2
votes
2answers
58 views

Dual Vector Space embedding

Is there an embedding of any vector space $V$ into $V^*$? As far as I know it is not true. The statement that I know of is that there is natural embedding of $V$ into $V^{**}$ Is there any ...
1
vote
1answer
11 views

Damped systems - deriving equation

I am having some troubles deriving the formula for the roots for different types systems.. I am not quite sure if they are correct (pretty sure they aren't). $y(s) = \frac{s+2\zeta\omega_n}{s^2 + ...
2
votes
1answer
8 views

Using primes to create unique character mappings for scrambled substring searching

Problem: given a string needle, and a string haystack determine if there is there an anagram of needle present as a substring of haystack? (Assume case doesn't matter). One solution is to map the ...
0
votes
1answer
10 views

Range of convergence for Taylor's series for e^(sin x)

Is there anything wrong with my method below? Also, is there an easier method? For $sin\,x = \sum^{\infty}_{k=0}\frac{(-1)^kx^{1+2k}}{(1+2k)!}$, $L_1 = \lim_{k\rightarrow\infty} \left| ...
1
vote
2answers
64 views

The Cantor set and integrability of $\frac{1}{x}$

Let $\chi_C$ be the characteristic function of the standard Cantor set fully contained in the interval $[0,1]$. The problem is to resolve if $\lim\limits_{\varepsilon\to 0^{+}} ...
1
vote
2answers
25 views

Laplace equation on semidisk

I am interested in the solution of the following boundary value problem on the semidisk $D=\{(r,\theta): 0<r<1, 0<\theta<\pi\}$: $$u_{xx}+u_{yy}=0 \mbox{ in } D, $$ $$u(1,\theta)=0 \mbox{ ...
0
votes
0answers
8 views

Find the values of the parameters for which the function admits an oblique asymptote…

can you please help me solve this exercise: Find the values of real parameters $a$ and $b$ so that the function $$\color{maroon}{f(x)={(ax^3+bx^2)}^{1/ 3}}$$ admits an oblique asymptote: ...
5
votes
4answers
206 views

if $AB\neq 0$ for any non zero matrix $B$ then $A$ is invertible

Question is to check that : If $A$ is an $n\times n$ matrix over a field $F$ and $AB\neq 0$ for any non zero matrix $B_{n\times n}$ over $F$ then, $A$ is invertible. This does make some sense to me ...
1
vote
1answer
27 views

Proof of Eckart-Young-Mirsky theorem

Could someone please explain why in http://en.wikipedia.org/wiki/Low-rank_approximation#Proof_of_Eckart.E2.80.93Young.E2.80.93Mirsky_theorem it says "we know that $\exists(k+1)$ dimension space ...
1
vote
0answers
25 views

How to solve this boolean algebra problem?

Given two expressions: $$A\bar{D}+A\bar{C}D +A\bar{B}C + ABCD = Y$$ and $$BD+A\bar{C}D=Z$$ is there a way to simplify this using the rules for Boolean Algebra? I tried different combinations, but if I ...
1
vote
1answer
20 views

Complex Taylor and Laurent expansions

Let $f(z):=\dfrac{1}{2-z-z^2}, z\in\mathbb{C}\setminus\left\{ {1, -2}\right\}$. i) Express $f$ in the form $\dfrac{A}{1-z}+\dfrac{B}{2+z}$. [Answer to this is $\dfrac{1/3}{1-z}+\dfrac{1/3}{2+z}$]. ...
1
vote
1answer
9 views

Change of variables in PDE

I need to use a change of variables in this PDE $f_{xx} - f_{yy} = 0$, using $s = (x + y)/2$ , $t = (x - y)/2$ I get $f_{ts} = 0$ But I'm asked to deduce that the general solution is of the form ...
1
vote
1answer
29 views

Roll 2 dice, what's the probability that at least one will come up 6?

Let $P(E)$ be the probability that at least one roll out of the two will come up as 6. I thought of doing $P(E)=1-P(E^{c})$, which is basically $1-P($neither of the 2 rolls are 6$)$. So ...
1
vote
2answers
118 views

Using implicit function theorem without using the inverse function theorem.

Let $f:U\rightarrow \mathbb{R}$ defined on the open set $U \subset \mathbb{R}^m$. It the function $g(x):U\rightarrow \mathbb{R}$, given by the expression $$g(x)= \int_{0}^{f(x)} (t^2 + 1)dt,$$ of ...
1
vote
0answers
10 views

Linear Fractional Transforms maps the upper half unit disc onto the first quadrant

Since the LFT(Linear Fractional Transform)preserves the angles, and since $\{|z|=1,\operatorname{Im} z>0\}$ intersects $[-1,1]$ at $-1$ and $1$. So we must map one of the two right angles to the ...
0
votes
0answers
14 views

Lambda Calculus using $\beta$-reductions

Use $\beta$ reductions to compute the final answer for the following $\lambda$ terms. Use a "fake" reduction step for "+" operator. Identify each redex for $\beta$-reduction steps. Does the order in ...
1
vote
1answer
44 views

question about range, quotient space and bounded linear operator

Suppose that $T$ is a bounded linear operator and let $N(T)$ and $R(T)$ be its kernel and range, respectively, and define $\tilde{T}$ be the induced one-to-one operator from the quotient space ...
2
votes
1answer
23 views

Inverse of the function $\frac{(1+x)^2-i(1-x)^2}{(1+x)^2+i(1-x)^2}$

It can be proved that the function $f:[-1,1]\to \mathbb{C}$ defined by $$f(x)=\frac{(1+x)^2-i(1-x)^2}{(1+x)^2+i(1-x)^2}$$ maps the interval $[-1,1]$ one to one onto the lower part of the unit circle. ...
1
vote
0answers
28 views

To show a function differentiable

Let $A \in \mathbb{R}^n$ be a fixed vector and $T : \mathbb{R}^n \rightarrow \mathbb{R}^n$ a linear transformation . Define $f : \mathbb{R}^n \rightarrow \mathbb{R}$ by $$f(x) = \langle ...
1
vote
1answer
24 views

$\iiint_Dz \;dxdydz$ where D is $z \ge 0 , z^2 \ge 2x^2+3y^2-1,x^2+y^2+z^2\le3.$

I asked a questions about regions and then tried to compute a tripple integral: $$\iiint_Dz \;dxdydz$$ D is $z \ge 0 , z^2 \ge 2x^2+3y^2-1,x^2+y^2+z^2\le3.$ I tried, but now I am stuck: how do I ...
1
vote
1answer
18 views

Can one conjugate any element in $S_3$ to any other element?

I've come across the following problem from herstein: Let $\varphi$ be an automorphism of $S_3$. Show that there is an element $\sigma \in S_3$ such that $\varphi(\tau) = \sigma^{-1}\tau\sigma$ ...
1
vote
1answer
41 views

Ring of fractions $S^{-1}A$ and localisation

I'd really appreciate if somebody could help me with the problem 6.4 Reid (Undergraduate commutative algebra), because I've been trying to get the solutions for days and I don't see it. (a) Give an ...
1
vote
2answers
18 views

maximized profit w/ a cost & demand function

I'm having trouble with this problem: If $C(x) = 14000 + 500x − 4.8x^2 + 0.004x^3$ is the cost function and $p(x) = 4100 − 9x$ is the demand function, find the production level that will ...
-4
votes
1answer
33 views

A question on combinatorics.

A factory with 7 machines, employs 7 operators and manufactures 7 different products. The factory operates 7 working days in a week. Only one operator works on a machine in a day and all operators ...
2
votes
1answer
31 views

Analysis Problem Help

Let $s_1 = \frac{1}{4}$ and $s_{n+1} = s_n(1-s_n)$ for all $n\geq2$. Show that $\lim_{n\to\infty} ns_n =1.$ Can someone please help me to solve this problem? I have couldn't figure out.
0
votes
1answer
51 views
+50

About the branch-cut in the complex logarithm

Say I have the function $log (f(z))$, then does the imaginary part of the value go down by $-i 2 \pi$ on crossing the branch-cut of the log function even if $f(z)$ is not crossing the branch-cut? ...
2
votes
2answers
22 views

Sub-modules of free modules

I'm going back through basic module theory notes, and I've come across a paragraph explaining that a sub-module of a finitely generated free module may not itself be free. In my course a free module ...
1
vote
0answers
11 views

Bounded on an union of squares

I would like to do this exercise : Let $\displaystyle h(z) = \pi \mathrm{cotan}(\pi z) = \pi \frac{\cos(\pi z)}{\sin(\pi z)}$. And for $q \in \mathbb{N}^{*}$, let $C_{q}$ be the square in the ...
2
votes
1answer
57 views

Does it make sense to multiply slopes?

Multiplying fractions is a regular occurance. If those fractions are considered slopes, does it make any sense? For example, if these fractions are slopes,$\frac{9}{8} \times \frac{49}{48},$ does the ...
5
votes
2answers
80 views
+50

Intuition/How to determine if onto or 1-1, given composition of g and f is identity. [GChart 3e P239 9.72]

9.72. $A,B$ are nonempty sets. $f: A \rightarrow B$ and $g: B \rightarrow A$ are functions. Suppose $g \circ f = $ the identity function on $A$. (♦) Are the following true or false? $1.$ $f$ ...
0
votes
0answers
8 views

Example for an entire function of finite order but of infinite type

I'm currently racking my brains for an example as described in the question. I have an example $$e^{e^z}$$ which is of infinite order and infinite type. Question is, does there exist an (entire) ...
3
votes
1answer
44 views

Mathematics or physics at university

I have a strong interest in maths, and I feel that advanced physics is cool too (although I've only studied classical mechanics at high school, which is kind of boring). So I'm not sure about which ...
1
vote
2answers
20 views

If $X$ is a continuous random variable uniformly distributed over $[a,b]$, then is $Y=2-4X$ uniformly distributed over $[c,d]$? Why?

I ran into this problem solving one of the problems on my course and if I knew that this applies and how to simply prove it, it would help me a great lot.
0
votes
1answer
362 views

Distribution of Sum of Discrete Uniform Random Variables

I just had a quick question that I hope someone can answer. Does anyone know what the distribution of the sum of discrete uniform random variables is? Is it a normal distribution? Thanks!
7
votes
1answer
83 views

How prove this sequence $a_{n}=\sqrt{n}+\frac{1}{2}-\frac{1}{8\sqrt{n}}+o\left(\frac{1}{\sqrt{n}}\right)?$

let sequence $\{a_{n}\}$ such $$a_{1}=1,a_{n+1}=1+\dfrac{n}{a_{n}}$$ show that: $$a_{n}=\sqrt{n}+\dfrac{1}{2}-\dfrac{1}{8\sqrt{n}}+o\left(\dfrac{1}{\sqrt{n}}\right)?$$ This result is china student ...
1
vote
0answers
30 views

No Angle Trigonometry problem

Need to find the length of h. No angles are known.
0
votes
1answer
28 views

measurable subset of nonmeasurable set

show that if E is measurable and E⊂P where P is nonmeasurable set in [0,1), then m(E)=0. Can one please tell how to start .. and I have one more question: is the union of m'ble set and non-m'ble set ...
0
votes
2answers
18 views

Some help needed with a geometry question

What is a formula for all integers n for which a regular polygon with n sides can be constructed using a ruler and compass construction?
0
votes
0answers
15 views

$C^\omega$ notation for real analytic functions

I've seen the notation $C^\omega$ used for the set of real analytic functions (e.g. on an interval). Where does it come from? What exactly does it mean? What is the reason behind it? Who first used ...
-2
votes
0answers
16 views

Boole's functions' domain is D = {1, 2, 3, 4}. Find ∃xF(x, 2), when F(x, y) = 1100 1111 0011 0101. [on hold]

The problem is, I actually do not understand this problem very well. When the logical function is given, making truth table is not a problem for me at all. I wonder, if this exercise requires to make ...
2
votes
0answers
26 views

Prove continuity of $\frac{x^3y+2xy^3}{x^2+y^2}$ using the definition

$f(0,0)=0$ and $f(x,y)=\dfrac{x^3y+2xy^3}{x^2+y^2}$ when $(x,y) \neq (0,0)$. Is $f$ continuous at $(0,0)$? I went to polar coordinates, $$ f(x,y)=g(r,t)=r^4(\cos^3t \sin t+2\cos t\sin^3t)/r^2=r^2 ...
0
votes
0answers
9 views

Show that there is only one conic passing through the five points $[0:0:1], [0:1:0],[1:0:0],[1:1:1]$ and $[1:2:3]$. Show that it is nonsingular

Show that there is only one conic passing through the five points $[0:0:1], [0:1:0],[1:0:0],[1:1:1]$ and $[1:2:3]$. Show that it is nonsingular
0
votes
2answers
48 views

Compact sets of real number

I don't know how to prove the following statement: If $K_1, K_2$ are non empty disjoint compact subset of the real numbers, prove that there exist $k_1\in K_1$, $k_2\in K_2$ such that $|k_1-k_2|= ...
0
votes
2answers
1k views

Maximum likelihood estimation when the density is $f(x;\theta) = \theta x^{\theta -1} $

Working through this given problem on maximum likelihood estimation (MLE). The density is given as $$f(x;\theta) = \theta x^{\theta -1} $$ transforming the above equation to MLE, we have ...
0
votes
1answer
23 views

Show that Q, as a Z module, is a direct summand in a direct product of copies of Q/Z.

Prove:Q, as a Z module, is a direct summand in a direct product of copies of Q/Z. This is a problem from P.J.Hilton&Stammbach's Homological Algebra. If this is true, then there exists a ...
4
votes
0answers
450 views

Different versions of functional central limit theorem (aka Donsker theorem)?

I have seen several versions of functional central limit theorem (see the end of this post). I am confused, and hope someone could help to clarify their relations and differences. For example, I am ...
0
votes
0answers
13 views

How to find multiple points and tangent lines at multiple points??

Find multiple points and tangent lines at multiple point of the following curve: $F(X,Y)=Y^3-Y^2+X^3-X^2+3XY^2+3X^2Y+2XY$. Now $F(X,Y)= (X+Y)^3-(X-Y)^2$ $F_x=3(X+Y)^2-2(X-Y)$ $F_y=3(X+Y)^2+2(X-Y)$ ...
0
votes
1answer
17 views

If $r_n\to r$ and $s_n\to s$, then $(r \star s)_M/M \to rs$.

I was going to ask this question, but I think I figured it out, so I thought I'd post my answer: In this question of mine, a user's answer makes the following claim: Suppose $r_n$ and $s_n$ are ...
0
votes
1answer
14 views

Probability of 2 of three independent events occuring

Three objects are thrown at a target. The probabilities the individual objects will connect with the target is .75, .85 and .90. Find the probability that at LEAST two of the objects hit the target? ...
3
votes
3answers
24 views

Finding a tangent to an ellipse parallel to a given line

Problem: Find the lines that are tangent to the ellipse $x^2 + 4y^2 = 8$ and parallel to $x +2y = 6$. I tried to find the derivative of $x^2 + 4y^2 = 8$ and I got: $$\frac{dx}{dy} = -\frac{x}{2y}.$$ ...

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