# Tag Info

1

okay The special number here is $1089$ . take a three digit number whose digits are distinct from each other ill give example and then proof.. e g $123$ reverse digits you get $321$ subtract smallest three digit number formed by these digits. $321-123=198$ reverse it $891$ and add both you get $891+198=1089$. $PROOF$: let the number be XYZ X is hunderth ...

2

Not exactly an solution to your question, but frankly it's too big to be fitting a comment. The distinction as to exactly what is being linear is very important. It is possible to approximate non-linear behaviours if we are allowed to make the linear space (as in linear algebra) large enough. Approximating solutions to smaller dimensional non-linear ...

0

I just recall a definition from calculus: Definition: A function $f : I \subseteq \mathbb{R} \to \mathbb{R}$ that is differentiable on the interval $I$ is linear if $\dfrac{\mathrm{d}f}{\mathrm{d}x} = c$ for a constant $c \in \mathbb{R}$.

1

Your last definition is (basically) as general as it gets (so far as I know). Definition. Let $S$ denote a semiring (with both $0$ and $1$.) Suppose $X$ and $Y$ are modules over $S$. Consider a function $f : X \rightarrow Y.$ Then: $f$ is linear iff for all $x,x' \in X$ and $s,s' \in S$, we have $f(sx+s'x') = sf(x)+s'f(x')$. $f$ is affine iff ...

4

Your question is very broad, and I'm not sure this fully addresses it; but this is too long for a comment, and hopefully you find it useful nonetheless. I think there's a couple false assumptions here. First, that there is a "better" way to approach studying mathematics. People vary wildly in how they learn, and ultimately I think it's best to find an ...

4

Your questions in order: Yes, it is highly advantageous to be exposed to more sophisticated ("graduate") subjects as early as one can tolerate it. Not clear that one should "study" them. Yes, some people do "read ahead". I myself found it very helpful. I would claim that the widely-believed sense of "deep and slow" versus "skimming ..." is a fake ...

0

Intuitively, the symbol $\text{d}W_{t}$ may be interpreted as an infinitesimal increment of a (one-dimensional) Wiener process, $W$. $$\text{d}W_{t} = W_{t+\text{d}t}-W_{t}\ .$$ Increments of a Wiener process are normal distributed random variables whose expected value is zero and whose variance is equal to the time-increment. Thus the symbol $\text{d}W_{t}$ ...

1

Personally, I was STUNNED by $$1+2+3+4\cdots=-1/12$$ This undoubtedly sparked my interest in mathematics. (Although I didn't know it then, this is a zeta-regularized sum)

0

I recommend 3Blue1Brown's Channel, just as an example, the last video: https://www.youtube.com/watch?v=cyW5z-M2yzw Also The Mathologer

1

I like the channel Tipping Point Math. Lots of interesting math made by a math professor.

3

As other answers already point out, isomorphism is just relabeling elements and renaming operation, but all relations you could think of such as subgroups, quotient groups, generators etc. are preserved, up to relabeling. Let me try to draw an analogy with similar but undoubtedly more familiar concept: what does it mean that two triangles (in a plane) are ...

5

It means that, even though the groups contain different elements and combine according to different rules, they are nevertheless from the perspective of group theory essentially identical in every important respect. The easiest example of this comes from elementary school arithmetic: We learn at an early age that adding two odds gives an even, etc. We ...

1

Let $p,q \in M$ and let $pq$ be a geodesic connecting them. $pq$ will be minimizing if and only if its intersection with the cut locus of $p$ is empty. The cut locus of $p$ is made of: (1) the conjugate points of $p$ and (2) those points $r$ such that there are multiple minimizing geodesics from $p$ to $r$. Since, by assumption, $pq$ contains no point ...

9

When we say that two groups are isomorphic, we are saying that they have the same structure and invariants as groups. An isomorphism between two groups do more than matching elements: it matches subgroups, normal subgroups, characteristic subgroups, conjugacy classes, $p$-subgroups, Frattini groups, ... In other words, two isomorphic groups can be ...

31

It means they are exactly the same except for the names of the elements and the name of the binary operation. An isomorphism between groups is a function that renames all of the elements. (Hence, it is bijective... each element in the first group gets renamed to be exactly one element in the second group.) The reason we care is that if you are only ...

13

In loose terms it means you can't tell them apart. They are the same except that the elements have different names. For example the group $Z_2 = \{0,1\}$ with the obvious rule for multiplication is isomorphic to the group {even, odd} with the usual rule for addition. @Dorabell 's confusion (see his comment below) was my fault for calling the operation on ...

0

From a comment above by littleO: Yes, they even use the notation $\langle \ell, x\rangle$ for $\ell (x)$ sometimes, to emphasize this viewpoint.

0

From comment; That's pretty much exactly the point. When the distributional derivative is representable as an Lp function, that is the weak derivative.

3

I'm not sure how much value I'm adding here, but perhaps this example will be interesting to kids of the modern age. Suppose we have the following two graphs of equal size: Graph 1: A group of youths with Facebook accounts. Some pairs of people in the group are friends on Facebook, some are not. Graph 2: A room full of pre-programmed old-school paging ...

1

First of all the distinction is not as important as one would think. When you use an informal formulation it can be considered a short hand description of how you would go about in creating the formal proof - just like a recipe for gingerbread is not a gingerbread, but it allows anyone to produce gingerbread if they're interested in doing so. Note that ...

2

There is a connection but to make it clear you have to be more precise about the boundaries of the inner integral on the right hand side of your second equation. The primitive function $F$ of $f$ is only determined up to a constant. Assuming that it is chosen so that $F(a)=0,$ then we have $$\int_a^bFg'=\int_{x=a}^b\int_{t=a}^xf(t)dt\ g'(x)dx$$ The ...

0

There's also a concept of 'simplicity' involved here, I think. From the purely formal point of view, it suffices to know (or be able to prove) that, for example, the operation '$+$' sends two natural numbers to a third natural number, as is the case with $3$ and $4$, which are 'sent' by the '$+$' operation to $7$. By using some properties of such an ...

1

You wrote "when is it allowed", but I think you are trying to ask something like "when is it a good idea"? If you are asking the latter, then I usually tell students something like "I want to do as little work as possible". Meaning I relate the problem to something I already know, like you did in the example. Another simple example is, I know that ...

1

This question is vague, but my general advice would be to understand what the mathematical objects and symbols you're working with mean. Then you will be able to determine for yourself whether a substitution is valid. If you're just manipulating symbols with no idea of what anything you're writing down actually means, that's when you run into trouble. For ...

2

If you've thought about the logarithm in the context of complex variables, you know it must be defined with branch cuts or else be a multivalued function. If you want to "graph" the multivalued function, what you have is a Riemann surface. Indeed, Riemann surface theory is a natural outgrowth of complex analysis when you want to see the topology behind ...

0

This is not a complete answer, but... A non-trivial fundamental group space means (roughly speaking) a more complicated space. Since the covering space share many properties with the original space, and because "good" spaces (with some reasonables hypothesis) always have universal covering (a covering with trivial fundamental group), to pass to the ...

3

I would argue, as you say, that knowledge is worth something even in absence of practical applications. However, that is not to say that theoretical mathematics has no practical applications. In my opinion the following is the main difference between applications of "theoretical" and "applied" mathematics: Pure mathematics is an investment for the far ...

2

TLDR : if the result is more important to you, go applied. If the road-to-result is more important, go pure. I'm sure you are aware that behind many world-changing inventions lie an accumulation of "pure math" results, that seemed useless at their time. Traditional examples include number theory results that are used in cryptography, or Turing's idea of a ...

1

Somewhere between a joke and an essay, is Impure Math. http://www.snowman-jim.org/science/humor/impure-math.html

0

The answer is basic functional analysis which rests on these theorem from topology. The follow up is harder, who knows what can be used to prove something.

-1

As mau mentioned, scale invariance is the key assumption in Benford's Law. I'll fill in the mathematical details on how that gets you to Benford's results for the probabilities (for more details see my "proof" in this article). Imagine writing all numbers in scientific notation, like $x\cdot10^y$. The $x$ is the only part we care about for Benford's law. ...

3

You have solved a mystery for me. I teach mathematics in a university, to bright, motivated students. They almost all have great difficulty in understanding the concept of equality, and in substituting expressions. For instance, if you write $f(z) = \frac{e^{iz} - e^{-iz}}{2i}$ and then ask them what is $f(3+4i)$, they often don't understand that you mean ...

2

Here is Doron Zeilberger's opinion bearing on this topic with a pointer to some feedback from me (which includes some remarks on Hagen's point about Principia Mathematica).

2

Comment You are right: in First-order theory of arithmetic (see : Peano axioms) a term can be : a variable, a constant ($0$) or a "complex" term built up with functional symbols : $S, +, \times$. Some examples of arithmetical terms : $0, S(0), S(x), S(0)+0, \ldots$ A term acts as a "name" for a number. We define : $1=S(0), 2=S(1)=S(S(0)), ... 4 What you defined could be called a valid formal proof. A valid mathematical proof (or a proof accepted by the mathematical community) on the other hand might be described as an informal(!) arrangement of arguments that the reader finds convincing in the sense that he/she strongly believes that it is possible to write down a valid formal proof reflecting the ... 1 My personal opinion goes more along: A proof is valid if it convinces a significant number of experts in the field to declare it valid. A prime example was Andrew Wiles proof of FLT, a long and complicated proof, with initial flaws as well, in a subject only a few were deep into. A PhD student in algebra told me at that time that he would need two ... 0 We don't need to change, neither extend the idea of a mathematical proof just because many fields uses informal language. This is true that informal language might be more clear to the wide majority of people, but such a language can only be used for an explanation of the idea of the proof. On the other side the formal language is understood by a very small ... 0 I once asked whether the fact that the visually rather arbitrary$\frac{3}{4} \log_e(2) - \frac{1}{2} \approx 0.01986$turned up as answers to two very different questions could mean there was some link between them. Nobody found a connection, suggesting that this value may be doubly meaningful, but less interesting. 5 Being successful at Olympiad mathematics is certainly correlated with being successful in later studies and research, but there is no implication in either direction. This is what you would expect a priori: coming up with creative ideas is a part of the work of a research mathematician, but by no means the only (and arguably not the most important) part. ... 8 I am not one thousandth the mathematician that Terry Tao is, but my own feeling is rather different. I had a college classmate who was far better at competition mathematics than I was, and when we went to grad school (together), he seemed good at following a prescribed path, but not so good at striking out on his own. In later professional life, he made no ... 9 Find Terry Tao's blog. He talks about his experience of learning at different levels of mathematics education. Among other insights, he writes how patterns from competition problems he later discovered to be examples of more general, deep and beautiful results. What I took away from all that was that while solving competition problems isn't directly ... 3 The following is just my opinion. I commend you on trying to further your brothers education but I would be careful as to how far you go and in what order. In my experience, we tend to forget the things we knew before we learnt harder stuff. Given that your brother is asking about something notational, it may be that he doesn't really understand why you are ... 2 John Nash struggled with significant mental health issues when he should have been in the twilight of his mathematical journey. He and his wife spent many difficult years battling with this illness. Slowly, Nash started to get back in touch with the mathematical community in Princeton; engaging with the students, his passion for mathematics never died. It ... 1 I'd like to offer a suggestion from personal experience with math troubles. Sign up for classes at a community college(2 yr). I managed to pay my way through my associates degree delivering pizzas for minimum wage + tips (Not much more the minimum wage). Community colleges are designed to be affordable and a helpful stepping stone to a 4 yr degree. I did ... -2 Grigori Perelman who possibly proved a generalization of the Poincare Conjecture - the Thurston Geometrization Conjecture ( has anyone claimed to show any flaw in the proof yet? ). That being said, it seems that it's not very uncommon with odd behaviour amongst great mathematicians. They have a huge uphill struggle. Anything they discover could overshine ... 0 Diving into what literature I could access, I have come up with an answer, in "story" form, which more or less satisfies my inquiry. (Perhaps it is an answer to a slightly different, or more well posed question.) In any case, it makes sense (so far) to me... As the 1960s dawned there must have been a certain air of anticipation in the mathematical world. ... 4 Andrew Wiles with his Fermat proof is an example of massive struggle. He worked on the topic for 9 years and got demolished when presenting an erroneous proof after 7 years. There were several additional problems, but I forgot the details. You can read about it on Wiki and in much detail in the very accessible "Fermat's Last Theorem" by Simon Singh. 8 The analytic number theorist Hua Loo-Keng overcame abject poverty, handicappedness, political persecution; for more information refer to one of his faithful biography. Charles Hermite overcame much too, but in different aspects; he failed nearly every math exam that he was to take. To supplement, the analytic number theorist Chen Jing-Run, the man closest ... 8 Srinivasa Ramanujan was such a mathematician. He failed to got admitted to college but he became one of the best mathematician of$20\$th century. Evariste Galois failed to enter to Ecole Polytechnique twice.

9

Some things to consider: To a various degree this happens to all of us. It is an essential skill, to be able to leave a problem behind, even if only for a few days. It is fine to skip a problem, esp. if you have solved almost all the others from the section. If it is really bugging you, get help from math.SE. Time is precious—get things done, and get done ...

Top 50 recent answers are included