0
votes
1answer
46 views

Point Set Topology Puzzle

Choose A, a subset of some topological space X. If we are given the closure and complement operations, how many distinct subsets of X can we create? How can we choose A to maximize this number? I ...
6
votes
1answer
68 views

Relation between different definitions of degree in complex geometry

Consider a holomoprhic map from a Riemann surface $$ f: \Sigma_g \to \mathbb{CP}^n. $$ This is given by some homogeneous polynomials in some variables. How can we show that the homogeneous degree $d$ ...
0
votes
1answer
97 views

Alternative proof of 'a linegraph of a Hamiltonian graph is Hamiltonian'?

So I've got an assignment to proof that the linegraph of a normal Hamiltonian graph is also Hamiltonian (but not Eulerian most of the time). I've found that this is a direct corollary of a theorem of ...
0
votes
2answers
38 views

Use of “for all” in definition of reflexive and symmetric relations.

My book says that a relation R on A is reflexive, if $\ (a,a) \in R, \ for\ every \ a \in A$ symmetric, if $\ (a_1,a_2) \in R \implies (a_2,a_1) \in R,\ for\ all\ a_1,a_2 \in A$ Although I ...
4
votes
1answer
137 views

Spectral decomposition of normal operator

Define $T$ from $L_{2}(R)$ into itself by $T(f)(t)=f(t+1)$. Show that $T$ is normal and finds its spectral decomposition. I've shown that $f$ is normal (in fact it's unitary) but how do I find its ...
1
vote
2answers
367 views

Find the next divisor without remainder

I divide a value and if the remainder is not 0 I want the closest possible divisor without remainder. Example: I have: $100 \% 48 = 4$ Now I am looking for the next value which divide 100 wihtout ...
0
votes
1answer
34 views

Find the polynomials wich are solution for a diferential equation

Find all the polynomials grade $\leqslant 2$ wich are solution for: $x^{2}y^{'}+(x+1)y=x^{3}$ I'm lost here, i don't understand what is asking for exactly. So mi attempt to do it is find the ...
1
vote
1answer
28 views

Correct equation

What is the correct equation which describes the following dates: 270, 180, 135, 112.25, 101.25. I've obtained data whit the equations: 180 + 90 = 270 135 + 45 = 180 112.5 + 22.5 = ...
0
votes
2answers
23 views

Help with this proof (Index Sifting)

Let $(x_j)^\infty_{j=1}$ be a sequence in $\mathbb{Z}$ and let $a, b, r \in \mathbb{Z}$ such that $a\le b$. Then $\sum\limits_{j=a}^b x_j = \sum\limits_{j=a+r}^{b+r} x_{j-r}$ I assume that I would ...
4
votes
0answers
133 views

Contour integration with 2 branch points

I need to compute a quite complicated Fourier transform, but I'm having problems due to the facts that I have two branch points. The integral I need to solve is $$\int_\infty^{-\infty} ...
2
votes
1answer
77 views

$\mathrm{Homeo}(S^1)$ and the Mapping Class Group

Is there a full description of $\mathrm{Homeo}(S^1)$ (i.e. the group of self-homeomorphisms of the circle)? By full description I mean a presentation/list of subgroups ect. Basically anything ...
0
votes
3answers
70 views

Undetermined Coefficients problem

Ok so I have this problem that I've been stuck on now for over an hour. I'm hoping someone can point out if I'm doing something wrong or what I am supposed to do next. It is an initial value problem ...
12
votes
3answers
305 views

Prove or disprove: $99^{100}+100^{101}+101^{99}+1$ is a prime number

Prove or disprove: $$99^{100}+100^{101}+101^{99}+1$$ is a prime number. My idea: let $100^{101}=x^{x+1}$,then $$99^{100}+100^{101}+101^{99}+1=(x-1)^{x}+x^{x+1}+(x+1)^{x-1}+1$$ is prime number? I ...
0
votes
2answers
99 views

Can you prove that $(a=b) \vdash (b=a)$ using only Fitch 'Elim' and 'Intro'?

My text says Proof: Suppose that a = b. We know that a = a, by Intro. Now substitute the name b for the first use of the name a in a = a, using Elim. We come up with b = a, as desired. However, ...
4
votes
0answers
214 views

Properties of a certain integer sequence

Consider the following sequence: $$a_1=1$$ $$a_n=\text{Number of subsets of } \{a_1,a_2,...,a_{n-1}\} \text{ that sum to } a_{n-1}$$ The first few elements of that sequence are ...
2
votes
4answers
78 views

What is the coefficient of $x^{50}$ in $\left(x + \frac1x\right )^{100}$?

I know I should use the Binomial Theorem, but I'm just having some trouble figuring this out. thanks!
1
vote
2answers
64 views

The inverse image of an open cover

I have been reading the proof of a continuous function mapping compact sets to compact sets. Let $f: X \to Y$ be continuous and onto where $X$ is compact. Let $\{U_{\alpha}\}_{\alpha}$ be an open ...
1
vote
1answer
58 views

Does $u_n \rightharpoonup u$ in $L^2(0,T;H^1)$ and $u_n' \rightharpoonup u'$ in $L^2(0,T;H^{-1})$ give something useful?

If $u_n \rightharpoonup u$ in $L^2(0,T;H^1)$ and $u_n' \rightharpoonup u'$ in $L^2(0,T;H^{-1})$ is there any way to extract a strongly convergent subsequence of $u_n$ in some useful space for PDEs? Or ...
5
votes
1answer
91 views

Automorphisms of elliptic curve

Consider an elliptic curve $y^2=x^3+b$ over $\mathbb{R}$. How to find all real automorphisms of this curve of order 3?
4
votes
1answer
129 views

Exercise from Etingof's notes on Representation Theory

I am reading through these notes of Etingof on Representation theory and I am stuck with one exercise (it's problem 4.69 in the notes). The problem is the following. Consider the space ...
2
votes
2answers
393 views

Delta function with imaginary argument

We have an integral representation for the Dirac delta function as $\delta(x) = \frac{1}{2\pi} \int_{-\infty}^\infty dk e^{ikx}$ . On the other hand, we have for delta function the property: ...
1
vote
1answer
66 views

A question to clarify the use of divergent series in calculating the casimir effect

I asked this question already on both Physics SE and quora, but I did not get an answer on either of these Q&A venues. I know this is strictly speaking not a mathematics question, but could the ...
1
vote
2answers
64 views

Find all the complex solutions to $\bar z$=z?

I am not really sure how to approach this question. Any pointers in the right direction would be great. Thanks
0
votes
1answer
109 views

Triangle inside a cylinder, surface times circumference - hard to visulize question

The surface of the triangle is $X$ and the circumfrence of the base of the cylinder is $Y$. What is the volume of the cylinder ? The answer is $XY$ but why ? If you'll take the ...
0
votes
1answer
96 views

Linear functional on the set of bounded functions

Let $S$ is non-empty set, set $$l^\infty(S)=\{f:S\rightarrow\mathbb{R}: \|f\|_\infty =:\sup_{x\in S} |f(x)|<\infty\}.$$ Suppose that $\psi:l^\infty(S)\rightarrow\mathbb{R}$ is a bounded linear ...
1
vote
0answers
21 views

Confidence bounds given random verification

[Edits made for clarification and brevity.] I'm working on an idea for a fault detection algorithm, and I've boiled it down (I think) to the following problem. A box contains 10 balls. The balls can ...
0
votes
1answer
29 views

Incidence function $\phi: E(G) \rightarrow V(G) \times V(G)$ of union of graphs $G = F \cup H$.

Incidence function $\phi: E(G) \rightarrow V(G) \times V(G)$ of union of graphs $G = F \cup H$. Suppose $F,H$ are graphs. Then according to two books, I've been studying, we can define the $G = F ...
0
votes
2answers
75 views

Is it always possible to transform one function into another?

Suppose f(x) = g(x) for all real x, and both f and g are sufficiently nice (perhaps we might limit them to be polynomials or analytic functions). Can we always manipulate one (with algebraic ...
0
votes
1answer
40 views

Prove $g$ is continous on metric space.

For:$(X,d)$ is a metric space , $f:X\to X$ is a continous function and $g:X\to R, x\mapsto g(x)=d(f(x),x)$. Prove that $g$ is a continous function. Definition: $f$ is continous at $x_0$ if only if ...
0
votes
0answers
91 views

Multivariable irreducible polynomials over finite fields

It is not difficult to prove the following result, and it seems that it should be already proved. I would appreciate it if someone offer me some reference to it. For any $f(x_1,\dots, x_n)=\sum ...
1
vote
3answers
133 views

How find this limit $\lim\limits_{n\to\infty}\int_{0}^{2\pi}\left(\frac{\sin{(nx)}}{x^2+n^2}+\frac{\cos{(nx)}}{1+x}\right)dx $

Find this limit $$\lim_{n\to\infty}\int_{0}^{2\pi}\left(\dfrac{\sin{(nx)}}{x^2+n^2}+\dfrac{\cos{(nx)}}{1+x}\right)dx $$ Maybe use Division of the integral?Thank you
1
vote
1answer
35 views

Normal distribution probability function definition

Up to now, I believed that k-dimensional normal distribution has probability function: $\frac{1}{\sqrt{(2 \pi)^k |\Sigma|}}e^{-\frac{(x-\mu)^T\Sigma^{-1}(x-\mu)}{2}}$ Recently I have read an article ...
2
votes
2answers
73 views

Is $\bigcup\limits_{n=1}^{\infty}[a+\frac{1}{n},\infty)=\bigcup\limits_{n=1}^{\infty}(a+\frac{1}{n},\infty)$?

Is $\bigcup\limits_{n=1}^{\infty}[a+\frac{1}{n},\infty)=\bigcup\limits_{n=1}^{\infty}(a+\frac{1}{n},\infty)$ There must be a difference, or not ? which is equal to $(a,\infty)$ ? RHS must be ...
0
votes
1answer
56 views

Is there a limit for how “good” a numerical method can be?

Multiplying two matrices $A \cdot B$ of size $n \times n$ in the trivial way requires $n^3$ computations. However, more efficient algorithms such as the Strassen algorithm have a lower complexity of ...
0
votes
1answer
58 views

Variations of Parameter problem

I am having trouble with this variation of parameters problem in my differential equations class: Find the general solution to $y''+9y'=3\tan(3t)$ Basically, I got down to the integrals where I had ...
1
vote
1answer
45 views

Evaluate the Series with certain property

I want to have any hint!! First i tried to replacement $x$ to $\frac{k}{n}$. But result is not good.
1
vote
0answers
28 views

Battery between liftimes

Suppose that the operating lifetime of a certain type of battery is an exponential random variable with $\theta$ $= 2$ (measured in years). Find the probability that a battery of this type will have ...
2
votes
2answers
54 views

evaluate $\lim_{x \to 0+} \frac{x-\sin x}{(x \sin x)^{3/2}}$

Evaluate $\lim_{x \to 0+} \frac{x-\sin x}{(x \sin x)^{3/2}}$ This is an exercise after introducing L'Hopital's rule. I directly apply L'Hopital's rule three times and it becomes more and more ...
5
votes
3answers
187 views

Conjectured closed form of $G^{2~2}_{3~3}\left(1\middle|\begin{array}c1,1;b+1\\b,b;0\end{array}\right)$

In my answer to this question, I come across the following case of the Meijer G-function: $$F(b)=G^{2~2}_{3~3}\left(1\middle|\begin{array}c1,1;b+1\\b,b;0\end{array}\right), b>0$$ and based on my ...
1
vote
2answers
297 views

Evaluate integrals to Riemann sum for $\sin x$ and $\ln x$

How can i evaluate to Riemann sum: i tried this $$\int_0^1 x^3 dx$$ I used $$ \sum_{n=1}^{n}\frac{i^3}{n^3}=1^3+2^3+..+n^3 = \frac{n^2(n+1)^2}{4}$$ $$\int_0^1 x^3 dx = ...
1
vote
1answer
50 views

Ordinal Arithmetic sufficient condition that a + c < b + c

I know that in general ordinal addition is not strictly increasing in the left argument (as in $0+ \omega = n+\omega$). Now I have a fixed countable ordinal $\delta$ and a natural number $k$. My ...
0
votes
6answers
53 views

$c \le n < m < c+1 \implies m-n<1 $

This should be simple, but I got stuck trying to manipulate the inequalities. Show that: $c \le n < m < c+1 \implies m-n<1 $, all are real numbers. I have that $m< c + 1 \implies ...
1
vote
3answers
599 views

Cauchy's argument principle, trouble working simple contour integral

I'm trying to teach myself Cauchy's argument principle by doing a simple example, but apparently I'm missing something, because every time I try to do the contour integral I get 0. Cauchy's argument ...
0
votes
1answer
52 views

Could someone explain this step in a proof?

Rudin PMA p.222 (Inverse function theorem) Let $E$ be open in $\mathbb{R}^n$. Let $f:E\rightarrow \mathbb{R}^n$ be a continuously differentiable function. Let $a\in E$ and assume that $f'(a)$ is ...
0
votes
1answer
244 views

Bipartite Graphs and Trees Questions

Which of the claims below is not equivalent to the rest? 1) Every cycle in a graph "B" has an even length 2) Graph "B" is bipartite 3) Graph "B" has two components that are connected. 4) Graph "B" ...
8
votes
4answers
826 views

Why should we get rid of indefinite integration?

It is the very symbol of "indefinite integral" that is flawed and confusing. It should be removed and kept only as a "guilt practice", like treating $dy/dx$ as a real fraction and things like that. ...
0
votes
3answers
178 views

Calculating the Location of a Point Relative to a Rectangle

Is it possible to calculate the location of a point (x, y) relative to a rectangle, knowing only only the differences between the distances from each corner of the rectangle to the point? In the ...
0
votes
1answer
36 views

Tangent space groups of compact manifolds

Reading a review, I have come across this sentence ...of the $SO(D)$ tangent space group of the manifold $K$ after some pages I have found ...for the case of a $D$-dimensional compact ...
1
vote
3answers
115 views

Solving an equality in 2 variables [duplicate]

I need to prove that $$\left(a + \frac{1}{a}\right)^2 +\left(b + \frac{1}{b}\right)^2 \gt \frac{25}{2}$$ if $a+b = 1$ and $a b \le 1/4$ I'd like a hint. Solve the equality first to $a$ or $b$, or ...
2
votes
0answers
81 views

Model selection: geometric mean of the standard deviation.

I have two models that represent a physical process. To determine which model is the best, I make some experiments and compare measured data with data predicted by each of the models. The model with ...

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