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0answers
39 views

Going to graduate school in applied mathematics without having taken a topology course?

Due to a critical course conflict, I won't be able to take any topology or geometry courses before I graduate. However, I plan to go to graduate school in [applied] mathematics. Will this hinder me ...
3
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2answers
63 views

How can Bayesian and Frequentist approach be different?

Let's say I am trying to add numbers, like say one to ten. I can either add them in order, or I can notice that I can group them into five groups of eleven, so I suppose which method to use depends on ...
1
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3answers
39 views

Concerning $Frac((Frac \space D)[x])$ and $Frac(D[x])$ for an integral domain $D$

Is the fraction field of $\mathbb Z[x]$ a proper subfield (or isomorphic to a proper subfield) of the fraction field of $\mathbb Q [x]$ ? In general , what can we say about $Frac((Frac \space D)[x])$ ...
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1answer
62 views

What are Bernoulli numbers?

In my calculus class, my teacher said that if one was to try to calculate the maclaurin or taylor series of $\tan x$ by strictly using the definition , then you would run into many problems and your ...
1
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1answer
22 views

Can we characterize those Euclidean domains $D$ for which $D/I$ is finite for any ideal $I \ne \{0\}$ of $D$?

Let $I$ be any ideal of $\mathbb Z[i]$ , then as $\mathbb Z[i]$ is euclidean domain , so $I=(z)$ for some gaussian integer $z$ ; so we can write every element of $\mathbb Z[i] / I$ as ...
3
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0answers
83 views

Examples of categories which appear naturally without objects

Regarding the morphisms-only-definition of a category (which is equivalent to the usual one dealing with objects and morphisms), I would like to ask: Which examples of categories in practice appear ...
2
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1answer
33 views

Nice application of fixed point theorems

I need some nice application of fixed point theorem for some kind of divulgative seminar. About Banach fixed point theorem a nice application is the definition of self-similar fractals (and I could ...
1
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1answer
57 views

Problem sets for Combinatorics?

I'm taking a class in Discrete Math at the moment and our prescribed text doesn't have much in the way of problem sets and solutions so I'm finding it difficult to practice. Does anyone have any ...
0
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0answers
15 views

Sorting algorithm based on the distance to the factors of the elements of the set

I am trying to understand better the basic concepts of modular-arithmetic and I was playing with a set of integers and decided to order them by using congruences, so I tried to define an algorithm to ...
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0answers
33 views

Looking for examples on Jordan Form

I am not asking a very specific question, but rather I am looking for any good examples that illustrate the following; $\mathbf{Theorem}: $ Let $T: V \to V$ be a linear operator with characteristic ...
1
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4answers
121 views

In math, when are there two equally strong, sound, but opposing, arguments? [closed]

Law abounds in 'equally strong, sound, but opposing arguments.' Does this happen in math? Are answers in math always clear-cut? I ask NOT about open, unsolved problems, or should I? Please advise ...
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15answers
9k views

Has lack of mathematical rigour killed anybody before? [closed]

One of my friends was asking me about tertiary level mathematics as opposed to high school mathematics, and naturally the topic of rigour came up. To provide him with a brief glimpse as to the ...
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0answers
28 views

What is the mathematical intuition of what an $ \mu-$incoherent matrix is?

Recall the definition of what a $\mu$-incoherent matrix is. The columns of $A \in \mathbb{R^{n \times m}}$ are $\mu$-incoherent for all $i \neq j$ if: $$ | \langle A_i , A_j \rangle | \leq ...
6
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1answer
123 views

Why are Lie algebras “rigid” objects?

I read the following motivation for quantum groups on wikipedia: The discovery of quantum groups was quite unexpected, since it was known for a long time that compact groups and semisimple Lie ...
0
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1answer
249 views

Does one necessarily need an MS in Math before taking a PhD in Math? [closed]

I finished bachelor's in mathematical finance and am nearly finished with master's in mathematical finance (I am already done with thesis), and I plan to pursue a PhD not in mathematical finance but ...
11
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1answer
137 views

Has the age at which we teach Mathematics changed over the last two centuries?

My experience of learning Advanced Trigonometry and Calculus is that it was done to 17 and 18 year olds (School Curriculum in Australia). I assumed that it was similar in the UK, US and Europe. In ...
7
votes
4answers
459 views

Does it make sense to learn any other language except English, being a mathematician? [closed]

Would it be an advantage for a committed modern day research mathematicians to learn any foreign language? Are there works in Russian, French, or any other foreign language being produced or that have ...
1
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2answers
37 views

What is the rigorous justification for using inner products as a function of similarity between two vectors?

In machine learning, it is a common thing to define similarity measures, specially using the so call Kernel function. Kernel functions are defined though through inner products of feature vectors: ...
2
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3answers
40 views

How might I read “$\cos\left(\theta\right):\sin\left(\theta\right):1::x:y:r$”?

In the book I'm reading, A Course in Pure Mathematics, the author writes the following when introducing polar coordinates in section 22: ...
5
votes
1answer
100 views

Being a good listener in math

I'm reasonably good at learning from textbooks, but I often find that when people explain mathematical ideas to me in person I don't understand what they are saying. I sense that there is actually a ...
1
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0answers
65 views

Examples of real world situations where mathemematical rigour is needed

I had a discussion with a friend about the need of mathematical rigour in the real world. He argues that little rigour is needed for the "application of mathematical results.Mathemticiians ...
11
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1answer
104 views

How to stay productive while you are studying math? [closed]

Not sure that this question is a good fit for this site, but I will try. When I am working through a chapter of a mathematical book first two hours are normally very productive (easily remember ...
1
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1answer
83 views

What is the difference (or relationship) between geometric length and arithmetic numbers?

In Abbott's Understanding Analysis there was a phrase like, "Ancient Greeks did not understand the difference (or relationship) between geometric length and arithmetic numbers." What is this ...
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2answers
54 views

Are there any databases for PhD dissertations? [closed]

I just know ProQuest which supplies some PhD dissertations. However, it's hard to find some the dissertations of French and German. So are there any other good databases for dissertations of French, ...
3
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1answer
136 views

Advice on mathematical cources and profession

I am a class 9 student studying in India. I deeply love mathematics and I am aspiring to become a mathematician ( I am aiming for research and development). My hobbies include learning and trying to ...
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1answer
108 views

Why does 2+2 equal to 4? [duplicate]

The question is in the title. I am very appreciative of any time and concern put into belaboring this relatively little problem.
4
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2answers
63 views

What is mathematical definition of a fluid?

I am searching the precise and mathematical definition of a fluid for a long time but I did not find it anywhere. What I mean by precise and mathematical can be understood by the following: There is ...
0
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0answers
33 views

Connections between probability theory and algebraic topology?

Are there any substantial connections between probability theory and algebraic topology? In particular, are there any current research areas in algebraic topology that involve the use of probabilistic ...
2
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0answers
25 views

On units in subrings ( or ideal ) and quotient ring of ring with unity

Let $R$ be a finite ring with unity and $S$ be an ideal ( or subring ) , let $R^*$ be the group of units of $R$ and $S^*:=R^* \cap S$ , then does $|S^*|$ divide $R^*$ ? Moreover , if $S$ is an ideal ...
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0answers
23 views

Why the generalized root test for series does not use $\liminf$ to get rid of the inconclusive part?

I wonder for what reasons the generalized root test for series (Roughly: $\limsup a_{n}^{1/n} < 1$ implies convergence; $\limsup a_{n}^{1/n} > 1$ implies divergence; $\limsup a_{n}^{1/n} = 1$ ...
52
votes
9answers
3k views

Why is the construction of the real numbers important?

There are a lot of books, specially in Real Analysis and set theory, which define the real numbers by Cauchy sequences or Dedekind cuts. So my question is why don't we simply define the Real numbers ...
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0answers
37 views

Best books about notation

I am reading A History of Mathematical Notations and I would love to read further about the trends and advantages of different notations in mathematics, pure or applied. Is there any good book or ...
3
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0answers
96 views

To prove $\phi(mn)\phi(d)=\phi(m)\phi(n)d$ without explicitly computing the phi function values

If $m,n$ are positive integers with g.c.d.$(m,n)=d$ , then we can show by explicitly computing respective totients that $\phi(mn)\phi(d)=\phi(m)\phi(n)d$, I want to know, is there any more elegant way ...
1
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0answers
33 views

Comparison of abstract algebra structures

I made a comparison table of the different abstract algebra structures to help myself refresh with these concepts. Can some kind souls here help cross check the basic correctness? (Any other ...
0
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1answer
18 views

Smallest integer $N(\epsilon)$ such that $K\subset \bigcup_{n=1}^{N(\epsilon)}B(x_i,\epsilon)$

In a metric space, a set $K$ is said to be totally bounded if for each $\epsilon>0$ there exist a finite number of balls $B_1,B_2\dots B_{N(\epsilon)}$ with radius $\epsilon$ which covers $K$. ...
2
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3answers
74 views

Infinite dimensional spaces other than functional spaces

"Functional analysis" is the study of infinite dimensional spaces equipped with inner product, norm, topology...etc. The most interesting spaces are the spaces of functions/operators and sequences. I ...
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2answers
66 views

Is there a proper term for these?

The 'square root' sign is formally the 'radical symbol'. 'Brackets' are formally 'parentheses' Is there a formal term for the 'fraction sign' or say the 'absolute value bars'? EDIT: I can see some ...
0
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0answers
36 views

Question about further studies

I am new here so I am not sure if these type of questions are allowed, or where exactly they should be posted. So if not my apologizes and just delete. I am studying math and some other subjects, in ...
0
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3answers
51 views

Working out a reverse formula

My math skills are getting rusty. I am trying to work out what the formula should be for calculating price, $P$, based on a formula I used to calculate margin, $\mu$, with a parameter, cost, $C$. ...
1
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1answer
52 views

Finding other problems similar to a math contest problems?

*I don't know if I can ask these type of questions here. Tell me and I will delete it right away if it's doesn't belong here. I'm preparing for a math contest, but I'm done answering all the previous ...
4
votes
1answer
65 views

Maximum number of points you can put on grid $ n\times m$ with no equidistant?

Assume we have a grid of $n\times m$ points. and the distance between two rows or two columns is 1 ( unit ). I have a couple of questions related to this grid:- What is the list of possible length ...
1
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1answer
22 views

Concerning ideals of $\mathbb Z[\sqrt m]$ and $\mathbb Z[\sqrt m] [x] $

For a given integer $m<-1$ or non-square integer $m>1$ , how do we calculate the quotient ring $\mathbb Z[\sqrt m]/I$ , for example its order or whether it is a field or has zero divisors or not ...
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0answers
29 views

References about algebraic/differential geometry in French

Aside from learning mathematics, I am learning French, so I would like to practise both at the same time if possible. Do you know of any good references about complex/algebraic geometry or ...
9
votes
1answer
170 views

Why are “algebras” called algebras?

There's a mathematical object called an "algebra" (e.g. an algebra over a ring), but why does this particular object have such an "important" name (which makes it sound like the most important concept ...
3
votes
1answer
117 views

A textbook for a rigorous introduction to Stochastic Analysis with emphasis on stochastic differential equations

I'm looking for a good textbook for an introduction to Stochastic Analysis, preferably one that focuses on rigour. I am familiar with measure theory and basic probability theory. The direction I am ...
5
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2answers
73 views

Examples of functors that preserves products but not equalizers, and vice versa.

What are simple examples, for student consumption, of A functor which preserves products (or at least finite products) but not equalizers. A functor which preserves equalizers but not products. ...
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0answers
48 views

Basic questions regarding group theory and symmetry in practice

I have for some months been interested in group theory. I was very fascinated by the level of abstraction I first met when working with groups. Another aspect that has fascinated me lately is ...
6
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1answer
31 views

Why are separable and normal field extensions so called?

To my understanding: A separable extension $K/F$ is one in which the minimal polynomial of every $\alpha\in K$ has no multiple roots. A normal extension $K/F$ is one in which some polynomial $f\in ...
2
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0answers
31 views

What does it mean when a field of mathematics has 'differential' in its name?

Topology and geometry, for example, have their 'differential counterparts' (as well as 'algebraic'). What is required of a mathematical discipline for there to exist such a subfield? For example, why ...
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0answers
46 views

Which books would you buy (Stochastic analysis Physical mathematics PhD)? [closed]

I have a soft question for you, any help will be appreciated. I have a large availability of money to buy mathematical text books (graduate level). I'm looking for suggestions with these topics: ...