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A Summer of Maths (ASoM) 2016

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How would you guys recommend learning Numbers and Sets? Reading a book or Dexters lecture notes (they're not that detailed)?

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Reply 441

drandy76

Original post by LondonGamer

How would you guys recommend learning Numbers and Sets? Reading a book or Dexters lecture notes (they're not that detailed)?

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Prefer books in this scenario

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Reply 442

LondonGamer

Original post by drandy76

Prefer books in this scenario

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Can you recommend a book?

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Reply 443

drandy76

Haven't actually covered this so don't really know, someone else should know a decent one though

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Reply 444

Insight314

Original post by LondonGamer

Can you recommend a book?

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I used "Numbers and Proofs" by Allenby last year, but I don't think it covers all of N&S. It does cover most of it though!

Original post by EnglishMuon

Also does anyone know of some Cambridge like 'example sheets' from other unis? Currently Im looking at the the Oxford ones but wandering if theres something more alike.

Why don't you use the Cambridge example sheets?

Reply 445

math42

Original post by EnglishMuon

Also does anyone know of some Cambridge like 'example sheets' from other unis? Currently Im looking at the the Oxford ones but wandering if theres something more alike.

Warwick is good; taking a glance at Cambridge they're not massively dissimilar but I'd have to sit down and bother with those exercises to really ascertain the style. But most questions are even too easy for me lol, though later ones can be more engaging.

Reply 446

username1162972

Original post by Insight314

I used "Numbers and Proofs" by Allenby last year, but I don't think it covers all of N&S. It does cover most of it though!

Why don't you use the Cambridge example sheets?

He probably wants to save them for term time or something.

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Reply 447

username1162972

Original post by 1 8 13 20 42

Coukd you drop a link for warwick ones?

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Reply 448

Insight314

Original post by 1 8 13 20 42

I swear the example sheets are made more difficult than Tripos questions to make the exam questions seem easier when you sit them. Maybe you only took a glance at the easier first ones?

Original post by physicsmaths

He probably wants to save them for term time or something.

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Yeah, I got that but I don't think it matters that much whether you do them now or then. You can re-do them during term time if you want, and then you get even more practice (or use that time to get more practice with some other course/do other exercises).

Reply 449

math42

Original post by Insight314

I swear the example sheets are made more difficult than Tripos questions to make the exam questions seem easier when you sit them. Maybe you only took a glance at the easier first ones?

No I meant Warwick sheets are mostly quite easy. I meant stylistically they didn't seem too dissimilar, from my very brief glances at the questions.

Original post by physicsmaths

Coukd you drop a link for warwick ones?

The only place I know to get them is password protected. Many are probably floating around unprotected somewhere (the analysis I booklets certainly are, and are easy to find) but I dunno where.

Reply 450

username1162972

Original post by Insight314

I swear the example sheets are made more difficult than Tripos questions to make the exam questions seem easier when you sit them. Maybe you only took a glance at the easier first ones?

Yeah, I got that but I don't think it matters that much whether you do them now or then. You can re-do them during term time if you want, and then you get even more practice (or use that time to get more practice with some other course/do other exercises).

I guess more different questions is what he wants rather then redoing them.Me personally will be doing oxford **** when i get around to it. Maybe Imperial if they are online. Then cambridge ones if i get in.

Reply 451

LondonGamer

Original post by Insight314

I used "Numbers and Proofs" by Allenby last year, but I don't think it covers all of N&S. It does cover most of it though!

Why don't you use the Cambridge example sheets?

Cheers, I'll take a look at that book. Can you recommend something that covers the rest of N&S?

By the way, are you a Cambridge student?

Reply 452

drandy76

Original post by LondonGamer

Cheers, I'll take a look at that book. Can you recommend something that covers the rest of N&S?

By the way, are you a Cambridge student?

Insight is a Maths and physics(theoretical physics nerd) applicant

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Reply 453

Krollo

Original post by Insight314

I was going to do D&R during summer, but was told it is best to self-study the fundamentals of high-level mathematics first, so I chose to do V&M and Groups during summer. How are you finding D&R so far? I don't think the content itself requires VC, but the example sheets do seem to include divergence and what not.

On a side note, are you planning on taking Maths with Physics at Cambridge? :tongue:

Spoiler

Almost all of the notes, so far as I can see, are manageable without VC; and it seems decently manageable from my perspective (except rotating frames, can't visualise them for s**t). The example sheets / exams sometimes need VC, so I'll probably look at VM and VC as my next couple (if I can be bothered).

(I probably ought to point out that my modus operandi is rather different to most other people's in these parts. For better or for worse, I'm trying to get a basic familiarity with the courses rather than learn one or two to exam standard, so I'm not doing a tremendous proportion of the example sheets at the moment. Feel free to judge me)

And I'm probably not doing maths with physics. Practicals would kill me.

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Reply 454

EnglishMuon

In some instances when I am showing two groups are isomorphic, would it be fine if I dont directly state the isomorphism if its trivial? For example I just did the question " If

G

is non-abelian group of order 6, prove that

G, S_{3}

are isomorphic". Basically I proved that G must be of the form

G= \{e, a, a^{2}, b, ab, ba= a^{2}b \}

and it is well known that S3 is of the same form, so can I just conclude that

G, S_{3}

are isomorphic or should I say "Define a mapping

\theta : G \rightarrow S_{3}

by ... " then verify it separately?

Reply 455

Insight314

Original post by Krollo

(I probably ought to point out that my modus operandi is rather different to most other people's in these parts. For better or for worse, I'm trying to get a basic familiarity with the courses rather than learn one or two to exam standard, so I'm not doing a tremendous proportion of the example sheets at the moment. Feel free to judge me)

That's completely fine, everyone is different, some people have decided to do exactly what you are doing now.

Original post by Krollo

And I'm probably not doing maths with physics. Practicals would kill me.
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Yeah, I was told that the workload for Maths with Physics is way more intensive than for those taking the only-Maths route; together with practicals work, you are also supposed to write some 2000 word essay (practical report I think) during Christmas, or I was told so at least.

It's pretty good if you are planning on specialising in theoretical physics in the next years, which is why I am happy to take it.

Original post by EnglishMuon

In some instances when I am showing two groups are isomorphic, would it be fine if I dont directly state the isomorphism if its trivial? For example I just did the question " If

G

is non-abelian group of order 6, prove that

G, S_{3}

are isomorphic". Basically I proved that G must be of the form

G= \{e, a, a^{2}, b, ab, ba= a^{2}b \}

and it is well known that S3 is of the same form, so can I just conclude that

G, S_{3}

are isomorphic or should I say "Define a mapping

\theta : G \rightarrow S_{3}

by ... " then verify it separately?

If it is not formal writing, or you just cba to do it due to time restrictions, you could leave that part but a formal proof would require you to give a rigorous reasoning, and I think that is when you should use the latter method you specified (define a mapping and show that it satisfies the properties for an isomorphism using the definition).

(edited 7 years ago)

Reply 456

InOrbit

I've seen a lot of book requests on this thread, so here are three lists that I've found helpful:
Undergraduate mathematics bibliography.
How to become a pure mathematician.
Archived pure mathematics for grad school.

There's probably a few people who want to get a head start in maths at university, or don't want to go to university and want to study maths themselves or just want to know what they actually learn. I'd say the bottom line should be:

•

Calculus: vector calculus, ordinary and partial DEs

•

Analysis: real analysis, complex analysis

•

Algebra: linear algebra, abstract algebra

•

Geometry/Topology: point-set topology, differential geometry.

•

Other things you might consider: number theory, combinatorics, logic and all of the applied areas.

All of this grounds you for more advanced maths. For instance in analysis, this includes: measure theory, probability theory, functional analysis, fourier analysis and PDEs.