# Questions tagged [filters]

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### On filters possessing a countable network

Let $\mathcal F$ be a free filter on $\omega$ and $$\mathcal F^+:=\{E\subset \omega:\forall F\in\mathcal F\;E\cap F\ne\emptyset\}.$$ A family $\mathcal N$ of subsets of $\omega$ is called a network ...
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### The property of the dense subfilter of a selective ultrafilter

Let us define the density of subset $A\subset\omega$ : $$\rho(A)=\lim_{n\to\infty}\frac{|A\cap n|}{n}$$ if the limit exists. Let $\mathcal{F_1}=\{A\subset\omega~|~\rho(A)=1\}$. $\mathcal{F_1}$ is the ...
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### Intuition for universal quotient maps

The universal quotient maps are precisely the descent morphisms in the category of topological spaces. In some papers of Janelidze, Tholen, Sobral, and Reiterman (see for instance Reiterman-Tholen), ...
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### Dense filter and selective ultrafilter

We say that $\rho(A)=\lim_{n\to\infty}\frac{|A\cap n|}{n}$ is the density of subset $A\subset\omega$ if the limit exists. Let us define the filter $\mathcal{F_1}=\{A\subset\omega~|~\rho(A)=1\}$. ...
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### A characterization of Cauchy filters on countable metric spaces?

Given a filter $\mathcal F$ on a countable set $X$, consider the family $$\mathcal F^+:=\{A\subset X:\forall F\in\mathcal F\;(A\cap F\ne\emptyset)\}.$$ The following characterization is well-known. ...
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### Theory of (definable) ideals on a multi-dimensional countable set

I'm wondering if there are any results about definable ideals/filters on $\omega\times\omega$, $\omega\times\omega\times\omega$, etc. To give a sense of the kind of results I might be looking for: ...
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### Is certain topology-related set a distributive lattice?

In my research the following problem appeared (and if it is true, this solves positively several my conjectures): Let $U$ be a fixed set (usually $U$ is infinite). Let $n$ be a fixed index set ($n$ ...
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### Why the intersection of a scott open (or \w the relatively compactness property) filter on a topology of a sober (and 2nd countable) space is compact?

Definitions and notations. Let $\mathcal{P}(X)$ the power set of $X$. Let $\tau_X\subseteq\mathcal{P}(X)$ a topology on X. We call $A$ irreducible if every time $A=B\cup C$ with $B,C$ closed set ...
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### Valuation Rings and Ultrafilters II

See my post here: Valuation Rings and Ultrafilters Let $K$ be a field, and let $\mathcal{S}$ be the set of pairs $(R, \mathfrak{p})$ of subrings $R$ of $K$ with designated prime ideals $\mathfrak{p}$ ...
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### Analogous filter to Kalman filter that maximized mode instead (as opposed to minimizing variance)

I may have a potential application where maximizing the mode (as opposed to typically minimizing the variance) would be useful for state estimates. The situation may arise from skewed distributions ...
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### Research in Algebraic Geometry involving Filters.

In Gillman and Jerrison's book, "Rings of Continuous Functions", they show a nice relationship between the set of z-ideals in C(X) and the set of filters on X. One can go for with this relationship; ...
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### Do filters complementive to a given filter form a complete lattice?

Really I should first ask this question here on MathOverflow and only then post it as an open problem in Open Problem Garden and propose it as a polymath problem. Indeed I did the reverse and now hope ...
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### MSE of measurable function is still conditional expectation

Motivation Then the usual stochastic filtering problem says that: $$\operatorname{argmin}_{Z \in L^2(\mathscr{G}_t)}\,\mathbb{E}[(Y_t-Z_t)^2],$$ where $\mathscr{G}_t$ is the $\sigma$-algebra ...
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### The trace of the filter on a big subset

Let $\scr{F}$ be free filter ($\cap\scr{F}=\emptyset$) on a countable set $X$ and $B\in\scr{F}$. We define the trace of $\scr{F}$ on $B$ as follows $\mathscr{F}_B=\{Y\cap B:~Y\in\scr{F}\}$. $\scr{F}$ ...
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### About ordering and equivalence of filters

Let $U$ is a set. I will speak about filters on this set. If $f$ is a function and $a$ is a filter then I define $f \left[ a \right]$ as the filter whose base is $\lbrace f[A] | A \in a \rbrace$. I ...
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### Posets which extend centered sets to filters

(Post cross-posted from math.se.) Suppose $(\mathcal O, \leq)$ is an arbitrary poset. Let us say that $\mathcal O$ is compact if every $\mathcal C\subseteq\mathcal O$ which is centered (any finite ...
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### Are the shapes of the $\mathbb{R}^2$ plane and a disk of infinite radius different? Or otherwise, why their areas differ by $\frac\pi{12}$?

The calculation of the area of the $\mathbb{R}^2$ plane depends on filtering used. I think, the most natural filtering is along the radius in polar coordinates: S_{\mathbb{R}^2}=\int_0^\infty 2\pi ...
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### About filters on real numbers

While trying to solve an open problem which I formulated earlier (currently "conjecture 1857" in this file), I met the following problem: Let $\Delta_{\geq}+\alpha$ be the filter on $\mathbb{R}$ ...
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### Digital Filters [closed]

Can somebody help with the constructing filter by amplitude and phase spectrum? Is it possible? I try to google it, but unsuccessufully. I have some thoughts about solving it by system of linear ...
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### Algorithm for calculating the sum-of-squares distance of a rolling window from a given line function

Given a line function $y = ax + b$, it is easy to calculate the sum-of-squares distance between the line and a window of samples $(1, y_1), (2, y_2), ..., (n, y_n)$ (where $y_1$ is the oldest sample ...
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### Magnitude spectrum of a cascade of filter

Given is a input vector $x=[x_1 x_2 x_3] \in \mathbb{R}^{3N}$ with 3 consecutive sub-blocks $x_1,x_2,x_3 \in \mathbb{R}^{N}$, which goes through a cascade of filtering operations defined as Step 1 (1-...
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### $\kappa$-translatability

I asked the following on MSE a few weeks ago but I did not get any answer : https://math.stackexchange.com/questions/1039593/carlsons-translatability-are-theses-characterisations-equivalent Reference ...
Let $U$ be a set. Let $N$ be a (possibly infinite) index set. Let $f$ be an $N$-ary relation on $U$ (that is $f$ is a set of functions $N\rightarrow U$). I denote $\mathcal{L}\in \upuparrows f \... 1answer 292 views ### Expressing a value related to an infinitary relation through ultrafilters Let$U$be a set. I denote$\mathfrak{A}$the lattice of filters on$U$ordered reverse to set theoretic inclusion of filters. I denote$\bigvee$and$\bigwedge$correspondingly the supremum and ... 1answer 1k views ### Filters and intersection of two binary relations Let$\mathfrak{F}$is the complete lattice of filters (including the improper filter) on some set, ordered inverse to set-theoretic inclusion. I will denote$\left\langle f \right\rangle \mathcal{X} =...
Product $a\times b$ of filters $a$ and $b$ is defined as the filter (on the set of binary relations) defined by the base $\{ A\times B | A\in a,B\in b \}$. I will denote the principal filter ...