# All Questions

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### What can we say about the integral curve of a vector field on the warped product manifold?

Let $Z=(X,Y)$ be a vector field defined on the warped product $M×_{f}N$ where $f$ is defined on $M$. The integral curve of $X$ on $M$ is $\alpha$ and the integral curve of $Y$ on $N$ is $\beta$. I ...
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### Find a differentiable $f$ such that $f'$ is not continuous. [duplicate]

I'm trying to solve this problem: Find a differentiable function $f:\mathbb{R} \longrightarrow \mathbb{R}$ such that $f':\mathbb{R} \longrightarrow \mathbb{R}$ is not continuous at any point of ...
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### Help needed for mathematical formulation

I'm trying to write a simple mathematical formulated which expresses the following: Let F={f1,f2,...,fn} be an ordered set of flights, each f being associated with a begin time begin(f) and an end ...
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### $\lim_{n\to\infty} e^{-n}\sum_{k=1}^n \frac{n^k}{k!}$ [duplicate]

How can be evaluated this limit: $$\lim_{n\to\infty} e^{-n}\sum_{k=1}^n \frac{n^k}{k!} .$$ Thank you.
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### Newton Raphson Method: approximating root

How do we start from approximating a root using this technique? I know of two, viz - a table of x vs f(x), and see where f(x) changes sign - plot a graph, and see where the graph cuts the x axis But ...
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### Two points of a vector

I have a source point of a vector (x, y), the vector's size, and the angle of it. What's the formula to calculate the X and Y values of the point the vector will get to from the source point? I tried: ...
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### Trigonometry equation $\sin(x)+\cos(x)-\tan(x)=0.4$

There's some way to find $x$ here ? $$\sin(x)+\cos(x)-\tan(x)=0.4$$
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### How to sketch $y = \sqrt{x-1} + \sqrt{6-x}$

How to sketch $y = \sqrt{x-1} + \sqrt{6-x}$ My solution: It does not have any roots Domain = [1,6] Increasing till 3.5 and then decreasing How to go on further? Please help.
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### Preservation of separatedness of a scheme of finite type over a field by shrinking the base field

This is a generalization of this question. Let $k$ be a field. Let $k'$ be an extension field of $k$. Let $X$ be a $k$-scheme of finite type. Suppose $X\times_k k'$ is separated over $k'$. Is $X$ ...
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### A generalization of Cayley-Bacharach Theorem

This is exercise 19.4.B on Ravi Vakil's notes. Let $C$ be a regular plane curve of degree $e>2$, and $D_1,D_2$ be two plane curves of same degree $d$ not containing $C$. By Bezout's theorem $D_i$ ...
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### The dimension of space of morhisms as the number of orbits

All groups are finite, all representations are over $\mathbb{C}$ (just in this context, of course), $G$ is a group, $K,H\subset G$ - its subgroups. By $\mathbb{C}$ we denote the trivial representation ...
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### The sum of Gaussian functions

Suppose there is a normal distribution and the Gaussian function is $F(x)=\exp(-c\|x-b\|^2)$ where $c$ is a constant and $x,b\in \mathbb{R}^N$, b means the mean value. ...
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### Example of rings with idempotent and non-zero Jacobson radical

I am looking for a simple (as simple as possible) example of a commutative ring (with identity) $R$ such that its Jacobson radical is non-zero but idempotent. (The simplest example that I know is ...
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### Generating function for lattice points in a sphere

This is a note in Sedgewick's Analytic Combinatorics: The number of lattice points with integer coordinates that belong to the closed ball of radius n in d-dimensional Euclidean space is ...
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### Existence of d-regular graphs

It is well known that if $0<d<n$ and $d\cdot n$ is even then there exist $d$-regular graphs on $n$ vertices. My question is: What is the easiest way to prove this?
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### Prove $1^{2007}+2^{2007}+\cdots+n^{2007}$ is not divisible by $n+2$

Prove that for any odd natural number $n$, the number $1^{2007}+2^{2007}+\cdots+n^{2007}$ is not divisible by $n+2$.
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### What's the pair of numbers made using 1-9 which has maximum product? [duplicate]

The task is to find a pair of numbers whose digits are 1-9, no digit repeated such that their product is maximum possible. eg. 123 and 456789 is one such pair. Question: Please suggest a way to do ...
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### Maclaurin Series confusion

Using the Maclaurin expansion formula: to find the Maclaurin series for $sin(3x)$, I can get the correct answer by using $x^n$ in the formula above (in the tail-end of the formula). Similarly, to ...
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### Dependence on parameters in propositional logic

Warning: my background is mostly in probability and analysis, and not in logic. When reading or writing a complex proposition, with long chains of "for all... there exists... for all...", I tend to ...
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### Figuring out whether a ring is a field

Given a ring, how do you test whether it is a field? What properties would you look at?
I am trying to construct all inequivalent $8\times 8$ matrices (or $n\times n$ if you wish) with elements 0 or 1. The operation that gives equivalent matrices is the simultaneous exchange of the i and ...