## Wednesday, June 23, 2021

### Carbon dioxide emission plumes from a large power station detected from space

Researchers at the Finnish Meteorological Institute developed a new methodology to derive source-specific NOₓ-to-CO₂ emission ratios using satellite observations. The method was applied to Matimba power station in South Africa. The results can be used to estimate carbon dioxide emissions.

Since the Paris agreement was adopted in 2015, the role of satellite observations in understanding anthropogenic CO2emissions has become increasingly important. Currently, the NASA’s CO2 instrument Orbiting Carbon Observatory-2 (OCO-2), launched in 2014, provides CO2 observations with the best coverage and resolution. However, the observations are obtained on a narrow swath (less than 10 km), which does allow the detection of the cross-sections of the emission plumes, but not the plumes in their entirety. Satellite observations of co-emitted species, such as NO2, facilitate the detection of the CO2 emission plumes. The European Commission is currently planning a new CO2 monitoring mission CO2M via the Copernicus Programme, which will observe both CO2 and NO2 over a larger swath (over 250 km).

 OCO-2 and TROPOMI observations near Matimba power station (red triangle) in South Africa between May 2018 and November 2020. Image: Hakkarainen et al. 2021. CC BY 4.0.

Estimating CO2 emissions from individual sources using satellite data can be challenging due to the large background levels, while it is easier for short-lived gases like NO2. In a recently published study, a new methodology to calculate source-specific NOₓ-to-CO₂ emission ratio from satellite observations is developed. This ratio provides information on how clean the employed technology is and can be used to convert NOₓ emission into CO2 emission. The method was tested for the Matimba power station in South Africa, which is an optimal case study as it is a large emission source with several satellite overpasses, and it is also well isolated from other sources.

The results are based on the CO2 observations from the NASA’s OCO-2 satellite and the NO2 retrievals from the European TROPOMI (TROPOspheric Monitoring Instrument), operating onboard the Sentinel 5 Precursor satellite since late 2017. During the 2018–2020 period, 14 collocations over Matimba enabled the simultaneous detection of the CO2and NO2 plumes. The mean NOx-to-CO2 emission ratio was estimated as (2.6 ± 0.6) × 10-3 and the CO2 emission as 60 kton/day. The obtained CO₂ emission estimates are similar to those reported in existing inventories such as ODIAC.

The research was carried on in the DACES project, which focuses on detecting anthropogenic CO₂ emissions sources by exploiting the synergy between satellite-based observations of short-lived polluting gases (such as NO₂) and greenhouse gases.

The full publication by Hakkarainen and co-authors can be found at the following link:https://doi.org/10.1016/j.aeaoa.2021.100110

## Monday, May 24, 2021

### Ihmisperäisiä hiilidioksidipäästöjä metsästämässä – apurina satelliitit

Ilmatieteen laitoksella kehitettiin uusi menetelmä laskea yksittäisten päästölähteiden, kuten kaupunkien ja voimaloiden, NOₓ/CO₂-suhde avaruudesta käsin. Tuloksia voidaan hyödyntää myös hiilidioksidipäästöjen arvioinnissa. Uutta menetelmää sovellettiin Etelä-Afrikassa sijaitsevaan Matimba-hiilivoimalaan, joka on yksi maailman suurimmista.

 Matimba-hiilivoimala Etelä-Afrikassa. Wikimedia commons. CC BY-SA 3.0

Vuonna 2015 solmitun Pariisin sopimuksen myötä ihmistoiminnasta peräisin olevien kasvihuonekaasupäästöjen rooli tutkimuksessa on noussut yhä tärkeämmäksi, sillä päästöjä ja niiden vähennyksiä halutaan seurata. Esimerkiksi Euroopan komissio suunnittelee uutta satelliittipohjaista CO2M-missiota ihmisperäisten hiilidioksidipäästöjen seuraamiseen osana Copernicus-maanseurantaohjelmaansa.

Ilmatieteen laitoksella on tutkittu ihmisperäisiä kasvihuonekaasuja avaruudesta käsin vuodesta 2016 alkaen. Tutkimuksessa on hyödynnetty erityisesti vuonna 2014 laukaistua NASAn OCO-2-satelliittia, joka on edelleen paras mittalaite tähän työhön. Ihmisperäisten hiilidioksidipäästöjen kartoittamisen kannalta sen kapea mittauskaista asettaa kuitenkin haasteita.

Eräs keskeisistä ongelmista ilmakehätieteissä on laskea päästöt yksittäisistä päästölähteistä kuten kaupungeista ja voimaloista. Tämä on erityisen haastavaa hiilidioksidin (CO₂) kohdalla, kun taas typen oksidien NOₓ-päästöjä on monitoroitu satelliiteista lähes rutiininomaisesti jo 1990-luvulta alkaen. Ilmanlaadun kannalta NOₓ- ja CO₂-päästöjen välinen suhde kertoo käytetyn tekniikan puhtaudesta.

Juuri julkaistussa tutkimuksessa kehitettiin uusi menetelmä, jolla voidaan laskea NOₓ/CO₂-suhde avaruudesta käsin. Näin laskettua suhdetta voidaan hyödyntää myös, jos halutaan kääntää NOₓ-päästöt hiilidioksidipäästöiksi. Tutkimuksessa tätä menetelmää sovellettiin Etelä-Afrikassa sijaitsevaan Matimba-hiilivoimalaan. Se on tutkimuksen kannalta erinomainen koelaboratorio, sillä se on voimakas pistemäinen päästölähde ja suhteellisen etäällä muista päästölähteistä.

Tutkimuksessa hyödynnettiin vuonna 2017 laukaistun eurooppalaisen Sentinel 5 Precursor S5P-satelliitin tekemiä mittauksia, jotka mahdollistavat ensi kertaa yksittäisten typpidioksidipilvien (NO₂-pilvet) kartoittamisen satelliitista käsin. Tutkimuksessa yhdistettiin S5P-satelliitin havaitsemat NO₂-pilvet ja OCO-2-satelliitin CO₂-havainnot. Vuosilta 2018–2020 löydettiin yhteensä 14 satelliitin ylilentoa, jossa kapealla kaistalla mittaavan OCO-2-satelliitin havainnot kyettiin kytkemään S5P-satelliittiin mittaamiin NO₂-pilviin.

 Kuvassa S5P ja OCO-2-satelliittien havaintoja vuosilta 2018–2020 yhdistettynä. Matimba-hiilivoimala on merkitty kuvaan punaisella kolmiolla. Hiilivoimalasta peräisin olevat NOₓ-päästöt näkyvät kuvassa punaisesta kolmiosta lähtevänä pilvenä lähialueella. OCO-2-satelliittin tekemät mittaukset hiilidioksidista näkyvät kapeina viivoina kuvassa. Nämä havainnot yhdistämällä voidaan laskea NOₓ/CO₂-suhde. Kuva: Hakkarainen et al. 2021. CC BY 4.0.

Ilmakehää mallinnettiin FLEXPART-mallilla ja tuloksena saatiin Matimba-hiilivoimalan NOₓ/CO₂-suhteeksi (2.6 ± 0.6) × 10-3 ja hiilidioksidipäästöiksi noin 60 kilotonnia päivässä, joka on suuruusluokaltaan noin puolet koko Suomen hiilidioksidipäästöistä. Tutkimuksessa lasketut arviot ovat yhteneviä aikaisempien päästöinventaarioiden (esim. ODIAC) tulosten kanssa.

Tutkimusta on tehty erityisesti Euroopan Avaruusjärjestö ESAn rahoittamassa DACES-projektissa, josta voit lukea lisää projektin verkkosivuilta.

Viite:
Janne Hakkarainen, Monika E. Szeląg, Iolanda Ialongo, Christian Retscher, Tomohiro Oda, and David Crisp: Analyzing nitrogen oxides to carbon dioxide emission ratios from space: A case study of Matimba Power Station in South Africa, Atmospheric Environment: Volume 10, 2021.

Lue tieteellinen artikkeli täältä: https://doi.org/10.1016/j.aeaoa.2021.100110

## Tuesday, April 20, 2021

### Analyzing nitrogen oxides to carbon dioxide emission ratios from space: A case study of Matimba Power Station in South Africa

Hi guys,

I just wanted to say that we have a new paper on Atmospheric Environment: X

The paper Analyzing nitrogen oxides to carbon dioxide emission ratios from space: A case study of Matimba Power Station in South Africa was written by myself, Iolanda Ialongo, Monika Szeląg, Christian Retscher, Tomohiro Oda, and David Crisp.

Highlights:
• A new methodology to derive source-specific NOx-to-CO2 emission ratios.
• The method is applied for TROPOMI and OCO-2 satellite observations.
• The mean emission ratio of (2.6±0.6)×10−3 is obtained for Matimba Power Station.
• The annual CO2 emissions for Matimba are ∼60 kt/d.
• The emission estimates are consistent with existing inventories such as ODIAC.

The Journal Pre-proof version is already available online: https://doi.org/10.1016/j.aeaoa.2021.100110

Stay tuned!

Janne

## Wednesday, March 17, 2021

### Pandigital π

Hah!

Yesterday I was watching this Numberphile video

and like Dr. James Grime, I did not like the pandigital formula for π. So I decided to play a little with pen and paper, and found the following
I think it is quite neat. Of course, I knew beforehand that 355/113 approximates π quite well, up to 6 decimal places. Actually, it is called Milü

Cheers,
Janne

## Saturday, February 13, 2021

### Faktoja ilmastonmuutoksesta: hiilinielu vai hiilivarasto?

Monilla tuntuu menevän käsitteet hiilinielu ja hiilivarasto lahjakkaasti sekaisin. Erityisen harmilliselta tämä tuntuu, jos kyseessä on ns. vakavasti otettava tutkija.

Fakta 1. Toisin kun yleensä luullaan vanhat metsät kuten Amazonin sademetsä ei ole voimakas hiilinielu. Nuoret metsät (alle 140 vuotta) taas ovat. Yleistajuisesti voi PNAS-lehdessä julkaistusta tutkimuksesta voi lukea täältä.

(Myös muut tutkimustulokset, kuten omani, ovat päätyneet samaan johtopäätökseen, mutta se ei nyt varsinaiset liity tähän.)

Tämä ei ole oikeastaan yllättävää. On selvää, että hyvin vanha metsä (yli 150 vuotta) ei enää toimi hiilinieluna vaan voi jopa toimia päästölähteenä. Toisaalta nuori parhaassa kasvuiässä oleva metsä toimii voimakkaana hiilinieluna eli se poistaa ilmakehästä hiiltä. Vanha metsä voi toki olla suurempi hiilivarasto kuin nuori metsä, mutta nieluna se ei enää toimi. Jos siis halutaan, että metsät toimivat hiilinieluina olisi oltava mahdollisimman paljon 20–80-vuotiasta metsää.

Fakta 2. Hiilinielu ei ole hiilivarasto. Nielu on prosessi, toiminta tai mekanismi, joka sitoo kasvihuonekaasun, aerosolin tai niiden esiasteen ilmakehästä. Lähde taas tarkoittaa prosessia, toimintaa tai mekanismia, joka vapauttaa kasvihuonekaasun, aerosolin tai niiden esiasteen ilmakehään. Hiilivarasto taas kertoo yksinkertaisesti paljonko johonkin on varastoitunut hiiltä. Yksinkertaisesti voi ajatella, että puuliiterissä olevat klapit ovat hiilivarastossa, mutta muuttuvat päästöiksi saunan pesässä. Kun saunan vieressä oleva puu kasvaa (ja poistaa ilmakehästä hiiltä), niin se toimii nieluna.

Fakta 3. Hiilivaraston muutos ei ole hiilinielu. Tehdään ajatuskoe. Käyt lähimetsässä ja poimit metsästä kilogramman edestä hiiltä. Nyt metsän hiilivarasto on pienentynyt kilogramman verran, mutta ilmakehä ei tiedä tästä mitään eli et ole aiheuttanut päästöjä. Vastaavasti jos viet hiilen takaisin metsään, niin metsä ei ole ymmärrettävästi toiminut ilmakehän hiilinieluna. Jos veistät löytämästä puusta lusikoita, se toimii hiilivarastona (ei nieluna!) ja jos taas poltat löytämäsi puun, aiheutat päästöjä.

Bonus. Puut eivät tulevat maasta vaan ilmasta kertoo Richard Feynman.

## Friday, December 18, 2020

### Virtual Inverse Days 2020

Virtual Inverse Days 2020

University of Helsinki and Finnish Meteorological Institute co-organized 26th Inverse Days of the Finnish Inverse Problems Society. This year the conference was organized virtually, and the chair of the Scientific committee was Tatiana Bubba from the University of Helsinki. The conference had altogether 59 scientific talks and more than 180 registered participants.

Inverse Days is the annual scientific conference of the Finnish Inverse Problems Society (FIPS). The first Inverse Days were organized at the University of Oulu in 1995. This year the conference was organized virtually for the first time due to the global COVID-19 pandemic. The conference was divided in 10 scientific session. The sessions covered both theoretical and applied inverse problems. Application areas included 3D X-ray tomography, electrical impedance tomography, forestry, uncertainty quantification and atmospheric inverse problems among others. The themes followed the themes of the Finnish Centre of Excellence in Inverse Modelling and Imaging. The session number 2 was dedicated to the memory of the late Mikko Kaasalainen (born 1965, died 12 April 2020), who was a professor of mathematics at the Tampere University and an important member of the Finnish Inverse Problems Society. The conference had 25 highlight talks, 29 regular talks and five plenary talks. The plenary talks were given by Chris Johnson (U. Utah), Silvia Gazzola (U. Bath), Valery Serov (U. Oulu), Simon Pfreunschuh (Chalmers U. Tech.) and Barbara Kaltenbacher (U. Klagenfurt). Number of registered participants was all-time record: 185.

In addition to scientific program, the conference also had a special session to celebrate the 60th birthday of Prof. Erkki Somersalo, the founding president of FIPS. The birthday program included scientific talks related to Erkki Somersalo’s research and career along with more humoristic ones. Master of the ceremony was Prof. Samuli Siltanen, the current president of FIPS. For the first time, the Inverse days also had virtual lab excursions. The lab excursion included:

·      X-ray Tomography Laboratory (UH), Alexander Meaney

·      Spectrometers in Atmospheric Measurements (FMI), Tomi Karppinen

·      Log X-ray Systems (Finnos Oy), Jere Heikkinen

·      Biomed. Optical Imaging and Ultrasound Lab (UEF), Aki Pulkkinen

·      Process Tomography Laboratory (UEF), Aku Seppänen

Please pay extra attention to the amazing short film by Aku Seppänen!

The Inverse Days week also included a special session “Women in FIPS”, and the annual meeting of the Finnish Inverse Problems Society.

The Finnish Inverse Prize, annual award of the Finnish Inverse Problems Society, was awarded to Jesse Railo who defended his PhD thesis “Geodesic Tomography Problems on Riemannian Manifold” with distinction at the University of Jyväskylä in 2019. In addition to University of Jyväskylä, Jesse has also worked at U. Tampere, U. Helsinki and the Finnish Meteorological Institute, and is now a Postdoctoral scientist at the ETH Zürich. Congratulations Jesse!

The scientific committee of the conference was

·      Tatiana Bubba (chair)

·      Janne Hakkarainen

·      Marko Laine

·      Matti Lassas

·      Samuli Siltanen

·      Johanna Tamminen

Special thanks to Antti Mikkonen for his work on putting together the Virtual Lab tours, Rashmi Murthy for taking care excellently of the technical arrangements for the conference and Lauri Ylinen for his work on the website. A job well done!

“I love math” logo: Joe Volzer

Conference website: https://www.fips.fi/id2020.php

## Abstract

Computer assisted searches of solutions of the Diophantine equation $x^3 +y^3 +z^3 = k$ have been made since 1954. Thanks to some intelligent people, modern super computing facilities and a YouTube channel, we finally now 66 years later have solutions for all $k < 100$. Here we report an interesting solution when $k=2^3$.

Keywords: Sum of three cubes, Diophantine equation, taxicab number.

## Introduction

In 2015, on a Numberphile video “The uncracked problem with 33” Prof. Tim Browning discussed the Diophantine equation \begin{equation*} x^3+y^3+z^3=k \end{equation*} that has interested mathematicians a quite some time now. As explained in the video, it is know that this equation has no integer solutions for $k \equiv 4$ or $5\,(\mathrm{mod}\,9)$, i.e., $4,5,13,14, 22, 23, \ldots$ For other values of $k$, it has been conjectured that there are infinitely many solutions [1]. In 1953, for $k=3$, Prof. Louis J. Mordell now famously wrote [2]: “I do not know anything about the integer solutions of $x^3+y^3+z^3=3$ beyond the existence of the four sets $(1, 1, 1)$, $(4, 4, -5)$, etc.; and it must be very difficult indeed to find out anything about any other solutions.” This led to the first computer assisted search for $k < 100$ in 1954 [3].

Since those days, for values $k <1\,000$, several searches have been made and more effective algorithms proposed. For example, in 2007, Andreas-Stephan Elsenhans and Jörg Jahnel searched systematically for solutions where the positive integer $k < 100$ is neither a cube nor twice a cube and $|x|,|y|,|z| k < 10^{14}$ [4]. These values “New sums of three cubes” are tabulated here. At this point, they reported 14 unsolved values below $1\,000$: 33, 42, 74,114, 165, 390, 579, 627, 633, 732, 795, 906, 921, and 975.

In 2016, motivated by the original Numberphile video, Sander Huisman extended the search of Elsenhans and Jahnel up to $10^{15}$ [5]. In his report “Newer sums of three cubes,” he found 966 new solutions. The most exciting one was the discovery of the first solution for $k = 74$: \begin{equation*} 74 = (-284650292555885)^3 + 66229832190556^3 + 283450105697727^3. \end{equation*} This result was discussed by Browning on a follow-up Numberphile video “74 is cracked.

It was this follow-up video that got mathematician Andrew Booker hooked. In 2019, he proposed a new algorithm [6] and searched solutions for unsolved values below $1\,000$. He found the first known solution for $k = 33$: \begin{equation*} 33 = 8866128975287528^3 +(-8778405442862239)^3 +(-2736111468807040)^3. \end{equation*} This results was, of course, reported on a Numberphile video “42 is the new 33” indicating that after the discoveries by Huisman and Booker, the only unsolved value below 100 was $k=42$. In his paper, Booker also searched for solutions for $k = 3$, addressing a question of Prof. Mordell, but found none. He also reported the first known solution for $k = 795$: \begin{equation*} 795 = (14219049725358227)^3+(14197965759741571)^3+(2337348783323923)^3. \end{equation*} At this point, also I got interested.

## Methodology

Albeit having a PhD degree in applied mathematics, I don't have any formal education on number theory, let alone, have no experience of computer searches of this type. It was quite obvious, that an exhaustive search for the range $10^{16}$ was out of my reach. The only way I thought I could participate was to randomly sample large values of $x$, $y$ and $z$ and then test if the sum of their cubes gives a small solution, say less than $1\,000$. With a little bit of online research, I wrote the Python code used in this study.

import random

for x in range(10**13):
a = random.randint(10**14,10**18)
b = random.randint(10**14,10**18)
c = random.randint(10**14,10**18)

if a > b:
if a > c:
val = a**3 -b**3-c**3
else:
val = c**3 -b**3-a**3
elif b > c:
val = b**3 -a**3-c**3
else:
val = c**3-a**3-b**3

if abs(val) < 1000:
print(a)
print(b)
print(c)
print(val)
Outa = open("a.txt","w")
Outb = open("b.txt","w")
Outc = open("c.txt","w")
Outval = open("val.txt","w")
Outa.write(str(a))
Outa.close()
Outb.write(str(b))
Outb.close()
Outc.write(str(c))
Outc.close()
Outval.write(str(val))
Outval.close()
break
elif (x % 10**6) == 0:
print(x)

I knew that I would need to sample many many times in order to find a new solution. As noted by Huisman [5], the number of solutions for each decade up to search bounds $B$ in the range from $10^2$ to $10^{14}$ have been roughly $1\,000$ solutions per decade, which is in accordance with [1]. This gave me some initial hope, but I soon realized, after discovering a little error in my calculations, that the probability of finding a new solution was virtually non-existent. I still decided to give it a go.

## Results

On 10 July 2019 I set a computer search for finding a new solution. At the meantime, Andrew Booker had teamed up with Andrew Sutherland and with computer resources from Charity Engine they were searching for new solutions, too. On September 2019, while I was waiting my first solution to appear, they announced a solution for $k=42$: \begin{equation*} (-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3=42, \end{equation*} and again, a new Numberphile video “The Mystery of 42 is Solved” was made. This marked the end of the journey that was started in 1954, as we now had solutions for all $k < 100$. This left the original question of Mordell still unanswered, but three weeks later, they also found: \begin{equation*} 569936821221962380720^3 + (-569936821113563493509)^3 + (-472715493453327032)^3 = 3, \end{equation*} and yet again a new Numberphile video was made “3 as the sum of the 3 cubes.” Booker and Sutherland report these results and two other new solutions on a preprint [7].

I left my computer search to go on for some months, but at some point the computer was rebooted or I stopped the script for running, I don't quite remember anymore which one was it. Anyways, no new solutions were found.

On 15 October 2020, during a Finnish Autumn school break, I decided to revisit this question. It was pretty obvious again, that the original code was not going to give me a solution, so I decided to check what kind of solution I would get if I would limit the search to $10^4$ and would also be satisfied with a solution of the same size. After running the script, it printed the numbers $7\,576, 4\,112, 7\,960$ and $4\,096$ which I arranged to a solution \begin{equation*} -7576^3-4112^3+7960^3 = 4096. \end{equation*} I immediately recognized $4\,096$ as a power of 2, i.e, $2^{12}$, so I decided to check the prime factors of the other numbers, too. I found that $7\,576 = 2^3 \times 947$, $4\,112 = 2^4 \times 257$ and $7\,960 = 2^3 \times 5 \times 199$, so the original solution given by the algorithm could be re-arranged to \begin{equation*} -947^3-514^3+995^3 = 2^3. \end{equation*} I found this solution quite fascinating, as also $k=2^3$ is a cube and the other numbers are relatively large. I decided to right away communicate this mesmerizing solution to the mathematical community via Twitter. At the time of writing this text, a day later, this tweet has already gotten zero likes and retweets combined.

The question that rises with $k$ being a cube is: Are there any others? At first glance, it would seem that requiring $k$ being a cube would make things more complicated, but this is not the case. For example, if we think about it a little bit (and even if we don't), the equations \begin{equation*} a^3+0^3+0^3 = a^3 \end{equation*} and \begin{equation*} a^3+a^3+0^3 = 2 a^3 \end{equation*} always give solutions. This is quite likely the reason why cubes and twice the cubes were not included in the original dataset of Elsenhans and Jahnel: here.

Moreover, if we read the article “Diophantine Equation--3rd Powers” on Wolfram MathWorld, we find that the general rational solution to $x^3+y^3+z^3=k^3$ exists. For example, famously, Plato's number $216 = 6^3$ is the sum of the cubes for the Pythagorean triple $(3, 4, 5)$ \begin{equation*} 3^3 + 4^3 + 5^3 = 6^3 \end{equation*} and is also the case $a=1, b=0$ of Ramanujan's formula: \begin{equation*} (3a^2+5ab-5b^2)^3 + (4a^2-4ab+6b^2)^3 + (5a^2-5ab-3b^2)^3 = (6a^2-4ab+4b^2)^3. \end{equation*} In fact, when $k$ is a cube the question is related to the famous story of Ramanujan and G. H. Hardy, and how the taxicab number $1\,729$ is the smallest integer that can be expressed as a sum of two positive integer cubes in two distinct ways: \begin{equation*} 1729 = 1^3 + 12^3 = 9^3 + 10^3. \end{equation*} If we allow also negative numbers, we can re-arrange the earlier example with Plato's number and have even smaller sum: \begin{equation*} 91 = 6^3 - 5^3 = 4^3 + 3^3. \end{equation*} The smallest sum that our re-arranged solution would give is the following: \begin{equation*} 135\,796\,752 = 995^3-947^3 = 514^3 + 2^3. \end{equation*} On the other hand, Euler conjectured that there were no positive integral solutions to \begin{equation*} a^4 + b^4 + c^4 = d^4. \end{equation*} In 1988, the smallest possible counterexample was found [8]: \begin{equation*} 95800^4 + 217519^4 + 414560^4 = 422481^4. \end{equation*}

## Acknowledgements

Most of my knowledge on this topic comes from the series of Numberphile videos “Sums of three cubes.” Do yourself a favor and go find the originals.

## References

[1] D. R. Heath-Brown, The density of zeros of forms for which weak approximation fails, Math. Comp. 59 (1992), no. 200, 613–623.
[2] L. J. Mordell, On the integer solutions of the equation x2 + y2 + z2 + 2xyz = n, J. London Math. Soc. 28 (1953), 500–510.
[3] J. C. P. Miller and M. F. C. Woollett, Solutions of the Diophantine equation x3 +y3 +z3 = k, J. London Math. Soc. 30 (1955), 101–110.
[4] Andreas-Stephan Elsenhans and J ̈org Jahnel, New sums of three cubes, Math. Comp. 78 (2009), no. 266, 1227–1230.
[5] Sander G. Huisman, Newer sums of three cubes, arXiv:1604.07746, 2016.
[6] A. R. Booker, Cracking the problem with 33, Res. Number Theory 5 (2019), no. 3, 5:26.
[7] Andrew R. Booker and Andrew V. Sutherland, On a question of Mordell, arXiv:2007.01209, 2020.
[8] N. D. Elkies, On A4 + B4 + C4 = D4, Math. of Comp. 51 (1988), 825–835.

## Wednesday, October 30, 2019

### Anthropogenic CO2 emission sources detected from space

Carbon dioxide (CO2) is the most important anthropogenic greenhouse gas and its increase in the atmosphere is responsible for the global warming. CO2 is emitted into the atmosphere by the burning of fossil fuels. Satellite-based observations provide information on the concentration of carbon dioxide (i.e., column-averaged CO2 dry air mole fraction, XCO2) around the globe.
Because of its long lifetime, carbon dioxide increases in the atmosphere and gets transported by the winds very far from the emission sources.
A recent study introduced the concept of XCO2 anomaly (i.e. the difference from the daily background over a certain area), which provides maps of the distribution of the CO2 emission areas worldwide.

Janne Hakkarainen, from the Finnish Meteorological Institute (FMI), comments: “The map shows positive XCO2anomaly over the major industrial areas: China, eastern USA, central Europe, India, and the Highveld region in South Africa. Also, we find positive anomalies over biomass burning areas (for example in Africa and Indochina) during different fire seasons. On the other hand, the largest negative anomalies correspond to the growing season in the middle latitudes”. The anomaly maps are based on the observations collected by the NASA’s OCO-2 (Orbiting Carbon Observatory-2) mission, which began operating in September 2014.
Comparing the XCO2 anomaly maps to short-lived polluting gases, such as nitrogen dioxide (NO2) observations derived from the Copernicus Sentinel-5P’s TROPOMI (TROPOspheric Monitoring Instrument), provides further insights on the spatial patterns of the carbon dioxide emission sources.
Using NO2 concentrations as indicator of anthropogenic fossil fuel combustion, helps in identifying anthropogenic XCO2 enhancement as visible, for example, over the Highveld region in South Africa,” continues Hakkarainen.
 Combining NO2 and CO2 observations enables the detection of CO2 emission sources in South Africa. See for example the plumes from Matimba Power Station.
Satellite observations available with such detail open new opportunities for societal applications, including urban and industrial emission monitoring. For example, satellite observations have been already used in the cleantech sector in order to evaluate the efficacy of their technology in reducing polluting emissions from metal smelting”, comments Iolanda Ialongo, from the Finnish Meteorological Institute. “Future greenhouse gas missions should be designed with a wider coverage than what is currently available, in order to improve the capabilities of monitoring man-made CO2 emissions”.
The results are achieved within the DACES project, which focuses on detecting anthropogenic CO2 emissions sources by exploiting the synergy between satellite-based observations of short-lived polluting gases (such as NO2) and greenhouse gases.
The full publication by Hakkarainen and co-authors can be found at the following link: https://www.mdpi.com/2072-4292/11/7/850

## Friday, May 31, 2019

### Finnish Inverse Problems Summer School 2019

Hi guys,

remember that the FIPS Summer School on Very Finnish Inverse Problems will start on next Monday. I found this picture when preparing for my mini course on Kalman filtering:

Now it looks so fun!

See you next week!

Janne

## Tuesday, March 5, 2019

### My book: Tarinoita matematiikasta

Hi guys,

I just wanted to advertise my new book “Tarinoita matematiikasta: Alkuluvuista Elämän peliin” (in English “Stories about mathematics: From prime numbers to Game of life”). It's a combination between science popularization and recreational mathematics.

Here's the link, if you can't see the widget below: https://www.bod.fi/kirjakauppa/tarinoita-matematiikasta-janne-hakkarainen-9789528007456

Check it out!

Janne

## Sunday, February 24, 2019

### Helsinki Inverse Problems Summer School 2019

Hi Guys,

this is just a short advertisement that “The Finnish Centre of Excellence in Inverse Modelling and Imaging and Finnish Inverse Problems Society (FIPS) are proud to organize a Summer School on inverse problems on June 3-7, 2019.

https://www.fips.fi/summerschool2019

(Summer School on Very Finnish Inverse Problems)

I will give a minicourse on Kalman filtering.

See you there!

Janne

## Sunday, October 21, 2018

### Conway's Game of Life

Hi guys,

today I wanted to try and code the Game of Life. So here it is:

Enjoy!

Janne

## Friday, October 5, 2018

### ILMApilot: Increasing the societal impact of satellite-based atmospheric observations for air quality monitoring

Hi guys,

I just wanted to share Iolanda's pitch from the Finlandia Hall:

See also her new paper on supporting Finnish cleantech sector with satellite data:

Ialongo, I., Fioletov, V., McLinden, C., Jåfs, M., Krotkov, N., Li, C., and Tamminen, J.:
Application of satellite-based sulfur dioxide observations to support the cleantech sector: Detecting emission reduction from copper smelters, Environmental Technology & Innovation, 12, 172-179,
ISSN 2352-1864, https://doi.org/10.1016/j.eti.2018.08.006, 2018.

Cheers,
Janne

## Wednesday, August 1, 2018

### Infinite Magic Squares

Magic squares have interested (recreational) mathematicians for hundreds, if not thousands, of years. A magic square is a $n\times n$ grid filled with integers such that the sum of integers in each row, column and diagonal is equal to a magic constant $M$.

There are various ways to construct magic squares. For odd integers, probably the most famous one is the Siamese method where one also requires that the grid is filled with distinctive positive integers in the range $1, \ldots, n^2$. Below is an example when $n=5$ (Du Royaume de Siam, 1693): \begin{equation*} \begin{array}{|c|c|c|c|c|} \hline 17 & 24 & 1 & 8 & 15 \\ \hline 23 & 5 & 7 & 14 & 16 \\ \hline 4 & 6 & 13 & 20 & 22 \\ \hline 10 & 12 & 19 & 21 & 3 \\ \hline 11 & 18 & 25 & 2 & 9 \\ \hline \end{array} \end{equation*} But what would happen if the grid would be infinite? The simplest “solution” to this problem would be setting all cells to zero \begin{equation*} \begin{array}{c|c|c|c|c} \ddots & \vdots & \vdots & \vdots & \kern3mu\raise1mu{.}\kern3mu\raise6mu{.}\kern3mu\raise12mu{.}\\ \hline \cdots & 0 & 0 & 0 & \cdots \\ \hline \cdots & 0 & 0 & 0 & \cdots \\ \hline \cdots & 0 & 0 & 0 & \cdots \\ \hline \kern3mu\raise1mu{.}\kern3mu\raise6mu{.}\kern3mu\raise12mu{.} & \vdots & \vdots & \vdots & \ddots \\ \end{array} \end{equation*} but this is not what we are really after here. We can obtain a slightly more interesting solution by subtracting the middle value from a Siamese magic square and adding zeros elsewhere: \begin{equation*} \begin{array}{c|c|c|c|c|c|c|c|c} \ddots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \kern3mu\raise1mu{.}\kern3mu\raise6mu{.}\kern3mu\raise12mu{.} \\ \hline \cdots & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \cdots \\ \hline \cdots & 0 & 4 & 11 & -12 & -5 & 2 & 0 & \cdots \\ \hline \cdots & 0 & 10 & -8 & -6 & 1 & 3 & 0 & \cdots\\ \hline \cdots & 0 & -9 & -7 & \mathbf{0} & 7 & 9 & 0 & \cdots\\ \hline \cdots & 0 & -3 & -1 & 6 & 8 & -10 & 0 & \cdots \\ \hline \cdots & 0 & -2 & 5 & 12 & -11 & -4 & 0 & \cdots\\ \hline \cdots & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \cdots \\ \hline \kern3mu\raise1mu{.}\kern3mu\raise6mu{.}\kern3mu\raise12mu{.} & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{array} \end{equation*} This procedure gives us an infinite magic square where the sum in each row, column and diagonal is equal to zero. This still does not feel quite right as the infinite square has nonzero elements only in the middle. But what about the infinite square below? $$\tag{\infty}\label{good} \begin{array}{c|c|c|c|c|c|c|c|c} \ddots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \kern3mu\raise1mu{.}\kern3mu\raise6mu{.}\kern3mu\raise12mu{.} \\ \hline \cdots & \mathbf{-1} & +1 & -1 & +1 & -1 & +1 & \mathbf{-1} & \cdots \\ \hline \cdots & +1 & \mathbf{-1} & +1 & -1 & +1 & \mathbf{-1} & +1 & \cdots \\ \hline \cdots & -1 & +1 & \mathbf{-1} & +1 & \mathbf{-1} & +1 & -1 & \cdots \\ \hline \cdots & +1 & -1 & +1 & \mathbf{-1} & +1 & -1 & +1 & \cdots \\ \hline \cdots & -1 & +1 & \mathbf{-1} & +1 & \mathbf{-1} & +1 & -1 & \cdots \\ \hline \cdots & +1 & \mathbf{-1} & +1 & -1 & +1 & \mathbf{-1} & +1 & \cdots \\ \hline \cdots & \mathbf{-1} & +1 & -1 & +1 & -1 & +1 & \mathbf{-1} & \cdots \\ \hline \kern3mu\raise1mu{.}\kern3mu\raise6mu{.}\kern3mu\raise12mu{.} & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{array}$$ It already looks quite magical with only +1 and -1 entries. But where would the series in each row, column and diagonal sum to? One can note that up, down, left and right from each diagonal cell we have Grandi's series $\sum_{n=1}^{\infty} (-1)^{n-1} = 1-1+1-1+1-1+\ldots$ Grandi's series is Cesàro summable, with Cesàro sum $\frac{1}{2}$. One way to justify this value is to set \begin{equation*} S = 1-1+1-1+1-1+\ldots \end{equation*} and then note that $S = 1-S$, and hence $S= \frac{1}{2}$. Now one may calculate \begin{align*} \ldots+1-1+1-1+\ldots &= -1 + \sum_{n=0}^{\infty} (-1)^n + \sum_{n=0}^{\infty} (-1)^n \\ & = -1 + \frac{1}{2}+\frac{1}{2}=0. \end{align*} Thus, the series in every row and column are Cesàro summable, with Cesàro sum $0$. But what about the diagonals? In both diagonals, after the center cell we have $-1-1-1-1-\ldots$ One can recognize this series as a specific case of the Riemann zeta function \begin{equation*} \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} \end{equation*} when $s=0$. We have that $\zeta(0)=-\frac{1}{2}$, thus one may write $-1-1-1-1-\ldots = -\zeta(0)=\frac{1}{2}$. In fact, this series is related to Grandi's series via the Dirichlet eta function \begin{equation*} \eta(s) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^s} = (1-2^{1-s})\zeta(s). \end{equation*} Now when $s=0$, we have that \begin{equation*} 1-1+1-1+\ldots = \eta(0)=-\zeta(0) = -1-1-1-1-\ldots \end{equation*} The diagonals are \begin{align*} \ldots-1-1-1-1-\ldots &= -1- \sum_{n=1}^{\infty}\frac{1}{n^0}-\sum_{n=1}^{\infty}\frac{1}{n^0} \\ &=-1-\zeta(0)-\zeta(0)=0. \end{align*} Now the infinite square (\ref{good}) is indeed an infinite magic square as the series in every row, column and diagonal are equal (in above sense) to the magic constant $M=0$. We note that by multiplying the infinite magic square (\ref{good}) with an integer $a$, we obtain another infinite magic square with $M=0$. If we set $a=-1$, we obtain the “evil twin:” \begin{equation*} \begin{array}{c|c|c|c|c|c|c|c|c} \ddots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \kern3mu\raise1mu{.}\kern3mu\raise6mu{.}\kern3mu\raise12mu{.} \\ \hline \cdots & \mathbf{+1} & -1 & +1 & -1 & +1 & -1 & \mathbf{+1} & \cdots \\ \hline \cdots & -1 & \mathbf{+1} & -1 & +1 & -1 & \mathbf{+1} & -1 & \cdots \\ \hline \cdots & +1 & -1 & \mathbf{+1} & -1 & \mathbf{+1} & -1 & +1 & \cdots \\ \hline \cdots & -1 & +1 & -1 & \mathbf{+1} & -1 & +1 & -1 & \cdots \\ \hline \cdots & +1 & -1 & \mathbf{+1} & -1 & \mathbf{+1} & -1 & +1 & \cdots \\ \hline \cdots & -1 & \mathbf{+1} & -1 & +1 & -1 & \mathbf{+1} & -1 & \cdots \\ \hline \cdots & \mathbf{+1} & -1 & +1 & -1 & +1 & -1 & \mathbf{+1} & \cdots \\ \hline \kern3mu\raise1mu{.}\kern3mu\raise6mu{.}\kern3mu\raise12mu{.} & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{array} \end{equation*} If we square the values of the magic square (\ref{good}), we obtain an infinite square full of ones: \begin{equation*} \begin{array}{c|c|c|c|c} \ddots & \vdots & \vdots & \vdots & \kern3mu\raise1mu{.}\kern3mu\raise6mu{.}\kern3mu\raise12mu{.} \\ \hline \cdots & 1 & 1 & 1 & \cdots \\ \hline \cdots & 1 & 1 & 1 & \cdots \\ \hline \cdots & 1 & 1 & 1 & \cdots \\ \hline \kern3mu\raise1mu{.}\kern3mu\raise6mu{.}\kern3mu\raise12mu{.} & \vdots & \vdots & \vdots & \ddots \\ \end{array} \end{equation*} In every direction we have \begin{align*} \ldots+1+1+1+1+\ldots &= 1+ \sum_{n=1}^{\infty}\frac{1}{n^0}+\sum_{n=1}^{\infty}\frac{1}{n^0} \\ &=1+\zeta(0)+\zeta(0)=0. \end{align*} Thus, it is also an infinite magic square. This new square, however, seems little bit less magical than the original one (\ref{good}).

### Global XCO2 anomalies as seen by Orbiting Carbon Observatory-2

I would like to advertise our latest work "Global XCO2 anomalies as seen by Orbiting Carbon Observatory-2." It was published yesterday as discussion paper in Atmos. Chem. Phys. Discuss. (ACPD). Here's a little preview:

Hakkarainen, J., Ialongo, I., Maksyutov, S., and Crisp, D.: Global XCO2 anomalies as seen by Orbiting Carbon Observatory-2, Atmos. Chem. Phys. Discuss., https://doi.org/10.5194/acp-2018-649, in review, 2018.

Janne

## Monday, April 30, 2018

### Pitch like a pro!

As you might remember I was in the Skolar Award finals at Slush 2017. This week we will have a little alumni meeting organized by the Kaskas Media. The 2016 Skolar award winner Virpi Virjamo listed her 5 tips for pitching your research idea like a winner.

So here are my 5+2 tips for pitching like a pro!

I think this was the best advice that I got. Scientists tend to oversimplify when they talk about their work to non-experts. But people are smart and they will understand. If you leave your science out, what is left in your pitch!? You are not a salesman!

2. Keep it simple!
Now this second point seems to contradict what I just said in the first point. It is not. You should keep it simple, but do not oversimplify! Easy as that.

3. Practice, practice, practice!
This may sound self evident, but it is important. You should learn your pitch by heart. Be a pro. I still remember my pitch! Practicing is a good advice also to your scientific presentations. (When I gave my pitch at Slush, I made a little mistake and skipped one slide. Since I knew the pitch, I could do it on autopilot and think how to back it up in the end. It was surreal feeling).

When people try to help you, you should listen. And make your pitch better. They will have good ideas. But remember, it is still your pitch and you have to deliver.

5. The structure! Problem, solution, vision.
This is a technical advice, but still very important. I think the structure should be a) problem, b) solution, and c) vision. So first tell what is the problem that you are solving, not the latest thing in your own research. Then give Your solution to this problem. Finally, tell how your solution will change the world.

6. Be yourself?
This is an advice that you are like to get for someone else but me. I say: Don’t be yourself, be a better version of yourself! In my daily life, I’m quite calm and low-energy guy. In a good way. In my pitch, I wanted to give 120% and be super energetic. It worked out for me.

7. Have fun!
Don’t think about  or the victory. It will show and you’ll regret it later. Have fun! This is a unique opportunity. Make the most of it!

## Friday, January 5, 2018

### 2018: Centre of Excellence of Inverse Modelling and Imaging

Happy New Year 2018!

They are plenty of new things in my academic life. I am officially back at Finnish Meteorological Institute, now with the title “Senior Research Scientist.” Because of the large organization change at FMI, the name of my group is now entitled “Greenhouse Gases and Satellite Methods” in Earth Observation Research Unit.

 Centre of Excellence of Inverse Modelling and Imaging: applications.
Probably the biggest news story is that now also our FMI team is part of the new 2018–2025 “Centre of Excellence of Inverse Modelling and Imaging.” The University of Helsinki team, the team where I had the pleasure of visiting for the last seven months, coordinates this Centre! We will have the kick-off meeting next week at Uunisaari. It will be fun!

Regarding this site, I updated my Biographical Sketch. Enjoy!

In the next post will recap my experiences from Slush 2017 Skolar Award.

Stay tuned!

Janne

## Thursday, November 30, 2017

### Slush 2017 Skolar Award Science Pitching

Hi guys!

I am very happy to say to that I am on of the ten finalists for the Skolar Award Science Pitching competition happening at Slush 2017 tomorrow! This means that I have amazing change to pitch my research idea in front of 2 000 people for three minutes! I will give everything I got!

Here’s my research idea in a nutshell:
As everyone should realize by now, climate change is one of the biggest threats to humanity. The main culprits are atmospheric greenhouse gases, GHGs, that increase the global temperature. This research aims to identify the main man-made areas of greenhouse gases with the help of space-based observations. Those offer a sustainable and cost-efficient tool to estimate the impact of human activities on our environment.
You can meet the finalist at http://www.slush.org/news/meet-science-pitching-2017-finalists/ and https://skolaraward.fi/finalists/

Yesterday morning I was in the Yle morning show talking about my idea. The interview went really well, and you can watch it from Yle Areena: https://areena.yle.fi/1-4299989

Yesterday evening we had The Skolar Award Premiere at the Kaskas Media HQ. Now I should be ready!

See You at Slush Central stage tomorrow at 13:10 Finnish time. You can follow the event online at http://www.slush.org/live/#central-stage

Crazy!

Janne