# All Questions

109 views

### Suggest a tricky method

Problem: If $a_1,a_2,a_3 \cdots a_n$ are in HP then find the value of $a_1 \cdot a_2 + a_2 \cdot a_3 + a_3 \cdot a_4 + \cdots + a_{n-1} \cdot a_n$ My initial approach,using the property of ...
53 views

### A small doubt on calculation

Problem: If the sum of the first $p$ terms of an arithmetic progression is $q$ and the sum of the first $q$ terms is $p$, then find the sum of $p+q$ terms. For the problem we can write (considering ...
207 views

### Series partial sums

I have a school assignment to do, but I have no idea, where to start. I hope you can help. Here it is: We have a finite sequence $A = (a_1, a_2,\ldots, a_n)$, length of $A$ is $n$, elements of $A$ ...
282 views

### If A is a subset of R with Lebesgue measure > 0 then are there a,b such that the measure of $[a,b]\cap A$ is b-a?

If $A$ is a subset of $\mathbb{R}$ with Lebesgue measure strictly greater than $0$, does it follow then that are there $a$ and $b$ such that the measure of $[a,b]\cap A$ is $b-a$? Thank you.
151 views

### Let $f\colon A\to A$; prove that if $f\circ f = \mathrm{id}_A$ then $f$ is a bijection

Let $f\colon A\to A$; prove that if $f\circ f = \mathrm{id}_A$ then $f$ is a bijection.
309 views

### Starting digits of 2^n

Prove that for any finite sequence of decimal digits, there exists an $n$ such that the decimal expansion of $2^n$ begins with these digits.
2k views

234 views

### Invariant Subspace Problem

Louis de Branges had a paper on his homepage claiming a solution for the Invariant Subspace Problem. But I don't see that paper anymore, though he still has a "proof" of Riemann Hypothesis on his ...
370 views

### Using Chazelle's simplicity test to verify simple polygons intersection

Is there a way to verify whether a non-empty intersection exists between two simple polygons (not necessarily convex) using the Chazelle's simplicity test ?
247 views

### What is the maximum number of primes generated consecutively generated by a polynomial of degree $a$?

Let $p(n)$ be a polynomial of degree $a$. Start of with plunging in arguments from zero and go up one integer at the time. Go on until you have come at an integer argument $n$ of which $p(n)$'s value ...
210 views

### Finding $\min_{\mathbf x} (\mathbf y - \mathbf G\mathbf x)^T(\mathbf y - \mathbf G\mathbf x)$

Let $\mathbf G$ be a given $m \times n$ matrix, $\mathbf y$ a given $m \times 1$ column vector and $\mathbf x$ an unknown $n \times 1$ column vector such that $\mathbf x \ge 0$. 1) How do you find ...
1k views

### No simple group of order 300

So Ive been trying to prove that there's no simple group of order 300. This is what I did and I was wondering if it was enough. Let |$G$|=2$^2$ . 3 . 5$^2$. Suppose $G$ is simple. Then there would be ...
281 views

### Soundness for $\forall$Introduction

I have some troubles proving soundness for $\forall$ Introduction. Let $T_1,...,T_n\vdash Q$. $\forall$ Intro gives that $T_1,...,T_n\vdash \forall x Q[x/t]$. Here, t is a constant symbol that does ...
239 views

### Prove in harmonic progression

Prove that if the system of equations $x+2ay + az = 0$,$x+3by+bz = 0$ ,$x+4cy+cz = 0$ has a non-zero solution then a,b,c are in harmonic progression. I am looking for a suitable approach for this ...
285 views

485 views

### deadline in math jobs application

In some math job advertisments, it shows "the deadeline is...", but to surprised some schools review applications before that, does that mean we submit application as earlier as possible could be in a ...
4k views

### How to show $e^{e^{e^{79}}}$ is not an integer

In this question, I needed to assume in my answer that $e^{e^{e^{79}}}$ is not an integer. Is there some standard result in number theory that applies to situations like this? Much later addendum: ...
370 views

### Is it problem of Mathematica or my own?

The following is a plot comparing Exp[Derivative[1,0][Zeta][0,x]+1/2Log[2 Pi]] and Gamma[x]: In theory the blue and the red ...
2k views

### $e^{e^{e^{79}}}$ and ultrafinitism

I was reading the following article on Ultrafinitism, and it mentions that one of the reasons ultrafinitists believe that N is not infinite is because the floor of $e^{e^{e^{79}}}$ is not computable. ...
325 views

### Ranges and the Fundamental Theorem of Calculus 1

I'm going over a chapter by chapter review for my calculus final and discovered this problem: $$y=\int_{\sqrt{x}}^{x^3}\sqrt{t}\sin{t}\;\mathrm dt$$ They split it up so that it became: ...
353 views

### binomial-Poisson/beta hierarchy

X|N,P ~ Binomial(N, P) N ~ Poisson(11) P ~ Beta(2,3) What is the moment generating function for X?
960 views

### Calculate the surface area of a solid of revolution

I have to calculate the surface area of the solid of revolution which is produced from rotating $f: (-1,1) \rightarrow \mathbb{R}$, $f(x) = 1-x^2$ about the $x$-axis. I do know there is a formula: ...
244 views

### AM-GM application

How to show that $$\displaystyle (a_1 + a_2 + a_3)(\frac{1}{a_1} + \frac{1}{a_2} + \frac{1}{a_3}) \ge 9$$ $a_1,a_2,a_3$ are all of same algebraic sign.
938 views

### Finitely axiomatizable theories

Let T1 and T2 be two theories having the same set of symbols. Assume that any interpretation of T1 is a model of T1 if and only if it is not a model of T2. Then: T1 and T2 are finitely axiomatizable. ...
240 views

### Are there problems that can't be expressed as languages?

OK, so I was told in CSTheory that I should be asking here. So my question is the following: I've taken my first course on Language Theory and we saw the "standard" classification of languages. We ...
989 views

2k views

### Tensors as matrices vs. Tensors as multi-linear maps

So I read the answers in this question, and don't feel that much closer to an answer about how tensors as multi-linear maps and tensors as "multi-dimensional" matrices are truly related. For ...
187 views

### What field is $\mathbb{F}_p(\zeta_{p-1}^{1/n})$

What order is the field: $\mathbb{F}_p(\zeta_{p-1}^{1/n})$? $\mathbb{F}_p^{\times}$ is size $p-1$ and cyclic (with a generator we shall call $\zeta_{p-1}$). Naively, it seems that by taking the ...
150 views

### Examples of advanced results and ideas explained in a down-to-earth way

Are there any advanced topics--preferably at the research frontier--which can nevertheless be explained accurately, if not efficiently, using very down-to-earth ideas which would be accessible to most ...
2k views

### Parallelograms & Axis of Symmetry

Consider a parallelogram which is neither a rhombus nor a rectangle. It is well-known that such shapes do not have an axis of symmetry. Is there a simple proof for this? I prefer a proof that I ...