# Questions tagged [filters]

The filters tag has no usage guidance.

**5**

votes

**1**answer

207 views

### 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 ...

**1**

vote

**1**answer

94 views

### Variance of convolution between filter $A$ and Ornstein-Uhlenbeck process $x_t$

If we consider $x_t$ an Ornstein-Uhlenbeck process (with $W_t$ the Wiener process), does anyone know what would be the variance of the convolution of $x_t$ with a given filter $A$ i.e. $V(x_t \star A)$...

**3**

votes

**0**answers

140 views

### 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.
...

**1**

vote

**0**answers

126 views

### 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 $\mathbb{R}$ ...

**1**

vote

**1**answer

69 views

### Convergence of a z-filtre to an outer point

Let $X$ be a completely regular topological space and let the set
of all continuous functions from the topological space $X$ into
the topological space $\mathbb{R}$ is denoted by $C(X)$. Let
$Z(X)=\{Z(...

**4**

votes

**1**answer

75 views

### SDE for conditioning on subfiltration

Setup:
Suppose $X_t$ solves the SDE
$$
dX_t = \mu(t,X_t)dt +\sigma(t,X_t)dZ_t,
$$
where $Z_t$ is a Lévy process on $\mathbb{R}^d$, $g(t,s,x):[0,T]\times[0,1]\times \mathbb{R}^d \rightarrow \mathbb{R}^...

**2**

votes

**0**answers

34 views

### Is there a well-developed theory on filtering on sub $\sigma$-fields?

Suppose we are given an observation process
$$
dY_t = \mu(Y_t)dt + \Sigma(Y_t)dW_t
$$
with valued in $\mathbb{R}^d$ (or potentially in an arbitrary separable Hilbert space). The classical filtering ...

**1**

vote

**1**answer

91 views

### 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 ...

**2**

votes

**1**answer

162 views

### Without Choice: Are there filters of cardinality continuum?

Is it provable, in ZF (without Choice), that every filter can be extended to one of cardinality continuum?
The extended filter is not requested to be an ultrafilter.

**15**

votes

**1**answer

924 views

### Duality between topology and bornology

I want to understand in what sense topology is dual to bornology at a most basic level. Therefore, I rephrased the definition of a bornology in the following way:
Let $X$ be a set and let $\mathcal{P}...

**4**

votes

**0**answers

85 views

### 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), ...

**4**

votes

**1**answer

148 views

### Convergent filters generated by (not necessarily countable) chains

Suppose $\langle X,\mathscr{O}\rangle$ is a topological space and let $\mathscr{O}_x$ be the family of all open neighbourhoods of $x\in X$. Let $\mathscr{F}$ be the filter generated from $\mathscr{O}...

**2**

votes

**2**answers

330 views

### 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 ($n$ ...

**0**

votes

**0**answers

187 views

### $\kappa$-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 ...

**-1**

votes

**1**answer

205 views

### Lowering from filters to ultrafilters for an infinitary relation

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 \...

**-2**

votes

**1**answer

244 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 ...

**2**

votes

**1**answer

158 views

### Interweaving two indexed families of filters

Conjecture
Let $U$ be an (infinite) set. Let $f$ be an $N$-ary (where $N$ is an
arbitrary index set) relation on $U$ (that is a set of functions $N
\rightarrow U$).
Let $\mathcal{L}_0$, $\...

**2**

votes

**2**answers

220 views

### 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 ...

**6**

votes

**0**answers

535 views

### Intersections of open sets and $\alpha$-favorable spaces

I would like to ask about some classes of topological spaces whether they have been studied, what are they called (if they have a name) and whether they have some interesting properties. For the ...

**2**

votes

**0**answers

44 views

### 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 ...

**7**

votes

**1**answer

599 views

### A characterization of the poset of filters on a set

For the lattices of all subsets of a given set it is known an axiomatic characterization: A poset is isomorphic to a set of all subsets of some set iff it is a complete atomic boolean algebra.
The ...

**4**

votes

**1**answer

342 views

### Connection between subnet and superfilter

Let's define a net and subnet in this way:
A net is any function of the form $n:(P,\le)\to X$ where $(P,\le)$ is a (preordered) directed set.
A net $m:(P',\le)\to X$ is a subnet of the net $n:(P,\le)\...

**2**

votes

**0**answers

283 views

### 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; ...

**3**

votes

**0**answers

136 views

### 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: ...

**0**

votes

**2**answers

1k views

### 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 ...

**1**

vote

**1**answer

156 views

### 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 ...

**-12**

votes

**1**answer

1k views

### Direct product of filters

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 ...

**-10**

votes

**2**answers

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} =...

**1**

vote

**1**answer

310 views

### 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 ...

**8**

votes

**6**answers

897 views

### Spaces of filters

This question arose more from curiosity than from an actual problem. There are situations when you embed some space $X$ in a set of filters on $X$, which inherits properties of $X$ or has even better ...

**1**

vote

**1**answer

370 views

### 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 ...