text:
# Propositiones ad Acuendos Juvenes/Problems/6 - De Duobis Negotiatoribus C Solidos Communis Habentibus

## Propositiones ad Acuendos Juvenes by Alcuin of York: Problem $6$

De Duobis Negotiatoribus $\text C$ Solidos Communis Habentibus
$2$ Wholesalers with $100$ Shillings
$2$ wholesalers with $100$ shillings between them bought some pigs with the money.
They bought at the rate of $5$ pigs for $2$ shillings, intending to fatten them up and sell them again, making a profit.
But when they found that it is not the right time of year for fattening pigs,
and they were not able to feed them through the winter,
they tried to sell them again to make a profit.
But they couldn't, because they could only sell them for the price they had paid for them, $2$ shillings for $5$ pigs.
When they saw this, they said to each other, "Let's divide them.
By dividing them, and selling them again at the rate they had bought them for, they made a profit.
How could they be divided to make a profit, which could not be made by selling them all at once?

## Solution

First there were $250$ pigs bought with $100$ shillings at the above mentioned rate, for $5$ fifties are $250$.

On division, each merchant had $125$ pigs.

One sold the poorer quality pigs at $3$ for a shilling.

The other sold the better quality pigs at $2$ for a shilling.

The one who sold the poorer quality pigs received $40$ shillings for $120$ pigs.

The one who sold the better quality pigs received $60$ shillings for $120$ pigs.

Then there remained $5$ of each sort of pig.

From these they could make a profit of $4$ shillings and $2$ pence.

## Historical Note

According to <NAME>, this is the first appearance of the type of problem known as the Applesellers' Problem, also known as the missing penny or marketwomen's problem.

It hinges on the fact that a common mean is erroneously being calculated of $2$ for a monetary unit plus $3$ for a monetary unit equals $5$ for $2$ monetary units.

<NAME> goes on to remark that, while this problem appears everywhere in European puzzle-books since its appearance here, he has never seen a non-European version.

However, <NAME> claims in his Mathematical Recreations that it appears in Mahaviracharya's Ganita Sara Samgraha, which dates from c. $850$. However, this has not been corroborated.


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# Electrical engineering for chemical engineers - Power supplies, how do they work?

28 Aug 2019

If you’re working with physical products, you probably have a standard that specifies a power supply type. You can use that and never have to worry about this topic. That being said, learning about power supplies will help in both your understanding of electronics and your ability to improvise in a pinch.

Power comes out of the wall in the United States at 120VAC, but most electronics require a lower DC voltage to run.💬USBs are 5VDC, industrial supplies are 24VDC, PoE is 48-54VDC, and many devices are 9-30VDC. We can get from one to another with an AC-DC regulator. There are options:

Switching Regulator, Low End Switching Regulator, High End Linear Regulator

All of these (any many more) can be used to produce a DC source. However, the results you get will vary.

# AC-DC conversion

As you may already know, AC current is often represented as a waveform:

where the voltage changes with time. However, you may not have thought about what this means. In short, it’s representing oscillating flow, moving back and forth 60 times per second. If you want more clarification, watch this 5min video on AC vs DC.

In order to convert AC to DC, we need to make all the current flow in the same direction.

As chemical engineers - think of the AC vs DC current as flow in a pipe. If you had an oscillating flow in a pipe, how would you make sure flow only went one direction?

One option would be a check valve, which is a diode in the electrical world. Forward flow would be allowed and backward flow would be prevented. This would create a flow pattern like this:

In the right direction but not quite there yet (also very stressful on the check valve).

You could improve this with a series of check valves. Check out this cleverness:

This is called the diode bridge and is conceptually at the heart of most AC-DC power supplies. It produces a waveform that looks like this:

It doesn’t look pretty, but we now have all the current flowing in the same direction so we can call it DC.

# Voltage reduction and smoothing

We need a specific DC voltage, and we want a more stable power supply.

Put your chemical engineer hat back on. Now that you have this pulsating flow, how would you go about reducing and smoothing it?

1. You could install a surge tank to buffer out the pulsating flow.
2. If you don’t need much flow, you could install a critical orifice. This would provide very clean flow, but you’d need to discard most of the flow to avoid back-pressure.
3. You could use a PID loop with a pressure indicator and a valve.
4. You could use combinations and multiples of everything above.

### Capacitor

A capacitor is the electrical version of a surge tank, smoothing out variations in input. A 10µF capacitor is sized to roughly correspond with AC oscillation frequency, and 0.1µF capacitors (“magic pixie dust” capacitors) are roughly sized to handle high-frequency noise (e.g. radio waves, wifi). Add one and you have an Amazon-worthy power supply:

Note that this provides a smoother current but doesn’t reduce the voltage!

### Linear Regulator

We could add a critical orifice to reduce the voltage, bleeding out the rest of the current and getting a smooth supply.

In electronics, this is a linear regulator (LR) and it does the same exact thing. It works well to provide a stable DC power supply, but it is ridiculously inefficient. LRs “bleed” out the excess current as heat, and they are prone to getting very hot.

These are useful for low current flows, like you would find on 4-20mA sensors. However, even in these situations, you probably don’t want to place an LR right after our power supply. We should have something else attempt to reduce the noise and take strain off the LR.

### Switching Regulator

A little more work, but how about a shutoff valve with a pressure transducer and PID loop? The valve turns on when the pressure is low, and the valve turns off when the pressure is high. Similar effect, but far less waste.

In electronics, this is a switching regulator. Contrasting an LR, this is not a component but rather a full circuit. It works reasonably efficiently but still has some wobble.

For more, read the Bald Engineer’s Switching Voltage Regulator Tutorial.

### A Combination

As you can imagine, the quality of switching regulators can vary greatly. If you’re looking to build a cheap “technically 24VDC switching regulator” power supply to sell on Amazon, you could just have a transistor on a timer. If you’re looking to build a good power supply, you’ll probably have multiple high-quality MOSFETs toggling based on voltage measurement data.

For a very good power supply, you would combine this circuit with many capacitors and linear regulators (although I believe LRs are less common today). You would copy-paste this circuit a few times to further smooth the current. While you’re at it, you would thicken up the wires and add over-current protection, ensuring the regulator will not emit much waste heat and will behave safely if wired improperly.

This is the difference between the cheap and the good power supplies.

# Conclusions

### When you can get away with cheap

There are plenty of cases where the cheap power supply works fine. Valves, heaters, and the like certainly don’t care whether their 24VDC is slightly off. Additionally, most instruments that communicate digitally won’t notice slight disturbances (many even have onboard LRs for their sensitive components).

### When you can’t get away with cheap

Analog sensors will not work well with an unstable power supply. Analog sensors send current and voltage out as their signal, and an unstable power supply will result in similarly unstable readings. If you can read your sensor fast enough, you’ll actually see the same AC oscillation in your sensory data.

### How to get away with cheap

If you only have cheap power supplies (or you’re making commodity equipment), then you can place a linear regulator after the power supply to create a separate low-noise circuit. Use the raw power supply for things that can take it, and reserve the regulated circuit for sensitive electronics.

If you’re going down this route, you should also place extra capacitors around your power supply. You might want to consider replacing the diode bridge with a transistor bridge, and then you can squeeze a lot of this onto a printed circuit board.

### Why you shouldn’t bother getting away with cheap

Good power supplies aren’t very expensive (maybe $80 vs Amazon’s$10). It may not perform noticeably better than the low-quality regulator with capacitors and an LR, but you don’t need to make that circuit each time you use something.

The bigger benefit is that good power supplies come with over-current protection. If you short a circuit on a cheap power supply, it will run until it explodes. If you short a circuit on a good power supply, a warning light will turn on and it will refuse to send more power. This is a great safety feature and should be included on every system you build.

format: plain


text:
# cos squared theta minus sin square theta equal to 1 minus 10 squared theta by 1 + 10 squared theta prove the identity

## cos squared theta minus sin square theta equal to 1 minus 10 squared theta by 1 + 10 squared theta prove the identity

Cos^2-Sin^2=1-Tan^2/1+Tan^2

Taking RHS, 1-Tan^2/1+Tan^2 ………..(i)

We Know that, Tan=SinCos, by putting in eq (i), we get

1-(Sin/Cos)^2/1+(SinCos)^2

1-(Sin^2/Cos^2/1+(Sin^2-Cos^2)

Cos2-Sin2Cos2Cos2+Sin2Cos2

Cos2-Sin2Cos2+Sin2

Cos2-Sin21

Cos2-Sin2=LHS

So, RHS=LHS


text:
# Math Help - Another Chi square - s^2 defintion

1. ## Another Chi square - s^2 defintion

Dear Matheagle,
You still there? Here's another one that I've been spending too much time on. Can you please help?

Let $X_1, X_2,..., X_n$ be iid r.v.s from $Normal (\mu, \sigma)$. Given the fact that the rv

$X^2_n = \sum_{i=1}^n \left(\frac{x_i-\mu}{\sigma}\right)^2$

has a chi square distribution with degrees of freedom n.

Show that

$\frac{(n-1)s^2}{\sigma^2} = \sum_{i=1}^n \left(\frac{x_i-\mu}{\sigma}\right)^2 - \left(\frac{\overline{x}-\mu}{\sigma/\sqrt{n}}\right)^2$.

Thanks a bunch!

2. Dear Poisson Ivy,

Use the definition of $s^2$

${(n-1)s^2\over \sigma^2} = {\sum_{i=1}^n(x_i-\bar x)^2\over \sigma^2}$

$= {\sum_{i=1}^n((x_i-\mu)+(\mu-\bar x))^2\over \sigma^2}$

$= {\sum_{i=1}^n (x_i-\mu)^2+2(\mu-\bar x) \sum_{i=1}^n(x_i-\mu)+n(\mu-\bar x)^2\over \sigma^2}$

$= {\sum_{i=1}^n (x_i-\mu)^2+2(\mu-\bar x) n(\bar x-\mu)+n(\mu-\bar x)^2\over \sigma^2}$

$= {\sum_{i=1}^n (x_i-\mu)^2-2n(\bar x-\mu)^2+n(\bar x-\mu)^2\over \sigma^2}$

$= {\sum_{i=1}^n (x_i-\mu)^2-n(\bar x-\mu)^2\over \sigma^2}$

$= {\sum_{i=1}^n (x_i-\mu)^2\over \sigma^2} -{n(\bar x-\mu)^2\over \sigma^2}$

$= \sum_{i=1}^n {(x_i-\mu)^2\over \sigma^2} -{(\bar x-\mu)^2\over \sigma^2/n}$

$= \sum_{i=1}^n \left(\frac{x_i-\mu}{\sigma}\right)^2 - \left(\frac{\overline{x}-\mu}{\sigma/\sqrt{n}}\right)^2$.


text:
Browse Questions

# One mole of a solute A is dissolved in a given volume of a solvent. The association of the solute take place according to $nA\leftrightharpoons A_n$.The van’t Hoff factor i is expressed as

$i=\large\frac{1-x+\Large\frac{x}{n}}{1}$


text:
# Triangles and circles but not geometry part -2

Discrete Mathematics Level 5

There is a $$193$$ sided regular polygon. Now triangles are formed by joining the vertices of the polygon. Find the number of obtuse angled triangles formed.

×


text:
## Sync rotation of a hadron collider shape to a rectangle with a moving offset (rotate origin)

Questions about the LÖVE API, installing LÖVE and other support related questions go here.
Forum rules
Before you make a thread asking for help, read this.
<NAME>
Posts: 89
Joined: Mon Dec 04, 2017 1:47 am
Location: Germany

### Sync rotation of a hadron collider shape to a rectangle with a moving offset (rotate origin)

Hello dear lövers,

I have a rotation problem when it comes to moving offsets and HC. I kinda have a solution to this, but I consider this not a smart solution, so I wonder what you think about it and maybe you know how to make it better.

Basically, when I draw a rectangle while the offset changes (moving rotate origin) and the rotation itself changes, too, the rectangle will move it's way like I want it to. Now this rectangle gets a hadron collider (HC) shape that I want to be synced on the rotation. For this I call shape:setRotation on update.

This will cause a problem, because I sync it after moving the rotation origin, so my HC shape (blue in my example) does not rotate and keep up with my rectangle (green in my example).

The only solution I came up with is resetting the rotation of my shape to 0 before modifying the origin to sync the rotation again. In summery I get another setRotation call that I really dislike, because I fear for some serious performance drop. I have two major questions now:

1. How expensive is a additional rotation-reset call (it's out-commented in the code below) on many game objects
2. If the answer to 1. is "expensive", is there a way (math is fun) to do this in only one shape:setRotation call?

Here is the code:

Code: Select all

hc = require "lib.hc"

local x, y, w, h -- rectangle location and size
local ox, oy -- rectangle absolute origin
local deg -- degree of rotation around origin
local vel -- speed of the animation

local collider -- HC collider
local shape -- HC shape

x, y, w, h = 250, 250, 100, 100 -- rectangle location starts centered on the left side, this won't be changed
ox, oy = 300, 300 -- absolute origin starts at center of the rectangle, this will be changed
deg = 0 -- no rotation on start
vel = 20 -- moderate speed

collider = hc(64)
shape = collider:rectangle(x, y, w, h)
end

function love.update(dt)
--shape:setRotation(0, ox, oy) -- reset the rotation of the shape before changing the offset !!! ANY WAY TO SOLVE IT WITHOUT THIS ADDITIONAL setRotation CALL?
ox = ox + vel * dt -- move absolute origin to right
deg = deg + vel * dt -- increase rotation
if deg <= 0 or deg >= 180 then vel = vel * -1 end -- invert velocity after half cycle
shape:setRotation(math.rad(deg), ox, oy) -- sync the rotation of the shape to the rotation of my rectangle
end

function love.draw()
love.graphics.push()
love.graphics.translate(ox, oy) -- move to absolute origin
love.graphics.rotate(math.rad(deg)) -- rotate by the changing rotation value

love.graphics.setColor({ 255, 0, 0 }) love.graphics.circle("fill", 0, 0, 3) -- draw a red dot for the absolute origin that is moving

love.graphics.setColor({ 0, 255, 0 }) love.graphics.rectangle("line", x - ox, y - oy, w, h) -- draw the greem rectangle with the proper rotation
love.graphics.pop()

love.graphics.setColor({ 0, 0, 255, 100 }) shape:draw("fill") -- draw the HC shape
end

Attachments
game.love

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────█░░░░░░░░░█────
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lachlaan
Prole
Posts: 30
Joined: Sun Jun 30, 2013 7:23 pm

### Re: Sync rotation of a hadron collider shape to a rectangle with a moving offset (rotate origin)

Not sure if still a relevant question, but i think you're meant to update the shape's origin/center and then omit rotation coordinates as it's meant to just rotate around its own center and translate along the trajectory. As it stands your green rectangle is both translating and rotating on itself while the shape is rotating itself around the center of the rectangle, as if trying to orbit it, but lagging behind.

### Who is online

Users browsing this forum: Bing [Bot] and 14 guests


text:
dc.contributor.author
<NAME>
dc.contributor.supervisor
<NAME>
dc.contributor.supervisor
<NAME>
dc.contributor.supervisor
<NAME>
dc.date.accessioned
2019-09-16T06:44:20Z
dc.date.available
2019-09-15T19:03:41Z
dc.date.available
2019-09-16T06:44:20Z
dc.date.issued
2019
dc.identifier.uri
http://hdl.handle.net/20.500.11850/364127
dc.identifier.doi
10.3929/ethz-b-000364127
dc.description.abstract
What makes people cooperate? How can one design mechanisms in order to incentivize players to contribute to public goods? These are the kinds of fundamental questions that are exemplified by analysis of the free-rider problem: The problem arises from the fact that, while an entire population benefits from the presence of a public good produced at some cost by cooperative individuals, free-riders (defectors) can benefit from the public good even when not producing any of it. Using the tools of Game Theory, Mechanism Design and Experimental Economics, one can identify and understand the underlying conflicting forces leading to such free-riders problems in human interactions. This understanding can then be used to design suitable mechanisms to avoid tragedies of the commons", i.e. convergence to socially sub-optimal outcomes. In this dissertation, I focus on Public Goods games, and in particular on understanding under which conditions the public good is successfully provided and sustained through voluntary contributions, when players interact in groups. I not only focus on how cooperation can emerge as a result of incentive mechanisms and/or behavioural regularities, but I also study the implications of said mechanisms in terms of the total welfare of the players. The aim is to assess how robust positive predictions obtained for Voluntary Contribution Games are, when transferred to more general models which are closer to real-world social dilemmas situations. I extend the Voluntary Contribution Game in several ways: by considering different strategy spaces and public good provision efficacies, by adding noise, and, crucially, by accounting for heterogeneity among players. Using a mixture of analytical, experimental and computational tools, I show that highly-efficient equilibria are enabled by so called grouping'' mechanisms but that they also often cease to exist when heterogeneity is taken into account. I identify under which conditions high cooperation can be achieved and determine what is the optimal mechanism in terms of social welfare, as a function of a social planner's preference. Finally, I also investigate mechanisms based on reputations expressed by scores", and show that the positive results obtained in pairwise interactions do not necessary apply to multiplayer prisoner’s dilemmas, regardless of how much information is provided about the past behaviour of the interacting partners.
en_US
dc.format
application/pdf
en_US
dc.language.iso
en
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dc.publisher
ETH Zurich
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dc.rights.uri
http://rightsstatements.org/page/InC-NC/1.0/
dc.subject
Game theory
en_US
dc.subject
Cooperation
en_US
dc.subject
Mechanism Design
en_US
dc.subject
Prisoner's Dilemma
en_US
dc.subject
Group behavior
en_US
dc.title
Cooperation in Groups: a Game-Theoretic Investigation of Behaviour, Mechanisms, and Dynamics
en_US
dc.type
Doctoral Thesis
In Copyright - Non-Commercial Use Permitted
dc.date.published
2019-09-16
ethz.size
215 p.
en_US
ethz.code.ddc
DDC - DDC::5 - Science::510 - Mathematics
ethz.code.ddc
DDC - DDC::3 - Social sciences::300 - Social sciences
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Zurich
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published
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ethz.leitzahl
ETH Zürich::00002 - ETH Zürich::00012 - Lehre und Forschung::00007 - Departemente::02045 - Dep. Geistes-, Sozial- u. Staatswiss. / Dep. of Humanities, Social and Pol.Sc.::03784 - Helbing, Dirk / Helbing, Dirk
en_US
ethz.leitzahl.certified
ETH Zürich::00002 - ETH Zürich::00012 - Lehre und Forschung::00007 - Departemente::02045 - Dep. Geistes-, Sozial- u. Staatswiss. / Dep. of Humanities, Social and Pol.Sc.::03784 - Helbing, Dirk / Helbing, Dirk
en_US
ethz.tag
doctoral dissertation
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COSS
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ethz.date.deposited
2019-09-15T19:03:55Z
ethz.source
FORM
ethz.eth
yes
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ethz.availability
Open access
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ethz.rosetta.installDate
2019-09-16T06:44:37Z
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2020-02-15T21:37:14Z
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ctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.atitle=Cooperation%20in%20Groups:%20a%20Game-Theoretic%20Investigation%20of%20Behaviour,%20Mechanisms,%20and%20Dynamics&rft.date=2019&rft.au=Duca,%20Stefano&rft.genre=unknown&rft.btitle=Cooperation%20in%20Groups:%20a%20Game-Theoretic%20Investigation%20of%20Behaviour,%20Mechanisms,%20and%20Dynamics
 Search print copy at ETH Library


text:
## ISPLC 2001, Proceedings of the 5th Int.Symposium on Power Line Communications and its Applications

Published in:
ISPLC 2001, Proceedings of the 5th Int.Symposium on Power Line Communications and its Applications, Malmo, Sweden, 15-21
Year:
2001
Publisher:
IEEE
Keywords:
Laboratories:


text:
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Lemma 15.95.4. Let $A$ be a ring and let $f \in A$ be a nonzerodivisor. There is an additive functor1 $L\eta _ f : D(A) \to D(A)$ such that if $M \in D(A)$ is represented by a complex $M^\bullet$ of $f$-torsion free $A$-modules, then $L\eta _ fM = \eta _ fM^\bullet$ and similarly for morphisms.

Proof. Denote $\mathcal{T} \subset \text{Mod}_ A$ the full subcategory of $f$-torsion free $A$-modules. We have a corresponding inclusion

$K(\mathcal{T}) \quad \subset \quad K(\text{Mod}_ A) = K(A)$

of $K(\mathcal{T})$ as a full triangulated subcategory of $K(A)$. Let $S \subset \text{Arrows}(K(\mathcal{T}))$ be the quasi-isomorphisms. We will apply Derived Categories, Lemma 13.5.7 to show that the map

$S^{-1}K(\mathcal{T}) \longrightarrow D(A)$

is an equivalence of triangulated categories. The lemma shows that it suffices to prove: given a complex $M^\bullet$ of $A$-modules, there exists a quasi-isomorphism $K^\bullet \to M^\bullet$ with $K^\bullet$ a complex of $f$-torsion free modules. By Lemma 15.59.10 we can find a quasi-isomorphism $K^\bullet \to M^\bullet$ such that the complex $K^\bullet$ is K-flat (we won't use this) and consists of flat $A$-modules $K^ i$. In particular, $f$ is a nonzerodivisor on $K^ i$ for all $i$ as desired.

With these preliminaries out of the way we can define $L\eta _ f$. Namely, by the discussion at the start of this section we have already a well defined functor

$K(\mathcal{T}) \xrightarrow {\eta _ f} K(\mathcal{T}) \to K(A) \to D(A)$

which according to Lemma 15.95.3 sends quasi-isomorphisms to quasi-isomorphisms. Hence this functor factors over $S^{-1}K(\mathcal{T}) = D(A)$ by Categories, Lemma 4.27.8. $\square$

[1] Beware that this functor isn't exact, i.e., does not transform distinguished triangles into distinguished triangles. See Example 15.95.1.

There are also:

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• ### Environnement-insensitive calibration for cryogenic broadband microwave reflectometry measurements(1801.00436)

Jan. 1, 2018 cond-mat.supr-con
The characteristic frequencies of a system provide important information on the phenomena that govern its physical properties. In this framework, there has recently been renewed interest in cryogenic microwave characterization for condensed matter systems since it allows to probe energy scales of the order of a few $\mu$eV. However, broadband measurements of the absolute value of a sample response in this frequency range are extremely sensitive to its environment and require a careful calibration. In this paper, we present an original in situ calibration method for cryogenic broadband microwave reflectometry experiments that increases the sensitivity of such measurements. The calibration references are here provided by the sample itself, at three reference temperatures where its impedance is assumed or measured, and not by external standards as is usual. We compare the frequency-dependent complex impedance of an a-Nb$_{15}$Si$_{85}$ superconducting thin film obtained through this Sample-Based Calibration (SBC) and through a Open-Short-Load Standard Calibration (SC) when working at very low temperature (T < 1 K). While the SC only provides a qualitative evolution of the impedance across the superconducting transition, the SBC allows us to obtain the absolute response of the sample, which we compared with the relevant theories. The higher sensitivity of the SBC, supported both by the experimental results and by numerical calculations of the error propagation, is due to the definition of the calibration planes - as close as possible to the sample - which allows to completely get rid of the sample's electromagnetic environment.
• ### Competition between electron pairing and phase coherence in superconducting interfaces(1704.03365)

April 11, 2017 cond-mat.supr-con
The large diversity of exotic electronic phases displayed by two-dimensional superconductors confronts physicists with new challenges. These include the recently discovered quantum Griffith singularity in atomic Ga films, topological phases in proximized topological insulators and unconventional Ising pairing in transition metal dichalcogenide layers. In LaAlO3/SrTiO3 heterostructures, a gate tunable superconducting electron gas is confined in a quantum well at the interface between two insulating oxides. Remarkably, the gas coexists with both magnetism and strong Rashba spin-orbit coupling and is a candidate system for the creation of Majorana fermions. However, both the origin of superconductivity and the nature of the transition to the normal state over the whole doping range remain elusive. Missing such crucial information impedes harnessing this outstanding system for future superconducting electronics and topological quantum computing. Here we show that the superconducting phase diagram of LaAlO3/SrTiO3 is controlled by the competition between electron pairing and phase coherence. Through resonant microwave experiments, we measure the superfluid stiffness and infer the gap energy as a function of carrier density. Whereas a good agreement with the Bardeen-Cooper-Schrieffer (BCS) theory is observed at high carrier doping, we find that the suppression of Tc at low doping is controlled by the loss of macroscopic phase coherence instead of electron pairing as in standard BCS theory. We find that only a very small fraction of the electrons condenses into the superconducting state and propose that this corresponds to the weak filling of a high-energy dxz/yz band, more apt to host superconductivity
• ### Ion irradiated YBa2Cu3O7 nano-meanders for superconducting single photon detectors(1612.07730)

Jan. 13, 2017 cond-mat.supr-con
We report the fabrication of few hundred microns long, hundreds of nanometers wide and 30 nm thick meanders made from YBa2CU3O7. Thin films protected by a 8 nm-thick Ce02 cap layer have been patterned by high energy (a few tens of keV) oxygen ion irradiation through photoresist masks. DC and RF characterizations outline good superconducting properties of nano-meanders that could be used as Superconducting Single Photon Detectors (SSPD). By mean of a resonant method, their inductance, which mainly sets the maximum speed of these devices, has been measured on a wide range of temperature. It compares favorably with expected values calculated from the geometry of the meanders and the known London penetration depth in YBa2CU3O7.
• ### Destruction of superconductivity in disordered materials : a dimensional crossover(1408.3673)

Aug. 15, 2014 cond-mat.supr-con
The disorder-induced Superconductor-to-Insulator Transition in amorphous Nb$_{x}$Si$_{1-x}$ two-dimensional thin films is studied for different niobium compositions $x$ through a variation of the sample thickness $d$. We show that the critical thickness $d_c$, separating a superconducting regime from an insulating one, increases strongly with diminishing $x$, thus attaining values of over 100 {\AA}. The corresponding phase diagram in the $(d, x)$ plane is inferred and related to the three-dimensional situation. The two-dimensional Superconductor-to-Insulator Transition well connects with the three-dimensional Superconductor-to-Metal Transition.
• ### Cryogenic Calibration Setup for Broadband Complex Impedance Measurements(1309.4321)

Reflection measurements give access to the complex impedance of a material on a wide frequency range. This is of interest to study the dynamical properties of various materials, for instance disordered superconductors. However reflection measurements made at cryogenic temperature suffer from the difficulty to reliably subtract the circuit contribution. Here we report on the design and first tests of a setup able to precisely calibrate in situ the sample reflection, at 4.2 K and up to 2 GHz, by switching and measuring, during the same cool down, the sample and three calibration standards.
• ### Axion searches with the EDELWEISS-II experiment(1307.1488)

July 4, 2013 hep-ph, astro-ph.CO
We present new constraints on the couplings of axions and more generic axion-like particles using data from the EDELWEISS-II experiment. The EDELWEISS experiment, located at the Underground Laboratory of Modane, primarily aims at the direct detection of WIMPs using germanium bolometers. It is also sensitive to the low-energy electron recoils that would be induced by solar or dark matter axions. Using a total exposure of up to 448 kg.d, we searched for axion-induced electron recoils down to 2.5 keV within four scenarios involving different hypotheses on the origin and couplings of axions. We set a 95% CL limit on the coupling to photons $g_{A\gamma}<2.13\times 10^{-9}$ GeV$^{-1}$ in a mass range not fully covered by axion helioscopes. We also constrain the coupling to electrons, $g_{Ae} < 2.56\times 10^{-11}$, similar to the more indirect solar neutrino bound. Finally we place a limit on $g_{Ae}\times g_{AN}^{\rm eff}<4.70 \times 10^{-17}$, where $g_{AN}^{\rm eff}$ is the effective axion-nucleon coupling for $^{57}$Fe. Combining these results we fully exclude the mass range $0.91\,{\rm eV}<m_A<80$ keV for DFSZ axions and $5.73\,{\rm eV}<m_A<40$ keV for KSVZ axions.
• ### Background studies for the EDELWEISS dark matter experiment(1305.3628)

May 15, 2013 hep-ex, physics.ins-det
The EDELWEISS-II collaboration has completed a direct search for WIMP dark matter using cryogenic Ge detectors (400 g each) and 384 kg$\times$days of effective exposure. A cross-section of $4.4 \times 10^{-8}$ pb is excluded at 90% C.L. for a WIMP mass of 85 GeV. The next phase, EDELWEISS-III, aims to probe spin-independent WIMP-nucleon cross-sections down to a few $\times10^{-9}$ pb. We present here the study of gamma and neutron background coming from radioactive decays in the set-up and shielding materials. We have carried out Monte Carlo simulations for the completed EDELWEISS-II setup with GEANT4 and normalised the expected background rates to the measured radioactivity levels (or their upper limits) of all materials and components. The expected gamma-ray event rate in EDELWEISS-II at 20-200 keV agrees with the observed rate of 82 events/kg/day within the uncertainties in the measured concentrations. The calculated neutron rate from radioactivity of 1.0-3.1 events (90% C.L.) at 20-200 keV in the EDELWEISS-II data together with the expected upper limit on the misidentified gamma-ray events ($\le0.9$), surface betas ($\le0.3$), and muon-induced neutrons ($\le0.7$), do not contradict 5 observed events in nuclear recoil band. We have then extended the simulation framework to the EDELWEISS-III configuration with 800 g crystals, better material purity and additional neutron shielding inside the cryostat. The gamma-ray and neutron backgrounds in 24 kg fiducial mass of EDELWEISS-III have been calculated as 14-44 events/kg/day and 0.7-1.4 events per year, respectively. The results of the background studies performed in the present work have helped to select better purity components and improve shielding in EDELWEISS-III to further reduce the expected rate of background events in the next phase of the experiment.
• ### Effect of annealing on the superconducting properties of a-Nb(x)Si(1-x) thin films(1304.4629)

April 16, 2013 cond-mat.supr-con
a-Nb(x)Si(1-x) thin films with thicknesses down to 25 {\AA} have been structurally characterized by TEM (Transmission Electron Microscopy) measurements. As-deposited or annealed films are shown to be continuous and homogeneous in composition and thickness, up to an annealing temperature of 500{\deg}C. We have carried out low temperature transport measurements on these films close to the superconductor-to-insulator transition (SIT), and shown a qualitative difference between the effect of annealing or composition, and a reduction of the film thickness on the superconducting properties of a-NbSi. These results question the pertinence of the sheet resistance R_square as the relevant parameter to describe the SIT.
• ### A search for low-mass WIMPs with EDELWEISS-II heat-and-ionization detectors(1207.1815)

We report on a search for low-energy (E < 20 keV) WIMP-induced nuclear recoils using data collected in 2009 - 2010 by EDELWEISS from four germanium detectors equipped with thermal sensors and an electrode design (ID) which allows to efficiently reject several sources of background. The data indicate no evidence for an exponential distribution of low-energy nuclear recoils that could be attributed to WIMP elastic scattering after an exposure of 113 kg.d. For WIMPs of mass 10 GeV, the observation of one event in the WIMP search region results in a 90% CL limit of 1.0x10^-5 pb on the spin-independent WIMP-nucleon scattering cross-section, which constrains the parameter space associated with the findings reported by the CoGeNT, DAMA and CRESST experiments.


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# Locally convex topological vector space

(Redirected from Locally convex space)

In functional analysis and related areas of mathematics, locally convex topological vector spaces or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.

Fréchet spaces are locally convex spaces that are completely metrizable (with a choice of complete metric). They are generalizations of Banach spaces, which are complete vector spaces with respect to a metric generated by a norm.

## History

Metrizable topologies on vector spaces have been studied since their introduction in M<NAME>'s 1902 PhD thesis Sur quelques points du calcul fonctionnel (wherein the notion of a metric was first introduced). After the notion of a general topological space was defined by <NAME> in 1914,[1] although locally convex topologies were implicitly used by some mathematicians, up to 1934 only <NAME> would seem to have explicitly defined the weak topology on Hilbert spaces and strong operator topology on operators on Hilbert spaces.[2][3] Finally, in 1935 von Neumann introduced the general definition of a locally convex space (called a convex space by him).[4][5]

A notable example of a result which had to wait for the development and dissemination of general locally convex spaces (amongst other notions and results, like nets, the product topology and Tychonoff's theorem) to be proven in its full generality, is the Banach–Alaoglu theorem which Stefan Banach first established in 1932 by an elementary diagonal argument for the case of separable normed spaces[6] (in which case the unit ball of the dual is metrizable).

## Definition

Suppose V is a vector space over K, a subfield of the complex numbers (normally C itself or R). A locally convex space is defined either in terms of convex sets, or equivalently in terms of seminorms.

### Convex sets

A subset C in V is called

1. Convex if for all x, y in C, and 0 ≤ t ≤ 1, tx + (1 – t)y is in C. In other words, C contains all line segments between points in C.
2. Circled if for all x in C, λx is in C if |λ| = 1. If K = R, this means that C is equal to its reflection through the origin. For K = C, it means for any x in C, C contains the circle through x, centred on the origin, in the one-dimensional complex subspace generated by x.
3. A cone (when the underlying field is ordered) if for all x in C and 0 ≤ λ ≤ 1, λx is in C.
4. Balanced if for all x in C, λx is in C if |λ| ≤ 1. If K = R, this means that if x is in C, C contains the line segment between x and x. For K = C, it means for any x in C, C contains the disk with x on its boundary, centred on the origin, in the one-dimensional complex subspace generated by x. Equivalently, a balanced set is a circled cone.
5. Absorbent or absorbing if the union of tC over all t > 0 is all of V, or equivalently for every x in V, tx is in C for some t > 0. The set C can be scaled out to absorb every point in the space.
6. Absolutely convex if it is both balanced and convex.

More succinctly, a subset of V is absolutely convex if it is closed under linear combinations whose coefficients absolutely sum to ≤ 1. Such a set is absorbent if it spans all of V.

A locally convex topological vector space is a topological vector space in which the origin has a local base of absolutely convex absorbent sets. Because translation is (by definition of "topological vector space") continuous, all translations are homeomorphisms, so every base for the neighborhoods of the origin can be translated to a base for the neighborhoods of any given vector.

### Seminorms

A seminorm on V is a map p : VR such that

1. p is positive or positive semidefinite: p(x) ≥ 0.
2. p is positive homogeneous or positive scalable: p(λx) = |λ| p(x) for every scalar λ. So, in particular, p(0) = 0.
3. p is subadditive. It satisfies the triangle inequality: p(x + y) ≤ p(x) + p(y).

If p satisfies positive definiteness, which states that if p(x) = 0 then x = 0, then p is a norm. While in general seminorms need not be norms, there is an analogue of this criterion for families of seminorms, separatedness, defined below.

A locally convex space is then defined to be a vector space V along with a family of seminorms {pα}αA on V. The space carries a natural topology, the initial topology of the seminorms. In other words, it is the coarsest topology for which all the mappings

$\begin{cases}p_{\alpha,y}:V\to\mathbf{R} \\ x\mapsto p_\alpha(x-y) & y\in V, \alpha\in A\end{cases}$

are continuous. A base of neighborhoods of y for this topology is obtained in the following way: for every finite subset B of A and every ε > 0, let

$U_{B, \varepsilon}(y) = \{x \in V : p_\alpha(x - y) < \varepsilon \ \forall \alpha \in B\}.$

Note that

$U_{B,\varepsilon}(y) = \bigcap_{\alpha\in B} (p_{\alpha,y})^{-1}([0,\varepsilon)).$

That the vector space operations are continuous in this topology follows from properties 2 and 3 above. The resulting TVS is locally convex because each UB, ε (0) is absolutely convex and absorbent (and because the latter properties are preserved by translations).

### Equivalence of definitions

Although the definition in terms of a neighborhood base gives a better geometric picture, the definition in terms of seminorms is easier to work with in practice. The equivalence of the two definitions follows from a construction known as the Minkowski functional or Minkowski gauge. The key feature of seminorms which ensures the convexity of their ε-balls is the triangle inequality.

For an absorbing set C such that if x is in C, then tx is in C whenever 0 ≤ t ≤ 1, define the Minkowski functional of C to be

$\mu_C(x) = \inf \{\lambda > 0: x\isin \lambda C\}.$

From this definition it follows that μC is a seminorm if C is balanced and convex (it is also absorbent by assumption). Conversely, given a family of seminorms, the sets

$\left \{x : p_{\alpha_1}(x) < \varepsilon, \cdots, p_{\alpha_n}(x) < \varepsilon \right \}$

form a base of convex absorbent balanced sets.

## Further definitions and properties

• A family of seminorms {pα}α is called total or separated or is said to separate points if whenever pα(x) = 0 holds for every α then x is necessarily 0. A locally convex space is Hausdorff if and only if it has a separated family of seminorms. Many authors take the Hausdorff criterion in the definition.
• A pseudometric is a generalisation of a metric which does not satisfy the condition that d(x, y) = 0 only when x = y. A locally convex space is pseudometrisable, meaning that its topology arises from a pseudometric, if and only if it has a countable family of seminorms. Indeed, a pseudometric inducing the same topology is then given by
$d(x,y)=\sum^\infty_n \frac{1}{2^n} \frac{p_n(x-y)}{1+p_n(x-y)}$
(where the 1/2n can be replaced by any positive summable sequence an). This pseudometric is translation-invariant, but not homogeneous, meaning d(kx, ky) ≠ |k|d(x, y), and therefore does not define a (pseudo)norm. The pseudometric is an honest metric if and only if the family of seminorms is separated, since this is the case if and only if the space is Hausdorff. If furthermore the space is complete, the space is called a Fréchet space.
• As with any topological vector space, a locally convex space is also a uniform space. Thus one may speak of uniform continuity, uniform convergence, and Cauchy sequences.
• A Cauchy net in a locally convex space is a net {xκ}κ such that for every ε > 0 and every seminorm pα, there exists a κ such that for all λ, μ > κ, pα(xλxμ) < ε. In other words, the net must be Cauchy in all the seminorms simultaneously. The definition of completeness is given here in terms of nets instead of the more familiar sequences because unlike Fréchet spaces which are metrisable, general spaces may be defined by an uncountable family of pseudometrics. Sequences, which are countable by definition, cannot suffice to characterize convergence in such spaces. A locally convex space is complete if and only if every Cauchy net converges.
• A family of seminorms becomes a preordered set under the relation pαpβ if and only if there exists an M > 0 such that for all x, pα(x) ≤ Mpβ(x). One says it is a directed family of seminorms if the family is a directed set with addition as the join, in other words if for every α and β, there is a γ such that pα + pβpγ. Every family of seminorms has an equivalent directed family, meaning one which defines the same topology. Indeed, given a family {pα}αI, let Φ be the set of finite subsets of I, then for every F in Φ, define
$q_F = \sum_{\alpha \in F} p_{\alpha}.$
One may check that {qF}F ∈ Φ is an equivalent directed family.
• If the topology of the space is induced from a single seminorm, then the space is seminormable. Any locally convex space with a finite family of seminorms is seminormable. Moreover, if the space is Hausdorff (the family is separated), then the space is normable, with norm given by the sum of the seminorms. In terms of the open sets, a locally convex topological vector space is seminormable if and only if 0 has a bounded neighborhood.

## Examples and nonexamples

### Examples of locally convex spaces

• Every normed space is a Hausdorff locally convex space, and much of the theory of locally convex spaces generalises parts of the theory of normed spaces. The family of seminorms can be taken to be the single norm. Every Banach space is a complete Hausdorff locally convex space, in particular, the Lp spaces with p ≥ 1 are locally convex.
• More generally, every Fréchet space is locally convex. A Fréchet space can be defined as a complete locally convex space with a separated countable family of seminorms.
$p_i \left(\left\{x_n\right\}_n\right) = \left|x_i\right|, \qquad i \in \mathbf{N}.$
The countable family of seminorms is complete and separable, so this is a Fréchet space, which is not normable. Note that this is also the limit topology of the spaces Rn, embedded in Rω in the natural way, by completing finite sequences with infinitely many 0.
• Given any vector space V and a collection F of linear functionals on it, V can be made into a locally convex topological vector space by giving it the weakest topology making all linear functionals in F continuous. This is known as the weak topology or the initial topology determined by F. The collection F may be the algebraic dual of V or any other collection. The family of seminorms in this case is given by pf (x) = | f (x)| for all f in F.
• Spaces of differentiable functions give other non-normable examples. Consider the space of smooth functions f : RnC such that supx|xaDbf | < ∞, where a and b are multiindices. The family of seminorms defined by pa,b( f ) = supx |xaDbf(x)| is separated, and countable, and the space is complete, so this metrisable space is a Fréchet space. It is known as the Schwartz space, or the space of functions of rapid decrease, and its dual space is the space of tempered distributions.
• An important function space in functional analysis is the space D(U) of smooth functions with compact support in URn. A more detailed construction is needed for the topology of this space because the space C
0
(U)
is not complete in the uniform norm. The topology on D(U) is defined as follows: for any fixed compact set KU, the space C
0
(K)
of functions fC
0
(U)
with supp( f ) ⊂ K is a Fréchet space with countable family of seminorms || f ||m = supx|Dmf (x)| (these are actually norms, and the space C
0
(K)
with the || ⋅ ||m norm is a Banach space Dm(K)). Given any collection {Kλ}λ of compact sets, directed by inclusion and such that their union equal U, the C
0
(Kλ)
form a direct system, and D(U) is defined to be the limit of this system. Such a limit of Fréchet spaces is known as an LF space. More concretely, D(U) is the union of all the C
0
(Kλ)
with the strongest locally convex topology which makes each inclusion map C
0
(Kλ) ↪ D(U)
continuous. This space is locally convex and complete. However, it is not metrisable, and so it is not a Fréchet space. The dual space of D(Rn) is the space of distributions on Rn.
• More abstractly, given a topological space X, the space C(X) of continuous (not necessarily bounded) functions on X can be given the topology of uniform convergence on compact sets. This topology is defined by semi-norms φK( f ) = max{| f (x)| : xK } (as K varies over the directed set of all compact subsets of X). When X is locally compact (e.g. an open set in Rn) the Stone-Weierstrass theorem applies—in the case of real-valued functions, any subalgebra of C(X) that separates points and contains the constant functions (e.g., the subalgebra of polynomials) is dense.

### Examples of spaces lacking local convexity

Many topological vector spaces are locally convex. Examples of spaces that lack local convexity include the following:

$\|f\|_p = \int_0^1 |f(x)|^p \, dx$
they are not locally convex, since the only convex neighborhood of zero is the whole space. More generally the spaces Lp(μ) with an atomless, finite measure μ and 0 < p < 1 are not locally convex.
$d(f, g) = \int_0^1 \frac{|f(x) - g(x)|}{1+|f(x) - g(x)|} \, dx.$
This space is often denoted L0.

Both examples have the property that any continuous linear map to the real numbers is 0. In particular, their dual space is trivial, that is, it contains only the zero functional.

• The sequence space p, 0 < p < 1, is not locally convex.

## Continuous linear mappings

Because locally convex spaces are topological spaces as well as vector spaces, the natural functions to consider between two locally convex spaces are continuous linear maps. Using the seminorms, a necessary and sufficient criterion for the continuity of a linear map can be given that closely resembles the more familiar boundedness condition found for Banach spaces.

Given locally convex spaces V and W with families of seminorms {pα}α and {qβ}β respectively, a linear map T : VW is continuous if and only if for every β, there exist α1, α2, ..., αn and M > 0 such that for all v in V

$q_\beta(Tv)\le M \left (p_{\alpha_1}(v) +\dotsb+p_{\alpha_n}(v) \right ).$

In other words, each seminorm of the range of T is bounded above by some finite sum of seminorms in the domain. If the family {pα}α is a directed family, and it can always be chosen to be directed as explained above, then the formula becomes even simpler and more familiar:

$q_\beta(Tv)\le Mp_\alpha(v).$

The class of all locally convex topological vector spaces forms a category with continuous linear maps as morphisms.


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Hierarchical bayesian model without packages

I'm attempting to build a hierarchical Bayesian model. For various reasons (including my own edification), I want to do this from scratch (i.e., without using the various packages and libraries written for this purpose). However, all the examples and tutorials I've come across make use of Stan, JAGS, PyMC3, etc. Does anyone know of or have an example of a working script that implements a simple hierarchical Bayesian model? Preferably it would be written in either python, R, or Matlab.

It's really good, and he even has a github repo called minimcmc which does some sampling. If you are interested in MCMC/HMC, I would start there.


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# Order of variables/constants

1. Apr 21, 2010

### Cspeed

Does anyone know if there is a list of the order that goes in writing a mathematical expression (i.e. 7 $$\pi$$ $$\sqrt{5}$$ vs 7 $$\sqrt{5}$$ $$\pi$$ )?

Is there a generally accepted method?

2. Apr 21, 2010

### Staff: Mentor

I don't know if there is an explicit rule other than by what looks better. For expressions with radicals, factors are usually collected in front of the radical, like this (using your example):
$$7 \pi \sqrt{5}$$

Tip: When you create expressions or equations in LaTeX, it's usually best to put everything inside one pair of tags rather than use a bunch of pairs of tags.

3. Apr 22, 2010

### HallsofIvy

The reason for that is that if you write something like $\sqrt{2}x$, rather than $x\sqrt{2}$ is that it can be hard to distinguish whether or not the "x" is supposed to be inside the square root or not. Of course, as long as you are careful about that, $x\sqrt{2}$ and $\sqrt{2} x$ mean exactly the same thing- it's not like the ordering of operations where a+ bc is different from (a+b)c.

4. Apr 22, 2010

### Tac-Tics

Speaking in (incredibly) general terms, the problem you have just described is that of finding the "normal" form of a term.

In the study of formal systems, equivalent terms usually have several different representations. For example, if we're working with polynomials, x^2 + 1 + 2x is equal to 2x + x^2 + 1, even though how they are written on paper ("exs squared plus one... etc" versus "two exs plus .... etc") are different.

So if you have two terms, how can you tell if they are equal if they can be written out different? One practical example of this would be: If you were a math teacher and the answer your student gave you was different from your answer key, how could you tell if their answer was incorrect? Your answer key may say the answer is "x^2 + 2x + 1", but the student put "1 + 2x + x^2". Obviously these terms are different, yet they are "equal" inside the system we're working with.

The answer is you convert them to normal formal. The definition on "normal form" differs from system to system. For polynomials, we usually use the following definition:

A polynomial is in normal form when it is written $$c_nx^n + c_{n-1}x^{n-1} + c_{n-2}x^{n-2} + \cdots + c_1x + c_0$$ for some (constant) integers $$c_0, c_1, c_2, \cdots, c_n$$ and (constant) integer n.

Some systems do not have normal forms. This was one of the big discoveries in the early 20th Century. For example, if instead of polynomials (a kind of mathematical object), we used computer programs in some programming language (another kind of mathematical object), we would find there is no useful way to put a program in any sort of "normal" form. The major consequence of this is that it's impossible to mechanically decide whether two programs are equal to each other. (To show two programs are equal, you need look for a custom proof -- it's effectively a very difficult guessing game to show two arbitrary computer programs are equal).

In your particular case, you're working with algebraic expressions. There are no hard-and-fast definitions for what constitutes a normal form (I wouldn't imagine there is one at all). However, there are lots of loose conventions. Generally, you keep integral constants on the far left. Variables on the right. But in the case of the radical, those rules can be bent to avoid confusion (since some people are sloppy with the "bar" of the radical). The two primary concerns are readability and ease of manipulation. If you can achieve those, it really doesn't matter how you write them.

5. Apr 22, 2010

### Cspeed

Thanks for the explanations, but I am a little surprised. I always felt that there was some "right" way to write algebraic expressions. For example, in published mathematical papers, I'm sure they strive for consistency, so that from article to article you don't have radicals coming first one place and last somewhere else.

I've been searching, but couldn't find a list like this (just made up):
1. Integers
2. Variables, in alphabetical order
3. Greek letters, in alphabetical order