6
votes
0answers
114 views

Educational software for graduate mathematics

Our university uses computer software to help automate and expedite the learning process for basic math classes (college algebra, trigonometry, precalculus, etc.). Software such as this provides ...
2
votes
0answers
99 views

Maximum Posterior: $ p(\bf{w}\mid\bf{x},\bf{t},\alpha,\beta) \propto p(\bf{t}\mid\bf{x},\bf{w},\beta)p(\bf{w}\mid\alpha) $ for Gaussian Distribution

At the moment I take a look at the book Pattern Recognition and Machine Learning from Christopher Bishop and as I try to understand the basics of the probability theory I get stuck trying to ...
2
votes
3answers
80 views

Proving a Sequence's Uniform Convergence

Another homework problem that's been giving me headaches for about a week now. Prove that the following sequence of functions $(f_n)$ converges uniformly on the interval $[1,2]$: $$f_n(x) = \frac ...
1
vote
1answer
105 views

Is there a power series which converges to $f(x) =| x|$ for all $x$?

I'm confused how to solve the following problem: "Is there a power series which converges to $f(x)$ = $\left| x\right|$ for all $x$?" Your help is greatly appreciated. Thanks a lot!
3
votes
3answers
157 views

How does $\mathrm {e}^z$ and $\log z$ look like as complex functions.

I want to visualize complex functions $\mathrm e^z$ and $\log z$ in $C$, here $z\in\Bbb C$. I want to know their behavior and zeros and singularities. Can anyone explain me in an easy way. Thank you ...
3
votes
1answer
107 views

Prove that if a $f '$is always rational, $f$ is a straight line [closed]

$f$ is a function from $\Bbb R \to \Bbb R$ such that $f'\left( x \right)$, the derivative of $f$, is always a rational number. Prove that the graph of $f$ is a straight line.
0
votes
1answer
100 views

Maths-Physics question, can I solve this situation for $x$?

So Let's say I have an object going at velocity $V$, initially. Each second, the current velocity $v$ is reduced by $v/x$ . After $250$ (arbitrary) seconds the velocity has been reduced to below/equal ...
0
votes
1answer
201 views

Proof of explicit formula for change of basis matrix [closed]

We are given two sets of (not necessarily orthonormal) basis vectors {a1,...,ak} and {b1,...,bk} of a nontrivial subspace of R^n. Define the matricies A, B to have their jth column be aj and bj ...
19
votes
1answer
301 views

How to prove $\sum_{n=1}^\infty\operatorname{arccot}\frac{\sqrt[2^n]2+\cos\frac\pi{2^n}}{\sin\frac\pi{2^n}}=\operatorname{arccot}\frac{\ln2}\pi$?

How can I prove the following identity? $$\sum_{n=1}^\infty\operatorname{arccot}\frac{\sqrt[2^n]2+\cos\frac\pi{2^n}}{\sin\frac\pi{2^n}}=\operatorname{arccot}\frac{\ln2}\pi$$
2
votes
1answer
84 views

Finding a limit , dyadic pavings

I need to show that the following limit equals $\pi/4$ : $$\lim_{k \to \infty}\sum_{n=1}^{2^k-1}\frac{ \left\lfloor\sqrt{4^k-n^2}\right\rfloor\ } {2^{2k}}$$ I don't know if it is even possible to do ...
1
vote
1answer
132 views

prove that if $L(f,P)=U(f,P)$ then $f$ is constant on $[a,b]$

Suppose that $f$ is a bounded function on $[a,b]$ and there exists a partition $P $of $[a,b] $such that $L(f,P)=U(f,P)$. Prove that $f$ is constant on $[a,b]$ I know that $L(f,P)=U(f,P)$ meaning $f$ ...
3
votes
1answer
62 views

Showing that $\int_0^{\pi/4} \frac{1-\cos{16x}}{\sin{2x}}\,\mathrm{d}x=\frac{176}{105}$

Wolfram Alpha tells me that $$\int_0^{\pi/4} \frac{1-\cos{16x}}{\sin{2x}}\,\mathrm{d}x=\frac{176}{105}$$ What are some quick/elegant ways of proving this?
1
vote
2answers
39 views

Show that this is one to one continuous and find its inverse which is continuous as well.

Let's define $\phi: \Bbb R^2 \to S$ for $S$ is subset of $\Bbb R^3$ For constant $a,b,c,d$ and $c\not =0$ $$\phi(x,y)=(x,y, \frac{d-ax-by}{c})$$ I want to show that the function $\phi$ is 1-1 ...
1
vote
1answer
78 views

Heat equation for a finite graph

Thank you for an interesting website. I would like to construct the heat equation for a finite graph $G$ using basic math concepts. For example, if $G = \mathbb{Z}/m\mathbb{Z}$ then I think of $G$ ...
1
vote
1answer
142 views

Linear isoparametrics with dual finite elements

The subject presented here is some content of the Wikipedia page about Platonic solids combined with my own experience on Finite Elements.To start with the latter, there is a standard piece of Finite ...
4
votes
2answers
99 views

Proofs from the Book - need quick explanation

I've been recently reading this amazing book, namely the chapter on Bertrand's postulate - that for every $n\geq1$ there is a prime $p$ such that $n<p\leq2n$. As an intermediate result, they prove ...
4
votes
3answers
560 views

Exact sequence of sheaves with non exact sequence of global sections

Let $X$ be some topological space. By $\mathcal{F}_i$ we denote some sheaves of abelian groups on $X$. The sequence of sheaves and morphisms $$\mathcal{F}_1\longrightarrow ...
0
votes
1answer
207 views

Can Fermat's Factorization Method Be used in any way to get the largest prime factor for a given number?

I have given a shot at trying to find the largest prime number for a given number, and thought of using Fermat's Factorization Method. I might be sitting the pot miss and I think I am going about this ...
2
votes
1answer
50 views

What is the hitting time distribution for white noise?

What is the distribution of the hitting time for a stochastic process $(W_t)_{t\in [0,T]}$, where $W_t$ are i.i.d. Gaussian random variables? How about in cases, in which $W_t$ are i.i.d. with a ...
2
votes
1answer
503 views

equivalent functions?

I have this two functions in ($1<x<50$) $y = -1/x$ and $ y = \frac{x - \sqrt{x^2+4}}{2} $ why this are very similar ?
1
vote
1answer
72 views

Remainder Question

What process do I use to show what is the remainder when 14 × 7^36 + 92 when divided by 8? Is it the same to show the remainder of ...
1
vote
1answer
57 views

Proving inequalities by induction

I'm having trouble understand the inductive when proving inequalities; Here's an example: Show that $2^n \gt n^2 $ for any integer $n \gt 4 $. Well for the basis $n=5$, it shows: $32>25$ Now, ...
0
votes
2answers
44 views

Prove that for any square matrix, an invertible matrix B exists, so that BA is triangular

I'm given a matrix A, its dimensions are n x n. I am required to prove that an invertible matrix B exists, such that the product of the matrices BA is triangular. Any help?
2
votes
1answer
144 views

An eigen-decomposition/diagonalization question

I'm data analyst without any good math background. I'm struggling to understand and to code myself eigen-decompositions. So far, I know QR algorithm of eigen-decomposition. My problem. Let $\bf A$ be ...
0
votes
2answers
72 views

$k$ colorings of the non empty subsets of $[n]$ gives the same color to two disjoint sets and their union.

This question was already asked but I didn't get enough information from the answer. Here is a link to the question. Here is the question restated. Show that for $n$ large enough, every $k$ coloring ...
1
vote
2answers
157 views

Show that $X = \{ (x,y) \in\mathbb{R}^2\mid x \in \mathbb{Q}\text{ or }y \in \mathbb{Q}\}$ is path connected. [duplicate]

How do I show that $X = \left\{ (x,y) \in \mathbb{R}^2 \mid x \in \mathbb{Q}\text{ or }y \in \mathbb{Q}\right\}$ is path connected? Note that $X$ is a topological space with subspace topology ...
2
votes
2answers
388 views

Is Complex Numbers the biggest field? If yes, is there any easy proof to understand it?

Is the Complex Numbers the biggest field? If yes, does anyone have a "simple"/"easy to understand" proof?
1
vote
2answers
71 views

Derivative of Integral help [closed]

$$\frac{d}{dx}\int_0^x\frac{1}{1+t^4}\,dt$$ Im not sure why but this is confusing me... Mathematica gives me one answer but I get another...
1
vote
0answers
122 views

(When) is this matrix positive definite?

I have a symmetric $n \times n$ matrix (say, $M$) with $[i,j]$ element \begin{equation} M_{[i,j]} = \int_{\mathbb{R}} [p_i(z)-g(z)][p_j(z)-g(z)]~dz, \end{equation} where $p_i(\cdot), p_j(\cdot),$ ...
1
vote
0answers
78 views

Gauss-seidel and implicit method

I have a matrix $\mathbf{X}$ and I want to apply a function $f_{ij}$ to each entry of it, until convergence is satisfied. If a value is known in this matrix, then the $f_{ij}$ at this point may be the ...
3
votes
1answer
43 views

Let $A$ and $B$ be fractals with box dimension of $x$ and $y$ respectively. Then prove:

Let $A$ and $B$ be fractals with box dimension of $x$ and $y$ respectively. Then prove that the Cartesian product $A \times B$ has box dimension $x+y$. Any hints to start out? (note that box ...
10
votes
1answer
306 views

Complexified tangent space

Let $M$ be a complex manifold of dimension $n$ and $p\in M$. So $M$ can be viewed as a real manifold of dimension $2n$ and we can consider the usual real tangent space at $p$, $T_{\mathbb{R},p}(M)$, ...
1
vote
1answer
130 views

Expressing a length in a triangle with no angles given

Let there be $ABC$ an isosceles triangle $(AB = AC)$. $D $ is a point on $AB$ such that $AD = 2BD$. $E$ a point on $BC$ such that $2EC = BE$ . Express $DE$ in terms of the base, $a$, and the sides, ...
0
votes
2answers
113 views

Convolution theorem for other transforms

The Fourier transform is an integral transform with turns any function into a superposition of sinusoidal waves. The convolution theorem states the astonishing property that if you convolve two ...
6
votes
2answers
283 views

What comes after Differential Equations?

First of all, please do excuse the lack of correct terminology, I've haven't learnt Differential Equations at school (yet) so this question comes from just a bit of research I did for my own ...
0
votes
2answers
56 views

Proof ceiling function monotonocity?

Is there any indication about the proof of the following statement: $\forall x,y. x \le y \Rightarrow \left \lceil x \right \rceil \le \left \lceil y \right \rceil$ Yes it's $\le$. Sorry. Thanks
6
votes
1answer
105 views

Can You Construct a Syndetic Set with an Undefined Density?

Let $A \subset \mathbb{N}$. Enumerate $A = \{A_1, A_2,...\}$ such that $A_1 \le A_2 \le ...$. We say that $A$ is syndetic if there exists some $M \geq 0$ such that $A_{i+1} - A_i \le M$ for all $i ...
1
vote
0answers
44 views

Constraints on eigenvalues due to the form (symmetry) of Matrix

So I have to deal with a square matrix with complex elements $A$ given by - $$A_{ij}(k) = \underbrace{\frac{1}{2}(c_{ij}+c_ji)}_{\text{symmetric under interchange of $i$ and $j$}} \exp(Ik(i-j))$$ ...
7
votes
2answers
397 views

Come up with some fun “equation Limericks”

We were discussing "Limericks" in my Calculus class. Specifically, "equation Limericks". A Limerick is a poem with five lines. The first, second, and fifth lines should have nine syllables each and ...
1
vote
1answer
40 views

stuck on a Cartesian question

we have a circle $(x-1)^2+(y-2)^2=9$ Point $P=(5,2)$ lies outside the circle. Solve the equation of the line which passes through $P$ and intersects the circle at two points whose mutual distance is ...
1
vote
1answer
189 views

Contour integral in complex plane as a Lebesgue integral

In an occasion, I'd like to use Fubini's Theorem to swap the order of integration of a countour integral with an integral with respect to a measure (show that $\int_{\gamma} \int_{\Omega} f(x,z) ...
1
vote
0answers
38 views

How do I prove the multiplicativity of separable degree in general?

Let $K/E/F$ be a tower of algebraic extensions. How do I prove that $[K:E]_s[E:F]_s = [K:F]_s$? This is done in all the books I searched for finite extensions (when it follows trivially from a ...
0
votes
1answer
34 views

Expressing a vector as the best linear combination of “random” vectors

Suppose I have something like: $\vec{v} = \langle 1, 2, 3, 4, 5 \rangle$ and I have a set of vectors (these are all just made up numbers): $\vec{w_1} = \langle 3, 7, -2, -4, 8 \rangle$ $\vec{w_2} ...
1
vote
1answer
105 views

Show that $V$ is a vector space when $V$ is the set of all continuous functions $\Bbb{S}^1 \to \Bbb{R}$.

Let $V$ (which is infinitely dimensional) be the set of all continuous functions $\Bbb{S}^1 \to \Bbb{R}$. Show that $V$ is a vector space. Define $\langle-,-\rangle: V\times V\to \Bbb{R}$ by $$\langle ...
1
vote
0answers
21 views

Largest classes of singularities

As seen, for instance, in "Survey on Singularities and Differential Algebras of Generalized Functions : A Basic Dichotomic Sheaf Theoretic Singularity Test", ...
0
votes
1answer
17 views

Number of equivalence relations on a set with fixed class

For A={a,b,c,d,e,f}, how much equivalence relations can we get if a,b and c are in relation? The total is: $\sum_{k=1}^6 S(6,k)$. But since a,b and c are already in the same class, i would say the ...
2
votes
1answer
76 views

Convergence in probability inverse of random variable

if $ X_n \to 1 $ in probability i need to prove that $X_n^{-1} \to 1$ under probability. I got till the point that i need to prove the following probabilities 0, but don't know how to prove them? i.e ...
0
votes
1answer
162 views

Degrees of interpolating polynomials

Given a collection of $m+1$ points $\{(x_0,y_0), (x_1,y_1), ..., (x_m,y_m)\}$, we can form the interpolating Lagrange polynomial $L(x)$: $$ L(x) = \sum_{i = 0}^{m} y_i l_i(x) \\ l_i(x) = \prod_{0 \le ...
1
vote
1answer
57 views

The intergral $I=\int _0^{\beta }f(x)dx$ is given,for $\alpha,\beta \in \mathbb{Z}$ ,how can we find $\int_0^{\alpha\beta}f(x)dx$ in terms of $I$

i am working with a gaussian normal distribution function in probability,i am given values for the integral when $z\le 4$ and i want to find a value $z=8$,in general if $z=4$ is given , how to find ...
3
votes
4answers
387 views

How can the trefoil knot be expressed in polar coordinates?

From Wikipedia, the parametric equations for a trefoil knot are \begin{align*} x(t) &= \sin t + 2\sin 2t \\ y(t) &= \cos t - 2\cos 2t \\ z(t) &= -\sin 3t. \end{align*} I am only ...

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