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3
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36 views

Line integral parametrization

We are given the field $\textbf{F}(x,y)=(x-y)\textbf{i}+xy\textbf{j}$ and C being $\frac{3}{4}$ of a circle of radius $2$ centered at the origin traversed from $(2,0)$ to $(0,-2)$. $$\textbf{F}(x,y) ...
3
votes
0answers
58 views

Show that a certain operator is symmetric

I am trying to prove that the operator $L^2 = -\partial_\theta^2 - \cot\theta\,\partial_\theta - \frac{1}{\sin^2\theta}\partial_\phi^2$ fulfills the following property: For $y_{l,m} = ...
3
votes
0answers
34 views

An undirected graph $G$ can be decomposed into simple edge-disjoint cycles if and only if all of its vertices have even degree.

Research effort: $\rightarrow)$ I think this is relatively easy. $\leftarrow)$ Let $G = (V,E)$, let $w$ be any vertex of $G$, given that all the vertex have even degree, I'm assured that I can ...
3
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0answers
35 views

Prove Differentiation Multivariable

Given $f(x,y) = \frac{ xy^2}{x^2 +y^2}$ From defintion we know it is differentiable if: $\lim_{h\to 0}\frac{F(X+h)-F(X)-c*h}{|h|}$ exists, where $c$ is the gradient of the function. I have ...
3
votes
0answers
48 views

Does this integral variable change makes sense to you?

I was Reading a book about calculus when I've found this part about variable substitution in integrals: Consider $f$ defined in na interval $I$. Suppose that $x =\phi(u)$ is inversible, and its ...
3
votes
0answers
50 views

Link between a cubic polynomial and a trig identity

Alright, so I am told to prove that: $$\tan (3A) = \frac{3\tan(A)-\tan^3(A)}{1-3\tan^2(A)}$$ This can be pretty easily done by applying the $\tan$ addition formula, taking the angles $2A$ and $A$, ...
3
votes
0answers
65 views

invariants of group action by algebra automorphism

I am trying to prove the following statement, but I'm having a lot of trouble with it: Let $k$ be an infinite field. Let $A$ be a commutative $k$-algebra. Let $G$, a group, act on $A$ by algebra ...
3
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34 views

Help proving(well, disproving) the convergence of $\sin^3(x)$

So I'm stuck on a question where it's asking for the power series and radius of convergence of $\sin^3(x)$ I've done the power series ok, but my problem is that when I apply the ratio test it's ...
3
votes
0answers
51 views

Bessel J function problem

Let $\xi_{0k}$ be the k-th positive zero of $J_{0}$ Bessel function. Determine the coefficients $c_k$, so that $1 = \sum^{\infty}_{k=1} c_kJ_0(\frac{x \xi_{0k}}{2})$. I don't see what to do, is ...
3
votes
0answers
55 views

Double lines in $X$, implies $X \cong \mathbb{P}^{n}$

Let $X$ be a smooth, complex, projective variety. How to prove that if through two general points of $X$ there exists a double line, then $X \cong \mathbb{P}_{\mathbb{C}}^{n}$?
3
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54 views

Prove that $e^{\ln{z}}=z$ from the power series

For my course in complex analysis we have to prove that the trivial relation $e^{\ln{z}}=z$. We are given the series for $\ln z$: $$f(w)=\sum_{n=0}^\infty (-1)^{n+1}\frac{w^n}{n}$$ $$\ln z = ...
3
votes
0answers
49 views

Prove that if $f:D^2\to D^2$ is a homeomorphism, then $f(S^1)=S^1$

I've already proved that set of points $z\in D^2$ such that $D^2-z$ is simply connected is precisely $S^1$. Now from this, I'm supposed to conclude that if $f:D^2\to D^2$ is a homeomorphism, then ...
3
votes
0answers
54 views

Application of Dirichlet Theorem in AP to elementary number theory problems.

I have learnt this theorem in my class, however, "elementary" examples are very limited. (focusing more on analytic machinery) Are there any interesting applications to elementary number theory that ...
3
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0answers
56 views

How do I solve this differential equation?

$y^{(7)}+4y^{(6)}+8y^{(5)}+9y^{(4)}+8y^{(3)}+8y^{(2)}+8y^{(1)}+4y=e^{-x} (5sinx-cosx) $ The characteristic equation $ \lambda ^7 +4\lambda ^6+8\lambda ^5+ 9\lambda ^4 +8\lambda ^3+8\lambda ...
3
votes
0answers
51 views

Fundamental Solution of a Nonlinear ODE (using Riccati Transformation & Wronskian)

I am given the differential equation: \begin{equation*} y^{\prime}(t) = y(t)^{2} + 2\sin(t)\cos(t) - \sin^{4}(t) \end{equation*} and one solution $y_{1}(t) = \sin^{2}(t)$. I wish to find a second ...
3
votes
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74 views

L'Hopital quicky

suppose L'Hopital applies and $$\lim_{x\to\infty}\frac{f(x)}{g(x)} = \lim\frac{f'(x)}{g'(x)}$$ under what conditions is it true then that $$\lim_{x\to\infty}\frac{\frac{f(x)}{g(x)} }{ ...
3
votes
0answers
45 views

Showing iterates of a complex function on the upper half plane converges uniformly on compact sets

The following is an irksome problem that my complex analysis class is having trouble solving: Let $*$ be an operator that takes a function $F:\mathcal{H}\to\mathcal{H}$ to a function ...
3
votes
0answers
132 views

Prove $f_\infty: A_\infty \rightarrow B_\infty$ is a bijection

Update: I was given some hints at how to approach this problem $A_\infty $ and $B_\infty$ are sets, not maps. (which is strange because there are function definitions coming into play here) The ...
3
votes
0answers
67 views

Differential Equation has a unique solution periodic

Let $A(t)$ continuous and periodic of period $S$ in $\mathbb{R}$. Suppose $x' = Ax$ has $\varphi \equiv 0$ as the only periodic solution of period $S$. Show that there exists $\delta> 0$ such that ...
3
votes
0answers
39 views

Exercise concerning logarithms…

I have such a problem: find all the values of real parameter "a", for which the following inequality is true for any "x" that belongs to R. I will show you my solution, and please can you verify ...
3
votes
0answers
60 views

Josephus Variant

I set myself the challenge of trying to solve a variant of trying to solve a variant of the josephus problem where instead of killing every second person, every third person dies. The formula for the ...
3
votes
0answers
67 views

An interesting system of equations

We have the following system with a and b, real numbers: $ax+y + z =4$ $x+2y+3z=6$ $3x-y-2z=b$ Show that $\forall a \in \mathbb{Z} $ there is a $b \in \mathbb{Z}$ such that the system admits a ...
3
votes
0answers
43 views

Has anyone used Complex Analysis in the Spirit of Lipman Bers as their textbook?

I have free access to many Springer books from my library, which includes Complex Analysis In the Spirit of Lipman Bers. From what I've seen, it's a decent book that introduces the subject. ...
3
votes
0answers
44 views

Conditional expectation and coupled set of ODEs

How to find a coupled set of ODEs and initial conditions for the deterministic functions $a$ and $b$ such that $$\mathbb{E}\left[e^{-\int_{t}^{T} W^2(u)du} | \mathcal{F(t)}\right] = e^{-a(T-t) - ...
3
votes
0answers
26 views

normal of a function

Function $$\mathbf{r}(u,v)=(2u+2v)\mathbf{i}+(-3+v^2)\mathbf{j}+(2u^2)\mathbf{k}$$ is a parametrization of a surface. What's the normal vector of this surface at the point $(u,v)=(1,-2)$. What I got: ...
3
votes
0answers
57 views

Are any solutions lost when solving non-exact differential equations?

I have just started studying differential equations, one of the problems I found while I was practicing is "Consider the equation $$ (y^2 + 2xy)dx - x^2 dy=0 $$ (a) Show that this equation is not ...
3
votes
0answers
57 views

Proof about sequences of functions.

Is this proof correct? If $\{f_{n}\}$ is a sequence of functions in $C(X,Y)$, $X$ compact, $Y$ complete, and the sequence converges, to $f$, then $K=(\bigcup\{f_n\})\cup \{f\}$ is closed. Proof. ...
3
votes
0answers
41 views

Notation of quotient groups

So I'm determining the quotient group of $(E,+)$ in $(Z,+)$ where E is even int. and Z is int. I know sort of what is happening, I split the group $Z$ into evens and odds (2 sets) and as we are under ...
3
votes
0answers
66 views

Conditions to make a function a metric on $\mathbb{R}$?

Let $f:\mathbb{R}\rightarrow \mathbb{R}$. What conditions ensure that $d(x,y)=|f(x)-f(y)|$ defines a metric on $\mathbb{R}$ Let $g:[0,\infty) \to \mathbb{R}$. What conditions on $g$ ensure that ...
3
votes
0answers
77 views

Combinatorial Proofs of Two Identities

I need a combinatorial proof for these two identites: (a) $\sum_{k=0}^{n} \binom{n}{k}^2 x^k = \sum_{k=0}^{n} \binom{n}{k} \binom{2n-k}{n} (x-1)^k$ (b) $\binom{n}{k} ^2 = \sum_{l=0}^{n-k} (-1)^l ...
3
votes
0answers
68 views

Understanding the setup for the probability that $Ax^2+Bx+C$ has real roots if A, B, and C are random variables uniformly distribted over (0,1).

Suppose that $A, B,$ and $C$ are independent random variables, each being uniformly distributed over $(0,1)$. What is the probability that $Ax^2 + Bx + C$ has real roots? First, I set $P(B^2 - 4AC ...
3
votes
0answers
49 views

Does metric on a vector space form a topologycal vector space?

Let $X$ be a vector space and that on $X$ there exists a metric $d$. Denote $\tau$ be the topology induce by metric $d$. Do we can conclude that $(X,\tau)$ is a topological vector space? I know that ...
3
votes
0answers
26 views

Check my proof: Prove that if $f$ is defined as having a positive disntinuity at $c$ and $0$ otherwise on [a,b], it is Darboux integrable

Prove that if $f$ is defined as having a positive discontinuity at $c$ and $0$ otherwise on [a,b], it is Darboux integrable and its integral is 0. $\forall \epsilon>0,$ choose ...
3
votes
0answers
73 views

Regularity of Daubechies wavelet

I am reading the book Wavelets: Theory and applications by A. K. Louis, D. Maass, A. Rieder ...
3
votes
0answers
48 views

Number of points over elliptic curve is p+1 given…

Suppose that -1 is not a square in $\mathbb{Z_p}$. Show that the number of points on the elliptic curve $y^2=x^3+ax$ over $\mathbb{Z_p}$ is $p+1$. Hint: Use the fact that $x^3+ax$ is an odd function. ...
3
votes
0answers
38 views

Quotient Spaces

Find a pair of topological spaces $X$ and $Y$ with $G$ acting on them such that $X/G \cong Y/ G$ while $X$ is not homeomorphic to $Y$. I know $\mathbb{R}/\mathbb{Z}_2$ and $[0,1)/\mathbb{Z}_2$ is ...
3
votes
0answers
115 views

Proving a language is not recognizable

I have the following question that I just want to verify I have done correctly. Let $L$, $L_1$, $L_2$ $\subseteq \Sigma^*$ such that $L = L_1 \cup L_2$, and $L_2$ is decidable. Prove that if $L$ is ...
3
votes
0answers
65 views

Find the minimum value of: $P=\dfrac{\sqrt{ab(a+b)}+\sqrt{bc(b+c)}+\sqrt{ac(c+a)}}{\sqrt{ab+bc+ca}}$

Let $a,b,c\ge0$ such that: $(a+b)(b+c)(c+a)=1$. Find the minimum value of: $$P=\dfrac{\sqrt{ab(a+b)}+\sqrt{bc(b+c)}+\sqrt{ac(c+a)}}{\sqrt{ab+bc+ca}}$$. I've tried many things but all failed. Please ...
3
votes
0answers
50 views

Prove that there is a real number $r>0$ such that…

Prove that there is a real number $r>0$ such that: There is no point in $\mathbb{R}^3$ with 3 rational coordinates, whose distance from $(0,0,0)$ equals $r$. In other words, if we build a sphere ...
3
votes
0answers
24 views

Solving a system of partial differential equations

We try to solve the partial differential equations for the unknown 2-variables function $f$: 1) $\dfrac{\partial f}{\partial y}=0$ 2) The system of equations $\dfrac{\partial^2 f}{\partial^2 y}=2xy$ ...
3
votes
0answers
38 views

Qualitative dependence of solution to second-order matrix differential equation on eigenvalues

Suppose we have a matrix differential equation in $\vec{x}(t)=\left(\begin{smallmatrix}x_{1}(t) \\ \vdots \\ x_{n}(t)\end{smallmatrix}\right)$, such that: ...
3
votes
0answers
89 views

Question about orthogonal transformation / orthogonal matrices

I have a question about orthogonal transformations. If $T$ is an orthogonal transformation from $V$ to $V$, should the representation matrix with respect to any orthonormal basis of any inner product ...
3
votes
0answers
77 views

Why are independent random variables always uncorrelated?

I'm trying to show that, if $X$ and $Y$ are two independent random variables, then $\mathbb E(XY)=\mathbb E(X)\mathbb E(Y)$. For the purposes of this question, assume that $X$ and $Y$ are ...
3
votes
0answers
100 views

A formula for the holomorphic sectional curvature.

I tried to compute the holomorphic sectional curvature of a hypersurface of ($\mathbb{C}^{n+1}$, std metric, i), but I failed. $$ V_{k}=\{(z_{0},...,z_{n})\in \mathbb{C}^{n+1} | \sum_{j}z_{j}^{k}=0\}- ...
3
votes
0answers
100 views

Knowing when to use Green/Stokes/Divergence theorem to evaluate line/surface integrals

$\newcommand{\mbf}{\mathbf}$ Evaluate $$ \iint \limits_{S} \mbf{F} \cdot d \mbf{S} $$ where $\mbf{F} = 3xy^2 \mbf{i} + 3x^2y \mbf{j} + z^3 \mbf{k}$ and $S$ is the surface of the unit sphere. I ...
3
votes
0answers
435 views

How to find probability distribution function given the Moment Generating Function

After searching, I found two questions like mine, but didn't see my answer to my question. Finding a probability distribution given the moment generating function Finding probability using ...
3
votes
0answers
42 views

Second derivative

I have this functional on $H^1_0$ defined by $J(u)=\frac12||u||^2-\int_0^1 F(t,u(t)) dt $ where $F(t,u(t))=\int_0^u f(t,s) ds $ and i have $J'(u)h= \int_0^1u'(t)h'(t) dt - \int_0^1f(t,u(t)) h(t) ...
3
votes
0answers
70 views

Products of homology groups

In the topology course I attend we work with the following, rather unusual definition of homology groups: For topological spaces $X,Y$ define the presheaf $\mathcal{F}_Y$ on $X$ as follows: Let ...
3
votes
0answers
54 views

How many elements are there in total?

Suppose that we have a collection of 12 sets, each of which has 8 elements. Every pair of sets shares 6 elements, any collection of 3 sets shares 4 elements, and no collection of 4 sets shares any ...
3
votes
0answers
43 views

Does convergence of $(a_{5n})$ and $(a_{n+1} - a_n)\rightarrow0$ imply convergence of $a_n$

As I wrote in the title: Does convergence of $(a_{5n})$ and the fact that $(a_{n+1} - a_n)\rightarrow0$ imply convergence of $a_n$. I understand that convergence of $(a_{5n})$ means that $(a_n)$can ...