# All Questions

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### model definitions for tautology, contradiction, and connectives quantify too much, no?

Occasionally I come across a definition based on what will happen in all models, for example, that a contradiction is a statement that is false in all models, that a tautology is a statement that is ...
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### Polar coordinates for vector field to find sticking flow

I am currently working on an impacting system which is basically just a spring damper and a circular enclosure. Because of the rotational symmetry of the problem I need the vector field in polar ...
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### Number of equivalence classes based on a relation regarding a non-principal ultrafilter

We have an equivalence relation on $\mathbb{N}^\mathbb{N}$ given by $$f\equiv g \iff \{n\in\mathbb{N}: f(n)=g(n)\}\in\mathbb{U},$$ where $\mathbb{U}$ is a non-principal ultrafilter on $\mathbb{N}$. ...
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### How small need it be to approximate integral as one area of product of initial value times length.

$$\left(\int_{t}^{t+\Delta t}a(t')dt'\right), a(t) \text{ is scalar}$$ How small need $\Delta t$ be to approximate $$\left(\int_{t}^{t+\Delta t}a(t')dt'\right)$$ as $$a(t)\Delta t$$ [ Just one ...
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### The set of all maximal ideals

I am trying to prove a proposition and in my proof I somehow need to find the set of all maximal ideals of a Banach Algebra. This is my working environment: Let $A(\mathbb{R}^2)$ be the (Wiener ...
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### Prisoners' Dilema

I started to learn about game theory just now. I am confusing about the prisoners' dilema, when 2 prisoners given a choice whether to keep silent or rat out the other guy. From what I red, if one rat ...
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### Proof regarding little o for vector field

I'm struggling with one part of a proof of a theorem. Let $\gamma(t) : A \subseteq \mathbb{R} \rightarrow \mathbb{R}^m$, with $\gamma \in \mathscr{C}^1(A)$ (hence $\gamma$ differentiable). ...
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### Prove without differentiation that $\log_{n+1}n$ is increasing

I want to show that the sequence $\{ \log_{n+1}n \}$ is increasing without differentiation. I don't have any idea. How can I prove that?
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### Questions on color theory, mostly linear algebra related

I'm reading into color theory and there were a few questions which I asked myself along the way, maybe you can put me forward to some source where I can find answers or give them directly. The human ...
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### Problem with proof of $H \cap K$ is of finite index if $H,K$ are finite index subgroups

I came across a proof earlier for a solution to a Herstein Topics in Algebra question earlier that I'm not convinced with, from AOPS site. If $G$ is a group and $H,K$ are two subgroups of finite ...
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### $A,B$ be countable dense subsets of $\mathbb R$ , let $A,B$ be given usual subspace topologies , then there exists a homeomorphism $f:A \to B$?
Let $A,B$ be countable dense subsets of $\mathbb R$ (with usual euclidean topology ) let $A,B$ be given usual subspace topologies , then is it true that there exists a homeomorphism $f:A \to B$ ?