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.

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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 ...
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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 ...
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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 ...
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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 ...
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58 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 ...
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Kalman Filter, accuracy of the current estimate

I am running experiments on the kalman filter, with a particular interest on the covariance Matrix. I am using the formulas given at http://en.wikipedia.org/wiki/Kalman_filter and something which ...
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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 ...
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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 ...
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An Explanation of the Kalman Filter

In the past 3 months I've been trying to understand the Kalman Filter. I have tried to implement it, I have watched youtube tutorials, and read some papers about it and its operation (update, ...
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envelope detection for audio transient suppression

I want to use exponential moving average filter( 1st order IIR) for envelope detection, to apply gain for transient suppression in audio. My requirement is that say at input x(n-1)[ when it starts ...
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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 ...
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Multiple echo FIR Filter in Matlab

This is my final homework for the semester, I would be glad if you can help me.
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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 ...
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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 ...
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26 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 ...
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29 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 ...
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1answer
37 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 ...
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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 ...
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55 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 ...
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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)
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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 ...
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38 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 ...
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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 ...
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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 ...
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36 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.
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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 ...
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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
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34 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 ...
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“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 ...
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74 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 ...
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73 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 ...
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38 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 ...
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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 ...
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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 ...
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24 views

A question about filters on a set

I will call a set $S\in\mathscr{P}U$ a free star (on some fixed set $U$) iff $X\cup Y\in S \Leftrightarrow X\in S\vee Y\in S$ for every sets $X,Y\subseteq U$ and $\varnothing\notin S$. Let ...
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Splitting Frechet filter into two proper filters

Let $\Omega$ be a Frechet filter (=cofinite filter) on an infinite set. Do there exist proper filters $\mathcal{A}$ and $\mathcal{B}$ such that $\mathcal{A}\cap\mathcal{B} = \Omega$ and ...
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22 views

Going down from filters to sets for specific relations

Let $\mathfrak{A}$ be a bounded lattice. I call a $2$-staroid a relation $f\in\mathscr{P}(\mathfrak{A}\times\mathfrak{A})$ such that $i\sqcup j \mathrel{f} b\Leftrightarrow i\mathrel{f} b\vee ...
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Certain constructs on filters and on principal filters

Let $\mathfrak{X}$ be a lattice. I will call a set $S\in\mathscr{P}\mathfrak{X}$ a free star when the least element of $\mathfrak{X}$ is not in $S$ and $X\sqcup Y\in S\Leftrightarrow X\in S\vee Y\in ...
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39 views

Amplitude Spectrum, Nyquist Frequency, mixed/min/max wavelets

The problem is here. Now I know the definition of mixed/max/min phase wavelets, whether the roots lie within the unit circle or not. Starting from n = 1, let $$ x_t = ( 5, 6) $$ $$ X(z) = 5 + 6z $$ ...
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Stone-Čech compactification using ultrafilters

Let $X = \omega \cup \{ x \}$ ne the Stone-Čech compactification of $\omega$. (I am viewing $X$ as a subspace of the set of ultrafilters over $\omega$). Let, $\mathcal A$, $\mathcal B$ be two disjoint ...
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94 views

“Hidden” axiom of choice?

Let $\mu$ be a measure on $S$ such that: $\mu\left(\emptyset\right)=0$ and $\mu(S)=1$ if $X\subseteq Y$, then $\mu(X)\leq\mu(Y)$ $\mu\left(\{a\}\right)=0$ for all $a\in S$ if $X_n$, ...
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Building normal filters around a stationary set

Recently I've been looking at connections between Laver functions on large cardinals and diamonds. While $\diamondsuit$-like principles tend to readily generalize to Laver function-like concepts, I've ...
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28 views

Equivalent definitions of P-Points (Ultrafilters)

It is stated in lots of places (including this one) that these two are equivalent definitons for an ultrafilter $u$ to be a p-point. If for every sequence $\left < A_n: n \in \omega \right>$ ...
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Questions related to intersections of open sets and Baire spaces

EDIT: I have reposted this question on MathOverflow. (The version posted there is more concise, with some details omitted. I have also added a question about pseudobases with similar property.) MO ...
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Non forking extensions of types as extensions of filters

Given a set of parameters $A$ a type in $S_n(A)$ may be thought of as a maximal filter on the monster model which can be constructed from $A$-definable subsets. Given a type $q\in S_n(B)$ saying that ...
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Different definitions of P-Points (ultrafilters)

Recently I have been exposed to the concept of P-Points. I have three definitions: A non-principal ultrafilter $u$ is a P-Point if: For every sequence $\left < A_n \right >_{n\in \omega}$ of ...
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An other question about filters and ultrafilters

Let $a$ be an ultrafilter on a set $\mho$. Let $\mathcal{L}$ be an $a$-indexed family of filters (that is $\mathcal{L}$ is a function from $a$ to the set of filters on $\mho$). Conjecture: For every ...
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A problem about filters and ultrafilters

Let $a$ be an ultrafilter on a set $\mho$. Let $\mathcal{L}$ be an $a$-indexed family of filters (that is $\mathcal{L}$ is a function from $a$ to the set of filters on $\mho$). Are the following two ...
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An open ultrafilter converges if and only if it has non-empty intersection?

If $(X,\mathcal T)$ is a topological space then $\mathcal F\subseteq\mathcal T$ is called an open filter on $X$ is (i) $X\in\mathcal F$, $\emptyset\notin\mathcal F$; (ii) $A,B\in\mathcal F$ ...
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Boolean-valued model and the use of generic ultrafilter in ZFC

So I asked the question about generic filter; but I was also reading http://math.mit.edu/~tchow/forcing.pdf which is a forcing (in ZFC) guide for dummies. Then I was struck with the part where it ...