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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different definitions of a subnet

The classical definition of subnet seems to be that $\Psi: J\to X$ is a subnet of $\Phi: I\to X$ if there exists a monotone, final map $h: J\to I$ s.t. $\Psi = \Phi\circ h$. I found another definition ...
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Ultrafilter and upper natural densities

It is straightforward to show that there is an ultrafilter $\mathcal{U}_0$ on the positive integers such that every element $A\in \mathcal{U}_0$ satisfies $$d^\ast(A):=\limsup_{n\to +\infty} ...
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Why do ultraproduct structures use a quotient as their universe?

For an $L$-structure $\mathfrak{A}$ with universe $A$, if we have an index set $I$, with an ultrafilter $U$, we create an ultraproduct structure having as its universe $\Pi_I \;A_i/U$. This is the set ...
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Assuming CH, can every tower be extended to a selective ultrafilter (or even a p-point)?

Assume CH. A tower is an almost decreasing family $(A_\alpha)_{\alpha\in\omega_1}$ with no pseudointersection. A selective (also called Ramsey) ultrafilter is one with the property that for every ...
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Characterisation of convergence of bounded sequences via ultra-filters

Let $\{a_n\}_{n\in\mathbb N}$ be a bounded sequence of real or complex numbers and $\mathscr F\subset\mathscr P(\mathbb N)$ be a non-principal ultra-filter. Then $a=\lim_{\mathscr F}a_n$ is ...
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Relation between polynomials and derivatives in Savitsky-Golay fitting

I am seeking a more thorough explanation of some of the properties of Savitsky-Golay (S-G) filters, that are maybe not intuitive to people who have worked with least squares in other contexts. To ...
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Filter without cluster point, then the clopen members have empty intersection

Consider a topological space $(X,\tau)$ and a filter $F$ on $X$ with no cluster point. The set $C$ of all clopen members of $F$ has the finite intersection property. Why has the intersection ...
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PIT implies: In a boolean lattice, every filter can be enlarged to a maximal one

I am working through this proof of Herrlich's Axiom of Choice: $(1)\Rightarrow(2)$: How do you define the quotient lattice $B$ modulo a Filter? And why is the preimage of the maximal filter ...
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54 views

How does filtration model information?

Lets say you have a probability space $(\Omega, \mathcal{F},P)$ And a stochastic process on this space $\{X_t, t \in T\}.$ Assume that our process takes vaslues in $\mathbb{R}$. T is a totally ordered ...
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Proof: Every lattice has a maximal filter iff AC

I'm working through a proof of Herrlich's book Axiom of Choice, p.58 (Google books): Equivalent are Every lattice has a maximal filter. Axiom of Choice. In this book, a lattice is ...
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Topological space in which the principal filters are the only filters that converge

Let $(X, \mathcal{T})$ be a topological space in which only the principal filters converge. Show that $\mathcal{T}$ is the discrete topology. It is similar to one of my previous questions (link: ...
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Topological space in which every filter in which every filter converges to every point

Let $(X, \mathcal{T})$ be a topological space in which every filter converges to every element $x \in X$. Show that $\mathcal{T}$ is the trivial topology. I'm kind of stuck on this one for a while ...
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Implementation of the LOWESS-algorithim (local regression data smoothing)

I need to implement the LOWESS-algorithm in a piece of software I am working on. The LOWESS-algorithm is a type of filter, which applies a locally weighted regression on each data point. In this ...
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State space representation for fractional order transfer function

What is the state space representation for the following filters? $H(s)=\frac{Y(s)}{U(s)}=\frac1{s^\frac12}$ $H(s)=\frac{Y(s)}{U(s)}=\frac1{s^\frac12+1}$ Where $u(t)$ is the input and $y(t)$ is ...
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Where has this common generalization of nets and filters been written down?

It is well-known that there are two different ways to generalize the theory of convergence of sequences to arbitrary topological spaces: nets and filters. They are of course essentially equivalent, ...
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37 views

How to extends a filter to a convergent ultrafilter?

I am new to the ultrafilters, so I apologise if the question is too elementary. Let S be a collection of sets with the finite intersection property, in a non-compact Hausdorff space. S can be ...
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$M$-amenable ultrafilters on $\kappa$ are $\kappa$-powerset preserving

Let $M$ be a transitive model of $\operatorname{ZFC-}$ and let $$ j \colon M \rightarrow N $$ be elementary with $\operatorname{crit}(j) = \kappa \in \operatorname{wfp}(N)$. Let $U_j$ be defined by ...
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60 views

The analogy between two Rudin-Keisler orders

Given a set $X$, an ultrafilter $U$ on $X$, and a function $f\colon X\to Y$, we can push forward $U$ along $f$ to obtain an ultrafilter $f_*U$ on $Y$, defined by $C\in f_*U$ if and only if ...
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Filters and their refinements vs nets and their subnets [duplicate]

True or false? a) Let $(x_{\alpha})_{\alpha\in A}$ a net over a space $X$ and $(x_{h(\beta)})_{\beta\in B}$ a subnet, where $h:B\to A$ is monotone and final. Let $\mathcal{F_1}$ be the filter ...
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Proof that no σ-complete non-principal filter on a countable set can exist

Jech (3rd edition, page 73) defines a principal filter as {X : Xo ⊆ X} with Xo a non empty subset of S and (page 77) indicates that there is no non-principal σ-complete (i.e. ω1 complete) filter on a ...
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compactness of the Stone-Čech compactification by ultrafilters

My question is about the proof of compactness of the Lemma 3.1(page 5) in this paper. Let $\beta \mathbb{N}$ be the set of all the ultrafilters on $\mathbb{N.}$ For each $A\subseteq \mathbb{N}$, we ...
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31 views

How does the sample space remain constant in filtered

I am trying to understand the concept of filtered event space from the axiomatic probability. From my reference (lecture script by Ramon Handel, Princeton) the filtered probability space looks like ...
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Ultralimit of an eventually constant generalized sequence

Suppose that $x_{i \in I}$ is a generalized sequence on a compact Hausdorff space $X$, indexed by the directed set $I$, and with the property that $\exists \, j \in I$ such that $\forall i \geqslant ...
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Recursive total least squares?

I have been reading up on total least squares, as well as ordinary least squares, and I've noticed that there is a pretty simple/well known set up for a recursive least squares algorithm. However, I ...
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Is a non-principal ultrafilter the same thing as a free ultrafilter?

Can someone please confirm if a non-principal ultrafilter is the same thing as a free ultrafilter. I keep finding conflicting definitions so am not sure.
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Show that every proper filter on a set X can be extended to a proper prime filter?

Are the following enough to complete the proof? The union of a chain of filters is a filter. A maximal filter is an ultra-filter. How I can use Zorn's lemma to find the maximal filter?
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Showing that the set of all cofinite sets is a filter [closed]

Let $F = \{B \subseteq\mathbb N : \mathbb N\setminus B\text{ is finite}\}$. Show that $F$ is a filter on $\mathbb N$. Let $A$ be a non-empty set. Let $a\in A$. Let $F = \{B \subseteq A : a \in B\}$. ...
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31 views

Two ways to describe image of a filter under a function

Let $f$ be a function from a set $A$ to a set $B$. Let $\mathcal{A}$ be a filter on $A$. (Note that I do not require that all filters are proper.) It is easy to verify that $\{Y\in\mathscr{P}B \mid ...
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What is a normal filter on a partially-ordered set?

I'm familiar with the concept of a filter F on a partially-ordered set: a non-empty downward directed (any two elements in F have a common lower bound) upper set (all elements above an element in F ...
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Principal Ultrafilters on natural numbers

Let $E$ be a countable set of subsets of $\mathbb{N} $. Show that the filter generated by $E$ cannot be a non-principal ultrafilter. My idea of solution is: Let $D$ be the filter generated by $E$. ...
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Characterization of the deductive closure of a set of axioms.

Thinking about the seemingly isomorphic nature of theories (I take a theory here as a deductively closed set of sentences) and filters has lead me to ask myself the following question: Can the ...
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Description of filter $F$ generated by a subset $E$ of $\mathcal P(W)$

I am trying to solve exercises from the book Modal Logic by Patrick Blackburn, Maarten de Rijke and Yde Venema. I am having a problem to solve one of the exercises in the section 2.5, the exercise is ...
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Is this use of Zorn's lemma in the proof that every infinite set has a non-principal ultrafilter correct?

I just want to quickly confirm that I am using Zorn's correctly in this short proof. There is a non-principal ultrafilter on any infinite set $X$. Consider the filter $\mathcal{F}$ of of ...
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Limiting the size of near-coherence classes in $\omega^*$

We say that two ultrafilters in $\omega^*$ (i.e., two non-principal ultrafilters on $\omega$) are nearly coherent if there is a finite-to-one mapping $\varphi:\omega\to\omega$ such that $\beta\varphi$ ...
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Why does multiplying by a wavelet equal to bandpass filtering?

The wavelet transform is given as $ \gamma(s,\tau) = \int f(t) \psi (\frac{t - \tau}{s})dt $ However when it comes to carrying out this transform it turns out that it has a "band pass" effect on ...
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What else does ZFC prove about the “spectrum” of a cardinal number?

An auxiliary definition: Definition 0. Given an infinite set $X$ and a filter $\mathcal{F}$ on $X$, let $\sim_\mathcal{F}$ denote the equivalence relation on $\mathcal{P}(X)$ defined as follows: ...
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Origins of Kalman filter Algorithms in his paper in 1960

Concerning Kalman's original paper published in 1960, "A New Approach to Linear Filtering and Prediction Problems", it seems the majority is to show the orthogonal projection is the optimal estimation ...
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45 views

Construction of a Ramsey ultrafilter

I am having difficulties with the proof in Jech Set Theory concerning the existence of Ramsey filters in case the continuum hypothesis holds. A similar question about the same proof was asked here ...
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62 views

Can this sequence be a hyperreal number? What would be its real part?

Consider the sequence $\{a_n\} = \{\sin(n) \mid n\in \mathbb N \}$. Can this sequence be viewed as a hyperreal number? What could be its real part? Any intuition would be highly appreciated :) ...
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On the definition of a filter: Isn't $\emptyset$ a subset of any set?

Beginning my study of nonstandard analysis, I have found this definition of a filter U on a set J, where A, B are subsets of J: Proper filter: $\emptyset \not\in U$, Finite intersection property: If ...
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Are trace of z-filter in dense z-embedded subset z-filter?

I found this article about z-filter, referring to Lemma 3 my question is: without the "every member of which meet Y" hypothesis and adding that Y has to be dense in X is it still true? EDIT: forgot ...
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71 views

Give an example of a filter that can not be generated by a sequence.

As the title I'm looking for an example of a filter that can't be generated by a sequence. If you took it from somewhere provide the source please. Expanding: Every sequence can generate a filter ...
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100 views

Stone-Cech-Compactification

In the lecture, we introduced the Stone-Cech-compactification via ultrafilters. More concretely, we defined $\beta X = \{\mathfrak{U}|\mathfrak{U}$ ultrafilter on $X\}$. This is possible for $X$ ...
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Filters, nets and Galois correspondence

In the lecture, our prof. mentioned that the correspondence between nets and filters is a Galois correspondence without giving any more details about that. In algebra, the proof of the Galois ...
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109 views

Non-Principal Ultrafilters Confused!!

I've just started learning about filters and non-principal ultrafilters. I'm getting confused on the requirement: $U$ contains no finite subsets of $J$; where $U$ is the ultrafilter and $J$ is a set. ...
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Duals of filters, an explicit formula for meet?

Fix some set $U$. Recall that filters on $U$ are nonempty sets $F$ such that $A\cap B\in F \Leftrightarrow A\in F\land B\in F$. Replacing every element of $F$ with its complement and simultaneously ...
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A filter concentrates on a set

Given a filter $\mathcal F$ on some nonempty set $X$ and some $Y \subseteq X$, people often say that "$\mathcal F$ concentrates on $Y$". Questions: Does this simply mean $$\forall Z \subseteq X ...
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Generalizing a theorem about filters on a boolean lattice

Let $\mathfrak{A}$ be a bounded distributive lattice with binary meet and join $\sqcap$ and $\sqcup$. I will denote $\partial F = \{ X\in\mathfrak{A} \mid \forall Y\in F: X\sqcap Y\ne \bot \}$ where ...
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An alternate definition of ideals

A filter on a poset if by definition its nonempty subset $F$ such that it does not contain the least element and $A, B \in F \Leftrightarrow \exists Z \in F : (Z \le A \wedge Z\le B)$ for every ...
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Characterization of continuity in terms of filters

Two characterizations of continuity are For all filters $\mathcal{F}$, $\mathcal{F} \to x \implies f(\mathcal{F}) \to f(x)$. $f(\overline{S}) \subseteq \overline{f(S)}$ for all $S$ ...