Chalmers Världsledande i Kampen mot KlimatFörnekelse: Avancez!

Chalmers Tekniska Högskola presenterar sig för världen med följande huvudbudskap: Klimatforskningen – vårt största bidrag:

• Vår forskning är vårt främsta och absolut viktigaste bidrag till klimatomställningen.
• På Chalmers bedrivs excellent forskning för klimatet från nya energikällor och tekniska innovationer till politiska system och mänskligt beteende.

Med forskningen inom politiska system och mänskligt beteende tar Chalmers ett nytt viktigt steg mot ett allomfattande universitet med världsledande roll i kampen mot olika former av (klimat)förnekelse:

• Chalmers har etablerat världens första, globala forskarnätverk om klimatförnekelse 
• som studerar hur högernationalismens framfart i Europa bidragit till att klimatförnekelse ökat.
• Martin Hultman har gjort sig känd internationellt för att ha visat på kopplingen mellan konservatism, främlingsfientlighet och klimatförnekelse​. 
• Han skiljer på organiserad och partipolitisk klimatförnekelse, men också responsförnekelse bland politiker och den vardagsförnekelse som människors dagliga agerande är uttryck för.

Många gamla Chalmerister förskräckes  över detta uttryck för Chalmers motto Avancez:  bl a Lars Bern och Elsa Widding:  Lyssna och begrunda. 

PS Som ett uttryck för den så berömda Göteborgsandan har GöteborgsPosten rankat Martin Hultman som mäktigast i Göteborgs universitetsvärld, kanske snart även globalt: Avancez!

DigiMat and the Multiplication Table

DigiMat is new reformed mathematics education. DigiMat starts by constructing the natural numbers (1, 2, 3, 4,.., in decimal notation) by repetition of the basic operation +1, starting from 0:

• 1 = 0 + 1
• 1+1 (= 2 )
• 1+1+1 (= 3 = 2+1)
• 1+1+1+ 1 (= 4 = 3+1)
• 1+1+1+1+1 (= 5 = 4+1)
• and so on

DigiMat starts with a binary representation instead of the usual decimal representation, because it is more basic and simpler (and therefore preferred by the computer): Instead of the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 of the decimal system, the binary system has only two digits 0 and 1, which when used in a position system gives the following representation 

• 1 = 1 (decimal)
• 10 = 2 (decimal)
• 11 = 3
• 100 = 4 
• 101 = 5
• 110 = 6
• 111 = 7
• 1000 = 8
• 1001 = 9
• ... 

In DigiMat for preschool different binary systems can quickly be mastered such as for example 

• one red (apple) is 1
• two red (apples) is one yellow (apple)
• two yellow is one green
• two green is one blue
• two blue is one violet
• ... 

with the following representation 

• 1 =  red (apple)
• 2 = yellow (apple)
• 3 = yellow and red (apples)
• 4 = green (apple)
• 5 = green and red (apples)
• 6 = green + yellow (apples)
• 7 =  green + yellow + red (apples)
• 8 = blue (apple)
• 9 = blue and red (apples)
• ... 

Alternatively, a one-bead abacus (column of beads with bead left = 0 and bead right = 1) is quickly mastered.  See Basics on DigiMat School.

With such binary representations any preschool child quickly learns to (without effort) perform the basic operations of addition, multiplication (as repeated addition) and subtraction of natural numbers, followed by division.  This is carefully described on the DigiMat School site.

This is to be compared with standard school mathematics, where the decimal multiplication table is both corner stone/top catch and greatest hurdle. The discussion on the necessity of learning the multiplication table by heart is filling endless texts on teaching school mathematics, with no consensus after 100s of years. Some experts claim that knowing by heart that 7 times 8 is 56, is immensely useful, while others claim this blocks true understanding, which is better expressed in a computation of the form 

• 7 x 8 = (10 - 3) x (10 -2) = 10 x 10 - 3 x 10 - 10 x 2 + 2 x 3 = 100 - 30 - 20 + 2 x 3 =  50 + 2 x (2 + 1) = 50 + 2 x 2 + 2 x 1 = 50 + 2 x (1+1) + 1 + 1 = 50 + 2 x 1 + 2 x 1 + 1 +1 = 50 + 1 + 1 + 1 + 1 + 1 +1 = 56  

requiring familiarity and understanding of in particular the distributive law. Here is a recent article presenting this conundrum with the shocking title:

• The Danger of Memorising the Multiplication Table (in Swedish) 

with references to authorities such as Arthur Baroody, Jo Boadler and Gina Kling.  

DigiMat lifts school mathematics out of this paralysis. Recall that the binary multiplication table is so simple, that it is not necessary to memorise: 0 x 0 = 0, 0 x 1 = 0, 1 x 0 = 0, 1 x 1 = 1.

Nor the (slightly more complex) addition table: 0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, 1 + 1 = 10.

DigiMat meets the World

DigiMat is new mathematics education for the digital society consisting of  

• Ada's World (app for pree-school introducing basic concepts for all ages)
• DigiMat School (from elementary school to university)
• DigiMat Pro (top academic and professional level)

DigiMat is now being launched on a broad international scale for school, university and continued education. Take a look and check out what DigiMat can offer to you, as student, teacher or professional. 

 

The Pressure Equation of Incompressible Flow

The standard mathematical model for incompressible flow of unit density with vanishing viscosity takes the form of the incompressible Euler equations:

• $\frac{\partial u}{\partial t}+u\cdot\nabla u + \nabla p = 0$,
• $\nabla\cdot u=0$,

where $u(x,t)$ is flow velocity, $p(x,t)$ is pressure and $(x,t)$ are space-time coordinates. 

The first equation expresses Newton's 2nd Law (conservation of momentum), while the second equation is not a physical law but a side condition as a stipulation that the velocity is to be divergence free. 

When solving the incompressible Euler equations computationally the second equations is typically replaced by an elliptic equation for the pressure of the form 

• $-\delta*\Delta p = - \nabla\cdot u$,      (1)
• where $\delta \approx h$ with $h$ mesh size.

In a variational setting this equation results from regularisation of the pressure as Lagrange multipler for the side condition $\nabla\cdot u=0$, which can be seen as a form of relaxation. The choice of $\delta =h$ brings $\delta\Delta$ in parity with $\nabla\cdot$. The equation (1) can be seen as a physical law connecting pressure to the divergence of velocity. 

Let us now give more perspective on the elliptic pressure equation by extending to compressible ideal flow where the equation for the internal energy $e$

• $\frac{\partial e}{\partial t}+\nabla\cdot (eu)+p\nabla\cdot u = 0$,

can be seen as an equation (expressing a law of physics) for the pressure $p=\gamma e$ with $\gamma >0$ a gas constant:

• $\frac{Dp}{Dt}=-\gamma p\nabla\cdot u$    (2)
• with $\frac{Dp}{Dt}=\frac{\partial p}{\partial t}+\nabla\cdot (pu)$,

With suitable normalisation the Mach number is $\sqrt{\gamma}$ with a switch from supersonic over transonic to subsonic flow as $\gamma$ increases beyond 1. 

This equation suggests a parabolic variant of the incompressible pressure equation of the form

• $\frac{\partial p}{\partial t}-\delta*\Delta p = - \nabla\cdot u$  (3)

where the right hand side of the compressible pressure equation $-\gamma p\nabla\cdot u$ is replaced by $-\nabla\cdot u$ for the incompressible case (and (2) is correspondingly regularised). Again, (3) can be seen as a law of physics connecting pressure to the divergence of velocity. It allows direct time stepping of the flow equations as an expression of physicality with finite speed of propagation/sound , compared with the elliptic equation (1) with infinite speed of pressure variation/sound.  

Choosing $\gamma$ small gives compressible flow with large variation of $\nabla\cdot u$ corresponding to moderate variation of density, while choosing $\gamma$ large will keep the variation in density small. 

We can thus view the compressible solver as a unified flow solver for both compressible and incompressible flow depending on the choice of the single parameter $\gamma$, with a natural connection to standard incompressible solvers. 

The unified flow solver can be explored as item 98 in the Model Workshop of DigiMat BodySoul including comparison a with standard incompressible solver.

Remark. Integrating (2) along streamlines gives

•  $p \sim \exp(-\gamma\int \nabla\cdot u\,dt)$ 

which expresses an exponential connection between the pressure $p$ and $-\gamma\nabla\cdot u$ compatible with $\nabla\cdot u$ being small for $\gamma >>1$. 

  

Philosophy of Science

 

To say that something is complicated can be easy. To say that something is easy may be complicated.

What is easy can be understood. Science means understanding, and so the objective of science is to make things easy. 

This connects to the observation (Arthus Clarke's 1st Law) that if someone (an elderly scientist) says that something is impossible (no solution possible), it is usually wrong, while if someone says that something is possible (there is a solution), then it is usually right. 

Can a Coordinate Transformation Contain Physics?

The Special Theory of Relativity SR is identical to the Lorentz coordinate transformation. The General Theory of Relativity involves as an essential step transformations of space-time coordinates. 

Many-Minds Relativity shows, in a careful mathematical analysis, that SR is empty as a theory about real physics. The argument is in brief that since SR is identical to the Lorentz transformation, and a coordinate transformation in itself does not contain any physics, SR is empty of physics. 

Why does a coordinate transformation not contain any physics? Consider for example the coordinate transformation of a pure translation from an $x$-axis to an $x^\prime$-axis, as 1d Euclidean coordinate systems, of the form $x^\prime = x - v$, with $v$ a constant.  One property of such a transformation is that distances are conserved, or are invariant, under transformation: If $x^\prime = x - v$ and $y^\prime = y - v$, then $x^\prime - y^\prime = x - y$. So far no physics has been introduced and so the coordinate transformation cannot have any physical meaning. A natural way to give physical meaning is to say the coordinate $x$ represents the position of a physical body $X$ and $y$ the position of a body $Y$. The positions of $X$ and $Y$ can then alternatively be expressed in terms of $x^\prime$ and $y^\prime$. 

Now, can we say that the transformation contains any physics of the position of the bodies? Can we say that since the transformation expresses that distances are conserved under the transformation, we can conclude a law physics expressing that translation of a rigid body does not change the shape of the body? Is this a law of physics, which may be false, or simply a law of logic or language and not physics which cannot be false? I guess, you we would agree with me that it is not a true law of physics. That a rigid body does not change shape is built in into the meaning of a rigid body, and has nothing to do with physics with the shape of a body determined by physical forces. It cannot be used to say that in a rigid body there are forces which maintain its shape. It would be like pulling a rabbit out of an empty hat.  

Choice of coordinate systems in physics directly connects to the choice of units in physics. If you believe that the choice of units contains physics, that using a meter stick instead of yard stick changes the size of physical objects, then you are well prepared to accept the wonders of SR. If you understand that this is not the case, then you will (like me and many others) have a hard time with SR. 

We can add a common time coordinate $t$ to the $x$ and $x^\prime$ systems into a Galilean space-time transformation of the form 

• $x^\prime = x-vt$, $t^\prime =t$,

between a $(x,t)$-system and a $(x^\prime ,t^\prime)$, with the $x^\prime$-axis translating with respect to the $x$-axis with speed $v$, but still with the same lack of real physics.

The Lorentz transformation of SR is a similar linear coordinate transformation from a a space-time $(x,t)$-system to a space-time system $ (x^\prime ,t^\prime )$-system of the form 

• $x^\prime =\gamma (x-vt)$, $t^\prime = \gamma (t-vx)$, $\gamma =\frac{1}{\sqrt(1-v^2)}$,
• $x =\gamma (x^\prime+vt^\prime )$, $t = \gamma (t^\prime+vx^\prime )$.

where $\vert v\vert <1$, with now also the time variable beeing subject to transformation, and in particular the property that a $x = t$ is transformed into $x^\prime  = t^\prime$ and back.

In order for the Lorentz coordinate transformation to contain physics, the coordinates must be given some physical meaning. Einstein did this by associating space-time coordinates $(x,t)$ to events (of some unspecified physical nature) or rather to observations of events (of some unspecified physical nature), like a body $X$ observed to be at position $x$ at time $t$. Einstein associated the system $(x^\prime ,t^\prime )$ to represent coordinates with the $x^\prime$ axis translating with speed $v$, like in the Galilean transformation, but now with also the time variable being subject to transformation.

Einstein then created SR by drawing conclusions about physics from the Lorentz transformation, like the fact that $x=t$ is the same as $x^\prime =t^\prime$ interpreted by Einstein to mean that a light signal always propagates with speed 1 in any (inertial) coordinate system. Einstein then proceeded to derive properties of space contraction and time dilation from the Lorentz transformation and boldly claimed that he thereby had discovered new truely astonishing real physics (apparently from mysterious forces twisting space and time). 

Many early critics said that this was empty physics, for the reasons exposed above, but somehow Einstein managed to twist the brains of the physical community into believing that he had revealed deep new truths about real physics. This came to serve as footstep into modern physics, as weird empty physics based on transformations of coordinates brought to full bloom in the General Theory of Relativity in curved space-time.  

Galileo of course understood that his coordinate transformation does not contain any real physics. Why do modern physicists insist that the Lorentz transformation contains real physics?  Lorentz did not believe so, but he was quickly side-stepped by not being modern enough. 

Modern physics is in deep crisis. Why? Is it time now for post-modern physics?

The crisis is expressed in the obsession of modern physicists to focus efforts on a Theory of Everything a ToE, with the aim of explaining the basic four forces (electromagnetic, weak and strong nuclear and gravitation) as different manifestations of one basic force/interaction. No progress towards an answer has been made and the mystery of the origin of the four forces is as mysterious today as ever.   

DigiMat: Physical Gravitational Collapse

We continue our DigiMat study of star formation by gravitational collapse into a high density hot spot. 

 

In a fluid at rest without self gravitation a hot spot cannot develop without external forcing by transfer of energy from a cold surrounding. This is because without forcing, heat energy can only transfer from hot to cold, by the 2nd law of thermodynamics.  

We now ask if in a fluid with self gravitation a hot spot can develop by surrounding cooling. We use DigiMat to give an answer: We thus simulate the motion of a fluid with self gravitation starting with the fluid at rest with constant  density, temperature and internal energy, assuming periodic boundary conditions for fluid variables. We set the gravitational potential to zero at the boundary and thus create a gravitational force directed to the center of the domain. Here is the result at a certain time:

What do we see? We see (crosscut of) central accumulation of mass/density (blue) combined with central heating and far-away cooling given by temperature (green), with black line the initial distribution. We thus effectively see a transfer of heat energy from cold to hot. This transfer is accomplished by gravitation, which from an initial state with zero fluid velocity attracts the fluid towards the center thereby increasing kinetic energy there, which then is transfered into heat energy as pressure finally retards the flow (and reverses the flow into expansion). 

We thus find that the presence of a gravitational field can be viewed as a form of external forcing making possible transfer of heat energy from cold to warm. Gravitational collapse thus appears possible thermodynamically.   

Gravitational collapse with transfer of heat energy from a cold exterior to a hot interior fits with the observation of interstellar dust at low temperture, about 20 K. The price of shining stars is chilly space between stars, by conservation of total energy. 

     

DigiMat: Unphysical Gravitational Collapse

DigiMat offers a rich playground to explore the World. In DigiMat Model Workshop (70-74) you find a cosmological model in the form of fluid/thermodynamics with self gravitation. As an example consider the following simulation of the phenomenon of gravitational collapse:

What do we see? We see unphysical gravitational collapse with 

1. gravitational potential peaking in the middle (light-blue)
2. high density middle (blue)
3. negative temperature middle (green)

with the unphysical nature signified by negative temperature. How do we know that the simulation is unphysical? This is because the negative temperature comes with negative Total Internal Energy E(t) (= -0.483 at a certain time t) with 

• $E(t) = \int \rho (x,t)T(t,x)\,dx$,
• where $\rho$ is density and $T$ temperature.

Now, total internal energy E cannot be negative, neither density nor temperature can be negative. This connects to the basic energy balance of a self gravitating fluid/thermodynamic system which takes the form 

• $TE(t) = E(t) + K(t)$ with 
• $TE(t) =$ Total Energy at time $t$
• $K(t) =\frac{1}{2}\int\rho (x,t) |v(x,t)|^2dx$ Kinetic Energy, with $v(x,t)$ fluid velocity.   

Notable is that Total Energy has no contribution from gravitation. 

What we see in the simulation is accelerated fluid motion from a gravitational potential with central peak (light-blue) accompanied by an increase of kinetic energy balanced by a decrease of internal energy/decrease of temperature (green), while density is increasing (blue). But there is a limit to this process, because temperature/internal energy cannot become negative. 

When the simulation passes this limit and temperture/internal energy becomes negative, the simulation turns unphysical and thus produces numerical artefacts, not physics. 

Before collapse of the numerics, density and kinetic energy are increasing while temperature decreases. This means that the concentration of mass towards the center does not come with an increase of temperature, but instead with a decrease of temperature.

This shows that a common idea of "stars being ignited by gravitational collapse" (see Wikipedia quote below) is incorrect in the sense that the ignition does not come from increasing temperature. The ignition of stars thus can only be the result of a concentration of mass (triggering nuclear fusion balancing further concentration), and not any fire ball sucking energy from gravitation.

And the other way around, if a black hole is what results if star ignition/fusion does not take place, it must be a very cold place. 

To see physical uncollapse run the code with larger gas constant $\gamma$ (e.g. 1 instead of 0.1)!

Summary: With Total Energy = Internal Energy + Kinetic Energy without contribution from gravitation, there is no physics of unlimited sucking of energy from gravitational collapse.  There is no physics of unlimited sucking of energy from internal energy (allowing negative internal energy).

More precisely, the energy balance takes the form 

• $\frac{dE}{dt} = -W+P+D,
• $\frac{dK}{dt} = W-P-D,  

where $p=\gamma\rho T$ and $\phi$ gravitational potential (with minus sign),  and

• $W(t) = \int p(x,t)\nabla\cdot v(x,t)\,dx$,
• $P(t)=\int\phi\nabla\cdot (\rho (x,t)v(x,t))\,dx$, where $-\Delta\phi =\rho$ (and thus $\phi >0$), 
• $D\ge 0$ turbulent dissipation, 

which as a 1st law of energy balance gives $E+K$ constant with $D\ge 0$ as a 2nd law. See Computational Thermodynamics.

Now you are ready to experiment yourself with DigiMat, and seek deeper understanding of how stars are formed...by thermodynamics, gravitation and fusion. 

Notice that fusion opens a new source of positive contribution to internal energy, which can be transformed into high temperature and high velocity. Without fusion this option is closed and the amount of internal energy is limited by given total energy.

We observe that energy without gravitation can be transferred from $E$ to $K$ (only) in expansion with $\int p\nabla\cdot v\, dx>0$ and vice versa from $K$ to $E$ (only) in contraction with  $\int p\nabla\cdot v\, dx<0$. Gravitation can transfer energy from $E$ to $K$ (only) in contraction with $\int\phi\nabla\cdot (\rho v)\,dx <0$ and the other way around. 

Is it true that 

• DigiMat unlocks the grand challenges in science, education and industry?

Compare with How are stars formed?

Compare with Wikipedia presenting the standard view:

• A star is born through the gradual gravitational collapse of a cloud of interstellar matter. The compression caused by the collapse raises the temperature until thermonuclear fusion occurs at the center of the star.

True?

  

DigiMat: BIG BANG

DigiMat is now being launched to a broad public. Here is an example of a DigiMat simulation of BIG BANG from Model Workshop of DigiMat BodyandSoul:

See expansion of BigBang initial data as central concentration of high density/temperature spreading out into low density/temperature space headed by a shock wave and followed by expansion with  scaling linearly with distance in accordance with Hubble's Law. DigiMat students master the coding of this example in early school years. Try yourself by clicking on the code link above.

Question: Is the shock wave beyond the observation horizon (since it is not observed)?

See also Cosmology 3d and Spiral Galaxy.

Mathematics of Swedish Corona Strategy

Consider the follow record as of June 20 by Folkhälsomyndigheten in charge of Corona in Sweden: 

We see a total of 2333 Corona patients in intensive care and a total of 5053 dead in Corona, in Sweden, with a display of numbers in each 10-year age category in middle an right columns. If we add the number in intensive care to the number of dead in each age category from 50-59 to 90+, we get roughly the same total number, seemingly as an expression of some mathematical law of conservation. This is not so surprising: With intensive care lives may be saved, while without intensive care death may follow. It is reported that 70% of patients in intensive care survived.

In the group 50-59 there are few dead and many in intensive care, while for 80-89 and 90+ the numbers are opposite with very few in intensive care and many dead. But how can it be that the distribution is so different from 50-59 to 80+? It appears that older people have been discriminated from intensive care by some active decision process on the sole basis of age without consideration of physical condition. Is it so?  

 Compare with DN June 22: Patients have been prioritised away