Let me here recall the derivation by a scaling argument of the law of Kolmogorov/Onsager stating that fully developed turbulent velocities are Hölder continuous with exponent 1/3.
If $dx$ is smallest scale in space and $du$ the corresponding variation of velocity u, then we have with $\nu >0$ the (small) viscosity:
$\nu (du/dx)^2 \sim 1$ (finite rate of turbulent dissipation)
$\frac{du\times dx}{\nu}\sim 1 $ (Reynolds number on smallest scale $\sim 1$).
We solve to get $dx\sim \nu^{\frac{3}{4}}$ and $du\sim \nu^{\frac{1}{4}}$ and so $du\sim dx^{\frac{1}{3}}$ showing Hölder continuity 1/3.
The idea is that the flow will by instability develop smaller and smaller structures until the local Reynolds number becomes so small ($\approx 1000$) that this cascade stops on a smallest scale generating the bulk of the turbulent dissipation.
We see that velocity gradients $\frac{du}{dx}\sim \nu^{-\frac{1}{2}}$ are large, since $\nu$ is small, and so velocities are non-smooth.
The official formulation of the Clay Navier-Stokes Prize Problem by Fefferman asks about existence of smooth solutions. By the above argument this question cannot have a positive answer and so the question does not serve well as a Prize Problem.
A pure mathematician may counter this argument by claiming that a velocity with very large gradients still can be smooth, just with very large derivatives. And so even a turbulent solution of the Navier-Stokes equations can be viewed to be smooth, just with very large derivatives, and so asking for existence of smooth solutions in fact can be meaningful and so the Prize Problem in fact is meaningful. I think this means twisting the logic and terminology, which is not in the spirit of meaningful mathematics, pure and applied.
In the last two posts a resolution of the Clay Navier-Stokes Prize Problem is presented, a resolution based on digital computation. I have tried to get some comment on our proposed resolution from the group of pure mathematicians in charge of the problem including in particular its official formulation: Charles Fefferman, Terence Tao and Peter Constantin, to whom I refer as the Problem Committee.
Sorry to say, I can only report silence from the Problem Committee: no comment whatsoever!
How can we understand this state of affairs? Is it so that our resolution lacks scientific substance? No, it represents a true break-through unlocking the main difficulties of mathematical modeling and simulation of fluid flow and it is world-leading. No doubt about that!
The reason behind the silence is thus not lack of scientific interest, but probably rather the opposite: Our resolution being based on digital computation brings in a new kind of mathematics, which is different from that envisioned in the official formulation expressed in the frame of classical analytical theory of partial differential equations. It appears that the Problem Committee does not know how to react to this new kind of mathematics in the form of digital computation, and so silence is the only possible reaction, so far at least.
This connects to a wider question of the role of mathematics in physics including fluid mechanics with particular focus on the new role of digital computation.
Now, mathematics can be seen as different forms of computation with classical pde-theory expressed as symbolic computation by pen and paper, and the new kind expressed by a computer executing the symbolic computation represented in the computer code.
So I again ask about the view of the Problem Committee on the possibility of resolving the Clay Problem by digital computation. Is it thinkable? Or can only a resolution in the form of symbolic computation with pen and paper be accepted? Is digital computation a form of mathematics?
Apparently Fefferman would be willing to give the Prize to Tao for a proof of blow-up towards infinite velocities, but so far Tao has not succeeded. But even if one day he would succeed, that would only mean that the mathematical model is no good as a model of real fluid flow, since no observation of infinite velocities has been made, and why give a Prize for a discovery that a model is no good? More meaningful maybe to give the Prize for a result about a mathematical model of physical significance, like the one we give?
PS1 A pure mathematician might say that digital computation cannot deliver an answer for all (smooth) data and so would lack the generality of an answer by symbolic computation valid for any (smooth) data. To meet this criticism we can add that our resolution exhibits a different form of universality: We show that lift and drag of a body only depends on the shape of the body for high Reynolds number flow beyond the drag crisis at Reynolds number around $5\times 10^5$, that is for a very wide range of flows. Lift and drag depending only on shape is a form of universality. And we can compute lift and drag of any given body, case by case, but of course we cannot get a result for all bodies in one computation.
PS2 The official problem formulation by Fefferman takes as a fact that a smooth unique solution can cease to exist only if velocities become unbounded (referred to as blow-up at some specific finite time). But this is probably a misconception, since smooth solutions may turn into non-smooth solutions because velocity gradients become unbounded, which is what happens as a shock forms in compressible flow and turbulence develops in incompressible flow, while velocities stay bounded.
The official problem formulation is thus filled with misconceptions, and requires reformulation to become meaningful as a Mathematics Prize Problem.
This is a continuation of the previous post proposing a resolution of the Clay Navier-Stokes Millennium Problem with further remarks on the aspect of wellposedness identified by
Hadamard in 1902 as being necessary in order for a mathematical model to have physical meaning and relevance. The Navier-Stokes equations serve as the basic mathematical model of fluid mechanics and the Clay Problem can be viewed to reduce to the question of wellposedness, since the existence of (weak) solutions was established by Leray in 1932.
And this is the question we give an answer: We show that weak solutions are computable (exist) and are non-smooth/turbulent with wellposed mean value outputs. We do this by solving a (dual) linearized problem with certain data and show a bound of the dual solution (here for lift of a jumbojet) in terms of the data, which we refer to as assessment of stability, and which translates to an error bound on output of a computed solution in terms of its Navier-Stokes residual, showing that the output is well determined under the presence of small disturbance.
The dual linearized problem has a reaction term with coefficient $\nabla u$ with $u$ a computed velocity. The reaction term drives both exponential growth and decay with its trace being zero by incompressibility. The wellposedness of computed turbulent solutions is reflected by cancellation effects from the reaction term with exponential growth balanced by exponential deacy from oscillations of turbulent solutions.
We thus argue that we have resolved the Clay Problem by showing that weak solutions are computable/exist and show to be non-smooth/turbulent with wellposed mean-value outputs. In particular we show that lift and drag are wellposed and thus reveal the secret of flight.
It remains to be seen if our resolution will be accepted by the group of pure mathematicians owning the problem including Charles Fefferman responsible for the official problem formulation, Peter Constantin and Terence Tao. One thing is notable: Fefferman’s formulation does not involve the aspect of wellposedness and so missses the heart of the problem, if Navier-Stokes is viewed as a mathematical model of fluid mechanics, which is clearly emphasized in the official problem presentation:
Waves follow our boat as we meander across the lake, and turbulent air currents follow our flight in a modern jet. Mathematicians and physicists believe that an explanation for and the prediction of both the breeze and the turbulence can be found through an understanding of solutions to the Navier-Stokes equations. Although these equations were written down in the 19th Century, our understanding of them remains minimal. The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations.
All of this is presented in detail in this book supplied as evidence to the Clay problem committee with complementing material listed in the previous post. In particular the book contains a study of the (dual) linearized Navier-Stokes/Euler equations, a topic which for some reason has not attracted the attention of mathematicians despite its fundamental importance from mathematical point of view. In short, we feel that we have made substantial progress toward a mathematical theory which unlocks the secrets vidden in the Navier-Stokes equations, including the Secret of Flight.
Concerning the view of the problem committee recall the opening statement in the opening article Euler Equations, Navier-Stokes Equations and Turbulence by Peter Constantin
(in this book):
In 2004 the mathematical world will mark 120 years since the advent of turbulence theory. In his 1884 paper Reynolds introduced the decomposition of turbulent flow into mean and fluctuation and derived the equations that describe the interaction between them. The Reynolds equations are still a riddle. They are based on the Navier-Stokes equations, which are a still a mystery. The Navier-Stokes equations are a viscous regularization of the Euler equations, which are still an enigma. Turbulence is a riddle wrapped in a mystery inside an enigma.
In other words, total confusion in the committee in charge of problem formulation and evaluation of proposed resolutions. In particular, Fefferman formulates the problem as the questions of existence and smoothness, forgetting wellposedness, and claims that his problem was solved by standard pde-theory long ago in the case of two space dimensions and evidently has in mind a similar resolution in three dimensions by som ingenious new estimate derived by a clever pure mathematician. But wellposedness is essential also in two space dimensions and so Fefferman exposes the gulf between pure mathematics and mathematics of fluid mechanics, which is not helpful to science.
Fefferman would probably say that wellposedness is a consequence of smoothness, but this is not necessarily so since assessment of smoothness may involve stability factors of arbitrary size and so may say nothing about wellposedness. But of course questions like this have to remain in the mist since the problem committee is not open to any form of discussion.
This is a reminder of the resolution of the Clay Navier-Stokes Millennium Problem which I have presented together with Johan Hoffman and Johan Jansson in different forms over the years:
Hopefully our suggested resolution will now be considered by the Clay Institute.
The Clay problem concerns existence of smooth solutions to Navier-Stokes equations as formulated by Charles Fefferman. No progress towards a solution using techniques of analytical mathematics has been reported in the literature since the problem formulation in 2000.
existence of smooth solutions for all smooth data. But Tao has not yet (fully) succeeded.
We suggest to seek an answer instead by a computational method in the form of Direct Finite Element Simulation DFS on a sequence of finite element meshes with mesh size tending to zero.
DFS is a Galerkin method stabilised by weighted least squares control of the Navier-Stokes residual R(U, P) with U velocity and P pressure. DFS introduces turbulent dissipation as an effect of residual least squares stabilisation and can be seen as a solver of the Euler equations (Navier-Stokes with vanishing viscosity) with an automatic turbulence model.
DFS produces on a given mesh a piecewise linear (U, P) with residual R(U,P) which is small in a weak sense (H-1) by construction (Galerkin orthogonality). The key point is then that the residual R(U, P) shows to be large in a strong sense (L2) as an expression of non-smoothness
of turbulent solutions.
DFS produces/constructs/computes solutions to Euler/Navier-Stokes which show to be non-smooth/turbulent even if initiated as smooth potential solutions.
DFS thus puts on the table for inspection a sequence of approximate solutions of Navier-Stokes equations with residuals tending to zero weakly in H-1, while showing blow up in L2 as an expression of non-smoothness of turbulent flow (finite rate of turbulent dissipation). DFS offers simulation/prediction of a very large range of important engineering applications in aero and hydro-mechanics of slightly viscous flow.
DFS shows Navier-Stokes equations to have non-smooth turbulent solutions and thus gives an answer to the Clay Problem.
Stability analysis in computational form through an associated dual solution gives the further information that mean-value quantities such as lift and drag are computable by DFS with error tending to zero as the square root of the mesh-size. But point-wise quantities are not computable to arbitrary precision.
Finally, stability analysis shows that any smooth solution is unstable (as conjectured by Birkhoff) and thus cannot persist over time. Potential solutions are examples of smooth solutions, which thus do not persist over time but turn into non-smooth turbulent solutions.
Our suggested resolution of the Clay Problem is based on computing approximate solutions to Navier-Stokes/Euler equations, which show to be non-smooth turbulent.
We thus compute solutions of Navier-Stokes equations and put them on the table for anyone to check that they are non-smooth turbulent.
Will this convince a jury of mathematicians used to analytical methods? Is it thinkable that for example Tao would give our argument a moment of scrutiny? We argue that we contribute the following basic elements to the scientific discussion of Navier-Stokes/Euler equations in the case of slightly viscous flow:
DFS computes solutions of Navier-Stokes/Euler without user-specied turbulence model. DFS thus solves the basic open problem of designing a mathematical model of turbulence.
Inspection of computed solutions shows them to be non-smooth/turbulent. It is concluded that solutions of Navier-Stokes/Euler for slightly viscous flow are non-smooth turbulent, which gives the Clay problem an answer.
Slightly viscous flow is identified as flow with Reynolds number larger than say $10^6$ associated with a reduction of drag resulting from delayed separation due to an effective slip boundary condition.
Remark 1Tao discusses Onsager's conjecture that Navier-Stokes solutions are (less smooth than) Hölder 1/3 resulting in finite rate of turbulent dissipation. DFS solutions typically show to be Hölder 1/3 with gradients $\nabla U$ blowing up like $h^{-0.5}$ with variations $h^{0.25}$ on scales of size $h^{0.75}$ consistent with weigthed least squares stabilisation $\int h\vert\nabla U\vert^2dx\sim 1$.
Here is one DFS Navier-Stokes solution put on the table for inspection showing turbulent flow with finite rate of turbulent dissipation around a jumbojet (with finite drag and lift):
Computer simulations and custom-built quantum analogues are changing what it means to search for the laws of nature.
Yes, computational techniques are changing the way physics is done and so also the mathematical physics of fluid mechanics and the related mathematics. The formulation, meaning and practical utility of a mathematical model for som physical phenomena, typically in the form a (differential) equation like Navier-Stokes equations, closely connects to techniques for computing solutions and thus it is natural to expect that questions concerning the nature of solutions can be answered by computation with thus the computer offering a powerful new tool for mathematical modeling and analysis. The Clay Navier-Stokes problem can be seen as the outstanding open problem of classical continuum physics. This is the problem of predicting turbulent flow, which can now be viewed to be solved by computation.
Remark 3 The Clay problem is officially presented in the following words:
Waves follow our boat as we meander across the lake, and turbulent air currents follow our flight in a modern jet.
Mathematicians and physicists believe that an explanation for and the prediction of both the breeze and the turbulence can be found through an understanding of solutions to the Navier-Stokes equations.
Although these equations were written down in the 19th Century, our understanding of them remains minimal.
The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations.
We see the connection with turbulence and we see that we can indeed solve the Navier-Stokes equations and thus predict turbulent flow representing world-unique breakthrough of making NASA Vision 2030 Grand Challenge into a reality already today. We are also proud to see that our New Theory of Flight indeed unlocks secrets hidden in the equations. We thus believe that we have something substantial to contribute which is worthy of consideration. But it is a new kind of science with new standards and so reviewers must be open-minded.
Remark 4 Recall that we argue that the problem formulation by Fefferman is unfortunate by not including the aspect of well-posedness, which isverywell understood by mathematicians since Hadamard to be a necessary quality for physical relevance. One can thus argue that the Navier-Stokes problem essentially concerns the question of well-posedness and that our resolution is to give this question a positive answer: Computed solutions show to be non-smooth/turbulent and as such show to be well-posed physical solutions with stable mean-value outputs (such as lift and drag persisting over time making flight possible). We also argue that a solution initiated as smooth does not have stable (mean-value) outputs persisting over time and thus is not well-posed.
We thus give a positive answer to the Clay problem formulated as a question of well-posedness.
In short: Computed solutions show to be non-smooth/turbulent and well-posed with stable outputs. A solution initialised as smooth (for example as potential flow) is unstable and develops over time into a non-smooth/turbulent solution.
It is the oscillating nature of turbulent solutions which make them well-posed as expressed by the dual solution which over an oscillating velocity field shows little growth. On the other hand, a smooth solution is not oscillating and thus can give the dual solution consistent growth into non-wellposedness.
Remark 6 DFS can be seen as an incompressible Euler solver with automatic turbulence model which produces a non-smooth turbulent solution with finite rate of turbulent dissipation. DFS applied to the compressible Euler equations typically produce solutions with dissipative (energy-consuming) shocks. DFS for Euler thus introduces dissipation from residual stabilisation as automatic turbulence/shock modeling.
Remark 7 Recall that the discussion involves the following elements:
Physics (of fluid particles): (i) Newton's 2nd Law + (ii) Incompressibility.
Mathematics: Formulation of (i) + (ii) in terms of Calculus = Navier-Stokes/Euler.
Computation: DFS as solver of Navier-Stokes/Euler.
Physical fluid particles move so as to satisfy (i) and (ii), while DFS computes motion of virtual fluid particles from a mathematical principle best possible satisfaction of (i) and (ii) in the form of Navier-Stokes/Euler.
DFS can be given an interpretation in physical terms through Navier-Stokes/Euler residuals and DFS can thus be viewed to be the physical model to be analysed, rather than the Navier-Stokes/Euler equations in conventional Calculus form which occurs in the problem formulation by Fefferman.
Concerning the mathematical formulation of an equation/model describing som physics the technique for solving the equation usually connects to the formulation of the equation, and so solution and formulation are intertwined and often cannot be separated.
This is the case for Navier-Stokes/Euler where the classical approach of first seeking to formulate a turbulence model and then solve the corresponding equations, has failed. With DFS we instead directly aim at solving Navier-Stokes/Euler in original formulation without turbulence model and where the computational technique of residual stabilisation automatically produces the turbulence model.
DFS expresses best possible satisfaction of (i)+(ii) on a given mesh with piecewise linear velocity-pressure, which is not exact satisfaction. Similarly, a physical fluid can be expected to seek best possible satisfaction of (i)+(ii), which may not mean exact satisfaction of e.g. incompressibility.
We argue that DFS can be a more meaningful object of study than the Navier-Stokes/Euler equations in a conventional strong or weak formulation asking for interpretations.
Remark 8 Our proposed solution of the Clay problem has the form of an open-source computer program (Unicorn FEniCS/HPC), which upon execution delivers a solution of Navier-Stokes/Euler showing to be non-smooth/turbulent. When executed on a sequence of automatically generated adaptive meshes with decreasing mesh size, mean-value quantities such as lift and drag are seen to converge to specific values within tolerances which can be estimated by duality, and can be made as small as desired. Residuals of computed solutions are seen to tend to zero weakly while becoming large strongly as expression of non-smoothness.
The FEniCS/HPC code is open-source and thus available for inspection, evaluation and testing by anyone. It uses the automated modeling of FEniCS and as such can be expected to be correct, or at least possible to be made correct.
It is thus possible to check if our claim of having solved the Clay problem is correct or not. The question is if anyone with connection to Clay is willing to make the check.
Remark 9 The standard following Prandtl is to specify the boundary condition for Navier-Stokes to be a no-slip condition $U=0$ asking both tangential and normal velocity to vanish, while for Euler only asking the normal velocity to vanish (non-penetration) leaving the tangential velocity free as a slip condition.
However, it is more natural from physical point of view to specify slip also for Navier-Stokes as an expression of small friction for slight viscosity, as mixed Dirichlet-Neumann condition. This is what we do, which opens a whole new perspective for computation without requirement of resolving thin boundary layers beyond the capacity of any forseeable computer. DFS thus solves Navier-Stokes /Euler with a slip boundary condition and so allows prediction of virtually any slightly viscous flow at affordable cost.
With slip we do not make a distinction between Navier-Stokes with slight viscosity and Euler with formally zero viscosity, since in the numerics it is the residual stabilisation which introduces the main viscosity/dissipation and not a vanishingly small standard viscosity term.
Altogether, we argue that we have given a resolution of the Clay problem by computation offering
Accurate prediction for arbitrary data (geometry and forcing) at affordable computational cost.
Understanding of the nature of solutions of Navier-Stokes/Euler from observations of computed solutions. In particular we observe that computed solutions can be described as non-smooth turbulent dissipative Euler solutions with Hölder continuity 1/3 in accordance with Onsager's conjecture and Kolmogorov's 4/5 law.
Remark 10 Recall that we consider the official formulation by Fefferman to be incorrect from mathematical point by not including the crucial aspect of well-posednedness (here and here). It may well be that the question posed by Fefferman (existence of smooth solution for all time for all smooth data) will be impossible to answer, since mathematical techniques for proving global smoothness will remain hidden to humans together with techniques for construction of blow-up.
We suggest to reformulate the problem into a question of wellposedness of weak solutions for which a positive answer is offered by DFS. This is the relevant question from physical point of view and then also from mathematical point of view, since Navier-Stokes is a mathematical equation with physical meaning.
Remark 11 We argue that only a notion of approximate solution of Navier-Stokes/Euler is meaningful, and this is what DFS delivers and which upon inspection shows to be non-smooth/turbulent with undetermined point-values of velocity and pressure, but with mean-value outputs such as lift and drag computable with quantitative error control. In particular,
we argue that it does not make much sense to ask about exact solutions in a situation where solutions are non-smooth without well determined point-values and thus have the form of distributions which to be defined require the specification of a wealth/infinity of integrals weighted with smooth test functions. In short, does it make any sense to ask for exact specification of mean-values requiring a wealth of information. Isn't it more reasonable to be satisfied with specification of a piecewise linear DFS velocity-pressure, which is an approximate solution with error controled output? What more you could you ask for?
Remark 12 Our resolution includes the following ingredients:
Open-source computer code FEniCS/Dolphin (about 100.000 lines) for automated discretization of the Navier-Stokes/Euler equations in standard analytical form into a system of algebraic equations in piecewise linear DFS velocity-pressure on a given finite element mesh (millions of mesh points) expressing asking residuals to vanish weakly combined with weighted least squares stabilisation.
Open-source computer code PETc (about 100.000 lines) for automated computation of DFS solution.
Open source computer code FEniCS/Unicorn for quality assessment of computed DFS solution as quantitative measure of accuracy of chosen (mean-value) output by computation of a dual solution expressing sensitivity of output with respect to DFS residuals.
Open-source code FEniCS/Unicorn for automated mesh adaptivity to reach specify output accuracy.
The codes express a massive volume of analytical mathematics and execution of the codes massive volume of computational work. Our resolution is the result of a combination of analytical mathematics and brute computational force with the goal/scope of delivering answers to "all that can be asked for". It is not to be expected that non-linear pde-theory within the frame of Fefferman's problem formulation, can deliver anything near this volume of information.
Remark 13 There is only one notable mathematical result for Navier-Stokes/Euler in the literature and that is the existence proof by Leray from 1934 of weak solutions, however without any information on uniqueness/wellposedness. Leray gives a short mathematically simple argument based on basic energy estimate everyone knows. And after Leray basically no progress! No existence of unique strong solutions and nothing about wellposedness of weak solutions.
What we do is to continue Leray's work by (i) computing weak solutions and (ii) assessing wellposed of weak solutions. From the pictures above of computed solutions it is clear that Fefferman's question about existence of unique smooth solutions has a negative answer, and so the remaining question concerns wellposedness of weak solutions, a question we answer.
PS For perspective browse this talk on the Clay problem by Titi, where at the end the question of computer-assisted proof is raised, and we learn that Titi believes it will take 5000 years to compute solutions to Navier-Stokes/Euler. We know that the reality today is that it takes hours.
The (super)human intellect of Euler (1707-83) formulated Euler's equation for fluid flow.
The information society is based on computational mathematical modeling with Automated Computational Mathematical Modeling ACMM now emerging as a form of Artificial Intelligence AI.
The FEniCS Project is software for ACMM in a setting of (partial) differential equations (mathematical models of physical systems) offering automation of discretisation and computational solution.
Unicorn/FEniCS is a unified solver for solid and fluid dynamics, which offers a solution to the major unsolved problem of mathematical modeling of turbulent fluid flow in the form of an automated turbulence model as the product of best possible computational solution of Euler's equations for fluid flow.
This is presented by Johan Jansson in edX courses on High Performance Finite Element Modeling Part I and Part II. Take the courses and see yourself! This is a high-light of the educational program DigiMat carrying from basic school to university bringing ACMM to the people!
The turbulence model is the result of AI in the form of a computational procedure for finding a best possible solution to a set of partial differential equations formed by the Human Intelligence HI of Euler as the sharpest mind of the scientific revolution. Here HI sets the goal and ACMM is the factual process to reach the goal.
The solution of the riddle of turbulence thus comes out from a combination of HI and AI in a setting where HI showed to be too weak to give an answer, as witnessed by
Werner Heisenberg:
When I meet God, I’m going to ask him two questions: why relativity? And why turbulence? I really believe he’ll have an answer for the first.
Horace Lamb:
I am an old man now, and when I die and go to heaven, there are two matters on which I hope for enlightenment. One is quantum electrodynamics and the other is the turbulent motion of fluids. About the former, I am really rather optimistic.
Turbulent flow (video) around a jumbojet in landing as best possible solution of Euler's equations.
We here view AI as an extension of HI and not (merely) as a replacement as in self-driving cars,
recalling that after all driving a car does not need much intelligence. But predicting turbulent flow does and requires both HI and AI.
Notice that AI is beyond full understanding of HI since AI is a self-learning system and not a system taught by HI. In particular the automatic turbulence model offered by ACMM is beyond full understanding by HI as a the result of a self-learning adaptive computational procedure, generating the turbulent viscosity which showed to be evasive for HI alone.
Notice further that once turbulent fluid motion can be simulated by ACMM, computational fluid environments can be set up for testing of airplane designs, training of pilots or self-learning of flying vehicles as a combined vehicle-environment application of AI.
Matematik-IT gör nu en nystart under namnet DigiMat med den nya web-editorn p5.js för JavaScript lanserad av Processing som ersättning/komplement till den tidigare använda plattformen Codea.
DigiMat kommer att lanseras som en ny matematikutbildning för skolan/högskolan relaterad till FEniCS Project som open source plattform för automatiserad matematisk simulering.
DigiMat är en konkret realisering av den nya läroplanen för grundskolan med programmering som en integrerad del av matematikämnet. DigiMat är det digitala samhällets matematikutbildning.
Huvudprincipen i DigiMat är att alla matematiska objekt konstrueras/beräknas genom exekvering av datorprogram. DigiMat är konstruktiv matematik med datorberäkning som motor.
DigiMat utvecklas av internationellt ledande forskare vid högskolan enligt de principer som driver den främsta forskningsfronten framåt (edX 1 och edX 2). DigitMat ger en sammanhållen utbildning över alla stadier från förskola till högskola förankrad i en topp-position och ger därmed nytt innehåll och mening åt matematikämnet, i dagens digitala samhälle.
DigiMat erbjuder:
begriplig, meningsfull och intresseväckande matematik för alla elever
flexibel bredd och djup efter olika elevers förutsättningar
high tech för elever med speciellt intresse för matematik och naturvetenskap
To the old alarm of global warming by CO2 emissions from burning of fossil fuel is now added a new alarm from the emission of light from light bulbs, issued by group of concerned climate scientists supported by UN. Above is a map showing the light emitted by light bulbs during night over Europe. You see that there is a lot!
Basic physics tells according to the scientists that the light from light bulbs is reflected back from space onto the surface of the Earth with warming effect which is 100 times more intense than that
from CO2.
Immediate steps have been taken by the Swedish government to initiate a total reconstruction of Sweden into light-free society in a process to be completed by 2030. By a successive reduction by 50% per year of the energy spent on illuminating light bulbs, a completely dark society can become reality in so little as 10 years, says concerned politicians determined to do something: With this action Sweden will take the lead in a global transformation into a light-free world. Of course that includes the light emitted by mobile and laptop screens. Sorry.
To survive the long dark winters in Sweden, light from burning wood in open fireplaces will be admitted during a period of transformation. On the other hand, in Summer time Sweden, in particular above the Polar Circle, will have an advantage of full light though the night which will give a much wanted boost to Swedish industry. It will also give Africa as an already dark continent an advantage.
Electrical vehicles of all kinds are currently being investigated with electrical airplanes as an novel option attracting eager investors. Let us seek an answer to the key question of the possible range of such a device powered with lithium-ion batteries.
The energy density of lithium-ion battery is about 1 MegaJoule per kilogram. Equipped with weightless wings with a lift-to-drag ratio of 15 a battery of weight 1 kp = 10 N would be able to fly a distance S determined by the equation
10/15 x S = 10^6
that is S = 1500 km. Not so bad. But that is the theoretical maximum which will be reduced adding weight for passengers, airplane structure and engine efficiency. How much?
Let us compare with kerosene fuel with energy density 40 MJ giving a jumbojet with again lift-to-drag of 15 a maximal range of 40 x 1500 km = 60 000 km, which we compare with the real range of about 10 000 km , thus with a reduction factor of about 6 to account for passenger+structure+engine efficiency.
The practical maximal flying distance of a battery driven airplane could then be about 250 km. In contrast to surface travel, the energy consumption for subsonic flight is independent of speed.
With a lift-to-drag of 30 the maximal distance would be up to 500 km, which may be what is needed
to make electrical flight economical. For a more information on flight take a look at The Secret of Flight.
In a graduate course on QM, which eventually turned into a book, I found that I was unable to explain the foundations of QM in a way that I found (entirely) satisfactory.
Weinberg recalls that there are two interpretations of Schrödinger’s wave functions as solutions of Schrödinger’s multi-dimensional wave equation as the basic model of QM:
instrumentalist (wave functions predict outcomes of experiments).
realist (wave functions describe all possible worlds = multi-versa).
The instrumentalist interpretation says nothing about the world we live in, only something about certain experiments.
In the realist or rather multi-realist interpretation, multi-d wave functions describe all possible worlds or multi-versa and thus the (quite special and particular) world we are living in gets little attention, if any.
With only these alternatives, it is not difficult to understand that Weinberg cannot make sense of QM. Weinberg also recalls that physicists agreeing that QM is correct, do not agree on what is correct.
Of course Lubos claims as usual that he can make sense to QM, or rather that there is no reason for him to do that, because this was done long ago. Not by Schrödinger or Einstein, but by Born who invented a probabilistic interpretation, which abhorred Schrödinger and Einstein.
Weinberg has a Nobel Prize while Lubos runs a blogg. Weinberg says there is a problem with QM, which is denied by Lubos. What do you think? Is there a problem or not? Weinberg is at the end of his career and can afford to tell the truth.
In a travesty to a statement by Clark (Someone saying that something is impossible, is mostly wrong. Someone saying that something is possible, is mostly correct), one might say:
Someone strongly claiming that there is a serious problem, is mostly correct.
Someone shouting that there are no problems whatsoever, is mostly wrong.
If you think there is a problem with QM in its current form, you may be motivated to browse Real QM offering a new realist interpretation, where the objective is the world we are living in and not all possible worlds. Take a look!
One may say that the current deep crisis of theoretical physics witnessed by so many, is rooted in the status of QM as a theory which the expert cannot make sense of. So to get out of the crisis it is necessary to come up with a QM that makes sense. Right?
There are several new books sending the message that QM does not make sense therefore is not understood by physicists:
Klimatförändringarna innebär att mänskligheten står i en rulltrappa på väg upp mot temperaturer som saknar slutstation. Det är hög tid att gåfrån protest till motstånd.
Andreas Malm skriver:
Lägg ned inrikesflyget.
Förbjud försäljningen av fossildrivna bilar.
Avlägsna fossilgasen från värmeverken, avbryt kolkonsumtionen i stålverken, stäng oljeraffinaderierna och bygg ut alternativen i den mest rasande takt en modern stat, med alla resurser den potentiellt förfogar över, kan åstadkomma.
Föreställningen om att vi tio miljoner individer saknar betydelse måste vändas i sin motsats. Varje ytterligare ton, varje extra kilo koldioxid driver rulltrappan vidare uppåt. Varje utsläppskälla måste täppas till, från matbord till garage, gruva till hamn – och om inte de folkvalda gör det den här gången heller är det upp till alla oss andra.
Den här veckan samlas återigen tusentals klimataktivister runt en ångande utsläppskälla, fossilgasfälten i holländska Gröningen, där utvinningen på samma sätt ska stoppas med obeväpnade kroppar.
Vi har tagit bussar från avfarter runt om i Europa. Vi har gjort det förr: år 1995 åkte vi till COP-1 i Berlin. På det första klimatmötets sista dag omringade vi konferensanläggningen, kedjade fast oss vid dörrarna och skanderade: ”no more bla bla bla, action now”. Ministrarna smög ut bakvägen. Sedan dess har världens årliga koldioxidutsläpp ökat med mer än 50 procent.
Det är hög tid att gå från protest till motstånd – från att önska sig mindre utsläpp till att fysiskt spärra dem – och ta kampen till nästa nivå.
DN refuserar varje debattartikel som på något ifrågasätter rådande svenska klimatpolitik med mål att leda Sverige in i det fossilfria samhället. Istället basunerar DN ut en extrem klimatalarmism med uppmaning till motstånd mot alla former av fossil energi, uppenbarligen med vilka medel som helst. Vet DN vad DN gör?
Andreas Malm är humanekolog och för en humanekolog är det humana underordnat det ekologiska. Utan fossil energi skulle mer än 80% av jordens befolkning stå utan möjlighet till fortsatt existens. För en humanekolog verkar det inte vara något större problem: Om det är ekologins villkor att samhället måste vara fossilfritt, så får det bli så.
Inlägg
