Filters (and ultrafilters) are used in various areas of mathematics, e.g. general topology, set theory, boolean algebras, model theory. In topology they can be used to study convergence in a more general way than just convergence of sequences.

learn more… | top users | synonyms

0
votes
0answers
17 views

Is the locale of filters on an arbitrary lattice compact?

A mathematician has claimed in a private email to me, that the lattice of filters on every lattice is compact. I have proved it only for distributive lattices. I need help for non-distributive case. ...
-2
votes
0answers
18 views

Limit of sum of stochastic increments [closed]

Let $(X_t)$ be a continuous local martingale with respect to a filtration $\mathcal{F}_t$. Show that the limit $ \lim_{n\to \infty} \sum_{i=1}^{2^n} (X_{i/2^n}^4 - X_{(i-1)/2^n}^4)^2$ exist in ...
1
vote
1answer
20 views

Generalization of a certain riddle and ultrafilters (?)

I was once told the following riddle: 100 dwarfs stand in a straight line, each wears a hat of the colour red, yellow or green and they can see only the hats of the dwarfs in front of them. A dwarf ...
1
vote
1answer
31 views

A filter $G$ on $P$ is generic iff whenever $W$ is a partition of $P$, $G \cap W \not = \emptyset$

I'm finally struggling a little bit with the notion of forcing, so I decided to do some of the exercises in Jech's book (2nd Edition) in order to sharpen a little bit my intuitions about the whole ...
2
votes
1answer
47 views

Existence of ultrafilters

STATEMENT: An “ultrafilter” is a filter that is not properly contained in any other filter. Use Zorn’s lemma to show that every filter is contained in an ultrafilter. PROOF: Let $F$ be the set of all ...
0
votes
2answers
43 views

Equality of two expressions describing a filter

Let $U$, $W$ be boolean lattices with order $\sqsupseteq$, and $U \supseteq W$. The top element of $U$ is the same as the top element of $W$. The bottom element of $U$ is the same as the bottom ...
0
votes
0answers
15 views

Why we multiply the the estimations in EKF filter by Jacobin and its transpose

p = J * Pi * J' + Q This is what we do in prediction step i know why we use Jacobean and what it does, but my question is why again we multiply the value by its transpose. What is the function of ...
0
votes
0answers
7 views

Finding a discrete Kalman-type process that produces a given Frequency spectrum

Given a power spectral density from f = -1/2 .. 1/2, is it possible to find a 1st order process that produces this series? In other words, x_i+1 = G x_i + W r_i ...
4
votes
1answer
96 views

Compositions of filters on finite unions of Cartesian products

Let $\Gamma$ be the lattice of all finite unions of Cartesian products $A\times B$ of two arbitrary sets $A,B\subseteq U$ for some set $U$. See this note for other equivalent ways to describe the set ...
0
votes
0answers
72 views

Another conjecture about filters and cartesian products

Now I present a similar but different (weaker) question (the difference is a different definition of the set $\Gamma$): Let $U$ be some set. Let $\Gamma$ be the set of all finite unions ...
1
vote
1answer
26 views

Is a Set a family of filters over itself?

Given a nonempty set $X$, define a family of sets $F$ over $X$ as $F:= \{X\}$ Since $F$ is nonempty and $$F\cap F=F\subseteq F \cap F$$ does that mean that $X$ is a family of filters over itself? ...
3
votes
1answer
130 views

A conjecture about filters and finite unions of cartesian products

Let $U$ be some set. Let $\Gamma$ be the set of all finite unions of cartesian products ($X_1 \times Y_1 \cup \dots \cup X_n \times Y_n$) of sets on $U$. Obviously, $\Gamma$ is a a distributive ...
2
votes
1answer
30 views

Nets and directed sets problem

I am trying to solve the following exercise: Let $\Lambda$ be a directed set, and for each $\alpha \in \Lambda$ let $\Gamma_\alpha$ be a directed set. Suppose that for each $\alpha \in \Lambda$ there ...
0
votes
0answers
12 views

How do I get around finding a digital filter which is a narrow bandpass with a small group delay?

I need to find a digital filter that meets the following criterion: Is a narrow bandpass (1/40 of normalized frequency width) As small as possible group delay (preferably less than 300 samples) ...
0
votes
1answer
29 views

Neighbourhood filter system exercise

Problem Let $X$ be a set. A neighbourhood filter system $\mathcal F$ on $X$ is a rule that assigns to each element $x \in X$ a family $\mathcal F_x \subset \mathcal P(X)$ such that (1) if $x \in X$, ...
0
votes
1answer
15 views

Feature selection of Gabor filter

I am working with Gabor filter with level set method here. As you known, gabor filter gave many filter output. Some of them is not necessary. Could you suggest to me some method that can choose best ...
0
votes
0answers
23 views

Discrete-time lowpass filter with rapid response to significative changes in the input

I have a signal that looks like this: To obtain an average, I apply this formula y[i] = α * x[i] + (1-α) * y[i-1] With a relatively small value of α the ...
0
votes
1answer
22 views

Gaussian filter: Which scales are smoothed for a given sigma?

I do have (I hope!) a simple question. Let's say I have a time series with a discretization of "$dt$" (in real world: yearly values in my specific case). Now I use a Gaussian Filter with $\sigma = 4$, ...
0
votes
0answers
6 views

How can I apply a median filter directly to a time-varying rotation matrix?

I need MatLab script which would take a series of rotation matrices (referring to an actual physical object's orientation) and apply median filter to it to eliminate speckle noise from it. The way ...
0
votes
0answers
26 views

How to interpret the process noise co-variance matrix Q in 1D tracking Kalman Filter example

I was watching Youbute about the 1D object tracking with Kalman Filter( the link is here: https://www.youtube.com/watch?v=NT7nYv9Ri2Y) The position and velocity of the moving object are described by ...
1
vote
0answers
39 views

Filters on a set of filters, are they equivalent to just filters?

Let $F(X)$ be the set of all filters (including the improper filter) on a poset $X$, ordered reversely to set-theoretic inclusion of filters. Let $U$ be a set. Is $F(F(\mathscr{P}U))$ order ...
0
votes
1answer
63 views

What is purpose of correlation kernel? IIs it high pass filter or low pass filter?

I am research about correlation kernel and I have some questions that need your help. Let see the pp. 3302-3303 in the http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6517250&tag=1 The special ...
0
votes
0answers
66 views

Estimation covariance of the Kalman filter state

I implemented Kalman filtering for a simplest 1D coordinate+velocity model. The prediction worked, but I wanted to estimate the prediction probability distribution. I.e. how likely it is that the ...
3
votes
1answer
65 views

An ultrafilter product topology

Suppose $X=\prod _{i\in\omega}X_i$ is the cartesian product of topological spaces $X_i$ and $u$ is a filter on $\omega$. Define a basis for $X$ by taking the collection of all sets of the form ...
3
votes
1answer
40 views

Asymptotic cones of reals

Let me begin with the definition. Suppose $u$ is a free ultrafilter on $\omega$. Theorem. If $(r_n)$ is a bounded sequence of real numbers, then there exists a unique $l\in\mathbb R$ such that ...
0
votes
1answer
41 views

Help to conceive a name

Filter $F$ is defined by the formula $$A\cap B\in F \Leftrightarrow A\in F\wedge B\in F.$$ Ideal $F$ is defined by the formula $$A\cup B\in F \Leftrightarrow A\in F\wedge B\in F.$$ In my book I ...
0
votes
0answers
36 views

Adaptive whitening / decorrelation

I have multidimensional data as a set of vectors. I am currently whitening this data and removing the mean vector. I end up with decorrelated data with zero mean and variance equal to 1. I'm using ...
-1
votes
1answer
52 views

Multiple echo FIR Filter in Matlab [closed]

This is my final homework for the semester, I would be glad if you can help me.
0
votes
0answers
19 views

Filtrations and sigma algebras [duplicate]

I have a doubt concerning the basilar aspects of the filtrations in the stochastic theory. A filtration is an increasing sequence of $\sigma$-algebras on a measurable space. That is, given a ...
0
votes
3answers
44 views

A non-principal ultra filter containing the even numbers, need hint now.

I posted a question about an exercise asking to prove that there exists a non-principal ultra filter on N containing the set of even numbers. My original post asked about a possible answer. It was ...
1
vote
0answers
27 views

Principal order filters on a POSET

I have another problem, but in this one I have no idea how to start. Let be $(X,\leq)$ a POSET with a first element and gifted with the topology $\{ (a,\rightarrow) : a \in X \}$ (principal order ...
1
vote
1answer
67 views

Determine frequency response from impulse response

I'm studying signal processing, using MATLAB to plot filter responses. So far, I understand I can use the impulse response to apply a filter to a signal. For example, the impulse response of an ...
1
vote
1answer
41 views

A non-principal ultra filter containing the even numbers

I just started trying to read through Lectures on the Hyperreals: An Introduction to Nonstandard Analysis, by Robert Goldblatt. I'm near the beginning in the section on filters. One of the exercises ...
3
votes
1answer
23 views

Language clarification in an article about filters

I started reading these notes. After enumerating four properties of a filter $\mathcal F$ in a topological space $(X,\tau)$ (1) $X\in\mathcal F$; (2) $V\in\mathcal F\wedge V\subseteq ...
1
vote
3answers
70 views

Definition of a principal ultrafilter

I just started trying to read through Lectures on the Hyperreals: An Introduction to Nonstandard Analysis, by Robert Goldblatt. I'm near the beginning in the section on filters. He's defined an ...
2
votes
2answers
43 views

how to find an omega-complete non-principal ultrafilter on omega?

Omega is the set of all natural numbers.Can anyboby give me an exact example of an omega-complete non-principal ultrafilter on omega?(I want an exact example,not just to prove the ultrafilter exist)
0
votes
1answer
41 views

Are ACF and Ultrafilter Lemma/BPIT equivalent?

$\mathsf{ACF}$ is the proposition that every set of nonempty finite sets has a choice function. It can be shown that $\mathsf{BPIT} \Rightarrow \mathsf{ACF}$, because $\mathsf{BPIT}$ implies that ...
2
votes
1answer
42 views

Are there any purely semantic proofs of the compactness theorem that don't use the full axiom of choice? [duplicate]

Using Godel's completeness theorem, it can be shown that the compactness theorem is equivalent to the ultrafilter lemma. The compactness theorem can also be proven using ultraproducts and Los's ...
3
votes
1answer
52 views

Is this an equivalent characterization of rapid filters?

A filter $\mathcal F$ on $\omega$ is called rapid filter if for every function $f\in\omega^\omega$ there exists $X\in\mathcal F$ such that $|X\cap f(n)|\le n$ for $n\in\omega$. In Lemma 4.6.2 in the ...
0
votes
0answers
21 views

Choosing from two index families of sets

Let $a$ be a (nontrivial) ultrafilter and $n$ be an infinite set. Let also $U$ be an infinite set. Define $n$-ary relation $\phi$ on $\mathscr{P}U$ by the formula $L\in\phi \Leftrightarrow \forall ...
1
vote
1answer
42 views

Example of an easy non-pseudotopological convergence

Recently, I have had an introduction to convergence spaces and I was wondering if there is an easy example of a non pseudotopological convergence space. Thank you for your help.
1
vote
1answer
17 views

Neighbourhood filter of an isolated point of a topological space

How can I prove this? Let $(X,\tau)$ a topological space, and $x \in X$. Then the neighbourhood filter $\mathcal{V}(x)$ is an ultrafilter if and only if $x$ is an isolated point of $X$ Thanks a ...
2
votes
1answer
46 views

Subsets and finer filters

Suppose $G$ is a finer filter than $F$ in a topological space $X$. Is the net base in $G$ a subnet of the net base in $F$? I am using the definitions of General Topology of Willard. Thank you
1
vote
1answer
41 views

Neighborhood vs. Neighborhood filter

Say I build some sort of Topology If $(X,\mathcal{T})$ is a topological space and $p \in X$, a $\textit{neighbourhood}$ of $p$ is a subset $V$ of $X$, in which $p \in U \subseteq V$, $U$ is open. We ...
1
vote
0answers
49 views

“Filters” of associative rings

Does there exist some good notion of "filters" for arbitrary associative rings with unity that would generalize filters of Boolean rings? For me, if $R$ is an associative ring with unity $1$, it ...
2
votes
1answer
76 views

Build an ultrafilter finer than the Frechet filter.

I need to build an ultrafilter finer than the Fréchet (Filter finite complements). In an infinite set $X=\mathbb{R}$, $\mathcal{F}_{c}=\{A\subseteq X\mid A^{c}\ \ \ \text{is a finite set}\}$ is the ...
3
votes
2answers
88 views

Ultrafilter Lemma implies Compactness/Completeness of FOL

Apologies if this has been asked somewhere before, but I didn't see what I was looking for after several pages of Google results. I was reading Jech's The Axiom of Choice and was introduced to the ...
2
votes
1answer
41 views

Why is there a bijection between the ultrafilters that converge and a topology

If we call $\mathcal{UF}(X)$ the set of ultrafilters on a set $X$. I read here that there is a bijection between topologies on a set $X$ and $\{0,1\}^{\mathcal{UF}(X)}$. As I am unfamiliar with ...
3
votes
1answer
53 views

Unique ultrafilter on $\omega$

We know that from axiom of choice (or just BPIT) we can deduce ultrafilter lemma, which states that every filter can be extended to an ultrafilter. From this lemma we can derive existence of at least ...
0
votes
0answers
26 views

One more question about filters

Let $S$ be a set of subsets of a set $U$ such that $\bigcup T \in S \Leftrightarrow \exists X \in T : X \in S$ for every set $T$ of subsets of the set $U$. Prove (or disprove) that there exists a set ...