0
votes
0answers
6 views

Formalizing a concept of vectors

I'm reading a book about Vectors. I was in a problem where he showed two vectors ($\overrightarrow{AB}$ , $ \overrightarrow{AD}$ | $\overrightarrow{AB} \neq \vec{0} , \overrightarrow{AD} \neq \vec{0} ...
-19
votes
0answers
36 views

I'm trying to solve the Riemann Hypothesis [on hold]

My teacher assigned me to solve the Riemann Hypothesis. I have been searching for hours on the Internet on "How to". And I found this very good cream called Fair and Handsome, and it claims that if ...
3
votes
7answers
48 views

how to show that f(x) = 0

I found this problem on the web: Let $f(x)$ be a real-valued, continuous function with the property that $$\int_a^bf(x)\,\text{d}x=0$$for all real numbers $a,b$. Prove that $f$ is identically $0$. ...
1
vote
0answers
14 views

An unusual two dimensional sum

Can anyone prove or reference a proof for the following bound (unless it's not true!) $$\sum_{|\underline{k}|_{\infty} > M} \frac{1}{((k_1)^2 + (k_2)^2 )^2} \leq \frac{C}{M^2}$$ where ...
0
votes
0answers
12 views

Vector Spaces: Tensor Product

Reference Foundation for: Hilbert Spaces: Tensor Product Problem Given a vector spaces $V$ and $W$. Take its algebraic tensor product: $\tau:V\times W\to V\otimes W$ How to prove that the image ...
6
votes
1answer
24 views

Differentiating a constant and switching order

Why does this work? $$\int x^2e^{ax}dx = \int \frac{d^2}{da^2}e^{ax}dx = \frac{d^2}{da^2}\int e^{ax}dx = \frac {d^2}{da^2} \frac {e^{ax}}a = \frac{e^{ax}(a^2x^2-2ax+2)}{a^3}$$ $a$ is a constant, so ...
1
vote
2answers
14 views

If we have $f(x)=e^x$, then what is the maximum value of $δ$ such that $|f(x)-1|< 0.1$ whenever $|x|<δ$?

If we have $f(x)=e^x$, then what is the maximum value of $δ$ such that $|f(x)-1|< 0.1$ whenever $|x|<δ$? I tried to solve this problem with delta-epsilon definition from the definition, 1 is ...
0
votes
0answers
7 views

a problem about finding an algorithm for a spanning tree in a 3-regular graph

I found a hard problem in one of my books about graphs and i have been thinking on it for 2 weeks. The problem is: "Find an algorithm that finds a spanning tree inside a connected 3-regular graph ...
2
votes
2answers
34 views

Theorem: the first positive number to have 500 divisors has to be even.

How can I get started on this proof? I was thinking originally: Let $ n $ be odd. (Proving by contradiction) then I dont know.
1
vote
4answers
24 views

Proving that $\int_0 ^1 \frac{\text{d}s}{\sqrt{1-s^2}}$ converges with no trig functions

Let $$\int_0 ^1 \frac{\text{d}s}{\sqrt{1-s^2}}$$ How to show that it converges with no use of trigonometric functions? (trivially, it is the anti-derivative of $\sin ^{-1}$ and therfore can be ...
2
votes
1answer
21 views

Simply connected and connected in complex analysis

Can some one please help me with this, why is third set in the picture not simply connected. The definition of simply connected (in space of complex numbers) is: A set is said to be simply ...
0
votes
0answers
20 views

What is the smallest positive period of the fractional part function $\mathbb R \to \mathbb R:x \to\{x \}$ ?

Is $1$ the smallest positive period of the fractional part function $\mathbb R \to \mathbb R:x \to\{x \}$ ?
1
vote
2answers
25 views

Find all intergers such that $2n^2+1$ divides $n^3+9n-17$

Find all intergers such that $2n^2+1$ divides $n^3+9n-17$. Answer : $n=(2 \ and \ 5)$ I did it. As $2n^2+1$ divides $n^3+9n-17$, then $2n^2+1 \leq n^3+9n-17 \implies n \geq 2$ So $n =2$ is ...
0
votes
1answer
10 views

Properties of Image and Inverse Image

Let $f:X\rightarrow Y,A\subset X$ and $B\subset Y$. If $f^{-1}(E)\subset A$, then $E\subset f(A)$ I cannot understand that why this statement is false. Any counterexample?
0
votes
0answers
5 views

Cycle of length k with no repeated edges

I need to figure out what is the minimum complexity class (L, NL, P, NPC etc..) of the following problem: Given an undirected graph G, is there exist a cycle (doesn't have to be a simple cycle) with ...
0
votes
1answer
9 views

A question about sum of angles in a non-positive curvature Riemannian manifold

Suppose on a non-positive curvature Riemannian manifold,we have a geodesic triangle $\triangle abc$ ,and counterpart edges donates $\alpha,\beta,\gamma$. If now I get $$ a^2 \geqq b^2+c^2-2bc ...
-1
votes
0answers
17 views

Proof of limit of a piecewise function, rational, irrational

Prove that: If $f(x) = 0$ for irrational $x$ and $f(x) = 1$ for rational $x$ then $\lim_{x \to a} f(x)$ does not exist for any $a$. So begin by the opposite assumption: Assume $\lim_{x \to a} f(x) ...
1
vote
0answers
12 views

need help with set notation

I have a set of Students: $S = \{s_1, \ldots, s_2 \}$. Now each student takes some class (doesn't matter what class). Now I need to have a set $X$ that contains all students that take the same class, ...
3
votes
1answer
23 views

Prove that the following polynomials are irreducible or not.

I want to show that: 1) $X^4+1$ is irreducible The roots are the elements of $$\left\{\frac{\pm 1+i}{\sqrt 2},\frac{\pm 1-i}{\sqrt 2}\right\}$$ therefore it's not a product of a polynomial ...
0
votes
0answers
11 views

Find probability density function of random vector

Random vector has continuous distribution of setA={(u,v), v>=0, u+v<=1, v-u<=1}. I need to find joint probability density function of this vector. In my ...
1
vote
0answers
10 views

Equilibrium points and linear stability

Consider the nondimensional amplitude equation for $A = A(t)$ where $t$ is time given by (1): $$ \frac{dA}{dt} = \sigma A - a_1 A^3 - a_3 A^5 = f(A) \text{ with } \sigma \in \mathbb{R}, a_1 < 0, ...
0
votes
0answers
10 views

Partial Correlation Coefficient

I have the following questions on computing the correlation coefficient. Let us say we have two discrete random variables $X_1$ and $X_2$, where $X_1$ has $n_1$ outcomes and $X_2$ has $n_2$ ...
0
votes
0answers
13 views

Graph Theory with Homology Diagram

In homological algebra we end up playing around with diagrams a lot. For example, when we prove that a whole diagram commutes (like a chain map between two chain complexes) we simply prove that each ...
0
votes
0answers
8 views

Limit distribution on return time $\tau = \inf\{k: X_k = X_m \text{ where }m<k\}$ [on hold]

Suppose there is a stochastic process ${X_i}_{i=1}^n$ where $X_i$ is distributed normally over $\{1,\dots,n\}$. As $n\rightarrow \infty$, the probability that any one value is repeated should go to ...
0
votes
0answers
21 views

advice for self studying [on hold]

I took up to ap calc bc, ap chem, and ap physics b until my sophomore year in highschool after that I quite school and came back to Korea and got a job I'm hoping to one day go to college here and ...
0
votes
0answers
11 views

Fast Fourier transfrom

What are the prerequisites for understanding the fast fourier transform for fast multiplication? What topics should I be familiar with first?
1
vote
3answers
27 views

Is ${\mathbb R}^n$ with the product topology the same as the metric topology

I have looked at several places into the definition for product spaces. Now all of the definitions I have seen, define the product space topology as generated from the product of sets $U_i$, for which ...
0
votes
0answers
15 views

On subgroups of the form $HZ(G)$ where $H$ is abelian subgroup of non-abelian group $G$ such that $H \subsetneq Z(G)$

Let $H$ be a an abelian subgroup of a non-abelian group $G$ such that $H \subsetneq Z(G)$ ; then I can prove that $HZ(G)$ is an abelian subgroup such that $Z(G) \subset HZ(G) \subset G$ ; also if $H$ ...
2
votes
3answers
45 views

Linear dependency of nilpotent matrices

I would like to prove that four $2\times 2$ nilpotent matrices are always linearly dependent, using the Cayley-Hamilton theorem or the minimal polynomial in some way. I think I have proved the ...
1
vote
1answer
15 views

A question about a perfect space and a linear order on it

Suppose I have a nonempty perfect Polish space $X$, and there's some linear order $<$ on it (it is not related to the topology on $X$ in any way). How can I prove that there is a point $y$ in $X$ ...
2
votes
2answers
26 views

Finding the limit of this integral: $\lim_{n\to\infty} \int_0^1 \dfrac{n x^p+x^q}{x^p+n x^q} dx$ if $q<p+1$

I am trying to find the following limit provided: $q<p+1$: $$ \lim_{n\to\infty} \int_0^1 \dfrac{n x^p+x^q}{x^p+n x^q} dx$$ Dividing by $n x^q$ so we have $$\dfrac{n x^p+x^q}{x^p+n ...
2
votes
3answers
27 views

How many different (circular) garlands can be made using $3$ white flowers and $6m$ red flowers?

This is my first question here. I'm given $3$ white flowers and $6m$ red flowers, for some $m \in \mathbb{N}$. I want to make a circular garland using all of the flowers. Two garlands are considered ...
-2
votes
0answers
31 views

Hiring someone to do cointossing lead experiments [on hold]

I want to commission somebody to do some statistical simulations of cointossing experiments. I am specifically interested in the phenomenon of Long Leads, which was discussed in Feller's classic ...
1
vote
7answers
102 views

Given matrix P such that $P^{102 } =0 $ , to show that $P^{2} = 0$.

P is given to be a 2×2 matrix such that $P^{102} = 0$. How to show that $P^{2} =0 $?
1
vote
1answer
53 views

2014 Fall OMO #28

Here is a problem from this year’s OMO: Let $S$ be the set of all pairs $(a,b)$ of real numbers satisfying $1+a+a^2+a^3 = b^2(1+3a)$ and $1+2a+3a^2 = b^2 - \frac{5}{b}$. Find $A+B+C$, where $$ A = ...
-5
votes
0answers
38 views

How to solve this horizontal multiplication $234\cdot 345\cdot 542 =?$ [on hold]

Kindly solve the following problem step-by-step How to solve this horizontal multiplication $234\cdot 345\cdot 542 =?$ Yours sincerely, RAJI REDDY. K
1
vote
2answers
33 views

Meaning of a symbol

I've seen the symbol "$B_\epsilon(a)$", but I don't know what it means. The context is limits of a subsequence. Here, $\epsilon>0$ is a real number, and the limit of subsequence $a_{n_k}$ is $a$, ...
0
votes
0answers
23 views

A combinatorial game theory problem

In details, Let, there are four bishops on a chessboard where every two bishops are in pair ( as there are 4 bishops that means 2 pairs and in each pair they sit in vicinal squares). How many ...
0
votes
0answers
5 views

Iso-density locus of Gaussian mixture distribution

I would like to known what is the equation of the iso-density locus of a Gaussian mixture distribution. Is such an iso-density locus a union of ellispoids? Let's say that this Gaussian mixture is in ...
1
vote
0answers
8 views

How to calculate convolution of function defining a measure

Given the function $F(t)=2-2e^{-t}$ defining a measure on $(\mathbb{R}_+,\mathfrak{B}(\mathbb{R}_+))$ and I want to calculate the convolution of this function with itself. I tried to do that by using ...
0
votes
0answers
14 views

indicial equation of a differential equation

The indicial equation for $x(1+x^2)y'' + (cosx)y' + (x^2-3x+1)y=0$ is $r^2=0$. How it is possible. I reduced the given diff eqn as: $x^2y'' + \frac{xcosx}{1+x^2}y' + \frac{x(x^2-3x+1)}{1+x^2}y=0$. ...
0
votes
0answers
8 views

Probability measure of rank-$r$ matrices

I have a question about the distribution of matrices with a specific rank. Consider $\mathcal{M}^{n\times m}$ the set of all $n \times m$ matrices with entries in some field $\mathbb{K}$. If I define ...
-5
votes
1answer
44 views

A big challenge on Number theory [on hold]

Let $N=\frac{60^{2014}}{7}$. What is the sum of the first $2014$ digit before the decimal point of $N$?
3
votes
3answers
61 views

How to compute $\sum_{n=1}^{\infty}\arctan{\frac{3n^2}{2n^4-1}}$

I find this problem on facebook group. Is it possible to find exact value of $\sum_{n=1}^{\infty}\arctan{\frac{3n^2}{2n^4-1}}$. I think this is not telescope sum. And wolfram alpha can not find it. ...
1
vote
0answers
9 views

$\int_0^{\infty} A( f(B(x)) ) + C(x) ) dx = \int_0^{\infty} f(x) dx$

I was thinking about $\int_0^{\infty} A( f(B(x)) ) + C(x) ) dx = \int_0^{\infty} f(x) dx$ The inspiration came from the following 2 integrals : Lemma If $f(x)$ is a bounded non-negative ...
1
vote
2answers
24 views

Is there any automorphism $f$ that satisfies these requirements? [on hold]

Suppose that $\Bbb R $ is the set of real numbers. Is there any automorphism $f$ from $(\mathbb R,+)$ to $(\mathbb R,+)$ of the following form? $$f(x)=kx, \quad k \neq 0,1 , \quad k \in \Bbb R$$
0
votes
0answers
38 views

Prove that $|Z(G)| < \infty$ [on hold]

Let $G$ be an infinite group. Suppose that $G/G^{\prime}$ is a finite group, where $$G^{\prime}= \left\langle xyx^{-1}y^{-1}\ \middle\vert\ x,y \in G\right\rangle.$$ Note that $G^{\prime}$ is a normal ...
1
vote
2answers
37 views

Absolutely convergent but not convergent

Here, Lemma $2.1$ states that A normed space $X$ is complete if and only if every absolutely convergent series is convergent. I would like to know a series which is absolutely convergent but ...
0
votes
1answer
8 views

Tree Traversal-Is the order ascending?

I have a question about the traversal of a tree. When we print the values of a binary search tree using in order traversal are the values printed in an ascending order??
2
votes
2answers
14 views

How to prove this statement about this relation:

Let $p$ be a prime. On $\mathbb{Z}_{>0}$ we define the relation $\sim$ as $a\sim b\iff [\forall n\in \mathbb{Z}_{>0}: p^n|a \iff p^n|b]$. Prove that $[\forall x,y \in \mathbb{Z}_{>0}: x\sim ...

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