For questions whose answers can't be objectively evaluated as correct or incorrect, but which are still relevant to this site. Please be specific about what you are after.

learn more… | top users | synonyms

1
vote
2answers
44 views

History of convolution

Let $f, g\in L^{1}(\mathbb R),$ we may define the convolution of $f$ and $g$ as follows: $f\ast g(x)= \int_{\mathbb R} f(x-y)g(y) dy, (x\in \mathbb R).$ It is well known that it can be defined on ...
2
votes
0answers
34 views

Calculus on Spaces other than $R$

I was thinking that all that the real numbers (and the complex numbers similarly) need for calculus to work is for them to be a field and a complete metric space, and probably have the usual field ...
6
votes
1answer
47 views

Topological space $\nRightarrow$ Metric space $\nRightarrow$ Normed space $\nRightarrow$ Inner product space (Examples)

If I have an inner product space, the hierarchy goes: Inner product space $\Rightarrow$ normed space $\Rightarrow$ metric space $\Rightarrow$ topological space. The reverse, however, is not always ...
3
votes
1answer
77 views

Difference between proof-based calculus and analysis?

When I asked a teacher at school what the difference between calculus and analysis he said that calculus is essentially analysis without proofs. So where does proof-based calculus lie? Is it somewhere ...
8
votes
1answer
133 views

Do Electrical Engineering Researchers Usually Know Higher Math? e.g. Measure and Distribution Theory, Functional Analysis

I am an EE undergraduate student, and I recently took two courses in signals and systems. We were exposed to things like the Dirac delta (without mention of distributions), Fourier series (without ...
3
votes
1answer
53 views

A negative third derivative implies a positive first derivative at a point.

Let $f$ be three times differentiable on $\mathbb{R}$ with $f(0)=f(1)$, and for all $x\in[0,1]$, $f'''(x)<0$. Prove that $f'(\frac{1}{2})>0$ I actually have a proof of this question using ...
2
votes
0answers
20 views

Mixed vs Equal Characteristic Local Rings

This is more of a vague, intuitive type of question, so perhaps there isn't anything too concrete anyone can offer. I am trying to get a sense of precisely why working with local rings of equal ...
33
votes
6answers
2k views

Is there something between summation and integration?

Let's take a general function $f(x)$, we can do a summation like: $$\sum_{k=m}^n f(k)$$ And we can do an integration like: $$\int_a^bf(k)dk$$ The basic difference between the two operation is that ...
1
vote
1answer
38 views

Is there a way to generate groups, rings, fields, etc.? [on hold]

There are ways to generate a list of numbers $a_1, a_2,...a,n$ such that no $a_i,a_j$ share any factors, mainly by letting $a_{n+1}=a_n^2-a_n+1$ with $a_0>1$, my question is: is there a way to ...
3
votes
1answer
107 views

What motivates the definition of a ring in abstract algebra? [on hold]

I've read a lot of questions about rings but I still don't know why they are useful/ I'm sure they are, but I don't know why. Are their properties somehow used in proofs or as foundations for ...
0
votes
0answers
20 views

Order of 'Strength' of inequalities

There have been times when we solve an inequality and we get the reverse sign of inequality. The reason is quite simple- we did not choose a strong inequality. So my question is- Is there an order of ...
2
votes
2answers
387 views

Changing from Positive to Negative

I may mess up a little bit...Sorry for that! When we want a summation to go negative in case of odd numbers and positive otherwise , we use: $$\sum\limits_{i=1}^{12} \color{red}{{(-1)}^i} 2x^3 $$ ...
-4
votes
0answers
65 views

Should I drop my Calculus III course? [closed]

S.E advisers, I am a college sophomore in US with major in mathematics and an aspiring algebraist. I wrote this email because I am having a great problem with my current Calculus III (vector ...
3
votes
1answer
59 views

Reference Book for Calculus

I've made this post as detailed as I can so as to give you a fair idea of my level. Calculus (particularly Integration) is my passion and frankly I spend all my free time learning as much of it as I ...
1
vote
3answers
76 views

Can we generalize Aleph numbers to non integer values? [duplicate]

I'm really new to those kind of arguments so don't call me mad but I was wondering if there is a way to define an infinite set which cardinality is an Aleph-number like: $\aleph_{\frac 12}$. I have a ...
3
votes
0answers
36 views

Classical geometry vs. p-adic geometry

I am interested in learning more about $p$-adic analysis and some of the geometric theories that are commonly used. However, it seems that many of the tools there are developed from analogous ...
1
vote
1answer
38 views

What is Skorohod's Represntation Theorem Saying?

From Wikipedia: Let $\mu_n, n \in N$ be a sequence of probability measures on a metric space S; suppose that $\mu_n$ converges weakly to some probability measure $\mu$ on S as $n \to \infty$. ...
9
votes
1answer
103 views

What are some easier books for studying martingale?

What are some easier books for studying martingale? They are defined to be comprehensive but easier than Roger and William's martingale book. For example, to study Q and F martingales? It should ...
3
votes
0answers
71 views

When can we use Differentiation under the Integral sign?

Let me elaborate, 'Feynman's' trick ranks up in the top ten on most people's list, right behind contour integration, for best ways to evaluate definite integrals. However, unlike contour integration ...
3
votes
4answers
112 views

Why memorize trig identities?

I want to be a mathematician or computer scientist. I'm going to be a junior in high school, and I skipped precalc/trig to go straight to AP Calc since I've studied a lot of analysis and stuff on my ...
5
votes
6answers
409 views

Elementary number theory - prerequisites

Since summer comes with a lot of spare time, I've decided to select a mathematical subject I want to learn as much as possible about over the next three months. That being said, number theory really ...
0
votes
0answers
22 views

Urban planning and maths [closed]

Sorry if my english is bad, it's not my mother tongue. I'm going to study Urban Planning and i'd like to know what math topics could be useful in my career. Regards!
0
votes
0answers
39 views

Is there any practical use of this algorithm?

Example The exact solution of a DE $\frac{dp}{dv}=-1.4\frac{p}{v} $ with initial condition $(P_1,V_1)=(1,1)$ can be obtain by solving the integral ...
0
votes
0answers
50 views

What can we do to help countries to recover/develop their mathematical strength? [closed]

I was reading "The antiscientifical revolution and mathematics" (1998), by V. I. Arnold. And I thought that the situation (decline) in Russia is really bad for mathematics in the world. A country that ...
1
vote
2answers
122 views

Is Fermat's Last theorem equivalent to $1 + 1 = 2$?

I got into a debate with someone concerning whether FLT is equivalent to $1 + 1 =2$. He said common sense tells us it isn't equivalent. However, I disagreed. Since both are provable statements, they ...
2
votes
1answer
42 views

Desk Reference Question: Linear Algebra and its Applications by Lay or Strang? Or Handbook of Linear Algebra?

I would like a good working desk reference for linear algebra for someone in applied mathematics (no proving abstract theorems). I saw the Handbook of Linear Algebra as well, but I am concerned it may ...
0
votes
0answers
84 views

How do i remove my Guilt? [closed]

When i see a theorem ,which i cannot prove in real analysis ,i think about it but still i couldn't figure it out .Then i look for its solution ,after understanding the proof i feel very guilt that i ...
1
vote
1answer
49 views

What is the significance of squaring a number? [closed]

I've always been baffled by the significance of squaring a number. I understand what it means ( $10^2 = 10 \cdot 10$) but what is the significance of doing this? Obvious examples are: $E=mc^2$ Area ...
0
votes
0answers
31 views

Does anybody know a good introduction to homology?

Essentially what the title says. I need something that will give me a decent introduction into homology theory. I don't need too deep of an understanding, just enough to get through a paper I'm ...
3
votes
0answers
41 views

Examples of calculus on “strange” spaces

I am interested in examples of calculus on "strange" spaces. For example, you can take the derivative of a regular expression[1][2]. Also the concept extends past regular languages, to more general ...
0
votes
0answers
28 views

Preparing for undergraduate mathematics exams [closed]

I have almost (I have my last exam tomorrow) finished the 3rd semester of my undergraduate mathematics degree, and have been reflecting on the study methods which have proven most effective. Most of ...
1
vote
1answer
44 views

Uses of stalks of sheaves and germs

I am trying to understand the motivation behind defining stalks of sheaves, but I suppose my complex geometry is a little weak. I know they are meant to represent germs of holomorphic functions at a ...
3
votes
0answers
66 views

Teaching to Learn [closed]

I am interested in using teaching as a way of learning, but I am uncertain as how to best start. At the moment, I am only a sophomore in university and am relatively new to studying math. Currently,my ...
0
votes
0answers
18 views

Convex functionals, min-max of linear functionals, etc

I was reading in this book on the well-known Courant-Fischer min-max Theorem, which states that if $\lambda_1(A)\geq \lambda_2(A)\geq \lambda_n(A)$ are the $n$ eigenvalues of a $n\times n$ Hermitian ...
-2
votes
0answers
45 views

Why Syracuse's problem so difficult to solve?

I'm really astonished that Syracuse's problem is very difficult to solve ! How it is possible, here the sequence : $U_{n+1}=\frac{U_n}{2}$ if $U_n$ is an even number and $U_{n+1}=3\times U_n+1 $ ...
5
votes
2answers
77 views

Is Keno a fair game?

This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with probability, which perhaps yields the shortest, simplest proofs, but other ...
59
votes
34answers
6k views

Easy math proofs or visual examples to make high school students enthusiastic about math [closed]

I'm a teacher in mathematics at a high school. Math has fascinated me for almost my entire life, so I would like to bring that enthusiasm to my students with beautiful yet easy to understand proofs or ...
0
votes
0answers
21 views

What does this statement mean exactly?

I would like some clarification about the following from Fejes Toth's paper "A stability criterion to the moment theorem" The setup is: For each positive integer $n$, let $r(H_n)$ and $R(H_n)$ ...
5
votes
0answers
56 views

How do you avoid getting rusty at applied math after univeristy [closed]

As a new postdoc working in a bio-math interface discipline, I often wish I had more formal math training than my math minor many years back. Compared to others who came from more of a ...
4
votes
0answers
65 views

Manifold in Milnors Morse Theory

While reading "Morse Theory" by Milnor, I noticed that certain arguments would not work, if the considered manifolds have nonempty boundary. Example: Proof of 3.5 I could not find the definition ...
3
votes
0answers
32 views

Where to study type theory?

I want to learn more about (homotopy) type theory, constructive mathematics and univalent foundations. To my knowledge, there are only few faculties with large type theory groups. In Europe, most of ...
0
votes
0answers
32 views

GRE Math Subject Test ,Please Check my Plan

I am to give test in october. I have planned to use Schaum series 3000 solved problems in calculus and 3000 solved problems in Linear Algebra for that .Also i will use Herstein for group theory .Since ...
2
votes
1answer
108 views

Is it feasible for a sophomore in high school (15 years old) to learn complex analysis? [closed]

I've been reading up on complex analysis and it seems an incredibly fascinating subject to me and one I'd like to learn more about. However, most of the books I've come across are for graduates, which ...
0
votes
1answer
55 views

General Inquiry on Mathematics syllabus [closed]

This isn't a technical math question but I would like to consult you masters as to what to study in math? I'm very interested and my knowledge level is at around Calculus II. I wanted to know more ...
1
vote
1answer
32 views

Complex integration: normally on a closed contour?

I have been studying complex integration for a few months now, and it seems my textbook mostly considers integration on closed contours. Is there no interest in integration on non-closed contours ?
3
votes
1answer
27 views

Skill plateau, overpracticing, and alternative practice methods

It's the summer holiday for me right now and I've been spending a lot of time doing math problems. I've done a bunch of Olympiad questions and the like recently, and I feel like I've hit a plateau ...
4
votes
4answers
84 views

Mathematical philosophical questions about the general theory of stochastic processes.

After 6 months spent on what is termed the "general theory" of stochastic processes and after having worked out many nuances of the field, I realized that: The general theory is beautiful ...
14
votes
4answers
201 views

How to write $\aleph$ by hand

So far, I've only seen the symbol $\aleph$ in its printed form and am wondering how this symbol could be written by hand on paper or on a board (in mathematical contexts, of course). Whenever I try to ...
4
votes
1answer
57 views

Difference between ,say, “At least 8” and “8 or more”

Are they not the same the thing? Just to be on the safe side I wanted to verify this with others. Sorry for the stupid question.
8
votes
1answer
103 views

Is there a “Coalgebra - Cogeometry” duality? Good opposite of a category of coalgebras?

So the category of affine schemes is dual to the category of commutative rings, Stone spaces are dual to Boolean algebras, localizable measurable spaces are dual to commutative Von Neumann algebras, ...