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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Where can I find material on polynomial filters?

Most students and mathematicians probably know a fair amount on roots-of-unity filters, or on Fourier analysis. The basic notion of this "filtering" is, given a polynomial, we can find the $n$th ...
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Stone Representation Theorem

Given two Boolean algebras $A$ and $B$ such that $A$ is a subalgebra of $B$. What is the relation between the Stone space of $A$ and the Stone space of $B$. The question maybe silly but I am getting ...
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About a function ranging filters

Let $U$ be an (infinite) set and $N$ be an (infinite) index set. I denote $\mathfrak{A}$ the set of filters on $U$ (including the improper filter). Let $f$ be an $N$-ary relation that is a set of ...
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Another way to express certain filter

Let $\mathfrak{A}$, $\mathfrak{B}$ be posets, $\mathfrak{A} \subseteq \mathfrak{B}$ (and $\mathfrak{A}$ is the induced order of $\mathfrak{B}$). Let $\mathcal{A}$ be a filter on $\mathfrak{A}$. Help ...
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An example of an ultrafilter

This is Theorem 3.5 (pp. 150) of the book "A course in universal algebra." http://www.math.uwaterloo.ca/~snburris/htdocs/UALG/univ-algebra.pdf Theorem 3.15. Let $\bf B$ be a Boolean algebra. (a) ...
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References on filter quantifiers

This post is primarily a reference request. In combinatorics and other areas, we use filter quantifiers to simplify the statements of various definitions, theorems and proofs. The general idea is ...
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Ultraproducts and Elementary Embeddings

Let $K= \{A_i: i\in \omega\}$ be a countable collection of $L-$structures. Suppose that for each $A_i, A_j$ in $K$, $\exists A_p \in K$ such that $i,j< p$ and $A_i \prec A_p $ and $A_j \prec ...
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Carlson's translatability - are theses characterisations equivalent?

Given a translation-invariant ideal $\mathcal{I}$ on a commutative group $G$ and it's dual filter $\mathcal{I}^*$, I am trying to show that $$ (\forall I \in \mathcal{I})(\exists I' \in ...
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Net and filter generated by it

Let {s(a)} -such that (a) belongs to order set (A)- is a net from the point of (X) , the net {s(a)} converges to (x) if and only if the filter that generated by it converges to (x)
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Prove that $\sigma (\cap_{i \in I} C_i)=\cap_{i \in I} \sigma (C_i)$

Do we have the following identity? $$\sigma (\cap_{i \in I} C_i)=\cap_{i \in I} \sigma (C_i)$$ Here $C_i$ is a subset of a set $\Omega$.
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Does ultralimit of sequence change after shift?

Let $(a_n)$ be a bounded sequence of numbers, $\omega$ be an non-principal ultrafilter on $\mathbb N$, then one can assign a limit along ultrafilter $(\omega-)\lim a_n$ to it as is said here. This ...
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Is $\mathscr{B}$ a filter on $X$, or only a filter base

Let $X_1$ be a nonempty set, $\mathscr{B}_1$ a filter on $X_1$, $X_2$ a nonempty set, $\mathscr{B}_2$ a filter on $X_2$; now we define $X=X_1\times X_2$ and $$\mathscr{B}=\{B_1\times B_2:B_1 \in ...
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Is there any filter space characterization for strong cardinals?

The theory of filter spaces is introduced by Apter, Diprisco, Henle & Zwicker, in their joint paper: Arthur Apter, Carlos Di Prisco, James Henle, and William Zwicker, Filter spaces: towards a ...
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Convergence of filters

Let (X,T) be a topological space. If F is a filter on X, then B:={G⊆X |G∈T and G∈F} is a basis for a filter F° on X. Prove that for x ∈ X the filter F° converges to x if and only if F converges to x. ...
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MATLAB implementation Spline Fitting

Check the attached problem please. I am a beginner in spline fitting and have a few questions: 1) How to find the coefficients c[n]. Is it by DTFT? 2) I understand how to find the derivative but ...
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Is the set of pseudo-complements of the elements of an ideal in a pseudocomplemented lattice a filter?

Let $L$ be a pseudocomplemented distributive lattice with $0$ and $1$, $I \subseteq L$ an ideal and set $F = \{\neg x \; | \; x \in I\}$, where $\neg x$ is the pseudocomplement of $x$. My question is: ...
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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. ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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? ...
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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 ...
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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 ...
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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) ...
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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 $x \in X$, ...
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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 ...
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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 ...
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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 $\sigma = 4$, ...
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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 ...
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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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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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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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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 [closed]

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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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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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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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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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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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)