0
votes
0answers
3 views

Simplify exponential sum over $\mathbb{F}_p$ to prove identity

I have a sum of powers of prime ($p>3$) roots of unity $\begin{align*} &d_{j,k}=\sum_{a,b,c \in ...
0
votes
0answers
6 views

Formally evaluating integral to calculate electric or gravitational field.

I never understood how such integrals are calculated, formally. In a line is easy, just a line integral. In a surface, sometimes is easy, like in a disc. But, some surfaces, like sphere, it gets ...
0
votes
0answers
8 views

Does anybody know about an ebook version of Saaty The Analytic Hierarchy Process?

I'm looking for Thomas L. Saaty The Analytic Hierarchy Process in pdf, but I only found hardcover versions to order.
0
votes
0answers
7 views

How do I find the relative extrema of a function in spherical coordinates?

I want to find the relative extrema for the following function. $f(\theta,\phi)=AR\cos\theta\sin\phi + BR\sin\theta\cos\phi + CR\cos\phi $ $A,B,C,R$ are constants In a function $g(x,y)$ using ...
0
votes
0answers
10 views

Two methods of solving the differential equation $y' = .75 -.005y$

I am working on a differential equation problem and I am stumped since two different methods seem to give me two different answers Method 1 Given $\frac{dy}{dx} = .75 -.005y$ ...
0
votes
0answers
10 views

Alegbraic Rearrangement

im going through an examination and the answer booklet and it gives the answer simply as " sk0.5 - δk = 0 rearrange to k* = (s/δ)² " could any explain how to get to this answer
0
votes
0answers
3 views

Does $\sum_B p(L|B)p(B|G) = \sum_B p(L,B |G) = p(L|G)?$

Does $\sum_B p(L|B)p(B|G) = \sum_B p(L,B |G) = p(L|G)?$ By chain rule, does $p(L,B |G) = p(L|B,G)p(B|G)$? Does $\sum_B p(L|B)p(B|G) = \sum_B p(L|B,G)p(B|G)$
1
vote
1answer
20 views

How can I find such a space which is not compact so the function f is not homeomorphism?

Let X be a compact space and Y be a Hausdorff space. Suppose that $f:X \rightarrow Y$ is a continuous bijection. Then f is homeomorphism. Prove that the compactness assumption is necessary.
0
votes
0answers
11 views

Find the Taylor series and prove it converges using the defintion

I'm studying for the FE Exam. A simple walk-through would be appreciated to help my understanding of how to solve similar problems. Find the Taylor series about $x=2$ for the function $f(x) = x^5 - ...
0
votes
2answers
20 views

Finding the change of variables to transform $u_{tt} - u_{xx} = 0$ into $u_{rs} = 0$

I'm just beginning to introduce myself to partial differential equations and one of the first problems presented in the textbook I have literally no idea how to do. I think the author intended the ...
0
votes
0answers
15 views

Looking for help with this elementary method of finding integer solutions on an elliptic curve.

In the post Finding all solutions to $y^3 = x^2 + x + 1$ with $x,y$ integers larger than $1$, the single positive integer solution $(x,y)=(18,7)$ is found using algebraic integers. In one of the ...
0
votes
1answer
10 views

Is this symmetric, block-diagonal matrix positive semi-definite?

I have a matrix of the following form, where $a,b,c>0$ \begin{align*} A = \left[ \begin{array}{cccccc} aM_{12}^2 & aM_{12}M_{13} & 0 & 0 & 0 & 0 & 0 \\ aM_{13}M_{12} ...
0
votes
2answers
37 views

If $\operatorname{rank}(A)=n$ then $\operatorname{rank}(AB)=\operatorname{rank}(B)$

I have looked here, but still I cannot understand how to get to equality. Let assume that the matrices are squared $\operatorname{rank}(AB) \leq \operatorname{rank}(B)$ is easy to show, but how can I ...
0
votes
0answers
10 views

Uniform convergence and Weakly convergence

Let $u_{k}\in l^2{\mathbb{(Z)}}$ be a sequence such that for every sequence $n_{k} \in \mathbb{Z}$ the sequence $n_{k}u_{k}\rightharpoonup 0$. Prove that $ u_{k} \rightarrow 0$ in $l^{q}(\mathbb{Z}) , ...
-1
votes
0answers
14 views

Kahler condition

Let $(M,\omega) $be a kahler manifold. Why the Kahler condition $\omega = 0$ is equivalent to: $\partial_kg_{i\bar j} = \partial_ ig_{k\bar j}$ ; for all $i; j; k$.
1
vote
0answers
10 views

How can I show that this space is not locally compact but normal?

Let E be the set of all ordered pairs (m,n) of non-negative integers. Topologize E as follows: For a point $(m,n)\neq (0,0)$ any set containing (m,n) is a neighbourhood of (m,n). A set U containing ...
1
vote
0answers
13 views

Integral form of Bessel's (n = 0)

While studying for a comprehensive exam, I have come across this old problem: Consider the Helmholtz equation in the $\mathbb{R}^2$ plane $$u_{xx} + u_{yy} + \omega^2u = 0$$ Derive an ...
0
votes
0answers
25 views

How many Geese were there before any flew away?

This equation represents geese flying away in one hour intervals. How many geese were there before any flew away? The first part x - [1/5 = x] represents the quantity of geese flying away at 1:00pm ...
4
votes
1answer
31 views

Characterization of measurability by closed sets.

If $E \subseteq \Bbb R$ is measurable, then for all $\epsilon > 0$, exists $F \subseteq \Bbb R$ closed such that $F \subseteq E$ and ${\frak m}^*(E \setminus F) < \epsilon$. I have already ...
1
vote
2answers
24 views

Finding a condition on the real $a$ such that $P$ is divisible by $(x-a)^2$

Let $P(x)=\frac{x^3}{6}+\frac{x^2}{2}+x+1$. I have to find a condition on the real $a$ such that $P$ is divisible by $(x-a)^2$. I tried to use Polynomial long division and solve a system (we need ...
1
vote
2answers
30 views

Why is $ A_1 x + … + A_n x^n $ a solution of $ \sum_0^{n} (-1)^n \frac{x^n}{n!} \frac{d^n y}{d x^n} = 0 $?

I was playing(/fiddling) around with some maths and I saw this pattern( where $ A_n $ is a constant.): $ A_1 x $ is a soultion of: $$ \frac{y}{x} - \frac{dy}{dx} = 0 $$ $ A_1 x + A_2 x^2 $ is a ...
2
votes
2answers
34 views

What is the simplest way to understand $\nabla \cdot (\nabla \times \textbf{A} ) =0$?

Of course one can prove it by brute force. But how to Understand it, in the simplest way? Another issue is $\nabla \times (\nabla f) =0 $.
0
votes
1answer
28 views

Example of three different points in $\mathbb{R}^3$ such that there are infinitely many planes

I am doing some extra work over the summer to get ready to go back to uni next month. Going over a number of questions there is one I can't get my head around. It reads Give an example of three ...
2
votes
3answers
40 views

Irreduciblility of $x^3 + 9x + 6 $ in $\mathbb{Q}[x]$

I am trying to prove the irreducibility of $x^3 + 9x + 6 $ in $\mathbb{Q}[x]$ without using Eisenstein's criterion. What I have done is -- Let assume it is reducible in $\mathbb{Q}[x]$, then it can ...
1
vote
4answers
41 views

Does $(m+1)+m2+(m-1)2^{2}…+2^{m}$ equal sth simpler?

Does $(m+1)+m2+(m-1)2^{2}...+2^{m}$ equal sth simpler, where $m\in \mathbb{N}$? Excuse me if it is too simple, I am bit tired. Thanks.
1
vote
5answers
26 views

If $a$ divides $bc$ and $\gcd(a,b) = d$ then $\frac a d$ divides c

I'm trying to prove that if $a$ divides $bc$ and $\gcd(a,b) = d$ then $\frac a d$ divides c. I tried using Bezout identity but couldn't get anywhere.
1
vote
2answers
29 views

$(a_1 a_2 \cdots a_n)^2=e$ in a finite abelian group

Let $G$ be a finite abelian group. Prove that $(a_1 a_2 \cdots a_n)^2=e$. My proof: $$\forall a \in \{a_1,a_2,\cdots,a_n\} \exists ! a^{-1} \in \{a_1,a_2,\cdots,a_n\}:a^{-1}a=aa^{-1}=e$$ hence, ...
0
votes
0answers
14 views

Finding the variable of a coordinate point on a circle

This might be a very simple question but I am having trouble figuring it out, so if anyone can explain: A circle is marked with three points A(-3,2),...
0
votes
0answers
11 views

Does the canonical morphism commute with direct image functor?

I am trying to prove the representability of the Quotient functor. I have the following problem. Let $\phi \colon T \to S$ be a morphism of noetherian schemes and let $F$ be a coherent sheaf on ...
-4
votes
1answer
26 views

The symetric operator is linear operator.

Definition: Let be $X$ unitary space. Operator $A:X\rightarrow X$ called symetric if $$ (Ax\vert y)=(x\vert Ay), (x,y\in X).$$ I saw some books functional analysis but can not find verification of ...
0
votes
0answers
17 views

Defining rotation without using angles, but as geometric transformations?

According to this article on angles, we can define rotation without using angles, and then use rotation to define angles. The relevant paragraph is at the very end: But what is a rotation? Is it ...
0
votes
4answers
75 views

Proving that $\int \frac{1}{x} dx = \ln(|x|) + c_1$

In all textbooks and online notes, there is always a table of antiderivatives and it always says $\int \frac {1}{x} dx = \ln(|x|)+c_1$ but there is nowhere a proof. I found some proofs online but ...
0
votes
2answers
29 views

Radius of convergence of powerseries $\sum_{n=1}^\infty \frac{(-1^n)}{n!}z^n$

$$ \begin{align} \sum_{n=1}^\infty \frac{(-1)^n}{n!}z^n \end{align} $$ Find the radius of convergence of this powerseries. To determine the radius of convergence should I split it into two separate ...
0
votes
1answer
31 views

What is missing? (Rudin's Principles of Mathematical Analysis - Theorem 2.30)

Let us first give a definition: Definition Given a metric space $X$, and a subset $Y\subseteq X$, we say a subset $E$ of $Y$ is open relative to $Y$ if for each $p\in E$ there is an associated ...
0
votes
1answer
38 views

Integral estimation

Show with Integral estimation that $$ \sum_{k=1}^{\infty} \frac {\ln k}{k^2} \le \frac {2+3ln2}{4}$$ $$f(k)=\frac {\ln k}{k^2} $$ For the integral it is : 1 But the other part is the estimation, ...
0
votes
0answers
5 views

A simple formula for pseudo-random binary string

I'm trying to illustrate the concept of Kolmogorov complexity by comparing two seemingly-random binary strings, but showing that one of them is in fact describable by a program shorter than its length ...
0
votes
0answers
6 views

Minimize Energy in Image processing - Geodesic active contours

I've read some papers in Geodesic active contours (Image processing), which use the minimization of an Energy, consist of Internal Energy and External energy, for example, in the paper of Kass (Snake: ...
0
votes
1answer
20 views

Inverse Matrix Multiplication

Let $A \in F^{n*n}$ a inverse matrix and $B\in F^{n*n}$ a none inverse matrix We can say that because A is row equivilate to $I_n$$ \implies $ $AB$ is none inverse matrix?
1
vote
3answers
68 views

Disproving the claim that the numbers 1+2+4, 1+2+4+8, 1+2+4+8+16… alternate between prime and composite

I am working through an elementary number theory book and I have come across the following problem. Show the following claims are wrong: Claim 1: The sequence 1+2+4, 1+2+4+8, 1+2+4+8+16, ...
1
vote
0answers
21 views

When can an infinite sum and complex integral be interchanged?

Are there some conditions under which the following two are equal? $$\displaystyle \oint_C \sum f_n(z)= \sum \oint_C f_n(z)$$ In the case of real valued functions, the condition $f_n(z) \geq 0$ ...
0
votes
4answers
41 views

Arithmetic Mean, Geometric Mean, Harmonic Mean and their relations

If $a$ be the arithmetic mean between $b$ and $c$, $b$ be the geometric mean between $c$ and $a$ then prove that $c$ is the harmonic mean between $a$ and $b$. I expressed $a$ as $$a=\frac{(b+c)}{2}$$ ...
1
vote
1answer
59 views

Inequality: $\left|x^3-y^3\right|<|x|^3+|y|^3$

Could anyone show me why $$\left|x^3-y^3\right|<|x|^3+|y|^3$$ for all real numbers (x,y) except 0? Im thinking of whether of how to remove the modulus sign on the left hand side of the equation ...
0
votes
0answers
12 views

CAS expression - Solve equations with $\sum$ and $\infty$

Can a TI89 or other CAS calculators solve this? I tried it on my classpad 330 did not work solve for p $$p = \sum_{j=0}^\infty p^j \frac{2^je^{-2}}{j!}$$ solve for p $$p = \lim_{z \to \infty} ...
-3
votes
1answer
52 views

Solving for $a^2+b^2+c^2$ given $\sqrt{a^2+1}+\sqrt{b^2+4}+\sqrt{c^2+9}$ and $a+b+c$. [on hold]

I am trying to solve the following problem involving three terms: $$\sqrt{a^2+1} + \sqrt{b^2+4} + \sqrt{c^2+9} = 8$$ $$a + b + c = 6$$ $$a^2 + b^2 + c^2 =?$$
0
votes
1answer
21 views

Arithmetic and Geometric Mean Inequalities [on hold]

Can someone help me to understand the logic of: $$\sqrt{ab} \le \frac{a+b}{2}$$ Proof: ?
1
vote
0answers
18 views

Sum/diff of matrix units

I understand what the product of matrix units means, but I don't understand what the sum/difference of two different matrix units represents. For example, what does ${e_{2,2}}-{e_{5,5}} $ equal? ...
0
votes
0answers
7 views

Form-invariant solution to PDEs

I'm trying to understand how to create form-invariant solutions to PDEs: $$\hat{L}u(x,t)=0$$ with the constraint $|u\big(x,t\big)|^2=|u\big(f(t)\cdot x + g(t),0\big)|^2$. During evolution, the ...
3
votes
3answers
45 views

Group of even order must contain $a:a=a^{-1}$ $ (a\not = e)$ [duplicate]

Let $G$ be a finite group. If the order of $G$ is even, prove that there is at least one element $a$ in $G$ such that $a\not= e$ and $a=a^{-1}$. Here's my idea: Suppose $\{x_1,\cdots,x_n\}$ is ...
1
vote
0answers
17 views

Adding a delta function to a differential equation

So say I have a differential equation of the form: $$ (\alpha \frac{d^2}{dx^2}+fx^2)y(x)=\lambda y(x) $$ Whose solutions are known (a Gaussian multiplying a Hermite polynomial.) I am now curious how ...
1
vote
2answers
56 views

what is the difference between average and expected value?

I have been going through the definition of expected value in Wikipedia (http://en.wikipedia.org/wiki/Expected_value) beneath all that jargon it seems that the expected value of a distribution is the ...

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