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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Two questions on completely regular filters in locales

I'm reading the exposition of the Stone-Čech compactification for locales in Johnstone's book Stone Spaces. In Chapter IV Paragraph 2.2, Johnstone constructs the Stone-Čech compactification of a ...
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Convergence of real numbers along an ultrafilter

Suppose we have an uncountable set $I$ and a non-principal ultrafilter $U$ on it. I am interested whether it is possible to conclude that if cardinality of $I$ is big enough, then every tuple $(x_i)_{...
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Can someone please offer a simple definition of “derived net”?

I was looking up a term called "derived net", however the Google search seems to conflate "net" with .NET programming language. (And filter with electronic filters, and "derived net from a filter" ...
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Show that if $x$ is an accumulation point of an ultrafilter on $X$, then the neighborhood filter is contained in the filter

Show that if $x$ is an accumulation point of an ultrafilter $\mathcal{F}$ on $X$, then the neighborhood filter $\mathcal{F}_x$ is contained in the filter i.e. $\mathcal{F}_x \subseteq \mathcal{F}$ ...
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Any subset of elements in an ultrafilter is in the ultrafilter

I am trying to prove the following fact: Given $\mathcal{F}$ an ultrafilter on $X$, suppose $ A \in \mathcal{F}$ and $B \subseteq A$, then either $B$ or $A\backslash B$ are in $\mathcal{F}$ ...
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Given $\mathcal{F}$ a filter on $X$, and either $A$ or its complement are in $\mathcal{F}$, then $\mathcal{F}$ is ultrafilter

I am trying to prove the following fact: Given $\mathcal{F}$ a filter on $X$, Suppose $\forall A \subseteq X$ where either $A$ or $X\backslash A$ are in $\mathcal{F}$, then $\mathcal{F}$ is ...
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Product of Ultrafilters: How to show $p,q <_{RK} p\otimes q$?

Let $p,q$ be two free ultrafilter on $\mathbb{N}$, i.e. elements of $\mathbb{N}^* = \beta\mathbb{N}\setminus\mathbb{N}$. The Rudin-Keisler order is defined as follows: $p \leq_{RK} q$ iff there is a ...
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need help with following butterworth filter unit sample response.

I have a problem for you. Can anyone help me to solve this. A system is represented by a0y(n)+a1y(n-1)+a2y(n-2)+a3y(n-3)=b0x(n)+b1x(n-1)+b2x(n-2)+b3x(n-3) B= [b0, b1 , b2 , b3 ]= [0.3151*10-3 0....
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Norms on an Ultraproduct

Suppose $X$ is a Banach space and $\mathcal{U}$ is a non-principal ultrafilter on $\mathbb{N}$. I am interested in the Banach space $(X)_\mathcal{U}$, where we consider sequences $(x_i)_{i \in \mathbb{...
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42 views

Ultrafilter invariant semigroups

This may be quite basic but I was unable to figure out the answer on an uncountable set. Suppose that $S$ is an infinite set and $U$ a non-principal ultrafilter on $S$. Does there exist a commutative ...
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Inital condition for 1st order derivative

I have a PT2 system filter, Y'' = (a0)*Y'+(a1)*Y+(b)*u. I want to apply this filter to my signal, i have a problem in deriving ...
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How to produce a continuous variation of a discontinuous function?

I have a differential equation that connects the "velocity" of a point in the FOV of a camera with the velocities of a robot's joints, that is $$\dot s=J(s) \dot q$$ where s is a vector with the $x$,$...
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31 views

Ultrafilter on $[0,1]$ consisting of closed sets

Today we learned about filters and ultrafilters in the General Topology course. I am trying to play around with these definitions. I wish to ask a question that I am unsure about. Let us say, we have ...
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72 views

Why are the ultrafilters on $\mathbb{N}$ precisely the minimal Cauchyfilters?

One can place an equivalence relation on the set of all Cauchyfilters in a uniform space: $\mathcal{U}\sim\mathcal{V}$ iff $\mathcal{U}\cap\mathcal{V}$ is a Cauchyfilter. In my topology course, we ...
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Gaussian filter. Formula for filtering sample

I need to apply gaussian filter to the sample of signal. Can you tell me the formula which I should use and say what is the purpose of gaussian filter?
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40 views

Existence of a particular element of an ultrafilter

I'm getting to know some ultrafilter theory. I'm stuck on the following exercise: Suppose $ \mathcal{U} $ is an ultrafilter on $ \omega $. Prove that there exists $ A \in \mathcal{U} $ such that $$\...
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44 views

Non-cofinite element of non-principle ultrafilters

Say, we have $n$ non-principal ultrafilters $\mathcal{U}_1,...,\mathcal{U}_n$ on an infinite set $X$. Obviously they all contain all cofinite subsets of $X$. But can they all contain some common ...
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Filters on $\omega$

I am currently reading the book "Set theory on the real line" by Bartoszynski and Judah and I do have problems to proof the following statement: Suppose $\mathcal{F}$ is a filter on $\omega$ including ...
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33 views

Filters and antichains in preorders

In a preorder $\mathbb{P}$ (reflexive and transitive), what means to say that a subset $C\subset\mathbb{P}$ is not contained in any filter of $\mathbb{P}$? For example, it is obvious that if $C$ ...
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52 views

Non-principal ultrafilters on a set [duplicate]

Let $X$ be a set. If $X$ is finite then all ultrafilters on $X$ are principal, i.e. have the form $\{A \subseteq X : x \in A\}$ for some $x\in X$. But now suppose $X$ is infinite, say $X=\mathbb N$. ...
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Sequential compactness implies compactness: what is wrong with this argument?

Definitions A filter is a poset $(I,\leq)$ such that for any $\alpha,\beta\in I$ there is $\gamma\in I$ such that $\gamma\geq\beta,\gamma\geq\alpha$. A net in a set $X$ is a function from a poset to ...
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Why can't we define intersection with all the elements of a filter?

Let's see the definition of a filter with the "$\subset$" order. Let $X$ be a set. We say a non-empty family $\mathcal{F}$ of subsets of $X$ is a filter if: $\emptyset \notin \mathcal{F}$...
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How to determine the transition probability in Sequential Importance Sampling (SIS) for Particle Filter

Given a state-space model \begin{align} x_k &= f_k(x_{k-1}, v_{k-1}),\\ z_k &= h_k(x_k, w_k), \end{align} where $x_k \in {\mathbb R}^{n}$ and $y_k \in {\mathbb R}^{m}$ are the system state ...
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Free filter on Wikipedia

(1) In the second example in Section 3.1 of the Wikipedia article on filter, the last sentence says: A nonprincipal filter on an infinite set is not necessarily free. On the other hand, Martin ...
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35 views

What it the intuition behind basis, filters as neighborhoods and neighborhoods?

I know every definition, but as hard as I think I can't get the intuition behind such objects. For example, the definitions I am familiar at most: Take $X$ a topological space with topology $\tau.$ ...
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34 views

First-countable uniform space: Is the filter definition of complete equivalent to the sequential definition?

This question is motivated mainly by the topology of first-countable spaces being uniquely determined by their notions of convergence of sequences: the closure of a set $A$ is the same as the set of ...
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53 views

Equivalent condition of convergence of Filter

Let $\mathcal F$ be a filter in a topological space $X$. Denote $E_\mathcal F$ the set of elementary filters that are refined by $\mathcal F$. Elementary filters are the filters associated to ...
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36 views

Is there a convenient way write sums and products with filters?

If we want to iterate a commutative and associative law of composition over $\{a_i\}_{i \in I}$ in some topological space for an index set $I$, there is a simple way to do it using the directed set of ...
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Convolution bounds

For $t\geq0$, let $g_\beta(t)=e^{-t}\sin(\beta t)$, where $\beta$ is a real number, and for $t<0$, $g_\beta(t)=0$. Find $h*g_\beta(t)$ for all $t\geq0$, where $h(t)=\begin{cases}1/d&t\in[0,d]\\...
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Convolution of a signal with the butterworth filter.

Let $f(t)$ be a signal that is $0$ when $t<0$ or $t>1$. Show that, for the Butterworth filter, one has $$Ae^{-\alpha t}\int_{0}^{\min\{t,1\}}e^{\alpha\tau}f(\tau)d\tau$$ My attempt: \begin{...
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Is a minimal Cauchy filter bounded?

Let $E$ be a $K$-vector space and $A, B$ two subsets of $E$. We say $A$ absorbs $B$ if there is a $\alpha>0$ such that $B \subseteq \lambda A$ for all $\lambda\in K$ such that $\lambda \geq ...
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A question on Cauchy filters

Let $\mathfrak{F}$, $\mathfrak{G}$ and $\mathfrak{G_1}$ be Cauchy filters in a uniform vector space $(X, \mathfrak{U})$. Let $c\in \mathbb{C}$. (1) If $\lim (\mathfrak{G}- \mathfrak{G}_1) = 0$, can ...
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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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52 views

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} \frac{\...
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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 $f:[\...
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76 views

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 well-...
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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 $\bigcap_{...
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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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65 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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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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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 $f^{-1}[C]\...