0
votes
1answer
10 views

Why isn't the identity $\sqrt{ab}$ = $\sqrt{a}$.$\sqrt{b}$ always true?

If we take $a=b=1$ the the L.H.S. is $1$ but the R.H.S. is $-1$. Is this identity not applicable for complex numbers? How to prove this and prove that this is not applicable for some complex ...
3
votes
1answer
13 views

Galois group of $X^{5}-2X+7$ over $\mathbb{Q}$

Is there any way to determine the Galois group of $X^{5}-2X+7$ over $\mathbb{Q}$ not using the discriminant? Thanks!
0
votes
0answers
6 views

Laurent series of $\psi(-2z)$ at positive integers and positive half integers.

The laurent series of $\psi(-z)$ at $z=n$ is $$\psi(-z)=\frac{1}{z-n}+H_n-\gamma-(H_n^{(2)}+\zeta(2))(z-n)+\cdots$$ However, may I ask how does one determine the laurent series expansion for ...
0
votes
4answers
29 views

A tricky complex numbers if and only if proof

For complex numbers $z$ and $w$ prove that $$|z|^2w -|w|^2z = z-w\quad \iff\quad z=w\quad\text{or}\quad z\bar{w}=1.$$ How would you go about solving this problem?
3
votes
1answer
22 views

Prove that $BA^{-1} B \not=-B$ if $A + B$ is invertible for $A$ invertible and $B$ non-zero matrix

Let $A$ and $B$ be $n×n$ real square matrices. Matrix $A$ is an invertible and $B$ is a non-zero matrix. a)Prove that $BA^{-1} B \not=-B$ if $A + B$ is invertible b) Let $B= uv^T$ for $u,v \in \Bbb ...
6
votes
4answers
42 views

process of showing that a number is irrational

How can I prove that the sum $\sqrt 2+\sqrt[3] 3$ is an irrational number ??
0
votes
0answers
12 views

Proving that the upper and lower Riemann sums converge to the integral

This is a question from Tom M. Apostol's Calculus, Volume 1 (Exercise 10.4): $f$ is monotonically increasing and bounded on $[0,1]$. Define the sequences $\{s_n\}$ and $\{t_n\}$ as follows: ...
0
votes
0answers
13 views

Is $\exp(-2\sin^2t)$ a characteristic function?

Is $\exp(-2\sin^2t)$ the characteristic function of some random variable?
0
votes
1answer
12 views

Finding if it is possible to protect all suburbs

Bob lives in a city that is circular in shape on a 2D plane. The city center is located at origin (0,0) and it has suburbs lying on the lattice points (points with integer coordinates). The city ...
0
votes
0answers
3 views

Finding Interpretations for First Order Logic

I'm currently looking at first order logic and I'm having a difficult time with the following question: Now I don't want answer cause that wont really help me. What I am looking for is help, with ...
1
vote
0answers
8 views

Integration against divergence free vector fields

Let $\chi:\Omega\to \mathbb{R}^n$ be a vector field on a bounded, smooth domain $\Omega \subset \mathbb{R}^n$. Assume that for any divergence free vector field $\eta:\Omega \to \mathbb{R}^n$ we have ...
0
votes
0answers
4 views

CDF of smallest eigenvalue of non-central Wishart matrix - how to evaluate the integral.

Does anybody know how to derive the distribution of the smallest root of a non-central Wishart matrix? I have got an integral expression that would give me the desired answer but cannot solve the ...
0
votes
0answers
8 views

Casio fx-83gt - help! Linear Programming

Hi guys I'm studying a module called Operational Research and in particular linear programming. I am doing the simplex method and as anyone who studies linear programming would know, you need to ...
0
votes
0answers
14 views

How to prove that all primitive polynomials are irreducible

Let $F$ be a finite field, and $F[X]$ set of all polynomials in $F$, how to prove that all primitive polynomials $\;$ $f \in F[X]$ $\;$ are irreducible.
0
votes
1answer
17 views

Finding the Product of $e_{i,j}$ and $e_{k,l}$

My teacher for calculus this year gave a handout on the first day with an excerpt from Rings, Fields, and Vector Spaces by B.A. Sethuraman. The reason for this is in the beginning of Sethuraman's book ...
0
votes
0answers
9 views

Lie algebra: If ad(g) is solvable then g solvable?

I'm trying to prove that if the image of the adjoint representation of a Lie algebra g is solvable then g is solvable, ie, if for some n (ad(g))^(n) = 0 then there exists a m such that g^(m) = 0 My ...
0
votes
0answers
9 views

Deriving equation in vector notation

I had some trouble deriving an equation from the book 'Elements of statistical Learning' p. 108 equation 4.9. This heavily relies on linear algebra, so I was wondering how the author came to his final ...
0
votes
1answer
14 views

The instantaneous value of current, i amp, at t seconds is given by: i = 15 sin(100π.t + 0.6) Find the value of;

Find the value of; amplitude period frequency initial phase angel value of i when t = 2.5s time when current first reaches maximum value Relevant equations $i=A*\sin(ωt+\phi)$ $i = ...
0
votes
0answers
8 views

How to find max and min bounds of a uncertain function

First I would like to say that I have searched the for uncertain fitting, robust fitting, linear optimization, convex optimization, etc. But I'm lacking the knowledge to solve this problem, and I need ...
0
votes
0answers
21 views

Integral Dependence relations for $x+y$ and $xy$

It is a question from Miles Reid's Undergraduate Commutative Algebra: Let $B$ be a $k$ -algebra and let $y,z \in B$ and both $y, z$ satisfies the following integral dependence relations over $k$ ...
0
votes
0answers
9 views

How to Prove an Algorithm is $O(n \log n)$ Using Substitution Method and a Substitution?

Consider the following: Show that $T(n) = 2T(\left \lfloor n/2 \right \rfloor + 17) + n$ is $O(n\log n)$. Here is what I have come up with: Restrict the domain of $n$ to $n \geq 1$ and define ...
0
votes
1answer
14 views

Re-arranging formula

I was asked to re-arrange the formula in terms of V for: NI = (SP-VC)(V)-FC These are the steps I have done and it was wrong. Can anyone explain (1) NI/(SP-VC)=V-FC (1) FC+[NI/(SP-VC)]
0
votes
2answers
21 views

Inverse of a Rotation matrix

If $R $ is a rotation matrix (determinant 1,orthonormal) can we say that $R^{-1}$ is also a rotation matrix? If yes how do we prove it?
0
votes
0answers
17 views

Why do we interpolate - no guarantee of success

this is somewhat of a general question about interpolation, I don't fully understand how can we be confident that our approximation is good, even if we know a lot of points. An example would be: ...
0
votes
2answers
28 views

Way to Show that a limit does not exist

how can I show that the limit of the function: $\displaystyle f(x)=2x\sin(1/x)-\cos(1/x)$ does not exist as $xto0$?
1
vote
2answers
12 views

Binomial Theorem on a Matrix

Does the expression follow binomial theorem? (A + I) ^n where A is matrix ,I is identity matrix. I know the binomial theorem but do not know whether it is applicable to matrices also.
3
votes
3answers
38 views

Is this statement meaningful if one of the elements is undefined?

Am I allowed to say a statement like $\max\left\lbrace a,b\right\rbrace$ if it turns out that the element $b$ is undefined, or simply does not exist? Would the result be $a$, or is the whole ...
1
vote
0answers
31 views

Alternative proof: Matrix $A$ is similar to $B$ iff $\lambda I - A$ is equivalent to $\lambda I - B$

We have this theorem for square matrices: If $\lambda I - A$ is equivalent to $\lambda I - B$, then $A$ is similar to $B$. ($A$, $B$ are matrices in $K^{n\times n}$, $K$ is a number field, ...
0
votes
0answers
16 views

Uniformly continuous function defined on open interval has limits defined at the endpoints.

How do you prove the statement below? Let $f:(a,b)\to\mathbb R$ be a function that is continuous on the bounded, open interval $(a,b)$. Then the two limits $$\lim_{x\to ...
0
votes
0answers
17 views

Can you graph equations with a negative discriminant? And how do you plot complex numbers both on a 2D complex plane and a 4D complex plane?

I don't understand the relationship between complex numbers and that way they are graphed. The equation I am working with is $2x^{2} - 6x + 5 = 0$ where my two roots are complex solutions: $x = ...
-1
votes
1answer
53 views

Can a finite value for $\int_1^\infty \exp(x^2)\,dx$ be defined?

Why should $$\int_1^{\infty}\exp(ix^2)dx,\int_1^{\infty}\exp(-ix^2)dx,\int_1^{\infty}\exp(-x^2)dx$$ converges but not: $$\int_1^{\infty}\exp(x^2)dx$$ Is there any way that assigns a value to ...
0
votes
0answers
2 views

How can I expand a zonal polynomial in a sum of two matrices? C(A+B) = C(A)*C(B)

I am trying to solve an integral involving zonal polynomials. TO do that I need to somehow separate out the zonal polynomial C(A+B) where my A varies but B is constant. They're all square matrices ...
0
votes
0answers
7 views

Calculating $\text{D}g$ of $g(x,y) = \int_\frac1x^1\frac1t\exp(t^3x^2y)\text{d}t$

Let $g:(1,\infty)^2\to\mathbb{R}$ be given by $$g(x,y) = \int_\frac1x^1\frac1t\exp(t^3x^2y)\text{d}t.$$ How can I calculate $\text{D}g$ using parameter-dependent integrals?
0
votes
2answers
31 views

The derivative of $z=x^2+xy+ y^2$

I have got confused about this problem, what I have thought was differentiating this with respect to $x$ gives - $\frac{dz}{dx} = 2x + x \frac{dy}{dx} + y + 2y \frac{dy}{dx}$ But, I came across an ...
1
vote
2answers
16 views

Deriving exponential distribution from geometric

Let $\lambda$ be the expected number of events in a unit time interval $[s,s+1]$ (events are independent of each other and of the time interval), and $T$ a continuous random variable that represents ...
0
votes
2answers
48 views

Questions on the empty set $\varnothing$

I came across through the following questions, i was able to answer some but confused in the others: $1. \varnothing \subseteq\varnothing$ $2. \varnothing \subseteq \{\varnothing\}$ $3. \varnothing ...
0
votes
1answer
30 views

Finding a formula for a pattern

I have this pattern which is an infinite sequence (I have placed commas so it's easy to see the pattern)... $1 ,1 2, 1 2 3, 1 2 3 4, 1 2 3 4 5 ...$ Is there any formula governing this sequence, ie, ...
1
vote
0answers
22 views

A second order differential equation

How does one solve the following differential equation $y^{"}+xy^{'}+(1-x^2)y=y\sin x$? I don't know how to proceed?
0
votes
1answer
15 views

Covariance of Ornstein - Uhlenbeck Process

I'm considering the Ornstein - Uhlenbeck process $ X(t)=x_{\infty}+e^{-at}(x_{0}-x_{\infty})+b \int_{0}^{t} e^{-a(t-s)} dW(s)$ where $a, b > 0 $ are given constants. I used the Itô Isometry to ...
1
vote
0answers
13 views

A question on Lagrange multipliers

The state of Megalomania occupies the region $x^4 + y^4 \leq 30,000.$ The altitude at the point $(x,y)$ is $\frac{1}{8}xy+200x$ meters above sea level. Where are the highest and lowest points in the ...
0
votes
4answers
26 views

Checking subspace

Let $B$ be a fixed matrix in $\mathbb{R}^{n\times n} $ and $W=\{{A \in \mathbb{R}^{n\times n} :AB=BA}\}$ Then is $W$ a subspace of $\mathbb{R}^{n\times n}$ ? I have tried this so far: a) The zero ...
0
votes
0answers
13 views

how to solve strum liouville problem of second order

$(1+x^2)y^{"}+2xy^{'}+\lambda x^2 y=0$ with $y'(1)=0$ and $y'(10)=0$. How do we solve this type of Sturm-Liouville problem?
1
vote
2answers
45 views

Does there exists a homomorphism for any groups $G$ and $H$

This is a question from Exercise 8.2 of Visual Group Theory which says:determine whether true or false. For any group $H$ and $G$,there is some homomorphism from $H$ to $G$. For any groups $H$ and ...
0
votes
1answer
25 views

Find total number of painted cubes [on hold]

Suppose that a wooden cube, whose edge is 5 inch, is painted red, then cut into 125 pieces of 1 inch edge. Find total number of painted cubes ?
0
votes
1answer
15 views

Two similar first order differential equations

Solve$$tx'^2-2xx'-t=0$$ and $$t^2-2txx'-x^2=0$$ These would be easy Riccati's equations if their middle terms werent multiplied by 2. In this case I have no idea how to solve that.
0
votes
3answers
31 views

Proving that $\sum \frac{n^{n+1/n}}{(n+1/n)^n}$ diverges

Show that the series $$\sum \frac{n^{n+1/n}}{(n+1/n)^n}$$ diverges The ratio test is inconclusive and this limit is not easy to calculate. So I've tried the comparison test without success.
0
votes
0answers
5 views

Frobenius norm and Gaussian noise

Why Frobenius norm is considered to a good tool for dealing with Gaussian noise?
0
votes
0answers
13 views

Maclaurin series for $\frac{1}{|1+x|}$

I believe that there is no Maclaurin Series for $\frac{1}{|1+x|}$ as the latter is not differentiable at $x=-1$. However, would it be appropriate for me to refer $\frac{1}{|1+x|}$ as 'not a smooth' ...
2
votes
1answer
35 views

Evaluating $\lim\limits_{(x,y)\rightarrow(1,1)} \frac {\sin(x) - \sin (y)} {x-y}$

I am taking a calculus exam in less than one week, and I've stumbled upon this expression. $$\lim\limits_{(x,y)\rightarrow(1,1)} \frac {\sin(x) - \sin (y)} {x-y}$$ Which I know to be cos(1), but ...
0
votes
1answer
6 views

Writing nonautonomous systems as autonomous systems

Apparently any mth order nonautonomous system is equivalent to a first order autonomous system in higher dimensional space. How does this work in practice? I would you write $\displaystyle ...

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