I just finished some chapters from munkres(on topology).Before that I had done real analysis from rudin.So in rudin all the proofs were done with respect to metric spaces(compactness,connectedness,continuity etc.).In Munkres the same theorems were proven with respect to a general topological space.Why did rudin base the book on metric space only and not include general topological spaces.For which spaces are the theorems in rudin's book valid?Any guidance is welcome.

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]]>I just finished some chapters from munkres(on topology).Before that I had done real analysis from rudin.So in rudin all the proofs were done with respect to metric spaces(compactness,connectedness,continuity etc.).In Munkres the same theorems were proven with respect to a general topological space.Why did rudin base the book on metric space only and not include general topological spaces.For which spaces are the theorems in rudin’s book valid?Any guidance is welcome.

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]]>Here in the sub we get somewhat frequently the question of why is it not possible to divide by zero or a common variant of people asking if it is possible to extend the (rational,real or complex) number system to something that allows this.

Among the usual answers people often mention the extended real line or the somewhat legendary theory of wheels, although these usually come up after people have explained why the value of dividing by zero is forced to be indeterminate which relies in the fact that 0x=0 for all x in our system.

Now this is what Id like to discuss a little bit. This easy result that 0x=0 uses the distribution of the product over the addition in a strong way. For a while Ive been a bit curious about systems where distributivity doesnt hold, or more precisely, holds in a weaker way.

There exist the theory of near-rings which are simply rings that distribute on one side but not the other ( my understanding is that this comes from trying to have a ring-like structure with noncommutative addition ). On these structures the equation 0x=0 holds ( if we are allowing distribution on the left ) but x0=0 doesnt. This is however still not exactly what Im looking for, here we still have the impossibility to divide by zero.

Googling around I stumbled into this paper which discusses something that the author calls w-rings and c-rings. Based on the appendix of that paper I found this MO question which discusses something called Kelley rings. If you are too lazy to click on the link, the point is that Kelley in his classic general topology book defined a ring as the usual except that instead of writing down distributivity, he wrote the equation (x+y)(z+w)=xz+xw+yz+yw which doesnt really boils down to the usual definition ( because we dont know that 0x=0! )

Apparently people wrote some papers on these "Kelley rings" but I havent really found anything by this name with a quick google search. I however found the general notion of (m,n)-distributive rings which satisfy an analogous condition of the form (a1+...+am)(b1+..+bn)=(a1b1+...+ambn).

Now what Id like to discuss or know is if there are other notions of weaker distributivity that come up naturally and if in these systems division by zero exists or if it is useful at all.

There seem to be also the notion of distributivy laws between monads, which sound sort of relevant but CS people kinda made studying monads for me a bit painful. As for why Id like to discuss this: I just thought it is a nice fun thought experiment which might turn out to have more on it than I suspect. Also winter break is approaching and my motivation to work on my stuff is decaying.

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]]>Here in the sub we get somewhat frequently the question of why is it not possible to divide by zero or a common variant of people asking if it is possible to extend the (rational,real or complex) number system to something that allows this.

Among the usual answers people often mention the extended real line or the somewhat legendary theory of wheels, although these usually come up after people have explained why the value of dividing by zero is forced to be indeterminate which relies in the fact that 0x=0 for all x in our system.

Now this is what Id like to discuss a little bit. This easy result that 0x=0 uses the distribution of the product over the addition in a strong way. For a while Ive been a bit curious about systems where distributivity doesnt hold, or more precisely, holds in a weaker way.

There exist the theory of near-rings which are simply rings that distribute on one side but not the other ( my understanding is that this comes from trying to have a ring-like structure with noncommutative addition ). On these structures the equation 0x=0 holds ( if we are allowing distribution on the left ) but x0=0 doesnt. This is however still not exactly what Im looking for, here we still have the impossibility to divide by zero.

Googling around I stumbled into this paper which discusses something that the author calls w-rings and c-rings. Based on the appendix of that paper I found this MO question which discusses something called Kelley rings. If you are too lazy to click on the link, the point is that Kelley in his classic general topology book defined a ring as the usual except that instead of writing down distributivity, he wrote the equation (x+y)(z+w)=xz+xw+yz+yw which doesnt really boils down to the usual definition ( because we dont know that 0x=0! )

Apparently people wrote some papers on these “Kelley rings” but I havent really found anything by this name with a quick google search. I however found the general notion of (m,n)-distributive rings which satisfy an analogous condition of the form (a1+…+am)(b1+..+bn)=(a1b1+…+ambn).

Now what Id like to discuss or know is if there are other notions of weaker distributivity that come up naturally and if in these systems division by zero exists or if it is useful at all.

There seem to be also the notion of distributivy laws between monads, which sound sort of relevant but CS people kinda made studying monads for me a bit painful. As for why Id like to discuss this: I just thought it is a nice fun thought experiment which might turn out to have more on it than I suspect. Also winter break is approaching and my motivation to work on my stuff is decaying.

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]]>The post Math Magic and Tricks appeared first on Nevin Manimala's Blog.

]]>the formula is [ C = floor(A/B)]

where:

A: is the number of times the walls are hit

B: is the number of times the corners are hit

C: is the max number of times the walls are going to get hit at the screensaver's position and direction

if you find ** FLAWS**, tell me

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]]>the formula is [ C = floor(A/B)]

where:

A: is the number of times the walls are hit

B: is the number of times the corners are hit

C: is the max number of times the walls are going to get hit at the screensaver’s position and direction

if you find ** FLAWS**, tell me

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]]>I’ve always felt like I don’t have a decent grasp on a topic until I can “play” with it. For example, I have lots of fun asking questions in elementary real analysis, and to some extent topology and linear algebra, which usually leads to nice problems and more fun questions.

But with certain topics, like complex analysis, abstract algebra, and basically anything at the grad level, I can’t play with them. I don’t know how to come up with interesting questions to ask, and I can’t create any problems. I can understand definitions on a surface level, and solve textbook exercises somewhat fluently, but I still don’t feel like I have enough intuitive understanding to play with the topic.

So, some questions:

1) Usually the best way to really get to know a subject is to play with it, but how do you get good enough to play when you can’t play yet?

2) Does anyone else share this view on playing? Are there also topics for you that you feel comfortable playing with, and some where you just feel stuck?

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]]>I’ve always felt like I don’t have a decent grasp on a topic until I can “play” with it. For example, I have lots of fun asking questions in elementary real analysis, and to some extent topology and linear algebra, which usually leads to nice problems and more fun questions.

But with certain topics, like complex analysis, abstract algebra, and basically anything at the grad level, I can’t play with them. I don’t know how to come up with interesting questions to ask, and I can’t create any problems. I can understand definitions on a surface level, and solve textbook exercises somewhat fluently, but I still don’t feel like I have enough intuitive understanding to play with the topic.

So, some questions:

1) Usually the best way to really get to know a subject is to play with it, but how do you get good enough to play when you can’t play yet?

2) Does anyone else share this view on playing? Are there also topics for you that you feel comfortable playing with, and some where you just feel stuck?

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]]>I recently finished reading "When Einstein Walked with Gödel: Excursions to the Edge of Thought" and found the book extremely interesting, especially the parts where Holt tries to go more in depth. The best parts of the book IMO where the ones about math, covering history around the topics of abstract algebra, category theory, calculus, logic, and set theory.

Are there any books about the history of math that is recommended? Preferably written for the purpose of reading casually (i.e. not a math theory book with historic elements secondary).

Also if you have any similar books involving math but not necessarily aimed at history I would be interested to know.

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]]>I recently finished reading “When Einstein Walked with Gödel: Excursions to the Edge of Thought” and found the book extremely interesting, especially the parts where Holt tries to go more in depth. The best parts of the book IMO where the ones about math, covering history around the topics of abstract algebra, category theory, calculus, logic, and set theory.

Are there any books about the history of math that is recommended? Preferably written for the purpose of reading casually (i.e. not a math theory book with historic elements secondary).

Also if you have any similar books involving math but not necessarily aimed at history I would be interested to know.

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]]>The post [Video] A conversation with Caucher Birkar, who this year won a Fields Medal for his work in algebraic geometry appeared first on Nevin Manimala's Blog.

]]>The post [Video] A conversation with Caucher Birkar, who this year won a Fields Medal for his work in algebraic geometry appeared first on Nevin Manimala's Blog.

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]]>I sometimes take a step back from whatever particular concept I'm learning about to get a bird's eye view of my present understanding of mathematics. One tool I have to help myself do this is to think about the components of the subject that really resonate with me.

A few things I love about the subject:

Different objects exist at different levels of abstraction, so it trains me to be able to think about both the specific and the general cases.

The notion of proof. Once a few ideas are taken as given, other ideas can be *certain* to follow logically from them. It helps me think more clearly about concepts I don't fully understand, since I'm always attempting to find more precise ways of defining them or identifying relationships to similar concepts.

The rich visual component. Many (if not most (or maybe even all)) objects in math can be represented geometrically or pictorially. The training I've had in mathematics has helped me so much in my understanding of physics. I can now visualize the airflow around a bird in flight, the forces acting on a turning car, the locking together of molecules as they form a crystal, or countless other things.

So what about mathematics really appeals to you? I'd imagine you could ask a thousand different mathematicians and get a thousand different answers, so I'm really curious about what others think.

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]]>I sometimes take a step back from whatever particular concept I’m learning about to get a bird’s eye view of my present understanding of mathematics. One tool I have to help myself do this is to think about the components of the subject that really resonate with me.

A few things I love about the subject:

Different objects exist at different levels of abstraction, so it trains me to be able to think about both the specific and the general cases.

The notion of proof. Once a few ideas are taken as given, other ideas can be *certain* to follow logically from them. It helps me think more clearly about concepts I don’t fully understand, since I’m always attempting to find more precise ways of defining them or identifying relationships to similar concepts.

The rich visual component. Many (if not most (or maybe even all)) objects in math can be represented geometrically or pictorially. The training I’ve had in mathematics has helped me so much in my understanding of physics. I can now visualize the airflow around a bird in flight, the forces acting on a turning car, the locking together of molecules as they form a crystal, or countless other things.

So what about mathematics really appeals to you? I’d imagine you could ask a thousand different mathematicians and get a thousand different answers, so I’m really curious about what others think.

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]]>The post This is the random pullback attractor of a stochastic Hopf normal form exhibiting shear induced chaos. It’s pretty pretty I think. appeared first on Nevin Manimala's Blog.

]]>The post This is the random pullback attractor of a stochastic Hopf normal form exhibiting shear induced chaos. It’s pretty pretty I think. appeared first on Nevin Manimala's Blog.

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