Questions tagged [filters]

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. For questions about filters in the sense of signal processing, do not use this tag and instead use (signal-processing) or another appropriate tag.

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When the restriction of a maximal filter is maximal? [closed]

Let P be a distributive lattice and Q one of its sublattices. Let F be a maximal filter in P, thus F interesecated with Q is a filter in Q. But, when it is maximal?
user826451's user avatar
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1 answer
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Not closed under ultraproduct, then not closed under "shorter" ultraproduct

The problem Let $L$ be a first-order logic, $K$ a class of $L$-structures closed under elementary equivalence and $\kappa$ the cardinality of the set containing all formulas of $L$. Prove that if $K$ ...
Dave the Sid's user avatar
1 vote
1 answer
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Class of countably join-complete lattices is not closed under ultraproduct

I need to prove the above claim by constructing a counterexample. My intuition is that $\mathcal{A}_n = \langle A, \leq \rangle$ where $A = (0, 1, \ldots, n)$ should work, but cannot verify why. Any ...
Dave the Sid's user avatar
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15 views

Probability of a Markov Chain value conditional on an observation of the next step

Let $X=(X_i)_{i \ge 1}$ be a Markov chain and $Y=(Y_i)_{i \ge 1}$ be such that $Y_i=\phi(X_i)$ for an arbitrary function $\phi$. In attempting to derive a recurrence relation for the expectations $\...
hegash's user avatar
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2 votes
1 answer
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Convergence of a net and its associated filter

I am reading "Topology: An Introduction" by Waldmann and I am trying to prove the result (iv) in Proposition 4.2.6: A net converges to $p$ $\iff$ its associated filter converges to $p$. I ...
NotNow11's user avatar
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Lossless steering Order of Magnitude estimation

This is an equation from Rau's book "Quantum theory-an Information Processing Approach". It's leading to "lossless steering" of a particle if you insert an infinite number of ...
N1otAn1otherN1ame's user avatar
0 votes
0 answers
34 views

Kaiser window equation for attenuation doesn't match the figure

In the article linked below, no paywall, their Figure 4 shows a Fourier transform of a Kaiser window with Alpha=8. In it, the attenuation is shown to be As ~ 30db. Below the figure, their Equation 20 ...
Ray J's user avatar
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Can the reals be embedded in an ultraproduct of finite fields?

Is it possible (using ZFC or any other axiom system which is known to be as consistent as ZF) to prove the existence of an ultrafilter $\mathcal F$ on the set of prime numbers s.t. there is an ...
Roee Sinai's user avatar
2 votes
1 answer
73 views

The equivalences between points in a locale in constructive mathematics

I am currently following the definitions of the book Frames and Locales and those in the articles Pointfree topology and constructive mathematics and Topo-logie. There are at least three ways of ...
Dylan Facio's user avatar
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Is the delay tracking a ramp for a unit gain stable causal LTI system equal to the group delay calculated at frequency (or pulsation) zero?

I am trying to get my head around the following problem: I have a unit-gain stable LTI system (we can assume it is a discrete-time one) of which I need to calculate the delay in tracking a ramp signal ...
Michele Martino's user avatar
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Find vectors a,b of causal discrete time filter, that correspond to moving average filter

I came across the following problem in a signals graduate class: Given the following general formula for a discrete causal filter, let's name (1): $y[n] + \sum_{j=1}^{M} a_j*y[n-j] = \sum_{i=0}^{N} ...
spiros's user avatar
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1D power spectrum of field filtered by 3D filters

Consider a 3D scalar function $u(x,y,z)$, in a numerical implementation represented by discrete values $u_{i,j,k}$ on a Cartesian grid. If this scalar field is filtered by a 1D filter, either ...
Vladimir F Героям слава's user avatar
2 votes
1 answer
65 views

Taking the ultrapower of the real numbers as an ordered field twice.

I've seen ultrapowers of $\mathbb{Z}$ and $\mathbb{R}$ discussed in this question, with the $\mathbb{R}$ case being described as the more straightforward case. I'm wondering what happens when you ...
Greg Nisbet's user avatar
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Ultraproduct with respect to a partially ordered set

I am interested in learning about limits with respect to ultrafilters on a poset (partially ordered set) and ultraproducts with respect to a poset. However, all I can find about this topic only ...
Nanoputian's user avatar
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Calculating the Jacobian with respect to pixel intensity for Gaussian Blur

I'm applying a Gaussian blur to an image where each pixel of the original image is being optimized for some purpose and the Gaussian blur is an intermediate transformation on the pixel intensities. It ...
R S's user avatar
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1 answer
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Prove that the forgetful functor $U: \textbf{Ab} \to \textbf{Set}$ preserves all filtered colimits

I realize that my question is exactly the same as this post here. However, I tried finding the book that was mentioned, Borceux's Handbook of Category Theory I, but my efforts to find the book here ...
love and light's user avatar
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1 answer
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How can I prove the existence of delta-incomplete / countable incomplete ultrafilters?

I am reading a W.A.J Luxemburg paper about nonstandard analysis (https://www.jstor.org/stable/3038221). He presents the following definition I am stuck trying to prove the existence of delta-...
DAGO's user avatar
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Equivalent statements of "weak" choice

We can state the axiom of choice ($\mathsf{AC}$) in the following way $$\forall x\Big(\neg(\varnothing\in x)\longrightarrow\exists f\big(f\text{ is a function from $x$ to $\bigcup x$ }\wedge \ \forall ...
Yester's user avatar
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5 votes
1 answer
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Does the term "free" in "free ultrafilter" have a meaning related to category theory?

I know that free ultrafilters are defined in contrast to principal/fixed ultrafilters. Nonetheless, is there some categorical way to view the use of the word "free" here (e.g. some pair of ...
Kellen Brosnahan's user avatar
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1 answer
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How to show this property of ultrafilter on a set $X$

Def: Filter: A filter on a set $X$ is a collection $F \subset P(X)$ such that satisfies $a,b,c$ Filter basis: A filter on a set $X$ is a collection $F \subset P(X)$ such that satisfies $a,b$ ...
Andrew_Ren's user avatar
1 vote
1 answer
44 views

Properties of proper maps using filters

I am reading a book on covering maps in Bourbaki-style with the following definitions: A map $f:X\to Y$ is separated if for every $x,x'$ in the same fiber, there exist open disjoint neighbourhoods of $...
Dr. Heinz Doofenshmirtz's user avatar
2 votes
1 answer
80 views

How to force character of ultrafilter be equal to $2^k$?

Let $k$ be an infinite cardinal. We already know there are exactly $2^{2^k}$ distinct non-principal ultrafilter on $k$. Here The set of ultrafilters on an infinite set is uncountable. And proof uses ...
Cezar's user avatar
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2 votes
1 answer
114 views

Ultrafilters arising from homomorphisms

Reading a paper from Bagaria and Magidor I stumbled upon a theorem which's probably well-known (but not to me at the time), the Łoś-Eda theorem. In the form stated: There exists a nontrivial group ...
interregno's user avatar
10 votes
1 answer
174 views

Is every "filter" of rings principal?

Let $\mathcal{F}$ be a class of rings* with the following property: If there is a homomorphism of rings $A \to B$ with $A \in \mathcal{F}$, then $B \in \mathcal{F}$. Moreover, $\mathcal{F}$ is closed ...
Martin Brandenburg's user avatar
3 votes
1 answer
63 views

Intersection of a filter of clopen sets is of pozitive measure?

Let $X$ nonempty set and $\tau$ a compact topology on it. Let $\mu$ be a measure on the Borel $\sigma $- algebra of $\tau$. Suppose we have a filter $\mathcal{F}$ of clopen sets such that $\mu(f)\geq0....
Cezar's user avatar
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2 votes
2 answers
78 views

Can an Ultrafilter be defined in terms of the Convergence of a Sequence?

We know that the set $\{\sin(0), \sin(1), \sin(2), ... \}$ is dense in the interval $(-1,1)$. So now consider the sequence $S$ = $\langle \sin(0), \sin(1), \sin(2), \ldots \rangle$. This sequence ...
Jonathan Hoyle's user avatar
4 votes
0 answers
95 views

Is there a robust version of the moving least squares or of the Savitzky–Golay filter?

Is there a name for the following type of filter? I want to filter a noisy signal $f(x) = f_0(x) + noise(x)$ (where $f_0$ is a noiseless signal), to get a filtered signal $f_\text{F}(x)$ while ...
HelloGoodbye's user avatar
1 vote
0 answers
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Explicit example of a discontinuous function between convergence spaces with the preimage of every open set open

The definition of continuity for convergence spaces is in terms of filters, but it’s still true that the preimage of any open set is open. Since this is no longer taken as the def of continuity, I ...
Rubaiyat Khondaker's user avatar
0 votes
1 answer
24 views

Small doubt concerning treatment of filters in v. Dalen's L&S

This is a rather miniscule —and most likely borne of misunderstanding— doubt concerning the treatment and definition of "filters" (in particular their characterization as "proper") ...
Sho's user avatar
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1 vote
0 answers
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Is this condition redundant (neighborhood filter for TVS in Trèves)?

In Trèves's Topological Vector Spaces, Distributions and Kernels, Theorem 3.1 is as follows. A filter $\mathscr F$ on a vector space $E$ is the filter of neighborhoods of the origin in a topology ...
WillG's user avatar
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4 votes
1 answer
250 views

A property related to cofinal similarity

A partial order $\le$ is reflexive, transitive and antisymmetric. It is directed, if for all $x, y$ in the set there is $z$ such that $x \le z$ and $y \le z$. Let $X, Y$ be partially ordered, directed ...
Ulli's user avatar
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0 votes
0 answers
44 views

Is every free ultrafilter on the natural numbers identical modulo permutation of the numbers? [duplicate]

Question Is every free ultrafilter on the natural numbers identical modulo permutation of the numbers? My thoughts I just thought this was interesting because knowing how much structure all the free ...
it's a hire car baby's user avatar
0 votes
1 answer
42 views

If a function preserves all filter limit points it is continuous

In Pete L. Clark's notes on convergence it is stated that (Proposition 5.14): For a set function $f:X \to Y$, if for every prefilter $F$ on $X$ with a limit point $x$, $f(F)$ has $f(x)$ as a limit ...
Fernando Chu's user avatar
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0 votes
1 answer
151 views

What does it mean to explicity exhibit something (modulo a proof of its existence) which cannot be explicitly exhibited?

I was reading this answer that no free ultrafilter can be exhibited on the natural numbers. I have as a theorem that if the Collatz conjecture is true then the following is a free ultrafilter on the ...
it's a hire car baby's user avatar
2 votes
0 answers
74 views

How many affine prime-quotient ultrafilters does a rational semiring have?

I know ultrafilters are considered powerful by more-learned mathematicians than I. I cannot profess to understand the reasons how and why although I can see the power of Zorn's Lemma and the axiom of ...
it's a hire car baby's user avatar
3 votes
2 answers
67 views

Quotient of infinite product of fields which is semi-artinian

Let $I$ be an infinite set (e.g. $I=\mathbb{N}$) and let $k$ be a field. The ring $R=\prod_I k$ is a notorious example of a ring which is absolutely flat (alternatively said von Neumann regular) and ...
N.B.'s user avatar
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2 votes
1 answer
69 views

A bounded increasing net converges – formulated using filters

We know the following monotone convergence theorem: If $n:X\to\mathbb R$ is a bounded increasing net, then $n$ converges. (Also for other spaces than $\mathbb R$.) I am trying to formulate this ...
Dominique Unruh's user avatar
0 votes
0 answers
18 views

No filter generated by a described family with the SFIP can be ultrafilter

Let $\mathcal{F} \subseteq \mathcal{P}(\mathbb{N})$ a "described", countable family of subset of $\mathbb{N}$ with the strong finite intersection property (that is, every nonempty finite ...
user1561017's user avatar
2 votes
0 answers
30 views

Extension of single zero-crossing property

Let $f\in\mathscr{C}^2(\mathbb{R},\mathbb{R})$ a strictly increasing function, striclty convex on $(-\infty,0)$, strictly concave on $(0,\infty)$ and let $\sigma_1>\sigma_2>0$ be two real ...
NancyBoy's user avatar
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Prove that if $\mathcal{F}[x_d]\subseteq \mathcal{G}$, then there's a subnet $(y_e)$ of $(x_d)$ such that $\mathcal{G}=\mathcal{F}[y_e]$.

Let $X$ be a set. Given any net $(x_d)_{d\in D}$ in $X$, we define $$\color{red}{\mathcal{F}[x_d]}:=\big\{F\in 2^X:(\exists d_0\in D)(\forall d\in D)\big(d_0\preceq _Dd\Rightarrow x_d\in F\big)\big\}.$...
rfloc's user avatar
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0 answers
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Why is there an asymmetry in the scale of these filter kernels?

In this video, the kernel for a low pass filter is given to be $[\frac{1}{2},\ \frac{1}{2}]$, while the kernel for a high pass filter is given to be $[-1,\ 1]$. Interpreting these as essentially ...
user10478's user avatar
  • 1,854
2 votes
1 answer
94 views

Measurable Cardinals, Elementary Embeddings, and building up to transitive collapse

I am looking at the proof that $\kappa$ a cardinal is measurable iff it is the critical point of an elementary embedding from $V$ to an inner model $M$, specifically the forward direction. I ...
abetray's user avatar
  • 95
1 vote
1 answer
51 views

Volatility as an Envelope for a Time Series

In time-series analysis, I keep running into graphs that look roughly like... Recalling a trig computation from an ODE book which explored the acoustic phenomena of beats, I noticed the orange ...
user10478's user avatar
  • 1,854
6 votes
1 answer
96 views

What length recursion constructs a free ultrafilter on $\Bbb{N}$?

Assume ZF together with some choice principle needed to make the choice in the following recursion that constructs a free ultrafilter on $\Bbb{N}$ $\mathcal{F}_0$ is the filter of co-finite subsets ...
Chad K's user avatar
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0 votes
1 answer
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Sufficiency for existence of $lim_{\mathcal{B}}f(x)$, Zorich

I am working on Mathematical analysis 1 by Zorich and i am stuck at the proof for the sufficiency of the cauchy criterion for the existence of a limit of a function (page 132) I am understanding all ...
cmatteo's user avatar
  • 334
2 votes
1 answer
94 views

If the ultrafilter space is Hausdorff, must the base space be discrete?

The question is in the title. Most of this post is a contextual preamble and some of my thoughts on the matter, I have not made much solid progress (and don't even know if this is true...) $\...
FShrike's user avatar
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2 votes
2 answers
132 views

Different definitions of filters in topology

So I have come across three different definitions for set theoretic filters that are used in topology, all of which only vary in their first axiom. The three different first axioms for a filter over a ...
guest1's user avatar
  • 351
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0 answers
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free ultrafilter question

I asked a question here Is this a free ultrafilter, and I'm attempting to write up the solution. $\mathbb{N}=\{1,2,...\}$ and for any $A\subseteq\mathbb{N}$, $A+1=\{a+1:a\in A\}$. Suppose $U$ is a ...
user124910's user avatar
  • 3,017
0 votes
1 answer
57 views

Is this a free ultrafilter

Suppose $U$ is a free ultrafilter on $\mathbb{N}$. I'm trying to understand what is the ultrafilter generated by the collection $$G=\{E\subseteq\mathbb{N}: E=A+1, \ for \ some \ A\in U\}$$. I think ...
user124910's user avatar
  • 3,017
3 votes
2 answers
153 views

What is the relationship between ultrafilters and propositional theories?

I am learning more about how to use ultrafilters by using them to prove several of the typical results which appear as applications of propositional (or first-order) compactness. Generally speaking, ...
John's user avatar
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