3
votes
1answer
150 views

cauchy schwarz inequality problemes

I have to prove that for all $x,y,z>0$, $$\left(\frac{x+y}{x+y+z}\right)^{0.5} + \left(\frac{x+z}{x+y+z}\right)^{0.5} + \left(\frac{z+y}{x+y+z}\right)^{0.5} \leq 6^{0.5}$$ using Cauchy-Schwarz ...
0
votes
2answers
96 views

Computing $\lim_{x\to 0}[(\sin x)^{\frac{1}{x}}+(\frac{1}{x})^{\sin x}]$

How can we compute the following: $$\lim_{x\to 0}\left[(\sin x)^{\frac{1}{x}}+\left(\frac{1}{x}\right)^{\sin x}\right]?$$ The expression looks rather daunting. If $x\to 0$, then $\sin x\to 0$ but ...
1
vote
4answers
121 views

A structure elementarily equivalent to $(\mathbb{N},0,\operatorname{S},<,+,\cdot)$

Given $\mathfrak{R} = (\mathbb{N},0,\operatorname{S},<,+,\cdot)$, Let $$\Sigma = \{ 0 < c, \operatorname{S}{0} < c, \operatorname{S}\operatorname{S}{0} < c, \ldots\}$$ By compactness ...
1
vote
1answer
53 views

Limit of a seqence $\{f_n \}_{n\in \mathbb N}$ of functions?

I don't really know how mathematicians talk about this concept. I try to explain better what I mean with limit of a sequence of functions: Given a countable set of functions $\{f_n \}_{n\in \mathbb ...
6
votes
2answers
68 views

Need to state “$p$ not equal to $61$” when solving $61p + 1 = n^2$?

In the pictures below, am I wrong to say that the 3 lines in the red box are not needed in the solutions? Regardless of whether 61 and p are distinct, it's still true that we have only the 2 possible ...
2
votes
1answer
227 views

Group theory - proof check about index and quotient group

I'm studying Cayley's Theorem on the Humpreys "A Course in Group Theory" and i did not understand a passage in a preposition. (pag 86 Corollary 9.23). It claims: "Let $H \leq G$ with finite index $n$. ...
0
votes
1answer
110 views

Probability theory problem

A bag contains $5$ white and $5$ black balls. A draws $5$ balls retain any that are white and returns any black ones to the bag. B then draws $5$ balls, retains any that are white and returns any ...
0
votes
2answers
736 views

Random walk with absorbing barriers

Consider a random walk with absorbing barriers at $0$ and $3$. $\mathbb P(S_{n+1}-S_n=1)=0.6$ and $\mathbb P(S_{n+1}-S_n=-1)=0.4$. What is the probability of eventual absorption at $0$, given that the ...
0
votes
1answer
37 views

Isomorphism canonical and Moebius bundle.

I have to prove that the canonical bundle over $\mathbb{R}P^1$ is isomorphic to Moebius bundle. We define the caninical bundle as $\xi=\{E^{\perp},p,S^1 \}$ where $E^\perp:=\{(l,v) \in \mathbb{R}P^{1} ...
1
vote
1answer
73 views

Differentiation in group space

In a few physics papers (lattice gauge theory papers, to be more specific) I've seen the following definition for differentiation on group space $$ \frac{\partial}{\partial U} f(U) = ...
2
votes
0answers
98 views

“Product” bundle notation.

Let $\newcommand{\Spin}{\operatorname{Spin}}M$ and $M'$ be two manifolds, equipped with a principal $\Spin_n$ and $\Spin_{n'}$ bundle called $P$ and $P'$, respectively. Then there is an induced ...
1
vote
1answer
1k views

Solving a second-order matrix differential equation

I have the differential equation $\frac{d^2 x}{dt^2}+Ax=0$ where $A$ is a matrix and $$\frac{d^2 x}{dt^2}=\left(\frac{d^2 x_1}{dt^2},\frac{d^2 x_2}{dt^2},\ldots,\frac{d^2 x_n}{dt^2}\right)^T\text{ for ...
0
votes
1answer
104 views

Complex function (find u and v)

I don't have an idea how to find the real and imaginary part of the following functions: $$\dfrac{1}{z},~\dfrac{2z-i}{iz+2},~\dfrac{z}{z^2+1},~\ln |z|$$ Can anybody help me?
0
votes
1answer
483 views

Parseval's Theorem Q

I have this question: I know Parseval's theorem is given by $2a_0^2 + \sum_1^{\infty} (a_n^2 + b_n^2) = \frac {2}{T} \int_{-T/2}^{T/2} f(x)^2 dx$, where T is the period. $f(x)$ is even, so I ...
2
votes
0answers
41 views

When compactness implies sequentially compactness [duplicate]

It is well known that if $ X $ is a metric space then sequentially compactness and compactness are equivalent.\ Now we consider a normed vector space $ E $ and its dual $ E^\ast $. From Banach ...
1
vote
2answers
66 views

Find all nonzero matrices $A\in M_2(\Bbb Z_3)$ which are not invertible.

Find all nonzero (with every element $\neq$0) matrices $A\in M_2(\Bbb Z_3)$ which are not invertible, and explain why they aren't invertible. It seems simple indeed, but I'm not sure how to solve ...
1
vote
1answer
90 views

What is the modulus of a number?

What is the exact definition of the modulus of a number? As far as I know, it is the distance between the origin and the point associated with this number. So if $z=a+bi \in \Bbb ...
1
vote
1answer
153 views

If N is a normal subgroup of G,decide whether np(N) | np(G) or np(G/N) | np(G)?

Here np(H) represents the number of sylow-p subgroups of H.Thanks in advance.
0
votes
2answers
71 views

Finding constrained optima of $f(x,y) = x^3 - y^3 -x$

For a function $$f(x,y) = x^3 - y^3 -x$$, the minimum and maximum under the constraint $$x^2 + y^2 =1 $$ is searched. So as usual, my approach is to set up the Lagrangian and the FOC: $$ L(x,y, ...
3
votes
3answers
1k views

Example of two norms on same space, non-equivalent, with one dominating the other

I am looking for an example (with proof) of two norms defined on the same vector space, such that the norms on the two spaces are NOT equivalent, but such that one norm dominates the other...
2
votes
3answers
695 views

Find a basis for the subspace $\mathbb{R}^3$ containing vectors

Let $v_1 = \langle 1,0,-1\rangle$ $v_2 = \langle -2,7,2\rangle$ $v_3 = \langle 3,-7,-3\rangle$ I found that these are linearly dependent since I have a free variable upon reducing. However, the ...
1
vote
1answer
63 views

Orthogonality & Adjoint Operator

I am trying to prove this simple statement left to the reader in Brézis's book. Let $A \colon D(A)\subset E \longrightarrow F$ be an unbounded operator. Let $G:=\operatorname{Graph}(A)$ and $L=E ...
12
votes
1answer
451 views

Tough Inverse Fourier Transform

In reference to this answer I gave the other day, I came across a very interesting function whose IFT would be nice to evaluate as part of completing the solution to the problem I answered. The ...
1
vote
2answers
389 views

Conformally Map Region between tangent circles to Disk

Suppose we are given two circles, one inside the other, that are tangent at a point $z_0$. I'm trying to map the region between these circles to the unit disc, and my thought process is the following: ...
0
votes
3answers
76 views

If $x = a + b$, and only $x$ is known, how to solve what is $a-b$?

If $x$ equals to $a+b$, how can I solve what is $a-b$, knowing only $x$? (approximation will do as well, if it cannot be solved exactly)
3
votes
2answers
137 views

Number of Abelian Groups of Order 256

I am trying to find the number of abelian groups of order 256. Is the following correct? We may write $256=2^8$ we then know that this may be represented in the form: $C_{n_1}\times.....\times ...
2
votes
1answer
77 views

Uniform convergence implies convergence of antiderivative

From this theorem, can we conclude that if $f_n'$ is a sequence of integrable functions that converge uniformly to $f'$, then the sequence of their antiderivatives $f_n$ converges pointwise to the ...
2
votes
1answer
126 views

Use difference quotient with not uniform bound to appoximate weak derivative

Suppose U is an open set,not necessarily bounded or has Lipschitz boundary, $f\in L^p(U)$ ,define the difference as usual: $$D^h_i f=\frac{f(x+he_i)-f(x)}{h},\ \ \forall x\in U'\subset\subset U$$ ...
7
votes
2answers
655 views

Why are (representations of ) quivers such a big deal?

Quivers are directed graphs where loops and multi-arrows are allowed. And we can talk about representations of quivers by assigning each vertex a vector space and each arrow a homomorphism. Moreover, ...
3
votes
1answer
121 views

Does convergence in $L^p$ and pointwise imply the same limit?

If $f_n\in L^p$ converge to $f$ pointwise and to $g$ in $L^p$. Does that mean $f=g$ almost everywhere?
3
votes
1answer
123 views

Exponential Distribution Function

If $X\sim \text{Exp}(X)$ then for all positive $a$ and $b$, $P(X>a+b\mid X>a)=P(X>b).$ So given independent random variables $X \sim \text{Exp}(\lambda)$, $Y \sim\text{Exp}(\mu)$ we would ...
1
vote
0answers
22 views

Is there a way to estimate the range of fitting coefficients from only the data?

Considering an approximation $f$ for a set of $N$ data points $(x,y)$ using, for example, $M$ radial basis functions at arbitrary sites in the domain $f_i = \sum_{j=1} ^M c_j\phi(||x_i-x_j||)$ where ...
2
votes
2answers
74 views

Finding Value, Related To Functional Equation

$f(x)$ is continuous for $\forall x \in R$ and $f(2x)-f(x)=x^{3}$ (1) $f(x)+f(-x)$ is constant ? (2) $f(0)=0$ ? I don't know how to use the continuity. especially for $f(0)=0$ ?
0
votes
1answer
634 views

How are binary operations used in the real world?

Not necessarily a mathematical question, but how could binary operations be used in the real world? What applies to it?
4
votes
1answer
72 views

Finiteness of ideal of given norm

I'm trying to prove that there are only finitely many ideals of any given norm in the ring of integers $\mathcal{O}_k$ over a numberfield $K$. I know there are "standard proofs" (eg How many elements ...
8
votes
2answers
128 views

$f: \mathbb{R}^2 \to \mathbb{R}$ a continuous open map, show that for each $x \in \text{range}(f)$, $f^{-1}(x)$ is always uncountable.

Let $f: \mathbb{R}^2 \to \mathbb{R}$ be a continuous open map. Show that in fact for each $x \in \text{range}(f)$, $f^{-1}(x)$ is always uncountable. I know that if this was simply a projection onto ...
1
vote
1answer
84 views

Parametric to Implicit ( {x(t),y(t)} --> P(x,y) == 0 )

I have this parametric equations: $x(\theta) = r Cos(\theta) - \frac{v_{0}^2Cos(\theta)Sin(\theta)}{g}$ $y(\theta) = \frac{v_{0}^{2}Cos^2(\theta)}{2g} + r Sin(\theta)$ This is for $\theta \in ...
5
votes
1answer
98 views

Showing that if $N \le G$ is finite minimal normal with every simple homomorphic image abelian, then it is abelian itself

I've been working on the following problem, with no success so far: "Let $N$ be a finite minimal normal subgroup of a group $G$, and suppose $N$ has the property that every simple homomorphic image ...
1
vote
1answer
52 views

what is the answer of this Homogeneous equation?

Please help me. Question:what is the answer of this Homogeneous equation? $m^4(m+1)^3(m+5)(m^2+1)^2$ My guess is: $m^4: m=0 \Rightarrow y_1=e^0 , y_2=xe^0 , y_3=x^2e^0 , y^4=x^3e^0$ ...
2
votes
2answers
89 views

Inverse Laplace by phase shifting

Can anyone help me out here? I have to find the inverse laplace of $$ \frac{s (1-e^{-s/2})}{s^2+\pi^2}$$ Sorry it looks bad, I just don't know how to format. Here's the wolframalpha link: ...
0
votes
1answer
55 views

Invariants of point sets in an affine space

A distance between a pair of points in an affine space is invariant under translation, rotation and reflection. An angle in a triangle whose corners are tree points is also invariant under scaling. ...
1
vote
1answer
83 views

How to show that $u$ and $v$ have continuous partial derivatives at $(x_0,y_0)?$

How to show that $f:D(\subset\mathbb C)\to\mathbb C:(x,y)\mapsto u(x,y)+iv(x,y)$ is differentiable at $z_0=(x_0,y_0)\implies u$ and $v$ have continuous partial derivatives at $(x_0,y_0)?$ Added: I ...
0
votes
1answer
42 views

Geometrical meaning of multiplying a non negative matrix

Given two square matrix $X,A \in \mathbb{R}^{N \times N}$ $Y = A^TX$ What is the geometrical meaning of $Y$ if $X$ is non negative? What properties can we claim from $Y$?
1
vote
2answers
598 views

Resonance Frequencies of Oscillator

I understand that resonance is when the force term increases the natural oscillation of the system. In the next equation the oscillator has a natural frequency $\omega_0=\sqrt{\frac{k}{m}}$. But I ...
2
votes
1answer
4k views

Time Average of Cosine squared function

I've carried out the steps for the time average for $\cos^2x$ for limits $0$ to $T$. I've gotten : $\frac{1}{T}\left[\frac{1}{2}[T+\frac{1}{4}\sin2T\right]$ I'm trying to find the average over a ...
0
votes
5answers
113 views

Prove That $|a +b| = |a| +|b|$ if $a$ and $b$ Have Same Signs, And $|a +b| < |a| + |b|$ if $a$ and $b$ Have Opposite Signs

My Proof: $|a +b| = |a| +|b|$ ..... $(i)$ $|a +b| < |a| + |b|$ ..... $(i)$ If $'a'$ and $'b'$ have same signs: Let $a$ and $b$ be equal to $-x$. Replacing $a$ and $b$ with $-x$ in the equation ...
3
votes
1answer
327 views

Uniform convergence and interchange or sum and integral in Cauchy integral formula

I am working with the Cauchy Integral Formula for a matrix $A$ over a closed contour $C$. I have the following calculation, I believe this is correct, but I don't understand why I am allowed to ...
1
vote
6answers
735 views

What is the value of $2^{3000}$ [closed]

What is the value of $2^{3000}$? How to calculate it using a programming language like C#?
2
votes
1answer
71 views

Bijection between hom sets of $k$ - algebras

Let $R:= k[x_1,\ldots,x_r]$, $S:= k[x_{r+1},\ldots,x_{r+s}]$ and $Q:= k[x_1,\ldots,x_{r+s}]$. Let $I \subseteq R$ and $J \subseteq S$ be ideals. I have in texts in algebraic geometry that for any $k$ ...
2
votes
2answers
63 views

Generalization of metric that can induce ordering

I was wondering if there is some generalization of the concept metric to take positive and negative and zero values, such that it can induce an order on the metric space? If there already exists such ...

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