# All Questions

1,099,532 questions
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### General solution methods for third-order linear differential equation with variable coefficients

I'm trying to solve the following IVP: $y''' + ay''+by'+cy=0$, with initial conditions $y(x_{0})=y_{0},y'(x_{0})=y_{1},y''(x_{0})=y_{2}$, where $y,a,b$, and $c$ are all functions of the ...
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### Difference between Compactness Proof Structure Creation and Creation of a Forcing Extension Structure

Having just started to learn Forcing and having looked though the large number of Forcing questions on StackExchange, I apologize if I overlooked the following 'overview' novice question: Enderton ...
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### Residue of $m$-order pole Computation

I'm having trouble with the following residue computation: Find the order of the pole and corresponding residue for $$\bigg(\frac{z}{2z+1}\bigg)^3$$ The solution is given as: order = $3$ and ...
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### largest known twin semiprimes and consecutive semiprimes

What are the largest known semiprimes differing by 2? What is the largest known pair of consecutive numbers that are both semiprimes? Has anyone computed this?
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### What are the elements of a fractional summation?

For example, what are the elements in: $$\sum_{i=0}^{2.5} i$$ Background: As part of a program I wanted to generate a sequence along $i^2$ in a range $[a,b]$ with a given sum $c$. So, \begin{align}...
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### Cauchy-Riemann equations with partial derivative being continuous

I am trying to understand a proof in complex analysis. And there is a step that I am not able to see why this ''re writing'' can be done. define $f(z) = f(x+iy) = u(x,y) + iv(x,y)$ two small ...
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### $x_{0}=?$ such that limit of $x_{n}$ is 2.

I have the following sequence $(x_n)_{n\geq0}; x_{n+1}=2^{\frac{x_n}{2}}; x_0$ is a real number.I need to find $x_0$ such that the limit of this sequence is $2$. Some hints?
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### How to integrate n-th tetration?

How can the following indefinite integral be computed ? $\int{x↑↑n} dx$ where $n$ = {$x$ $\in$ $N^+$ : ${x > 2}$} Here ${x↑↑n}$ refers to $n$th tetration of $x$. I tried searching over the ...
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### Prouhet-Thue-Morse sequence and Arithmetic Progressions

This question is a fragment from a question posted by @Mathphile for which the link will be provided below. Let $(x_i)$ be an arithmetic progression of length $M$. Let $(t_i)$ be the $i$th element in ...
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### Reference request: networks of morphisms where each morphism goes from a product to a product

I am studying networks of morphisms where each morphism goes from a product to a product. Does anyone know if there is literature or references on this? In particular I study the following structure:...
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### How to solve this integral with exponential?

$$\int_\mathbb R \frac{e^{ixy}}{y^2+1} dy - \int_\mathbb R \frac{e^{ixy}}{y^2+4} dy$$ I am wondering if it is feasible by hand or if my examinator does not want that I calculate it explicitly
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### compute double integral$\iint_{|x-y|\leq r}{\rm d}x{\rm d}y$?

I want to compute the following double intergal but I do not know how to do $$\iint_{|x-y|\leq r}{\rm d}x{\rm d}y$$ where $(x,y)\in \mathbb R^3 \times \mathbb R^3$ and $r$ is a positive constant.
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### Polynomial rings and congruence classes

Let's consider the polynomial $m(x)$ over a field $\mathbb{Z}_{3}$. We know that $[m(x)]_{m(x)}=m(a)=0$. Now $m(x)=x^{3}+1$; in my lectures slide, it's said at this point that: $m(a)=0$ implies that ...
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### Calculation of digits

Suppose that an array of numbers from $1$ to $10^{23}$ is given. Let's calculate the sum of digits of each number. What kind of numbers among them are more with a $3$-digit sum or a $2$-digit sum? ...
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### 11 x  = 297 & 792 ÷ 11 =  …why is that?

Another example of a reversed factor giving you a reversed multiple is 11 x  = 583 & 385 ÷ 11 =  Is that pattern unique to the number 11 ~ google wouldn't say :( & what is the name ...
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### Existence of Rotation Matrix

Given vectors $v_1, v_2 ... v_n \in \mathbb{R}^n$ satisfying $||v_i||_2 = 1$ for all $i$ and $0 \leq\langle\ v_i,v_j\rangle \leq 1$ for all $i, j$. (Not all of them are 0 or 1). Does there always ...
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### Identity of tan(x)

I came across the following formulas for analytical expressions of fundamental modes of asymmetric dielectric waveguide. $$\tan(x) = x\frac{\pi^2-x^2}{\pi^2-4x^2}$$ This approximation is not present ...
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### How to find a line field in a vector field

I have a vector field described by $$\mathbf{F}(x,y) = y \hat{x} -x\hat y$$ I am trying to find the field lines for it, ideally by choosing a starting point and then having a parameter t that will ...
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### Formal solution diffusion equation

I'm not a mathematician, so please bear with me if I write things down in a non-rigorous manner. I read in either a mathematical finance or physics book (can't remember) that the formal solution of a ...
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### Check of $f(x)=\sum_{n=1}^{\infty}\frac{1}{x^2+n^2}$ properties

For function defined as $$f(x)=\sum_{n=1}^{\infty}\frac{1}{x^2+n^2}$$ check if $f$ is continuous and differentiable function. My approach: I would like to use the connection between this sum and ...
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### Completeness + Totally Bounded vs Compactness for subsets

We know that A metric space is compact if and only if it is complete and totally bounded. Does this hold for its subsets? I mean, is a totally bounded subset of complete metric space a ...
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### 25% chance occurring 7/11 times.

If an event has a probability of happening 25% of the time. How do you calculate the chances of this happening 7 out of 11 times. If A is 25% and B is 75%. What is the probability of A occurring 7 ...
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### Poker chance calculation with unknown cards

I'm trying to recreate a poker chance calculation like they do on television or poker broadcasts, but without knowing the opponent cards. On the internet you can find probabilities of having this kind ...

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