0
votes
0answers
16 views

Is it sensible to define the absolute value of the integral and the derivative with measures in this way?

Is it sensible to define the absolute value of the integral and the derivative with measures in this way? $\mu$ and $\nu$ are measures (functions that take a shape [a set of points] and give the ...
0
votes
0answers
14 views

Quadratic Approximation Using Chebyshev Economization

For the quadratic approximation of given function, we use following: $\ Q(f) = f(a) + f'(a)*x + f''(a)*(x^2)/2$ In the question, it wants me to find quadratic approximation using chebyshev ...
2
votes
0answers
17 views

Do local Galois representations always lift?

Suppose $G:=G_F$ is the absolute Galois group of a local (residue char. $\ell$) or global field $F$, and $\bar{\rho}$ a (linear) representation of $G$ on the $\mathbb{F}_q$-module $\mathbb{F}^d_q$, a ...
1
vote
1answer
29 views

Are every measure on $\mathbb{R}^{n}$ borel and/or regular?

I saw the above question in an exam paper and I am not sure how to even start. The question is true for the case of Lebesgue measure but I am not sure for arbitrary measures. I have tried looking at ...
0
votes
1answer
27 views

finding general solutions of second order diffrential equation

find the general solution of $$\frac {d^2y}{dx^2} +9y =18$$ I am not sure how to write it in its complementary form because of the roots one being positive and ...
0
votes
1answer
19 views

Nearest singular matrix

Let the SVD of $A \in \mathbb R^{{n}*{n}} $ be given as $A=\sum_{i=0}^n \sigma_{i}u_{i}v_{i}^{T}$ where $\sigma_{1}\gt \sigma_{2}>{...}>\sigma_{n-1}=\sigma_{n}>0 $ Compute a matrix $B$ such ...
0
votes
2answers
35 views

H prime order, normal subgroup of group G. Prove H in center Z(G).

I am looking at the following question: "Let H be a normal subgroup of prime order p in a finite group G. Suppose that p is the smallest prime that divides the order of G. Prove that H is in the ...
1
vote
0answers
14 views

Khovanov polynomial twisted unknot. [on hold]

I am reading bar nathans paper about khovanov polynomials and am having a lot of trouble. so if we construct the chain complex we get: $0 \rightarrow V \otimes V \rightarrow V\{1\} \rightarrow 0$ I ...
1
vote
2answers
35 views

How to prove that $\sin x = x \, _{0}F_{1}(-;\frac{3}{2};-\frac{x^{2}}{4})$?

How to prove that $\sin x = x \, _{0}F_{1}(-;\frac{3}{2};-\frac{x^{2}}{4})$? where $_{0}F_{1}$ is the hypergeometric series?
1
vote
0answers
18 views

Solution to second order PDE $u_{xy}-xu_x+u=0$

Given second order partial differential equation $u_{xy}-xu_x+u=0$, where $u=u(x,y)$ find the general solution. I tried to use $u(x,y)=v(ξ(x,y),η(x,y))$ substitution to get ...
0
votes
0answers
20 views

Change of Basis between linear Transformations

I am trying to get a better understanding in change of basis with matrices and linear transformations, therefore I am using several linear Transformations $^{i-1}A_i=\begin{bmatrix} \cos\theta_n ...
1
vote
0answers
15 views

A Question about the Intersection Multiplicity

In a recent lecture the lecturer defined the local ring of an irreducible affine variety $V$ at $P\in V$ as $$ \mathcal{O}_{V,P}=\{\phi\in K(V)\mid\phi\text{ is defined at }P\}. $$ Then he defined ...
1
vote
2answers
16 views

Proving Rank of a matrix is greater than its sub matrix

How can I show that the rank of a matrix is always greater than or equal to the rank of every square matrix thereof.. I mean it is self evident to anyone who knows anything about rank of matrices but ...
1
vote
0answers
9 views

On those integers $n>1$ such that any commutative ring with identity having exactly $n$ ideals is a PIR

Convention : All rings are commutative with unity unless stated otherwise . By ideals we will mean to include $\{0\}$ and $R$ also . Let us call an integer $n>1$ a "principal number " if any ring ...
0
votes
1answer
24 views

Sending bits and parity bits over noisy channel

Consider a sender is trying to send three information bits $a_1$, $a_2$, and $a_3$ over a noisy channel with error probability $$p = 0.001$$ That is with probability $p$ each bit may be flipped ...
1
vote
0answers
21 views

integral of a vector field in $\mathbb{R}^n$

I'm wondering the definition of the integral of a vector field on a hypersurface in R^n. Here is what I guess, but I did not found it on the internet. Let $v$ be a vector field on $\mathbb{R}^n$ and ...
0
votes
2answers
46 views

What's wrong with my Taylor -Maclaurin- Series? $e^{x^2+x}$

Here's what I have: We know: $$e^x = 1 + x + \frac{1}{2!}x^2+\frac{1}{3!}x^3 +\frac{1}{4!}x^4$$ Now I can calculate the Taylor Series for $e^{x^2+x}$: ...
0
votes
1answer
19 views

Supremum infimum on more general space

What is the definition of supremum and infimum on more general spaces, say $\mathbb{R}^2$, $\mathbb{R}^n$.
0
votes
0answers
22 views

Mathematical conjectures for which Numerical Search promising

I am fascinated by mathematical conjectures, especially those that are believed to be false. Now I wonder, are there some mathematical conjectures for which the numerical search would be promising? ...
2
votes
1answer
30 views

How can I find the $n^{th}$ 'reversible prime'?

I just thought of an interesting problem when discussing prime numbers with a friend. Some numbers are prime, but even fewer numbers preserve their primality when we reverse their digits. So for ...
1
vote
1answer
10 views

Prove that a semigroup which satisfies a certain conditions is a group

This is an exercise from "Abstract Algebra" by P.A.Grillet(p.12, ex.2). Let $S$ be a semigroup(that is, a set with an associative binary operation) in which there is a left identity element( $\exists ...
0
votes
0answers
15 views

Prove that if $H$ is an abelian subgroup of a group $G$ then $\langle H, Z(G)\rangle$ is abelian.

Prove that if $H$ is an abelian subgroup of a group G then $\langle H, Z(G)\rangle$ is abelian. So I started out trying to show that $H$ is a subset of $Z(G)$ but then I realised that $Z(G)$ is the ...
-1
votes
1answer
17 views

What can you do with a Frobenius Monad? [on hold]

A frobenius monad is an endofunctor that has natural transformations making it both a monad and a comonad. What are the most interesting aspects to Frobenius monads?
0
votes
2answers
47 views

Without calculating them determine whether $36^2+1$ and $154^2+1$ are prime and find the prime factors if not prime

I know that $36^2+1$ is prime, $154^2+1$ is not, both are equal to 1mod4. The prime divisors of $154^2+1$ should also be of the form 1mod4. Tried showing this by Wilson's theorem however i don't feel ...
2
votes
1answer
12 views

Surface Normal from Cross Product

Given an equation (in this case $x^2-y^2+z=0$) how would I find the surface normal using a cross product at a certain $(1,2,3)$ point? I know how do it with $grad(f)$ but I presume that isn't what ...
-2
votes
2answers
15 views
1
vote
4answers
44 views

What would be the value of the limit $\lim _{x\to \infty} (\frac{3x +1}{3x-1})^{4x}$?

What would be the value of the limit $\lim _{x\to \infty} (\frac{3x +1}{3x-1})^{4x}$? My initial idea was to divide by x in the numerator and denominator. However that would only solve the inner ...
0
votes
0answers
10 views

Finding the fundamental Pell solution from a system of Pell-like equations

Assume $d$ is a non-square integer, and $r,s,t,w$ are integers, and $n$ and $m $ are integers with $n,m \neq 0,\pm 1$, satisfying the system of Pell-like equations \begin{align} r^2-ds^2 &= m, \\ ...
4
votes
1answer
28 views

What definition of Reed-Solomon code is correct?

In the discuss with @Evinda we have the contradiction with the definition of Reed-Solomon Codes (not generalized case) over the finite field $\mathbb{F}_q$. We have two papers and ...
2
votes
2answers
32 views

What is the motivation for normed division algebras?

The famous Hurwitz theorem states that the only normed division algebras are $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$. What is some good pedagogical motivation why we should think ...
4
votes
4answers
67 views

Is the natural logarithm actually unique as a multiplier?

The Wikipedia page on the natural logarithm says: 'Logarithms can be defined to any positive base other than 1, not only e. However, logarithms in other bases differ only by a constant multiplier from ...
1
vote
0answers
14 views

How can I find the distance between two points within a triangle if I have the distance between each point and each vertex of the triangle?

Title says it all. It would be useful to extend the question to finding the distance if any of the points is outside of the triangle, but I'm trying to figure out the basic problem first.
1
vote
1answer
14 views

Assume $T$ is a complex operator

Assume $T$ is a complex operator such that $T^{2}=T$. Prove that $Tr(T)$ is a non-negative integer. There is a remark in my book, Suppose the characteristic polynomial $\chi_{T}(x)$ factors intro ...
1
vote
2answers
20 views

Give a regular expression

Let Σ be {0, 1} Give a regular expression generating words over Σ containing an even number of 1’s or with a length which is multiple of 3. i came up with this solution: ε ( ((0*(10*10*)) + ((0+1) ...
0
votes
1answer
22 views

Counterexample to Maximum modulus principle [on hold]

How can I show a counterexample to the maximum modulus principle? The Maximum modulus principle states, suppose $f$ is holomorphic and nonconstant in a region $G$. Then $|f|$ does not attain a weak ...
2
votes
5answers
71 views

Example 3.40 (b) in Baby Rudin: How to find $\lim_{n\to\infty} \sup \frac{1}{\sqrt[n]{n!}}$?

Here is Theorem 3.39 in the book Principles of Mathematical Analysis by Walter Rudin, third edition: Given the power seires $\sum c_n z^n$, put $$\alpha = \lim_{n\to\infty}\sup\sqrt[n]{\vert c_n ...
0
votes
0answers
5 views

Identification of Infinite Dimensional State in Hidden Markov Model

Consider a hidden markov model (HMM) where the state, $X_t(\alpha)$, is a stochastic distribution over $\alpha \in \mathbb{R}_+$ and one observes a signal $Y_t$, which is simply a moment of this ...
0
votes
2answers
16 views

On those integers $n>1$ such that there exist a coommutative ring with identity with exactly $n$ ideals

Let $n>1$ be an integer ; we call $n$ a " ring number " if there exist a commutative ring $R$ , with identity , having exactly $n$ ideals ( including $\{0\}$ and $R$ ) ; now since for every ...
0
votes
2answers
18 views

Specific solution for ODE

Can somebody explain step-by-step, as I can't understand, how to find the particular solution of the ODE? 1) $y' + y = 1$ 2) $y' + 2y = 2 + 3x$
2
votes
0answers
10 views

Integral inequality with $f(x,a)$ and $f(x,a)_a$

A function $f(x,a)$ has the following properties (denoting with $f_x$ the partial derivative w.r.t. $x$) along a known interval $[\underline x, \bar x]$: $$f_x > 0\\ f_a = \begin{cases} > 0 ...
0
votes
0answers
12 views

Justify the use of a random variable as an estimator

I have a set of independent random variables $U_i\sim \mathcal{N}(\mu,1)$. With them, I create the new set $X_n=\sum\limits_{i=1}^n U_i$. Thus $X_n\sim \mathcal{N}(n\mu,n)$, but they are not ...
0
votes
0answers
10 views

Is the mixture of Exponential family distributions an Exponential family distribution too?

Consider we have a mixture of multinomials or in a broader sense, a mixture of $f$s where $f$ is an distribution of exponential family type and the membership components are known with the sum of 1. ...
2
votes
0answers
16 views

about minimal non-nilpotent groups

Newman and Wiegold have studied the AN-groups i.e. the locally nilpotent groups which are not nilpotent but every proper subgroup is nilpotent. I was asking why the notion locally nilpotent was added ...
2
votes
2answers
25 views

Checking if “continuous” when $x$ is 1 and reaches 1

I have $$f(x) = x \left| x - 1 \right|$$ Here my given value for $x$ is 1 And I need to test if the function is "continuous" when $x$ is $1$ and also when reaching $$ f(1)$$ $$ \lim\limits_{x \to ...
1
vote
1answer
24 views

Prove the monotone convergence theorem for sequences of Lebesgue-integrable functions

I'm trying to prove the monotone convergence theorem for $L^1$ functions: Suppose $(f_n)$ is a sequence of $L^1$-functions (i.e Lebesgue-integrable functions) over a measure space $(X,\sigma (X), ...
0
votes
1answer
19 views

Convergence of $f_n (x)=\sqrt[3]{\rvert{x}\rvert ^3+\frac{1}{n}} \qquad x \in [-1,1]$

$(f_n)$ is a succession of functions $$ f_n : [-1,1] \rightarrow \mathbb{R} \\ \\ f_n (x)=\sqrt[3]{\rvert{x}\rvert ^3+\frac{1}{n}} \qquad x \in [-1,1] $$ Punctual convergence $\forall x \in ...
0
votes
0answers
32 views

Understanding the solution of this integral

The following integral represents an expected value of a geometric brownian motion for $S_T>K$: $$\int_{z^*} ...
0
votes
1answer
19 views

Rationnal canonical form of the matrix $A$

Let the matrix \begin{equation} A=\begin{bmatrix} 2 & 1 & 2 \\ -2 & -1 & -4 \\\ 1 & 1 & 3 \end{bmatrix}. \end{equation} So far I found the characteristic polynomial ...
-2
votes
2answers
39 views

how can ı solve this problem? [on hold]

Suppose $f$ is in $L_1$ space of $μ$, where $μ$ is the Lebesgue measure. Prove that to each $ϵ>0$, there exists a $δ>0$ so that the Lebesgue integral of the absolute value of $f$ is less than ...
1
vote
0answers
13 views

How to make $S\otimes_R M$ into a left $S[G]$-module

I have a basic question on tensor products which might have been asked before-sorry if this is the case. Let $G$ be a finite group and $R\subset S$ any (unital) ring extension. Let $M$ be a finitely ...

15 30 50 per page