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1,698,696 questions
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Prime ideal of infinite height in a Noetherian ring
Let $R$ be a ring. Is it possible that there is a chain of prime ideals in $R$, $$(*)\ \ \ \ \cdots\subsetneq\mathfrak p_i\subsetneq \cdots\subsetneq \mathfrak p_2\subsetneq \mathfrak p_1\subsetneq \...
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Generalized “incenter map” is symmetric with respect to permutations of the four input lines
$ABCDEF$ is the complete quadrangle determined by the four lines $AB,BC,CD,DA$ so that $E=AD\cap BC$ and $F=AB\cap CD$.
Reflect $D$ across the line $AC$ to a point $D'$, and reflect $F$ across the ...
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Difference between arcsin and inverse sine.
I first learned that arcsin and inverse sine are two ways of saying the same thing.
But then I was thinking about the inverse sine function being a function, so it must be limited in it's range from -...
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Div and curl operators on matrix multiplied vectors
Let u(x) be a vector field of n dimensions and M(x) be a matrix of dimension mxn. Are there any elegant general identities for div and curl of :
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Number of solutions to $x^2-y^2 = n$ in some rings
I am interested in studying the number of solutions to $x^2-y^2 \equiv n$ in the ring $(\mathbb{Z}/{p^q}\mathbb{Z})[t] / (t^2-\delta)$. Can anyone please help me find resources that could be of great ...
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Is there a $(3,3)$-windmill graph with $19$ vertices?
$(19,6,1,2)$ satisfies the basic relationship for the parameters of a strongly regular graph but such a graph does not exist - see 1st reference for the definitions and 2nd reference for this case.
If ...
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Finding more closed form results for $\sum_{n=0}^{\infty}\frac{\alpha^n}{n!}\Gamma^2\bigl(\frac{2n+1}{4}\big)$
I have the following equation, which is valid for $|\alpha|\leq2$. $K(k)$ is the complete elliptic integral of the first kind.
$$\sum_{n=0}^{\infty}\frac{\alpha^n}{n!}\Gamma^2\biggl(\frac{2n+1}{4}\...
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Sufficient and Necessary Conditions for a Set that Proves $\mathbb{Q}$ is Measure $0$
$\renewcommand{\epsilon}{\varepsilon}$
The standard demonstration that the Lebesgue measure of $\mathbb{Q}$ (or any countable set, but we'll take $\mathbb{Q}$) is zero is as follows:
Enumerate $\...
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Area identities for a triangular "prism" circumscribed about a sphere
Consider a convex triangular "prism" in $\mathbb{R}^3$ circumscribed about a sphere.
The three lateral faces are planes tangent to the sphere whose outward normals lie in the horizontal ...
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Mexican Hat Function that Converges to 1
The Ricker wavelet, also known as the Mexican Hat function, converges to 0 for negative and positive infinity.
I'm looking for a function with the same shape in the center, but that converges back to ...
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Finding a moment generating function for a continuous random variable defined by its pdf.
The pdf of a continuous random variable $X$ is defined by
$$f_X(x)=\begin{cases}
\frac1{10} (5-2x)&,\text{ if } x \in [0,2)
\\ \frac1{10}(x-1)&,\text{ if } x \in [2,4]
\\ 0 &,\text{ ...
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How to find all polynomials that map integers to integers?
How can one find all polynomials of degree k that map integers to integers? In other words, how to get all combinations of coefficients
$a_0,...,a_k \in \Bbb R$
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How to find the necessary and sufficient conditions for a polynomial to map integers to integers? [duplicate]
I am trying to find necessary and sufficient conditions on the real coefficients of a polynomial such that it maps integers to integers. More precisely:
Let
$$
f(x) = \sum_{k=0}^{n} a_k x^k, \quad a_k ...
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Multivariate Hypergeometric Cumulative Distribution Function
I think my problem is unique in that it hasn't been posed here.
Starting with a simple case to which I think I have an answer:
I have 11 cards, 3 of which are bad. These cards are used in a game ...
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Do universal properties like that of products hold in categories of categories?
I am trying to prove that certain expressions between product categories are functors, to this end I think it would be quite easy if we had something of universal propertiies, say of products in the ...
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