tisdag 31 maj 2016

Samtal med Kodcentrum om Matematik-IT

Jag har idag haft ett konstruktivt och mycket trevligt samtal med Jessica Berglund och Lisa Söderlund på Kodcentrum om eventuellt samarbete vad gäller att sprida programmeringens evangelium till svenska elever och svensk skola.

Kodcentrum har hittills satsat på Scratch som introduktion till programmering och verkar ha behov av att kunna leverera vidareutbildning,  och kanske Matematik-IT därvid kan vara ett alternativ. Vi får se om Kodcentrum vill utnyttja denna möjlighet under nästa läsår. Chansen finns...

Vad gäller programmeringsplattform har jag alltså använt Codea (programmering på iPad för iPad), men det finns många alternativ, tex Corona för PC/Mac som använder samma språk som Codea (Lua). Codea kostar några kronor, medan Corona är gratis.

Sen finns det ju många andra möjligheter som Xcode, Swift, Python, JavaScript, Perl....Till slut måste man välja något specifikt språk, om man vill göra något konkret eller säga något med någon mening... på samma sätt som med kärleken, som är ju evig och det bara är föremålen som växlar...

måndag 30 maj 2016

New Theory of Flight: Time Line

Potential flow around circular cylinder with zero drag and lift (left). 
Real flow with non-stationary turbulent 3d rotational slip separation and non-zero drag (right). 

The New Theory of Flight published in J Mathematical Fluid Mechanics, can put be into the following time line:

1750 formulation by Euler of the Euler equations describing incompressible flow with vanishing viscosity expressing Newton's 2nd law and incompressibility in Euler coordinates of a fixed Euclidean coordinate system.

1752 d'Alembert's Paradox as zero drag and lift of potential flow around a wing defined as stationary flow which is
  1. incompressible
  2. irrotational
  3. of vanishing viscosity
  4. satisfies slip boundary condition 
as exact solution of the Euler equations.

1904 resolution of d'Alembert's paradox of zero drag by Prandtl stating that potential flow is unphysical, because 4. violates a requirement that real flow must satisfy
  • no slip boundary condition.
1904 resolution of d'Alembert's paradox of zero lift by Kutta-Zhukovsky stating that potential flow is unphysical, because 2. violates that a sharp trailing edge in real flow creates 
  • rotational flow.
2008 resolution of d'Alembert's paradox of zero drag and lift by Hoffman-Johnson stating that potential flow is unphysical, because 
  • potential flow it is unstable at separation and develops into non-stationary turbulent 3d rotational slip separation as a viscosity solution of the Euler equations with substantial drag and lift.  
Recall that d'Alembert's paradox had to be resolved, in one way or the other, to save theoretical fluid mechanics from complete collapse, when the Wright brothers managed to get their Flyer off ground into sustained flight in 1903 with a 10 hp engine. 

Prandtl, named the Father of Modern Fluid Mechanics, discriminated the potential solution by an ad hoc postulate that 4. was unphysical (without touching 2.) and obtained drag without lift.

Kutta-Zhukovsky, named Fathers of Modern Aero Dynamics, discriminated the potential solution by an ad hoc postulate that 2. was unphysical (without touching 4.) and obtained lift without drag. 

Hoffman-Johnson showed without ad hoc postulate that the potential solution is unstable at separation and develops into non-stationary turbulent 3d rotational slip separation causing drag and lift. 

The length of the time-line 1750-1752-1904-2008 is remarkable from scientific point of view. Little happened between 1752 and 1904 and between 1904 and 2008, and what happened in 1904 was not in touch with reality. For detailed information, see The Secret of Flight.

1946 Nobel Laureate Hinshelwood made the following devastating analysis:
  • D’Alembert’s paradox separated fluid mechanics from its start into theoretical fluid mechanics explaining phenomena which cannot be observed and practical fluid mechanics or hydraulics observing phenomena which cannot be explained.
The only glimpse in the darkness was offered by the mathematician Garret Birkhoff in his 1950 book Hydrodynamics, by asking if any potential flow is stable, a glimpse of light that was directly blown out by a devastating critique of the book from fluid dynamics community, which made Birkhoff remove his question in the 2nd edition of the book and to never return to hydrodynamics.

The 2008 resolution of d'Alembert's Paradox leading into the New Theory of Flight by Hoffman-Johnson,  has been met with complete silence/full oppression by the fluid mechanics community still operating under the paradigm of Hinshelwoods analysis. 

söndag 29 maj 2016

Restart of Quantum Mechanics: From Observable/Measurable to Computable

                Schrödinger and Heisenberg receiving the Nobel Prize in Physics in 1933/32..

If modern physics was to start today instead of as it did 100 years ago with the development of quantum mechanics as atomistic mechanics by Bohr-Heisenberg and Schrödinger, what would be the difference?

Bohr-Heisenberg were obsessed with the question:
  • What can be observed?
motivated by Bohr's Law:
  • We are allowed to speak only about what can be observed.
Today, with the computer to the service of atom physics, a better question may be:
  • What can be computed?
possibly based on an idea that
  • It may be meaningful to speak about what can be computed. 
Schrödinger as the inventor of the Schrödinger equation as the basic mathematical model of quantum mechanics, never accepted the Bohr-Heisenberg Copenhagen Interpretation of quantum mechanics with the Schrödinger wave function as solution of the Schrödinger equation interpreted as a probability of particle configuration, with collapse of the wave function into actual particle configuration under observation/measurement. 

Schrödinger sought an interpretation of the wave function as a physical wave in a classical continuum mechanical meaning, but had to give in to Bohr-Heisenberg, because the multi-dimensionality of the Schrödinger equation did not allow a direct physical interpretation, only a probabilistic particle interpretation. Thus the Schrödinger equation to Schrödinger became a monster out of control, as expressed in the following famous quote: 
  • If we have to go on with these damned quantum jumps, then I'm sorry that I ever got involved.
And Schrödinger's equation is a monster also from computational point of view, because solution work scales severely exponentially with the number of electrons and thus is beyond reach even for small $N$.

But the Schrödinger equation is an ad hoc model with only weak formal unphysical rationale, including the basic ingredients of (i) linearity and (ii) multi-dimensionality.

Copenhagen quantum mechanics is thus based on a Schrödinger equation, which is an ad hoc model and which cannot be solved with any assessment of accuracy because of its multi-dimensionality and thus cannot really deliver predictions which can be tested vs observations, except in very simple cases.

The Copenhagen dogma is then that predictions of the standard Schrödinger equation always are in perfect agreement with observation, but a dogma which cannot be challenged because predictions cannot be computed ab initio.

In this situation it is natural to ask, in the spirit of Schrödinger, for a new Schrödinger equation which has a direct physical meaning and to which solutions can be computed ab initio, and this is what I have been exploring in many blog posts and in the book (draft) Many-Minds Quantum Mechanics.

The basic idea is to replace the linear multi-d standard Schrödinger equation with a computable non-linear system in 3d as a basis of a new form of physical quantum mechanics. I will return with more evidence of the functionality of this approach, which is very promising...

Note that a wonderful thing with computation is that it can be viewed as a form of non-destructive testing, where the evolution of a physical system can be followed in full minute detail without any form of interference from an observer, thus making Bohr's Law into a meaningless limitation of scientific thinking and work from a pre-computer era preventing progress today.

PS It is maybe wise to be a little skeptical to assessments of agreement between theory and experiments to an extremely high precision. It may be that things are arranged or rigged so as to give exact agreement, by changing computation/theory or experiment.

lördag 28 maj 2016

Aristotle's Logical Fallacy of Affirming the Consequent in Physics

One can find many examples in physics, both classical and modern, of Aristotle's logical fallacy of Affirming the Consequent (confirming an assumption by observing a consequence of the assumption):
  1. Assume the Earth rests on 4 turtles, which keeps the Earth from "falling down". Observe that the Earth does not "fall down". Conclude that the Earth rests on 4 turtles.
  2. Observe a photoelectric effect in accordance with a simple (in Einstein's terminology "heuristic") argument assuming light can be thought of as a stream of particles named "photons" . Conclude that light is a stream of particles named photons. 
  3. Assume light is affected by gravitation according the general theory of relativity as described by Einstein's equations. Observe apparent slight bending of light as it passes near the Sun in accordance with an extremely simplified use of Einstein's equations. Conclude universal validity of Einstein's equations.
  4. Observe lift of a wing profile in accordance with a prediction from potential flow modified by large scale circulation around the wing. Conclude that there is large scale circulation around the wing. 
  5. Assume that predictions from solving Schrödinger's equation always are in perfect agreement with observation. Observe good agreement in some special cases for which the Schrödinger equation happens to be solvable, like in the case of Hydrogen with one electron. Conclude universal validity of Schrödinger's equation, in particular for atoms with many electrons for which solutions cannot be computed with assessment of accuracy.
  6. Assume there was a Big Bang and observe a distribution of galaxy positions/velocities, which is very very roughly in accordance with the assumption of a Big Bang. Conclude that there was a Big Bang.
  7. Assume that doubled CO2 in the atmosphere from burning of fossil fuel will cause catastrophic global warming of 2.5 - 6 C. Observe global warming of 1 C since 1870. Conclude that doubled CO2 in the atmosphere from burning of fossil fuel will cause catastrophic global warming of 4 - 8 C.
  8. Assume that two massive black holes merged about 1.3 billion years ago and thereby sent a shudder through the universe as ripples in the fabric of space and time called gravitational waves and five months ago washed past Earth and stretched space making the entire Earth expand and contract by 1/100,000 of a nanometer, about the width of an atomic nucleus. Observe a wiggle of an atom in an instrument and conclude that two massive black holes merged about 1.3 billion years ago which sent a shudder through the universe as ripples in the fabric of space and time called gravitational waves...
  9. Observe experimental agreement of the anomalous magnetic dipole moment of the electron within 10 decimals to a prediction by Quantum Electro Dynamics (QED). Conclude that QED is universally valid for any number of electrons as the most accurate theory of physics. Note that the extremely high accuracy for the specific case of the anomalous magnetic dipole moment of the electron, compensates for the impossibility of testing in more general cases,  because the equations of QED are even more impossible to solve with assessment of accuracy than Schrödinger's equation.
The logic fallacy is so widely practiced that for many it may be difficult to see the arguments as fallacies. Test yourself!

PS1. Observe that if a theoretical prediction agrees with observation to a very high precision, as is the case concerning the Equivalence Principle stating equality of inertial and gravitational (heavy) mass, then it is possible that what you are testing experimentally in fact is the validity of a definition, like testing experimentally if there are 100 centimeters on a meter (which would be absurd).

PS2 Books on quantum mechanics usually claim the there is no experiment showing any discrepancy whatsoever with solutions of the Schrödinger equation (in the proper setting), which is strong evidence that the Schrödinger equation gives an exact  description of all of atom physics (in a proper setting). The credibility of this argument is weakened by the fact that solutions can be computed only in very simple cases. 

fredag 27 maj 2016

Emergence by Smart Integration of Physical Law as Differential Equation

Perfect Harmony of European Parliament: Level curves of political potential generated by an empty spot in the middle.

This is a continuation of previous posts on a new view of Newton's law of gravitation. We here connect to the Fundamental Theorem of Calculus of the previous post, allowing a bypass to compute an integral by tedious laborious summation, using a primitive function of the integrand:
  • $\int_0^t f(s)ds = F(t) - F(0)$  if  $\frac{dF}{dt} = f$.
This magical trick of Calculus of computing an integral as a sum without doing the summation,  is commonly viewed to have triggered the scientific revolution shaping the modern world.

The magic is here computing an integral $\int_0^t f(s)ds$ in a smart way, rather than computing a derivative $\frac{dF}{dt}$ in a standard way.

The need of computing integrals comes from the fact that physical laws are usually expressed in terms of derivatives, for example as an initial value problem: Given a function $f(t)$, determine a function $F(t)$ such that
  • $DF(t) = f(t)$ for $t\ge 0$ and $F(0) = 0$,
where $DF =\frac{dF}{dt}$ is the derivative of $F$. In other words, given a function $f(t)$, determine a primitive function $F(t)$ to $f(t)$ with $F(0)=0$, that is, determine/compute the integral
by the formula
  • $\int_0^t f(s)ds = F(t)$ for $t\ge 0$. 
Using the Fundamental Theorem to compute the integral would then correspond to solving the initial value problem by simply picking a primitive function $F(t)$ satisfying $DF = f$ and $F(0)=0$ from a catalog of primitive functions, allowing to in one leap jump from $t=0$ to any later time $t$. Not very magical perhaps, but certainly smart!

The basic initial value problem of mechanics is expressed in Newton's 2nd Law $f=ma$ where $f$ is force, $m$ mass and $a(t)=\frac{dv}{dt}=\frac{d^2x}{dt^2}$ is acceleration, $v(t)=\frac{dx}{dt}$ velocity and $x(t)$ position, that is,
  • $f(t) = m \frac{d^2x}{dt^2}$.           (1)
Note that in the formulation of the 2nd Law, it is natural to view position $x(t)$ with acceleration $\frac{d^2x}{dt^2}$ as given, from which force $f(t)$ is derived by (1) . Why? Because position $x(t)$ and acceleration $\frac{d^2x}{dt^2}$ can be observed, from which the presence of force $f(t)$ can be inferred or derived or concluded, while direct observation of force may not really be possible. In this setting the 2nd Law acts simply to define force in terms of mass and acceleration, rather than to make a connection with some other definition of force.

Writing Newton's 2nd law in the form $f=ma$, thus defining force in terms of mass and acceleration, is the same as writing Newton's Law of Gravitation:
  • $\rho = \Delta\phi$,                          (2)
thereby defining mass density $\rho (x)$ in terms of gravitational potential $\phi (x)$ by a differential equation. 

With this view, Newton's both laws (1) and (2) would have the same form as differential equation, and the solutions $x(t)$ and $\phi (x)$ would result from solving differential equations by integration or summation as a form of emergence. 

In particular, this reasoning gives support to an idea of viewing the physics of Newton's Law of Gravitation to express that mass density somehow is "produced from" gravitational potential by the differential equation $\rho =\Delta\phi$. 

To solve the differential equation $\Delta\phi =\rho$ by direct integration or summation in the form
  • $\phi (x) = \frac{1}{4\pi}\int\frac{\rho (y)}{\vert x-y\vert}dy$,
would then in physical terms require instant action at distance, which is difficult to explain. 

On the other hand, if there was a "smart" way of doing the integration by using some form of Fundamental Theorem of Calculus as above, for example by having a catalog of potentials from which to choose a potential satisfying $\Delta\phi =\rho$ for any given $\rho$, then maybe the requirement of instant action at distance could be avoided.

A smart way of solving $\Delta\phi =\rho$ would be to use the knowledge of the solution $\phi (x)$ in the case of a unit point mass at $x=0$ as
  • $\phi (x)=\frac{1}{4\pi}\frac{1}{\vert x\vert}$ 
which gives Newton's inverse square law for the force $\nabla\phi$, which is smart in case $\rho$ is a sum of not too many point masses. But the physics would still seem to involve instant action at distance.

In any case, from the analogy with the 2nd Law we have gathered an argument supporting an idea to view the physics of gravitation as being expressed by the differential equation $\rho =\Delta\phi$ with mass density $\rho$ derived from gravitational potential $\phi$. Rather than the opposite standard view with the potential $\phi$ resulting from mass density $\rho$ by integration or summation corresponding to instant action at distance.

The differential equation $\Delta\phi =\rho$ would thus be valid by an interplay "in perfect harmony" in the spirit of Leibniz, where on the one hand "gravitational potential tells matter where to be how to move" and "matter tells gravitational potential what to be".

This would be like a Perfect Parliamentary System where the "Parliament tells People where to be and what to do" and "People tells Parliament what to be".

PS There is a fundamental difference between (1) and (2): (1) is an initial value problem in time while (2) is a formally a static problem in space. It is natural to solve an initial value problem by time stepping which represents integration by summation. A static problem like (2) can be solved iteratively by some form of (pseudo) time stepping towards a stationary solution, which in physical terms could correspond to successive propagation of effects with finite speed of propagation.

torsdag 26 maj 2016

Fatal Attraction of Fundamental Theorem of Calculus?

Calculus books proudly present the Fundamental Theorem of Calculus as the trick of computing an integral
  • I=$\int_a^b f(x)dx$,
not by tedious summation of little pieces as a Riemann sum
  • $\sum_i f(x_i)h_i$
on a partition $\{x_i\}$ of the interval $(a,b)$ with step size $h_i = x_{i+1} - x_i$, but by the formula
  • $I = F(b) - F(a)$, 
where $F(x)$ is a primitive function to $f(x)$ satisfying $\frac{dF}{dx} = f$,

The trick is thus to compute an integral, which by construction is a sum of very many terms, not by doing the summation following the construction, but instead taking just one big leap using a primitive function.

On the other hand, to compute a derivative no trick is needed according to the book; you just compute the derivative using simple rules and a catalog of already computed derivatives.

In a world of analytical mathematics, computing integrals is thus valued higher than computing derivatives, and this is therefore what fills Calculus books.

In a world of computational mathematics, the roles are switched. To compute an integral as a sum can be viewed to be computationally trivial, while computing a derivative $\frac{dF}{dx}$ is a bit more tricky because it involves dividing increments $dF$ by small increments $dx$.

This connects to Poisson's equation $\Delta\phi =\rho$ of Newton's theory of gravitation discussed in recent posts. What is here to be viewed as given and what is derived? The standard view is that the mass density $\rho$ is given and the gravitational potential $\phi$ is derived from $\rho$ as an integral
  • $\phi (x) = \frac{1}{4\pi}\int\frac{\rho (y)}{\vert x-y\vert}dy$,
seemingly by instant action at distance. 

In alternative Newtonian gravitation, as discussed in recent posts, we view instead $\phi$ as primordial and $\rho =\Delta\phi$ as being derived by differentiation, with the advantage of requiring only local action.

We thus have two opposing views:
  • putting together = integration requiring (instant) action at distance with dull tool.
  • splitting apart = differentiation involving local action with sharp tool. 
It is not clear what to prefer?

Connection between Neo-Newtonian and Einsteinian Gravitational Theory

                                                 Hen laying eggs by local action. 

If you are a strong supporter of Einstein's general theory of relativity, like almost all modern physicists, then maybe you would be open to see the following connection with the Neo-Newtonian
theory I have been exploring in recent posts, with the gravitational potential $\phi$ viewed as primordial and matter density $\rho =\Delta\phi$ as derived by local action in space of the Laplacian $\Delta$ and with the gravitational potential playing the same role as the "space-time curvature" of Einstein:
  • space-time curvature tells matter to move along geodesics
  • gravitational potential tells matter to move according to Newton's 2nd Law
  • space-time curvature connected to matter by Einstein's equation
  • gravitational potential connected to matter by Poisson's/Newton's equation.
This connects to the iconic summary of general relativity by John Archibald Wheeler:
  • Spacetime tells matter how to move; matter tells spacetime how to curve,
where the "telling" goes both ways. 

But maybe it is enough that the "telling" only goes one way, maybe it suffices that the gravitational potential tells where matter will be and how it is to move. After all, it is only the equality $\rho =\Delta\phi$ that counts and thus has to be established is some way, and then possibly through one-way communication from $\phi$ to $\rho$ in some form of local action. 

Maybe it is enough to understand/explain how a hen can lay an egg by local action in a poultry yard, and leave out the much more difficult problem of how a hen can come out of an egg by global action outside the poultry yard.

onsdag 25 maj 2016

Newton's Genius and New View on Gravitation

Newton computed the gravitational attraction of a planet as a spherically symmetric distribution of matter, to be equal to that of a point mass of the same total mass at the center of the planet, away from the planet. 

This made it possible for Newton to model gravitational interaction of planets as gravitational interaction of point masses, as a much simpler problem from computational point of view. 

Newton thus could simplify the computationally impossible problem of instant gravitational interaction at distance of all the individual atoms of one planet with all the individual atoms of another planet, to the interaction between two point masses of the same total mass. Genial and absolutely necessary to make the theory useful and thereby credible.

But let us reflect a bit about the physics of instant individual interaction at distance of each atom of one planet with each atom of another planet, which we agreed is computationally impossible. We now ask if it is physically possible?

Is it thinkable that each atom can instantly at distance exchange details about position and mass with all others atoms by using some form of world wide web? Think about it! 

The answer can only be NO. Atoms cannot have access to such technology. It is unthinkable.

The result is that we have to view gravitation in a different way, not as individual instant attraction between small pieces of matter at distance, and then why not in the other way as suggested in recent posts: 

What is primordial is then a gravitational potential $\phi$ with associated gravitational force $\nabla\phi$, to which matter density $\rho$ is connected by $\rho =\Delta\phi$ through the local operation in space of the Laplacian $\Delta$.

With this view there is no instant action at distance between atoms to explain, but instead local production of matter without demand of atomistic resolution into pieces, which at least is thinkable.

It would be interesting to listen to Newton's reaction to this idea.

PS Business Insider reports that:
  • Earth's core is 2.5 years younger than its crust due to some eerie physics
the eerie physics being Einstein's general theory of relativity claiming that clocks slow down with increasing gravitation. Yes, maybe your feet are a bit younger than your head...

The Value of Compulsory (Climate) Science?

                                  Sunset over fossil free state without CO2 polluting people and welfare.

Tim Ball asks for Compulsory Courses for Any Curriculum; The Science Dilemma:
  • Science is pervasive directly and indirectly in every phase of modern life. 
  • This knowledge must be a fundamental part of any school curriculum. 
Tim suggests that compulsory science could save people from a meaningless ban on fossil fuel: 
  • Climate skeptics struggle with getting the majority of people to understand the problems with the UN Intergovernmental Panel on Climate Change’s (IPCC) anthropogenic global warming (AGW) story. The problem is much wider because it relates to the lack of scientific abilities among a majority of the population.
Tim is not happy with the present situation:
  • I was involved in many curricula fights, few of them ever resolved much.
  •  Ever subject area and discipline considered theirs essential to an education. They failed in achieving curricula useful to the student and society. 
  • This was because they were controlled by people ensuring what interested them or what ensured their job, rather than what the student needed to become an effective informed citizen.
  • Students are not given the tools to avoid being exploited. Indeed, sometimes I think the system keeps them ignorant so it can exploit them as adults. 
  • Peoples of the Rainforest teach their children what they need to survive in the real and dangerous world in which they live.
  • We don’t do this at any level. For most North American university or college students the experience is a socially acceptable and ridiculously expensive form of unemployment. Most of them learn more about life and themselves in part-time and summer jobs. 
I agree with Tim about the incitaments behind curricula, but I am not sure compulsory science would be beneficial. The trouble with anthropogenic global warming is that it is massively backed by scientists and academic institutions, and compulsory science could just mean more backing of fake science.

The real Science Dilemma is maybe rather to distinguish real science from fake science.

PS Our new social democratic Minister of Climate and Vice Prime Minister Isabella Lövin today proudly announces that

tisdag 24 maj 2016

Wallström: Förmån (utan Tjänstesamband) Ingen Muta

Chefsåklagare Alf Johansson vid riksenheten mot korruption meddelar till TT:
  • Det begicks inget mutbrott när utrikesminister Margot Wallström fick en lägenhet av Kommunal. Åklagaren lägger ner förundersökningen.
  • Det är också uppenbart att Wallström har mottagit en förmån, anser Johansson, men det kan alltså inte bevisas att hon har fått den just i egenskap av minister.
  • Sammanfattningsvis har jag inte kunnat finna att det föreligger ett så kallat tjänstesamband, det vill säga en förmån i form av ett hyreskontrakt som har lämnats med anledning av statsrådets uppdrag.
Lawline gjorde an analys när lägenhetsaffären uppdagades i januari med titeln MARGOT WALLSTRÖM UPPFYLLER REKVISITEN FÖR TAGANDE AV MUTA, med bl följande motivering:
  • Av kontraktet parterna emellan framgår att hyresförhållandet gäller så länge Margot Wallström har sin nuvarande anställning.
Chefåklagaren anser alltså att Wallström tagit emot förmån, men anser, i synbar strid med kontraktet, att denna förmån inte har haft samband med Wallströms anställning som minister. 

Eller kanske har tjänsten haft visst samband, men jämfört med det rena vänskapssamband som Johansson vet måste ha funnits (ja, vad annat skulle det ha kunna vara?) mellan Wallström och Kommunal (dock utan att ha hört Wallström), måste väl tjänstesambandet anses som försumbart?
Så klart att Wallström säger "Jag älskar Kommunal" och så klart att kärleken är besvarad!

Under alla förhållanden är Sverige det land i världen som har minst korruption, om man inte räknar vänskapskorruption förstås, och det är ju inte detsamma som den riktiga rejäla korruption som florerar i alla andra länder...

Det rimliga är väl nu att Wallström åter flyttar in i Kommunals lägenhet, eftersom som Wallström säger "inget fel är begånget". Det är ju trots allt en ganska trevlig lägenhet, stor, möblerad, central, låg hyra...

Se också tidigare post.

Jämför också med SVT: Wallströms lägenhet direkt kopplade till hennes politiska uppdrag.