# All Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ?
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### 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 ...
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### 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, ...
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### 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?
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### 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 ...
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### $k$ 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 ...
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### 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 ...
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### 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?
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### 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...
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### (When) is this matrix positive definite?

I have a symmetric $n \times n$ matrix (say, $M$) with $[i,j]$ element $$M_{[i,j]} = \int_{\mathbb{R}} [p_i(z)-g(z)][p_j(z)-g(z)]~dz,$$ where $p_i(\cdot), p_j(\cdot),$ ...
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### 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 ...
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### 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 ...
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### 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 $p$, $T_{\mathbb{R},p}(M)$, ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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
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### 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 ...