# All Questions

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### Becoming a Mature Mathematician

I am currently a sophomore in my undergraduate mathematics program. It has taken me a while to take school seriously; I was one of "those" students who just skated by without studying until Linear ...
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### Are these moduli problems of curves “well-behaved”?

Let X be a smooth projective surface over $\mathbb C$, and let $d\geq 3$ be an integer. Suppose that all smooth hypersurfaces of degree $d$ are of genus $g\geq 2$. Let $H_{X,d}$ be the Hilbert scheme ...
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### applications that involves the Legendre Polynomials [on hold]

i am requested to make a basic research about the applications that involve the Legendre Polynomials.. thanks in advance
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### harmonic analysis on $p$-adic $x^2 + y^2 + z^2 = 1$?

Are there p-adic analogues to spherical harmonics? In the case of $K = \mathbb{R}$, the spherical harmonics form a basis to $L^2 [SO(3)]$ where What happens in the $p$-adic case? Is there sphere ...
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### History of the Vertex Disjoint Cycle Cover with Minimal Edgeweight Sum

Questions: who first posed the problem of determining a collection of (directed) cycles, whose edgeweight sum is minimal and, for which each vertex belongs to exactly one of the cycles? who came up ...
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### Is there any infinite set which is Dedekind finite and weakly Dedekind finite?

Is there any infinite set which is Dedekind finite and weakly Dedekind finite? If $X$ is a weakly Dedekind finite amorphous set, can we show that $\mathcal P(X)$, the power set of $X$, is also weakly ...
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### How are $\alpha$ and $f(\alpha)$ obtained from the calculated values of $h(q)$ in the Multifractal analysis? [on hold]

In most papers they mention some kind of Legendre transform but I'm not entirely clear about the process, sine they don't mention enough details, at least for me, I tried reading the cited references ...
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### Points of continuity of a lower semicontinuous function have non empty interior

Let $X\subset \mathbb R^d$ having non empty interior and let $f:X\to\mathbb{R}$ be lower semicontinuous. I know that the set of discontinuities of such a function is contained in a meager set, and ...
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### Solving polynomial equations modulo $1$

Let $P\in \mathbb{R}\lbrack x\rbrack$ be given. (In practice, the coefficients could be given as, say, decimals to sufficient precision.) Let $M\geq 1$, and let $I$ be an interval in ...
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### Closed subgroups of $\mathrm{SO}(4)$

My question is quite simple : we know all closed subgroups of $\mathrm{SO}(3)$; is it also known what are the closed subgroups of $\mathrm{SO}(4)$?
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### Let $f:R\to S$ be a local finite monomorphism .If $M$ is an Artinian $S$-module, is it an Artinian $R$-module?

Let $(R,m)$ and $(S,n)$ be local rings (commutative Noetherian with 1). Let $f:R\to S$ be a local homomorphism/monomorphism ($f(m)=n$), such that the natural induced homomorphism $R/m\to S/n$ is an ...
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### On property of monic polynomial with integer coefficients

For a monic polynomial with integer coefficients $f$ where $\partial f = 2$, we have $$\textrm{inf}(f(x)) > 0 \implies \textrm{inf}(f(x)) \geq \frac{3}{4} .$$ Could we generalize this (for ...
Consider a continously differentiable non-constant function $f:\mathbb{R}^2\to\mathbb{R}$. Define $$K=\{x\in\mathbb{R}^2|f(x)=0\}.$$ I wish to know whether there is a continuously differentiable ...