3
votes
3answers
163 views

Evaluate the infinite series $\sum_{r=1}^{\infty} \frac{2^{-r}}{r^{2}}$

My teacher posed an infinite series question to me today and I'm not quite sure how to start to go about it. $$\sum_{r=1}^{\infty} \dfrac{2^{-r}}{r^{2}}$$ Any hints would be much appreciated.
-1
votes
3answers
346 views

Good books on combinatorics

I have a math Ph.D. but my knowledge of combinatorics sucks and I simply don't know how to compute anything more complicated, i.e. what happens when we put restrictions on the allowed configurations ...
0
votes
1answer
201 views

least square solution for obtaining a line 3d by intersecting many planes

If I have been given 4 planes and I know only a point (just a point lie on the plane e.g. o1,o2, o3 & o4) and normal vector of each plane (n1, n2, n3 & n4). Then, by intersecting all 4 (or ...
1
vote
1answer
94 views

Prove that if R and S are nonzero rings then $R\times S$ is never a field.

This question has been asked on here before but I'm looking for some additional insights. The question comes from the section of the Dummit and Foote textbook on the Chinese Remainder Theorem. All of ...
1
vote
3answers
36 views

Find Simple Convergent sum

We have a sum $\sum_{0}^{\infty} 2^{n+1}(x+1)^{3n+1}$ I am asked to find all values of x s.t this sum converges then compute the sum. I have no idea what im doing so i did this. $\frac ...
6
votes
2answers
431 views

Relationship between connectedness and continuity

Let $f:\mathbb R^n\to \mathbb R$. $f$ is continuous, The graph of $f$ if connected in $\mathbb R^{n+1}$ We define "connected" to be cannot be separated by 2 disjoint non-empty open set. My ...
2
votes
1answer
88 views

Does $\operatorname{MSE}(\hat{\theta}) = \operatorname{Var}(\theta)+ \left(\operatorname{Bias}(\hat{\theta},\theta)\right)^2$?

We know that $\operatorname{MSE}(\hat{\theta})=\operatorname{E}\left[(\hat{\theta}-\theta)^2\right]$ and $\operatorname{MSE}(\hat{\theta})=\operatorname{Var}(\hat{\theta})+ ...
1
vote
0answers
85 views

How to prove this set has volume 0?

Let $A \subset \mathbb R^n$ be an open set and the function $g: A\to \mathbb R^n$ be a $C^1$ function. Let $$B=\{x\in A: Jg(x)=0\}$$ ($Jg$ means the Jacobian of $g$), then $g(B)$ is of volume zero. ...
6
votes
1answer
258 views

Colored balls puzzle

Imagine you have $n$ balls in a bag that are colored from $1$ to $n$. At each turn you take two balls at random out that have different colors and color one the color of the other. You then put them ...
0
votes
1answer
57 views

Eigenvectors of a Matrix Homework

Find eigenvectors of: \begin{pmatrix}4 & 2 \\ 2 & 3 \end{pmatrix} $$\det(A-\lambda I) = \lambda^2-7 \lambda+8=0 \iff \lambda_1=\frac{7+\sqrt{17}}{2} \ \lor \ \lambda_2= \frac{7-\sqrt{17}}{2}$$ ...
1
vote
0answers
35 views

Is this series of hermite polynomials $\sum_{n=0}^\infty \frac{(-1)^n}{\sqrt{2^n n!}} H_n(x)$ a known function?

Is this series of hermite polynomials $\sum_{n=0}^\infty \frac{(-1)^n}{\sqrt{2^n n!}} H_n(x)$ a known function? Thank you!
0
votes
0answers
92 views

Conditional probability measurement with n-grams

Let's say you have an alphabet with two words, a,b and a long string of these letters. You measure the probability of finding an $a$ or $b$ to be $p(a)=3/4$ and ...
1
vote
3answers
51 views

How can I solve this recurrence?

I have a weird recurrence relation and don't know how to solve it: $$a_n = pa_{n-1} + qa_{n+1} + cb_n$$ $$b_n = p'b_{n+1} + q'a_n$$ $$a_0 = 1$$ $p,q,c,p',q' \in [0,1]$ and $p+q+c=1,p'+q'=1$. Thanks ...
0
votes
1answer
86 views

How to calculate this integral?

Define $$F=(x^2+y-4,3xy,2xz+z^2)$$ Compute the integral of Curl F over the surface $x^2+y^2+z^2=16, z\geq 0$
9
votes
5answers
600 views

Matrices with eigenvalues 0 and 1

How can you describe all $2\times 2$ matrices whose eigenvalues are 0 and 1? My attempt: I know that 0 and 1 has to be solutions of the characteristic polynomial. And I've considered some examples ...
7
votes
5answers
294 views

Find the following integral: $\int {{{(\ln x)}^2}} dx$ by using the method of integration by parts

Find $\int {{{(\ln x)}^2}} dx$ by using the method of integration by parts. My attempt: $$\eqalign{ & \int {{{(\ln x)}^2}} dx = \int {2\ln x} dx \cr & u = \ln x,{\rm{ }}{{du} \over ...
23
votes
6answers
529 views

Examples of Diophantine equations with a large finite number of solutions

I wonder, if there are examples of Diophantine equations (or systems of such equations) with integer coefficients fitting on a few lines that have been proven to have a finite, but really huge number ...
5
votes
3answers
648 views

What free software can I use to solve a system of linear equations containing an unknown?

Question: What free software can I use to solve a system of linear equations $M\mathbf{x}=\mathbf{y}$ where the entries of $\mathbf{y}$ vary with an unknown quantity $n$? Presumably I could do ...
15
votes
4answers
501 views

What are the important theorems in the theory of dynamical systems?

I recently stumbled over the section about dynamical systems in my physics textbook. I noticed that, although most of the rest of the book was very rigorous, this part contained nearly no firm ...
3
votes
2answers
61 views

$\frac{S}{mS}$ is isomorphic to $\frac{R}{mR}$, where m is coprime to n.

I am trying to prove the following statement: Let R be a commutative ring with a unit element, and S be a subring of R of finite index n. Then $\frac{S}{mS}$ is isomorphic to $\frac{R}{mR}$, where m ...
2
votes
3answers
49 views

Testing for Orthogonality

When testing for orthogonality, why is it that when $\vec{v} \cdot \vec{w}=0$ do we say that these vectors are perpendicular? What does the dot product do for us that we end up with that conclusion?
2
votes
0answers
82 views

The closed orientable surface of genus 2

I have a very simple question to ask. What is a closed orientable surface of genus 2? Thank you in advance for helping me.
2
votes
1answer
142 views

Proof that differential of differential form $=0$ i.e $d(df) = 0$.

Let $f$ be a differentiable function on an open space $U \subset \mathbb{R}^n$. Proove that $d(df) = 0$. So my proof is: Let $$f = \sum c_{i_1, \cdots i_k}(x_*)dx_{i_1} \wedge \cdots ...
1
vote
1answer
44 views

Taking a derivative of the same equation in different form produces different results

I have a feeling this question is going to have an obvious answer, but I'm left a bit puzzled. I have the following equation: $$\tag 1 \frac{1}{S(x)} = \frac{W}{12}\cdot(1 - U(x)^2)$$, where $W$ is a ...
-1
votes
1answer
67 views

Does $S^1 \times S^2$ nontrivially covers itself?

I am not sure how to show whether or not $S^1 \times S^2$ nontrivially covers itself. Some help would be appreciated. Thanks
3
votes
1answer
110 views

$AB - BA = I$ in Hilbert Space [duplicate]

Let H be a Hilbert space and $A$ and $B$ be bounded operators in $H$. How can I prove that $AB - BA = I$ is not possible ? Probably this is as easy as in the matrices case, but I couldn't prove it. ...
0
votes
1answer
48 views

Prove that the $2$ form defines a symplectic structure

Prove that the $2$ form $$\omega = -2[(1+x_2^2)dx_1 \wedge dx_2 + dx_1 \wedge dx_3 + dx_3 \wedge dx_4]$$ defines a symplectic structure on $\mathbb{R}_x^4$. My definition of as ...
3
votes
1answer
353 views

Integrating a partial fraction with multiple quadratic denominators

When integrating a real rational fraction, you first break it into partial fractions. You then end up with fractions with linear denominators $\frac{A}{(x-b)^n}$, which are easy. You also end up with ...
0
votes
1answer
18 views

Clues to prove average in T is minor or equal than average in a smaller inner interval.

Suppose I want to prove (or disprove) this assertion Let $f$ be a discrete function, $T,h,k$ are constants So these terms are averages over $T$ and over $h$ $\sum\limits_{i=0}^{T}\frac {f(i)}{T}$ ...
2
votes
1answer
86 views

Breaking a contour integral into 3 separate contours?

We can try to integrate the following function around a counter-clockwise circular contour: $$\frac{x^3}{(x-1)(x-2)(x-3)}$$ Can someone show how to use the Cauchy–Goursat theorem (explained here and ...
3
votes
1answer
78 views

Stability under supremum of sets of social choice function with single peaked preferences

Here is a question emerging from reading Moulin, H. (1980). On strategy-proofness and single peakedness. Public Choice, 35(4), 437–455. The setting is as follows: A non-empty finite set of ...
4
votes
2answers
100 views

Is the validity of measuring area by approximation an assumption of calculus?

The assumption that if you subdivide an area into more and more sub intervals, the approximation gets better and better. Has this been formally proved, or is it just intuition? Thanks!
2
votes
1answer
154 views

Partial derivative chain rules.

$$x= \cosθ -r\sinθ$$ $$y= \sinθ + r\cosθ $$ Show that, $$ \frac{\partial^2θ }{\partial x^2}= \frac{\cosθ }{ r^3} (\cosθ -2r\sinθ)$$ Please Help :) I used the chain rule: $$ \frac{\partialθ ...
0
votes
1answer
91 views

General solution and intervals (over which the general solution is defined)

So I did a homework and I got x^2/(x+1)e^x + C/(x+1)e^x for all R (except -1), but they say it's for -1 < x < oo+. I don't understand how that makes any sense. Is there something I am not ...
3
votes
1answer
44 views

Jacobian of $f(|x|)x$

Suppose one has a $C^1$ function $f:\mathbb{R} \rightarrow [0,\infty)$. A throwaway line in a paper I'm reading claims \begin{equation} \det \left[ \nabla \left( f(|\vec{x}|)\vec{x} \right) \right] = ...
0
votes
1answer
96 views

Multivariable integral

What is the result of the following integral? $$ 2 \cdot \int_0^{\infty} \frac{1}{\sqrt{2\pi s}}e^{-\frac{b^2}{2s}} \int_{1-s}^{\infty}\frac{b}{\sqrt{2 \pi u^3}}e^{-\frac{b^2}{2u}} du db$$ where $0 ...
7
votes
2answers
186 views

How to prove $n$ is prime?

Let $n \gt 1$ and $$\left\lfloor\frac n 1\right\rfloor + \left\lfloor\frac n2\right\rfloor + \ldots + \left\lfloor\frac n n\right\rfloor = \left\lfloor\frac{n-1}{1}\right\rfloor + ...
3
votes
2answers
100 views

PDE: Why do they have the wrong units?

Take a look, for example, at the telegrapher's equations (let's look at the voltage one). They have the wrong units. Equation $u_{x} = Li_{t} + Ri$ *where $u$ is potential in volts $V$, $L$ is ...
1
vote
1answer
58 views

Homeomorphism on Identification Space

Let $\sim$ be and equivalence relation on the unit line $X=[0,1]$ defined by $x\sim y$ if either $x=y$ or $\textbf{both}$ $x$ and $y$ $\in$ {${0,\frac{1}{2},1}$}. Construct a homeomorphism ...
3
votes
5answers
239 views

Why isn't this limit equal to $0$?

$f(2)=4$, $g(2)=9$, $f'(2)=g'(2)$. $ \displaystyle \lim_{x \to 2} \frac{ \sqrt{f(x)}-2} { \sqrt{g(x)}-2} $. Why isn't this limit equal to $0$? Since $f$ and $g$ are differentiable at $x=2$, that ...
1
vote
0answers
60 views

How does the Tarski axiom relate to the Grothendieck universe?

It's claimed, e.g. in the formulation of Tarski–Grothendieck set theory, that the Tarski axiom implies that for every set, there is a Grothendieck universe containing it. However, I can't see how ...
1
vote
3answers
77 views

Solve the following integral using substitution only?

Can you solve the following integral using only substitution? $$\int \dfrac{dx}{\left(\sqrt{x^2-4}\right)^3}$$ I saw a solution to this which began with $x=2\sec(u)$, but is there another way to solve ...
3
votes
0answers
80 views

why is an open faithfully-flat morphism fpqc?

Why is an open faithfully-flat morphism fpqc? In other words, why must an open faithfully flat morphism $X\rightarrow Y$ have the property that around every $x\in X$, there is an open nbhd $U$ of ...
0
votes
1answer
17 views

normal vector condition on 1 component

If you have two normal random variables $Z_1$ and $Z_2$ possibly correlated namely you have a multivariate normal distribution $Z= (Z_1, Z_2)$ What is the conditional distribution $(Z_1|Z_2 =a)$? ...
2
votes
1answer
120 views

Finding distributional limit

How to find $\lim_{\varepsilon\rightarrow 0+}f_{\varepsilon}$ in $D'(R)$, if $f_\varepsilon$ is defined as: $f_\varepsilon(x)=\frac{1}{\varepsilon^3}$ for ...
3
votes
3answers
239 views

Topological boundary vs geometric boundary

Let $M_1=B((0,0),1)=\{(x,y) \mid x^2+y^2<1\}$ $M_2=\{(x,y) \mid x^2+y^2\le1\}$ What are the interior of $M_1$ and $M_2$ ? And What are the boundary of $M_1$ and $M_2$ ? How to find them? ...
2
votes
1answer
126 views

Sequential criterion for differentiability

Is there a sequential criterion for differentiability,just like there is one for continuity ? If not then,why so ? I'm studying undergraduate real analysis and haven't really come across one. ...
3
votes
1answer
608 views

Change of variables in a complex integral

I want to evaluate this integral using Residue Theorem $$\int_C^\ \frac{4z} {z^4 +6z^2 +1} dz = $$ $$ C : |z| = 1 $$ so I substitute letting $$\ W = z ^ {2 } $$ $$ dw = 2z dz $$ and the ...
1
vote
1answer
245 views

Continuity of a Function in Terms of Closure and Interior

I've managed to show that the following are equivalent (where $f^*$ and $f_*$ are the preimage and image of f respectively): $\bullet$ $f:X \rightarrow Y$ is continuous $\bullet$ $f^*(S^{\circ}) ...
1
vote
1answer
206 views

Weight enumerator of the Hamming code

Let $H_r$ be the usual Hamming code of length $2^r-1$. What is the weight enumerator of $H_r^\perp$? Using this find an expression for the weight enumerator of $H_r$. (we are in binary case)

15 30 50 per page