0
votes
1answer
386 views

Dimension of generalized eigenvector space

From Wikipedia For a matrix A, there may not always exist a full set of linearly independent eigenvectors that form a complete basis – a matrix may not be diagonalizable. This happens when the ...
1
vote
4answers
313 views

Transform $\; (\lnot q \lor r) \land ( q \lor \lnot r)\;$ into a disjunctive normal form

I was working on an examples from my textbook concerning transforming formulae into disjunctive-normal form (DNF) until I found an expression that I cannot solve. I hope somebody can help me transform ...
2
votes
2answers
270 views

Proving non satisfiability of the barbers paradox with tableau method

The barbers paradox: In a town there is only one barber. For every man in town, either the barber shaves him or he shaves him self. I need to formalize this: The barber shaves exactely those who ...
3
votes
4answers
7k views

Derive the expected value for a Pareto distribution?

X is a random value that is Pareto distributed with parameter $a>0$, if $\Pr(X>x)=x^{-a}$ for all $x≥1$. Show that $EX=a/(a-1)$ if $a>1$ and $E(X)=∞$ if $0< a \le1$. I can derive the ...
2
votes
3answers
204 views

Superball total bounce distance

I am asked to explain how to calculate total bounce distance: A "super" rubber ball is one that is measured to bounce when dropped 70% or higher that the distance from which it is dropped. You are ...
1
vote
1answer
165 views

If $N$ is normal, show that $\begin{Vmatrix} Nx \end{Vmatrix}$=$\begin{Vmatrix} N^{H}x \end{Vmatrix}$ for every vector $x$

If $ N $ is normal, show that $\begin{Vmatrix} Nx \end{Vmatrix}$=$\begin{Vmatrix} N^{H}x \end{Vmatrix}$ for every vector x. Deduce that the ith row of N has the same legth as the ith column. In here, ...
4
votes
1answer
92 views

Showing that $ (1-\cos x)\left |\sum_{k=1}^n \sin(kx) \right|\left|\sum_{k=1}^n \cos(kx) \right|\leq 2$

I'm trying to show that: $$ (1-\cos x)\left |\sum_{k=1}^n \sin(kx) \right|\left|\sum_{k=1}^n \cos(kx) \right|\leq 2$$ It is equivalent to show that: $$ (1-\cos x) \left (\frac{\sin ...
1
vote
1answer
189 views

Asymptotic stability of fixed point

$f'(t)=af(t)(K-f(t))-bf(t)g(t)$ for $a,b,c,d,t,K>0$ $$g'(t)=cf(t)g(t)-dg(t)$$ This system has 3 fixed points (You can evaluate them if you set the 2 equations = 0). One point is ...
1
vote
2answers
226 views

Second order ODE with product of two functions on RHS

Find the general solution of the ODE $y′′ +16y=64x\cos4x.$ If $y(0)=1, y′ (0)=0,$ what is the particular solution? Attempt: I am just needing some help with the particular integral. I have tried ...
1
vote
1answer
119 views

Polynomial rings over rational numbers

We know that polynomial rings over $\mathbb{Q}$ is a vector space over $\mathbb{Q}$. It has a well-known basis $1, x, x^2,\ldots$ but can we classify all bases?
2
votes
2answers
588 views

Obtaining cumulants using the characteristic function

If a random variable $x$ has a characteristic function $\phi(\omega)$, then the $n^{\mathrm{th}}$ moment of the distribution of $x$, $\mu_n$ can be calculated as: $$\mu_n = ...
3
votes
2answers
488 views

Find a diagonal matrix M

Find a diagonal M, made up of 1's and -1's, to show that $A=\begin{pmatrix} 2 & 1 & & \\ 1 & 2 & 1 & \\ & 1 & 2 & 1 \\ & & 1 & 2 \end{pmatrix}$ is ...
2
votes
2answers
186 views

isomorphism $\Bbb Q$ to $\Bbb Q \cap (0,1)$

I've got a question in my homework: Prove that $\langle \mathbb{Q},< \rangle$ and $\langle \mathbb{Q}\cap (0,1),< \rangle$ are isomorphic. I have tried to find a bijective function without any ...
2
votes
1answer
872 views

Visualization of Singular Value decomposition of a Symmetric Matrix

The Singular Value Decomposition of a matrix A satisfies $\mathbf A = \mathbf U \mathbf \Sigma \mathbf V^\top$ The visualization of it would look like But when $\mathbf A$ is symmetric we can do: ...
1
vote
2answers
202 views

Testing a hypothesis with significance level

Prove the hypothesis that the average content of containers of a particular lubricant is $10$ liters, if the contents of a random sample of $10$ containers are: \begin{array}{|c|c|c|c|c|} \hline ...
-1
votes
2answers
163 views

Irrationals can be separable by finding a countable dense subset. [duplicate]

Possible Duplicate: Is the set of irrationals separable as a subspace of the real line? Prove the irrationals are separable directly by finding a countable dense subset.
4
votes
1answer
170 views

Centre of a quadric

I found the following sentence in my linear linear algebra book (affine and projective geometry): $Q:V \to \mathbb{K}$ is a quadric (quadratic function) and $\alpha\in Aff(V)$. $Aff(V)$ is the set of ...
5
votes
2answers
280 views

Using combinatorial argument prove that $\frac{(3n)!}{2^n\times 3^n}$ is an integer.

Using combinatorial argument prove that $\frac{(3n)!}{2^n\times 3^n}$ is an integer. If we arrange $3n$ objects where there are 3 objects of one kind, another 3 objects of second kind $\cdots$ ...
7
votes
1answer
291 views

Showing that $ \sum_{n=1}^{\infty} \arctan \left( \frac{2}{n^2} \right) =\frac{3\pi}{4}$

I would like to show that: $$ \sum_{n=1}^{\infty} \arctan \left( \frac{2}{n^2} \right) =\frac{3\pi}{4}$$ We have: $$ \sum_{n=1}^N \arctan \left( \frac{2}{n^2} \right) =\sum_{n=1}^N \arctan ...
4
votes
3answers
638 views

Integrating using Laplace Transforms

$$\int_{0}^\infty {\cos(xt)\over 1+t^2}dt $$ I'm supposed to solve this using Laplace Transformations. I've been trying this since this morning but I haven't figured it out. Any pointers to push me ...
1
vote
1answer
339 views

Geometrico-Harmonic Progression

Like Arithmetico-geometric series, is there anyway to calculate in closed form of Geomtrico-harmonic series like $$\sum_{1\le r\le n}\frac{y^r}r$$ where $n$ is finite. We know if $n\to \infty,$ the ...
1
vote
0answers
1k views

Hellinger distance between Gaussians - multivariate and univariate forms

On pages 46 and 51 of the book Statistical Inference based on divergence measures By Leandro Pardo Llorente there is a derivation for the Hellinger distance between two multivariate Gaussian ...
9
votes
4answers
555 views

Coproducts in $\text{Ab}$

I am currently trying to understand why finite products and coproducts in the category $\text{Ab}$ coincide. In fact, I'm not even sure I can show it. My question is the following: Is there an ...
4
votes
3answers
161 views

How to prove that $Ax = e^x$ has two solutions when $e < A < \infty$

How to prove that $Ax = e^x$ has two solutions when $e < A < \infty$? This is easy to visualise graphically, but how can it be shown with algebra?
3
votes
2answers
260 views

The smallest ring containing square root of 2 and rational numbers

Can anyone prove why the smallest ring containing $\sqrt{2}$ and rational numbers is comprised of all the numbers of the form $a+b\sqrt{2}$ (with $a,b$ rational)?
0
votes
1answer
67 views

Convex function plus $v e^{-x}$

If $f(x)$ is strictly convex, and $$\lim_{x\to \infty}\left(f(x) - x - ue^{x}\right) = w$$ for some $u\ge 0$ and $w$ then what can be said about: $$g(x) = ve^{-x} + f(x)$$ on $x\ge0$ where $v$ is ...
3
votes
1answer
2k views

Amount of transitive relations on a finite set

In counting the amount of relations on finite sets, we can quite easily count the amount of reflexive and symmetric relations on a finite set. We just consider (in accordance with the definition of a ...
1
vote
0answers
93 views

Defining things in a non-recursive manner

In this question I asked, if it was possible to define certain functions without the use of the recursion theorem. The answer, among other things, indicated that it theoretically would be possible, ...
184
votes
14answers
7k views

Can every proof by contradiction also be shown without contradiction?

Are there some proofs that can only be shown by contradiction or can everything that can be shown by contradiction also be shown without contradiction? What are the advantage/disadvantages of proving ...
3
votes
2answers
237 views

Trigonometric equation

I have been trying to solve this equation for over a week now: $$\tan5x-2\tan3x=\tan3x\tan5x$$ I found one solution $x=k\pi$ but I cannot prove that this is the only solution. It is equivalent to: ...
2
votes
3answers
213 views

Prove that if $ |a_n-a_{n-1}| < \frac{1}{2^{n+1} }$ and $a_0=\frac12$, then $\{a_n\}$ converges to $0<a<1$

I try to solve this question but I don't know how. given $ a_0 = \frac12 $ and for each $n\geq 1$: $$ |a_n-a_{n-1}| < \frac{1}{2^{n+1}} $$ show that $\{a_n\}$ converges and the limit is $a$ such ...
1
vote
1answer
201 views

Restriction Of Parametric Functions Domain

The problem I am working on is, "Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the ...
7
votes
3answers
301 views

$f$ is an entire function with Im $f\geq 0$

$f$ is an entire function with $\operatorname{Im}f \geq 0$. Then which of the followings are true: 1) $f$ is constant. 2) $\operatorname{Re}f$ is constant. 3) $f = 0$. 4) $f'$ is a non-zero constant. ...
1
vote
1answer
54 views

Probability of a problem with a bride and a groom

A group of 7 people including a bride and a groom is to be photographed. In how many ways can you arrange them in 2 rows where one row is a level higher than the other such that the bride and the ...
1
vote
1answer
187 views

Complex injective function

I'm trying to see if the function: $$z \mapsto z^n+\exp(ia) \cdot nz$$ is an injective function at the open unit circle. Please help.
2
votes
1answer
322 views

Extension of scalars

Let $A\rightarrow B$ be a ring homomorphism, $M$ and $N$ - modules over $A$. How to prove, that $$ (M \otimes_A N) \otimes _A B = (M \otimes_A B) \otimes_B (N\otimes_A B) $$ as $B$-modules in the ...
0
votes
1answer
134 views

Compactness is closed-hereditary - significance of closed property?

Here's a proof that any closed subset of a compact metric space is compact - Let $(X, d)$ be a compact metric space and let $F$ be a closed subset of $X$. Let $U = \{U_i : i \in I \}$ be an open ...
1
vote
2answers
215 views

metrics, open sets

i am studying for my exam and trying to solve some questions. I have got a problem about proving the following. Let $X$ be a set, and let $d_1$ and $d_2$ be two metrics on $X$. Suppose that $d_1$ and ...
1
vote
1answer
151 views

Linear programming - task formulation

I have a question concerning the formulation of a linear programmign task. I am trying fo find $x^* \in argmax_{x \in R^n}\{ a_1x_1 + a_2x_2, a_2x_2 + a_3x_3 + a_4x_a, a_4x_4 + a_5x_5 \}$, subject to ...
3
votes
1answer
431 views

Counterexample of Sobolev Embedding Theorem?

Is there a counterexample of Sobolev Embedding Theorem? More precisely, please help me construct a sobolev function $u\in W^{1,p}(R^n),\,p\in[1,n)$ such that $u\notin L^q(R^n)$, where ...
2
votes
1answer
263 views

Pointwise sup of step functions is lower semicontinuous (a.e.)

I've found this problem while I was reading a paragraph about Riemann integration on some notes a mate gave me a long time ago. Let $f \colon [a,b] \to \mathbb R$ be a bounded function. Suppose ...
1
vote
2answers
137 views

How can i prove that the Hyperbolic space is complete by using Divergent Curves?

Let $$H_+^2=\{(x,y)\in\mathbb{R}^2:\ y>0\}$$ and consider the Lobatchevski metric on $H_+^2$: $$g_{11}=g_{22}=\frac{1}{y^2},\ g_{12}=0$$ How can one prove the completeness property of this space ...
2
votes
2answers
369 views

Density in sobolev spaces

Is $H^{s+1} (\Bbb R^n)$ dense in $H^s(\Bbb R^n)$ for $s = 0,1,2, \cdots$ ? ($H^s$ : general sobolev space)
15
votes
1answer
378 views

Is there any holomorphic version of the tubular neighborhood theorem?

This question arised when I was studying Beauville's book 'Complex Algebraic Surfaces'. Castelnuovo's theorem says that a smooth rational curve $E$ on an algebraic surface $S$ is an exceptional ...
0
votes
2answers
112 views

Calculating variance, how to determine when to use 1/n or 1/(n-1)?

I'm learning multivariate analysis. I am asked to calculate covariance of $$X=\begin{pmatrix} 3&7 \\ 2&4 \\ 4&7 \end{pmatrix}$$ According to P8 of Applied Multivariate Statistical ...
1
vote
0answers
26 views

group law on weil-chatelet group

Is there a reference for the gemetric definition of the group law on the Weil-Châtelet group of an Abelian variety more recent than the original Weil's paper ("On algebraic groups and homogeneous ...
7
votes
2answers
2k views

What are the advantages of dual of a problem

I am studying linear programming and I came across primal-dual algorithm in Linear Programming. I understood it but I am unable ...
1
vote
1answer
721 views

probability of choosing the same color twice in 5 tries

I'm sure this question has been asked before but I couldn't find an answer. There are 8 balls: 3 red, 2 blue and 3 black. Whats the probability of choosing at least 2 red balls if we pick 5 balls? I ...
0
votes
2answers
251 views

Discontinuity of differential functions

There is a corollary in Rudin analysis. But I am not able to understand it. Can someone help to understand it? The Corollary is: Let $f$ be a real differential function on $[a,b]$, then $f'$ ...
1
vote
3answers
98 views

misinterpreting logic expression

Represent in first order logic "Every person who dislikes all vegetarians is smart" My solution $\forall x,y$ $person(x) \land vegetarian(y) \land dislike(x,y) \rightarrow smart(x) $ People say to ...

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