0
votes
0answers
7 views

How to show that the $\phi and \varphi$ satisfy the Cauchy Riemann equation

when u=(u,v)=($\frac{\partial\varphi}{\partial y},-\frac{\partial\varphi}{\partial x}$) and u(x)=grad$\phi$(x)=$\nabla\phi$(x) how can you use the equations above to prove that the $\phi$ and the ...
0
votes
0answers
17 views

Why is P or not P is unsatisifiable by construction?

A proof of predicate logic inability to express graph reachibility (page 63) involves a formula which can be interpreted as (there is no path, no matter what is the length) or (there is some path). ...
1
vote
0answers
16 views

If $a$ is a root for $f(x):=5x^3-7x^2+3x+6$, then find $a_{0}\in \mathbb{Z}$ s.t. $|a-a_{0}|_{7}\leq 7^{-4}$

The problem is: By Hansel's lemma I found that f has root $a\in \mathbb{Z}_{7}$ s.t. $|a-1|_{7}<1/7$. So next I am asked to find $a_{0}\in \mathbb{Z}$ s.t. $|a-a_{0}|_{7}\leq 7^{-4}$. One approach ...
0
votes
0answers
4 views

soft (standardness of definition): “Regular sequence” in non-ommutative rings

Is it non-standard to call a sequence $x_1,..,x_n$ of commutating elements in a possible non-commutative ring "regular" if multiplication by $x_{k+1}: R/(x_1,..,x_k) \mapsto R/(x_1,..,x_k)$ is a ...
0
votes
4answers
36 views

Topological spaces X and Y and a continuous bijection $f : X → Y$ while $f^{-1} : Y → X $ is not continuous

Give an example of topological spaces X and Y and a continuous bijection $f : X → Y$ such that $f^{-1} : Y → X $ is not continuous.
0
votes
0answers
12 views

An MLE Exercise.

Consider $$ f(x;\theta)=\frac{1}{\pi} \frac{e^{\theta x}cos(\theta \pi/2)}{cosh(x)}, x\in{\mathbb{R}} $$ be a family of densities and which is clearly exponential family. Then what is the Maximum ...
-2
votes
1answer
51 views

Reflexive. What does it mean?

I would like to know the definition for reflexive. I have not found anything on the internet or in my book.
0
votes
0answers
15 views

set functions and relation multiplication

$\ M$ is the set of all relations on $\ A = \{1,2,3\}$ $\ K$ is the the following relation on A $\ K=\{(1,1),(2,1),(3,1)\}$ let there be $\ f :M\rightarrow M$ $\ f(R) = RK$ is f injective? ...
0
votes
1answer
25 views

a question about convergence of sequecce!I have tried cauchy method, but it doesn't work

suppose $a_n>0$,and$\sum_{i=0}^\infty a_i$ is convergent,so we need to prove $\sum_{n=1}^\infty{ {1\over n}(a_n+a_{n+1}+\cdots+a_{2n})}$ is also convergent! I have tried cauchy method, but maybe ...
1
vote
1answer
36 views

Calculus and infinitesimals

In the definition of reimann integral, why do we put a 'dx' inside the integral sign when practically it serves no purpose except maybe telling what variable you are talking about. Then in some ...
0
votes
0answers
17 views

How to show uniqueness of $\nabla\phi$ using Green's theorem when the value of Neumann problem exists.

It says getting function $\phi(x,y,z)$ that satisfies the following conditions $\frac{\partial\phi(x)}{\partial n}$=h(x), x$\in$S is called the Neumann problem. The problem is to show the uniqueness ...
0
votes
2answers
13 views

Regular expression translation.

Given a set {1,2,...9} how can I construct a regular expression starts with a 3 has no 8's and has even number of 6's? Here's what I tried: $$$$ Define a new set no8 = {1,2,3,4,5,6,7,9} ...
0
votes
0answers
18 views

Simplification of Double Integral with Independent Parameters

I am trying to find a posterior distribution and the hint is that the double integral in the denominator should simplify because $p1$ and $p2$ are independent. $\displaystyle \int$$\displaystyle ...
0
votes
1answer
10 views

Moment Generating Function of the Chi-Squared Distribution

The questions wants us to show that the MGF for the chi-squared distribution is equal to I know that to show that I need to evaluate this integral. I'm not sure where to begin to evaluate it. ...
0
votes
0answers
7 views

Differential Equation for the Asymptotic Directions of a given Surface

I am doing a differential geometry problem, and after some work, I boiled it down to the differential equation presented below (assuming sufficient niceness). $u$, $v$ are local coordinates; $e$, $f$, ...
0
votes
0answers
16 views

Bounds for being very far from the mean

If I toss $n$ coins each with probability $1/\sqrt{n}$ of getting a head, I would like to know bounds for the probability of getting $n/2$ or more heads. Clearly the mean number of heads is ...
0
votes
1answer
32 views

Verifying the cosine rule

Verify the following system of linear equations in cos A, cos B , and cosC. Triangle cannot be shown. Then use Cramer’s Rule to solve for cosC , and use the result to verify the Law of Cosines: ...
0
votes
0answers
20 views

How many expansion methods exist in math?

For now I know about polynomial expansion and fractional expansion, but what other methods exist that I can use to rewrite and maybe simplify an algebraic expressions ? Is there something strictly ...
1
vote
0answers
16 views

Verifying divergence theorem for a unit sphere

Verify the div. Theorem for the vector field $\underline{A} = (x+y)\underline{i} + (x^2+xy)\underline{j} + z^2\underline{k}$ and a unit radius ball centred at $(1,1,1)$. The question gives a hint to ...
2
votes
1answer
32 views

How do you pronounce Korteweg–de Vries

As stated in the title, how do you pronounce Korteweg-de Vries? I've always just heard it referred to as "KdV" but I have to give a talk on it so I'd like to know how to pronounce it properly.
0
votes
1answer
16 views

2 dimensional Laplace's equation in polar coordinates

The problem asks you to get Laplace's equation in 2 dimensions in polar coordinates using the fact that $\operatorname{div}(\cdot)$ in two dimensional vector field could be written as $$\nabla \cdot u ...
1
vote
1answer
28 views

Find the number of real solutions

Let $$f(x)=\frac{1}{2}( |x-a|+|x-b|),$$ where $x$ is a real number ; no information is given on $a$ and $b$. Study the differentiability of this function and determine how many real solutions does ...
1
vote
0answers
15 views

Differential Equation for the spread of rumors

Let $y(t)$ be the number of people who have heard the rumor at time t and assume that everyone who has heard the rumor passes it to r others in unit time. Thus, from time t to time (t+h) the rumor is ...
1
vote
1answer
10 views

$\mathbb{E}[B_t-B_s], \mathbb{E}[\exp(\sigma(B_t-B_s))]$ etc.

This may be a duplicate but I cannot find the corresponding question. I have been asked to show: $\mathbb{E}[\exp(\sigma(B_t-B_s))] = \exp\left(-\dfrac{\sigma^2}{2}(s-t)\right)$ As a side note I ...
0
votes
0answers
18 views

measure $\lambda(E)=0$ or $\lambda(E)=+\infty$

Let $\mu$ be a finite measure, let $\lambda<<\mu$ ($\lambda$ is absolutely continuous wrt. $\mu$) let $P_n$,$N_n$ be a Hahn decomposition for $\lambda-n\mu$. Let $P=\cap P_n$ and $N=\cup N_n$. ...
0
votes
0answers
9 views

Inequality in a recurrence relation.

Let $(y_n)_{n=1}^\infty$ be a strictly increasing sequence of positive integers such that $2^{y_n} - 3^n > 0$, and let $(m_n)_{n=1}^\infty$ be the sequence of positive integers such that $m_1 = 1$ ...
0
votes
0answers
6 views

question about division algorithm described in handbook of applied crypto

http://cacr.uwaterloo.ca/hac/about/chap14.pdf#page=9 gives the following as a division algorithm: So step 1 is making it so that $yb^{n-t}$ is the same length as x and then step 2 loops until the ...
-1
votes
1answer
21 views

order of finit group which has even elements [duplicate]

I have a question, prove that a finite group has an even number of elements, if and only if the group consists of an element of order $2$.
0
votes
1answer
23 views

Proving Limits of f(x) and f(a+h) are equal

The question asks me to prove that the equality of these two expressions $\lim_{x\to a} f(x)$ and $\lim_{h \to 0}f(a+h)$ provided their limits exist. My answer: Let $x=a+h$ so this $\lim_{h \to ...
0
votes
1answer
13 views

A multiple choice question on the system of linear equations

Consider the following system of linear equations. $x + y + z + w = b_1$; $x - y + 2z + 3w = b_2$; $x - 3y + 3z + 5w = b_3$; $x + 3y - w = b_4$. For which of the following choices of $b_1$, $b_2$, ...
1
vote
0answers
26 views

($\cos^4x$)($\sin^2x$) in terms of first power of cosine

I believe that I have his correct but if someone could check it and see that'd be great. Here's a pic! [IMG]http://i58.tinypic.com/2dgm5ic.jpg[/IMG]
1
vote
1answer
24 views

The interval $(0,\infty)$ is an open set.

I want to prove this using interior points, $\epsilon$-neighborhoods and interior sets. The interior of a set A is denoted $A^o$. To show that $(0,\infty)$ is an open set, we must show that ...
0
votes
1answer
10 views

Compostion of functions in taylor series

I was attempting to help answer a question, but I am curious about the following. Suppose we have an analytic function $f(z)\equiv\sum\limits_{k=0}^{\infty} a_k(z-z_0)^k $ on some suitable disk of ...
0
votes
2answers
30 views

Does $f^{(n)} = 0$ imply that complex $f$ is a polynomial?

Let $f$ be a complex function with the property that $f^{(n)} = 0$. Does this imply that $f$ is a polynomial? If so, why? Upon thinking about this problem myself, I can easily observe that every ...
0
votes
0answers
7 views

Find the intersection of three bisection lines

Let $p_1 = (a_1, b_1), p_2 = (a_2, b_2), p_3 = (a_3, b_3)$ be three, non-colinear points in the plane. For each pair of these points, let $L_{ij}$ denote the line segment from $p_i$ to $p_j$. (a) For ...
1
vote
0answers
27 views

A Simple yet interesting “function of a random variable” question

Given continous density functions $f_0,f_1$ on $\mathbb{R}$ and $Y$, a random variable following the density $f_0$, I am able to calculate the density function $h$, of $\ln l(Y)=\ln(f_1/f_0(Y))$ as ...
1
vote
3answers
63 views

prove or disprove: $\lim_{x\to \infty} \frac{f(x)}{g(x)}=\lim \frac{f'(x)}{g'(x)}=0 \implies \lim \frac{\frac{f(x)}{g(x)}}{\frac{f'(x)}{g'(x)}}\ne 0$

my attempt $$\lim_{x\to \infty}\frac{\frac{f(x)}{g(x)}}{\frac{f'(x)}{g'(x)}}\text{ yields }\frac{0}{0}$$ then use l'hopital on this $$\lim_{x\to \infty}\frac{\frac{f(x)}{g(x)}}{\frac{f'(x)}{g'(x)}} ...
0
votes
1answer
15 views

denumerables: prove or disprove the following

Prove or disprove: a. if $A \subseteq B$ and A is denumerable, then B is denumerable. b. $J \cup K$ isdenumerable, where J is the set of all linear functions with slope 1 and retional y intercept, ...
0
votes
1answer
15 views

PDF of Sum of Two Random Variables [on hold]

$X$ and $Y$ are uniformly distributed on the unit disk. Thus, $f_{X,Y}(x,y) = \begin{cases} \frac{1}{\pi}, & \text{if} ~ x^2+y^2 \leq 1,\\ 0, &\text{otherwise.}\end{cases}$ If $Z=X+Y$, find ...
1
vote
0answers
7 views

Characteristic curves of 2nd-order PDEs under invertible coordinate transformations

First off, I'm not very experienced with the subject and English is also not my first language, so if there are any inaccuracies in the following text, let me know. Given a linear, scalar, ...
0
votes
0answers
15 views

Acceptance Probability [on hold]

10 Students are applying for postdoctoral position. The employer will only choose one candidate. Student Data 1 (PhD) 2 (MSc) 3 (PhD) 4 (BSc) 5 (BSc) 6 (MSc) 7 (MSc) 8 (BSc) ...
1
vote
1answer
15 views

Painting a grid of squares.

Consider a $9\times9$ block of squares where each square is painted either black or white. If each square is adjacent to at most three black squares, what is the maximal number of black squares? Here ...
0
votes
1answer
8 views

Defining a region as a data structure

Is there a way for one to define a curve or region (such as a closed, 2-d disk) as a data structure into the computer, and make an algorithm which detects if a point is a boundary point, limit points, ...
0
votes
2answers
37 views

Show that $(1 – \cos θ – \sin θ )^2 – 2(1 – \sin θ )(1 – \cos θ ) = 0$.

Show that $(1 – \cos θ – \sin θ )^2 – 2(1 – \sin θ )(1 – \cos θ ) = 0$. What kind of formulas should I use?
1
vote
1answer
19 views

Ambiguity definitions - accumulationpoint

The literature is a bit ambiguous in my point of view. Limit points and accumulation points seems to be the same. I can accept that; that's just two names for the same. But I've seen different ...
0
votes
0answers
6 views

Statistics: When to use pooled vs non-pooled vs paired

I'm looking for a simple method for knowing when to use pooled vs non-pooled vs paired tests in a given statistics problem. For example, if the standard deviations are exactly the same, I'm told I can ...
0
votes
1answer
15 views

Matlab Cropping Plots

I want to adjust the amount of whitespace that is shown when plots are generated. I found this command to do just that: ...
3
votes
1answer
11 views

Alexander–Briggs notations for the links or knots of $N^3_m$

We can use Alexander–Briggs notations for the links or knots. For example, is three separate loops with no links. And there are many other examples of Alexander–Briggs notations for three ...
0
votes
0answers
11 views

Expanding sum of negative power using log

Example expands the series with natural logarithm but I do not know this rules. $\sum_{a} a^{-n} = \sum_{a}1 - n \sum_{a}log(a) + \cdots $ Anybody understand this expansion? Thank you!
0
votes
0answers
7 views

probability and applied statics 4 [on hold]

In cairo 30% of residents listen to the local fm radio . ten residents are chosen at random? a) state the distribution of the random variable b) find the smallest value of s so that P (x >or equal ...

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