel was in transit, so it was cheaper that either natural gas of Number 2 fuel oil. No problems.

BJC
The fuel was for the boat, but if I'm low, the pump flows what the pump flows

BJC

#### Rik-

##### Well-Known Member
different additives will destroy different seals.
Science and politics promote the use of different additives in the different named fuels.

It is my experience that at times all three fuels delivered to my village originate from the same pump, however if they have a customer requiring the newer diesel (with additives for newer diesel engines) that they have to rinse their tanks before delivering jet fuel as it will destroy the seals in our turbine engines.

For the most part I understand the underlying fuel will burn for most of the 3 main purposes (jet/turbine, diesel engine and heaters) but the wrong one in any given system could destroy the seals and or pumps. Those delivering the fuel know what is on the work order as well as the real life end use for the fuel and should ensure that the fuel they deliver is compatible with its end use as well as communicating the reality of the standards of the actual fuel being delivered.

(I am not directly involved in any of this except to receive the fuel for our village generators, so accept my experience with a grain of salt)
If you want an "insured fuel" Jet A is going to be the route if you are flying. If you are land based application then it doesn't matter as it's just a price issue not a product difference.

There are a LOT of industrial back up generators that use a Turbine instead of a C18 Cat Diesel Gen Pack. The turbines are cheaper and require less maintenance and the burn the exact same fuel or any fuel you care to put into them. As long as the Specific Gravity is not to far off.

I've heard that for road trucks/cars that you can use the Kerosene if you add some marvels mystery oil but I simply didn't care. Lasted for years and that's saying a lot if you understood how I drive.

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Comunicazione

# Novel topological states in a thin strip of Haldane model.

##### Traverso S.
Giovedì 15/09   09:00 - 13:00   Aula D - Marianna Ciccone   II - Fisica della materia
In 1988 F.D.M. Haldane conceived the first proposal for a Chern insulator: his model $\mbox{---nowadays}$ known as the Haldane model--- describes spinless electrons on a honeycomb lattice, and it achieves a topological phase with non-zero Chern number by breaking time-reversal symmetry without net magnetic flux. The topological phase of the model is characterized by the presence of one-dimensional chiral edge states, with energy in the bulk gap and connecting the valence and conduction bands. Interestingly, if one considers a thin enough strip of the honeycomb lattice, then the chiral modes on the two edges will hybridize with each other, causing the opening of a gap in the edge spectrum. We found that, for a specific choice of the strip edge type, this gap can host a pair of zero energy modes, corresponding to topological states localized at the two ends of the strip. Moreover, we discovered that for a strip of appropriate width the model manifests an even richer phenomenology. Indeed, in such special configuration, the topological phase described above is only achievable after a topological phase transition, which can be induced through the introduction of an on-site staggered potential.


text:
# Thread: cos x = sqrt(3)/2

1. ## cos x = sqrt(3)/2

Determine all solutions to the trigonometric equation

in the range 0,2pi

i alredy got the first answer it is pi/6
but the second i thought i would do pi/6+2pi =S? plz help need fast help!

2. ## Re: cos x = sqrt(3)/2

Originally Posted by Petrus
Determine all solutions to the trigonometric equation

in the range 0,2pi

i alredy got the first answer it is pi/6
What about $2\pi-\frac{\pi}{6}~?$

3. ## Re: cos x = sqrt(3)/2

solved it :P can some1 explain for me this
for tan its x+kpi
what is it for sin?

4. ## Re: cos x = sqrt(3)/2

solved it 11pi/6

,

,

,

,

,

,

,

,

,

,

,

,

,

,

# the equation cosx=square root 3 over 2 is an identity

Click on a term to search for related topics.


text:
# Major head loss in pipe

## Homework Statement

why the author use major loss due to fluid friction is fL(Q^2) / 3(d^5) ???
according to darcy weisbech , the major loss due to friction has the formula of fL(v^2) / 2gD

why the author wrote it as fL(Q^2) / 3(d^5) ???
I dont understand

## The Attempt at a Solution

since v= Q/ A , we can rewrite it as
fL(Q^2) / 2g(π^2)(d^5) , am i right ?

Simon Bridge
Homework Helper
1. are the authors using the same models?
2. maybe (Q^2) / 3(d^5) = (v^2) / 2gD ?
... do you know what the variables mean and how the two formulas were derived, what assumptions they use?

since v= Q/ A , we can rewrite it as
fL(Q^2) / 2g(π^2)(d^5) , am i right ?
... that's the idea. Does this make the two equations the same?

foo9008
1. are the authors using the same models?
2. maybe (Q^2) / 3(d^5) = (v^2) / 2gD ?
... do you know what the variables mean and how the two formulas were derived, what assumptions they use?

... that's the idea. Does this make the two equations the same?
what do you mean by same models here ?

ia my idea correct ? or the author's idea is correct ?

Simon Bridge
Homework Helper
The mathematical equations you use are based on a set of assumptions about how the world works, which is described, in turn, by other maths.
The combinations of assumptions and maths is called a physical model.
Make different assumptions you get different equations.

You also get equations that look different but are actually the same ... for instance, the volume of a sphere could be written ##\frac{4}{3}\pi r^3## where ##r## is the radius, or it could be ##\frac{4}{3\sqrt{\pi}}A^{3/2}## ... where ##A## is the area of the biggest circular slice through the sphere, or it could be ##\frac{4}{16}\pi d^3## where ##d## is the diameter. You may use one equation instead of another because is is convenient to the information you have: i.e. it is easier to measure diameter than radius, but easier to build a sphere by fixing the radius rather than the diameter.

So do the two authors equation look different because they are actually different or is it more superficial than that?
The question is asking why the authors use the different equations ... that will be related to what they want to use the equations for as well as the assumptions they are making about reality.

I cannot tell if your idea is correct because I don't know what it is supposed to do. You didn't say.
However there is quite a lot going on which you have yet to address.


text:
# Standard coupling unification in SO(10), hybrid seesaw neutrino mass and leptogenesis, dark matter, and proton lifetime predictions

@article{Parida2016StandardCU,
title={Standard coupling unification in SO(10), hybrid seesaw neutrino mass and leptogenesis, dark matter, and proton lifetime predictions},
author={M. K. Parida and Bidyut Prava Nayak and Rajesh Satpathy and Ram Lal Awasthi},
journal={Journal of High Energy Physics},
year={2016},
volume={2017},
pages={1-58}
}
• Published 13 August 2016
• Physics
• Journal of High Energy Physics
A bstractWe discuss gauge coupling unification of SU(3)C × SU(2)L × U(1)Y descending directly from non-supersymmetric SO(10) while providing solutions to the three out-standing problems of the standard model: neutrino masses, dark matter, and the baryon asymmetry of the universe. Conservation of matter parity as gauged discrete symmetry for the stability and identification of dark matter in the model calls for high-scale spontaneous symmetry breaking through 126H Higgs representation. This…
23 Citations
Triplet leptogenesis, type-II seesaw dominance, intrinsic dark matter, vacuum stability and proton decay in minimal SO(10) breakings
• Physics
Journal of Cosmology and Astroparticle Physics
• 2020
We implement type-II seesaw dominance for neutrino mass and baryogenesis through heavy scalar triplet leptogenesis in a class of minimal non-supersymmetric SO(10) models where matter parity as
Unification predictions with or without supersymmetry
• Physics
• 2020
Supersymmetric (SUSY) grand unified theories (GUTs) appear to be best motivated for understanding strong, weak and electromagnetic interactions of nature. We briefly review emergence of new formulas
Neutrino Mass, Coupling Unification, Verifiable Proton Decay, Vacuum Stability, and WIMP Dark Matter in SU(5)
• Physics
Advances in High Energy Physics
• 2018
Nonsupersymmetric minimal SU(5) with Higgs representations 24H and 5H and standard fermions in 5¯F⊕10F is well known for its failure in unification of gauge couplings and lack of predicting neutrino
Asymmetric dark matter models in SO(10)
• Physics
• 2016
We systematically study the possibilities for asymmetric dark matter in the context of non-supersymmetric SO(10) models of grand unification. Dark matter stability in SO(10) is guaranteed by a
High Scale Type-II Seesaw, Dominant Double Beta Decay within Cosmological Bound and LFV Decays in SU(5)
• Physics
Advances in High Energy Physics
• 2019
Very recently novel implementation of type-II seesaw mechanism for neutrino mass has been proposed in SU(5) grand unified theory with a number of desirable new physical phenomena beyond the standard
Neutrinoless double beta decay and Sterile dark matter in extended left right symmetric model
• Physics
• 2022
We have studied a ﬂavor symmetry based extended left-right symmetric model(LRSM) with dominant type-II seesaw mechanism and have explored the associated neutrino phenomenology. The particle content
Vanishing threshold uncertainties and prediction for proton decay in non-supersymmetric $E_6$ GUT with intermediate trinification symmetry
• Physics
• 2020
We consider a class of non-supersymmetric $E_6$ Grand Unified Theories (GUTs) with trinification symmetry $SU(3)_C \times SU(3)_L \times SU(3)_R\times D$ ($D$ denoted as D-parity for discrete
Importance of generalized μτ symmetry and its CP extension on neutrino mixing and leptogenesis
• Physics
Journal of High Energy Physics
• 2019
Abstract Within the framework of residual symmetry, two ℤ 2 type associate μτ inter- change symmetries robustly constrain the Dirac CP phase δ in a model independent way. Both of them predict

## References

SHOWING 1-10 OF 409 REFERENCES
A neutrino mass-mixing sum rule from SO(10) and neutrinoless double beta decay
• Physics
• 2017
A bstractMinimal SO(10) grand unified models provide phenomenological predictions for neutrino mass patterns and mixing. These are the outcome of the interplay of several features, namely: i) the
Unification of gauge couplings in radiative neutrino mass models
• Physics
• 2016
A bstractWe investigate the possibility of gauge coupling unification in various radiative neutrino mass models, which generate neutrino masses at one- and/or two-loop level. Renormalization group
Minimal SUSY SO(10) model and predictions for neutrino mixings and leptonic CP violation
• Physics
• 2003
We discuss a minimal supersymmetric SO(10) model where $B\ensuremath{-}L$ symmetry is broken by a 126 dimensional Higgs boson multiplet which also contributes to fermion masses in conjunction with a
Fermion masses and mixings in SO 10 models and the neutrino challenge to supersymmetric grand unified theories
• Physics
• 2006
We present a detailed study of the quark and lepton mass spectra in a SO(10) framework with one 10H and one 126H Higgs representations in the Yukawa sector. We consider in full generality the
Inverse Seesaw Mechanism in Nonsupersymmetric SO(10), Proton Lifetime, Nonunitarity Effects, and a Low-mass Z' Boson
• Physics
• 2012
Recently realization of TeV scale inverse seesaw mechanism in supersymmetric SO(10) framework has led to a number of experimentally verifiable predictions including low-mass W_R and Z' gauge bosons


text:
# Latin Capital Letter Q

Q
Symbol Format Data
Q Code Point
U+0051
TeX
Q
SVG

## Usage

The capital Latin letter Q is used in mathematics to represent the set of natural numbers. Usually, the letter is presented with a “double-struck” typeface when it is used to represent the set of rational numbers.

## Related

The set of rational numbers is denoted with the Latin Capital letter Q presented in a double-struck type face.

The latin letter q.


text:
4
Q:

# The ratio of investments of two partners is 11:12 and the ratio of their profits is 2 : 3. If A invested the money for 8 months , find for how much time B invested his money.

Q:

Vishal and Raghu jointly started a business. Vishal invested three times as Raghu did and invested his money for double time as compared to Raghu. Raghu earned Rs. 4000. If the gain is proportional to the money invested and the time for which the money is invested then the total gain was  ?

 A) Rs. 28,000 B) Rs. 18,000 C) Rs. 14,000 D) Rs. 8,000

Explanation:

Investments ratio is = 3:1
Time period ratio is = 2:1

As they are proportional to gain

------
Gain ratio of Vishal and raghu = 6:1

But given Raghu got Rs. 4000,

=? 1 ----- 4000
7 -----    ?

=> Rs.28,000

The total gain = Rs.28,000

2 15
Q:

The ratio of investments of two partners A and B is 7:5 and the ratio of their profits is 7:10. If A invested the money for 5 months, find for how much time did B invest the money ?

 A) 11 months B) 9 months C) 7 months D) 10 months

Explanation:

7x5: 5xk = 7:10
k = 10 months

1 17
Q:

A, B, C and D enter into partnership. A subscribes 1/3 of the capital B 1/4, C 1/5 and D the rest. How much share did A get in a profit of Rs.2490 ?

 A) Rs. 820 B) Rs. 830 C) Rs. 840 D) Rs. 850

Explanation:

Let the total amount in the partnership be 'x'.
Then A's share = x/3
B's share = x/4
C's share = x/5
D's share = x - (x/3 + x/4 +x/5) = 13x/60

A : B : C : D = x/3 : x/4 : x/5 : 13x/60 = 20 : 15 : 12 : 13

A's share in the profit of Rs. 2490 = 20 (2490/60) = Rs. 830.

3 37
Q:

Rs.1200 divided among P, Q and R. P gets half of the total amount received by Q and R. Q gets one-third of the total amount received by P and R. Find the amount received by R ?

 A) Rs. 1100 B) Rs. 500 C) Rs. 1200 D) Rs. 700

Explanation:

Let the amounts to be received by P, Q and R be p, q and r.
p + q + r = 1200
p = 1/2 (q + r) => 2p = q + r

Adding 'p' both sides, 3p = p + q + r = 1200
=> p = Rs.400

q = 1/3 (p + r) => 3q = p + r

Adding 'q' both sides, 4q = p + q + r = 1200
=> q = Rs.300

r = 1200 - (p + q) => r = Rs.500.

2 37
Q:

K and L start a business jointly. L invests Rs.16000 for 8 months and L remains in the business for 4 months. Out of the total profit B claims 2/7th share. How much money is contributed by B  ?

 A) Rs. 9040 B) Rs. 10,400 C) Rs. 12,800 D) Rs. 6350


text:
# Why isn't freefall based on the mass of the object falling?

1. Oct 12, 2004

### stunner5000pt

We all know that with newtonian physics that it doesnt matter in freefall, what the mass of an object is, the only thing that matters is the height from which it dropped (given no external force and all)

But why is that true? Why doesn't it depend upon mass?

i have read in a book or seen on television that general relaitivity answers this kind of question.

Last edited: Oct 12, 2004
2. Oct 12, 2004

Staff Emeritus
Thought Experiment

Suppose you do a Galileo and drop two identical weights from a tower. They fall side by side and being identical (and of course always neglecting air resistance!) they maintain that side by side till they hit the ground. You can even suppose they are touching each other as they fall. Now add the tiniest drop of glue to fasten them together so they are now ONE object, twice as heavy as either one of the twin weights. Are they going to fall differently? Is one little tad of glue going to make them fall twice as fast?

Now do it in reverse, drop a single weight, then carefully bisect it and drop the two halves in touching position. You see that weight doesn't make the difference.

3. Oct 12, 2004

### stunner5000pt

i understand that

but Why

4. Oct 12, 2004

### JasonRox

General Relativity showed that the Inertial Mass and Gravitational Mass balance each other out.

5. Oct 12, 2004

### geometer

Plain old Newtonian physics answers this question.

Here's why:

Consider a mass we will call m1.

Then we can write the equation for the gravitational attraction between m1 and the earth:

F1 = Gm1Mearth/r^2

By Newton's Second Law: F1 = m1a

Substitute this in for F1 in the gravitational equation:

m1a = Gm1Mearth/r^2

Notice we can divide m1 out of both sides leaving:

a = GMearth/r^2

In other words the acceleration (or the rate at which a body falls) is independent of its mass. And assuming we drop objects from the same height; G, Mearth and r are all constants so all masses will fall at the same rate.

Last edited: Oct 12, 2004
6. Oct 13, 2004

### blue_sky

We can find, at best, the laws of the universe.
The "why the laws are like they are" it's something God should answer.
But I don't think He will.

blue

7. Oct 14, 2004

### cepheid

Staff Emeritus
WHY is this question being in two or three places on PF? It's not hard to figure out! To further simplify what geometer illustrated already:

Normally when a force acts on each of two objects, the more massive one will have a smaller acceleration by Newton's second law.

However, what if the magnitude of the force acting on the two objects is proportional to the mass of the object?!? If the objects are falling bodies, the more massive one will also have a greater force acting on it! i.e. it will weigh more. It will weigh more by just the right amount to compensate for the fact that it is more massive and has more inertia...because the greater weight is due to the fact that it has that much more mass in the first place! The two effects (from Newton's second law and Newton's law of Gravitation) will cancel out:

$$F = ma$$ Newton's second law

$$F = mg$$ Magnitude of force acting on object in gravitational field

$$a = \frac{F}{m} = \frac{mg}{m} = g$$

The assumption that both m's are equivalent hinges on what JasonRox said about the equivalence between inertial and gravitational mass, which I believe was verified experimentally even before GR, right?

Now, for an object twice as massive:

$$a = \frac{F}{2m} = \frac{2mg}{2m} = g$$

8. Oct 16, 2004

### Eyesaw

That might be true if the gravitational field was a single value, g, throughout the universe but obviously this is not the case in the real world. If g is proportional to mass in the first place (the moon has a different g for example than the earth), two objects with different masses must accelerate at different rates toward the earth since each mass generates a different g field than the other with the Earth.

The acceleration rates of two masses then should include the rate of acceleration of the earth towards the mass so that the total freefall
rate is g + G(the acceleration of the Earth towards the mass). So
although g is the same in any two mass system with the Earth as
one mass, the G is different for different masses. The freefall
rate of a baseball to the Earth then is g + b while the free fall rate
of a bowling ball, g +B- they are different.

Galileo jumped to conclusions then when he couldn't find a difference
in freefall rates between a bowling ball and a baseball. The contributions
of G to the Earth's g field between a bowling ball and a baseball is what?
Pretty pretty small.

And his critique of Aristotle's logic of gravity is also flawed for the same reason- he forgot about the gravity of the Earth. Trying to detect the free fall rate differences between a baseball and a bowling ball is like trying to measure the difference in displacement of the water in the Pacific Ocean caused by a flea and a frog. It's probably much worse actually. Even more absurd would be if one measured the dispacement of the Pacific Ocean when the flea and the frog are in the water at the same time and used that data to conclude there is no difference in weight between a flea and a frog, kind of reminds me of dropping two balls of different mass at the same time in Galileo's freefall experiment
of gravity.

Last edited: Oct 16, 2004
9. Oct 16, 2004

### cepheid

Staff Emeritus
I'm sorry Eyesaw, but I don't follow any of your above argument.

What do you mean by "the" gravitational field? I'm well aware that every particle or body in the universe that has mass generates its own gravitational field. In this case, I was referring to Earth's gravitational field!

What do you mean "generates a g field with the Earth"? The objects do not generate combined gravitational fields. Each of the three objects (Earth, the bowling ball, and the baseball) generates its own distinct gravitational field. However, since we are considering the accelarations of the two balls, only the forces acting on the balls are relevant. In this case there is one, and it is due to Earth's gravitational field. In detail:

Let $m_1$ be the mass of the bowling ball, $m_2$ be the mass of the baseball, and $M_E$ be Earth's mass. Let $m_1$ = $2m_2$ (the bowling ball has twice the mass of the baseball). Earth's gravitational field strength is:

$$g = \frac{GM_E}{r^2}$$

This gravitational field strength is the force exerted per unit mass on any body under the influence of the field. So the force experienced by the baseball is:

$$F_2 = G\frac{m_2 M_{E}}{r^2}$$

The force experienced by the bowling ball is:

$$F_1 = G\frac{(2m_2) M_{E}}{r^2} = 2G\frac{m_2 M_{E}}{r^2}$$

The acceleration of each ball from Newton's 2nd Law is:

$$a = \frac{F}{m}$$

For the baseball:

$$a_2 = \frac{F_2}{m_2} = G\frac{m_2 M_{E} }{m_{2} r^2} = \frac{GM_E}{r^2} = g$$

For the bowling ball:

$$a_1 = \frac{F_1}{m_1} = G\frac{(2m_2) M_{E} }{(2m_{2}) r^2} = \frac{GM_E}{r^2} = g$$

10. Oct 16, 2004

### Eyesaw

That's acceleration of a mass to the Earth caused by the G force of the Earth, you ignored the acceleration of the Earth towards the second mass caused by the g force of the second mass, which is different for two different masses. The total freefall rates then are different for different mass systems. g+l is not equal to g + m.
In other words, a baseball does not fall at the same rate to the Earth as a bowling ball. By ignoring the acceleration of earth to claim UFF you contradict a precept in both Newton and GR- that mass is a cause of gravitational force (in Newton) or space-time distortion (in GR).

Last edited: Oct 16, 2004
11. Oct 16, 2004

### Eyesaw

That argument is flawed since the mass of the objects you are dropping is so small in comparison to the Earth. That's like arguing two drops of water in an ocean should cause the sea level to rise twice as much as one drop, then failing to observe such a change, to conclude that one drop of water weighs as much as two drops of water.

12. Oct 16, 2004

### geometer

Actually, I think we are talking about two different things here. All objects accelerate toward the earth at the same rate regardless of their mass. However, the total closing rate of the earth/object system varies with the mass of the other object in the system.

By the way, with respect to the bowling ball and the baseball, the correction would be on the order of 10 exp (-24) meters per second squared - I would consider this negligible.

13. Oct 16, 2004

### Eyesaw

Since gravitational field strength falls off with squared distance (or conversely increases with squared proximitiy), the larger acceleration on the Earth caused by the larger mass would bring the Earth closer to it at every point in time when compared with the situation with the smaller mass so at no time is the g caused by the Earth on the larger mass even equal to the g caused by the Earth on the smaller mass. So two different masses never fall at the same rate toward the Earth.

E.g.:

Baseball :

a = mg/m = g.

Soccer Ball:

A = SG/S = G.

G is not g because the r^2 is different in time for
the soccer ball than the baseball. The g of the Earth on the baseball would
grow larger as time passes than the G of the Earth on the soccerball due to
the larger mass of the baseball causing the r^2 to decrease
faster than in the case with the Earth and the soccer ball in this example.

The Universality of Free Fall rates is a myth created by an unreasonable
expectation on extant technologies from Galileo on to perform a task that is likened to an attempt to finding the differences in displacement of the Pacific Ocean caused by one drop of water and by two drops of water falling on it.

Though the myth has helped generate much public interest in gravity with its mistique, it has done nothing but damage to the science of gravity due to its defiance of the logical methods.

Last edited: Oct 16, 2004
14. Oct 16, 2004

### franznietzsche

I just wasted fifteen minutes trying to refute this and failed. It then took me 30 seconds to figure why: The problem arises from the assumption that the eart is an inertial frame. If the earth is treated as an inertial frame, as is generally the case in solving newtonian problems, then the fall times will be equal. This is because an inertial frame is in uniform motion--not accelerating, so the earth could not be accelerating towards the two objects.
Because the coordinate system is designated as fixed to the earth, there is no incorporation of the earth's acceleration--the origin should not be moving.

So to make that shorter: you are right, because the earth is not an inertial frame, and technically it is a bad thing to fix a coordinate system. Anyone who tries to show that the two do will meet the earth in the same time with respect to a seperate inertial frame not fixed to either the earth or the balls, will faile. Further, if you attach the inertial frame to the balls, who have notably different masses (proportionally) you will get different accelerations, obviously.

15. Oct 16, 2004

### franznietzsche

No, it is just erroneously based on the assumption of the earth as an inertial frame of reference. It is not some horrific conspiracy, just an over-simplification, that for all practical purpose makes little significant difference in terms of the calculated values because the earth is so large. Ignoring air resitance is far more significant than the mislabeling of the earth as an inertial frame.

16. Oct 19, 2004

### Eyesaw

Yes, the moon landing demonstrated quite dramatically the error of attaching an inertial frame to a mass, aka the error of Galileo's two spheres. They demonstrated that the moon, with a mass 1/6th the Earth, "freefell" at a rate 1/6th that of the Earth towards the astronauts.

17. Oct 19, 2004

### franznietzsche

Umm. Not sure what you meant by that. Given that, assuming an inertial frame attached to the alrger mass we get: g= GM/r^2 , then the acceleration is obviously going to be 1/6 if the mass of the larger object is 1/6. But that is based on the attachment of an inertial frame to the larger mass M. So your post is a little confusing.

18. Oct 19, 2004

### Eyesaw

When discussing the Universality of Freefall what usually happens
is we see a calculation like the one made in a post above :

$$a_1 = \frac{F_1}{m_1} = G\frac{m_1 M_{E} }{m_{1} r^2} = \frac{GM_E}{r^2} = g$$
for a baseball,

and
$$a_2 = \frac{F_2}{m_2} = G\frac{m_2 M_{E} }{m_{2} r^2} = \frac{GM_E}{r^2} = g$$

for a bowling ball,

demonstrating that two different masses fall at the same rate.

I think we came to the agreement above that this result was achieved by ignoring the acceleration of the Earth towards the smaller masses. (The
actual total accelerations should be g+baseball and g+bowling ball.

My comment about the moon landing is that it demonstrated the folly, in
real spectacular fashion, of neglecting the acceleration of 'the other mass' in free fall experiments.

For if we but replaced the $$M_{E}$$ symbol in the above equations
to mean the mass of say, an astronaut, and the $$m_1$$ and $$m_2$$ to mean the mass of the Earth and the mass of the Moon, respectively, we arrive at the same result of $$a_1$$ and $$a_2$$ both being equal to $$g$$ with 'g' having the meaning of the gravitational field generated by the astronaut. But we all saw on TV that it was not the case that the Moon "free-fell" at the same rate of $$g$$ towards the astronaut as when the Earth "free-fell" to him when he was on it. And surely this "experiment" is no different in principle than the usual consideration of two different masses falling
toward the Earth?

The landing on the moon then was truly a giant step for mankind for
it really stood the Universality of Free Fall principle on its head.

Last edited: Oct 19, 2004
19. Oct 19, 2004

### Phobos

Staff Emeritus
Explain how the "moon's free fall" was measured during the Apollo missions.

As noted earlier, the difference in gravitational attraction between a planetary body and small masses like a baseball vs. a bowling ball is negligible for the purposes at hand.

20. Oct 19, 2004

### Eyesaw

When you see the footage of the astronauts, it's obvious when they
jumped up, it took longer for them to land than if they made the
same jump on Earth. Their steps were more like leaps. This longer "landing"
time on the moon equates to a large difference in the "free-fall" rates
between the moon and the Earth toward the astronaut . That is,
if we focused on the Astronaut instead of the planetary masses as the cause of the g (neglecting the G caused by the planetary masses on the Astronaut as we neglect the G nowadays of the smaller mass in "regular" free-fall experiments) we would obtain a "free-fall" rate that is dramatically different
for different masses (Earth having a much greater "free-fall" rate toward
the astronaut than the Moon).

Maybe the above would be clearer if we spoke in terms of two different spheres being "dropped". In Galileo's experiment, one sphere, a baseball
is dropped from a height on Earth, then another sphere, a bowling ball
is dropped from the same height. The "identical" rate is used as experimental
support for the claim of Universality of Free Fall. If we focused on the baseball instead and compared the "free-fall" rate of the Earth to the baseball
with the "free-fall" rate of the Moon to the baseball (first dropping the baseball on the Earth from a height and then going to the moon and dropping
it from the same height), we see that the Universality of Free Fall claim doesn't hold water.

The former is like trying to find the differences in the dispacement of the
water level of the Pacific Ocean between one and two drops of water falling on it, then finding none, concluding that all masses cause the same displacement on a body of water, while the latter is like comparing the differences in water level by filling the Grand Canyon with all the water in the Red Sea, draining it and then filling it again with all the water in the Yellow River and noting that since the larger mass of water filled a larger volume
of the Grand Canyon, differences in mass must cause a difference in displacemnt on a body of water. Now, which experiment do you think is more instructive?

Last edited: Oct 19, 2004


text:
Dictionary Learning Strategies for Compressed Fiber Sensing Using a Probabilistic Sparse Model

We present a sparse estimation and dictionary learning framework for compressed fiber sensing based on a probabilistic hierarchical sparse model. To handle severe dictionary coherence, selective shrinkage is achieved using a Weibull prior, which can be related to non-convex optimization with p-norm constraints for 0 < p < 1. In addition, we leverage the specific dictionary structure to promote collective shrinkage based on a local similarity model. This is incorporated in form of a kernel function in the joint prior density of the sparse coefficients, thereby establishing a Markov random field-relation. Approximate inference is accomplished using a hybrid technique that combines Hamilton Monte Carlo and Gibbs sampling. To estimate the dictionary parameter, we pursue two strategies, relying on either a deterministic or a probabilistic model for the dictionary parameter. In the first strategy, the parameter is estimated based on alternating estimation. In the second strategy, it is jointly estimated along with the sparse coefficients. The performance is evaluated in comparison to an existing method in various scenarios using simulations and experimental data.

Authors

• 3 publications
• 18 publications
• NOODL: Provable Online Dictionary Learning and Sparse Coding

We consider the dictionary learning problem, where the aim is to model t...
02/28/2019 ∙ by Sirisha Rambhatla, et al. ∙ 0

• Observable dictionary learning for high-dimensional statistical inference

This paper introduces a method for efficiently inferring a high-dimensio...
02/17/2017 ∙ by Lionel Mathelin, et al. ∙ 0

• Self-expressive Dictionary Learning for Dynamic 3D Reconstruction

We target the problem of sparse 3D reconstruction of dynamic objects obs...
05/22/2016 ∙ by Enliang Zheng, et al. ∙ 0

• Learning Hybrid Representation by Robust Dictionary Learning in Factorized Compressed Space

In this paper, we investigate the robust dictionary learning (DL) to dis...
12/26/2019 ∙ by Jiahuan Ren, et al. ∙ 9

• Globally Variance-Constrained Sparse Representation for Image Set Compression

Sparse representation presents an efficient approach to approximately re...
08/17/2016 ∙ by Xiang Zhang, et al. ∙ 0

• Example Selection For Dictionary Learning

In unsupervised learning, an unbiased uniform sampling strategy is typic...
12/18/2014 ∙ by Tomoki Tsuchida, et al. ∙ 0

• A Dictionary Learning Approach for Factorial Gaussian Models

In this paper, we develop a parameter estimation method for factorially ...
08/18/2015 ∙ by Y. Cem Subakan, et al. ∙ 0

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1 Introduction

Fiber sensors are versatile devices with broad applicability [38, 23, 46, 63]. They are of high interest in smart structures to sense and react to the environment [42, 58]. For quasi-distributed sensing based on wavelength-division multiplexing (WDM), fiber Bragg grating (FBG) sensors are often employed due to their sensitivity to strain or temperature [38, 23]. An FBG describes a local variation of the refractive index and reflects light at a certain wavelength, called Bragg wavelength. Typically, a number of detuned FBGs is imprinted into the core of an optical fiber. Fiber interrogation is performed using broadband light sources or wavelength-tunable lasers. The latter feature higher local signal-to-noise ratios (SNRs) [46, 63]. However, in order to monitor time-varying perturbations, the laser has to sweep quickly through the tuning range. This requires high-speed analog-to-digital converters (ADCs) and produces large amounts of data.
Compressed sensing (CS) [5, 15, 27] can help to alleviate these problems by taking samples in form of projections into a low-dimensional subspace. The original signal can be reconstructed by exploiting the sparsity of the signal with respect to an adequate dictionary [14, 11]. This task strongly resembles the sparse synthesis problem with redundant dictionaries in [57, 53]. Besides greedy methods, such as Orthogonal Matching Pursuit (OMP) [49], -minimization is a popular method to solve the sparse reconstruction problem [56, 24, 12]. It relies on the restricted isometry property (RIP), which essentially states that unique sparse solutions can be recovered by restricting the -norm instead of the -norm [25, 13]. Redundant dictionaries can yield highly sparse representations, that allow for estimating quantities at high resolution directly in the sparse domain [24, 41]. However, redundancy causes inter-column coherence and it is likely that the required RIP conditions are no longer fulfilled [24, 52, 12]. The -norm, with , offers a trade-off to avoid an NP-hard combinatorical problem imposed by the -norm, while a unique solution might still be retrieved [17, 18].

Dictionaries can be classified as parametric or non-parametric. Non-parametric dictionaries are typically learned from training data and often used if no analytical model is available

[53]. While they can yield sparser representations of certain data realizations [40], non-parametric dictionaries usually lack an interpretable structure and are inefficient in terms of storage [53]. Parametric dictionaries, in turn, rely on an analytical model for the observed signal. Their analytic form offers an efficient implementation and a means to obtain optimality proofs and error bounds [53]. They are also favorable in terms of scalability and storage-efficiency [4, 53]. Translation-invariant dictionaries represent an important sub-class of parametric dictionaries, that can be used to estimate the translation coefficients of localized signals [36, 9, 29]. Nonetheless, due to the complexity of natural signals, some model parameters might be unknown or contain uncertainty. Parametric Dictionary Learning (DL) addresses this problem with the aim of estimating these parameters from the measured data. Herein, statistical DL methods, such as maximum likelihood (ML) or maximum a posteriori (MAP) estimation, are commonly employed [53]. In order to solve the resulting optimization problem, alternating estimation (AE) is a frequently pursued sub-optimal paradigm, that iteratively optimizes a local objective function [43, 6, 2]

. In a Bayesian setting, the Expectation Maximization (EM) algorithm is a popular variant of AE-based estimation

[53].
A model for the sparse coefficients can be of deterministic or probabilistic nature. While the deterministic case is often assumed in sparse estimation [24, 12], a probabilistic model offers high flexibility to take model deviations and measurement errors into account. Moreover, a hierarchical structure can be used to incorporate additional uncertainty in prior assumptions. Sparsity can either be promoted by continuous distributions, resulting in weakly sparse models, or by discrete mixtures, leading to strongly sparse models [43]. A prominent example of discrete mixtures are Spike & Slab models [34]

. They are based on binary activations and yield strongly sparse representations. Continuous sparse priors, such as a Gaussian or double-exponential (Laplace) prior, feature high excess kurtosis with heavy tails and a narrow peak around zero

[50, 43]. Besides sparsity, additional knowledge of the signal, e.g. correlation, can be incorporated [26, 66].
For many practical models, evaluating the posterior distribution is not feasible and approximate methods, such as Markov Chain Monte Carlo (MCMC) or variational Bayes methods, have to be used to accomplish inference [8, 48, 54]. Variational methods use rather simple analytic functions to approximate the posterior distribution by factorization, which is favorable in terms of scalability and computational costs but leads to a deterministic approximation [54, 8]. MCMC methods attempt to sample the posterior distribution, where subsequent samples form a Markov chain [8]. The Hamilton Monte Carlo (HMC) method is a powerful technique, that is especially suitable for sampling high-dimensional spaces in the presence of correlation [47]. However, MCMC performance is generally limited by the available computation time, thereby relying on a stochastic approximation. Another application of MCMC is found in non-convex optimization, where Stochastic Gradient (SG) MCMC has gained popularity for large-scale Bayesian learning [22, 20, 19].
In the present work, we consider the problem of Compressed Fiber Sensing (CFS) with highly coherent translation-invariant dictionaries and imperfectly known parameters. For the sparse coefficients, a weakly sparse hierarchical model is considered. We also establish a relation between this model and non-convex optimization with -norm constraints for . In order to alleviate the problem of dictionary coherence, we leverage additional structure of the dictionary and achieve augmented sparsity by establishing a Markov random field (MRF) relation among the sparse coefficients. For dictionary learning, we pursue two different strategies: In the first strategy (S1), we consider a deterministic dictionary parameter, that is estimated using a Monte Carlo EM algorithm. In the second strategy (S2), a probabilistic hierarchical model for the dictionary parameter is considered, leading to a full Bayesian formulation and joint estimation of the sparse coefficients and the dictionary parameter. In both strategies, approximate inference is accomplished using a hybrid MCMC method based on Gibbs sampling and HMC. Finally, we use simulations and real data to compare the proposed methods to previous work in [62], where a deterministic model is considered for the sparse coefficients and the dictionary parameter. For the deterministic case, we derive the Cramér-Rao bound (CRB) to assess the performance gain achieved by a probabilistic model.

1.1 Contributions

• We propose a probabilistic model for the sparse coefficients, where a Weibull prior is used to promote (weak) sparsity. Additional collective shrinkage is achieved by establishing an MRF-relation among the sparse coefficients based on a bivariate kernel function in the joint prior density. This helps to moderate the impact of severe dictionary coherence and can be used in general sparse synthesis problems with similar dictionary structure. We also establish a relation to non-convex optimization with constraints on the -norm for .

• For dictionary learning, we investigate two conceptually different strategies, assuming either a deterministic (S1) or a stochastic (S2) dictionary parameter. In both strategies, the noise level can be jointly estimated along with the sparse coefficients. We further highlight advantages, disadvantages and limitations to offer support in choosing an adequate method for practical systems.

• To accomplish inference in these models, we use a hybrid MCMC method, combining HMC and Gibbs sampling, We show its applicability and efficacy in the considered sampling problem for CFS.

• We use simulations to evaluate the performance of the proposed sparse estimation and DL methods for various scenarios of different CS sample sizes, SNRs and CS matrices. These results are compared to an existing method in [62], where the sparse coefficients and the dictionary parameter are assumed to be deterministic. In addition, we provide a real-data example to verify the practical applicability of S1 and S2.

• We derive the Cramér-Rao bound for jointly estimating the sparse coefficients and the dictionary parameter in the deterministic case. It is a valid bound for the competing method in [62], and serves to assess the achieved performance gain of our probabilistic approach.

2 Related Work

There exists little work addressing the combination of CS and DL for the application of WDM-based distributed fiber-optic sensing [60, 61, 62]. In [60], a model for the received sensor signal is presented, from which a redundant shift-invariant parametric dictionary is created. The works in [61, 62] focus on the aspect of CS and sparse estimation in the case of uncertain dictionary parameters. The authors use AE-based estimation to determine the dictionary parameters, where a pre-processing routine accounts for severe dictionary coherence. Unlike our approach, these works use a deterministic model for the sparse coefficients and dictionary parameters.
Weakly sparse models have been widely used in the literature. A comprehensive analysis of different hierachical sparse prior models is provided in [44]. The general problem of choosing the prior in weakly sparse models for sparse regression is addressed in [50], where the authors describe various properties of different shrinkage priors and illuminate the selection problem from two perspectives: prior distributions and penalty functions. The work in [43] also investigates Bayesian methods with different sparse models in comparison to classical -minimization. Seeger [54] found that the Laplace prior is able to shrink most components close to zero, while allowing for selected components to become sufficiently large. This effect, termed selective shrinkage in [35], is most noticeable for heavy-tailed priors, e.g. the Student’s -prior [54] or the horseshoe prior in [16, 50]. Based on these findings, we select a sparsity prior that resembles a positive version of the horseshoe prior. Other works, that focus on the perspective of penalized regression, report higher sparsity levels by penalizing the -norm with instead of the -norm [30]. The authors in [17] show that the RIP requirements for the dictionary can be relaxed in this case. It is also pointed out in [17, 18] that non-convex CS with -norm penalization requires less measurements than standard CS, which is based on the -norm. We rely on these results and show a relation between the considered sparsity prior and non-convex optimization with -norm constraints.
There exist several approaches to exploit additional structure of the signal. One example is block sparsity [26]. A block sparse Bayesian learning framework is proposed in [66], pointing out how correlation can be exploited in regularization algorithms. Wakin et al. [59] introduce the concept of joint sparsity for signal recovery in distributed CS theory. In [41], temporal correlation across subsequent CS measurements is considered, while the authors in [21] use correlation to achieve smoothness. Another related concept is proposed in [3], where a truncated multivariate Ising MRF model is used to describe the correlation between adjacent pixels for image processing. Different from these works, we use the ideas of MRFs [45] and exploit correlation to achieve collective shrinkage among the sparse coefficients.
A comparative analysis in [43] suggests that MCMC methods are powerful for inference in sparse models. In [47], the benefits of HMC and Gibbs sampling in hierarchical models are outlined. It is also shown, that HMC can be more effective than a Gibbs sampler for sampling high-dimensional spaces in the presence of correlation. According to these results, we consider a hybrid MCMC method that combines HMC and Gibbs sampling for inference in our hierarchical model, where the sparse coefficients are high-dimensional and correlated. For parametric DL, the Monte Carlo EM algorithm in S1 represents one variant of the frequently applied AE-based estimation technique [39, 51]. Comparable to S2 is the Bayesian framework for sparse estimation and DL in [31]. However, the authors use a Gaussian prior without correlation.

2.1 Outline

In Section 3, the signal model for CFS is introduced, and in Section 4, the CRB for joint estimation of the deterministic sparse coefficients and dictionary parameters is derived. Section 5 details the sparsity and local similarity model, while Section 6 describes the hybrid MCMC method for approximate inference in this model. The parametric DL strategies S1 and S2 are described in Section 7. Section 8 shows the working principle along with a performance analysis of the proposed and an existing method based on simulations and experimental data. A discussion of the results and findings is given in Section 9. Section 10 concludes this work.

3 Signal Model

In order to determine the quantity and nature of impairments at the FBGs in a WDM-based fiber sensor, the time delays of the reflections from the individual FBGs need to be estimated. We adopt the model in [60, 61, 62], where CS-based acquisition is employed to reduce the number of samples to be stored and processed. The CS measurements are described by

 y=ΦA(θ)x+n, (1)

where is the CS sampling matrix and is a Gaussian noise component with independent and identically distributed (i.i.d.) entries, ,

. The vector

is sparse with significant components, and is a scalar dictionary parameter. The matrix represents a redundant shift-invariant dictionary and its columns, called atoms, represent FBG reflections on a dense grid of delays. The indices of the significant components in indicate the desired reflection delays. They are collected in the set . We can write the full data likelihood function for this model by

 (2)

The -th dictionary atom, , is defined by [62]

 [ai]l(θ)=r(lTd−iδt,θ),  l=1,…,L, (3)

where the generating function, , describes the reflection from a single FBG, incrementally shifted by and sampled with a design sampling period, . In order to specify the dictionary parameter in CFS according to [62], we write

 r(t,θ)=∫∞−∞ej2πftH\tiny LP(f,θ)iph(f)df. (4)

Herein, is the received photocurrent in the frequency domain, and is the transfer function of a lowpass filter, that models a limited effective bandwidth of the receiver circuitry. This bandwidth is described in terms of a positive dictionary parameter, . As an auxiliary parameter, it accounts for different indistinguishable sources of uncertainty, that all contribute to the broadening in the temporal response of the FBG reflections. A detailed model for is provided in [62].

4 The CRB for joint estimation of (x,θ) in CFS

We derive the CRB for jointly estimating the deterministic parameters (). This is a valid bound for the model considered in [62] and can be used to assess the relative performance gain achieved by the proposed probabilistic sparse model and DL strategies. Although the Bayesian CRB in [10] can be empirically determined, we found that this bound is very lose, due to the high information content in the considered sparsity prior. Therefore, and in regard of the comparative analysis with the deterministic case in [62], the non-Bayesian CRB is more useful in this case.
The constrained CRB for estimating with sparsity constraints has been derived in [7]. However, this derivation does not assume uncertainty in the dictionary. It is based on locally balanced sets and involves the projection of the Fisher Information matrix (FIM), , onto a low-dimensional subspace spanned by the so-called feasible directions. Any estimator, , for which the constrained CRB is a valid lower bound, must be unbiased with respect to these directions. The projection matrix can be created from the unit vectors corresponding to the non-zero coefficients in , that is with , . For a Gaussian likelihood as in (2), the FIM can be derived from the expected value of the Hessian matrix of the log-likelihood function, i.e. [37, 7]

 I(x)=−Ey∇2xlogp(y|x,θ)=1σ2nB⊤B, (5)

with . Further, we define the reduced FIM by . Then, given that is exactly -sparse, the constrained CRB for a known dictionary becomes [7]

 Cov(^x)⪰UI−1KU⊤,∥x∥0=K. (6)

Based on these results, we derive the CRB for the joint parameters . First, we derive the Fisher information for , given that is known. It is given by

 I(θ) = −Ey∂2∂θ2logp(y|x,θ) (7) = Ey∂2∂θ212σ2n(y−ΦA(θ)x)⊤(y−ΦA(θ)x) = 1σ2nx⊤A′(θ)⊤Φ⊤ΦA′(θ)x.

Herein, denotes the (element-wise) derivative of with respect to . Next, we have to take into account that and share some mutual information. Therefore, we define the combined FIM:

 I(γ)=(I(x)−Eyu−Eyu⊤I(θ)), (8)

where and , . Since the partial derivatives can be interchanged, the off-diagonal elements are identical. In order to complete the definition of , we determine

 −Eyui = −Ey∂2∂xi∂θlogp(y|x,θ) (9) = 1σ2nx⊤A′(θ)⊤Φ⊤Φai(θ).

The reduced FIM is obtained by appending the set of feasible directions, such that the coordinate is included, i.e. . Hence, . To obtain the inverse, we apply twice the matrix inversion lemma [32]

 I−1K+1=⎛⎜ ⎜⎝(IK−vv⊤I(θ))−1−1˘bI−1Kv−1˘bv⊤I−1K1˘b⎞⎟ ⎟⎠, (10)

where , and

 (IK−vv⊤I(θ))−1=I−1K+1˘bI−1Kvv⊤I−1K. (11)

The constrained CRB for the joint parameters in becomes

 Cov(γ) ⪰ ~UI−1K+1~U⊤,∥x∥0=K. (12)

Finally, a lower bound for the mean squared error (MSE) in the joint setting is obtained by updating the individual estimation errors to account for the information shared between and :

 MSE(^x) ≥ (Tr I−1K)+1˘bv⊤I−1KI−1Kv (13) MSE(^θ) ≥ 1˘b= I(θ)−1+v⊤IKvI(θ)(I(θ)−v⊤IKv). (14)

5 Probabilistic sparse model

Regarding the model in (1), the data can be explained in different ways. On the one hand, many non-zero components in and a large bandwidth, , result in many narrow temporal peaks that can yield a good approximation of the observed reflections. On the other hand, it is known that the sensing fiber contains FBGs, so we expect exactly reflections. Therefore, a more useful explanation is given by significant elements in with a smaller value of , such that correctly indicates the reflection delays. Nevertheless, even for a suitable value of , the signal is usually not exactly sparse but contains many small elements close to zero, e.g. due to measurement noise. In a strongly sparse model, these contributions are not taken into account, which impacts the positions of non-zero elements in . Hence, it may lead to incorrectly estimated reflection delays. This motivates a weakly sparse model, where the most significant components indicate the reflection delays. When and are both unknown, the reflections delays can only be estimated when prior information of sparsity is incorporated, since depends on and vice versa. Severe dictionary coherence aggravates this problem and results in several non-zero components with moderate amplitudes around the true significant elements. The coherence level is even stronger when the dimensionality of the acquired data is further reduced by CS. Thus, an adequate sparse model for must compensate for this effect. Classic -minimization can be interpreted as an MAP estimation problem, where has i.i.d. entries with Laplace priors [43]. However, the required performance guarantees for -minimization, essentially the RIP [14, 11], are no longer fulfilled in the case of strong dictionary coherence. According to [17, 18], the RIP conditions can be relaxed for -minimization, when . Therefore, we use a prior with stronger selective shrinkage effect, that can be related to constraints on the -norm in non-convex optimization. Yet, specific characteristics of the signal have to be considered. The measured reflection signal is proportional to the optical power, and the dictionary atoms essentially model the optical power reflected from the individual FBGs. Thus, the prior must also account for the non-negativity of the data. Due to these restrictions, we choose a Weibull prior that resembles a positive version of the horseshoe prior in [50] and induces the required selective shrinkage effect:

 xi∼p(xi)=W(xi|λw,kw), xi>0, i=1,…,N, (15)

where are the scale and shape parameters, respectively. Then, the joint prior density of is given by

 p(x|kw,λw)=kwλkwwN∏i=1xkw−1iexp(−λ−kwwN∑i=1xkwi). (16)

Fig. 1 (top left) shows qualitatively the shape of the considered prior in the bivariate case.
Based on (16) and (2), we can relate the problem to constrained ML estimation. First, let us consider an interpretation in terms of MAP estimation as in [44], by calculating or, equivalently,

 argminx∥y−ΦA(θ)x∥22+μ1N∑i=1log(xi)+μ2N∑i=1xkwi, (17)

where and with and . In order to formulate a related constrained ML problem, let us define two functions,

 g1=N∑i=1xkwi−λkw1  and  g2=N∑i=1log(xi)−λ2, (18)

where are related to the coefficients , respectively. The functions in (18) can represent inequality constraints of the form and , that account for the impact of the prior by restricting the search space. Hence, a constrained version of the ML problem can be formulated by

 argminx≻0 ∥y−ΦA(θ)x∥22 (19) s.t. ∥x∥kw ≤ λ1 (20) and N∑i=1log(xi) ≤ λ2. (21)

In this non-convex problem, denotes the -norm with

. The hyperparameters

control the shrinkage effects. Fig. 1 (top right) depicts the search space restricted by the constraints (20)-(21). The borders are shown for a fixed value of and in the bivariate case.

5.1 Local covariance model for augmented sparsity

In analogy to the concept of block sparsity [26], we can use the specific sparse structure of the signal with respect to the shift-invariant dictionary for CFS to exploit sparsity among groups of variables. The signal contains only reflections that arrive at temporally separated delays, indicated by the significant components in . Therefore, we can assume that a significant coefficient is always surrounded by larger groups of non-significant coefficients and any two significant components are always well separated. Also, it is likely that the amplitudes of adjacent non-significant coefficients are similarly close to zero. Borrowing from the ideas of MRFs [45], such local similarity can be modeled by a prior on the differential coefficients, , where . It restricts the variation of adjacent amplitudes and establishes a MRF relation between neighboring coefficients in . Then, non-significant coefficients with larger amplitudes are pulled down to match the majority with almost-zero amplitudes, which promotes additional collective shrinkage. However, if a significant coefficient follows a non-significant one (or vice versa), the model should allow for larger changes. Therefore, the differential variation must be locally specified, dependent on the respective amplitudes, in order to avoid undesired shrinkage or equalization.

To this end, we define a kernel function for all adjacent pairs of sparse coefficients, i.e. , with hyperparameter :

 K(xi,xi+1|λ\tinyΔ) = exp(−λ\tinyΔ|xi+1−xi|f\tinyK(xi,xi+1)). (22)

The bivariate function controls the similarity level between adjacent coefficients. Within the scope of this work we consider cases, where this function takes the form , , with positive constants , . They can be incorporated in to yield a modified joint prior density,

 ~p(x|kw,λw,λ\tinyΔ)  =  1Z\tinyKW(xN|kw,λw)
 (23)

with normalization constant . For any , it holds that and

 ~p(x|kw,λw,λ\tinyΔ)∣∣Z\tinyK=1 ≤ p(x|kw,λw) (24)

is bounded. Hence, there exists a positive constant that normalizes (23) to make a proper density. Fig. 1 (bottom) visualizes the function and its impact on the original prior in the bivariate case.
In the view of constraint ML estimation, the modified prior density in (23) can be related to the optimization problem in (19)-(21) by imposing additional constraints

 |xi+1−xi|f\tinyK(xi,xi+1)≤μi,i=1,…,N−1. (25)

Fig. 1 (top right) depicts a bivariate example. In order to show the MRF relation between the coefficients, we calculate the conditional densities . To this end, we conveniently define and get

 ~p(xi|x∖i,kw,λw,λ\tinyΔ) = ~p(xi|xi−1,xi+1,kw,λw,λ\tinyΔ% )

 ∝ W(xi|kw,λw)K(xi−1,xi|λ\tinyΔ)K(xi,xi+1|λ\tinyΔ), (26)
 ~p(x1|x∖1,kw,λw,λ\tinyΔ)   ∝ W(x1|kw,λw)K(x1,x2|λ\tiny% Δ), (27)
 ~p(xN|x∖N,kw,λw,λ\tinyΔ) ∝W(xN|kw,λw)K(xN-1,xN|λ\tinyΔ), (28)

where dependencies appear only between directly adjacent coefficients.
In order to account for deviations from prior assumptions, we consider randomization of the hyperparameters and assign conjugate inverse Gamma priors to the scale parameters and . Finally, given and a normalization constant , the shape parameter,

, is assigned the conjugate prior distribution according to

[28]:

 p(kw|a′,b′,(d′)kw,λw)=ka′wZkwexp(−b′kw−(d′)kwλw), (29)

Fig. 2 shows a factor graph for the complete sparsity model with randomized hyperparameters.

6 Approximate Inference: Hybrid MCMC

In order to accomplish inference in the sparse model, we apply a hybrid MCMC technique, i.e. HMC within Gibbs sampling. The reasons for using HMC are twofold: Firstly, it only requires an analytic expression for the posterior density to be sampled. Secondly, it is efficient in sampling high-dimensional spaces in the presence of correlation. However, as pointed out in [47], it can be more efficient to sample the hyperparameters separately, as their posterior distributions are often highly peaked and require a small step size in the HMC algorithm, which limits the general performance. Therefore, we employ an outer Gibbs sampler for approximate inference of the latent variables. In each iteration,

is sampled using HMC, while all other variables are fixed. Since we are also interested in estimating the noise variance,

, it is assigned an inverse Gamma () prior and sampled along with the other variables. The resulting model is summarized below:

 x|kw,λw,λ% \tinyΔ ∼ ~p(x|kw,λw,λ\tinyΔ)  in (???), λw ∼ Inv-Γ(λw|a,b), kw|λw ∼ p(kw|a′,b′,(d′)kw,λw)in (???), λ\tinyΔ ∼ Inv-Γ(λ\tinyΔ|a′′,b′′) σ2n ∼ Inv-Γ(λ\tinyΔ|aσ,bσ). (30)

We also define as a representative variable with corresponding positive, real-valued parameters and , that belong to the respective density functions. Further, the set denotes the set without the respective variable . Fig. 3 shows a graphical model that helps to visualize the dependencies in this model. Herein, and are only valid for strategy S2, which is discussed in Section 7. For the particular model in (30), we assume that the variables and are mutually independent. Gibbs sampling requires the full conditional distributions for each parameter of interest. Based on these assumptions, we obtain the relation

 ∝ p(y|x,C)p(ζ|x,C∖ζ) (31) ∝ p(y|x,C)p(ζ|C∖ζ)~p(x|C). (32)

Since the prior distributions are all conjugate to the Gaussian likelihood function in (2), a simple calculation yields the posterior distributions of the parameters involved in the Gibbs sampling procedure. For , we obtain

 ζ|y,x,C∖ζ∼ Inv-Γ(ζ|aζ+\scriptsizeM2, bζ+\scriptsize12) ~p(x|C), (33)

and for , we obtain

 kw|y,x,C∖kw ∼  p(kw|~a′,~b′,~c′) ~p(x|C), (34)

with parameters , , and . Samples of the posterior variables can be obtained using Metropolis Hastings [8] or HMC.

The sparse coefficients are sampled using HMC. We briefly describe the idea of this method according to [47], adapted to our model for :
Within the framework of HMC, the sampling process is described in terms of Hamilton dynamics, a concept known from classical physics. It is used to describe the trajectory of a physical system in phase space, based on its potential and kinetic energy. HMC assigns to every sparse coefficient, , an associated momentum variable, , , that is responsible for the sampling dynamics. The posterior density to be sampled is related to the potential energy, given by [47]

 U(x|y,C) = −log~p(x|y,C)−log(Zu), (35)

where is a suitable normalization constant. Since and are fixed, we may drop them and write instead. The kinetic energy, , depends only on the auxiliary variables . A standard choice for corresponds to independent particles in free space with mass , i.e. . The dynamics of the sampling process are governed by the Hamiltonian function, which is given by and represents the total system energy. The joint density of is defined by [47]

 p(x,ξ)=1Zce−H(x,ξ)T\scriptsize sys= ~p(x|y,C)N∏i=1N(ξi|0,mi). (36)

Herein, is called the system temperature and is a normalization constant. The last equation is obtained by setting and , while the Gaussian density arises from the special choice of the kinetic energy term. In HMC, a proposal for a new sample is obtained by the final points () of a trajectory described by Hamilton’s equations of motion. They are calculated , [47]:

 dxidt=ξimi,dξidt=−∂∂xi~p(x|y,C)~p(x|y,C). (37)

A Metropolis update decides, whether a proposed sample is accepted or rejected, with acceptance probability

[47]

 P(accept)=min(1, exp(−H(x∗i,ξ∗i)+H(xi,ξi)) ). (38)

7 Parametric DL strategies for CFS

In this section, we present two strategies for parametric dictionary learning in CFS. In the first strategy (S1

), we follow the ideas of hybrid Bayesian inference

[64, 65] and AM-based DL [6], where is a deterministic parameter, that is estimated using the Monte Carlo EM algorithm in [8]. In the second strategy (S2), we pursue a full Bayesian approach and consider a probabilistic model for . Herein, approximate inference is accomplished by extending the Gibbs sampler in Section 6 to jointly estimate . Fig. 3 depicts the dependency graph for both strategies, where belong exclusively to S2.
As pointed out in [64, 65], hybrid and full Bayesian strategies have their individual advantages in certain situations. For small sample sizes, Bayesian methods can be superior if good prior knowledge is available [64]. Nonetheless, they are often computationally more complex and insufficient prior information can lead to a small-sample bias, even if a non-informative prior is used [64]. In CFS, the sample size is small and only vague prior knowledge of is available. Therefore, we investigate the performance of both DL strategies based on our probabilistic sparse model. The computational complexity of both strategies is comparable. It is dominated by HMC, i.e. by sampling the high-dimensional vector in each iteration of the Gibbs sampler. Regarding , the following prior knowledge is assumed: In S1, we roughly restrict the range of values that can take, while in S2, we define a non-informative prior over the same range. Recall that effectively describes the filter characteristics of the lowpass filter . To create the dictionary for a certain value of using (3

), the inverse Fourier transform in (

4) has to be evaluated for each atom. Thus, the dictionary is not a simple function of and we restrict ourselves to a discrete set of parameters, with lower and upper bound, and , respectively. Since the bandwidth should be positive and bounded, we have and . Then, the set contains the discrete values ,

7.1 Hybrid DL: iterative estimation of θ and (x,C) (S1)

The dictionary parameters in the CFS problem can be iteratively estimated using a Monte Carlo EM algorithm. First, an initial value, , has to be chosen. In subsequent iterations with indices , we obtain joint samples , by Gibbs sampling and HMC according to Section 6. Then, we determine the posterior expectation of , using the previous estimate :

 ^ζ(d) = ∫dom(ζ)ζp(ζ|y,^θ(d−1))dζ (39) ≈ 1L\tiny MCL\tiny MC∑l=1 ζ(d)lp(ζ(d)l|y,^θ(d−1)), (40)

where is the domain of . The current estimates of the reflection delays, , are determined by identifying the indices of the largest elements in the posterior mean of , denoted by . It is obtained by exchanging with in (40). Besides, we also estimate the amplitudes of the significant components in . They can be useful to assess the sparsity level of the solution and to determine the amount of optical power reflected from the FBGs. Since the posterior of is multimodal with one narrow peak around zero and another peak at some larger amplitude, the MAP is more suitable for this task. It is given by

 {^x,^C}(d)\tiny MAP = argmaxx,C logp(x,C|y,^θ(d−1)) (41) ≈ argmax{xj,Cj}∈{xl,Cl}(d)l=1,..,L\tiny MClogp({xj,Cj}(d)|y,^θ(d−1)). (42)

However, the estimates of obtained from are less accurate than those obtained by the posterior mean. Therefore, the empirical MAP solution is only used to estimate the reflection amplitudes. Next, we calculate the current estimate by taking the expected value over given (E-step):

 (43) = ∫RN+∫Ψlogp(y,x,C|θ)p(x,C|y,θ)dCdx ≈ 1L\tiny MCL\tiny MC% ∑l=1 logp(y,{xl,Cl}(d−1)|θ) ≜ Q(θ|^θ(d−1)). (44)

Herein, is the product space formed by the individual domains of all variables in . In the -step, a locally optimal value, , is obtained by maximizing over the set , i.e.

 ^θ(d) = argmaxθ∈Θ Q(θ|θ(d−1)). (45)

7.1.1 Initialization of θ via bisectional search

An adequate initialization, , can alleviate the problem of local optima in the EM algorithm. In CFS, the desired sparsity level is known to be the number of reflections, . Hence, a good choice for yields a solution for with significant non-zero elements. Starting at an arbitrary value , a bisectional search within can quickly determine a suitable initial value. After choosing the first value at random, is subdivided into two parts, containing all larger and all smaller values, respectively. When the number of peaks is too high, the next trial is chosen as the median of the lower division. If it is too low, the next trial is the median of the upper division, and so on. For a properly selected , S1 converges faster and is more likely to approach (or even attain) the global optimum.

7.2 Bayesian DL: joint estimation of (x,C,θ) (S2)

In strategy S2, we treat

as a random variable. Due to its discrete nature, each element

is assigned a probability mass, , where . Then, is categorically (Cat) distributed over the set of discrete dictionary parameters, , with corresponding probability masses in . Uncertainty in the a priori assigned probability masses is taken into account in terms of a prior on . The Dirichlet (Dir) distribution can be used as the conjugate prior with parameters , i.e.

 p(Ξ)= 1B(ν)RΘ∏r=1pνrr, (46)

where denotes the Beta function and the variables , , describe the number of occurrences of the values in . When a new element, , is sampled, a posterior count is assigned to that value. After sampling another value in the next iteration, this count is reassigned to the new value. Let indicate the current sample, i.e. for one index , while all other elements are zero. A non-informative prior is obtained if all values are equally likely and each element is assigned a single count. Then, and a new sample has a strong impact on the posterior distribution. In contrast, for large values, e.g. , a new count leaves the distribution almost invariant. The complete model is then given by (30) and, in addition,

 Ξ ∼ Dir(Ξ|ν) (47) θ|Ξ ∼ Cat(θ|RΘ,Ξ). (48)

To accomplish approximate inference in this model, the variables and are included in the Gibbs sampling procedure of Section 6. Therefore, the conditional distributions must be determined. Based on the dependencies in Fig. 3, and since and are assumed to be mutually independent, we find

 Ξ|θ = ˜Ξ ∼ Dir(Ξ|ν+˘c), (49) θ|y,˜Ξ ∼ Cat(θ|RΘ,˜Ξ). (50)


text:
| | | |

## Simple continuous modules

#### Yongduo Wang [1]

A module $M$ is called a simple continuous module if it satisfies the conditions $(min-C_{1})$ and $(min-C_{2})$. A module $M$ is called singular simple-direct-injective if for any singular simple submodules $A$, $B$ of $M$ with $A\cong B\mid M$, then $A\mid M$. Various basic properties of these modules are proved, and some well-studied rings are characterized using simple continuous modules and singular simple-direct-injective modules. For instance, it is shown that a ring $R$ is a right $V$-ring if and only if every right $R$-module is a simple continuous modules, and that a regular ring $R$ is a right $GV$-ring if and only if every cyclic right $R$-module is a singular simple-direct-injective module.
simple continuous module, singular simple-direct-injective module, $V$-ring, $GV$-ring
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Birincil Dil en Matematik Matematik Orcid: 0000-0002-0756-3899Yazar: Yongduo Wang (Sorumlu Yazar) Yayımlanma Tarihi : 8 Ekim 2019
 Bibtex @araştırma makalesi { hujms629829, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe Üniversitesi}, year = {2019}, volume = {48}, pages = {1336 - 1344}, doi = {}, title = {Simple continuous modules}, key = {cite}, author = {Wang, Yongduo} } APA Wang, Y . (2019). Simple continuous modules. Hacettepe Journal of Mathematics and Statistics , 48 (5) , 1336-1344 . Retrieved from https://dergipark.org.tr/tr/pub/hujms/issue/49321/629829 MLA Wang, Y . "Simple continuous modules". Hacettepe Journal of Mathematics and Statistics 48 (2019 ): 1336-1344 Chicago Wang, Y . "Simple continuous modules". Hacettepe Journal of Mathematics and Statistics 48 (2019 ): 1336-1344 RIS TY - JOUR T1 - Simple continuous modules AU - Yongduo Wang Y1 - 2019 PY - 2019 N1 - DO - T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 1336 EP - 1344 VL - 48 IS - 5 SN - 2651-477X-2651-477X M3 - UR - Y2 - 2018 ER - EndNote %0 Hacettepe Journal of Mathematics and Statistics Simple continuous modules %A Yongduo Wang %T Simple continuous modules %D 2019 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 48 %N 5 %R %U ISNAD Wang, Yongduo . "Simple continuous modules". Hacettepe Journal of Mathematics and Statistics 48 / 5 (Ekim 2019): 1336-1344 . AMA Wang Y . Simple continuous modules. Hacettepe Journal of Mathematics and Statistics. 2019; 48(5): 1336-1344. Vancouver Wang Y . Simple continuous modules. Hacettepe Journal of Mathematics and Statistics. 2019; 48(5): 1344-1336.


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• 1
Electronic Resource
Springer
The European physical journal 1 (1998), S. 123-130
ISSN: 1434-6052
Source: Springer Online Journal Archives 1860-2000
Topics: Physics
Notes: Abstract. Measurements of open charm hadro-production from CERN and Fermilab experiments are reviewed, with particular emphasis on the absolute cross sections and on their A and $\sqrt{s}$ dependences. Differential $p_{\mathrm{T}}$ and $x_{\mathrm{F}}$ cross sections calculated with the Pythia event generator are found to be in reasonable agreement with recent data. The calculations are scaled to nucleus-nucleus collisions and the expected lepton pair yield is deduced. The charm contribution to the low mass dilepton continuum observed by the CERES experiment is found to be negligible. In particular, it is shown that the observed low mass dilepton excess in S-Au collisions cannot be explained by charm enhancement.
Type of Medium: Electronic Resource
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• 2
Electronic Resource
Springer
The European physical journal 1 (1998), S. 123-130
ISSN: 1434-6052
Source: Springer Online Journal Archives 1860-2000
Topics: Physics
Notes: Abstract Measurements of open charm hadro-production from CERN and Fermilab experiments are reviewed, with particular emphasis on the absolute cross sections and on their A and √s dependences. Differentialp T andx F cross sections calculated with thePythia event generator are found to be in reasonable agreement with recent data. The calculations are scaled to nucleus-nucleus collisions and the expected lepton pair yield is deduced. The charm contribution to the low mass dilepton continuum observed by the CERES experiment is found to be negligible. In particular, it is shown that the observed low mass dilepton excess in S-Au collisions cannot be explained by charm enhancement.
Type of Medium: Electronic Resource
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• 3
Electronic Resource
Springer
The European physical journal 16 (2000), S. 519-525
ISSN: 1434-6052
Source: Springer Online Journal Archives 1860-2000
Topics: Physics
Notes: Abstract. We consider the charmonium production in the thermalized hadronic medium created in ultrarelativistic heavy ion collisions at LHC energy. The calculations for secondary $J/\psi$ and $\psi^{\prime}$ production by $D\bar D$ annihilation are performed within a kinetic model taking into account the space-time evolution of a longitudinally and transversely expanding medium. We show that secondary charmonium production appears almost entirely during the mixed phase, and that it is very sensitive to the charmonium dissociation cross section with co-moving hadrons. Within the most likely scenario for the dissociation cross section of the $J/\psi$ mesons their regeneration in the hadronic medium will be negligible. The secondary production of $\psi^{\prime}$ mesons however, due to their large cross section above the threshold, can substantially exceed the primary yield.
Type of Medium: Electronic Resource
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