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$$x\neq y\in S$$

## Infinite Hausdorff space that is not homeomorphic to any proper quotient

https://mathoverflow.net/questions/200613/infinite-hausdorff-space-that-is-not-homeomorphic-to-any-proper-quotient
Let S be a set and \vartheta be an equivalence relation on S . We say that \vartheta is proper if there are x\neq y\in S with (x,y)\in\vartheta . Is there an infinite Hausdorff space (X,\tau) such that for every proper equivalence relation \vartheta on X we have X\not\cong X/\vartheta ?...
$$x^p = y \in S$$

## $k[[x]]$ as a $(k[[x]])^p$ module for ugly fields

The answer to the question is no. Let R = k[[x]] , S = k[[y]] , and f:R \to S is the absolute Frobenius. We must show that S is not a free R -module. By assumption on k , we know that S must have infinite rank if it were free. On the other hand, S is x -adically complete (since x^p = y \in S ). Hence, the claim follows from: Lemma: The R -module M := R^{\oplus I} is x -adically complete if and only if I is finite. Pr...
$$y\lt x\in S\to y\in S$$

## A principle of mathematical induction for partially ordered sets with infima?

Something very close to François' conditions achieves the desired if-and-only-if version of the theorem for partial orders, providing an induction-like characterization of the complete partial orders, just as Pete's theorem characterizes the complete total orders. Suppose that (P,\lt) is a partial order. We say that it is complete if every subset A has a least upper bound sup(A) and a greatest lower bound inf(A) . This implies th...
$$x,y\in S.$$

## Cancellable elements of a power semigroup

https://mathoverflow.net/questions/192157/cancellable-elements-of-a-power-semigroup
For a semigroup S, its power semigroup P(S) is the semigroup of all non-empty subsets of S with the operation given by AB=\{ab\,|\,a\in A,b\in B\}. I would like to know about the cancellable elements of P(S) given some knowledge of cancellability in S. If s\in S is left-cancellable in S, then \{s\} is also left-cancellable in P(S) because if sA=sB and a\in A, then sa=sb for some b\in B, and a=b\in B follows...
$$xy \in S.$$

## Arithmetic closed subsets

https://mathoverflow.net/questions/67618/arithmetic-closed-subsets
Let S, S_1 be subsets of the positive numbers \mathbb{N} . We say (as usual) that S is multiplicative closed if x \in S and y \in S implies xy \in S. We say also that S_1 is arithmetic closed if x \in S_1 and y \in S_1, and \gcd(x,y)=1, implies xy \in S_1. Let T be a subset of the positive integers containing 1 and at least another element. The smallest multiplicative closed set that contains T (say C(T) ) i...
$$v=xy \in S_{\alpha}$$

## Characterizing subfields $\mathbb{C}(u,v) \subseteq \mathbb{C}(x,y)$ invariant under an involution

https://mathoverflow.net/questions/354155/characterizing-subfields-%24%5Cmathbb%7Bc%7D%28u%2Cv%29-%5Csubseteq-%5Cmathbb%7Bc%7D%28x%2Cy%29%24-invariant-under-an-involution
Let \iota be an involution on \mathbb{C}(x,y) , namely, a \mathbb{C} -algebra automorphism of \mathbb{C}(x,y) of order two. Examples of involutions: \alpha: (x,y) \mapsto (y,x) , \beta: (x,y) \mapsto (x,-y) , \epsilon: (x,y) \mapsto (-x,-y) . Observe that \alpha and \beta are conjugate, while \epsilon is not conjugate to them. (The Jacobian of \alpha and \beta is -1 , while the Jacobian of \epsilon is 1 ). Assume ...
$$x,y\in S$$

## Idempotents in Green J classes

Any regular J -class with a joined zero is a 0-simple semigroup. So your question is reduced to the following: who many idempotents has a 0-simple semigroup? In particular, if S is finite, a J -class (with 0) is completely 0-simple semigroup, so it has just one idempotent \ne 0 iff it is a group. Moreover, if a 0-simple semigroup S with 1 has no other idempotents, then it is a group with 0. Proof: Let G be its subgroup of inve...
$$x,y\in S$$

## An extremal property of points on the unit sphere of a 2-dimensional Banach space

https://mathoverflow.net/questions/344612/an-extremal-property-of-points-on-the-unit-sphere-of-a-2-dimensional-banach-space
Let (X,\|\cdot\|) be a 2-dimensional real Banach space and S=\{x\in X:\|x\|=1\} be its unit sphere. Assume that S is smooth in the sense that for any y\in S there exists a unique functional y^*:X\to\mathbb R such that y^*(y)=1=\|y^*\| . This unique functional y^* will be called the supporting functional at y . Let x,y\in S be points such that \|y-x\|+\|y+x\|=\max\{\|s-x\|+\|s+x\|:s\in S\}. Question. Is y^*(x)=0 ?...
$$x,y \in S$$

## Example that a finite collection of contractions can't approach the set of all contractions well enough

https://mathoverflow.net/questions/224237/example-that-a-finite-collection-of-contractions-can%27t-approach-the-set-of-all-contractions-well-enough
I'm looking for an example of a seperable and complete metric space (S,d) such that there exist some \varepsilon > 0 and P a probability measure on the Borel open sets \mathcal{B}_S for which \int d(x,a) P(dx) < \infty for some a \in S such that for any finite collection F of non-expansive maps from S to \mathbb{R} , there exists some probability measure Q on B_S for which there exists some a \in S: \int d(x,a) P(d...
$$x,y \in S$$

## Li-Yorke chaos: the non compact case

https://mathoverflow.net/questions/299083/li-yorke-chaos%3A-the-non-compact-case
1) Is there any notion of Li-Yorke chaos for non compact (metric) spaces X and non continuous transformation f:X \rightarrow X ? Could you bring some references? 2) I mean, why are so important the compactness of the space X and continuity of the function f:X \rightarrow X in the Li-Yorke chaos definition? Just to remember, a pair x,y \in X is called scrabled if \liminf d(f^{n}(x),f^{n}(y)) = 0 and \limsup d(f^{n}(x),f^{n}(y...
$$xy\in S$$

## A prime ring which is or is not clean

So, if any element of R which is a unit in E is also a unit in R , then R would be clean too. Well, that is easy to prove, right? Let 1+x\in R be a unit in E ( x\in S of course.) Then there exists y\in E such that (1+x)y=1 . Then y=1-xy\in R since xy\in S ....
$$(\forall y \lt x \,.\, y \in S) \Rightarrow x \in S.$$

## (Types of) induction on infinite chains

You seem to be asking about well-founded induction. It generalizes many forms of induction, including the usual induction on numbers and transfinite induction on ordinals. Consider a relation < on a set A . Say that S \subseteq A is < -progressive when, for all x \in A , (\forall y < x \,.\, y \in S) \Rightarrow x \in S. In words, an element is in S as soon as all of its predecessors are. There is a logical counter-part: sa...
$$\Omega:=\{x=r \theta: 0 \lt r \lt 1, \theta \in S\}$$

## improved Sobolev embedding

https://mathoverflow.net/questions/368241/improved-sobolev-embedding
This is probably not a research level question but I am struggling with the geometry. My question is related to whether some monotonicity can increase the range of exponents in the Sobolev embedding. For instance on the unit ball, nonnegative radially symmetric functions which are nondecreaing in the radial direction should satisfy a Sobolev embedding for an improved range of exponents: Indeed such functions must be large on the full bo...
$$x,y \in a$$

## What is a hypergraph minor?

In a note concerning a version of Hadwiger's conjecture for hypergraphs, I once defined what a (complete) minor map of a hypergraph is, and probably this can be used to define a notion of "minor of hypergraph". (If you are interested in the note, which I did not publish or put on arXiv, I am happy to send it to you.) For the following definitions assume that H=(V,E) is a hypergraph. A subset \emptyset \neq S \subseteq V is said to b...
$$\mathbb{x},\mathbb{y}\in\mathcal{S}\Rightarrow|\langle \mathbb{x},\mathbb{y} \rangle| \lt \epsilon$$

## Designing almost orthogonal vectors in a deterministic manner

https://mathoverflow.net/questions/150892/designing-almost-orthogonal-vectors-in-a-deterministic-manner
Consider the vector space \mathbb{R}^n , the standard inner product \langle \cdot,\cdot \rangle:\mathbb{R}^n\times\mathbb{R}^n\rightarrow \mathbb{R} , and some 0<\epsilon\leq \frac{1}{\sqrt{n}} . Is it possible to generate a set of M vectors, say \mathcal{S} , in a deterministic fashion (not random), such that the vectors in \mathcal{S} satisfy the following properties: all the entries of the vectors come from \{\frac{-1}{\sqrt...
$$x\in S(i), y\in S(j), i+j\lt d$$

## What is matrix A such that Hamming weight of [x, Ax] is maximal ? (Min distance of 1/2 block code?)

This is, indeed, an open question for most values of n . A.D. Brouwer maintains a database of the best known lower and upper bounds, and everything that is known for small n can be found there. A minute of googling points me at http://mint.sbg.ac.at/desc_CBrouwerTable-Bound.html Asymptotically the best bound is usually the linear programming bound due to McEliece-Rodemich-Rumsey and Welch, but the asymptotic bounds may not be very go...
$$(\forall y\lt x)(y\in S) \Rightarrow x\in S$$

## Strong induction without a base case

Any constructively valid proof by well-founded induction, specialized to the natural numbers as a particular well-founded set, will be an example. Recall that a set A with a relation < is well-founded if for any S\subseteq A , if (\forall y\lt x)(y\in S) \Rightarrow x\in S , then S=A . A proof by well-founded induction proceeds by proving that the set S of "all x\in A such that blah" satisfies that condition, i.e. that if blah ...
$$X=\{(x,y)\in S\times S\mid x \lt y\textrm{ and }(x+y\in S \textrm{ or}n-x-y\in S)\}.$$

## A Freiman-type of question

https://mathoverflow.net/questions/160068/a-freiman-type-of-question
Recently I have bumped in the following question. This is not my field of research, but it looks to me very much related to Frieman-type of theorems. Let n be a positive integer and let S be a subset of \{1,\ldots,n\} of cardinality \ell with s
$$x\preceq y$$

## Extending a partial order while preserving an automorphism

$$x\in S$$