-1
votes
1answer
67 views

Using Abel's test for negative coefficients

For the power series representation of $$\log(1-z)=\sum_{n=1}^{\infty}=\frac{-z^n}{n},$$ is Abel's test still valid to conclude that the $\log(1-z)$ will converge everywhere on the unit disk, ...
3
votes
4answers
2k views

Prove that the real vector space consisting of all continuous, real-valued functions on the interval $[0,1]$ is infinite-dimensional.

Prove that the real vector space consisting of all continuous, real-valued functions on the interval $[0,1]$ is infinite-dimensional. Clearly it's infinite dimensional, because if you consider say $P ...
0
votes
1answer
44 views

What is the name of this delta operator

In Euler-Lagrange Equation: $${\delta \over \delta y}F \equiv {\partial F \over \partial y}- {d \over dx} ({\partial F \over \partial {y'}})$$ What is the name of operator $\delta$ here?
0
votes
1answer
69 views

How to simplify the following modulo operation

I am trying to find the modulo of an expression. All I know is that (a+b) mod N = ((a mod N) + (b mod N)) mod N How do I use it to simplify the following modulo ...
3
votes
1answer
116 views

If $f(x)$ is positive and decreasing, can $xf(x)$ have more than one maxima?

Assume that $f(x)$ is positive and decreasing on $[0,1]$ with $f(0)=1$ and $f(1)=0$. We see that $xf(x)$ is 0 at 0 and 1 and is positive in between so it must have a maxima. Is it possible that ...
1
vote
1answer
52 views

Prove the theorem: if $ (a_{n}) \rightarrow \infty $ and $ d < 0 $ then $ (da_{n}) \rightarrow -\infty$

I've attempted this question and my solution is below, just wanted to check that this is correct and that it makes sense. Let c < 0 be arbitrary, and let $N = \frac{c}{d}$. Then $\forall n > ...
0
votes
1answer
175 views

Is my alternative proof that the intersection of nested compact sets is nonempty valid?

Let $\{A_n\}_{n=1}^\infty$ be a collection of nested closed sets in a compact space $X$. Since $A_n$ is closed, it is compact, and consequently limit point compact. Let $\varepsilon > 0$ and define ...
1
vote
3answers
7k views

Given $f(x)$ its inverse function, domain and range

$f(x) = \frac{{2x + 3}}{{x - 1}},\left[ {x \in {R},x > 1} \right]$ I've got the inverse function to be: ${f^{ - 1}}(x) = \frac{{x + 3}}{{x - 2}}$ How would I go about working out the range and ...
-1
votes
2answers
1k views

If a linear system has no free variables, then it is consistent: Why is the statement false?

I recently got this question wrong on a test, and I have no clue why. How can a system be inconsistent if there are no free variables?
2
votes
1answer
63 views

Doubt in proof of special case of implicit function theorem

I've been studying the implicit and inverse functions theorems and I've started with one special case of the implicit function theorem. The book I'm reading states the theorem as follows: ...
2
votes
1answer
811 views

Why is the matrix norm $||A||_1$ maximum absolute column sum of the matrix.

By definition, we have: $||V||_p=\sqrt[p]{\displaystyle \sum_{i=1}^{n}|v_i|^p}$ $||A||_p=sup\frac{||Ax||_p}{||x||_p}$, $x\not=0$ and if $A$ is finite, we change sup to max. However I don't really ...
1
vote
1answer
116 views

Zariski tangent space and $K[\epsilon]/(\epsilon^2)$ [duplicate]

I want to prove that the Zariski tangent space at $x\in X$ ($X$ is an affine scheme) is isomorphic to $Hom_K(X,K[\epsilon]/(\epsilon^2))$ (K is the residue field at $x$). I want to say that ...
2
votes
1answer
151 views

spanning trees of an edge transitive graph

Let $G$ be an edge transitive graph. Let $t(G)$ be the number f spanning trees on $G$. Show that each edge lies in exactly $\tfrac{(n-1)t(G)}{m}$ spanning trees. Where $|V(G)|=n$ and $|E(G)=m$. ...
2
votes
1answer
105 views

A question about an epsilon-delta proof

Currently, I am stuck on a question: Let $ g : [ 0 , \infty ) \mapsto \mathbb{R} $ be defined by $g(x)= \left\{ \begin{array}{ll} x^2 & \mbox{if } 0 \leq x \leq 1 \\ 3x & \mbox{if } x ...
5
votes
1answer
106 views

Maximize $W(x) - (\ln(x) - \ln{\ln{x}})$

How can you maximize $f(x) = W(x) - (\ln(x) - \ln{\ln{x}})$ for $x\geq 2$? Numerically the answer seems to be at around $x \approx 41$ where you get $f(41) \approx 0.31$. Mathematica suggests the ...
2
votes
3answers
98 views

Suggestions for the optimal estimator in “one-shot” prediction problems?

Assume you have a prediction distribution for a quantity. What point on this distribution should you use if the process you are predicting will end after the next observation and you want to be within ...
1
vote
0answers
54 views

Question on Fourier Transform

Fourier transform on $f$: $$\hat{f}(\xi)=\int_{\mathbb{R}^d}f(x)e^{-2\pi ix\cdot\xi}dx$$ $\xi\in\mathbb{R}^d$. How to show that $$\hat{f}(\xi)=\frac{1}{2}\int_{\mathbb{R}^d}[f(x)-f(x-\xi')]e^{-2\pi ...
0
votes
2answers
74 views

Why can a polynomial that has only one repeated solution be factorised?

For example $f(x)=2x^2+11x+12$ can be factorised as: $$(2x-3)(x+4)$$ but $f(x)=0$ has only one solution, $x=0$. This confuses me because $(2x-3)(x+4)=0$ has two, namely $x=\frac{3}{2}$ and $x=-4$. ...
0
votes
1answer
33 views

topological space in which compace set is closed

I saw the bellow example in this site that "The product of two $T_B$ spaces need not be $T_B$. Let $\Bbb Q^*$ be the one-point compactification of the rationals; then $\Bbb Q^*$ is $KC$, but $X=\Bbb ...
0
votes
2answers
111 views

Lattice property of coprime integers

I was reading on the Wikipedia page for coprime numbers that (for $a \gt b$), gcd($a,b$)$=1$ if and only if the diagonal connecting $(0,0)$ and $(a,b)$ does not cross through any lattice points ...
0
votes
1answer
24 views

How to test non-negativity of linear functions on homogeneous semi-algebraic sets?

Fix a homogeneous semi-algebraic set $S \subset \mathbb{R}^{n}$, by which I mean that the set $S$ is defined by inequalities $f_{1},...,f_{k} \geq 0$, where $f_{1},...,f_{k}$ are homogeneous ...
3
votes
1answer
58 views

Differentiation of $x!$, where $x\in \mathbb{N}+\{0\}$

Calculation of $\displaystyle \frac{d}{dx}(x!) = $, where $x\in \mathbb{N}+\{0\}$ My Try:: We Know that $x! = (x)\cdot (x-1)\cdot(x-2)...........(3)\cdot(2)\cdot(1)$ Now Taking $\bf{\ln}$ on both ...
0
votes
3answers
66 views

How can a subset be disjointed?

I have the proof: Suppose A,B, and C are sets. Prove that C⊆A∆B iff C⊆A∪B and A∩B∩C=∅. If I suppose that C ⊆A∆B, how can the three sets be disjointed?
5
votes
1answer
126 views

Good upper bound for $(1-x)^r$

The Bernoulli's inequality gives a lower bound on numbers of the form $(1-x)^r$: $$(1-x)^r\geq 1-rx$$ for integer $r\geq 0$ and real number $0<x<1$. Is there a corresponding upper bound for ...
1
vote
0answers
53 views

How to solve linear system of equations in 1 and inf-norm?

I have the problem to find a linear program that is equivalent to solving the problem that finds a minimum for $||Ax-b||_1$ and $||Ax-b||_{\infty}$. We defined a linear program as follows: $min_{x} ...
0
votes
1answer
43 views

Permanent of a linear transformation

Permanent of a linear transformation. I'm guessing that there is not as changing the basis would change it? Is the permanent of a matrix changed by being acted on by change of basis matrices?
0
votes
1answer
59 views

Computing the limit.

Studying for a midterm. Compute the following limit: $$\lim_{x\to 4} \frac{x+4}{x^2+3x-4}$$ Factor the denominator: $$\lim_{x\to 4} \frac{x+4}{(x+4)(x-1)}$$ The $(x+4)s$ cancel out: $$\lim_{x\to ...
0
votes
1answer
27 views

Does for $u\in L^1(\Omega)$ and every $t$ also hold $\nabla u \cdot 1_{\{u=t\}}=0$ a.e.?

The result is known if $u$ is more regular e.g. $u \in W^{1,1}(\Omega)$. Is it also possible to extend such an result to mere integrable or even just measurable functions? Unfortunately the result ...
0
votes
2answers
342 views

If $f$ is a function of moderate decrease then $\delta \int f(\delta x) dx = \int f(x) dx$

A function of moderate decrease is a map from $\mathbb{R}$ into $\mathbb{C}$ such that there exists $A \in \mathbb{R}$ such that $\forall x\in \mathbb{R}, \ |f(x)| \lt \frac{A}{1 + |x|^{1+\epsilon}}$. ...
0
votes
1answer
33 views

Given a set of numbers $x_1, x_2, \ldots, x_k$, what is the largest number $h$ such that $x_i \bmod{h} = 0$ for all $i$?

I am solving a system of differential equations with respect to length, let's say 0 to $x_{max} = 10$ meters. Now, I want to choose an integration step such that my step will land on each of the ...
0
votes
0answers
126 views

Find vertex of a parallelogram/parallelepiped/parallelotope with minimum distance to a point

Suppose you have a parallelogram and a point. It's easy to tell which of the parallelogram's vertices is closest to the point (Euclidean distance) by checking the distance for every vertex - but this ...
2
votes
1answer
69 views

A question from Eisenbud, Commutative Algebra

On page 35, the proof of corollary 1.8: If k is an algebraically closed field and A is a k-algebra, then A = A(X) for some algebraic set X iff A is reduced and finitely generated as a k-algebra. In ...
1
vote
2answers
173 views

Using the Squeeze Theorem, Find the Limit of $\lim_{x\to 3} (x^2-2x-3)^2\cos\left(\pi \over x-3\right)$

Using the Squeeze Theorem, how do I find: $$\lim_{x\to 3} (x^2-2x-3)^2\cos\left(\pi \over x-3\right)$$ I thought I knew the Squeeze Theorem, but I haven't encountered anything like this yet, so I ...
0
votes
1answer
26 views

Let $( X,\tau)$ be a countable Hausdorff space

Let $( X,\tau)$ be a countable Hausdorff space. Is there a topology $ \sigma \subset \tau$ s.t $( X, \sigma)$ is a second countable Hausdorff space?
0
votes
2answers
75 views

Prove the identity

$$\cos \frac{x}{2} \cdot \cos \frac{x}{4} \cdot \cos \frac{x}{8} = \frac{\sin x}{ 8\sin \frac{x}{8}}$$ Conjecture a generalization of this result and prove its correctness by induction. Ps: I have ...
1
vote
2answers
39 views

finding range of function of three variables

Three real numbers $x$, $y$, $z$ satisfy the following conditions. $x^{2}+y^{2}+z^{2}=1~$, $~y+z=1$ Find the range of $~x^{3}+y^{3}+z^{3}~$ without calculus. I solved this problem only with ...
1
vote
2answers
54 views

Finding the limit $\lim_{x\to-\infty} (2x)/(2x-1)^2$.

Studying for a midterm: Let $f(x)=\frac{2x}{(2x-1)^2}$ Then $\lim_{x\to-\infty} f(x)$ is: Now keep in mind I'm shaky on how to do infinity limits. I have $f(x)=\frac{2x}{(2x-1)^2}$ Remove x by ...
0
votes
1answer
27 views

Why does $\exp\left[W\left(b\left(\ln{n}\right)^2\right) - \ln{b} - \ln{\ln{n}}\right] = \frac{\ln{n}}{W(b\ln^2{n})}$?

Why does $$\exp\left[W\left(b\left(\ln{n}\right)^2\right) \; - \; \ln{b} - \ln{\ln{n}}\right] = \frac{\ln{n}}{W(b\ln^2{n})}\;?$$ $W$ is the Lambert-W function and all variables are real and positive. ...
1
vote
2answers
261 views

Find the general solution of the given second order differential equation.

Find the general solution of the given second order differential equation. $$4y''+y'=0$$ This was my procedure to solving this problem: $\chi(r)=4r^2+r=0$ $r(4r+1)=0$ $r=0, -\frac14$ $y_1=e^{0x}, ...
2
votes
1answer
123 views

Is the Cartesian product of an infinite number of $\mathbb{Z}^+$ countable?

We know the Cartesian product of a finite number of $\mathbb{Z}^+$ is countable, for example, $\mathbb{Z}^+ \times \mathbb{Z}^+ \times \mathbb{Z}^+$, because, let $m, n, p$ be from each of the ...
0
votes
1answer
63 views

Proving that $T(B(x,2\epsilon))\cap B(y,2\epsilon) \neq \emptyset $

$H$ Hilbert space. $x,y \in H$ and $T\in L(H)$ 1) $T(B(x,\epsilon))\cap B(0,\epsilon) \neq \emptyset $ 2) $T(B(0,\epsilon))\cap B(y,\epsilon) \neq \emptyset $ 3) $T(B(x,2\epsilon))\cap ...
1
vote
2answers
51 views

Ideals in $F[X]$ are of the form $(f(x))$ where $f$ can be chosen to be monic. How?

I am reading a statement whereby it says that In $F[X]$, where $F$ is a field, any ideal is of the form $(f(x))$ where $f$ can be chosen to be monic. I don't get this part of the statement '$f$ can ...
2
votes
1answer
406 views

Best book to learn Affine Geometry?

I'm going to learn Affine plane as well as affine Geometry. Unfortunately, my text book (not in English) is not good at all, so please recommend some book you think it's good for self-learning (and ...
0
votes
1answer
270 views

How do I proove the symmetry of Metric space?

I'm looking for the proof of that: d(x,y) = d (y,x). I know that I have to use the "non-negativity" and "triangle inequality" but I don't know how to combine them to get the result.
1
vote
1answer
81 views

A question about Pearson correlation coefficient

Suppose that we have two vectors $x=(x_1,\ldots,x_n),y=(y_1,\ldots,y_n)$ is the following correct about their Pearson correlation coefficient? $\operatorname{corr}(x,y)=\operatorname{corr}(x+a,y+b)$ ...
0
votes
2answers
728 views

Dot Product and Cross Product: Solving a set of Simultaneous Vector Equations

I was wondering if $\mathbf{A}\times\mathbf{C}=\mathbf{B}$ and $\mathbf{A}\cdot{\mathbf{C}}= p$, is it possible to express C in terms of A, B, and p? Note: A, B, C are vectors and p is a scalar. ...
1
vote
1answer
61 views

Problem similar to the birthday problem

a biased coin is tossed $n$ times (each toss is independent) with probability $h$ for heads. I need the smallest $n$ that lets the probability of at least one head to be $0.9$. I found p (no ...
0
votes
2answers
51 views

Simplify $\frac{x^{-6}+y^{-6}}{\left( x^{2}+y^{-2} \right)y^{-4}}$

Simplify $\frac{x^{-6}+y^{-6}}{\left( x^{2}+y^{-2} \right)y^{-4}}$ Attempt: I start by multiplying with $x^6y^6$ on both numerator and denominator. Then substitute x^2=a and y^2=b for an easier ...
1
vote
1answer
253 views

Matrix vector spaces isomorphic to column vector spaces?

my question is a basic linear algebra question, so hopefully someone can answer without too much trouble. My question was motivated by a problem I was doing about a linear transformation from the ...
1
vote
3answers
165 views

What is a good way to think of Factor Groups?

I'm having a hard time thinking about factor groups. I just don't understand what notation like $\mathbb{Z}_{60}/\langle 12 \rangle$ means. Furthermore, when asked about giving the order $26 + \langle ...

15 30 50 per page