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Table of Contents

Based on a traditional teaching philosophy and a non-traditional writing approach the stair-step method , this book …. This book provides a comprehensive, in-depth overview of elementary mathematics as explored in Mathematical Olympiads around the world. It expands on topics usually encountered in high school and could even be used as preparation for a …. This unique collection of new and classical problems provides full coverage of algebraic inequalities. Many of the exercises are presented with detailed author-prepared-solutions, developing creativity and an arsenal of new approaches for solving ….

This second volume in a two-volume series provides an extensive collection of conjectures and open problems in graph theory. It is designed for both graduate students and established researchers in discrete mathematics who are searching for …. This unique collection of new and classical problems provides full coverage of geometric inequalities. Many of the 1, exercises are presented with detailed author-prepared-solutions, developing creativity and an arsenal of new approaches for …. Versatile and comprehensive in content, this book of problems will appeal to students in nearly all areas of mathematics.

The text offers original and advanced problems proposed from to at the Mathematics Olympiads. Essential for …. This text records the problems given for the first 15 annual undergraduate mathematics competitions, held in March each year since at the University of Toronto. Problems cover areas of single-variable differential and integral calculus ….

This is the first in a series of volumes, which provide an extensive overview of conjectures and open problems in graph theory. The readership of each volume is geared toward graduate students who may be searching for research ideas. However, the …. Sam Penrose. Mirzakhani math superwoman Turing Machine.

Theorems and Problems in Functional Analysis

It is interesting to observe that the two biggest scientific revolutions were creates by young still unknown students outside of academy:. Andrew Krause. Note that in both cases, these two had substantial training at excellent Universities, and hence had a substantial exposure both to research results and methodologies, and many opportunities to learn from easier problems.

Far more great results have been produced by hard work and perseverance over decades than by lone genius, and I think it is this fact which Terence Tao is trying to emphasize — the modern problems of mathematics and physics will require not just sheer intellectual effort, but also some familiarity with the countless avenues which have thus far been unsuccessful in answering them. The problem is that academic researchers are somewhat reluctant to share really brilliant ideas among the group members — preferring to develop it alone and get full academic credit for it.

It seems that the academic system should encourage scientists to share freely more brilliant ideas and make more polymath-like research projects by giving more academic credit to scientists working as a team on such projects. I have read about many conjectures in number theory, and for many of them, the numerical evidence is enormous. This state of affairs is profoundly frustrating to my mind; I find it very distressing that we cannot be sure that these statements are true in the infinite. Yet my efforts produced only one inequality, and presumably it is already known. Using a simple sieve-theoretic argument, I was able to prove that.

And that they are allowed spending their time in the heavenly palace of mathematical thought, whereas all others have to confine themselves to the dark and misty forests of mundane day-to-day work? Now I myself would have every intention of resolving some of the mysteries surrounding the prime numbers, but, especially after reading this article, I am afraid that the old bit of humanistic wisdom that says that everyone who tries really hard at something will necessarily succeed might be wrong. Indeed, there might have been loads of mathematicians who unsuccessfully tried to prove all the large conjectures in number theory.

Many of them even might have been brilliant minds. And yet, most of them failed and all we have is the prime number theorem and some other results that are not even asymptotically optimal let alone the error term. Perhaps the reason why so many people including myself give it a shot is that we are forced to rely on other mathematicians, who, presumably, are much smarter and much more gifted than ourselves, yet even they fail to produce the results that we long for.

Thus, we are on our own. Even the great mathematician Ramanujan made a false claim in a letter to Hardy to have an approximation to with bounded error term. As the thoughts expressed in the above post might indicate, I am uncertain as to whether that is supposed to make me happy or sad. After all, Ramanujan was a human like myself.

And if he, being a genius, makes a mistake, what is that supposed to say about myself? Perhaps I would be happier if number theory was in a better state than it is, because it would mean that at least, members of my species are somehow able to do number theory. But we are talking about something that frankly seems too hard for us.

That is not a reason for joy. You are commenting using your WordPress.

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Subscribe to feed. What's new Updates on my research and expository papers, discussion of open problems, and other maths-related topics. While it is true that several major problems have been solved, and several important theories introduced, by precisely such an obsessive approach, this has only worked out well when the mathematician involved had a proven track record of reliably producing significant papers in the area already; and had a secure career e.

I recommend asking yourself the following questions regarding the paper: What is the key new idea or insight? How does it differ from what has been tried before? Is this idea emphasised in the introduction to the paper? How does the arguments in this paper relate to earlier partial results or attempts on the problem? Are there clear analogues between the steps here and steps in earlier papers?

Does the new work shed some light as to why previous approaches did not fully succeed? Is this discussed in the paper? What is the simplest, shortest, or clearest new application of that idea? A related question: what is the first non-trivial new statement made in the paper, that was not able to have been shown before by earlier methods? Is this proof-of-concept given in the paper, or does it jump straight to the big conjecture with all its additional and potentially error-prone complications?

In the event that there is a fatal error in the full proof, is there a good chance that a deep and non-trivial new partial result can at least be salvaged? Any major problem comes with known counterexamples, obstructions, or philosophical objections to various classes of attack strategies e. Do you know why your argument does not encounter these obstructions?

Is this stated in the paper? Do you know any specific limitations of the argument? Are these stated in the paper also?

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What was the high-level strategy you employed to attack the problem? Was it guided by some heuristic, philosophy, or intuition? If so, what is it? Is it stated in the paper? Can you state, in high-level terms i. Does the proof come with key milestones — such as a key proposition used in the proof which is already of independent interest, or a major reduction of the unsolved problem to one which looks significantly easier? Are these milestones clearly identified in the paper?

How robust is the argument — could a single sign error or illegal use of a lemma or formula destroy the entire argument? Good indicators of robustness include: alternate proofs or heuristics, or supporting examples of key steps, or analogies between key parts of the argument in this paper and in other papers in the literature.

Problem Books in Mathematics

How critically have you checked the paper and reworked the exposition? Have you tried to deliberately disprove or hunt for errors in the paper? One expects a certain amount of checking to have been done when a major paper is released; if this is not done, and errors are quickly found after the paper is made public, this can potentially be quite embarrassing. Note that there is usually no rush when solving a major problem that has already withstood all attempts at solution for many years; taking the few extra days to go through the paper one last time can save oneself a lot of trouble.

How much space in the paper is devoted to routine and standard theory and computations that already appears in previous literature, and how much is devoted to the new and exciting stuff which does not have any ready counterpart in previous literature?

How soon in the paper does the new stuff appear?

A Sequence of Problems on Semigroups |

Are both parts of the paper being given appropriate amounts of detail? Like this: Like Loading Why global regularity for Navier-Stokes is hard. Top Posts Career advice Almost all Collatz orbits attain almost bounded values Does one have to be a genius to do maths? Open question: effective Skolem-Mahler-Lech theorem The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation On writing Books Work hard The Collatz conjecture, Littlewood-Offord theory, and powers of 2 and 3 Solving mathematical problems.

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ST 6 non-technical admin 44 advertising 36 diversions 4 media 12 journals 3 obituary 12 opinion 30 paper book 18 Companion 13 update 19 question polymath 84 talk 65 DLS 20 teaching A — Real analysis 11 B — Real analysis 21 C — Real analysis 6 A — complex analysis 9 C — complex analysis 5 A — analytic prime number theory 17 A — ergodic theory 18 A — Hilbert's fifth problem 12 A — Incompressible fluid equations 5 A — random matrices 14 B — expansion in groups 8 B — Higher order Fourier analysis 9 B — incompressible Euler equations 2 A — probability theory 6 G — poincare conjecture 20 Logic reading seminar 8 travel Professor Terence Tao, I am about your age, I am moved by your humility and sincerity evident in your posts given your own unusual status in the mathematical world.

Hug you! Not important what multifuncoids are for the general idea of this question. Hm, if I will first deal with finitary case I will get two articles and a citation.

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Regards, -Larry. Best regards, dr. Nico Benschop nfbenschop-at-onsbrabantnet. It is interesting to observe that the two biggest scientific revolutions were creates by young still unknown students outside of academy: 1.