A **locally compact group** is a type of topological group that has the property of local compactness in addition to the group structure. Let's break down the definitions: 1. **Topological Group**: A group \( G \) is equipped with a topology such that both the group operation (multiplication) and the inverse operation are continuous.

Hodge theory is a central area in differential geometry and algebraic geometry that studies the relationship between the topology of a manifold and its differential forms. It is particularly concerned with the decomposition of differential forms on a compact, oriented Riemannian manifold and the study of their cohomology groups. The key concepts in Hodge theory are: 1. **Differential Forms**: These are generalized functions that can be integrated over manifolds.

In algebraic geometry and related fields, a **coherent sheaf** is a specific type of sheaf that combines the properties of sheaves with certain algebraic conditions that make them suitable for studying geometric objects.

The Riemann–Roch theorem is a fundamental result in algebraic geometry and complex analysis that provides a powerful tool for calculating dimensions of certain spaces of sections of line bundles on smooth projective curves.

The projective tensor product is a construction in functional analysis and tensor algebra that generalizes the notion of the tensor product of vector spaces to arbitrary topological vector spaces. It is particularly useful when dealing with dual spaces and various types of convergence in topological spaces.

A torus knot is a special type of knot that is tied on the surface of a torus (a doughnut-shaped surface). More formally, a torus knot is defined by two integers \( p \) and \( q \), where \( p \) represents the number of times the knot winds around the torus's central axis (the "hole" of the doughnut) and \( q \) represents the number of times it wraps around the torus itself.

As of my last knowledge update in October 2021, "Georgia Rodgers" does not refer to a widely recognized public figure, event, or concept. It's possible that the name could refer to a private individual or perhaps has gained prominence after my last update.

Poor countries don't have a lot of money.

Therefore, you have to pick a few key the next big thing deep tech bets, and invest on those enough.

These have to be few, because your country is poor, and so you can't invest on everything.

Therefore, the bets have to be well selected, because it is useless to make several insufficient investments: you have to pick a few ones, and put enough time and money into each one of them for them to stand any chance. These bets should be made and reevaluated on 5/10 year horizons.

The key things that you have to select are:

And then you only tax those companies heavily when the start to bring in real money. These are startups remember! You only need 5 unicorns a year to call it a success. And countries should not be greedy and invest through equity, but rather recoup their investment through taxation alone.

Ciro's second removed uncle, who was a physicist at the University of Campinas, one of the best universities in the country, told him an anecdote. He had moved from fusion energy research to solar cell research. At some point, there was a research lab that needed 10 million to buy a machinery critical for their experiment. They asked and asked, and finally the government gave them only 2 million. So in the end they spent those 2 million in random ways, but of course did not achieve their research goal and no money came out of it.

He also explained how as a result of the insufficient investments, he felt clearly that some of the semiconductor production facilities related to solar power he saw simply were not able to control the production process adequately to produce consistent silicon. As a result, everything failed sooner or later as people found more and more bugs that they did not have the time to solve.

Another key investment is enticing back experienced exchange-students who have learnt new techniques to be heads of laboratory/founders to back in your country.

A fantastic initiative from Brazil for example is BRASA, which aims to put together Brazilian exchange students to make a difference back in Brazil.

Do not try to forbid external companies from selling in your country. Instead, fund your own companies to be able to fight the external market off. And if they can't, let them die and pick a different bet. Video "How Taiwan Created TSMC by Asianometry (2020)" has a good mention. Protectionism is something that Brazil notably tried to do, and look at what it led, not a single international success.

www.tritoncycles.co.uk/urban-bikes-c4/hybrid-c30/trek-fx-3-equipped-urban-bike-2021-p39860/s102315 XL color Dnister Black. £675.00 plus £35.00 delivery. Disc breaks, 2x9 gears. Bought 2023 on a bit of a sale.

The 2020 one on official website: www.trekbikes.com/gb/en_GB/bikes/hybrid-bikes/fitness-bikes/fx/fx-3-disc/p/28474/ archive.ph/wip/jMDdw TODO find 2021.

Specs upon buying:

Weight after bought measured differentially on bathroom scale: 13.5 kg

Gears: 11-36T on back, 46-30T on front. So min speed is 30/36 = 0.83 and max is 46/11 = 4.18, so range is 4.18 / 0.83 ~= 5. Compared to Kross which was 11-32 on back vs 48-28 on front, so min 28/32 = 0.875 and max 48/11 = 4.36. So the ranges are very comparable! This confirms Ciro's suspicion that 3 speeds on cranckset is just useless.

I was sized just at the lower limit of XL, but decided to go for it. The seatpost goes all the way into the tube, which is a good thing, might help with noises?

Frame serial under bottom bracket: WTU173C0374S. Registered at: www.bikeregister.com/

Initial impressions:

I'm unable to place a U-lock carrier on this bike in a way that doesn't block one of the two bottles inside frame, or hits leg on if on side. Tried both Kryptonite and Abus ones I have lying around. Shame. Also the Topeak MTS Trunkbag does not fit well on the pannier rack, it is not wide enough to strap: 11cm outer width, 9cm inner width, same geometry as that of the Liv bike.

Log:

2026-04-15: was hearing back wheel hit something periodically. First I separated the disk breaks with the tool but didn't solve. Then I noticed broken spoke, 9th clockwise from tube valve. I replaced it, but unfortunately spreading the disk break made me lose back breaks so I had to go into break fluid territory. Some conclusions: you can't do break fluid without the break pads and disk break in the break. This opens it up for contamination. The correct thing to do is to put a plastic protection on top of them. I spilled fluid on break at one point, it was horrible, I ended up changing the break pad which improved things a bit. Overall break felt slightly worse than before but passable.

2026-02-13: chain reched .7, changed it to new KMC X9. x9 9 speed chain 114 links. www.amazon.co.uk/dp/B07J38GX7L Started slipping like crazy on cassette when I pedaled hard, changed it to www.amazon.co.uk/dp/B073H3DST7 SHIMANO CS-HG400 9-speed cassette (11-36 teeth) which internet said was compatible and it seems to be. Didn't manage to remove it and needed help from bike shop as usual. After that it stopped slipping.

2025-09-30: kickstand broken while taking it on car. Proceeding to buy Bontrager Rear Mount Adjustable Kickstand www.trekbikes.com/gb/en_GB/equipment/bike-accessories/bike-kickstands/bontrager-rear-mount-adjustable-kickstand/p/36015/ 20 pounds + 4 delivery. The new kickstand is slightly different from the previous one, it is less slick looking, but perhaps a bit more sturdy, notably the little arm is shorter which might help prevent the previous failure mode. It came with 3 types of screws, but the two of correct diameter were a bit shorter or longer than the old ones so I just kept the old ones.

2025-09-13: chain reached 0.75, rebuying KMC X9. x9 9 speed chain 114 links. www.amazon.co.uk/dp/B07J38GX7L

2025-08: rear mudgard broke in the exact same way as last time, so I'm saying fuck this bullshit and I just fixed it up with some wire which will likely be more resistant than the shitty metal thing they had.

2025-06-26: I suddenly lost the ability to shift the front gear, cable seemed fine, upon inspection I noticed that the pedal could move sideways. Maybe linked to screeching sound that has been coming from that area for a while. As per www.youtube.com/watch?v=cPQyQnNdews leading to www.youtube.com/watch?v=QqBtB8Kyl2U this is a different system from the Kross one, two piece compression slotted rather than 3 piece. The side to side issue is mentioned precisely at: youtu.be/QqBtB8Kyl2U?t=357 Buying the standalone removal tool: www.amazon.co.uk/Professional-Extracto-Compatible-Hollowtech-Maintenance/dp/B0DP6QF11X Fixing it fixed the shift issue, but I noticed that the little plastic piece was broken. Also a few days later it became loose again during a ride, but thankfully I had the tool and put it back mid rid. TODO wanted to rebuy it but it's really hard to find... Going to try this one: www.bikeparts.co.uk/products/shimano-altus-fc-mt210-chainset-46-30-9-speed-black-170-mm-w-o-chainguard 175 mm and see if it comes with both arms. Bought it. has both arms. But I can't fit it, appears to be 1 or 2 mm short compared to the other cranck, the new arm does go on the old crank. There is a small rubber seal that is perhaps not too well inserted which might be to blame? Fuck my life! On the old one, it seems to be fully inserted into a tight hole. I didn't manage to insert the new one enough to make it fit, but if I remove it then it fits... just removing it for now then, how important could that be? Famous last words. Web says 12-14 nM torque on holding screws.

2025-06: rebuying 2x Shimano B05S brake pads, breaking is not too great and I don't want to repeat last year's episode of on-road failure. Will also be bleeding for the second time, God help me. The old pads weren't too bad, but I replaced them anyways. The bleeding was a failure and the brakes were too soft to be useful at the end, fuck. Went to two bike shops, both said bleeding not too bad, one said likely problem was contamination/friction issues. Tried cleaning rotors further, improved. Buying sandpaper and new rotors as well 2x SHIMANO SM-RT26 6-Bolt Disc Rotor and also got two more brake pads, totalling 3 new before I start putting them in with new rotor.

2025-04: rear shift cable snapped near shifter when going over a cattle grid, was already a bit off in the previous days after last adjust I then noticed. I'll buy a SHIMANO Gear Inner Wire Y60098070 on Amazon and might pass by bike shop as well I'll see.

2025-03: rear middle metal connection broke off read mudguard. Rebuying £21.04 from: www.tritoncycles.co.uk/frames-forks-c6/frame-fork-spares-c152/eurofender-snello-700c-fenders-46mm-rear-p33330/s93941 Quite disappointed with that, didn't come with any screws, though perhaps these are frame dependant. And the drilling was a bit different, this model seems to be intended to have some wiring passing inside via some rubber tubes that I removed, and unfortunately the hole on front was too high up and didn't match up with the screw hole on frame. I just removed the broken metal piece from it and put it on the old mudguard.

2024-09: buying CHUMXINY Brake Bleed Kit: www.amazon.co.uk/dp/B0CXDQ2S1S I hope I don't regret this. But I can't find an original one on Amazon either so... £12.74. Quality is a bit crap but seems to work. The 7mm wrench however is so bad that it can't unscrew a 7mm nut.

2024-09: chain reached 0.75, rebuying KMC X9. x9 9 speed chain 114 links: www.amazon.co.uk/dp/B07J38GX7L

2024-09: when going down a hill in intense breaking, noise started coming from back disk brake. Tried to adjust, but made it worse, had to bail out on nearby bike shop. Readjusted, told me to replace back pads and likely bleed. Bought two Shimano B05S brake pads to replace the B01S that came with the bike, Internet says compatible. Brake maintenance finally caught up with me, it was a great summer and I kept putting it off. Changed just the back one for now, front still has life it seems.

2024-06: noticed a sideways lump on back tyre between spoke 23 and 24 when washing. The only sign of damage was internal near grip part, I could see some clear shearing... God those tyres are crap, or at least crap to what I've subjected them. Inner tube and outside seem perfect, so weird. If only the Schawlbe had worked! But OK now I'm forced to get something new and better, thinking:They had a Schwalbe Marathon Plus 700x35c (non-Tour, as that one didn't fit as per previous experiment) on the shop and I went for it. They said 35c was the maximum recommended for that mudgard which is a bit of a shame, but good to know. Also getting one from Amazon to match on front as shop only had one.

2024-04: micropuncture on front tube? Could not identify during ride, refilled and was OK to get home. Almost no pressure in middle of ride. Passed through some deep water which may have made things worse? Had a small pice of glass-like material stuck in tyre, but didn't seem to penetrate? One day and a half later, was full empty again after full fill without riding, so confirming micropuncture theory. Buying 2x Conti Tube Cross 28 (700C) presta inner tubes.

Tried to put a spare Schwalbe Men's Marathon Plus Tour 700x40C on the bike to get better stability. It seemed to fit the wheel quite well, but unfortunately failed to go well in the fork because it was too tall and hit the mudgard. Without mudgard it might work. Measured height was 6cm for Schwalbe vs 5.3cm for the stock one.

2024-02: another broken spoke... what crappy weak wheels!!! Bought Bavel 290 mm spokes 36pcs 14G J Bend Silver. Broken position was 22nd clockwise from inner tube valve hole.

2023-12: chain reached 0.75, rebuying KMC X9. x9 9 speed chain 114 links.

2023-12: broken spoke! Also had to replace nipple because broke inside nipple. At least learned how to change tire at last.

2023-05: buying another pair of Shimano 324 combination pedals for it: www.wiggle.com/p/shimano-m324-combination-pedals

2023-05: rebuying another Topeak Aero Wedge saddle bag for this new bike

2023-05: 2x Elite custom race cage water bottle frame cages, felt great, much better than those on the Kross! www.elite-it.com/en/products/bottle-cages/fibre-reinforced-material-frp/custom-race

2023-05: ParkTool TR1 tyre levers + GP-2 super patch kit

2023-05: Kask Rapido Medium Black/Yellow fluo helmet, got a more racy helmet too! 69 pounds, way more any previous helmet I've owned: www.kask.com/en/cycling/helmets/rapido_24_2.htm

2023-05: Lezyne hand pump, I think Sport Drive HP? ride.lezyne.com/collections/hand-pumps/products/sport-drive-hp

2023-05: Lezyne V-Pro 10 multi-tool ride.lezyne.com/products/v-pro-tool No spoke wrench unfortunately.

Initial equipment:

Annotated copypasta:Weight limit: This bike has a maximum total weight limit (combined weight of bicycle, rider and cargo) of 136 kg (300 lb)

Fatty acids are carboxylic acids with long hydrocarbon chains, which can vary in length and degree of saturation. They are essential components of lipids (fats and oils) and play crucial roles in biological processes. ### Key Features of Fatty Acids: 1. **Structure**: - **Carboxyl Group**: Each fatty acid has a carboxyl (-COOH) group at one end, which gives it acidic properties.

This shows that viewing electromagnetism as gauge theory does have experimentally observable consequences. TODO understand what that means.

In more understandable terms, it shows that the magnetic vector potential matters where the magnetic field is 0.

Similar to quantum jump in the Bohr model, but for the Schrödinger equation.

The idea the the wave function of a small observed system collapses "obviously" cannot be the full physical truth, only a very useful approximation of reality.

Because then are are hard pressed to determine the boundary between what collapses and what doesn't, and there isn't such a boundary, as everything is interacting, including the observer.

The many-worlds interpretation is an elegant explanation for this. Though it does feel a bit sad and superfluous.

Mirror images.

Key exmaple: d and L amino acids. Enantiomers have identical physico-chemical properties. But their biological roles can be very different, because an enzyme might only be able to act on one of them.

A more concrete and easier to understand version of it is the more photon-specific Poincaré sphere, have a look at that one first.

Rational representation can refer to different concepts depending on the context, but it is most commonly associated with mathematics, particularly in number theory and algebra. 1. **In the context of numbers**: A rational representation usually refers to the expression of a number as a ratio of two integers.

Adds special relativity to the Schrödinger equation, and the following conclusions come basically as a direct consequence of this!

Experiments explained:

Experiments not explained: those that quantum electrodynamics explains like:See also: Dirac equation vs quantum electrodynamics.

The Dirac equation is a set of 4 partial differential equations on 4 complex valued wave functions. The full explicit form in Planck units is shown e.g. in Video 1. "Quantum Mechanics 12a - Dirac Equation I by ViaScience (2015)" at youtu.be/OCuaBmAzqek?t=1010:Then as done at physics.stackexchange.com/questions/32422/qm-without-complex-numbers/557600#557600 from why are complex numbers used in the Schrodinger equation?, we could further split those equations up into a system of 8 equations on 8 real-valued functions.

A random field you add to make something transform locally the way you want. See e.g.: Video "Deriving the qED Lagrangian by Dietterich Labs (2018)".