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Mathematical
Services. 2024-09-08 13:42:13 - Paul D. Foy - AMathematician I note with interest that the latest Guardian UK University department league tables, from the student's perspective (that is a bottom up view) list Cambridge University mathematics' department as having slipped to 4th place. Oh no what's happening at my Alma Mater's. Perhaps small variations in position are not unusual (particularly in this case where Cambridge was the only institution not to publish student satisfaction with teaching or assessment statistics ;)), but what I did notice was the dwindling number of mathematics departments full stop, in contrast to the blossoming number of Computer Science departments and also the large numbers of marketing and public relations departments. I think mathematics has got a bit of an image or marketing problem itself, and is shunning some of the tools for students discovering and enjoying mathematics and the relevance of it in their daily lives. Speaking personally and practically - I have used computers to learn and discover mathematics: For example computing problems led me to realise and discover the singularity of a solution which lead me to ask questions into the theory of the area and discover the mathematics behind this - I didn't wait to be taught it. Developments of guttering software opened my eyes (again) to networking and graph theory. What mathematician tells you of his experiences in setting a gutter. (not used it in car maintenance much though). No mathematics is about life, of about understanding it and of understanding the tools of life such as computers. It's true that my experience on my specialism might be difficult to make a case out for choosing a course - that is a PhD to be completed more fully 30 years later with no intervening work, but you never know where these things my lead (and they are interesting). The challenges of computing in the areas of formations are yet to come I feel (computational group theory is in its infancy in this area) - "so there is a lot to do" - and what better reason for a course! People will never be poorly served by knowing the underlying techniques or theory behind things - how to write the algorithm and WHY it will be successful - to be able to demonstrate these things formally (a proof). Mathematics is English as well (if you're English) - "it's just the words that count". This is a rambling post, but I though I'd say these things. 2024-12-09 15:23:31 - Paul D. Foy - I will also take inspiration, in my course of action, from the Siberian Mathematical Journal, an article of which I had to purchase to furnish myself information on Dr Burichenko's counterexample to the conjecture of Gaschutz. It's a pity some state's can't learn the example of Prof Gaschutz who finally busied himself with these peaceful endeavours rather than engaging his fanciful emotions of glory and aggrandizement even if his private opinion was difficult to change. Everybody else was richer for his course of action. Indeed there appear to be many of his jewels that can still prove worthwhile and far more worthy to master for an aspiring individual mathematician and nation state equally. Having mastered them you will find you have slayed the voices within, and the insecurities that come from not knowing how to do something better. 2024-12-03 10:39:54 - Paul D. Foy - The articles that I have written have been tightened up and I would say they are worthy of journal articles (although not quite of that format as they are presented differently and are serving a purpose of clarifying a book, which is probably aimed at a graduate student). They will be offered for sale in a paper back booklet form, and I'm not thinking of subsequently making the pdfs available. I do not think it is quite right to simply make them freely available at no cost, as though they have no value, even though I enjoyed producing them. That is de-valuing all those that have had an effort in carefully producing them (e. g. the printers), as well as myself. I think there as appropriate an endeavour as documenting the houses in Addingham between 1750 and 1850 for example - and that has a small cost. So there will be a small charge to cover materials, printing, postage and a small profit to myself. 2024-11-09 16:15:27 - Paul D. Foy - I'm getting a bit confused by the powers of AI now. When I asked Google: are the terms of the socle series of a group characteristic subgroups, it correctly answered yes. But when I asked it what are the calories in 100ml of port it wouldn't give me the answer, hedging its bets by saying that it depends on the sugar content of the wine. So it can now do maths of which I know the answer, but not those questions which I don't know the answer. Confused. 2024-10-19 20:16:38 - Paul D. Foy - Perhaps it was John G Thompson who was the first person to show that mathematics is not just a collection of tricks making a result a couple of pages long but a body of study. This would be the 250 pages or so of showing that a group of odd order is soluble (I wonder how many pages (and what you count as a page) in describing the sub formations of a group (when they are finite)). That is a discipline worthy of taking seriously that you can get your teeth into rather than waiting for divine inspiration. More like an engineering discipline where a clever turn is just part of the solution. One of my lectures (in statistics actually) did advise the class not to view the subject as compartmentalised into exam course classes. But getting good grades doesn't reward taking such a broad view as the student and admittedly assessor knows. 2024-10-19 12:38:00 - Paul D. Foy - I was intrigued more by Google AI. So I asked: What is the Fitting subgroup of the symmettric group S4. It replied: The Fitting subgroup of a finite group is the largest normal nilpotent subgroup of that group. The Fitting subgroup of a group is itself a Fitting group if and only if the group is a Fitting group. � Fitting subgroup - Groupprops - Subwiki20 Jul 2009 � A group is its own Fitting subgroup if and only if it is a Fitting group. For finite groups, this is equivalent to the . GrouppropsThe Fitting subgroup - Peter Cameron's Blog - WordPress. com14 Sept 2020 � I have talked a bit about the Frattini subgroup. Time for its big brother. The definition of the Fitting subgroup F(G)Peter Cameron's Blog The normal subgroups of the symmetric group \(S_{4}\) are: The whole group, The trivial subgroup, A_4, and V_4. � Subgroup structure of symmetric group:S4 - Groupprops - Subwiki20 Nov 2023 � Quick summary. maximal subgroups have order 6 (S3 in S4), 8 (D8 in S4), and 12 (A4 in S4). There are four normal subgro. Groupprops The maximal subgroups of \(S_{4}\) have orders 6, 8, and 12: S_3 has order 6, D_8 has order 8, and A_4 has order 12 Not bad I'd say - it didn't actually know the answer but it told me a lot, and I can see it's just relaying what someone else told it (which appears to have been a mathematician thankfully). I still think I can do better than it in a Tripos exam (if I try hard), especially if it comes to character tables. So I'm now in the same position as professor Brian Hartley. Getting nervous. 2024-10-18 06:26:28 - Paul D. Foy - I tried out Google's generative AI. I asked: are two uniserial groups isomorphic. The answer was "Yes, two uniserial groups can be isomorphic if they have the same order and there exists a bijective homomorphism (isomorphism) between them". Not bad I'd say. Two uniserial groups of the same order need not be isomorphic - see cyclic group of order 6 and the symmetric group on 3 elements. So the AI was hedging its answer pretty well ('can'), and providing me with the definition of isomorphic and one of the necessary condition (same order). When we have a 'will' we'll have a mathematician :). 2024-10-12 13:18:47 - Paul D. Foy - My advice, early on, as an applied mathematician was that the underlying concepts behind things, vectors, div, grad, curl, the differential equations, their convergence etc etc can just be noted and then glossed over as you just use them freely. I was never all that comfortable with this, yet I learned to do it successfully both in pure and applied mathematics. Yet, with pure mathematics, in the final analysis it does not work - there has got to be certainty. It may be with maturity and experience you can omit things knowing that they are straightforward, so you don't need to be quite so explicit. It is a fine art to portray something which is believable, yet which does not bog the reader down in obfuscating details which do not contribute to showing the truth. I tend to err on the side of an elegance of exposition, rather than complete notational clarity to show something. More and more, and even in pure mathematics (and this I've done throughout my career) I think - how would you write a computer program to do this. It's not that I'm a machine it's that it give mathematics reality, a real envisagement of the truth of something because it can be realised in mechanical (Charles Babbage) or electronic instructions. And even if that can't be done in applied mathematics, then is the mathematics correct any more so than the pure mathematics? The mind boggles. 2024-10-09 19:42:48 - Paul D. Foy - I've tidied up my first paper a bit - tightened up the notation, made it more clear that I know what I'm talking about, through choice words and references. I've only got the soluble case of a group of socle length 2 to do. On the surface there doesn't seem that much to it. What could be simpler 2 layers of abelian groups stacked on top of each other. We know they can't get very involved (its been shown that there is only a finite number of formation critical groups). My hunch is that the allowed structures are very similar to the other cases, involving one at most 2 groups. We'll see. 2024-10-08 19:58:42 - Paul D. Foy - When I've made a bit of progress I'll upload it to my server as though I'm chatting with mathematicians in a Senior Common Room. :). 2024-09-29 07:29:55 - Paul D. Foy - According to The Times UK Universities survey, Cambridge has also slipped to fourth place :(. The student perspective was that "no two weeks are the same". I can vouch for that - there's always something catching your eye, trying to make things happen. 2024-09-27 12:46:13 - Paul D. Foy - Mathematics to me seems to be about so many facets of life. One of the things universities (and classically this would be mathematics departments certainly) is about exams and assessment. I tend to find there is two areas in what I'm doing (research if you like, I would certainly say its mathematics). There's the 'seeing' of a result, a plan of attack to establish it, and the actual doing of the nitty gritty details of the proof. I certainly consider myself good at the former (sometimes I do it when I'm dreaming!), the denouement can take time and I would say I can be slower at that. So I'm probably poorer at exams. If I can't see it, or scribble something down to show incite I might not be able to elucidate all the steps in time - rum lot in life. I like the working man I'm considered to be on someone's production line, measured for my time of performance of which I'm not very good. There's no reward for a patient quality study at something exploring my own incite. And he never gets off this treadmill because his lot is to have to do his weakness, so he's not rewarded well for it. 2024-09-18 20:14:57 - Paul D. Foy - As I approach 60 in a few days and engage in more leisurely pursuits I venture into enumerating the sub-formations in the formations generated by a finite group (of those that have a finite number). Yet now I am conscious of saving the subject from obsolescence, and the dwindling number of mathematics departments, when one realises that everything can be deduced from little more than Galois's axioms of 1830 or so. So I have half in mind thinking how would a computer, or a computer scientist, go about calculating these sub-formations. How is my proof turned into computer code, in the way a numerical studies mathematician write code to calculate an SVD matrix decomposition. Can I rely on a computer that can calculate the subgroups of a given group, the socle series, the maximal normal subgroups, the minimal normal subgroups. I wasn't around when computers were being used early on in the classification of finite simple groups. What did they do? So many areas I don't know - I only know my furrow :(. How does a pure mathematician introduce a computer into his courses?. 2024-09-12 06:36:14 - Paul D. Foy - I think the image problem comes out in other guises as well. Mathematics is often perceived as the province of the exceptionally gifted - why should this be the case for a subject which is applicable in all aspects of life and by everybody. I've often heard (by people (students) at University) that mathematics (particularly pure mathematics) is all about seeing a 'trick' or clever technique or device. One can certainly perceive a good idea in this way but my experiences is that these tricks on a more thorough exploration and understanding of the subject become seen in a different context - that is of the natural and straightforward theory of the subject which is starring you in the face when you gain the proper perspective. It's a bit hard to advocate a subject as one for the spotters of tricks but more for a person who loves to "develop theory" if they can't see how to do something (as I was advised admittedly). Perhaps a 'slow' student is one who is good at developing theory and understanding and not as good at seeing the tricks. It takes all in life and perhaps the subject should mirror life. I've found inspiration for diving into mathematics in all areas of life, subject areas such as history and onomastics. This is again a bit of a rambling comment. Post a comment: |