0
votes
0answers
8 views

Is the density of an absolutely continuous distribution necessarily unique?

Is the density of an absolutely continuous distribution necessarily unique?
0
votes
1answer
14 views

Is $W^{1,2}_0$ a Hilbert space?

I came across the term $W^{1,2}_0$. Just a quick question on what the 0 means and is it a Hilbert space? I know $W^{1,2}$ is a Hilbert space. Thanks!
0
votes
2answers
29 views

Is the sum of $z_1 = a + ib$ and $z_2 = a + i(-b)$ real or complex?

$z_1 + z_2 = 2a + i(0)$ Is $z_1 + z_2$ real or complex?
-1
votes
0answers
14 views

How to find the inverse of the Haar (4) matrix?

H4 = (1,1,1,1)(1,1,-1,-1)(1,-1,0,0)(0,0,1,-1)
1
vote
1answer
17 views

prove limit of exponential function without concept of logarithm

The question is, prove that if $x>1$, then $\lim_{n\to\infty} = \infty$, where $n \in \mathbb N$, without using the logarithmic concept. I came up with a proof, but I'm not so sure about it. So I ...
1
vote
1answer
9 views

Enough Projectives in Category of Groups

Working on homology and completion a question has arisen in my head. I know that $R$-mod as a category has enough projectives in it, and as such the category of abelian groups has it as they are in ...
1
vote
2answers
26 views

A good book for beginning Group theory

I am new to the field of Algebra and so far it's looking to me quite tough. So far I have encountered the following books in group theory - Contemporary abstract algebra by Joseph Gallian and Algebra ...
4
votes
2answers
21 views

Isomorphism between $G$ and $\mathbb{Q}^{*}$

Let $\{G_{n}\}_{n\in \mathbb{N}}$ be a family of additive groups with $G_{1}=\mathbb{Z}_{2}$ and $G_{n}=\mathbb{Z}$ for $n\geq 2$ $$G=\bigoplus_{n\in \mathbb{N}}G_{n}$$ I want to prove that $G\cong ...
1
vote
0answers
9 views

$G$ be a group such that every maximal subgroup is of finite index and any two maximal subgroups are conjugate

Let $G$ be a group such that every maximal subgroup is of finite index and any two maximal subgroups are conjugate and any proper subgroup is contained in a maximal subgroup . Then is $G$ cyclic ? I ...
0
votes
0answers
4 views

Does irreducibility of a representation imply irreduciblity of all restricted representations?

Let $G$ be a group with a subgroup $H$. Then any representation of $G$ can be restricted to $H$. If the $G$ representation is irreducible then should the $H$ representation also be irreducible? If ...
0
votes
0answers
9 views

Coupled differential equation

$\ddot{y}=\omega \dot{z}$ $\ddot{z}=\omega (\frac{E}{B}-\dot{y})$ These are the coupled differential equation i came across . They have already been asked here How to Solve the Coupled Differential ...
3
votes
2answers
20 views

Intuitive understanding of vector / matrix calculcation

I am currently dealing with calculations done on vectors and matrices. For some parts I have gained an intuitive understanding, for others I didn't. E.g., when we are adding two vectors, you can ...
2
votes
2answers
29 views

Find all $n$ such that $\varphi(n) \equiv 2 \pmod 4$

My question is Find all $n$ such that $\varphi(n) \equiv 2 \pmod 4$, where $\varphi(n)$ is the Euler totient function. I am no where to start so any hint or help ? and if we are given such a ...
0
votes
2answers
24 views

Unsure why ODE non-exact equation solution is wrong?

The question I'm trying to solve is $$\left(y-4y^6\right)=\left(y^4+5x\right)y'$$ where $y(0)=1$ I want to find the solution explicitly for $x$. I found the integration factor to be $u=y^-6$. ...
0
votes
1answer
23 views

Need help with a second-degree Taylor polynomial

It says to let T2(x) be the second degree polynomial for the functionf(x) = 6 + xe4x where a=0. I need to find T2(1). I thought it was just a taylor expansion and look at the second term, which I ...
3
votes
2answers
52 views

Is $\rho(A^2) = \rho(A)^2$?

How can I show that $\rho(A^2) = \rho(A)^2$? Is that even true? I´ve tested it with matlab for random matrices, and the equation was always true. I´m pretty sure that even $\rho(A^n) = \rho(A)^n$ ...
-1
votes
0answers
12 views

finding Lipschitz constant for a discrete system

can anyone help me to find the lipschitz constant of a discrete system in the following form: y(k+1)=y(k)+`[phi_1(k) phi_2(k)...phi_ly(k)...phi_ly+lu(k)][y(k-1) y(k-2)...y(k-ly) u(k)....u(k-ly+lu)] ...
0
votes
1answer
30 views

Prove Expression cannot be factored

I am currently working on primes which can be expressed in form of a polynomial. For eg, Find all primes which can be expressed in form $n^4-52n^2+595$ It is very essential to tell whether a ...
0
votes
0answers
19 views

Distinct real roots .

Problem : If $|\log(x)| - px = 0$ has three distinct real roots then the range of $p$ will be ? My attempt : I tried to see the problem graphically and made the graph. So I am able to see that ...
-1
votes
1answer
14 views

Picture of vector in $R^3$ and vector in $R^2$ reflected across plane

I have trouble imagining what reflecting a vector in $R^2$ and a vector in $R^3$ across x-y plane and y-z plane look like. Would you please draw me a picture?
0
votes
1answer
11 views

How to derive the volume of a tetrahedron with the following data?

The vertices of a tetrahedron are:- A - (0, 0, 0) B - (0, 0, a) C - (0, b, 0) D - (c, 0, 0) Prove that the volume is:- 1/6 abc. A figure will be helpful.
2
votes
4answers
44 views

Find $\lim_{h \to 0}\frac{\tan\sqrt{x+h}-\tan\sqrt{x}}{\log(1+3h)}$ without L'Hospital's rule

What is the limit of $$\lim_{h \to 0}\frac{\tan\sqrt{x+h}-\tan\sqrt{x}}{\log(1+3h)}$$ I am confused with square root function and L'Hospital's rule is prohibited . The professor requires a detailed ...
0
votes
0answers
18 views

Specific example of a space that is separable but not second countable.

A toplogical space $X$ is said to be second countable if there exists a countable basis for the topology. $X$ is separable if there exist a countable dense subset. Show that a second ...
0
votes
0answers
8 views

Converting all arcs to polygonal arcs in a plane graph

I am trying to understand a proof on the conversion of arcs to polygonal arcs in plane graphs, in the book "Graphs on Surfaces" by Mohar and Thomassen. In the book, an arc joining two points $x,y \in ...
0
votes
0answers
16 views

EM algorithm for objective HMM M-step

How did step 2 in the derivation below arrive at step 3?
1
vote
1answer
21 views

Are these two definition of boundedness equivalent?

Definition 1: A set $S \subset M$ is bounded if $\forall x \in M, \exists r > 0,$ such that $S \subset B_r(x) = \{y \in M | d(x,y) < r\}$ Definition 2: A set $S \subset M$ is bounded if ...
3
votes
0answers
23 views

On the definition of commuting self adjoint operators.

I'm reading Mathematical Methods in Quantum Mechanics by Gerald Teschl and I came across the following exercise whose statement is causing me some troubles. It goes like this: Let $A$ and $B$ two ...
4
votes
2answers
46 views

Looking for details on historical math anecdote

My memory is very sketchy here so bear with me. A fairly prominent 19th or 20th century mathematician was captured by a military force, probably invaders. He claimed that he was just a civilian, a ...
0
votes
0answers
7 views

Efficiency of quasiconvex optimization

Summary: Could we minimize quasiconvex objectives in polynomial time? Whenever an objective function of an optimization problem can be formed as a convex function, this is considered as victory. ...
1
vote
2answers
32 views

Are planes in $3$-dimensions two-dimensional?

Are planes in $3$-dimensions two-dimensional? The reason I ask is because mathematically the $xy$-plane exists in $3$D space but appears to be $2$D, but how can something $2$D be in $3$D space? I ...
1
vote
1answer
27 views

Is it possible to write $\cos \left( \frac{1}{3}\arccos \frac{37}{64}-\frac{\pi }{3} \right)$ as a radical expression of real number

$\cos \left( \frac{1}{3}\arccos \frac{37}{64}-\frac{\pi }{3} \right)=\frac{{{\left( -37-3\text{i}\sqrt{303} \right)}^{1/3}}+{{\left( -37+3\text{i}\sqrt{303} \right)}^{1/3}}}{2}$, the number inside the ...
0
votes
1answer
36 views

Limit of a derivative is 1/2

How do I show that $$ \lim_{x \rightarrow b} \frac{d}{dx} \frac{xn^x-bn^b}{n^x-n^b} = \frac{1}{2}$$ where n and b are constants and $n>1$. I saw that it is 1/2 graphing it but I think i still ...
1
vote
1answer
23 views

If $v$ is a valuation and $v(x)<v(y)$ then $v(x+y)=v(y)$.

I was studying from some book and I came across something I haven't been able to justify. Let's suppose se have a field $K$ and an ordered group $G$ (with multiplicative notation) with an extra ...
1
vote
2answers
33 views

f: R → R and $|f'(x)| ≤ |f(x)|$

Let $f: R → R $ be a function such that $f'(x)$ is continuous and $|f'(x)| ≤ |f(x)|$ for all $x ∈ R$ , if $f(0)=0$ the maximum value of $f(5)$ is My Attempt: I proved that $f'(x)=0$ for $x ∈ [0,1]$ ...
1
vote
1answer
14 views

prove that $MN \parallel BC$ in an equilateral triangle

$\Delta ABC$ is equilateral with $M$ and $N$ being interior points. if $\angle MAB=\angle MBA=40^{\circ}$ $\angle NAB=20^{\circ}$ and $\angle NBA=30^{\circ}$. Prove that $MN \parallel BC$ from ...
1
vote
0answers
10 views

Relation between error of estimate and rate of convergence

How is bounds on estimated error of an iterative algorithm related to rate of convergence?
0
votes
2answers
30 views

Simplify the expression and leave answer in terms of $\sin x$ and/or $\cos x$

$1-\sin^2 x = \cos^2 x$. However, $1-\sin^2 x$ can also be factored using the difference of two squares. I am stuck on whether $1- \sin^2 x$ should turn into $\cos^2 x$ or be factored by using the ...
0
votes
1answer
17 views

Integral, partial fractions, need explanation for how to get from one step to another.

Can someone explain how they go from the red step to the blue one?
-1
votes
0answers
14 views

Difference quation to state space form

I have a difference equation that I am trying to convert to a linear state space. The equation is $$ y(k)=\frac{1}7{(u(k)+u(k-3))} $$ It would be nice if someone can go through the steps. I'm using ...
1
vote
1answer
32 views

Prove that the following function has a unique maximum?

I was working on a problem and reduced it to showing $$f(\alpha)=n\ln \alpha-\ln \left(\sum_{i=1}^n t_i^\alpha+\int_a^b x^{\alpha+\beta-1} e^{-\lambda x^\beta} \, dx \right) + (\alpha-1)\sum_{i=1}^n ...
1
vote
2answers
67 views

Show that $a_n = 1 + \frac{1}{2} + \frac{1}{3} +\dotsb+ \frac{1}{n}$ is not a Cauchy sequence

Let $$ a_n = 1 + \frac{1}{2} + \frac{1}{3} + \dotsb + \frac{1}{n} \quad (n \in \mathbb{N}). $$ Show that $a_n$ is not a Cauchy sequence even though $$ \lim_{n \to \infty} a_{n+1} - a_n = 0 ...
1
vote
1answer
36 views

Is this proof correct? Show $\mathbb{Q}$ is dense in $\mathbb{R}$

I like proof by contradictions in showing that $\mathbb{Q}$ is dense in $\mathbb{R}$. But I can't understand this one> https://math.dartmouth.edu/archive/m54x12/public_html/m54densitynote.pdf ...
0
votes
0answers
72 views

I want to self study systematically pure mathematics? Where do I start? [on hold]

I am an undergraduate student in Mechanical Engineering and I am highly interested in studying pure mathematics systematically.I have a fair amount of knowledge on real and complex analysis, ...
0
votes
0answers
20 views

Convergence or divergence of infinte power towers of complex numbers

Let $s$ be any complex number, $t = e^s$ and $z = t^\frac{1}{t}$. Define the sequence $a_n$ by $a_0 = z $ and $a_{n+1} = z^{a_n} $. I want to show that the sequence $a_n$ converges to $t$ if and ...
1
vote
1answer
43 views

The Triple Number Game (n + n + n)

I work at the Science Museum on weekends, and I sometimes present the following puzzle: $$0\;0 \; 0 = 6$$ $$1 \; 1 \; 1 = 6$$ $$2 \; 2 \; 2 = 6$$ $$3 \;3 \; 3 = 6$$ $$\vdots$$ $$n\;n\;n=6$$ The idea ...
0
votes
1answer
14 views

Laplace Transform Injectivity

Intuitively how can the Laplace transform be injective? You are taking an integral with limits $0$ and $\infty$. So you don't care about the function before $0$. Define $g(x)=x^2$ for $x>0$ and ...
1
vote
0answers
54 views

Evaluating a Difficult limit!

I have to evaluate a very complicated limit, I've done this task already but I wanna make sure I did it right. The function I have in my hands is $$ F(\omega)= \tanh \Big[a\cdot ...
2
votes
0answers
45 views

Are all finite sets measurable?

In my textbook, it says: "Let E be any set with m*(E) < $\infty$. Then E is measurable if and only if there exists a measurable set B with m(B) = m*(E)." There always exists a measurable set of ...
0
votes
0answers
8 views

Comparing distributions against expected to determine the one fitting better

I have a 4 sets of observed absolute frequencies for a categorical variable and the expected frequence for each category (not normal distributions). Would it be correct to use the Chi-square goodness ...
0
votes
0answers
14 views

Partitions of unity subordinate to open cover of manifolds with boundary?

I am attempting to adapt Lee's proof of the fact that open covers of manifolds without boundary always admit smooth partitions of unity to the case in which the manifold does have boundary. The ...

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