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# Have you solved problems in your sleep? [closed]

I have hit upon major (for me—relative to my trivial accomplishments) insights in my research in various sleep-deprived altered states of consciousness, e.g., long solo car-drives extending through the night into the morning. But I have never actually solved a problem in my sleep. I have awakened thinking That's it!, but never was it actually it.

Q. Can anyone report an actual significant advance in their research that occurred during and emerged from their sleep?

Of course this is entirely subjective, but you would know it if it happened to you.

Poincaré's famous step onto the bus in 1908 ("At the moment when I put my foot on the step the idea came to me...") indicates significant unconscious processing, and his insomnia account (quoted below) adds further credence to such "background" processing. But I am not aware of first-hand reports of significant and accurate reasoning occurring during sleep.

One evening, contrary to my custom, I drank black coffee and could not sleep. Ideas rose in crowds; I felt them collide until pairs interlocked, so to speak, making a stable combination. ... (Link)

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## closed as off-topic by <NAME>, Suvrit, <NAME>, R W, <NAME>Apr 28 '14 at 15:37

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question does not appear to be about research level mathematics within the scope defined in the help center." – <NAME>, Suvrit, <NAME>, R W
If this question can be reworded to fit the rules in the help center, please edit the question.

Yes, this happened to me. My very first pusblished result in fact (so at least then it was major for me :-)). I thought I knew how to do it while brushing my teeth in the evening but then realized it did not work that way. Yet, the next morning I knew how to actually prove it. –  quid Apr 27 '14 at 0:37
What result would that be, quid? :) –  Lucia Apr 27 '14 at 0:59
I have come up with proofs in dreams, but when I wake up they've always turned out to be nonsense. –  <NAME> Apr 27 '14 at 1:24
@ToddTrimble: If the idea was not present in consciousness prior to sleep, but is present upon awakening...? –  <NAME> Apr 27 '14 at 1:45
I'm surprised nobody's mentioned the Thomason-Trobaugh theorem, which is the only instance I can think of where a dream character was bestowed coauthorship. –  <NAME> Apr 27 '14 at 3:37

On several occasions it has happened that I have made a key insight while sleeping or drifting in and out of sleep.

For example, one of the critical ideas in my paper

• <NAME>, <NAME>, Israel J. Math. 125 (2001), 237--252,

came to me this way, and waking up with the mathematical idea, I tore myself out of bed to work it out more fully on paper. It was totally right and formed the basis of later work. I remember sitting in my night attire in the bare moonlight at the table in my apartment, looking out at the empty sidewalk at Wall Street and Williams, where I lived at the time, pondering the approximation property applied to ultrafilters.

Because this has now happened several times, I now quite regularly try to prime myself, by intensionally focusing on a particular mathematical issue just as I am going to sleep. My mind floods with mathematical ideas just as I drift off. On welcome rare occasions, the problem is solved in the hypnagogic state, and having awoken I lay in bed pondering it, trying to check it, and wondering if it really is right (sometimes, of course, what seems right is later found to be mistaken). More often, though, when there is welcome news it consists not of a full solution but rather of a new perspective, which later forms the framework of a solution. That is, the result of the unconscious thought is a new way of thinking about the problem, rather than a complete logical proof.

At times, naturally, it is an interesting (or obsessive) MathOverflow question that I set myself to thinking about as I lay myself down. But let me say categorically that it has never been the case (ahem, cough, cough) that an hour or two after going to bed, I would wake with an answer and crawl out to my computer to type up an MO answer in the dark, while the rest of the household is sleeping, only to realize at that point, right before clicking "Post Your Answer" that the solution was totally flawed or wrong. What a downer that would be, to be sitting in the dark in the middle of the night, tired, with nothing to show for it but a wrong mathematical idea. That has NEVER happened... :-)

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+1. $1-\epsilon$ of which is for the last paragraph. –  <NAME> Apr 27 '14 at 13:38

In a dream I saw such a vivid image of a sort of vortex at a crossroads that I woke up convinced that it was an important geometric message to myself. In a half-waking state I free-associated and became certain that I knew roughly what question this was the answer to and roughly what direction it was pointing me in. For about two days I was fully expecting this to lead to a little breakthrough, muttering to myself about PL cotangent microbundles; but it all came to nothing.

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"Use the fork, Tom." –  The Masked Avenger Apr 27 '14 at 15:47
Well, it looked more like a crossroads than a fork in the road, but I'll bear that in mind. I haven't given up on this. –  <NAME> Apr 27 '14 at 16:58
@TheMaskedAvenger: Is the fork quote a "literary" reference? Do explain for the less literate... –  <NAME> Apr 27 '14 at 19:25
Star Wars episode IV (1977). Except the advice is given to Luke by Kenobi. –  The Masked Avenger Apr 27 '14 at 20:09
And except that the original line is "use the force". It appears that it has been repeatedly parodied as "use the fork". –  <NAME> Apr 27 '14 at 20:56

Hadamard investigates these kinds of issues at length in his book The psychology of invention in the mathematical field. He gives several examples of famous mathematicians dreaming about solutions, including Poincare. His conclusion is that the unconscious definitely plays a decisive role in mathematics, and that sleep often has to do with it, but that it differs from person to person how to tap in to it.

It is (necessarily) a bit pseudoscientific, but has some great tidbits. For example, did you know <NAME>, who was a psychologist into the then-popular phrenology, actually went around measuring mathematicians' heads, trying to locate the "bump" in the skull where mathematical ability should lie?

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Have you solved problems in your sleep?

$\quad$ <NAME>, a former accountant general of Madras, would tell of Ramanujan's insistence that after seeing in dreams the drops of blood that, according to tradition, heralded the presence of the god Narasimha, $\,$ the $\,$ male $\,$ consort $\,$ of $\,$ the $\,$ goddess $\,$ Namagiri, “scrolls containing the most complicated mathematics used to unfold before his eyes.”

<NAME>, a friend of Ramanujan from Madras days, said that Ramanujan even attributed his early interest in mathematics to a dream-a dream about, of all things, a street peddler hawking pills.

$\qquad\qquad\qquad\qquad$ — <NAME>, The Man Who Knew Infinity: A Life of the Genius Ramanujan.

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An interesting piece to read which tells the story of a dream leading, not to a single result, but a fundamental shift in a mathematician's work (leading to the proof, with collaborators, of Local Langlands for $p$-adic fields and of the Sato-Tate conjecture) by <NAME>:

http://www.math.jussieu.fr/~harris/1992_dream.pdf

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I had a "near miss" while I was a PhD student. In a dream, I proved something about definability in some fragment of first-order logic with generalized quantifiers. I woke up straight away and wrote the proof down before going back to sleep. In the morning, it still looked like a proof so I went and discussed it with my advisor. After a little while, he found a flaw but the proof was still correct in a special case.

I'm not sure if this meets your criterion of "significance". It wasn't trivial (the proof was about a third of a page) but it was hardly ground-breaking. It was something we'd been thinking about for a couple of weeks and, as far as I'm aware, the problem is still open.

As I recall, the problem was something like this. Let $\mathrm{FO}(\mathrm{GI})$ be the extension of first-order logic with a quantifier for graph isomorphism. The syntax is generated by the normal rules for $\mathrm{FO}$ plus the rule that, if $\varphi(\bar{x},\bar{y})$ and $\psi(\bar{x},\bar{y})$ are formulas in which the variables $\bar{x}=x_1\dots x_k$ and $\bar{y}=y_1\dots y_k$ are free, then $\mathrm{GI}_{\bar{x},\bar{y}}(\varphi,\psi)$ is a formula in which $\bar{x}$ and $\bar{y}$ are bound. The semantics is as follows. In any relational structure with domain $V$, we can consider each of $\varphi$ and $\psi$ as defining a directed graph with vertex set $V^k$ and an edge between any pair of vertices that satisfy the formula. Now, $\mathrm{GI}_{\bar{x},\bar{y}}(\varphi,\psi)$ is true if, and only if, the two graphs are isomorphic.

Question: In the vocabulary with only equality (and no constant symbols), is $\mathrm{FO}=\mathrm{FO}(\mathrm{GI})$ on finite structures?

The case $k=1$ is trivial, as there are only four graphs you can define. I think my proof worked for $k=2$ but not for anything larger.

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The article "Do androids prove theorems in their sleep?" available at http://www.math.jussieu.fr/~harris/androids.pdf describes an instance in which a dream inspired a mathematical result. It describes the details of the Thomason-Trobaugh theorem mentioned above in the comments.

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Reading Harris's article makes me realize how sorely we miss <NAME>'s wit: he would have found the right words for debunking the quoted article's pretentious nonsense. –  <NAME> Apr 28 '14 at 15:36

This BBC article mentions the Ramanujan story, with other examples from the arts and science, and is followed by readers giving their own examples. One conclusion to draw from this, is that quite often the "discovery in a dream" is just made up. Kekulé's well-known story how he discovered the ring structure of Benzene in a dream may be just an anecdote.

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## Algebra 2 (1st Edition)

$30$
We know that if we want to select $r$ objects out of $n$ disregarding the order, we can do this in $_nC_r=\frac{n!}{r!(n-r)!}$ ways. This gives: $_{5}C_2\cdot_{3}C_1=\frac{5!}{3!2!}\frac{3!}{2!1!}=10\cdot3=30$

format: plain


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## Superconducting magnesium diboride

WARNING The following example aims at providing physically meaningful results. Calculations can therefore take a significant amount of time. For quick calculations, look at the EPW/tests/ folder.

Running EPW to calculate the anisotropic superconducting gap of MgB2

This example is based on the papers Phys. Rev. B 87, 024505 (2013) and arXiv:1604.03525, which you should check for the physical interpretation of the results and a description of the main equations used in this tutorial.

The MgB2 example is located inside the EPW/examples/mgb2/ directory. Within this directory you will find three subdirectories: pp/ contains the Mg and B pseudopotentials; phonon/ contains the input files needed for calculating the phonon dispersions of MgB2; epw/ contains the input files to run epw.x on MgB2.

Once epw.x has been compiled, we are ready to run the example.

Calculating phonons

The phonon code of QE requires a ground-state self-consistent calculation:

../../../../bin/pw.x < scf.in >& scf.out &

The electronic bandstructure of MgB2 is shown in Figure 1 below.

We will now proceed with a sequential run. Let us compute the phonon frequencies and eigenmodes using ph.x:

../../../../bin/ph.x < ph.in >& ph.out &

This will provide us with the files containing the dynamical matrices and the variations of the self-consistent potentials at 28 irreducibles q-points. We now need to copy the .dyn and .dvscf files, as well as the _ph0/diam.phsave folder, inside the save/ folder. These files contain all the information that EPW needs from Quantum Espresso.

EPW expect these files to be labelled according to the number of the irreducible q-point in the list. In order to label the files correctly you can use the python script pp.py. If you decide to proceed this way, just issue the statement:

python pp.py

from within the phonon/ folder. The script will ask you the prefix used for the QE calculations as well as the number of irreducible q-points computed. The script will rename all the required files and place them in the folded save/. At this point we are done with QE and can move to the epw folder.

The phonon dispersion relations of MgB2 are shown in Figure 1 below.

Figure 1: Electronic band structure and DOS of bulk MgB2 (top), as well as phonon dispersion relations and PDOS (bottom). The calculations are executed using the experimental lattice parameters. The inelastic X-ray scattering data (black dots) at 300~K are from A. Shukla et al..

Running EPW

We first have to perform one scf and one nscf calculation. To do this, we go inside the epw directory and issue:

../../../../bin/pw.x < scf.in >& scf.out

../../../../bin/pw.x < nscf.in >& nscf.out

You can then launch the EPW calculation:

../../../bin/epw.x < epw.in >& epw.out

The calculation of superconducting properties is triggered by the flag eliashberg = .true. and will be accompanied by significant I/O. In the following we describe the various output files, and we will use XX to indicate the temperature at which the anisotropic gap equations are solved. Please note that the figures shown below have been generated using denser Brillouion zone grids than those indicated in the example input files.

• MgB2.a2f
This file contains the Eliashberg spectral function as a function of frequency \omega (meV), calculated using various smearing parameters (see Figure 2 below).

Figure 2: Calculated isotropic Eliashberg spectral function \alpha^2 F(\omega) of MgB2 (blue solid line), and cumulative electron-phonon coupling strength \lambda (black dashed line). The results are integrated on a homogeneous fine k-point grid containing 60x60x60 points, and a fine homogeneous q-mesh with 30x30x30 points.

• MgB2.a2f_iso
This file contains the Eliashberg spectral function as a function of frequency \omega (meV) . The second column is the Eliashberg spectral function corresponding to the first smearing value in MgB2.a2f . The remaining columns in MgB2.a2f_iso (3 x number of atoms) contain the mode-resolved Eliashberg spectral function (there is no specific information on which modes correspond to which atomic species).
• MgB2.phdos
This file contains the total phonon density of states as a function of frequency \omega (meV) calculated using various smearing parameters. The smearing values are the same as in MgB2.a2f .
• MgB2.phdos_proj
This file contains the phonon density of states as a function of frequency \omega (meV). The second column is the total phonon density of states corresponding to the first smearing value in MgB2.phdos . The remaining columns in MgB2.phdos_proj (3 x number of atoms) contain the mode-resolved phonon density of states (there is no specific information on which modes correspond to which atomic species).
• MgB2.imag_aniso_XX
This file contains 5 columns: the frequency along the imaginary-axis \omega (eV), the Kohn-Sham eigenvalue E_{n{\bf k}} for band n and wavevector {\bf k} relative to the Fermi energy (eV), the quasiparticle renormalization Z_{n{\bf k}}, the superconducting gap \Delta_{n{\bf k}} (eV), and the quasiparticle renormalization Z_{n{\bf k}} in the normal state. The gap is shown in the top panel of Figure 3 below.
• MgB2.pade_aniso_XX
This file contains similar information as in MgB2.imag_aniso_XX, but the superconducting gap is along the real axis and is obtained via Pade approximants. This file is written if iverbosity=2 in the input file, and the result is shown in the lower panel of Figure 3 below.

Figure 3: Calculated energy-dependent superconducting gap of MgB_2 from the anisotropic Eliashberg theory, at 10~K (blue), 38~K (green) and 43~K (red). The upper panel shows the gap along the imaginary energy axis, the lower panel is along the real energy axis (approximate analytic continuation via Padé functions. Only the KS states nk with an energy within 0.1~eV from the Fermi level are shown.

• MgB2.imag_aniso_gap0_XX
This file contains the distribution of the anisotropic superconducting gap \Delta_{n{\bf k}}(\omega=0) (eV) on the Fermi surface, as obtained from the imaginary-axis calculation.
• MgB2.pade_aniso_gap0_XX
This file contains similar information as in MgB2.imag_aniso_gap0_XX, but in this case the gap is calculated on the real axis using Padé approximants. This is shown in Figure 4 below.

Figure 4: Calculated anisotropic superconducting gap \Delta_{n\mathbf{k}}(\omega=0) of MgB_2 on the Fermi surface, evaluated as a function of temperature. The superconducting gaps are obtained from the aproximate analytic continuation using Padé functions, and the Coulomb potential is set to \mu_c^*=0.16. For each temperature the histograms indicate the number of states on the Fermi surface with that superconducting gap energy. The dashed lines are guides to the eye. The two superconducting gaps vanish at the calculated critical superconducting temperature T_{\rm c} of 51~K [note this value is about 30\% higher than in the experiment].}

• MgB2.qdos_XX
This file contains the quasiparticle density of states in the superconducting state, relative to the DOS in the normal state, N_{\rm S}(\omega)/N_{\rm N}(\omega) as a function of frequency \omega (eV). This is shown in Figure 5 below.

Figure 5: Quasiparticle DOS in the superconducting state for various temperatures [obtained using \Delta(\omega=0)]. The dashed black line is the normal state DOS taken from the top of Figure 1 and normalized to 1 at the Fermi level. The superconducting DOS N_{\rm S}(\omega) was scaled so that its high energy tail becomes 1 beyond the highest phonon energy. The two-gap nature of MgB_2 is clearly seen, and can be associated with the \sigma and \pi sheets on its Fermi surface, as shown in Figure 6 below.

• MgB2.imag_aniso_gap_FS_XX
This file contains the superconducting gap on the Fermi surface. The first three columns contain the 3 Cartesian coordinates of each k-point, the 4th column is the band index within the chosen energy window, the 5th column is the energy difference with respect to the Fermi level. Only states within 0.2 eV of the Fermi energy are considered. The 6th column is the superconducting gap in eV. A 3D rendering of superconducting gap on top of the Fermi surface sheets is shown in Figure 6 below.
• MgB2.imag_aniso_gap_XX_YY.cube
This file contains the same information as MgB2.imag_aniso_gap_FS_XX, in a format compatible with VESTA. This is suitable for mapping the gap on the Fermi surface. Here YY represents the band number within the chosen energy window during the EPW calculation.

Figure 6: The superconducting energy gap of MgB_2, calculated at T=10~K, mapped on the Fermi surface. The Fermi surface consists of two \sigma sheets along the \Gamma-\Gamma lines, and two \pi sheets along the K-M and the H-L lines.

• MgB2.lambda_aniso
This file contains 4 columns: KS eigenvalue E_{nk} relative to the Fermi level (eV), \lambda_{n{\bf k}}, the wavevector {\bf k}, and the band index n.
• MgB2.lambda_k_pairs
This file contains the normalized distribution of the anisotropic electron-phonon coupling strength \lambda_{n{\bf k}} on the Fermi surface. This quantity is shown for example in Fig. 5b of Phys. Rev. B 87, 024505 (2013).
• MgB2.lambda_pairs
This file contains the normalized distribution of the anisotropic electron-phonon coupling strength \lambda_{n{\bf k},n'{\bf k}'} on the Fermi surface.
• MgB2.fe_XX
This file contains the free energy in the superconducting state.
• MgB2.lambda_FS
This file contains \lambda_{n{\bf k}} on the Fermi surface. The first three columns represent Cartesian coordinates of the k-point, the 4th column is the band index within the specified energy window, the 5th column is the energy relative to the Fermi level. Only states within 0.2 eV of the Fermi energy are considered. The 6th column is \lambda_{n{\bf k}}.
• MgB2.lambda_YY.cube
This file contains the same information as MgB2.lambda_FS but can be read by VESTA for direct visualization. Note that YY is the band index within the specified energy window.

Note for expert users

If you wish to reproduce Figure 6 above (showing the map of the superconducting gap on the Fermi surface) you can use two python scripts from the following tar file

The following video explains how to use the scripts in conjunction with VESTA to obtain Figure 6.

This video tutorial can also be found here.


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# How many square meters is 17000 square feet?

## How many square meters is 17000 square feet?

17000 square feets = 1579 square meters So, 17000 square feets = 17000 × 0.09290304 = 1579.35168 square meters.

## How many square feet is 17000?

Square feet to square feet conversion chart near 17000 square feet

Square feet to square feet conversion chart
14000 square feet = 14000 square feet
15000 square feet = 15000 square feet
16000 square feet = 16000 square feet
17000 square feet = 17000 square feet

How many acres is 17 000 square feet?

Square feet to acres conversion chart near 17000 square feet

Square feet to acres conversion chart
17000 square feet = 0.39 acre
18000 square feet = 0.413 acre
19000 square feet = 0.436 acre
20000 square feet = 0.459 acre

How many square feet is 14000 square meters?

14000 square meters = 150700 square feets Formula: multiply the value in square meters by the conversion factor ‘10.763910416714’.

### What is the size of an acre?

43,560 square feet
Acre, unit of land measurement in the British Imperial and United States Customary systems, equal to 43,560 square feet, or 4,840 square yards. One acre is equivalent to 0.4047 hectare (4,047 square metres).

### How do I conceptualize my square footage?

To find square feet, multiply the length measurement in feet by the width measurement in feet. This yields a product called the area, which is expressed in square feet (or square inches if you are calculating a much smaller space, such as a dollhouse).

How large is a square meter?

The area equal to a square that is 1 meter on each side. Used for measuring areas of rooms, houses, blocks of land, etc. Example: A typical car parking space is about 12 square meters.

How do I calculate sq ft?

for short), determine the length and width of the area you are working with, measured in feet. Multiply the length by the width and you’ll have the square feet. Here’s a basic formula you can follow: Length (in feet) x width (in feet) = area in sq.

## How big is a sqft?

A square foot is basically a square that’s 12 inches (30.5 cm) on each side. To measure the approximate square footage of a space, you measure the length and width of a space. Then, you multiply the numbers to get the total square feet.

## How do I measure an acre?

To calculate acres by hand, multiply your length and width (in feet) to get square feet. Try our length conversion tools, if needed. Then divide by 43,560 to determine the size of the land in acres.

Is 1100 sq ft small?

1100 Square Feet: A family of 4 to 6 people could comfortably occupy an 1100 square foot space. In an 1100 square feet apartment, you can have three standard-sized bedrooms and two full bathrooms. You’d still have plenty of room in the kitchen and living room areas.

How many sq ft is 10×10?

100 square feet
How many square feet is a 10×10 room? The square footage of a room 10 feet wide by 10 feet long is 100 square feet. Find the square footage by multiplying the width (10 ft) by the length (10 ft).