The weights tag has no usage guidance.

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### Robust weighted estimator of location

Let $X = (x_1, \ldots, x_n)$ be a sample of i.i.d values. There are several robust estimators of sample location, most notably sample median and Hodges-Lehmann estimator.
Now let $W = (w_1, \ldots, ...

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### Embeddings between weighted Besov spaces

Consider the Besov spaces $B_{p,q}^s(\mathbb{R}^d)$ for parameters $0<p,q\leq \infty$ and $s\in \mathbb{R}$. The weighted Besov space $B_{p,q}^s(\mathbb{R}^d;\mu)$ is defined for $\mu \in ...

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### Logarithmic weights on number theoretic sums

Suppose we are interested in the sum
$\sum _{n\leq x}a_n.$
The study of the sum
$\sum _{n\leq x}a_n\log (x/n)$
may be easier.
What can one say about the first sum from knowing the behaviour of ...

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### When is the identity Hilbert-Schmidt between weighted Sobolev spaces?

Set $w(x) = (1 + |x|^2)^{1/2}$ with $|\cdot|$ the Euclidian norm on $\mathbb{R}^n$. For $s,\mu \in \mathbb{R}$, we define the Sobolev space
$$H_2^{s}(\mathbb{R}^n) = \left\{f : \lVert f \rVert_{s} := ...

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### Intrinsic definition of the weight filtration

Let $X$ be a smooth quasiprojective complex variety. Then Deligne (Theorie de Hodge II) defined a weight filtration on the Betti cohomology of $X$. The general philosophy is quite simple: express the ...

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### What sorts of weights for perverse sheaves were or can be computed?

I am studying certain weights for (triangulated categories of relative) motives. Those are interesting; yet one can hardly say that they are very much explicit or effectively computable. So, I would ...

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### Convergence of weighted double sum of random variables

I'm looking for convergence results of particular weighted sum:
$$S_n=\frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{n}a_{i,j}X_i X_j.$$
when random variables $X_i$ ar i.i.d. Are there any investigation ...

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333 views

### “Weight-monodromy” for open varieties

Suppose that $X/\mathbb{Q}_p$ is a smooth, projective variety, and choose a prime $\ell\neq p$. Then the weight-monodromy conjecture says that the graded pieces $\mathrm{Gr}_k^M$ of the monodromy ...

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### Weight multiplicities for some particular representations of SO(2m).

I am looking for explicit formulas for the weight multiplicities of some particular irreducible representations of $SO(2m)$.
It is possible that they have been already computed; in this case I will ...

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### Is semistability of smooth Weil sheaf preserved under tensor product?

Let $X_0$ be a smooth, geometrically connected scheme over $\mathbb{F}_q$. As usual, let $\tau : \bar{\mathbb{Q}}_{\ell} \simeq \mathbb{C}$ be a fixed isomorphism. Let $\mathcal{C}$ be the category of ...

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### Hodge structure versus Weight structure

This is a naive question.
One is told that, somehow, Hodge theory for varieties over complex numbers, is an analog of weight theory for varities over finite fields. In weight theory, one considers ...

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### Tiling a rectangle with weighted cells (min-max problem)

I have been struggling with a research problem. The problem can be formalized as follows:
Given a $n\times m$ matrix $A$ containing cells with non-negative integer values, partition it in $J$ ...

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462 views

### A_infinity structure on cohomology and the weight filtration

Let $X$ be a complex algebraic variety. The rational cohomology of $X(\mathbb{C})$ carries a canonical filtration called the weight filtration. It also carries a canonical equivalence class of ...

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### Weights of restricted modules of some Cartan type Lie algebras

Let $L$ be a simple Lie algebra of Cartan type of absolute toral rank 2 over an algebraically closed field $\mathbb{F}$ of characteristic $p\geq 5$.
Denote by $L_{[p]} $ the minimal $p$-envelope of ...

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### Universal Correlation measure — ranking correlations

I have time series data of experimental observations for two related processes. I want to measure correlation for use in further analysis.
Correlation of the series changes over time and across ...

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### weighted to centre mean

I'm not even sure if what I want to do is a good idea but I figure I'll experiment and see.
I have two predicted ratings in the range of 1-5 based on two different algorithms for predicting movie ...

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### Which (reducible) projective varieties could be presented as 'relatively smooth' hyperplane sections of irreducible (normal) ones?

Are there any restrictions known on a (complex reducible) projective variety $Y$ that can be presented as $Z\cap H$, where $Z$ is a normal (or just irreducible) closed subvariety of a smooth ...

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### A Weyl invariance constructed from Clebsch-Gordan Coefficients.

Let $V$ and $\tilde{V}$ be irreducible representations of SU(N) with tensor decomposition:
\begin{equation}
V \otimes \tilde{V} = \bigoplus_i U_i
\end{equation}
\noindent were $U_i$ are also irreps ...

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### Does Artin's vanishing hold for '$E_2$-weight pieces' for (torsion) cohomology of affine varieties?

Let $X$ be an affine variety of dimension $n$ (say, over complex numbers). Then Artin's vanishing theorem yields: both singular and etale cohomology $H^i(X):=H^i(X,\mathbb{Z}/l\mathbb{Z})=0$ for ...

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### Does intersection pairing on `$IH^*(X)$` agree with cup-product on `$H^*(X)$`?

Let $X$ be a proper singular variety over $k=\overline{\mathbb F}_p,$ irreducible of dimension $d.$ Let $H^*(X)$ and $IH^*(X)$ be the $l$-adic cohomology groups and $l$-adic intersection cohomology ...

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### Smooth in codimension-k and the weight filtration

Let $X$ be an algebraic variety. Then $H_{et}^k(X)$ has a filtration whose associated graded pieces are labeled by "weights", certain integers between $0$ and $2k$. If $X$ is smooth, then the weights ...

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### A Hodge substructure with ''nice' weight factors that does not correspond to a mixed submotif?

Suppose that we have a mixed motif $M$ with only two (subsequent) weights, and have a subobject for each weight factor. How we can control whether there exists a submotif of $M$ with these two ...

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### Do Deligne's decalage and two filtrations for mixed Hodge complexes have a conceptual explanation?

As far as I understood, there are two way to assign weights to a mixed Hodge complex (a way that does not depend on shifts, and another one that does). There is a clever way to relate these to ways ...

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### Cohomology of Zariski neighborhoods

Do there exist smooth compact (=complete) connected complex algebraic varieties $X\subset Y$ and a Zariski neighborhood $U$ of $X$ in $Y$ such that the image of $H^{\ast}(U,\mathbf{Z})$ in ...

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### Which statements in section 5 of BBD will fail if we consider $\mathbb{Q}_l$-adic sheaves there?

A stupid question: which statements in section 5 of BBD will fail if we replace $\overline{\mathbb{Q}_l}$-sheaves by just $\mathbb{Q}_l$-ones? I am especially interested in Proposition 5.1.15.
BBD = ...

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### In what setting does one usually define mixed sheaves and weights for them?

In BBD mixed sheaves and weights for them were only defined for ($\overline{\mathbb{Q}_l}$-)sheaves over a variety $X_0$ defined over a finite field $F$. Weights start to behave better when one ...