0
votes
1answer
94 views

Verification of simple proof of coercivity

I just want to check the following simple proof: Consider the operator $A: X \rightarrow X^{*}$. Then A is called coercive if and only if $$\lim\limits_{\Vert u \Vert_{X} \rightarrow ...
4
votes
2answers
123 views

Existence of an operation $\cdot$ such that $(a*(b*c))=(a\cdot b)*c$

When we can define a binary operation $\cdot:M\times M\rightarrow M$ on an algebraic structure $(M,*)$ such that $$a*(b*c)=(a\cdot b)*c$$ If $*$ is associative then $\cdot=*$ even if I'm not sure ...
1
vote
1answer
20 views

How to form the equation of a line from a gradient?

I am given that the gradient of a curve is $dy / dx = 10x^4 - 6x^2 + 5 $ And I need to find the equation of the curve. I started by integrating this (as it is the reverse of differentiation) and got ...
10
votes
3answers
89k views

What five odd integers have a sum of $30$?

I've been asked the following question: What five odd integers from the set $\{1, 3, 5, 7, 9, 11, 13, 15\}$ that when summed together equals to $30$? Note that any integer can be used more than ...
2
votes
0answers
99 views

The space C(X) and uniformly convergence on compact subsets of X

Let $X$ be a completely regular space. For every compact subset $K$ of X, define a seminorm $p_K: C(X)\to {\Bbb C}$ such that $p_K(f):=\sup_{x\in K}|f(x)|$. Then $\{p_K;K ~is ~compact \}$ is a ...
2
votes
4answers
300 views

How do you work out $\sqrt[4]{16^3}$ without a calculator.

$$\sqrt[4]{16^3}$$ I just don't know what to do when I get to $4096$. The original equation was $16^{3/4}$.
0
votes
1answer
91 views

Ramanujan (LPS) graph construction

In the original LPS article they mention two constructions for Ramanujan graphs, one with order $q^3$ nodes and the other in order $q$. My question is regarding the second construction (which is ...
5
votes
1answer
137 views

The complex function $\log(1+e^{iz})$

G.H. Hardy states the following: The function of a complex variable $z$ $$ e^{i p z} \log(1 \pm e^{iz})\frac{1}{z^{2} \pm \theta^{2}} = f(z), \ (-1 < p <1, \theta >0),$$ is ...
1
vote
1answer
116 views

Assume that $(\text{X}, T)$ is compact and Hausdorff. Prove that a comparable but different topological space $(\text{X},T')$ is not.

Say that a topological space is CH if it is both compact and Hausdorff. Let $T$ and $T'$ be two topologies on the same set X that are comparable but different, i.e., $T$ is either strictly ...
2
votes
1answer
107 views

Show: $M\subset\mathbb{R}^n$ Jordan-measurable, iff $vol^*(\partial A)=0$

Show that a bounded subset $A\subset\mathbb{R}^n$ is Jordan-measurable iff and only if $\partial A$ is a Jordan null set, i.e. $vol^*(\partial A)=0$. Here Show some properties of the ...
0
votes
1answer
63 views

How the weak convergence is related with trivial topology?

I hope my question is not going to be silly, but I am really confused, and will appreciate any help. How the weak convergence is related with trivial topology? Weak convergence and pseudo-metric ...
2
votes
1answer
50 views

showing that $f=\chi_A$ for some measurable set $A$!

hi need some hints with this question: If $f^n$ is integrable for each $n$ and $\int f^n dµ = c$ for some constant c then show that $f(x) = \chi_A(x)$ for some measurable set $A ⊂ X$. I know that we ...
1
vote
2answers
40 views

Let $p(x)$ have a zero $a\in \mathbb{Q}$ …

"Let $p(x)$ have a zero $a\in \mathbb{Q}$..." Where $p(x)$ is a polynomial. I came about this part of a statement and I was not entirely sure what it meant. Although, I assumed that it meant that ...
0
votes
1answer
45 views

A Binomial Expansion (Sum of Coeffients)

If $(1+x+x^2)^n = a_{0}+a_{1}x + a_{2}x^{2} +\cdots +a_{2n}x^{2n}$, then find the value of $a_{0}+a_{3}+a_{6}+\cdots $.
1
vote
1answer
53 views

Utility optimization question

Having trouble with how to put this together. I have an answer key, but the individual steps I am struggling with. Two period economy with a representative consumer that maximizes the utility ...
12
votes
1answer
304 views

Evaluating $\int_0^\infty \frac {\cos {\pi x}} {e^{2\pi \sqrt x} - 1} \mathrm d x$

I am trying to show that$$\displaystyle \int_0^\infty \frac {\cos {\pi x}} {e^{2\pi \sqrt x} - 1} \mathrm d x = \dfrac {2 - \sqrt 2} {8}$$ I have verified this numerically on Mathematica. I have ...
2
votes
1answer
66 views

Show that a function is well-defined and continuous

Prove that $$ h(x) = \sum_{n \ge 1} \frac{\exp\left(-nx^2 [\arctan(x^n)]^2 \right)}{|x| + n^2}$$ is well-defined and continuous on $\mathbb{R}$. I think using the M-test could work here, but not ...
2
votes
3answers
625 views

Example of a bounded lattice that is NOT complete

I know that every complete lattice is bounded. Is there a simple example for a bounded lattice that is not complete? Thank you
6
votes
1answer
92 views

Galois group of $X^5-n$

In the following situation I want to find the Galois group of a specific polynomial: Let $n > 1$ be a square-free integer, $f =X^5 - n \in \mathbb{Q}[X]$, $x:=\sqrt[5]{n}$ and $\zeta=\text{e}^{2 ...
2
votes
1answer
111 views

Question about the second fundamental form

I am studying Riemannian geometry and have a question understanding something. I use Do Carmo's book. In the book, a vector field is defined for isometric immersions: for an immersion $$ ...
0
votes
0answers
231 views

Set theoretic identities involving the Cartesian product

In "Naive Set Theory" by Halmos (amazon link) there is a chapter involving ordered pairs which eventually mentions the Cartesian product and many of it's properties. For the sake of completeness I ...
5
votes
2answers
421 views

Submartingale example: proof

I am trying to prove if the process $M_t = e^{W_t^2-t}$ is a submartingale ($W_t$ is the Wiener Process). The proof becomes a bit difficult, to the point where I am unsure how to move forward. Let ...
1
vote
0answers
44 views

What is the number of x-intercepts in this graph of sine?

The function : $y=3-4\sin(2\pi x-3\pi)$ .. how many $x$-intercepts over the interval $[0,2]$? I am confused if they're 3 or 5 because there are 3 $x$-intercepts that are really intercepting ...
0
votes
1answer
31 views

4 points not separable by SVM

We know in a support vector machine: Considering we have a linear feature mapping $\phi(x_n)=x_n$ and the XOR problem. We have 2 classes in $R^2$, class 1 $ t_+=+1$ and class 2 $t_-=-1$ and 4 ...
0
votes
2answers
57 views

Uniform convergence and boundedness

Assume that $(f_n)$ is a sequence of continuous functions on $[a,b]$, which converge uniformly. Prove that $(f_n)$ is uniformly bounded, i.e., there exists $M \ge 0$ such that, for any $n \in ...
0
votes
1answer
62 views

Cocycles vector bundles and metrics

It is well known, and not difficult to prove that a vector bundle $E$ over a (smooth) manifold $M$ together with a metric gives rise to orthonormal frames (by Gram-Schmidt). An consequnece is that the ...
0
votes
0answers
87 views

Writing down the KKT optimality conditions

Consider the problem Minimize $(1/2)\times{x}^{T}\times Q\times x+{P}^{T}\times x$ Subject to $(1/2)\times {x}^{T}\times P\times x+{d}^{T}\times x≤r$ Where Q and P are n×n matrices, P is ...
6
votes
3answers
111 views

How to prove these two limits $\displaystyle \lim_{n\to\infty}\frac{a_{n}}{n}$ and $\displaystyle\lim_{n\to\infty}\frac{a_{n+1}-a_{n}}{n}$ exist

Assume that $$\lim_{n\to\infty}(a_{n+2}-a_{n})=A$$ show that $\displaystyle \lim_{n\to\infty}\dfrac{a_{n}}{n}$and $\displaystyle\lim_{n\to\infty}\dfrac{a_{n+1}-a_{n}}{n}$ exist and find these limits. ...
4
votes
2answers
130 views

Double integral: $\int_{y=0}^1\int_{x=y}^1 e^{\large x^2}\ dx\ dy$

Could someone help me with this question? I am stuck on it. Compute the following double integral: $$\int_{y=0}^1\int_{x=y}^1 e^{\large x^2}\ dx\ dy.$$ How to compute the integral when the inner ...
2
votes
1answer
28 views

Broken trucks at a road

If three trucks break at locals random distributed of a road with lenght $L$, find the probability that $2$ of those trucks are not at a greater distance than $d$, fot $d \leq \frac{L}{2}$ My ...
1
vote
2answers
68 views

Generate base 2 numbers that add up to $2^n-1$ when left-shifted

I am trying to generate such odd numbers $p$ that satisfy $$\sum_{i=0}^n{2^{ik}p} = 2^m-1$$ for some $m, n, k, p \in \mathbb N$. In other words, numbers that can be left-shifted (multiplied by ...
1
vote
0answers
40 views

$A$ matrix is diagonalisable if $\exists S : S^{-1}AS $ is a diagonal matrix, how can I find S?

Per definition a matrix $A$ is diagonalisable if there exists a matrix S such that $S^{-1}AS$ is a diagonal matrix. My question is how do I find the matrix $S$? Is it always the combination of the ...
2
votes
0answers
58 views

Hypersphere packings from hypercubic graphs?

Consider a $D$-dimensional hypercubic lattice, i.e. a graph $H$ embedded in ${\mathbb R}^D$ where the vertices have integer coordinates $ (x_1,...,x_D) \in {\mathbb Z}^D$ and edges are between all ...
0
votes
1answer
173 views

A proof with a lim sup ratio test

Let us recall the ratio test: If $\lim_{n\to \infty}\left|{b_{n+1} \over b_n}\right | < 1$ then $\sum b_n$ converges. Now let $\sum a_n x^n$ be a power series and let $L = \limsup_{n \to ...
0
votes
2answers
64 views

Cubic factoring question

I'm trying to figure out how a colleague factored an expression. I don't get how: $$a^3+a^2b-(b+1)=(a-1)[a^2+a(b+1)+(b+1)]$$ Multiplying the result I see it's true, but not sure how he got there..is ...
1
vote
1answer
17 views

Normally distributed data or not

Can I say that the datas are normally distributed? I would say yes, but I am not entirely sure.
1
vote
2answers
42 views

Unique solution?

If I have the function $f \colon \mathbb{R} \to \mathbb{R}$ with $ f'(t) = k \cdot f(t) $ how can I argue that this solution has to be of the form $f(t) = Ce^{kt} $ and can't look any different? ...
1
vote
0answers
38 views

Matrix calculus: rules for partial traces

I'm trying to understand a paper and have trouble seeing why the following can be written: $Tr_E\{[ \rho,V] \} = \sigma Tr_E\{\rho_E V\} - Tr_E\{ V \rho_E \} \sigma$, when we know the following ...
1
vote
1answer
40 views

a trig question

Because my book doesn't have solutions to these problems, I'm checking here if I solved them correctly (I know it's all probably wrong): 1) $$\tan(\pi+\frac{x}{3})>0$$ What I noticed first is that ...
1
vote
1answer
122 views

Find the primitive of a particular complex function

Let $D_1=\{|z-a|<r_1\}$,$D_2=\{|z-b|<r_2\}$,$D_3=\{|z-c|<r_3\}$ such that $D_1,D_2,D_3$ are disjoint.Let $f$ be an analytic function in $\mathbb{C}\diagdown(D_1\cup D_2\cup D_3)$.Prove that ...
0
votes
1answer
49 views

invariants of a Lie algebra

What does it mean by "constructing invariants" in algebraic topology or algebra in general? How to define a "invariant" in algebra? What does it mean by the "invariant of a Lie algebra"?
1
vote
1answer
35 views

if $g: \mathbb{R}^k \rightarrow \mathbb{R}$ continuous, $f_i=X \rightarrow \mathbb{R}$ measurable prove $h(x) = g(f_1(x),…,f_k(x))$ is measurable.

if $g: \mathbb{R}^k \rightarrow \mathbb{R}$ is continuous and $F_i=X \rightarrow \mathbb{R}, i = 1,2,...,k$ is measurable Prove that $h(x) = g(f_1(x),f_2(x),...,f_k(x))$ is measurable. So far I ...
2
votes
2answers
81 views

What is $\lim\limits_{x\to 0}\left(\dfrac{x}{e^{-x}+x-1}\right)^x$

What is $$\lim_{x\to 0}\left(\frac{x}{e^{-x}+x-1}\right)^x$$ Using the expansion of $e^x$, I get that the function $$y=\left(\frac{x}{e^{-x}+x-1}\right)^x$$ is not defined for negative ...
1
vote
1answer
65 views

Integrate $\int_a^b e^{- \cos(t)} dt$

I am looking for an explicit representation of $\int_a^b e^{- \cos(t)} dt$. The only way I could imagine to find the antiderivative is to expand this function in spherical harmonics or use the taylor ...
4
votes
3answers
82 views

Is it possible that $N_G(H)=H$ and $N_G(K)=K$ where $K\subsetneq H$?

Is it possible that $N_G(H)=H$ and $N_G(K)=K$ where $K \subsetneq H$ and $H,K$ are proper subgroups of $G$ ?
0
votes
2answers
27 views

A question about uniform convergence of $g_n=f\left(\frac xn\right)$

Could you give me some hint how to prove this statement: Suppose $f(x)$ is some function on R. Prove: If $g_n=f\left(\frac xn\right)$ converges uniformly to zero on R than $f(x)=0$ for all x. I ...
1
vote
1answer
91 views

52-card trick for a larger deck?

Long ago someone demonstrated the following card trick with a standard 52-card deck: (1) A volunteer selects 5 cards from a shuffled deck, which the performer does not see. (2) The assistant puts ...
0
votes
2answers
38 views

Is this answer right?

There is a pentagon inscribed in a circle with a diameter of 10. What is the area and perimeter? Is the answer 20 and 25? I tried using examples and applying them to the problem.
2
votes
0answers
48 views

Can a continuous function on a compact group $G$ be interpreted as the sum in $C(G)$ of its Fourier series?

For a given function $f\in C(G)$ on a compact group $G$ its Fourier transform is defined as the family of operators $$ \widehat{f}_\sigma=\int_Gf(t)\cdot\sigma(t^{-1}) \ \text{d}\ t,\quad ...
0
votes
0answers
37 views

A conjecture on the product of digits of a number

Define $(m,n)$ to be a special pair if $n=m \cdot Pd(n)$. Where $Pd(n)$ is the product of digits $n$. Then I have the following conjecture - For every $m$ with no digit of $m$ being $0$ , there ...

15 30 50 per page