1
vote
1answer
73 views

An application of Schwarz lemma to $af(0)+bf'(0)$ [closed]

Let $f$ be holomorphic and $|f(z)|\le 1$ for all $|z|\le 1$. If $a,b \in \mathbb{C}$, show that $$|af(0)+bf'(0)| \le (|a|^2+|b|^2)^{1/2}$$
0
votes
1answer
55 views

Boundedness of continuous summable function

Let $f\colon\mathbb{R}\to\mathbb{C}$ be a continuous function. If we suppose that $f$ is a $L^1(\mathbb{R;C})$ function too, then can we conclude that $f$ is bounded? ADD: I asked the preceding ...
1
vote
1answer
63 views

Derivative of a function of vector parameter. Problem with notation.

I have an error function $$err=\frac{1}{N}\left[\textbf{y}^T\ln{\textbf{x}}+(\textbf{1}-\textbf{y})^T\ln{(\textbf{1}-\textbf{x})}\right]$$ I need to find the gradient $\bigtriangledown_x{}err$, such ...
0
votes
1answer
82 views

How to find the supremum of this?

I would like to know how to find this supremum $$ \sup_{x \in [1,\infty)} \left| n\left( \sqrt{x+\frac{1}{n}}-\sqrt{x} \right) - \frac{1}{2\cdot \sqrt{x}} \right|=?$$ where $ n \in \mathbb{N} $. I ...
1
vote
0answers
39 views

Idempotents in a ring of fractions of the tensor product of Gaussian integers [duplicate]

Let $S=\{x^0,x^1,x^2,...\}\subset \mathbb{Z}$ be the multiplicatively closed subset generated by $x$. What are the nontrivial idempotents in the total quotient ring ...
1
vote
2answers
68 views

Does this figure represent a cumulative distribution function?

Is this a c.d.f.? I have no problem for random variable $X$ at $-\infty<X<x_2$. But if p.d.f. were continuous in interval $x_2\leq X<\infty$ , then c.d.f. should have been continuous. If ...
3
votes
1answer
89 views

A sequence of random variables $(X_n)$ such that $\mathbb E(X_n)\to -\infty$ but $X_n\to +\infty$ a.s.

Let $\xi_{1},\xi_{2},\dots$ be random variables (i.e measurable functions) such that $\mathbb{P}(\xi_{n}=-3^{n})=2^{-n}$ and $\mathbb{P}(\xi_{n}=1)=1-2^{-n}$ Let ...
0
votes
3answers
147 views

Why is the group of units mod 8 isomorpic to the Klein 4 group?

I recently learned that $U_8\cong \mathbb Z/2\mathbb Z\oplus \mathbb Z/2\mathbb Z$. I can see, through a bit of computation, that this is the case, but I was wondering if this is just a coincidence ...
3
votes
2answers
160 views

Writing real invertible matrices as exponential of real matrices

Every invertible square matrix with complex entries can be written as the exponential of a complex matrix. I wish to ask if it is true that Every invertible real matrix with positive determinant ...
2
votes
1answer
183 views

When are $\mathbb Z_m$ and $\mathbb Z_n$ homomorphic?

Let $m$ and $n$ be two given positive integers such that $m<n$. Then what are the necessary and sufficient conditions for the groups $(\mathbb Z_m,+_m)$ and $(\mathbb Z_n,+_n)$ to be homomorphic ...
4
votes
1answer
152 views

Determining the automorphism group of a disconnected graph

There is this know formula for determining the automorphism group of a graph $G$: let the connected components of $G$ consist of $n_1$ copies of $G_1$, $\dots$, $n_r$ copies of $G_r$, where $G_1, ...
1
vote
2answers
43 views

If $N$ is a complete linear subspace of $H$ and $\inf_{f \in N} |f-g| = d$, prove that $\exists f'$ such that $f' \in N$ and $|f'-g|=d$

The problem as stated is Let $H$ be a Hilbert space. If $N$ is a complete linear subspace of $H$ and $\inf_{f \in N} |f-g| = d$, prove that $\exists f'$ such that $f' \in N$ and $|f'-g|=d$. I ...
8
votes
0answers
132 views

Is the solution of $e^x \log(x)=1$ transcendental?

Let $u$ be the solution of the equation $$e^x \log(x)=1$$ Is $u$ rational, irrational algebraic or transcendental? $u$ seems to be transcendental, but I cannot prove it. Perhaps, someone has an ...
1
vote
3answers
123 views

Any counterexample to answer this question on elementary geometry?

Question: See the figure below. If AB=BC and DE=EF, is the line DF parallel to the line AC? This should be an elementary problem. But I can't construct a counterexample to disprove the above ...
0
votes
1answer
83 views

Question about linear transformations?

So suppose a linear transformation U maps a vector space into itself. Is the rank of U necessarily equal to the rank of its transpose? I know the transpose maps the dual space the same dual space, but ...
5
votes
2answers
133 views

Solve differential equation $y'' = -a y +\frac by$

I am trying to solve for y(t): $$y'' =-ay + \frac by$$ I have tried a lot, but haven't succeeded so far. Actually I am not sure there is a 'nice' solution. Do any of you have ideas of how to solve ...
2
votes
1answer
60 views

Four term exact sequence for Artin algebras

Suppose $\Lambda$ is an Artin algebra, i.e. an algebra finitely generated as $R$-module for a commutative Artin ring $R$, $A$ is a finitely generated left $\Lambda$-module, and $X$ a finitely ...
0
votes
1answer
89 views

Determine the convergence/divergence of the following series

I tried using DCT and LCT, but in both cases the comparison test fails. Below is the series: $$\sum_{n=1}^{\infty} (-1)^n \frac {\sin(n)}{n^{0.9}}$$. Using the absolute convergence test is the first ...
1
vote
2answers
122 views

Why, precisely, is $\{r \in \mathbb{Q}: - \sqrt{2} < r < \sqrt{2}\}$ clopen in $\mathbb{Q}$?

Just going over some old notes and I realized I always took this for granted without actually fleshing out exactly why it is true. The set of all r is closed in $\mathbb{Q}$, because the set of all ...
0
votes
1answer
38 views

How to find a countable set satisfying some property

Let $\tau=\{G\subset \mathbb R: 0\in \mathbb R\setminus G$ or $\mathbb R\setminus G$ is finite $\}$ and let $\tau_u$ be the usual topology on $\mathbb R$. I want to show that if $f:(\mathbb R,\tau)\to ...
0
votes
1answer
103 views

Perpendicular at a defined distance from point on line intersects another line in coordinates?

It approximately looks like the following picture The figure may be rotated at any angle. I know the coordinates of points $A$, $B$, $C$, $D$ and the length of $BF$. $\angle ABD$ and $\angle CBD$ ...
4
votes
1answer
75 views

Existence of solution of singular ODE

Suppose $f$ is a Lipschitz continuous function defined on $\mathbb{R}$. How can one prove that the following ODE admits at least one solution. \begin{equation} y'' + \frac{1}{x}y' + f(y) = 0 ...
1
vote
3answers
176 views

Computing an inverse modulo $25$

Supposed we wish to compute $11^{-1}$ mod $25$. Using the extended Euclid algorithm, we find that $15 \cdot 25 - 34 \cdot 11 =1$. Reducing both sides modulo $25$, we have $-34 \cdot 11 \equiv 1$ mod ...
1
vote
0answers
19 views

Isolating clusters in data

This is a problem that I keep encountering in one form or another every so often. Given a large N dimension space with a sufficiently large (M) sample of points. [1] If I were to be asked to find a ...
1
vote
0answers
120 views

How to prove: $GH\cdot DJ=GD\cdot JH$

In any $\Delta ABC$, and the incenters of the triangle $ABC$ is $I$, and let $D$, $E$, and $ F$ be the points on $BC$, $AC$, $AB$, respectively, and the point $M$ is on $AD$, such that $AD$, $BM$, ...
0
votes
2answers
510 views

Given that $gcd(a,b)=1$, prove that $gcd(a+b,a^2-ab+b^2)=1$ or $3$, also when will it equal $1$? [duplicate]

It is an exercise on the lecture that i am unable to prove. Given that $gcd(a,b)=1$, prove that $gcd(a+b,a^2-ab+b^2)=1$ or $3$, also when will it equal $1$?
0
votes
1answer
62 views

finding factors for gcd

To compute $gcd(25, 11)$, Euclid's algorithm would proceed as follows: $$\underline{25} = 2 \cdot \underline{11}+3$$ $$\underline{11} = 3 \cdot \underline{3}+2$$ $$\underline{3} = 1 \cdot ...
0
votes
1answer
189 views

Find the irreducible factors of $f(x)=x^4-5x^2+6$ over $\mathbb{Q}$ and over $\mathbb{R}$ individually.

Find the irreducible factors of $f(x)=x^4-5x^2+6$ over $\mathbb{Q}$ and over $\mathbb{R}$, individually. I have a bit of a start, but need some help.
8
votes
3answers
614 views

If a group $G$ has odd order, then the square function is injective.

Suppose $G$ has odd order, show the function $f:G\rightarrow G$ defined by $f(x)=x^2$ is injective. This proposition is easily provable if we assume $G$ is Abelian, but I don't know how to start this ...
1
vote
1answer
287 views

Why left continuity does not hold in general for cumulative distribution functions?

Definition: The c.d.f. $F$ of a random variable $X$ is a function defined for each real number $x$ as follows:$$F(x)=Pr(X\leq x) \text{ for } -\infty<x<\infty$$ Let ...
3
votes
2answers
153 views

Show that the even and odd terms of the sequence described by $x_1=\frac{1}{3}, x_{n+1}=\frac{{x_n}^2+1}{5x_n}$ are monotonic and bounded.

Let $(x_n)_n$ be a sequence of real numbers such that $$x_1=\dfrac{1}{3}, x_{n+1}=\dfrac{{x_n}^2+1}{5x_n}.$$ Show that the even terms are monotone decreasing and bounded and the odd terms are monotone ...
2
votes
1answer
61 views

Find the real vector $x$ which satisfies all this?

I got this after applying KKT conditions to an optimization problem. Let $\mathbf{h}$ be a given $N\times 1$ real vector. Let $\alpha$ be a real constant. We need to find $\lambda$ and the $N\times ...
0
votes
1answer
56 views

Transforming rectangle to disk-shape

The problem is formed when I am doing image processing: transforming the left image to the right. I haven't found any proper method to transform the image directly, so an algorithm or function is ...
0
votes
2answers
51 views

Find asymptotes of $(2x)/(x-1)^2$

What are the asymptotes of $$\frac{2x}{(x-1)^2}$$ ? I have problems already on domain.
0
votes
1answer
192 views

Prove that $\sup \{-x \mid x \in A\} = -\inf\{x\mid x \in A\}$

I need to prove that $\sup \{-x \mid x \in A\} = -\inf\{x \mid x \in A\}$ and am having trouble moving the $-x$ out of the $\sup$ to $\inf$. Another thing is that I don't quite know how to prove $b = ...
0
votes
3answers
195 views

How to solve $9\sin 2x-40\cos 2x=\frac{41}{\sqrt{2}}$

Solve the following question: \begin{eqnarray} \\9\sin 2x-40\cos 2x&=&\frac{41}{\sqrt{2}}\\ \end{eqnarray} I know that there is a formula for solving the above question like : ...
-1
votes
1answer
74 views

Using Inclusion Exclusion [closed]

How many integer solutions are there to the inequality $$y_1+y_2+y_3+y_4\lt184$$with $y_1\gt0$, $0\lt y_2\le10$, $0\le y_3\le17$, and $0\le y_4\le 19$? How do we solve this using Incl. Excl. ...
1
vote
2answers
62 views

For a Gaussian Random walk where $x_n$ is the sum of $n$ normal random variables, what is $P(x_1 >0, x_2 >0)$?

I know that the events $x_1 >0$ and $x_2 >0$ are not independent, but I can't think of a way to find a conditional probability so I can solve this. Thanks!
0
votes
1answer
103 views

Poisson Compound Process

$N(t)$ is a Poisson process with parameter $\lambda> 0$, and $X_1,X_2,...$ are independent and identically distributed random variables with a common mean and positive variance. Let ...
0
votes
1answer
62 views

Integrating Linear Differential Equations

If I have a differential equation in terms of velocity such as $$2\left(\frac{d^2 x}{dt^2}\right) - 3\left(\frac{dx}{dt}\right) + 4 = 0$$ why am I not allowed to think of integration as an operator ...
2
votes
1answer
2k views

Calculate number of password combinations

An eight-character password consisting of upper- and lowercase letters and at least one numeric digit (0–9): My working: Passwords including digits: 62^8 passwords with no digits: 52^8 Passwords ...
1
vote
4answers
98 views

Proof of sets operations equivalency having a hypothesis

This is the exercice: Let $A, B$ and $C$ be sets. Suppose that $C \subseteq A$. Prove that $A - (B - C) = (A - B) \cup C$. This is what I do: $x \in A-(B-C) \Longleftrightarrow x \in A \neg \wedge ...
0
votes
1answer
110 views

Predicting the Nth combination in sequence of A and B

First I need a way of generating every possible combination of As and Bs in an array of 2048 items. Once I have that would it be possible to row N without action generating every row between 0 and N? ...
3
votes
1answer
166 views

A set that is transitive but not well-ordered by $\in$?

I am trying to find a transitive set which is not well-ordered by $\in$. This question raises when I read Jech's Set Theory, in which an ordinal number is defined as a transitive and ...
6
votes
1answer
356 views

Where to get help with Homotopy type theory?

I'm trying to understand the Homotopy Type Theory book. I find myself completely lost in Chapter 2, especially when it starts using higher groupoids. What is the recommended background for this book? ...
2
votes
2answers
54 views

In the field $\mathbb{Z}_7[x]/\langle x^4+x+1\rangle$, find the inverse of $f(x)=x^3+x+3$.

In the field $\mathbb{Z}_7[x]/\langle x^4+x+1\rangle$, find the inverse of $f(x)=x^3+x+3$. I know how to find the inverses of elements within sets, rings, and fields. I know what to do if the field ...
1
vote
0answers
100 views

About quadratic form and its discriminant

There are 3 parts of the problem. Let d be a perfect square, possibly 0. Show that there is a quadratic form $ax^2+bxy+cy^2=0$ of discriminant d for which a=0. Let a,b,c be integers with $a\ne0$. ...
0
votes
1answer
107 views

$GL_n(F)$ is not abelian for $n\ge 2$ counter example?

So I am asked to prove that for $n \ge 2$, the group $GL_n(F)$, where $F$ is any field, is non-abelian. I figure this amounts to finding a counter-example for all such $n$. It wasn't hard, but I'm ...
1
vote
1answer
193 views

Logarithm Problem : Find the number of real solutions of the equation $2\log_2\log_2x+\log_{\frac{1}{2}}\log_2(2\sqrt{2}x)=1$

Find the number of real solutions of the equation $2\log_2\log_2x+\log_{\frac{1}{2}}\log_2(2\sqrt{2}x)=1$ My approach : Solution : Here right hand side is constant term so convert it into log ...
1
vote
1answer
105 views

Are powers of primes the sum of two primes?

I was thinking about this and am wondering if it is true. Currently trying to look for a counter example, but haven't found anything yet. Conjecture: $p^\alpha$ can be written as the sum of two ...

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