1
vote
1answer
36 views

Finding the range of $f(x) = 5-x$

Let $a < b$ and $f: (a, b] \to R$, $f(x) = 5 - x$. What is the range of the function? How do you find the range of an equation with unknowns?
0
votes
1answer
69 views

Limit set of Kleinian group

Let $\Gamma \subset PSL_2 (\mathbb{C})$ a Kleinian group coming from a discrete faithful representation $\rho : \pi_1(M) \to PSL_2 (\mathbb{C})$ of the fundamental group of a closed connected ...
1
vote
1answer
44 views

Sturm Liouville form

How do you put $u'' +c u' +d =0$ into regular SL-form? Can not see how it's an eigenvalue problem without a first order term. But the theorem states EVERY second order operator can be put into SL ...
1
vote
0answers
155 views

Primal-dual subgradient method

In these notes, an extension of the subgradient method is presented in Section 8 (page 30). The method is described so quickly and neither convergence analysis (compared to classical subgradient for ...
1
vote
1answer
94 views

2-dimensional Cohen-Macaulay domain

I am searching for a $2$-dimensional Cohen-Macaulay (normal or not) domain. Thanks in advance for any suggestion.
1
vote
2answers
125 views

How do I differentiate ${(e^e)}^x$?

I know how to differentiate $e^x$ (it's just $e^x$), but how do I differentiate ${(e^e)}^x$? any hints would be welcome.
2
votes
1answer
183 views

Is this module noetherian?

Let $k$ be a field, and let $A$ be a commutative $k$-algebra. Assume that $A$ is a noetherian ring, and let $I\subseteq A$ be a proper ideal. Consider the ideal $I\otimes_k A \subseteq A\otimes_k A$....
0
votes
1answer
130 views

Are random walk variations Markov-Chains?

Let $S_{n}:= S_0 + \sum_{i=1}^{n}X_i$ be a simple random walk, $X_i$ are independent random variables with $P[X_i=1] = p, P[X_i = -1] = 1-p$. Let $M_n:=\max\{S_0, \dots, S_n\}$. The task at hand is ...
1
vote
1answer
28 views

Sin along a vector?

I have a 2D vector that I currently use to plot a line in its direction. I calculate the next point using: p_new = p_old + v * i Were i going from 0 to some ...
2
votes
1answer
155 views

If there exists an integrable function that is not zero a.e., then the measure is $\sigma$-finite

Suppose $f\in L^1(\Omega,\mathcal{A},\mu)$ and $f(x)\neq 0$ for almost every $x\in \Omega$. How to prove $\mu$ is $\sigma-$finite? I only got that $\Omega=\cup_{n=1}^\infty \{x\in \Omega:|f(x)|\geq \...
0
votes
0answers
61 views

Geometric question (middle line of a parallelogram)

Let $ABCD$ be a parallelogram and $I,J$ are two points such that $AI = ID$ and $BJ = JD$. Show that ($IJ$) is parallel to ($AB$).
3
votes
1answer
84 views

Suppose $X$ is a Hausdorff Lindelöf scattered space. Is $\xi(X)$ a successor ordinal?

Let $K$ be a (Hausdorff) scattered topological space and for each ordinal $\alpha$ denote by $K^{(\alpha)}$ the $\alpha$th derivative of $K$ by the Cantor-Bendixson derivation (i.e., define ...
7
votes
1answer
220 views

What is semantics of “type”? Do “types” of “type theory” semantically differ from “set” of set theory?

"To be of a (certain type)" is a fundamental relationship for ontology and the computer science "ontologies" are in the core of Semantic Web (which is my interest). But I did not encounter a ...
2
votes
1answer
98 views

Why the space of probability measures is a subset of the measure space

Consider $\mathcal M (X)$ the measure space of a metric, compact space $X$ allowed of the weak-* topology induced by the semi-norms $\mu \in \mathcal M (X) \mapsto |\int_X f ~d\mu| \in \mathbb R \cup\...
0
votes
3answers
138 views

Deduct percentage from x to reach desired value

Let's suppose I need \$200,000 to pay a debt, no more, no less. So I placed a property to sell with an agent, but I know the agent will deduct 6% for himself, which will amount to \$12,000. Since ...
0
votes
2answers
135 views

In Dijkstra algorithm, it takes the source, what about the sink?

I'm studying the Dijkstra algorithm, but in my book, the algorithm takes as input only the graph and the source. Why it doesn't ask for the destination vertex? How can it work? Thanks a lot.
0
votes
1answer
76 views

Proving closure of unit space of a Hausdorff groupoid

For Hausdorff topological groups, the set $\{e\}$ containing only the identity is closed. This is because Hausdorff implies T1 which implies singletons are closed. For topological groupoids, defined ...
2
votes
3answers
303 views

Bayes Probability Problem

I need a small confirmation regarding a probability problem: We estimate that 5% of Americans spent their holidays in Texas, this proportion reaching 40% among Texans. Texans represent 2% of the ...
0
votes
1answer
49 views

How do I form these quadratic equations

A car travels a distance of $1200 \ km$ at a speed of $a \frac{km}{h}$, while a train travels the same distance at $(a – 20) \frac{km}{h}$. If the time taken by the train is $5$ hours more than the ...
1
vote
2answers
72 views

Show that, $Z$ is $\mathcal N(0,1)$

If $Y\sim\mathcal N(0,1)$ and let $a>0$. Let $$Z=\ \begin{cases} Y&\text{if } |Y|\le a\\ -Y &\text{if }|Y|> a\\ \end{cases}\ $$ Show that $Z\sim\mathcal N(0,1)$ I ...
1
vote
1answer
75 views

Collection vs set in this textbook about category theory, and some related questions.

What is the meaning of collection in this context ? Is it here a synonym of set ? Can someone please explain what the author means by "A moment's though shows that, as sets of functions, these two ...
1
vote
1answer
166 views

How to show an inequality involving a geometric series?

I am trying to show the following: For any number $a\geq2$ and any integers $r_0,r_1,\ldots,r_{n-1}$ with $0\leq r_i<a$ ($i=0,\ldots,n-1$), show that $$r_0+r_1a+r_2a^2+\ldots+r_{n-1}a^{n-1}&...
1
vote
1answer
95 views

Calculate area enclosed by curve

Calculate the area of the bounded surface enclosed by the curve $(x+y)^4 = x^2y$ with the help of the coordinate transformation $x = r\cos^2 t, y = r\sin^2 t$. As I see it the area is unbounded, so ...
1
vote
2answers
139 views

Why does the additive subgroup of $\mathbb{R}$ generated by $1$ and $\sqrt{2}$ contain arbitrary small elements? [duplicate]

Let $G\subset \mathbb{R}$ be the additive subgroup of $(\mathbb{R},+)$ defined by $G=\mathbb{Z}+\sqrt{2}\mathbb{Z}$. I want to prove that for every $\epsilon>0$ there exists an element $g_\epsilon\...
2
votes
3answers
211 views

Proof that $\int \frac{1}{x}$ is $\ln(x)$

When I was learning Calculus AB and Calculus II/III at my high school, I noticed that our textbooks never gave a full fundamental proof that $\int \frac{1}{x}$ is $\ln(x)$, and rather said that when ...
0
votes
3answers
42 views

Quadratics with unknowns

If $5x^2 – t = 4x$, and $x$ and $t$ are both positive real numbers. What is $x$ equal to? How do you find $x$? Is there a specific formula?
0
votes
2answers
129 views

If $f(x)$ is increasing, and $g(x)$ is decreasing, $f(x)g(x)$ must be decreasing?

Can the multiplication of an increasing function, say $f(x)$, with a decreasing function in $x$, say $g(x)$, be increasing? If it is possible, can you give me an example please? Constraint: $x > 0$...
0
votes
1answer
79 views

Prove that $\iint_{|x|+|y| \leq 1} f(x+y) \, dx \, dy = \int_{-1}^{1} f(t) \, dt$

Prove: $$\iint\limits_{|x|+|y| \leq 1} f(x+y) \, dx \, dy = \int_{-1}^{1} f(t) \, dt$$ I suppose we need to substitute $x+y$ with $u$ and continue the job with four divided parts, but I got $\...
1
vote
2answers
85 views

How many times the digit 9 occurs in the sequence?

A sequence is given say : 100, 101, 102, 103, 104, 105………, 364, 365. Now, How many times the digit 9 occurs in the above sequence? How do I approach this kind of problems. I am preparing for my ...
2
votes
1answer
38 views

Geometric characterization of an Euclidean norm

Show that $N$ is an Euclidean norm if and only if the intersection of the unit ball with any plane is an ellipse. I'm stuck on this one. I do not see how can I connect the definition of an Euclidean ...
1
vote
1answer
134 views

Probability density function with squares and triangles

Suppose we have a square with corners at the points $(0,0), (0,1), (1,0)$, and $(1,1)$, and we choose a fifth point arbitrarily from inside this square (i.e. both the $x$ and $y$ coordinates of the ...
1
vote
2answers
68 views

Solve $(2z-1)^5 - i = 0$

Solve $(2z-1)^5 - i = 0$ I started by saying that $(2z-1)^5 = i$ $(2z-1) = \sqrt[5]i$ $z =$ $(\sqrt[5]i +1) \over 2$ $z^5 =$ $(i +1) \over 32$ $z^5 =$ $1 \over32$$ *(i +1)$ From there, it'...
3
votes
6answers
153 views

If $ x^2+y^2+z^2 =1$ for $x,y,z \in \mathbb{R}$, then find maximum value of $ x^3+y^3+z^3-3xyz $.

If $ x^2+y^2+z^2 =1$, for $x,y,z \in \mathbb{R}$, what is the maximum of $ x^3+y^3+z^3-3xyz $ ? I factorize it... Then put the maximum values of $x+y+z$ and min value of $xy+yz+zx$... But it is ...
1
vote
0answers
80 views

Name of an algebraic structure $(A,*,\cdot)$ weaker than semirings.

I have a set $A$ with two binary operations on it $(A,*,\cdot)$ STRUCTURE A $(A,*)$ is not commutative, is not associative, it has not an identity $(A,\cdot)$ is a commutative group $(a*b)\cdot c=(...
4
votes
2answers
131 views

Determinant involving recurrence

Evaluate $$\left| A \right| = \left| {\matrix{ {x + y} & {xy} & 0 & \cdots & \cdots & 0 \cr 1 & {x + y} & {xy} & \cdots & \cdots & 0 \cr 0 ...
2
votes
1answer
126 views

Norm on unitisation of a $C^\ast$ algebra

In the theory of $C^\ast$ algebras there exists the following theorem: If $A$ is a $C^\ast$ algebra and $\widetilde{A}$ denotes its unitisation then there exists exactly one norm that extends the ...
4
votes
4answers
307 views

question on limits and their calculation

In taking each of the limits $$\lim_{x\to -\infty}\frac{x+2}{\sqrt {x^2-x+2}}\quad \text{ and } \quad \lim_{x\to \infty}\frac{x+2}{\sqrt {x^2-x+2}},$$ I find that both give the value $1$, although it ...
1
vote
1answer
127 views

Mathieu differential equation

Given the operator $T (\psi)(x):= \psi''(x)-2q \cos(2x)\psi(x)$ with $T : D(T) \subset L^2[0,2\pi] $ I was wondering: What is the right domain $D(T)$ for this operator if we want to solve the ...
1
vote
2answers
68 views

Whats wrong with my proof

I am trying to find the angle $BCB'$ Here's my solution: But doesn't match with the answer given in the book. Its got to be $27.5^\circ$. Whats wrong with my solution?
1
vote
2answers
56 views

double integral of $\sin \frac{y}{x+y}$

Calculate $$\iint \sin \left(\frac{y}{x+y}\right)dxdy$$ over the region surrounded by lines: $x+y=1$, $x=0$ and $y=0$ Since it seems difficult to directly calculate, I think substitution would ...
4
votes
2answers
84 views

Homework problem - Ways to test if a density function is cumulative density function

I have a problem that states: Let $F : \mathbb R \to R$ be defined by $$F(x) =\begin{cases}e^{\frac{-1}{x}} &\text{if } x > 0\\ 0&\text{if } x \leq 0\end{cases}$$ Is $F$ a ...
0
votes
1answer
54 views

Subtracting terms from a Fourier series

It is known that $\sum_{n=1}^{\infty}\frac{\sin(nx)}{n}=\frac{\pi-x}{2}$ in $]0,\pi]$, mostly because this is a way of evaluating $\zeta(2)$. Knowing this, is there a way to evaluate $\sum_{n=1}^{\...
0
votes
1answer
30 views

For a complex matrix $A$ that preserves the inner product, prove that $A^*=A^{-1}$

I was interested in the proof of the following: Let $A$ be a matrix which preserves the inner product; i.e. $\langle x,y\rangle=\langle Ax,Ay\rangle$ Prove that $A^*=A^{-1}$ I have read ...
4
votes
1answer
38 views

Calculate integral with $\Gamma$ and $B$

The integration is like: $$\int_{a}^{b}\left(\frac{b-x}{x-a}\right)^{p}dx$$ with $0<p<1$ Answer is $(b-a)p \frac{\pi}{\sin p\pi}$ Apparently, we can reversely construct $$\Gamma(1-p) \Gamma(...
0
votes
1answer
29 views

Conditional probability and distribution

Let Y ∼ Exp(1/5). Find P(Y ≤ 18|Y > 13). Could anyone give me any hints?
2
votes
2answers
57 views

How to resolve multiply differentiation function algorithms?

My simple function is $f(x)=\frac{1}{2}e^{-x}\sin(2x)$; Can I resolve for multiply differentiation $f^{(n)}=?$ algorithm? Thx for answer.
2
votes
2answers
82 views

Prove $\dim W \ge 2$

Let $U_1, U_2, W$ subspaces of a finite dimensional vector space, such that: $U_1 \cap U_2 = \{0\}$ $U_1 \cap W \ne \{0\}$ $U_2 \cap W \ne \{0\}$ Show that $\dim W \ge 2$. My ...
1
vote
2answers
46 views

Combination problem technique

How many numbers can be formed from 1, 2, 3, 4, 5 (without repetition), when the digit at the units place must be greater than that in the tenth place? It can be easily solved that, the total ...
2
votes
1answer
126 views

Minimum score for winner and maximum score for loser in a round-robin tournament.

I have just correctly solved this programming problem. The problem is the following: $N$ teams play a round-robin tournament, i.e. each pair of teams plays exactly one game and the winner gets 3 ...
2
votes
0answers
45 views

Angles between adjacent roots in a reduced root system.

Let $R$ be a reduced root system. ($R$ is a finite set spanning $V$, $\alpha \in R \rightarrow -k\alpha \in R$ iff $k=1$, $s_{\alpha}(R)=R$, $s_{\alpha}(\beta)-\beta=k\alpha$ whit $k$ integer). ...

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