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1
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0answers
15 views

Abbreviating the definition of a tangent vector field?

Let $A \subset \mathbb{R}^{n}$ be open in $\mathbb{R}^{n}$ and let $F: A \to \mathbb{R}^{n} \times \mathbb{R}^{n}$ be continuous. Then $F$ is called a tangent vector field on $A$ if and only if $F(x) ...
1
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1answer
21 views

Notation and name for this function?

Let $k \geq 1$; let $V,W$ be vector spaces; and let $T: V \to W$ be linear. Then how do we call and denote the function $(v_{1},\cdots, v_{k}) \mapsto (T(v_{1}), \cdots, T(v_{k})): V^{k} \to W^{k}$?
2
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1answer
11 views

Definition of linear independence in R-module

I am revising module theory over commutative rings with 1 and I have a "soft" question regarding to why don't we define linear independence as follows: "$v_1,...,v_n$ are linearly independent if ...
0
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0answers
8 views

Number of possible non crossing paths on a grid of $m$ by $n$ size?

Given two points on 2 dimensional m by n grid, moving in units of 1 in either direction, how many non intersecting paths exist between the two points? in other words, with taxi cab metric, on a m by ...
-4
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2answers
69 views

What is the most complex mathematical topic? [on hold]

I'm a simple man living my simple life and often I like to learn more about math and science. Today my daughter asked me about how many numbers are there and I explained that there are infinite ...
1
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1answer
38 views

The sections of the projection $\bigsqcup_{i:I} X_i \rightarrow I.$

I just noticed something funky. Let $X$ denote an $I$-indexed family of sets. There is a projection $$\pi_X: \bigsqcup_{i:I} X_i \rightarrow I.$$ It isn't necessarily surjective, of course, because ...
2
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0answers
13 views

What is the purpose of continuous and differentiable dependence

In learning Gronwall's inequality you also get to learn about continuous an differentiable dependence. I know the theorems but I have no idea about their application. What is the big idea of ...
8
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1answer
118 views

What did Lagrange, Euler, Gauss etc. learn in order to know what they knew?

What did the great mathematician, like Cauchy, Lagrange, Euler and Gauss, learn in order to know what they knew? It seems that they were extremely good in the most basic rules/structures/issues of ...
0
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0answers
32 views

Is math, in the end, only geometry [on hold]

When thinking about the Universe, or "reality", Isn't every part of mathematics a tool for expressing something geometrically further down the line? Yes, every part of math is related, but isn't ...
1
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0answers
23 views

How important is it that I study Probability if I like Analysis/Algebra much more?

Is it crucial to a student's undergraduate studies in Math that he/she takes a course in Probability and/or Statistics? I am much more interested in Analysis/ Algebra and I was wondering if it would ...
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0answers
16 views

Can we build a DFA less than 5 state for word length 4( 1100)? [on hold]

========================================================== 1 if possible kindly, help me with this question.
3
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1answer
61 views

What solution would you come up with for this problem?

So the question is: put numbers $1, 3, 5, 7, 9, 11, 13$ and $15$ into gaps in the following expression: $$\_\_ + \_\_ + \_\_ = 30$$ The most naive approach to use summation in the group of integers ...
0
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4answers
58 views

Why in general there is no systematic way to find counterexamples? What kind of property do they all break that lead to this? and other things

We came across counterexamples in many areas of mathematics: For example Sum of irrational numbers not necessary being irrational The "Windmill blade" function (for lack of a better name of one of ...
1
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2answers
29 views

Encyclopedia of Mathematics?(non-Alphabetical)

Do you know any Encyclopedia of Mathematics which is in non-alphabetical order, like it starts from basic mathematics and then goes up to very advanced level. And what's the difference between say, ...
3
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2answers
223 views

Can I follow a graduate course in PDE without having studied ODE

Hi I am considering taking the first course on Partial differential equations at my university next semester. I have already taken a first course on functional analysis . I haven't taken a proof based ...
1
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0answers
56 views

How to invent mathematics to find solution to real world problems?(without high level mathematical knowledge) [on hold]

Let's say I want to know how waves are formed when a stone is dropped in water, how trees are deformed by the wind, etc, and I want to invent the mathematical equation to predict the behaviour of ...
1
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1answer
31 views

How do we call a pair of sets $A,B$ such that there is some injection $f: A \to B$?

Let $A,B$ be sets and let $f: A \to B$. If $f$ is a surjection, then we may simply write $f(A) = B$ or say in a more laborious way that $f$ maps $A$ onto $B$, to mean the same thing. However, if $f$ ...
4
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2answers
221 views

How do we call a pair of sets between which there is a bijection that need not have additional property?

Let $A,B$ be sets and let $f: A \to B$. Then we say that $A,B$ are isomorphic under $f$ if $f$ is a linear function that maps $A$ onto $B$ in a one-to-one manner; that $A,B$ are homeomorphic under $f$ ...
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0answers
78 views

Mathematicians who didn't study mathematics in college or university [on hold]

I would like to know mathematicians born after 1900 who didn't study mathematics in college or university. I posted a similar question recently but it was closed as opinion based. So I will define ...
7
votes
3answers
158 views

A reason for the value of $\int_{0}^{1}\log{(x)}\log{(1-x)}\,\mathrm{d}x$

In this .pdf document, which is just a list of Putnam-style undergraduate-level problems from various sources, the third question is as I have stated it below (up to a change of notation). ...
1
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3answers
96 views

Good algebra book to cover these topics?

I will be studying two algebra modules next year and I am looking for a comprehensive book that will cover both of them, however due to having very minmal exprience with algebra I am looking for your ...
0
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3answers
81 views

How would you explain Functional Integration to an 8 year old?

I get the definition of the Functional Integral, but what heuristic interpretations are available to better understand the integral? For instance, what motivates the definition? How is it related to ...
0
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1answer
43 views

What are some easier papers/books I can read? [on hold]

I'm trying to improve my ability in reading mathematics papers. My field is more related to biological sciences, but there are a lot of interesting papers I'd like to read that use more mathematical ...
5
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1answer
40 views

Does it matter if you use big $L$ or little $l$ when talking about $L$-norms?

I was reading a post on Quora regarding the application of "$l_1$", "$l_2$" norms for convex linear programming when I became very confused at which $L$-norm the posters are actually referring to. I ...
2
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3answers
81 views

Fun logic puzzles to teach logic/proof-writing to students

Forgive me if this is too soft of a question, but I am looking for some fun, quick, and interesting logic puzzles to give to my students. I'm teaching an honors calculus course, and this will be their ...
2
votes
1answer
59 views

Intuition for Burnside's Lemma (aka Cauchy-Frobenius Lemma)

Here is the theorem: Lemma: Let a group $G$ act on a set $S$. Define $\text{Fix}(g)$ as the set of all elements in $S$ fixed by $g$ under this group action. Then the number of distinct orbits of ...
0
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0answers
31 views

In what ways would a course on convex optimization be useful in game theory?

From talking to several other people in the past, and referencing Quora, it seems that convex optimization is really a tiny subset of game theory in that it only models the behavior of one single ...
2
votes
3answers
73 views

How do mathematician make sense of “outcome” and “events” in probability?

One of the biggest challenge for me to understand probability is to make sense of this concept of outcomes and events. To put it plainly, it just doesn't feel like mathematics anymore when we talk ...
5
votes
1answer
48 views

Stating the induction hypothesis

I would like to ask about the best way to state the induction hypothesis in a proof by induction. Just to use a concrete example, suppose I wanted to prove that $n!\ge 2^{n-1}$ for every positive ...
-1
votes
1answer
15 views

Zeitz's ACoPS vs Larson's PSTP

Which of the following books is better to prepare for a mathematical competition at the undergraduate level? The art and craft of problem solving (ACoPS) or Problem solving through problems (PSTP). ...
1
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2answers
43 views

How to document solutions for future use? [closed]

I'm taking courses of math at university level, it's kind of the equivalent of master degree in mathematics, I'm from Argentina. The way to learn mth in my university is this: We attend lectures, we ...
1
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0answers
34 views

Is there anything called kernel space?

Here I am referring kernel as an integral operation.The wikipedia link is this https://en.wikipedia.org/wiki/Integral_transform My question is: consider the function insider the integral $f(t)$ is ...
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0answers
53 views

Magic of the number $2000+15$ [closed]

What is the most clever way of getting the number $2015$ using only addition, subtraction, and multiplication?
4
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3answers
67 views

Motivations for Hyperbolic Geometry

Why would one study hyperbolic geometry? I am only aware of the motivation where you give axioms for elementary euclidean geometry and then start to wonder wether the parallel axiom is necessary. You ...
1
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1answer
77 views

Struggling to stay at current course.

Disclaimer: I'd like to say I've been a member of this site for over a year, so I know this may be a nonstandard question, however, for personal reasons, I'd like to keep this anonymous. I'm a second ...
1
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3answers
66 views

Acronyms and words as variables and in mathematical notation

I am unsure if this question is warranted. Often mathematical symbols and objects are represented by a single character, e.g. variables are most often single characters like $x$, and often to ...
0
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0answers
24 views

About Carl Meyer's matrix analysis

I have taught some part of it to myself when i was an engineering student. but now i changed my major to the pure math so now i am studying math as an undergraduate student. i thought the book is ...
2
votes
1answer
31 views

$f(x)$ be the characteristic polynomial of a matrix $A \in M_n(\mathbb R)$ ; then is it true that $f(1)=1+\operatorname{trace}(A)+O(\|A\|^2)$?

Let $f(x)$ be the characteristic polynomial of a matrix $A \in M_n(\mathbb R)$; then is it true that $f(1)=1+\operatorname{trace}(A)+O(\|A\|^2)$ ? I need a proof if it is true ; or any modification ...
2
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0answers
62 views

Measuring the set-theoretical complexity of sets/spaces encountered in general analysis

In analysis, it is common to encounter subsets of $\mathbb R$ (or even $\mathbb R^n$) which appear to be "well-behaved", especially with regard to properties like being measurable, compactness, etc. ...
2
votes
1answer
49 views

Can anyone please help to clarify the sentences “ into a fat tail part in L2 plus a fat body part in L1.”

In the link https://en.wikipedia.org/wiki/Fourier_transform#On_Lp_spaces what does this sentences mean: into a fat tail part in L2 plus a fat body part in L1? Would anyone please help?
5
votes
3answers
453 views

What is the pre-requisite knowledge for generating my own integer sequence?

I've recently come across the On-Line Encyclopedia of Integer Sequences and I'm completely fascinated by it; something about how easy integers are to grasp and yet how complex the sequences are. I ...
1
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2answers
74 views

Conceptual differences between the notations $\int_{a}^{b}f$ and $\int_{[a,b]}f$

Let $[a,b] \subset \mathbb{R}$ and let $f: [a,b] \to \mathbb{R}$ be continuous. Then $f$ is Riemann-integrable. What are the conceptual differences between the two notations $\int_{a}^{b}f$ and ...
2
votes
4answers
113 views

Is $i^i$ mathematically valid? [duplicate]

WARNING: SLIGHT NSFW http://www.smbc-comics.com/index.php?db=comics&id=2934#comic Uhh...guys, mathematically speaking, how accurate is this comic. From what I remember in High School $$a^b= ...
3
votes
0answers
77 views

May Algebraic Geometry be appropriate for me? [closed]

I am a student of Mathematics who have to choose its area of specialization. I am trying to obtain as more information as possible, by asking a lot of questions to more experienced people, trying to ...
7
votes
1answer
120 views

What is $\varphi(0)$? [duplicate]

$\varphi$ is Euler's totient function. My question is: When/if $\varphi$ is defined at $0$, what is it usually defined as? Is there a "most natural" or more commonly accepted definition of ...
4
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0answers
100 views

Why I can't use Latex editor? [closed]

I cant use latex editor because I need 10 reputation, so what should I do?
1
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0answers
42 views

Can we have different methods to estimate elements from Lp spaces?

Sorry if my question is vague. Consider I have some time samples and it is known to be summation of sinusoidal. Problem is to estimates the frequencies. Generally, Fast Fourier transform (FFT) is the ...
6
votes
2answers
244 views

What is the interest of duality in algebra, and in general in mathematics?

Before to ask my question I precise I'm a chemist, I ask this question because it makes me crazy to don't understand something I learnt in school. So I had two years ago a small chapter about ...
0
votes
3answers
26 views

Quick and simple funtamental question about sets

Please feel free not to read this its just a prelude: Ok I am sorry for making this question since as far as I can tell the level of the other questions here is higher by far compared to mine, but I ...
0
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0answers
22 views

Does a ratio of PDFs have any usable meaning?

I'm calculating the probability that a standard Brownian motion path will cross a boundary. I have $A$ and $B$ representing the PDFs for the Brownian motion going above a boundary function $a$ and ...