4
votes
4answers
58 views

Prove that $a_n$ is a perfect square

Let $(a_n)_{n \in \mathbb{N}}$ be the sequence of integers defined recursively by $a_1 = a_2 = 1$, $a_{n+2} = 7a_{n+1}-a_n-2$ for $n \geq 1$. Prove that $a_n$ is a perfect square for every $n$. We ...
1
vote
0answers
8 views

$-\Delta u = u^p$ in bounded domain

In my PDE lecture we had the following theorem and I am wondering how strong it is: Theorem Let $\Omega \subset \mathbb{R}^n$ be a bounded domain (with $\partial \Omega$ sufficiently smooth, lets ...
0
votes
1answer
718 views

How to calculate t-value, given degrees of freedom and $\alpha$.

While solving problems, we can look up physical t-tables or use a statistical analysis software like R to calculate t-values. But how do we actually calculate these values ? What is the algorithm ...
4
votes
6answers
3k views

Applications of Complex Numbers

For my Complex Analysis course, we are to look up applications of Complex Numbers in the real world. The semester has just started and I am still new to the complex field. I want to get a head start ...
0
votes
0answers
6 views

What is the most general context to study Laplace transform?

In undergraduate course, one learns the fourier transform of continuous absolutely integable functions using Riemann integrals. Then, one learns the fourier transform in the context of measure theory ...
0
votes
1answer
10 views

How does one plug radicals with non-perfect squares and variables into the Pythagorean theorem formula?

I am working on the following integral $$\int { { \left( 7{ x }^{ 2 }-3 \right) }^{ \frac { 5 }{ 2 } } } dx$$ I want to use the $\sqrt{u^2 - a^2}$ $u = aSecθ$ I know in order to get it into ...
14
votes
3answers
2k views

Does writing a bachelor thesis make sense?

I am a math student in my fourth semester. At my university, it is common to write a bachelor-thesis in the end of the bachelor program in almost all subjects while in the math undergraduate program ...
2
votes
1answer
96 views

How do you use reference books?

Reference books at the research levels often does not include any problem or exercise. While you can't read these books like novels(you normally need to work on other sheet of paper), I'm just ...
5
votes
0answers
141 views
+50

solve this 1999 problem with geometry

if $\bigodot P\bigcap \bigodot Q=A,B$,and the common tangent is $C,D$,and $E\in BA$,and $EC\bigcap \bigodot P=F,ED\bigcap \bigodot Q=G$,and if $\angle FAH=\angle HAG$ show that $$\angle FCH=\...
6
votes
1answer
313 views

Recommended research topics for high school student

I am a high school senior and I am interested in doing a math research. I hope someone can recommend areas or topics of research that are challenging, rewarding, and yet do not exceed my capability. (...
36
votes
6answers
2k views

Olympiad Inequality $\sum_{cyc} \frac{x^4}{8x^3+5y^3} \geqslant \frac{x+y+z}{13}$

$x,y,z >0$, prove $$\frac{x^4}{8x^3+5y^3}+\frac{y^4}{8y^3+5z^3}+\frac{z^4}{8z^3+5x^3} \geqslant \frac{x+y+z}{13}$$ Note: Often Stack Exchange asked to show some work before answering the ...
23
votes
11answers
3k views

Gap year to study math

This is a plan in its earliest and thus least concise stage, so either bear with me or don't read the following babble (I bolded some of the important stuff): I am a high school graduate who is about ...
2
votes
2answers
319 views

Solving Induction $\prod\limits_{i=1}^{n-1}\left(1+\frac{1}{i}\right)^{i} = \frac{n^{n}}{n!}$

I try to solve this by induction: $$ \prod_{i=1}^{n-1}\left(1+\frac{1}{i} \right)^{i} = \frac{n^{n}}{n!} $$ This leads me to: $$ \prod_{i=1}^{n+1-1}\left(1+\frac{1}{i}\right)^{i} = \frac{(n+1)^{n+1}}...
48
votes
6answers
10k views

What is mathematical research like?

I'm planning on applying for a math research program over the summer, but I'm slightly nervous about it just because the name math research sounds strange to me. What does math research entail exactly?...
0
votes
0answers
13 views

ODE, Lipschitz condition

Under the given hypotheses the solution is 'unique' by the existence and uniqueness theorem in ODE. I couldn't fathom why they asked to show it again. Any help is much appreciated.
-3
votes
2answers
50 views

Redundant proof in Math paper

Recently, I read a published math paper and I found that in the excessive argument in the proof of one of its theorem. In fact, in my opinion, the redundant part is not even correct, because it ...
2
votes
1answer
110 views

Undergraduate Project Suggestions

A student of mine has expressed interest in doing an independent project next quarter with me. This would not be for credit and it is purely for her own educational stimulation. She wants to study ...
2
votes
0answers
21 views

'2nd order' Picard Iteration

I'm self-studying differential equations using MIT's publicly available materials. One of the problem set exercises deals with what I'm calling a second order Picard Iteration. To be explicit, we ...
14
votes
3answers
575 views

Prove $\frac{xy}{5y^3+4}+\frac{yz}{5z^3+4}+\frac{zx}{5x^3+4} \leqslant \frac13$

$x,y,z >0$ and $x+y+z=3$, prove $$\tag{1}\frac{xy}{5y^3+4}+\frac{yz}{5z^3+4}+\frac{zx}{5x^3+4} \leqslant \frac13$$ My first attempt is to use Jensen's inequality. Hence I consider the function $...
0
votes
1answer
2 views

Find intersecting points on rectangle edges for line drawn inside it

Draw a rectangle ABCD. Draw a line inside it connecting any two edges GF. Draw a perpendicular bisector to line GF. At what points does the perpendicular bisector intersect the edges of the ...
3
votes
0answers
20 views

Question about arc-connected property in a continuum

Suppose $X$ is metric, compact, connected, and $p\in X$. An arc is a copy of $[0,1]$. Is it possible that every two points in $X\setminus \{p\}$ can be joined by an arc, but there is no arc in $X$ ...
25
votes
2answers
1k views

Collections of undergraduate research projects [closed]

I would like to compile a "big list" of undergraduate research projects in the following areas of mathematics: calculus; analysis; abstract algebra; linear algebra; number theory; geometry; ...
1
vote
1answer
44 views

self-homeomorphism of the circle

$|z|=1$ is the unit circle in the complex plane. Suppose $g$ is a self-homeomorphism of this circle of order $n$, $n \in \mathbb{N}$, and $g$ acts freely. Is that true that $g$ must be defined by $z ...
2
votes
0answers
32 views

$A,B$ be countable dense subsets of $\mathbb R$ , let $A,B$ be given usual subspace topologies , then there exists a homeomorphism $f:A \to B$?

Let $A,B$ be countable dense subsets of $\mathbb R$ (with usual euclidean topology ) let $A,B$ be given usual subspace topologies , then is it true that there exists a homeomorphism $f:A \to B$ ?
0
votes
1answer
125 views

Researching in Mathematics [closed]

I am presently pursuing Engineering, but I want to make my career in the field of mathematics. How do I come to know of the specific topic in math in which I would like to research, in which I would ...
3
votes
0answers
204 views

Inquiry about My Self-Study Plan for Real Analysis (associated with my undergraduate research) [closed]

S.E advisers, I am a college sophomore in US with a major in mathematics and an aspiring mathematician in the computation theory and cryptography. I recently got an undergraduate research in the ...
1
vote
2answers
113 views

Advice for an undergraduate math major [closed]

I've recently switched into the mathematics department at my university preceding my second sophomore semester. My question is the following: What should I study over the summer to best aid my ...
3
votes
5answers
88 views

Recommended books that discuss the Fundamental Theorem of Algebra?

I've been assigned to do a project on the Fundamental Theorem of Algebra and in particular discuss it's proofs and applications. I was wondering if anyone could recommend books that would aid me in my ...
1
vote
0answers
41 views

Error Correcting Code and Graph Theory

I am currently in an introductory graph theory class, and we are supposed to give a short presentation by the end of the semester. Recently, I've learned (a very small amount) about error correcting ...
-1
votes
2answers
55 views

probability of getting an erdos number once published [closed]

Can I know that I don't have an erdos number once I published, what the probability is of getting an erdos number with "random" coauthors or can I formulate the probability of having a finite erdos ...
18
votes
4answers
617 views

Problems from the Kourovka Notebook that undergraduate students can fully appreciate

The Kourovka Notebook is a collection of open problems in Group Theory. My question is: could you point out some (a "big-list" of) problems [by referencing them] presented in this book that are, ...
0
votes
0answers
23 views

Dynamical system in a square

I am considering a problem that is asking me to explore a deceptively simple dynamical system and discover some of surprising properties. I want to consider the motion of four particles A,B,C and D in ...
4
votes
1answer
229 views

Research: Looking for a sequence that produce variation's of Pascal's triangle

Prologue I am an undergraduate so if my terminology or approach seem inappropriate/confusing please explain in the comments. I created a notation where $$F(0 \rightarrow n,x) = [\hspace{1mm}F(0 ,...
0
votes
0answers
5 views

Non empty interier in the image implies open map

I am looking at the proof showing that $L^2(0,1)$ is meager in $L^1(0,1)$. Define $B_n = \{f\in L^2 : \|f\|_2 \leq n\}$. With the continuous identity map $T:L^2 \rightarrow L^1$, if one of $T(B_n)$ ...
1
vote
0answers
38 views

A generalization for Binomial Theorem, Leibniz General Rule other like functions.

$n$ & $m :=$ any value in $\{0,1,2\ldots\}$ $\Omega$ & $\beta :=$ any value in $\{1,2,3\ldots\}$ [1] If there is a function $F_\beta$ such that for some value $\Omega$ and some function $T_{...
0
votes
0answers
26 views

Differentiating a Unique Multi-variable Function

I found this interesting property during my research were $\beta$ is some function of x and the a,e,n are all dependent on x; If $\frac{d \hspace{1pt} \beta(a,e,n)}{dx}=\beta(a+\phi,e,n)+\beta(a,e+\...
1
vote
2answers
41 views
+50

Proportionally Distributing $N$ items across $B$ bins.

My question is similar to this: Proportional Distribution My problem follows: I have $N$ items that cannot be broken up into fractional components, but should be distributed across $B$ bins where ...
0
votes
0answers
18 views

distribution and density of maximum minus element

I am a bit rusty in probability, and for a project I am studying the random variable $Z = \max(X_1, \ldots, X_n) - X_i, i = 1, \ldots, n$ where the $X_i$ are positive independent random variables. In ...
0
votes
0answers
7 views

Can we find exact factor of inert prime ideals?

$K=\mathbb{Z}[\sqrt{m}]$ with $m$ being square free. I studied the proof of the statement that a prime ideal $\mathfrak{p}=\langle p \rangle$ of $\mathbb{Z}$ stays inert in $\mathcal{O}_K$ if $p=2$...
0
votes
0answers
9 views

Counting the number of Eulerian trails in a connected, directed graph

I can't find anything about this online, and I'm beginning to suspect it's a hard problem. I know that counting the number of circuits is #P-complete, but I don't need the number of circuits; I need ...
0
votes
1answer
18 views

Text explanation: Ellipses and their intersection points

Given two ellipses $E_1,E_2$ of equation $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=4$$ prove that for all $p\in E_1$ there exist a unique ellipse $F_p$ that meets $...
1
vote
4answers
83 views

Show that $\log(1+y) \approx y- \frac {y^2}2 + \cdots$ without Taylor Series

For small $y$, prove that $\log(1+y)\approx y -\frac {y^2}2 + \cdots $ I have no idea to solve it.
1
vote
2answers
14 views

Considering the complex number $z = m+i$ for which values of $m$ do we have $ \left|\overline{z}+\frac{2}{z}\right| \ge 1 $

Good evening to everyone. I have the following problem that I tried to solve but my mathematical instinct tells me that I didn't solve it right: Considering the complex number $z = m+i$ for which ...
0
votes
0answers
8 views

Deformation complex of Lie algebra structures

I am learning about deformation theory, e.g. through The unbearable lightness of deformation theory by Szendröi. There the standard example of deformations of a structure of associative algebra, ...
0
votes
0answers
14 views

Calculus for Proving Properties of Discrete Objects

I posted a question earlier about a proof in graph theory I was trying to figure out. In my attempt I used Calculus to prove a part of the theorem. In the comments people kept saying how you shouldn't ...
0
votes
0answers
30 views

The Spacing of $e$ and $\pi$ Segments Within the Decimal Expansion of $\pi$

I discovered something seemingly very improbable today when I was searching for segments of $e$ and $\pi$ within the decimal expansion of $\pi$. I searched for $314159265$ and found it starts at the ...
0
votes
1answer
28 views

Poisson's equation solution explicitly

How can I write poisson's equation $\partial_{xx} u = f$ solution in 1d explicitly? I have seen somewhere I can write $u(x) = \int^{x}_{0}\int^{y}_{0} f(z) dz dy - \int^{1}_{0}\int^{x}_{0}\int^{y}...
3
votes
2answers
43 views

Are there $a,b \in \mathbb{N}$ that ${(\sum_{k=1}^n k)}^a = \sum_{k=1}^n k^b $ beside $2,3$

We know that: $$\left(\sum_{k=1}^n k\right)^2 = \sum_{k=1}^n k^3 $$ My question is there other examples that satisfies: $$\left(\sum_{k=1}^n k\right)^a = \sum_{k=1}^n k^b $$
2
votes
3answers
52 views

Baker-Campbell-Hausdorff/Zassenhaus formula to first order in one matrix

Is there a closed-form expression for the term of $e^{t(c \hat{X} + d \hat{Y})}$ that is first-order in $d$, where $t$, $c$, and $d$ are scalars and $\hat{X}$ and $\hat{Y}$ are finite-dimensional ...
2
votes
0answers
28 views

Is the tensor product of BAOs a kind of extended BAO?

I've been reading "Boolean algebras with operators. Part I." (Jonsson, Tarski) where, given a subalgebra of a Boolean Algebra, they define its perfect extension. As far as I understand it can be ...

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