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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pointwise convergence of a filter on $\mathbb{R}^\mathbb{R}$

In my topology lecture we have defined pointwise convergence for filters on function spaces, say $\mathbb{R}^\mathbb{R}$. A filter $\varphi$ on $\mathbb{R}^\mathbb{R}$ converges pointwise to ...
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Why do we need ultrafilters to make sense of the cartesian product of $\mathcal{L}$-structures

I'm trying understand why we need ultrafilters in model theory. Here is how I see things. Could someone tell me if this is correct ? Further explanations are always welcome. Let $\mathcal{L}$ be a ...
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Filter output of a signal

So I have a filter $$H(z) = 0.5 + 0.5z^3 = (1/2, 0, 0, 1/2)$$ and need to find the output of it on a cyclical signal $$x = (..., 3, -1, 2, 1, 5, 2, 3,-1, 2, 1, 5, 2, 3,...) $$ Would the output be ...
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Literature about Ultrafilters

I am in the early stages of planning my senior project and was wondering if anybody had some recommendations of literature about the applications of ultrafilters in social choice theory, along with ...
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Every nonprincipal ultrafilter on $\omega$ is uncountable.

I imagine this is true as it is easy to prove that any principal ultrafilter on $\mathbb{N}$ is uncountable, and nonprincipal ultrafilters seem in a way bigger. (My proof is $\mathcal{P}(\mathbb{N})= ...
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Proving that every filter is $ℵ_0$–complete

I know that every filter is $ℵ_0$-complete from the definition of being k-complete where k is an infinite cardinal. However I don't know how to prove it specifically.
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Iterated Limits Along an Ultrafilter

Setting: Let $\mathfrak{U}$ be an ultrafilter on an index set $I$. Let $G$ be a compact group with identity $e$, and let $\mathbb{T}$ denote the unit circle in the complex plane. For each $i\in ...
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Connection between ultrafilters and maximal consistent sets

I have been told by reputable sources that ultrafilters over the set of all formulas in a given logic corresponds to a maximal consistent set of formulas in that logic, and I am trying to wrap my head ...
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An elementary proof about filters

I my book draft I have proved a theorem which is equivalent to the following. My proof uses ultrafilters, Galois connections, and the cofinite filter. Let $S$ be a set of filters on some set $U$. ...
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Strengthening of a theorem about filters vs a counter-example

Let $S$ be a non-empty set of filters on a meet-semilattice. If our semilattice is a distributive lattice, then the supremum (on the poset of filters ordered by set-theoretic inclusion) of $S$ is the ...
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Let $\mathcal{U}$ now be an ultrafilter on $\mathbb{N}$ such that $\mathcal{U}\subseteq 2^\mathbb{N}$ is non-principal.

Let $\mathcal{U}$ now be an ultrafilter on $\mathbb{N}$. View $\mathcal{U}$ as a subset of $2^\mathbb{N}$. Question: If $\mathcal{U}$ is non-principal then $\mathcal{U}$ does not have the Baire ...
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Are continuous functions on uniform spaces Cauchy continuous?

Suppose $X,Y$ are uniform spaces. Since uniform structures give topologies $X,Y$ are naturally topological spaces, so we can consider continuous functions $X \to Y$. What I am wondering is if $f ...
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If $U$ is an ultrafilter on $\mathbb{N}$, then $U$ limits exist.

This is rather silly, I expect Asaf will point out what I am missing immediately. Let $U$ be a filter on $\mathbb{N}$. If $\{a_n\}_{n=1}^\infty$ is a sequence of reals, we write $\lim_U a_n = a$ if ...
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When the preimage of the family of all open neighborhoods of a point is cofinal

Let $f \colon X \to Y$ be a continuous map of topological spaces. Denote by $O_y(Y)$ the family of all open neighborhoods of a point $y \in Y$. Define $$ f^{-1}(O_y(Y)) = \bigl\{ f^{-1}(U) \colon U ...
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Non-measurability of ultrafilter on $\omega$

It is well-known that any non-principal ultrafilter on $\omega$ is non-measurable regarded as a subset of $2^{\omega}$. My question is "how well-known" is this fact? Here is the only proof I know: ...
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Prove that $\mathscr F^+ = \mathscr J^+$ and $a \in \mathscr J^+$ iff there exists a filter

Let $\mathscr F$ a filter and let $\mathscr J = \mathscr F^c$ be an ideal dual to $\mathscr J$. Denote $\mathscr F^+ = \{a : (\forall x \in \mathscr F)[a \land x \neq 0]\}$ and $\mathscr J^+ = \{a:a ...
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Distributivity of lattices and prime ideals.

I need help proving that, if $L$ is a lattice with the property that for every nonempty proper filter $F$ and every ideal $I$ such that $F \bigcap I = \emptyset$ then there is a prime filter $G$ such ...
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If a set X has the finite meet property, then there is an ultrafilter such that X is a subset of it.

I need to prove that if $X \subseteq B$ is a set with the finite meet property, then there exists an unltrafilter $U$ of $B$ such that $X \subseteq U$. I know that the finite meet property means that ...
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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 ...