Likely came in a bundle: www.gak.co.uk/en/yamaha-p45-digital-piano-bundle/908294

Two ways to see it:

2019-11 www.amazon.co.uk/B00I8JC4WC Mr Muscle 5 in 1 Shower Shine, 500 ml

2019-11 www.amazon.co.uk/dp/B00FGOY51A Lakeland Moth Stop Moth Killer Carpet & Fabric Spray, 500ml

2019-10 Mykal Sticky Stuff Remover 250ml. Helped remove sticky tape adhesive from surfaces. Still required a lot of elbow grease, but worked. www.amazon.co.uk/dp/B000TAT4GM (archive).

30L tall pedal bin light blue. Liner type: "Tall Bin Liners 30 L".

"Rail Riders" can refer to various things depending on the context, but it generally can relate to the following: 1. **Recreational Activity**: In some contexts, "Rail Riders" refers to people who engage in activities on railroads, often using specialized equipment to ride on railway tracks. This can include a variety of activities, like rail biking, where individuals ride bikes or carts specifically designed to travel on rails.

$M_n({\mathbb{F}}_q)$ accounts for them all, which we know how to do due to the classification of finite fields.

So we see that the classification is quite simple, much like the classification of finite fields, and in strict opposition to the classification of finite simple groups (not to mention the 2023 lack of classification for non simple finite groups!)

A ring where multiplication is commutative and there is always an inverse.

A field can be seen as an Abelian group that has two group operations defined on it: addition and multiplication.

And then, besides each of the two operations obeying the group axioms individually, and they are compatible between themselves according to the distributive property.

Basically the nicest, least restrictive, 2-operation type of algebra.

Examples:

One of the defining properties of algebraic structure with two operations such as ring and field:This property shows how the two operations interact.

These are apparenty an important part of transcriptional regulation given the number of modifications they can undergo! Quite exciting.

There's exactly one field per prime power, so all we need to specify a field is give its order, notated e.g. as $GF(n)$.

Every element of a finite field satisfies $x^{order}=x$.

It is interesting to compare this result philosophically with the classification of finite groups: fields are more constrained as they have to have two operations, and this leads to a much simpler classification!

2019-12-26: applied Ronseal mould killer on external north east living room walls, had widespread light brown mould spots, and more localized black mould spots. The water insulation here is bad, possibly due to being on the last floor. Also applied next to window sills on those walls.

A function:

A vector field with a bilinear map into itself, which we can also call a "vector product".

Note that the vector product does not have to be neither associative nor commutative.

Examples: en.wikipedia.org/w/index.php?title=Algebra_over_a_field&oldid=1035146107#Motivating_examples

There are 3: real numbers, complex numbers and quaternions.

Notably, the octonions are not associative.

Soundness is a term that can have different meanings depending on the context in which it is used. Here are a few common interpretations: 1. **Logic and Mathematics**: In the context of formal logic and mathematics, soundness refers to a property of a deductive system (like a proof system or a formal language). A system is considered sound if every statement that can be derived within that system is also true in its intended interpretation.

2021-01: kitchen extractor right lamp burnt. Replaced with another Eveready.

2021-01: left back bulb burnt, put in another LE GU10 LED Bulbs

2020-11: kitchen extractor left lamp burnt again. Did I forget to report a right side burn earlier, or is there something wrong with the left one? Also can't find the second bulb that was likely bought last time, so likely the right one burnt and the other bulb was used for it. Buying www.amazon.co.uk/dp/B00LOBUDSS "4 x Eveready 20 W 12 V G4 Dimmable Halogen Capsule Light Bulbs Pack of 4" for 3.66 pounds. Worked, felt a bit more yellow than the previous one. Packet documents 2000 hours, 2800K color, 8000+ switches.

2020-08: kitchen right back had been out for a while, bought: www.amazon.co.uk/dp/B00LN0RPA4 LE GU10 LED Bulbs, Warm White 2700K, 50W Halogen Lamp Equivalent, 4W 350lm, 120° Wide Beam Angle, Recessed or Track Light Bulb, Pack of 5 (Energy Class A++), 5 for 10 pounds.

2019-11: front right kitchen lamp burnt, Sylvania GU10 50W. 50W was likely some useless "incandescent equivalent" measure. Replaced: GE LED 5W 345 lumens 2700K Warm 15k hours, which looks exactly the same.

2019-05: kitchen extractor left lamp burnt. Was: DURA G4 12V20W: www.amazon.co.uk/Halogen-Light-Bulbs-Lamps-5watt/dp/B003IVP12A Replaced: Sainsbury's halogen G4 20W12V.