# All Questions

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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
141 views
+50

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?...
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.
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 ...
110 views

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 ...
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 ...
575 views

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 $f:A \to B$ ?
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 ...
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 ...
113 views

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 ...
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 ...
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 ...
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 ...
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, ...
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 ...
229 views

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 ,... 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) ... 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_{... 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+\... 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 ... 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 ... 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... 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 ... 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$$prove that for all p\in E_1 there exist a unique ellipse F_p that meets ... 4answers 83 views ### Show that \log(1+y) \approx y- \frac {y^2}2 + \cdots without Taylor Series For small y, prove that \log(1+y)\approx y -\frac {y^2}2 + \cdots  I have no idea to solve it. 2answers 14 views ### Considering the complex number z = m+i for which values of m do we have  \left|\overline{z}+\frac{2}{z}\right| \ge 1  Good evening to everyone. I have the following problem that I tried to solve but my mathematical instinct tells me that I didn't solve it right: Considering the complex number z = m+i for which ... 0answers 8 views ### Deformation complex of Lie algebra structures I am learning about deformation theory, e.g. through The unbearable lightness of deformation theory by Szendröi. There the standard example of deformations of a structure of associative algebra, ... 0answers 14 views ### Calculus for Proving Properties of Discrete Objects I posted a question earlier about a proof in graph theory I was trying to figure out. In my attempt I used Calculus to prove a part of the theorem. In the comments people kept saying how you shouldn't ... 0answers 30 views ### The Spacing of e and \pi Segments Within the Decimal Expansion of \pi I discovered something seemingly very improbable today when I was searching for segments of e and \pi within the decimal expansion of \pi. I searched for 314159265 and found it starts at the ... 1answer 28 views ### Poisson's equation solution explicitly How can I write poisson's equation \partial_{xx} u = f solution in 1d explicitly? I have seen somewhere I can write u(x) = \int^{x}_{0}\int^{y}_{0} f(z) dz dy - \int^{1}_{0}\int^{x}_{0}\int^{y}... 2answers 43 views ### Are there a,b \in \mathbb{N} that {(\sum_{k=1}^n k)}^a = \sum_{k=1}^n k^b  beside 2,3 We know that:$$\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 
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 ...