0
votes
2answers
68 views

Bernoulli's inequality by induction

I'm proving Bernoulli's inequality by induction but I noticed something strange. See wikipedia proof: http://en.wikipedia.org/wiki/Bernoulli's_inequality Notice how they multiply both sides of the ...
4
votes
1answer
43 views

Integer solutions to an ellipsoid surface

Given the equation $$x^2+2y^2+5z^2+xz =n$$ where $n$ is any positive integer, what is the smallest odd integer for which no integer solution $(x,y,z)$ exists (i.e. $x,y,z$ are integers)? I know that ...
1
vote
1answer
107 views

Maclaurin Series Complex Numbers

I'm having trouble getting to the right solution on the function ${z^2\over (1+z)^2}$ ${z^2\over (1+z)^2}$ = ${z^2}$${1\over (1+z)^2}$ = ${z^2}$${1\over (1+z)(1+z)}$ = ${z^2}$${A \over (1+z)}$ + ...
1
vote
1answer
264 views

Parametric equation to find distance from origin to plane

The equation $2x_1 - 3x_2 -6x_3 = -4$ defines a plane in $\mathbb{R}^3$ I know the normal vector $\bf a$ for this is $(2,-3,-6)$ I am trying to use the parametric equation of the line that pass ...
2
votes
1answer
282 views

Proving that diophantine equation has no solutions

I am trying to show that the equation $x^5y + 5x^3 - xy^5 = 1$ has no solutions. Anyone has an idea on this?
0
votes
3answers
123 views

Show that $G\{m\}$ is finite for all non-zero $m\in \mathbb{Z}$

Let $G$ be an abelian group, and let $m$ be an integer, then we define $G\{m\} := \{a\in G:ma=0_G\}$. Now, suppose that $G$ is an abelian group that satisfies the following properties: (i) For all ...
1
vote
1answer
95 views

Prove Gaussian Elimination Preserves this Matrix Property

Prove or disprove that if a matrix has the property $0 \neq |a_{ii}| \leq \sum_{\substack{j=1 \\j \neq i }}|a_{ij}|$ Then Gaussian elimination without pivoting will preserve this property I have ...
1
vote
4answers
59 views

$\dfrac{dy}{dx}=\dfrac{y^2-1}{x^2-1}$, with the initial condition $y(2)=2$.

Solve the following differential equation: $\dfrac{dy}{dx}=\dfrac{y^2-1}{x^2-1}$, with the initial condition $y(2)=2$. My attempt: I notice that this is a separable differential equation, so I ...
1
vote
1answer
33 views

Bounding a Bilinear Map $||A(v,w)||\leq M||v||||w||$

In a normed vector space I know that for a linear map $L:E\rightarrow F$ that there exists an $M\in \mathbb{R}$ such that $\forall x\in E$ $||L(x)||\leq M||x||$. The proof is this is quite ...
1
vote
1answer
151 views

Prove Newton's method converges…

How would you prove that Newton's Method applied to $f(x) = ax + b$ converges in one step? Would it be because the derivative of $f(x)$ is simply $a$?
2
votes
0answers
53 views

Fast solution to problems involving Lagrange multipliers

Suppose we have a function $f:\mathbb{R^n} \rightarrow \mathbb{R}$ subject to the constraint $g(x_1,...,x_m)=0$ for some natural $m$. We can find the local maxima and minima of $f$ on $g$ by setting: ...
1
vote
2answers
104 views

Game theory reference for a beginner

I need to use game theory to model interaction in a network. What are some books or lectures that a beginner in game theory could use to understand the theory well?
0
votes
1answer
123 views

What are the simple Heesch-2 polyforms?

At the Tiling Database: There are 3, 20, 198, 1390 non-tiling polyominoes of order 7 to 10. There are 4, 37, 381, 2717 non-tiling polyhexes of order 6 to 9. There are 1, 0, 20, 103, 594, 1192, 6290 ...
2
votes
1answer
56 views

Converses to a commonplace proposition about binomial distributions

I suspect I'll post my own answer to this question shortly, but it may be of interest to see what answers others post. A theorem found in Feller's famous book and elsewhere says that if $X,Y$ are ...
1
vote
1answer
64 views

Any useful tips and tricks when trying to determine if one expression is greater than or equal to another expression?

I tend to get stuck on questions that ask to determine if a given expression is greater than another expression. For example: Let: $MSE_\alpha = \frac{1}{5}\pi(1-\pi)$ $MSE_\beta = ...
-1
votes
1answer
107 views

$a^3+b^3+c^3 + 21abc \geq 3$ for $(a+b)(a+c)(b + c) = 1$ and $a,b,c>0$

$a, b, c \gt 0$ and $(a+b)(a+c)(b + c) = 1$ Prove that $a^3+b^3+c^3 + 21abc \geq 3$ In this problem I spotted one trick $(a+b)(a+c)(b + c) = 1 \Leftrightarrow \\(a \sqrt{b+c})^2+(b\sqrt{a+c})^2+(c ...
2
votes
0answers
101 views

Pointwise convergence on a complete metric space

Let $ f_n :X \rightarrow R $ be sequence of continuous function on complete metric space $(X,d)$, which convergence pointwisely to $ f: X\rightarrow R $ mean that $ f_n(x)\rightarrow f(x)$ $ ...
1
vote
1answer
44 views

Define a branch of $\frac{z^\alpha}{z^2+1}$

Define a branch of $\frac{z^\alpha}{z^2+1}$. $\alpha$ is considered real and in the interval $(-1,1)$ Sketch the branch cut and the poles in the complex plane. I have that the poles are $z=i, ...
1
vote
0answers
184 views

Choosing the vector that minimizes this sum related to the rearrangement inequality

The rearrangement inequality states that, for two sets of real numbers $x_1\leq\dots{}\leq x_n$ and $y_1\leq\dots{}\leq y_n$, the sum $\sum_{i=1}^n x_{\sigma(i)}y_i$ is minimized for the particular ...
1
vote
2answers
38 views

Is the series absolut convergent?

I would like to solve the two following expressions for absolut convergent series: 1.$$\sum_{n=0}^{\infty}\frac{sin\cdot n}{n(n+1)}$$ 2.$$\sum_{n=0}^{\infty}{n}\sqrt{n}$$ For the 1., I would say ...
5
votes
1answer
399 views

Linear Programming with One Quadratic Equality Constraint

I have a problem which can be formulated as a Linear Programming with One Quadratic Equality Constraint: where variable x is n-dimensional vector and H is a Semi-Positive Definite n-by-n matrix. I ...
0
votes
2answers
48 views

3-space viewer?

Is there a software package that would allow visulaizing/rendering some example structures in 3-space? Specifically, I'm thinking of something that would provide a 3-D rendering of, say, 3-vectors ...
1
vote
2answers
73 views

What is $\operatorname{arccot}(-1)$?

According to me $\operatorname{arccot}(-1)$ should be equal to $3\pi/4$ because $\operatorname{arccot}(-x) = \pi - \operatorname{arccot}(x)$. But in my book it is given to be $-\pi$/4... also on ...
2
votes
0answers
107 views

clique number of generalized Johnson graph $J(4n-1,2n-1,n-1)$

The generalized Johnson graph $J(v,k,r)$ is defined to be the graph whose vertex set is the set of all $k$-element subsets of $\{1,2,\ldots,v\}$, and with two vertices adjacent iff their intersection ...
1
vote
1answer
206 views

Is this topology metrizable?

Is the irrational sequence topology metrizable? The irrational sequence topology is generated by the clopen basis $$ \big\{\{x\}\,:\,x\,\text{is irrational}\big\}\cup\big\{A_n(x) \cup ...
0
votes
2answers
195 views

Solving a limit using MacLaurin series

I want to find $$ \lim_{x\to0} \frac{(e^{-x^2}-1)\sin x }{x\ln(1+x^2)}$$ using a Maclaurin series and not using the l'Hôpital's rule. However I can't seem to get it right. Thanks for any possible ...
0
votes
1answer
86 views

Problem in harmonic analysis

suppose $p$ be a fixed psitive real number and $f$ is an entire function with $$\lvert f(0) \rvert^p=\int_\mathbb{C}\ \lvert f(z)\exp(-\alpha\lvert z \rvert ^2) \rvert^p dA(z) $$ where $\alpha ...
1
vote
1answer
38 views

Relating to $( a ^ b ) + c = d$ and $( a ^ b ) - c = d$

Is there a way of deducing the smallest integer values for $a, b$ and $c$ that satisfy either $( a ^ b ) + c = d$ or $( a ^ b ) - c = d$ such that the addition $( a + b + c )$ is the smallest ...
1
vote
2answers
91 views

How to solve $(1+i\sqrt{3})^{-1+i}$??

Good morning, I want to solve this... but I lose my way. I hope somebody help me... I show you my calculus $(1+i\sqrt{3})^{-1+i}=e^{(-1+i)\log(-1+i)}$ $(1+i\sqrt{3})^{-1+i}=e^{(-1+i)(\log ...
1
vote
3answers
154 views

Calculate Laurent series for $1/ \sin(z)$

How can calculate Laurent series for $$f(z)=1/ \sin(z) $$ ?? I searched for it and found only the final result, is there a simple way to explain it ?
2
votes
3answers
93 views

The limit of sequence tends to $0$

I am trying to show that if $0<x<1$, $$ \lim_{n\to \infty} {n^2 x^n (1-x)}=0 $$ I can't think of a clever way to show it.
0
votes
1answer
198 views

Formula for compound interest with N withdrawals

I have a calculator that allows users to see how much they need to save per period (month, year, etc) when putting money into a savings account. There are N withdrawals made in the end, with N ranging ...
0
votes
2answers
114 views

Find a non diagonalizable matrix that commutes with a given matrix

We are asked to find a non diagonalizable matrix that commutes with $\begin{pmatrix} 0 & 0 & -1 \\1 & 1 & 1 \\0 & 0 & 1\end{pmatrix}$. What I tried: My first thought was the ...
0
votes
2answers
63 views

On finding the zeros of a polynomial

What is the zero (real) of the polynomial $$x^{k+1}-2x^{k}+1=0$$ If there is such, how can I find it or what method can I use?
1
vote
2answers
59 views

Compactness of the solution operator

Let $\Omega$ be a smooth open bounded subset of $\mathbb{R}^n$. The bilinear form $$a(u,v)=\int_{\Omega}\frac{\partial u}{\partial x}\frac{\partial v}{\partial x}dx$$ is elliptic on ...
3
votes
2answers
566 views

Induction: show that $\sum\limits_{k=1}^n \frac{1}{\sqrt{k}} < 2 \sqrt{n}$ for all n $\in Z_+$

The question: show by using induction that $\sum\limits_{k=1}^n \frac{1}{\sqrt{k}} < 2 \sqrt{n}$ for all n $\in Z_+$ My attempt at a solution: The base case $n = 1$ is true. First we use the ...
0
votes
1answer
20 views

Proving formally

$((\exists x : X.P) \Rightarrow (\forall x: X.Q)) \vdash (\forall x: X. PvQ) \Rightarrow((\forall x: X.P) \vee(\forall x: X. Q)$ exist stands for the existential quantifier all stands for for-all ...
3
votes
2answers
157 views

What will I be doing? [closed]

I'm a freshman studying discrete mathematics B.sc. It's easy to google ''typical mathematician jobs'' and get an idea of what mathematicians do for a living, but what sort of jobs can I expect to work ...
3
votes
1answer
100 views

What is the expectation of the product of two random variables of Dirichlet distribution?

I understand that the expectation of a random variable $X_i$ a Dirichlet distribution is $E[X_i] = \frac{\alpha_i}{\sum_k \alpha_k}$ and $E[\ln(X_i)] = \psi(\alpha_i) - \psi(\sum_k \alpha_k)$ I read ...
1
vote
1answer
60 views

ruin of the gambler with probability to die

Consider a random walk on $\mathbb{Z}$ starting from $i >0$. With probability $p$ it moves to the nearest neighbor on the left, with the same probability it moves to the nearest neighbor on the ...
5
votes
2answers
103 views

Ways to calculate $\int_0^1 \frac{-\log x}{1+x}\ \mathrm dx$

I came across the integral $$ \int_0^1 \frac{-\log x}{1+x}\ \mathrm dx = \frac{\pi^2}{12}, $$ which can be calculated as $\frac 1 2 \zeta(2)$ using analytic number theory. I'm interested if this ...
0
votes
1answer
46 views

Modular exponentiation with Straightforward method

I would like to understand who it works the modulo with Straightforward method. For example I try to test this: 290 mod 1009 = 257^x mod 1009. Which is the "x"? ...
0
votes
2answers
42 views

Computing eigenvalues of a matrix

I am given the following matrix $$\left( \begin{matrix} 0 &1&0\\0&0&1\\1&-1&0 \end{matrix} \right).$$ Now I computed the characteristic polynomial of $A$ to be $p_A(\lambda) ...
1
vote
0answers
45 views

Approximating a probability distribution given multiple datasets

I am beginner in mathematics/statistics and apologise in advance for my faulty use of language. I am working on a problem in statistical genetics. Given a dataset, $D$, I obtained probabilities for ...
1
vote
1answer
29 views

Financials Maths- Credited Interest

You invest £1,000 in an account for 5 years at 9% pa nominal. How much will you get at the end of the 5 years if the interest is is credited: a) annually; b) 6 monthly; c) 3 monthly; ...
5
votes
2answers
170 views

Extending weak solution to global weak solution of parabolic PDE

Fix $T > 0.$ Let $V \subset H \subset V^*$ be a Gelfand triple. Consider the linear parabolic PDE $$u_t - Au = f\quad\text{in $L^2(0,T;V^*)$}$$ $$u(0) = u_0$$ where $u_0 \in H$ and $f \in ...
0
votes
1answer
56 views

Trouble with fixed point iteration.

Find the fixed point(s) of $g(x) = (1/2)x^2 + (1/2)x$. Does the fixed point iteration(s) converge(s) to the the fixed point(s) if you start with a close enough approximation? Then choose $x_0 ...
0
votes
3answers
46 views

Question about the characteristic subgroup.

Let G be a group. A subgroup H is called characteristic if $f(H)\subset H$ for all auto morphism $f$ of G.Pick out true statements. (a) Every characteristic subgroup is normal. (b) Every normal ...
1
vote
1answer
399 views

Eliminating Epsilon Production for Left Recursion Elimination

Im following the algorithm for left recursion elimination from a grammar.It says remove the epsilon production if there is any I have the following grammer ...
0
votes
3answers
89 views

Evaluate the integration?

Find: $$\int_{-\infty}^\infty\int_{-\infty}^\infty e^{-(3x^2+2\sqrt{2}xy+3y^2)}\,dxdy$$ I have no idea how to solve this,I would be thankful, if someone help me to solve this Thank you.

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