text:
Question

# $$60 g$$ of ice at $$0^0$$ is mixed with $$60 g$$ of steam at $$100^0$$. At thermal equilibrium, the mixture contains:(Latent heats of steam and ice are $$540 cal { g }^{ -1 }$$ and $$80 cal { g }^{ -1 }$$ respectively, specific heat of water $$= 1 cal { g }^{ -1 } { }^{ -1 }$$)

A
80g of water and 40g of steam at 1000
B
120g of water at 900
C
120g of water at 1000
D
40g of steam and 80g of water at 00

Solution

## The correct option is A $$80 g$$ of water and $$40 g$$ of steam at $$100 ^0$$Heat energy required to melt $$60 g$$ of ice $$= 60\times 80=4800 Cal$$.Heat energy required to raise the temprature of water at $$0$$ degree centigrade to $$100$$ degree centigrade $$= 100\times 1\times 60=6000 Cal$$Total energy required $$= 4800+6000=10800 Cal$$This energy is given by the steam by getting converted into water.Amount of steam getting converted into water is $$\dfrac{10800}{540}=20g$$Thus, finally total $$80 gm$$ of water will be there and $$40 gm$$ of steam. Physics

Suggest Corrections

0

Similar questions
View More

People also searched for
View More

format: plain


text:
Let $f: A \to B$ and $g: B \to C$ be uniformly continuous. What can be said about $g\circ f: A \to C$?
We have only briefly covered uniform continuity so it is unclear what I should see about $g\circ f$.
I guess A, B and C are metric spaces. Fix $\varepsilon>0$. Then use the uniform continuity of $g$ on $B$: you will find a $\delta$ such that $|g(x)-g(y)|\leq\varepsilon$ if $d_B(x,y)\leq \delta$. Now, can you find a $\delta'$ such that if $a,a' \in A$ and $d_A(a,a')\leq \delta'$ then $|g \circ f(a)-g \circ f(a')|\leq \varepsilon$?


text:
# MATLAB: How to suppress answer in Matlab

MATLABsuppress

Hi, I'm trying to calculate e^x for some value of x using the taylor expansion approximation
T_n(x) = Sum (x^k/k!)
And I have managed to do this using the following m-file
function T = findexp(x,n) %findexp, function to evaluate e^x T(1)=1; for i = 2:n+1 T(i) = T(i-1)+x^(i-1)/(factorial(i-1)); if i == n+1 fprintf(1,'T(%d) = %1.15e\n',x,T(i)); end end
However, when I then type findexp(1,25) for example the output is
T(1) = 2.718281828459046e+000
ans =
Columns 1 through 8
    1.0000    2.0000    2.5000    2.6667    2.7083    2.7167    2.7181    2.7183
Columns 9 through 16
    2.7183    2.7183    2.7183    2.7183    2.7183    2.7183    2.7183    2.7183
Columns 17 through 24
    2.7183    2.7183    2.7183    2.7183    2.7183    2.7183    2.7183    2.7183
Columns 25 through 26
    2.7183    2.7183
How do I suppress the columns and just show the T(1) part.
Sorry I realise this is a really amateurish question I just have done very little Matlab programming before.
findexp(1,25);


text:
# Parametric plot / I just need to check my work (quick)

1. Jun 10, 2005

$$x(t)= \left( u\cos A \right) t$$

and

$$y(t)= \left( u\sin A \right) t - \frac{gt^2}{2} + h$$

represent the horizontal and vertical coordinates of a batted or thrown baseball. $$A$$ is the initial angle of elevation and $$u$$ is the initial speed of the ball. I need to plot $$x(t)$$ and $$y(t)$$ parametrically. I'm given the following

$$u = 125 \mbox{ ft/s}$$

$$g = 32 \mbox{ ft/s}^2$$

$$h = 3 \mbox{ ft}$$

$$A$$ may take several different values. Here follows the random choices I made and the corresponding graphs:

$$A=\frac{\pi}{3}$$

http://img25.imagevenue.com/img.php?loc=loc93ℑ=663_parametricplot_1.jpg

$$A=\frac{4\pi}{9}$$

http://img6.imagevenue.com/img.php?loc=loc123ℑ=26c_parametricplot_2.jpg

$$A=\frac{\pi}{2}$$

http://img6.imagevenue.com/img.php?loc=loc123ℑ=26c_parametricplot_2.jpg

Could you please take a quick look at those graphs? In my opinion, there seems to be something wrong.

Thank you

Last edited: Jun 10, 2005
2. Jun 10, 2005

### NateTG

I can't see the images. You should be getting parabolas for the first two, and a vertical line for the third (since the ball goes straight up and down.)

3. Jun 10, 2005

Oh, I'm sorry about that. I actually tested the links, but a minute later they just wouldn't work. If you still wanna take a quick look, I placed them at:

photos.yahoo.com/thiago_j

They're the only ones there. What you said about the shapes sound right to me, but that's not what I have. They are pretty much the same. I used the command (mathematica 5):

ParametricPlot[{x(t), y(t)}, {t, t_min, t_max}]

Anyway, that's definitely stange.

4. Jun 10, 2005

### whozum

The discrepancy is probably because you cite g in feet per second squared and your initial velocity in meters per second. Your equation forms are correct.

5. Jun 10, 2005

Oh, no!!! That's a typo. u = 125 ft/s. The units are incorrect in my 1st post.

6. Jun 10, 2005


text:
# How does one prove:$\nabla(\vec{\mu_m}\cdot\vec{B})\cdot\vec{dr}=0$?

Work done by a magnetic force(even over an infinitesimally short displacement)=0

Net Force on a current loop in an external magnetic field is given by: $$\vec{F}=\nabla(\vec{\mu_m } \cdot \vec{B})$$

How does one prove: $$dW=\nabla(\vec{\mu_m} \cdot \vec{B})\cdot\vec{dr}=0$$

$\vec{\mu_m}$: Magnetic Moment of the current loop.

-
Magnetic force cannot perform work but electric field can do work.Such a field may result from the spatial variation of B from the curl B equation[comment to the answer may be considered] – Anamitra Palit Aug 14 '12 at 3:23

You can think of redistribution of charges and currents elsewhere(gadgets producing the field) to produce an electric field due to variations of magnetic field at some fixed point. But a pure magnetic field which does not vary with time cannot perform any work.What would be your mechanism for accounting for such variations of $\vec{B}$ in evaluating:$\nabla (\vec{\mu_m}.\vec{B}).\vec{dr}$. What would be the mechanism of including electric field in the said evaluation of elementary work? – Anamitra Palit Aug 13 '12 at 6:53
Let the magnetic field be in the x direction.If the observer moves along the y direction with a uniform speed v,we have $\vec{E}=\gamma(\vec{v}\times\vec{B})$. The electric field accelerates the charged changing its speed and doing work. Initially there was a acceleration normal to the direction of motion. But now we have acceleration due to change of speed also . If the net acceleration changes the radiating power should be different from the two frames.But energy is not an invariant wrt frame transformation though it is a conserved quantity. It might have different values in differentframes – Anamitra Palit Aug 13 '12 at 10:51
Spatial variation of $\vec{B}$ can produce an electric field:$\nabla \times \vec{B}=\mu_0 \vec{j}+\mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}$.If $\vec{E}\cdot\vec{dr}=\nabla(\vec{\mu_m}\cdot\vec{B})\cdot\vec{dr}$ to produce non-zero work then,$\vec{E}=\nabla(\vec{\mu_m}\cdot\vec{B})$. How do you prove that by direct calculation?[here you have a view of the problem through the differential equations which are supposed to represent the physical conditions accurately.] – Anamitra Palit Aug 14 '12 at 3:16


text:
# All Questions

6,912 questions
12 views

### explain the inlet valve mach index and the mean piston ratio

this is the attire i wear during practical day in the university. The white shirt is an overcoat which I wear and the goggle is worn to protect my eyes from any particles from entering my eyes during ...
9 views

### Glass used in biometric machines

I would like to know what kind of glass is used in biometric attendance machines over which we press our thumb? Is there something special about these glasses or their manufacturing?
15 views

18 views

### Solving parallel without GUI in batch mode not working as expected (Linux)

I'm currently trying to run my solutions in batch mode because in the future I'll be running simulations which I want to be done in the background. I've tried to adapt the code from here: https://www....
20 views

### How is the liquid refrigerant distributed throughout an AC system before start [on hold]

When an air conditioning system is at rest (eg before starting) how is the liquid refrigerant distributed throughout the system? I would think that the expansion valve would remain slightly open, ...
14 views

### What is the pitch angle squared representative of?

I am working with a sensor which gives me a tilt reading in degrees and I have been observing the waveform in the frequency domain using MS Excel 16. I was curious as to what the units would be for ...
58 views

### Viscoelasticity and Hyperelastic model, history and difference

Actually, I see some papers use viscoelasticity or hyperelastic model for the same materials. It seems to me that those models were invented in different eras, and the hyperelastic is a newer model ...
30 views

### what is the most efiicient way to mill/ machine a spherical and pancake ish structure as seen in the figure?

The overall dimensions for rectangular block: 8” x 3” x 2”. Ball should stay the longitudinal center line of rectangular block Bottom pancake feature should stay lateral center line of rectangular ...
15 views

### Does a closed loop biomass-fired cogeneration/CHP make sense?

I'm working on a project concerning electricity production using biomass, and we have seen that making use of cogeneration is very attractive in order to increase the efficiency of the installation. ...
39 views

### Pathway for drone racing

How can I build a v shaped pathway for drones? It would be a sort of racing path and there would be multiple drones flying over the pathway at the same time. The race would be a relay race. So there ...
34 views

### Flow of hot gasses around the pipe

I have the following problem: there is a copper pipe getting through a wooden wall. On one side of the wall (hot side), I have elevated temperature which is defined by radiation and convection. On the ...
29 views

### Meshing Complex models in abaqus

I'm trying to mesh a model of a morphing wing in Abaqus and have been running into issues. It was recommended that I partition the model into smaller parts, but it seems that the more I try to ...
51 views

### Which container orientation, if any, can hold more weight?

These are cross sections of identical shipping containers, except one is rotated 45 degrees. The cross sections are squares, 3 meters to a side. I drew dashed lines to show where they're suspended by ...
565 views

### Dimensions for CNC machining

Creating a drawing for a part that will be CNC milled. Should all of my dimensions come from a baseline on the part, like an edge or is it ok to use Centerlines? See yellow highlighted dimensions ...
20 views

### Solving 1D Unsteady Homentropic Flow using Method of Characteristics

I am looking to solve the 1D Unsteady Homentropic Flow equations using the Method of Characteristics (MOC). I am using Zucrow and Hoffman's Gas Dynamics Volume 1 as a reference and I am currently ...
20 views

### Maintaining constant speed of a CVT engine during a slope

I am not sure if this question is related to this community. But nevertheless. I work on CVT engines, I had a new idea of modifying my controller during a slope or a gradient on the road so that the ...
606 views

### Why are tunnel boring machines not using cone-shaped drills?

The surface area of the head of a tunnel boring machine is usually flat. A cone-shaped head would increase the surface area. The question is if it could speed up the boring process. I know that with ...
53 views

### Easily compute a good approximation of heat conductivity of cooled semi-vacuum (water vapor + air)

I am preparing a classroom demonstration about heat, including a demo of how a satellite loses heat only radiatively, since it's flying through a vacuum. I'd like to make the vacuum-production ...
36 views

### What thread to use in tapping a hole for a 2-56 Helicoil insert?

Helicoil from McMaster-Carr 18-8 Stainless Steel Helical Inserts 91732A201 calls out a specific tap to use, but I cannot find out what thread this tap is putting into the hole for the insert. I want ...
39 views

### 90 degree pivot steering system to allow vehicles to move sideways

Are there any restrictions that would not allow a steering system to cause a vehicle to be able to move sideways - at 90 degrees from the convention direction of travel?
36 views

### Can secondary beams be supported to columns occaisionally?

If there are secondary beams running between primary beams, and columns support the primary beams, then is it allowable to have the occasional secondary beam pinned to a column? Otherwise it seems to ...
113 views

### Can I use a wooden flat screw head with a washer to mimick a pan head screw?

For example - I am installing a fold up table from Ikea, it needs to hang from the wall, as shown below. It explicitly says to use pan head instead of flat screw heads like shown in the picture, I ...
16 views

### Two wire tandem gas metal arc welidng (GMAW-TW)

Please explain how two wire tandem gas metal arc welding (GMAW-TW) process requires less heat input. Appreciated if reference is given.
38 views

### How to add radial protrusions to a disc in Autodesk Inventor

First post here, trying to get a little help with Autodesk Inventor. I am trying to make a part that is basically a wheel with little nubs on the end (think of a lego with tabs on the outer ...
16 views

### Virtual work method for a disk with axisymmetric distributed and concentrated loads

I'm trying to determine all forces/reactions on a disk with a hole in the center. The disk is loaded with a (known) uniform distributed load from A to C (all around the disk), a concentrated (known) ...
15 views

### Gear Ratio and Wheel Diameter [on hold]

I have been told that vehicles with larger wheels are less economical. How is that? Also how is it that Gear Ratio is a function of Wheel Diameter?
73 views

### Which is the best extendable pole?

Which of these poles would be straighter and stronger then the others? 1) STEM ( Storable Tubular Extendible Member), a pre-stressed piece of thin metal that could be rolled up like a carpenters ...


text:
# Why does the dipole moment order of group 15 hydrides increase down the group?

The order given in my book is

$$\ce{NH3} > \ce{SbH3} > \ce{AsH3} > \ce{PH3}$$

Phosphorous is more electronegative than arsenic and antimony. Then shouldn't the dipole moment be more in case of $$\ce{PH3}?$$

And why does the anomaly of $$\ce{NH3}$$ occur?

• Geometry/symmetry also has a role in determining the dipole, for example, if the molecules were planar as is $\ce{BF3}$ the dipole would be zero. Aug 16, 2019 at 16:56
• @porphyrin could you convert this into a proper answer for this question
– Stan
Aug 16, 2019 at 17:59

E.N. of $$\ce{P}$$ is $$2.19$$, while that of $$\ce{As}$$ is $$2.18$$ and of $$\ce{Sb}$$ is $$2.05$$
Whereas H as an E.N. of $$2.20$$. So hence now you can see clearly that your order is justified, as in actuality, H becomes the negative end of the dipole.
Another factor that further widens this gap is that the dipole vector in chemistry notations is from + to-, which means all vectors add up in case of $$\ce{As}$$ and Sb


text:
Change search

Cite
Citation style
• apa
• ieee
• modern-language-association-8th-edition
• vancouver
• Other style
More styles
Language
• de-DE
• en-GB
• en-US
• fi-FI
• nn-NO
• nn-NB
• sv-SE
• Other locale
More languages
Output format
• html
• text
• asciidoc
• rtf
The bumpy light curve of Type IIn supernova iPTF13z over 3 years
Stockholm University, Faculty of Science, Department of Astronomy. Stockholm University, Faculty of Science, The Oskar Klein Centre for Cosmo Particle Physics (OKC).
Stockholm University, Faculty of Science, Department of Astronomy. Stockholm University, Faculty of Science, The Oskar Klein Centre for Cosmo Particle Physics (OKC).
Stockholm University, Faculty of Science, Department of Astronomy. Stockholm University, Faculty of Science, The Oskar Klein Centre for Cosmo Particle Physics (OKC).
Stockholm University, Faculty of Science, Department of Astronomy. Stockholm University, Faculty of Science, The Oskar Klein Centre for Cosmo Particle Physics (OKC).
2017 (English)In: Astronomy and Astrophysics, ISSN 0004-6361, E-ISSN 1432-0746, Vol. 605, article id A6Article in journal (Refereed) Published
##### Abstract [en]

A core-collapse (CC) supernova (SN) of Type IIn is dominated by the interaction of SN ejecta with the circumstellar medium (CSM). Some SNe IIn (e.g. SN 2006jd) have episodes of re-brightening (''bumps'') in their light curves. We present iPTF13z, a Type IIn SN discovered on 2013 February 1 by the intermediate Palomar Transient Factory (iPTF). This SN showed at least five bumps in its declining light curve between 130 and 750 days after discovery. We analyse this peculiar behaviour and try to infer the properties of the CSM, of the SN explosion, and the nature of the progenitor star. We obtained multi-band optical photometry for over 1000 days after discovery with the P48 and P60 telescopes at Palomar Observatory. We obtained low-resolution optical spectra during the same period. We did an archival search for progenitor outbursts. We analyse the photometry and the spectra, and compare iPTF13z to other SNe IIn. In particular we derive absolute magnitudes, colours, a pseudo-bolometric light curve, and the velocities of the different components of the spectral lines. A simple analytical model is used to estimate the properties of the CSM. iPTF13z had a light curve peaking at Mr <~ -18.3 mag. The five bumps during its decline phase had amplitudes ranging from 0.4 to 0.9 mag and durations between 20 and 120 days. The most prominent bumps appeared in all the different optical bands, when covered. The spectra of this SN showed typical SN IIn characteristics, with emission lines of Hα (with broad component FWHM ~ 103 - 104 km s-1 and narrow component FWHM ~ 102 km s-1) and He I, but also with Fe II, Ca II, Na I D and Hβ P Cygni profiles (with velocities of ~ 103 km  s-1). A pre-explosion outburst was identified lasting >~ 50 days, with Mr $\approx$ -15 mag around 210 days before discovery. Large, variable progenitor mass-loss rates (>~ 0.01 MSun yr-1) and CSM densities (>~ 10-16 g cm-3) are derived. The SN was hosted by a metal-poor dwarf galaxy at redshift z = 0.0328. We suggest that the light curve bumps of iPTF13z arose from SN ejecta interacting with denser regions in the CSM, possibly produced by the eruptions of a luminous blue variable progenitor star.

##### Place, publisher, year, edition, pages
2017. Vol. 605, article id A6
##### Keywords [en]
supernovae: general, supernovae: individual: iPTF13z, galaxies: individual: SDSS J160200.05+211442.3
##### National Category
Astronomy, Astrophysics and Cosmology
Astronomy
##### Identifiers
ISI: 000412231200072OAI: oai:DiVA.org:su-147273DiVA, id: diva2:1143033
Available from: 2017-09-20 Created: 2017-09-20 Last updated: 2019-08-08Bibliographically approved
##### In thesis
1. Bumpy light curves of interacting supernovae
Open this publication in new window or tab >>Bumpy light curves of interacting supernovae
2017 (English)Licentiate thesis, comprehensive summary (Other academic)
##### Abstract [en]

A supernova (SN) is the explosive destruction of a star. Via a luminous outpouring of radiation, the SN can rival the brightness of its SN host galaxy for months or years. In the past decade, astronomical surveys regularly observing the sky to deep limiting magnitudes have revealed that core collapse SNe (the demises of massive stars) are sometimes preceded by eruptive episodes by the progenitor stars during the years before the eventual SN explosion. Such SNe tend to show strong signatures of interaction between the SN ejecta and the circumstellar medium (CSM) deposited by the star before the SN explosion, likely by mass-loss episodes like the ones we have started to observe regularly. The complex CSM resolved around certain giant stars in our own galaxy and the eruptions of giant stars like η Car in the 19th century can be seen in this context. As the SN ejecta of an interacting SN sweep up the CSM of the progenitor, radiation from this process offers observers opportunity to scan the late mass loss history of the progenitor. In this thesis, interacting SNe and eruptive mass loss of their progenitors is discussed. The SN iPTF13z (discovered by the intermediate Palomar Transient Factory, iPTF) is presented. This transient was followed with optical photometry and spectroscopy during 1000 days and displayed a light curve with several conspicuous re-brigthenings ("bumps"), likely arising from SN ejecta interacting with denser regions in the CSM. Around 200 days before discovery, in archival data we found a clear precursor outburst lasting >~ 50 days. A well-observed (but not necessarily well understood) event like SN 2009ip, which showed both precursor outbursts and a light curve bump, makes an interesting comparison object. The embedding of the (possible) SN in a CSM makes it hard to tell if a destructive SN explosion actually happened. In this respect, iPTF13z is compared to e.g. SN 2009ip but also to long-lived interacting SNe like SN 1988Z. Some suggestions for future investigations are offered, to tie light curve bumps to precursor events and to clarify the question of core collapse in the ambiguous cases of some interacting SNe.

##### Place, publisher, year, edition, pages
Institutionen för astronomi, Stockholms universitet, 2017. p. 61
##### Keywords
iPTF13z, supernova, SN, circumstellar medium, CSM, SN Type IIn, intermediate Palomar Transient Factory, iPTF
##### National Category
Astronomy, Astrophysics and Cosmology
Astronomy
##### Identifiers
urn:nbn:se:su:diva-147289 (URN)
##### Presentation
2017-05-18, FB42, AlbaNova, Roslagstullsbacken 21, Stockholm, 13:00 (English)
##### Supervisors
Available from: 2017-09-27 Created: 2017-09-22 Last updated: 2017-09-27Bibliographically approved
2. Supernova surroundings on circumstellar and galactic scales
Open this publication in new window or tab >>Supernova surroundings on circumstellar and galactic scales
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

Some stars cease to be in a bright and destructive display called a supernova. This thesis explores what we can learn about supernovae (SNe) by studying their immediate surroundings, and what the SNe can teach us about their environments. The work presented is mostly based on the rich harvest of observations from 2009-2017 by the Palomar Transient Factory (PTF) and its successor, the intermediate PTF (iPTF). The PTF/iPTF was an untargeted sky survey at Palomar Observatory, aimed at finding and following up astronomical transients, such as SNe. During its existence, a massive star typically loses several solar masses of material. If much mass is lost in the decades or centuries before the SN, this material around the star (the circumstellar medium, CSM) will be quickly swept up by the ejecta of the eventual SN. This interaction can contribute strongly to the luminosity of the SN and make the light curve of an interacting SN carry signs of the progenitor star mass loss history. SNe with a hydrogen-rich CSM are called SNe Type IIn. A SN of this type, iPTF13z, found and followed by iPTF, had a slowly declining lightcurve with at least 5 major rebrightenings ("bumps") indicating rich structure in the CSM. Archival images clearly shows a precursor outburst about 210 days before the SN discovery, demonstrating the iPTF13z progenitor to be restless before its demise. Type IIn supernovae are heterogeneous, but only limited statistics has been done on samples. From PTF/iPTF, a sample of 42 SNe Type IIn was therefore selected, with photometry allowing their light curve rise times, decline rates and peak luminosities to be measured. It was shown that more luminous events are generally more long-lasting, but no strong correlation was found between rise times and peak luminosities. Two clusters of risetimes (around 20 and 50 days, respectively) were identified. The less long-lasting SNe Type IIn dominate the sample, suggesting that stars with a less extended dense CSM might be more common among SN Type IIn progenitors. Thermonuclear SNe (SNe Type Ia) are useful as standardisable candles, but no secure identification has yet been made of the progenitor system of a SN Type Ia. Using a late-time spectrum from the Nordic Optical Telescope of the nearby thermonuclear SN 2014J, a search for material ablated from a possible non-compact companion gave the upper limit of about 0.0085 solar masses of hydrogen-rich ablated gas. One likely explanation is that the SN 2014J progenitor system was a binary white dwarf. Supernovae are also useful tracers of the star formation history in their host galaxies, with SNe Type Ia tracing earlier epochs of star formation and exploding massive stars tracing more recent. For active galactic nuclei (AGN, the luminous centres of galaxies harbouring accreting supermassive black holes) SNe allows the so-called unification model to be tested. The unification model assumes that the main distinction between the two types of AGN is the viewing angle towards the central black hole, and that other properties (e.g. star formation history) of the host galaxies should be the same for the two AGN types. Matching 2190 SNe from PTF/iPTF to about 89000 AGN with spectra from the Sloan Digital Sky Survey, a significantly higher number of SNe in the hosts of AGN type 2 was found, challenging the unification model.

##### Place, publisher, year, edition, pages
Stockholm: Department of Astronomy, Stockholm University, 2019. p. 116
##### Keywords
supernovae, circumstellar medium, SNe Type IIn, iPTF13z, intermediate Palomar Transient Factory, active galactic nuclei, SN 2014J
##### National Category
Astronomy, Astrophysics and Cosmology
Astronomy
##### Identifiers
urn:nbn:se:su:diva-171451 (URN)978-91-7797-803-9 (ISBN)978-91-7797-804-6 (ISBN)
##### Public defence
2019-09-23, sal FB42, AlbaNova universitetscentrum, Roslagstullsbacken 21, Stockholm, 13:00 (English)
##### Note

At the time of the doctoral defense, the following paper was unpublished and had a status as follows: Paper 4: Manuscript.

Available from: 2019-08-29 Created: 2019-08-08 Last updated: 2019-10-09Bibliographically approved

#### Open Access in DiVA

No full text in DiVA

Publisher's full text

#### Search in DiVA

##### By organisation
Department of AstronomyThe Oskar Klein Centre for Cosmo Particle Physics (OKC)
##### In the same journal
Astronomy and Astrophysics
##### On the subject
Astronomy, Astrophysics and Cosmology

doi
urn-nbn

#### Altmetric score

doi
urn-nbn
Total: 99 hits

Cite
Citation style
• apa
• ieee
• modern-language-association-8th-edition
• vancouver
• Other style
More styles
Language
• de-DE
• en-GB
• en-US
• fi-FI
• nn-NO
• nn-NB
• sv-SE
• Other locale
More languages
Output format
• html
• text
• asciidoc
• rtf


text:
# Productivity and Income … Again

Today I’m going to revisit a topic that a month ago I committed to stop writing about — the productivity-income quagmire. Neoclassical economists argue that income is proportional to productivity. The problem is that they have no way of measuring productivity that is independent of income. So in practice, they test their theory by assuming it’s true. They show that one form of income — value added per worker — is correlated with another form of income — wages.

In No, Productivity Does Not Explain Income, I pointed out this circularity. And I haven’t heard the end of it since. The problem, critics say, is that my argument is flawed. Yes, value added and wages are related by an accounting definition. But that doesn’t ‘guarantee’ (in strict mathematical terms) a correlation between the two. In an update post, I admitted that the critics were correct. It is possible for wages and value added per worker to be uncorrelated.

Does this mean my argument is fatally flawed? No. But I will admit that the language in the original post was slightly overheated. The accounting definition between wages and value added doesn’t guarantee a correlation between the two. It virtually guarantees one.

I’m going to show you here that yes, it’s possible for wages and value added per worker to be uncorrelated. But it’s so unlikely that it’s not worth considering. The reality is that because of the underlying accounting definition, it’s virtually guaranteed that we’ll find a ‘statistically significant’ correlation between wages and value added per worker.

### The accounting definition

Let’s revisit how neoclassical economists test their theory of income. What they should do is measure the productivity of workers and see how it relates to income. But for a host of reasons, economists don’t do this. The most basic problem is that workers do different activities. This makes it impossible to compare their outputs. How, for instance, do you compare the output of a musician with the output of a farmer? The only logical answer is that you can’t — at least not objectively.

Realizing this problem should have caused neoclassical economists concern. It means that their theory is untestable. But instead of admitting defeat, neoclassical economists devised a loophole. To test their theory, they assume that one form of income (value added) actually measures productivity. Then they compare this income to another form of income (wages). The two types of income are correlated! Productivity explains income!

No.

The problem is that economists never actually measure productivity. Instead, they show that two types of income are correlated. But these two types of income are related by an accounting definition. And this accounting definition, I claim, (virtually) guarantees a correlation.

Let’s dive into the math.

Economists want to show that wages correlate with productivity — a fundamental tenet of their theory of income. To do this, they measure productivity in terms of value added per labor hour (or sales per labor hour, if data for value added is unavailable).

Let’s put this test in math form. Let w be the average wage of workers in a firm. Let Y be the value added of the firm. And let L be the labor hours worked by all workers in the firm. Economists look for a correlation between average wages (w) and value-added per labor hour (Y/L):

\displaystyle w \longleftrightarrow \frac{Y}{L}

The problem is that the left-hand and right-hand sides of this comparison aren’t independent from each other. Instead, they’re connected by an accounting definition. Let’s look at it.

Value added is, by definition, the sum of the wage bill (W) and profits (P):

Y = W + P

Now let’s calculate value added per labor hour. We take value added Y and divide it by labor hours L. To maintain the equality, we also divide the right-hand side by L:

\displaystyle \frac{Y}{L} = \frac{W + P}{L}

Now let’s distribute the L to the two terms in the numerator:

\displaystyle \frac{Y}{L} = \frac{W }{L} + \frac{P}{L}

What is W / L? It’s the wage bill divided by the number of labor hours. This happens to be the average hourly wage, w. So value-added per worker is equivalent to the average wage plus some noise term:

\displaystyle \frac{Y}{L} = w + \text{noise}

Given this accounting definition, it’s unsurprising that wages correlate with value added per labor hour. But do the two terms have to correlate? In strict math terms, no. If the noise term is large, it will drown out the correlation between wages and value added.

The problem is that the noise term is itself likely to correlate with value added per worker. Why? Because the noise term is actually profit per labor hour (P/L). It shares the same denominator (L) with value add per labor hour (Y / L). This codependency increases the chance of correlation.

So yes, it is possible for wages and value added to be uncorrelated. But it is also extremely improbable.

### Throwing numbers into the accounting formula

I want to show you just how probable it is that we’ll find a correlation between wages and value added per worker. Here’s how I’m going to do it. I’m going to randomly generate values for the wage bill (W), profit (P) and labor hours (L) of imaginary firms. Then I’ll throw these values into the accounting definition. Finally, I’ll look for a correlation between the average wage and value added per labor hour.

Just so I’m clear, here’s the steps:

1. Draw random values for the wage bill W and profit P of firms
2. Use these numbers to define value added: Y = W + P
3. Randomly generate a value for the number of labor hours (L) worked in each firm
4. Compare value added per labor hour (Y/L) to the average wage (w = W/L)

To generate the random numbers, we have to assume some sort of distribution. I’ll assume that W, P and L come from a lognormal distribution — a distribution that is common among economic phenomena. Because I want a general test, I’ll also assume that the parameters of this lognormal distribution are themselves random. See the notes for the math details and code.

If you want a simple metaphor for this process, think of defining W, P and L by rolling a dice. But each time we roll, the number of dice varies. This means that W, P and L can vary over a large range.

### Judging correlation

With our random numbers in hand, we plug them into our accounting definition and look for a correlation between value added per worker and the average wage. Here things get slightly more complicated.

It’s easy to measure correlation — just calculate the correlation coefficient (r) or the coefficient of determination (R2). The problem is that correlation exists on a scale. To say that two variables are ‘correlated’, we need to decide on some arbitrary threshold. This entails a value judgment.

For better or for worse (mostly for worse), the standard practice in econometrics is to hide this value judgment in more math. We define something called the p-value’ of the correlation. Then we judge correlation (or lack thereof) based on an arbitrary threshold in the p-value.

What’s a ‘p-value’? It’s a probability. The p-value tells you the probability of getting your observed correlation (or greater) from random numbers. The lower the p-value, the more ‘statistically significant’ your correlation.

Using p-values depends on a host of assumptions, many of which are violated when we study economic phenomena. Worse still, there are many ways to rig the game so you get better (i.e. lower) p-values. It’s called p-hacking, and it’s a huge problem in the social sciences.

Despite these problems, I’m going to use p-values to judge the correlation between simulated wages and value added per worker. I do so not because I like p-values (I don’t), but because using them is the standard practice in econometrics. Getting a low p-value means your results are publishable. Your correlation is ‘significant’! (If you want to read about how silly this is, check out the book The cult of statistical significance.)

### Correlation from randomness

With our p-values in tow, here’s the question we want to ask. If we plug random numbers into our accounting definition, how likely is it that the correlation between wages and value added per worker will be ‘statistically significant’?

The answer, it seems, is very likely. Although we’re dealing with random numbers, a ‘significant’ correlation between simulated wages and value added per worker seems to be in the cards. Table 1 tells the story.

Let’s unpack the results. I have a model that throws random numbers into our accounting definition. For each set of random numbers, I calculate the p-value of the correlation between wages and value added per labor hour. Then I look at how often these p-values are below some critical value. The left-hand column in Table 1 shows various thresholds for the p-value. The right-hand column shows the portion of the simulations in which the p-value is below this critical value for ‘statistical significance’.

By throwing random numbers into our accounting definition, we get a ‘statistically significant’ correlation 99.9% of the time. Note that this holds no matter how stringent our level of statistical significance. Even for the very low (in the social sciences) p-value of 0.0001, some 99.89% of the results are ‘statistically significant’. It seems that by throwing numbers into our accounting definition, a ‘statistically significant’ correlation between wages and value added per worker is virtually guaranteed.

Let’s look at the simulation results another way. Figure 1 shows how the p-values are distributed across all of the simulations. Most of the p-values are so small (meaning the correlation is so ‘significant’) that I have to plot p on a log scale. The x-axis shows the logarithm of the p-value. The y-axis shows the relative frequency of each p-value. To get some perspective, the vertical red line shows the standard threshold for statistical significance, a p-value of 0.05. Virtually all of the results are below this value, meaning the correlation is ‘statistically significant’.

Figure 2 shows yet another way of looking at the simulation results. Instead of plotting p-values, here I look at the distribution of the f-statistic. The f-statistic is another way of measuring ‘statistical significance’ (p-values are actually derived from the f-statistic). But whereas a lower p-value is ‘more significant’, a higher f-statistic is ‘more significant’.

As with p-values, the f-statistics are so extreme that I need to plot their logarithm. The vertical red line in Figure 2 shows the threshold for statistical significance at the 5% level. Results with an f-statistics above this value are deemed ‘statistically significant’. Again, we see that the vast majority of results are ‘statistically significant’.

It seems that by throwing random numbers into our accounting definition, we can’t help but find a correlation between wages and value added per worker.

### Manufacturing correlation

The charge that I’ve laid against neoclassical economists is that when they test their theory of income, they’re fooling themselves. Their method (virtually) guarantees a positive result. They regress two types of income — wages and value added per labor hour — that are related by an accounting definition. The problem is that this accounting definition (virtually) guarantees a statistically significant correlation.

Now the degree to which this correlation is guaranteed depends on the specifics of how the wage bill, profit and labor hours are distributed. But what I’ve shown here is that across a huge class of numbers, a ‘statistically significant’ correlation is almost unavoidable. It’s manufactured by our accounting definition.

This result highlights a problem with how economists use p-values. The use of p-values has turned into a production function: run a regression → get a low p-value → get published. Rarely do economists question whether the assumptions behind p-values are actually met in the real-world data.

In my simulation, the assumptions behind p-values are systematically violated. To use them, we must assume that our data is ‘statistically independent’. Here, this means that wages and value added per labor hour can be treated as independent, random variables. The problem is that they’re not independent. Wages and value add per labor hour are related by an accounting definition. This renders them highly dependent. So the use of p-values is moot.

Still, p-values are the standard by which economists judge correlation. By this standard, our accounting definition virtually guarantees a ‘statistically significant’ correlation between wages and value added per labor hour.

### The larger problem

The larger problem here is that the marginal productivity theory of income is untestable. Its core components — productivity and the ‘quantity’ of capital — cannot be measured objectively. If you want to know more about these problems, I recommend John Pullen’s book The Marginal Productivity Theory of Distribution: A Critical History.

The reality is that when it comes to explaining income, there is a long and sordid history of political economists fooling themselves. Neoclassical economists may be the most visible fools, but they’re by no means the only ones. Marxists too test their theory of income in circular terms. If you’re interested in Marxist theory, check out the debate between Jonathan Nitzan, Shimshon Bichler and the Marxist Paul Cockshott. What appears to be evidence for the labor theory of value, Nitzan and Bichler show, is actually mathematical foolery.

This issue of circular testing cuts to a core problem in both neoclassical and Marxist theories of income. Both explain income in terms of quantities that are unobservable. Unsurprisingly, tests of these theories resort to circular logic. Such tests invariably show that two forms of income are correlated. Then they claim that one form of income is something other than what it seems.

Sadly, this foolery has been standard practice for a century. And that’s not really surprising. There is perhaps no topic in which objectivity is more difficult than the distribution of income. Still, if we want a scientific theory of income, we need to do better. We need to stop fooling ourselves.

### Notes

Here’s my model. I assume that profit (P), the wage bill (W) and the number of labor hours (L) in firms are random variables that are lognormally distributed. If you’re not familiar, the lognormal distribution looks like a bell curve when you take the logarithm of its values. If the variable x is lognormally distributed, log(x) is normally distributed. Many quantities in economics are lognormally distributed, which is why I use this function here.

The lognormal distribution has two parameters, the ‘location’ parameter mu and the ‘scale’ parameter sigma. I’ll denote the lognormal distribution with the notation used in R. If x is lognormally distributed with parameters mu and sigma, I denote it as:

x = lnorm(mu, sigma)

To assume almost nothing about the distribution of P, W and L, I let the parameters of the lognormal distribution vary randomly over a uniform distribution. I’ll denote the uniform distribution using the notation used in R. If x is a uniformly distributed over the range 0 to 1, we write:

x = runif(0, 1)

I let the parameters of the lognormal distribution vary between 0 and 10. If you’re familiar with the lognormal distribution, you’ll know that this is a huge parameter space. So the values of P, W and L are:

P = lnorm( mu = runif(0, 10), sigma = runif(0, 10))

W = lnorm( mu = runif(0, 10), sigma = runif(0, 10))

L = lnorm( mu = runif(0, 10), sigma = runif(0, 10))

I take these values and throw them into the accounting definition. Average wages are then W / L. Value added per worker is (W + P) / L. To see how the correlation between wages and value added per worker varies, I run the algorithm several thousand times.

### Code

Here’s the R code for the model. Run it for yourself and see what you find.

library(doSNOW)

n_test = 20000
n_firms = 10^4

# cluster
cl = makeCluster(4, type="SOCK")
registerDoSNOW(cl)
clusterSetupRNG (cl, type = "RNGstream")

# progress Bar
pb = txtProgressBar(max = n_test, style = 3)
progress = function(n) setTxtProgressBar(pb, n)
opts = list(progress = progress)

test = foreach(i = 1:n_test, .options.snow=opts, .combine=rbind)  %dopar%  {

# wagebill
mu = runif(1, 0, 10)
sigma = runif(1, 0, 10)
wagebill = rlnorm(n_firms, mu, sigma)

# profit
mu = runif(1, 0, 10)
sigma = runif(1, 0, 10)
profit = rlnorm(n_firms, mu, sigma)

# value added (sum of wagebill and profit)

# labor hours
mu = runif(1, 0, 10)
sigma = runif(1, 0, 10)
labor_hours = rlnorm(n_firms, mu, sigma)

# hourly wage
hourly_wage = wagebill / labor_hours

# valued added per labor hour

# regress value added per hour and hourly wage
r = lm( log(va_per_hour) ~ log(hourly_wage) )

f = summary(r)$fstatistic f_stat = f[1] p = pf(f[1],f[2],f[3],lower.tail=F) r2 = summary(r)$r.squared

output = data.frame(f_stat, p, r2)

}
stopCluster(cl)

# portion that are significant
sig_frac =  length(test$p[ test$p < 0.05 ]) / length(test$p) # f statistic hist(log10(test$f_stat), breaks = 100, xlim = c(0, 10))
abline(v = log10(3.85), col = "red" )

# export
write.csv(test, "test_data.csv")

Cockshot, P., Shimshon, B., & Nitzan, J. (2010). Testing the labour theory of value: An exchange. http://bnarchives.yorku.ca/308/02/20101200_cockshott_nitzan_bichler_testing_the_ltv_exchange_web.htm

Pullen, J. (2009). The marginal productivity theory of distribution: A critical history. London: Routledge.

Ziliak, S., & McCloskey, D. N. (2008). The cult of statistical significance: How the standard error costs us jobs, justice, and lives. University of Michigan Press.

#### Support this blog

Economics from the Top Down is where I share my ideas for how to create a better economics. If you liked this post, consider becoming a patron. You’ll help me continue my research, and continue to share it with readers like you.

1. You spend a lot of time attacking theory instead of assumptions, but it is the assumptions that tell you everything about what economists seek to abstract into oblivion.

The problem here is the assumption that “the iron clad law of supply and demand” sets prices. That’s empirically false for anything other than commodities. In most situations, prices are set by the sellers based on how much profit they want to make. In this sense, wages fully reflect the value provided by labor, from the perspective of the owner of the business. Ironically, then, the insistence that wages reflect the actual value addded by labor is pretty much an admission that the “iron clad law of supply and demand” is a dead letter.

• Hi Tao,

I agree that the most important way to criticize a theory is to show that it’s assumptions are flawed. This is true of the core of a theory. But it’s also true of tests of a theory.

What I’m doing here is criticizing how economists test marginal productivity theory. They claim that the correlation between wages and value added per worker is evidence for their theory. But this is based on flawed assumptions.

2. […] [3] There are many other blows to human capital theory that are fatal. Most importantly, the theory posits that productivity explains income. But economists never measure productivity independently of income. Instead, all of their evidence for productivity is, in fact, circularly tied to income. For details, see these posts: No, Productivity Does Not Explain Income, Productivity Does Not Explain Wages, Debunking the ‘Productivity-Pay Gap’, and Productivity and Income … Again. […]

3. Dave C says:

I’m an engineer so I understand your math and there’s nothing wrong with your theory.

But there is another theory that looks correct to me.

Most people and companies operate in a competitive market. I got 3 bids to paint my house. I chose the one I thought was the best value. I considered price and quality.

The owner of this painting company will go broke unless he hires productive painters and pays them as much as they could make elsewhere. He competes for my business and he competes for his labor supply. He cannot take money from anyone unfairly.

In a competitive market, people will be paid approximately according to their productivity.

How can we compare a musician to a farmer? In a competitive market, millions of people will bid for their services, and the price will be approximately correct.

• You’re arguing by assumption. To ‘prove’ your point (that people earn their marginal product) you’re assuming that prices are ‘correct’. And by ‘correct’, you mean that prices reveal productivity. It’s a circular operation. You can’t assume what you want to prove.

4. I do not understand why you keep going at it. Your weird “neo classics are dumb as a rock and missed the most obvious flaw” is just… it’s just weird.

1. The simulation you run is just silly:

The question you raise boils down to this “is the sum of two **independent and identically distributed** random variables correlated with each of these random variables”. There is no need for a simulation, since three lines of algebra based on the definition of correlation will give you the answer: Yes. It’s a common question in first year stats problem set. Even ignoring the weird decision to randomly generate L, which of course does absolutely nothing (since the correlation of X and Y is the same as the correlation of X/Land Y/L), the only thing this elaborate simulation you ran tells us is that you have less than rudimentary understanding of what correlation is.

Your statement that “The reality is that because of the underlying accounting definition, it’s virtually guaranteed that we’ll find a ‘statistically significant’ correlation between wages and value added per worker” is sloppy logic at best.

What the (silly, unnecessary) simulation shows is this:
The existence of a correlation between wages and value added cannot tell us which of two theories of wages is more correct: (a) the neo-classical theory (b) the theory that profits are determined randomly, and wages are also determined randomly, and that the two are independent of each other (in the statistical sense of independence).
Since both these theories share a prediction – that there will be a correlation between wages and value added – the realization of this prediction in the data cannot tell us anything about which theory of these two is more correct.

Of course, there is no theory that wages and profits are determined independently and randomly, so your simulation adds no useful information to the debate. Your use of “The reality is” is just wrong. The correlation is not because in “reality” the accounting definition virtually guarantees anything. The correlation you find is due to the way you assumed the relationship between the variables (namely, independently distributed).
If, “in reality”, wages are held at subsistence level because of the “army of unemployed” then there would be no correlation between wages and value added – even though the accounting is the same. In reality, something determine wages, and something determines profits. Assuming they are random and independent and calling that “in reality” is flat out wrong.

To sum up, your unnecessary simulation only shows that you can come up with another “theory” that predicts a correlation between wages and value added. This is a theory in which profits and wages are determined randomly and independently. I can come up with a hundred more before breakfast. This does not at all mean anything like “The reality is that because of the underlying accounting definition, it’s virtually guaranteed that we’ll find a ‘statistically significant’ correlation between wages and value added per worker.”

Show some humility. You clearly have very limited understanding of the issues, not even understanding what correlation is. There’s no shame in that – some of the smartest and most insightful people I know don’t understand any of these things. Just maybe you want to be a bit more cautious before declaring all of neo classical economics an obvious fraud.

• on second read, I see that you divide by a different L for each firm. This isn’t quite as meaningless, but still silly. What’s the point? It only means you draw the numbers from a different distribution, where of course the distribution itself does not matter. The correlation is basically determined from the definition of correlation and from your assumption of independence.

• Welcome back assafzim,

I was waiting for you to troll this post. As usual, you do not disappoint. Sadly, there’s very little to respond to here besides ad hominem attacks. It seems that you don’t understand how null hypotheses work. They are not “theories”. They are an assumption by which we judge correlations. We assume the variables are independent and randomly distributed. Then we see how likely it is to find the given correlation.

• you are wrong, and while I did let a couple of ad hominem slip, it’s only b/c I am truly amazed by the audacity: you trash a whole discipline as obviously flawed, trash a whole community of scholars as frauds, when you don’t even understand the basics. Anyways –

Minor point: it is incorrect that “we assume the variables are independent and randomly distributed”. While this is common, null hypothesis often assume different things. They may assume a perfect correlation, the may assume negative correlation, they may assume correlation of whatever magnitude. But that’s a minor issue.

Major point: what you do is NOT hypothesis testing.
Hypothesis testing would be to ask “if **wage and value added** were NOT in fact correlated, how likely is it that we would observe the correlation between **wage and value added** that we see in the data”.
This is clearly rejected.

Alternatively, hypothesis testing would be to ask if **wage and profits** were not in fact correlated, how likely is it that we would observe the correlation between **wage and profits** that we do in the data”. This too, btw, is clearly rejected.

What you do is assume **wage and profit** are independently distributed (and hence not correlated), and then ask how likely is it, given that **wage and profit** are not correlated, that we will find in the data that **wage and value added** are correlated. This is just sloppy thinking. It is certainly not hypothesis testing.

Also, it’s silly. If wage and profits are independent, then with three lines of first year algebra you can show that wage and value added are correlated. It’s just shocking that you would think this was somehow missed by generations of scholars.
And that, pointing out this trivial mathematical fact, is all your simulation does. It shows that if you assume that wages and value added are correlated (which is what you assume as it follows directly from your assumption that wages and profits are not correlated), and then generate a whole lotta samples of wages and value addeds, in the vast majority of them you will get that the sample of wages is correlated with the Sample of value added. Really. That is literally what your simulation does.

I am trying to give a coherent meaning to your point that the accounting makes it “virtually guarantee[d]” that a correlation between wages and value added will be found. It’s hard, but the only sense I can make of this statement is that “regardless of what is the actual process that determines wages and value added, the accounting virtually guarantees the data will show a correlation”. This is obviously false. Only because of your assumption that wage and profit are independent you find a correlation between wage and value added. If the process that determines wages and value added in reality was not such (and, btw, of course it is not such) then it is not at all guaranteed that there will be a correlation between wages and value added.

This is a good point to continue to the non-technical comment

5. Here are things that would follow if your logic was correct:

1. Call a person expenditure on cigarettes W, and his expenditure on everything else P. His total expenditure is therefore W+P, which we can call Y. This virtually guarantees that expenditure on cigarettes is positively correlated with total expenditure.
But of course that is false. You can run this empirical test many times over, and in the vast majority of cases you will find a negative correlation.

2. Marx theory of wages, according to which wages are kept at subsistence, should have been empirically rejected without any need to actually do any empirical work. Since, by Marx theory, wages are not correlated with value added, and since the accounting of it all virtually guarantees we will find a correlation, there’s no need for empirical work to reject Marx theory.
Again – this is obviously a ridiculous statement.

3. I have a sample of only startups in Silicon Valley (zero value added as there are no sales yet, wages are just the negative of profits), and diamond mines in Africa (very low wages, very high value added). All the accounting is the same as in your post here. The accounting virtually guarantees that there is a positive correlation between wages and value added.
Again – this is false. In this example there is a negative correlation. Again – even though the accounting of wages, profits, and value added is the same as in your post.

ok, I think the point is clear. The accounting does not guarantee, not does it virtually guarantees, nor does it even imply that “in reality” there will be a correlation between wages and value added. The underlying process that generates the data is what determines the existence of the correlation or its absence. Therefore, the existence of the correlation or its absence does indeed teach us about the underlying process that generates the data.

6. ok, I had time to organize my points, so let me offer the readers this clearer explanation of what this post is doing.

1. It uses a statistical software to generate many random values for wages and for profits. It defines wages and profits to be statistically uncorrelated with each other.

2. It then defines (correctly) added value as the sum of wages and profits. BY DEFINITION, a sum of two independent random variables (value added) is correlated with each of its components (wages, profits). So, by defining wages and profits to be independent, the post here defines value added to be correlated with wages.

3. The statistical software, obviously, works based on the instructions it is given. And it was given the instruction to generate wages and value added so that they are correlated.

4. Now, because there is some randomness in the process, not 100% of the samples the software will generate will indeed show a correlation. But of course – of course!!! – if the instructions to the statistical software were “please generate samples of wages and value added that are correlated”, which is exactly the instruction that it was given, the vast majority of these samples will indeed show a correlation.

5. Astonishingly, the post then argues that this exercise – telling a software to generate variables that are correlated, and then finding that they are indeed correlated in most cases – teaches us something about reality. It is saying that this exercise shows that “in reality” most samples of wages and value added have to be correlated, because of the accounting. This is just silly in the extreme.

6. The correlation in the samples that the statistical software created comes from – who would have thunk it – the fact that it was told to generate correlated variables, i.e. from the assumption that wages and profits are independent (which leads, mathematically, to the correlation between wages and value added).

7. In reality, what determines whether or not there is a correlation between wage and value added is the process that generates wages and value added. Even though they are related to each other by accounting, in reality, they may be positively correlated, uncorrelated, or negatively correlated. All depending on the process that generates the data. Numerous examples from reality show this very clearly.

8. Hypothesis testing, which seems to be the source of confusion in this post, deals with something entirely different. Hypothesis testing asks, for example, if X and Y in reality are not correlated in the whole population, how likely is it that we would see a correlation in a specific sample. It has absolutely nothing to do with what’s in this post.

9. Based on this sloppy thinking and misunderstanding of basic concepts, the author wants you to believe he has found a fatal flaw in decades of empirical research. sigh…

7. Marcos says:

Blair, I tremendously appreciate the mission you are undertaking in this blog. It is refreshing to discuss economics and the social sciences critically and openly without submitting to the back-slapping rituals of incestuous academic peers. I wish your blog thrives and encourages many other like-minded initiatives.

After your posts and the discussions they generated, it is crystal clear that the correlation between wages and value added per worker is a miserably poor piece of evidence in support of the neoclassical theory of income. Now we are free to go a step further in terms of substantive explanations of wages and income.

Clearly, thousands of stories could be told that would yield the correlation between wages and value added per worker as a prediction. The neoclassical story tells us that the correlation emerges because competitive firms tend to reward workers according to their productivity. But here is another story that leaves us with the same result.

Workers are paid just enough to reproduce their workforce according to socially agreed minimum standards of living. On the one side, because it costs more to reproduce the workforce of better qualified workers, they are paid more whatever the minimum standards are. On the other, because those minimum standards of living are socially agreed, they can vary widely in time and space. As a general rule, however, as societies produce more value per worker, workers devote more of their collective effort as a class not to lag too far behind the classes of exploiters who do not work for their living. As a result of (i) the social need to reproduce the workforce and (ii) workers’ collective efforts to keep their living standards in pace with the rest of society, we can generally expect to find a correlation between wages and value added per worker across and within capitalist societies. And this has nothing to do, of course, with the idea that exploitative classes reward workers for their productivity.

For those who did not know, this story is a rough approximation of the story Marx tells us in the Capital. Forget all the nonsense about “subsistence wages” that neoclassic economists deceitfully pass on together with their other capitalist publicity stunts.

The Marxist story is superior for a host of reasons, among which:

– It predicts a correlation in wages and value added per worker without assuming foolish metaphysical entities (such as “perfectly competitive markets”) that exist in foolish metaphysical states (such as “equilibrium”)

– It doesn’t need ad hoc patches to explain how worker heterogeneity affects wage differences.

– It makes sense of social unrest, union gatherings, workers’ strikes, labor laws, minimum wages – just to name a few institutions and social phenomena that most of us observe in our daily lives.

– It makes sense of why exploitative classes won’t spare resources in their attempt to persuade themselves and the rest of society that workers are paid what they produce.

• Hi Marcos,

Thank-you for the kind words. I’m glad my writing is proving useful!

8. […] Productivity and Income … Again […]

9. […] Productivity and Income … Again […]

10. […] Productivity and Income … Again […]

11. […] Productivity and Income … Again […]

12. […] of income (usually sales and wages) and then call one of the incomes ‘productivity’. Since they always find a correlation, they always ‘confirm’ their theory of income. […]

13. […] (usually sales and wages) and then call one of the incomes ‘productivity’. Since they always find a correlation, they always ‘confirm’ their theory of income. […]


text:
Home

>

English

>

Class 12

>

Physics

>

Chapter

>

Jee Mains 2020

>

ABC is a plane lamina of the s...

# ABC is a plane lamina of the shape of an equilateral triangle. D , E are mid points of AB,AC and G is the centroid of the lamina . Moment of inertia of the lamina about an axis passing through G and perpendicular to the plane ABC is I_0 If part ADE is removed, the moment of inertia of a remaining part about the same axis is (NI_0)/16 where N is an integer . Value of N is

Khareedo DN Pro and dekho sari videos bina kisi ad ki rukaavat ke!


text:
# Inverse Laplace transform of two-sided and one-sided Laplace transform

As I read in Wikipedia, there are two types of Laplace transforms

• One-sided Laplace transform: $$F(s) = \int_{0}^\infty e^{-st} f(t) dt$$

• Two-sided Laplace transform: $$F(s) = \int_{-\infty}^\infty e^{-st} f(t) dt$$

But they give only one formula for Inverse Laplace transform:

$$\hspace{3.0cm} f(t) = \frac{1}{2\pi i} \lim_{T \to \infty} \int_{\gamma - i T}^{\gamma + i T} e^{st} F(s) ds$$

My question is that, does the type of Laplace transform I use affects the Inverse formula ?

### p.s:

I've proved the Inverse Laplace transform above corresponding to Two-sided Laplace transform using Fourier transform. But I've not come up with any idea of proving the correctness of the Inverse Laplace transform corresponding to One-sided Laplace transform.

According to my proof, the Inverse transform above is correct for One-sided transform if $$f$$ satisfies $$f(t) = 0$$ $$\forall t < 0$$. In other words,

$$\hspace{3.0cm} f(t) = \frac{1}{2\pi i} \lim_{T \to \infty} \int_{\gamma - i T}^{\gamma + i T} e^{st} F(s) ds$$, $$\forall t \geq 0$$

$$f(t)=\frac{1}{2\pi j}\int_{\alpha-j\infty}^{\alpha+j\infty}F(s)e^{st}ds\tag{1}$$
The difference is in the choice of the constant $$\alpha$$. The line $$\textrm{Re}\{s\}=\alpha$$ must be inside the region of convergence (ROC). For causal functions (i.e., functions for which $$f(t)=0$$ for $$t<0$$), the ROC is to the right of the pole with the most positive real part, whereas for non-causal functions, the ROC is a vertical strip between two poles.
• Does the choice of $\alpha$ affects the result of the Inverse transform ? – HOANG GIANG Jan 16 '19 at 13:34
• @HOANGGIANG: Yes, because one function $F(s)$ can have different inverse transforms $f(t)$ depending on the ROC. – Matt L. Jan 16 '19 at 13:50


text:
# Could a “nuclear turbo-jet” be built?

Being familiar with jet engines myself, I wonder whether a similar device could be built to provide high $$I_{\text{sp}}$$ and high thrust in vacuum and space. Using the tried-and-true turbojet formula as an analogy --

• Inlet. This would be the storage of propellant, say hydrogen, always at local $$M=0$$ for a rocket engine. Because we aim for a high thrust and a high $$I_\text{sp}$$, the mass flow rate is very low, so an electric drive will be sufficient to drive a propellant pump. The very low-density gaseous propellant is subsequently by an RF antenna ionized to a low-temperature plasma.
• Multiple-stage compressor. The jet engine needs this to establish a stable direction of flow, plus it'a necessary part of the Brayton cycle. So I imagine for plasma this is more or less the same. For a nuclear turbo-jet, this would be cascading stages of magnetohydrodynamic drives with converging cross-section and diffusers which is simply a section with a diverging cross-section, I assume this rather than the other way because the plasma is hot and the absolute number of local speed of sound must be very high so compression happens subsonic. Or this could be done in a simple stage for plasma.
• Fuel injection and burn. Here is the part that distinguishes nuclear turbo-jet from conventional ones. For fission to happen the fuel has to be squeezed as well. I don't know if you can do this simply by jetting ionized Plutonium or Uranium and squeezes it or you need an ionized tamper around it to contain it for a more complete burn, and that if the burn time for fission can be made less than the dwell time in the combustion section. As with jet engine, pressure peaks before this stage, and only goes downhill from here, so exhaust comes out from the backend. I also don't know if there's a magnetic equivalence of a flame holder to induce vortices inside the plasma to help the combustion.
• Turbine. This would be a magnetohydrodynamic generator that feeds the combuster magnetohydrodynamic drive.
• Nozzle. A variable de Laval magneto nozzle.

A jet engine is streamlined, has very low vibration compared to its pulsed peer the piston engine, and has the amazing ability to burn nearly every hydrocarbon on the market provided that you can jet it out of the fuel line. I imagine plasma is not that different than gas and a nuclear turbo-jet could offer similar advantages while providing very high exhaust speed and thus strong thrust at a low mass flow rate.

Is this engine feasible, if we handwave the superconducting coil generating the magnetic field part, and the exact pattern of the magnetic field part?

• – uhoh Mar 26 '19 at 13:52
• This question is probably more aviation related than space. en.wikipedia.org/wiki/Nuclear-powered_aircraft – JCRM Mar 26 '19 at 14:02
• You're confusing the requirements for a fission chain reaction and those for a magnetically confined fusion reaction – Steve Linton Mar 26 '19 at 14:48
• The system uses stored hydrogen as a reaction mass and a fuel that doesn't need an oxidizer, and so it would work in space (if it worked). It's on-topic here. – uhoh Mar 26 '19 at 14:50
• @SteveLinton No, the continuous fuel jet must stay subcritical in the fuel line and go supercritical when it's supposed to burn (or the engine will just end up nuking itself), so squeezing is the proper way, as in a nuke you squeeze using dynamite to make it go boom and it's safe when stored. – Meatball Princess Mar 26 '19 at 15:03

No, it won't work.

• Fuel injection and burn. Here is the part that distinguishes nuclear turbo-jet from conventional ones. For fission to happen the fuel has to be squeezed as well. I don't know if you can do this simply by jetting ionized Plutonium or Uranium and squeezes it or you need an ionized tamper around it to contain it for a more complete burn, and that if the burn time for fission can be made less than the dwell time in the combustion section...

Squeezing a little bit of fissile material through the engine at a time will not produce much fission. The neutrons come out fast and need to be moderated by a lot of intervening material to slow them down before they are captured by the next nucleus.

To make the chain reaction, you need a nuclear pile with a lots of fuel and lots of moderator mass all in one place. This is why reactors are big and heavy and don't really fit into aircraft or spacecraft very well.

Neutron damage would make your fan blades brittle quickly if there was a lot of fission, which there won't be.

Compact sources of nuclear power like RTGs use spontaneous nuclear decay rather than chain reactions. But that means you can't induce "ignition" inside the engine.

A really interesting idea was proposed based on a nuclear isomer and is discussed at length in this excellent answer to the question How did Northrop Grumman propose to make the Global Hawk nuclear powered?.

From the New Scientist's article Nuclear-powered drone aircraft on drawing board

Tightly controlled reaction

The reaction works because a proportion of the hafnium nuclei are “isomers” in which some neutrons and protons sit in higher energy levels than normal. X-ray bombardment makes them release this energy and drop down to a more stable energy level.

Here the idea was that an X-ray generator (which can be compact) is used to stimulate isomeric states to decay and release energy promptly. It's a little bit like the X-ray laser concept except the excited state lasts days or years rather than microseconds, and there's no optical gain. It's not a LASER, it's only a SER (stimulated emission of radiation) without the LA.

• There have been some small reactors within spacecrafts, for instance the TOPAZ reactors. Of course not as "turbo jets". – Uwe Mar 26 '19 at 14:27
• @Uwe but that wasn't used for propulsion. 300 kg for 5kW would make a ridiculously small thrust to weight ratio. – uhoh Mar 26 '19 at 14:31
• @uhoh What if we use weapon-grade HEU and add a dense vapor or plasma tamper around it? – Meatball Princess Mar 26 '19 at 14:53
• Wait, the Hafnium X-ray simulation actually works? It has been demonstrated experimentally? I thought it had been discredited. – Eth Mar 26 '19 at 15:31
• @Eth no, don't wait, go ahead and re-read what I've written to see that there's no suggestion of that. – uhoh Mar 26 '19 at 15:40


text:
# I don't see the relation

Algebra Level 2

$\begin{eqnarray} \large \color{blue}2^{\color{brown}x} - \color{blue}2^{\color{brown}y} & = & \large 1 \\ \\ \large \color{green}4^{\color{brown}x}- \color{green}4^{\color{brown}y} & = & \large { \frac 5 3} \\ \\ \\ \large {\color{brown}x} - {\color{brown}y} & = & \large \, ? \end{eqnarray}$

Details and Assumptions:

• $$x$$ and $$y$$ are real numbers.
×


text:
# I Lie Algebra of Lorentz group

1. May 17, 2017

### Silviu

Hello! I read that the for the lie algebra of the Lorentz group we can parametrize the generators as an antisymmetric tensor $J^{\mu \nu}$ and the parameters as an another antisymmetric tensor $\omega_{\mu \nu}$ and a general transformation would be $\Lambda = exp(-\frac{i}{2} \omega_{\mu \nu} J^{\mu \nu})$. So based on Einstein notation, this would mean $\Lambda = exp(-\frac{i}{2} (\omega_{00} J^{00} + \omega_{01} J^{01} + ... ))$. But for example $\omega_{00}$ and $J^{00}$ are numbers, so in the end we will end up with a complex number in the exponential and now with a (4x4 most probably) matrix. Can someone explain to me what I do wrong in reading this notation? Thank you!

2. May 17, 2017

### Orodruin

Staff Emeritus
The $J^{\mu\nu}$ are not tensors, they are the basis of the Lie algebra.

3. May 17, 2017

### Silviu

So they are in the case of Minkowski space 4x4 matrices?

4. May 17, 2017

### Orodruin

Staff Emeritus
If you use the (usual) matrix representation of the Lorentz group, yes.

5. May 18, 2017

### vanhees71

6. May 18, 2017

### dextercioby

They are tensor operators with respect to the Lorentz group.

7. May 18, 2017

### Orodruin

Staff Emeritus
They can be represented by tensor operators. This still does not make them tensors in the sense implied by the OP.

8. May 20, 2017

### samalkhaiat

We do not ‘parameterize’ the generators. The generators of any Lie algebra are independent of the parameters of the corresponding Lie group. This $\omega_{\mu\nu}$ is not a tensor. It is a set of six real numbers, $\{ \omega_{a}, \ a = 1, ..., 6 \}$, (3 rotation angles $\theta_{i} \equiv \frac{1}{2}\epsilon_{ijk}\omega_{jk}$, and 3 boosts $\beta_{i} \equiv \omega_{0i}$) grouped together under the condition $\omega_{\mu\nu} = - \omega_{\nu\mu}$. So, when you use the symbol $\omega_{\mu\nu}$ to represent the (six) parameters, you should keep in mind that $\omega_{00} = \cdots = \omega_{33} = 0$. This means that there are only six generators ($M^{a}, \ a = 1, ... , 6$) which (also) can be grouped to satisfy $M^{\mu \nu} = - M^{\nu\mu}$. Since Lorentz group is a connected Lie group, then (almost all) its elements can be represented by the exponential map $$g = e^{\vec{\omega}} = e^{\omega_{a}M^{a}} \equiv e^{\frac{1}{2}\omega_{\mu\nu}M^{\mu\nu}} ,$$ where $\vec{\omega}$ is an element of 6-dimensional vector space (the Lorentz algebra) and $\omega_{\mu\nu}$ are its components (local coordinates on the group manifold) “along” the basis vectors $M^{\mu\nu}$. This should remind you with the school-days vector equation $\vec{x} = x_{i}\hat{e}^{i}$.
Now, the question whether the generators are Lorentz tensors or not is a different story. To answer it, we need to study the representation of the Lorentz group. Basically, in the representation theory of Lorentz group, you encounter three objects. These are: the coordinates transformation matrix $\Lambda$, with matrix elements denoted by $\Lambda^{\mu}{}_{\nu}$; the finite-dimensional representation matrix $D(\Lambda) = \exp \left(\frac{1}{2}\omega_{\mu\nu}\Sigma^{\mu\nu}\right)$ which acts on the index space of geometrical objects (classical fields) on Minkowski space-time, i.e., under a Lorentz transformation $\Lambda$, the components of a given field $\varphi_{a}(x)$ get mixed by the matrix $D(\Lambda)$; and the third object is the infinite-dimensional unitary representation operator $U(\Lambda) = \exp \left( \frac{i}{2} \omega_{\mu\nu}J^{\mu\nu}\right)$ which acts on states and operators in the Hilbert space. In QFT, the above three objects appear in one fundamental (transformation) equation, that is $$U^{\dagger}(\Lambda) \varphi_{a}(x) U(\Lambda) = D_{a}{}^{c}(\Lambda) \ \varphi_{c}(\Lambda^{-1}x) .$$ From this equation we understand that on the one hand the field transforms under the $SO(1,3)$ transformation like a classical geometrical object in the Minkowski space-time, $$\varphi_{a}(x) \to \bar{\varphi}_{a}(x) = D_{a}{}^{c}(\Lambda) \ \varphi_{c}(\Lambda^{-1}x) ,$$ on the other as an operator-valued distribution in the Hilbert space, $$\varphi_{a}(x) \to \bar{\varphi}_{a}(x) = U^{\dagger}(\Lambda) \varphi_{a}(x) U(\Lambda) .$$ Also, the generators $\Sigma^{\mu\nu}$ of the $D(\Lambda)$ transformation are given by constant numerical (spin) matrices (i.e., not a tensor). To see that consider the transformation law in the case of a vector field $A^{\mu}(x)$, $$\bar{A}^{\mu}(x) = D^{\mu}{}_{\nu}(\Lambda) \ A^{\nu}(\Lambda^{-1}x) = \Lambda^{\mu}{}_{\nu} A^{\nu}(\Lambda^{-1}x) .$$ This tells you that for a vector field, the finite-dimensional representation $D(\Lambda)$ is given by the Lorentz matrix $\Lambda$: $$D^{\mu}{}_{\nu}(\Lambda) = \Lambda^{\mu}{}_{\nu} .$$ To first order in the parameters, this becomes $$\delta^{\mu}_{\nu} + \frac{1}{2} \omega_{\rho\sigma} \left( \Sigma^{\rho\sigma} \right)^{\mu}{}_{\nu} = \delta^{\mu}_{\nu} + \omega^{\mu}{}_{\nu} .$$ This leads to $$\frac{1}{2} \omega_{\rho\sigma} \left( \Sigma^{\rho\sigma} \right)^{\mu}{}_{\nu} = \omega^{\mu}{}_{\nu} ,$$ which can be rewritten as $$\frac{1}{2} \omega_{\rho\sigma} \left( \Sigma^{\rho\sigma} \right)^{\mu}{}_{\nu} = \eta^{\mu\rho} \delta^{\sigma}_{\nu} \omega_{\rho\sigma} = \frac{1}{2} \omega_{\rho\sigma} \left( \eta^{\mu\rho} \ \delta^{\sigma}_{\nu} - \eta^{\mu\sigma} \ \delta^{\rho}_{\nu} \right) .$$ Thus
$$\left( \Sigma^{\rho\sigma} \right)^{\mu}{}_{\nu} = \eta^{\mu\rho} \ \delta^{\sigma}_{\nu} - \eta^{\mu\sigma} \ \delta^{\rho}_{\nu} .$$ This tells you that the generator $\Sigma^{\rho\sigma}$ represents a set of six antisymmetric, $4 \times 4$ constant matrices with matrix elements given by $(\Sigma^{\rho\sigma})^{\mu}{}_{\nu}$. Of course, you may regard the object $\Sigma^{\rho\sigma\mu}{}_{\nu}$ as a rank-4 invariant tensor, but this does not mean that $\Sigma^{\rho\sigma}$ itself is a rank-2 tensor.
However, Hermitian generator $J^{\mu\nu}$ of the $U(\Lambda)$ transformation is a rank-2 Lorentz tensor operator. Indeed, under Lorentz transformation, it is easy to show that $$J^{\mu\nu} \to U^{\dagger}J^{\mu\nu}U = \Lambda^{\mu}{}_{\rho} \ \Lambda^{\nu}{}_{\sigma} \ J^{\rho \sigma} ,$$ which is the correct transformation law for rank-2 tensor operator.