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  two queens
 
 
    mathematics and death
    not cheerful not sad
    guide me through the land
    pat me on my head
 
    they live
    in my one man nation
    i follow them
    to my destination
 
 
wh,
(long ago)



  polynomials over finite Galois field
  spread so evenly across their finite affine space
  i wish for a network of friends
  to count on


wh,
1996-03-05/06



Jun
10
comment Looking for techniques of How to measure the Similarity/Dissimilarity between two images?
@RajeshD is right. Similarity with respect to what?
Jun
7
comment If $A>B>0$, can we always find a positive real number $\alpha$, $0<\alpha < 1$ such that $\alpha A \geq B $?
The general definition considers the complex field, not real.
Jun
7
comment If $A>B>0$, can we always find a positive real number $\alpha$, $0<\alpha < 1$ such that $\alpha A \geq B $?
The real part the relevant expression related to $\ A - B\ $ is greater than certain $\ \epsilon > 0\ $ when evaluated on the unit sphere. If you change $\ A-B\ $ a little, i.e. if you change $\ A\ $ a little then it'll be fine.
Jun
7
comment If $A>B>0$, can we always find a positive real number $\alpha$, $0<\alpha < 1$ such that $\alpha A \geq B $?
Naturally. Thank you. Can't you prove a little sharper result: $\ \alpha\cdot A\ >\ B\ $ ?
Jun
4
comment Extremely messy proofs
Banach used Baire category--simple and elegant.
Jun
4
comment Extremely messy proofs
@ToddTrimble -- yes, and more, it's a characterization: a topology $\ T\ $ in $\ Y\ $ is maximal (among non-discrete topologies) $\ \Leftarrow:\Rightarrow\ $ there exists a (unique) point $\ v\in Y\ $ such that all points of $\ X := Y\setminus\{v\}\ $ are isolated, and the family $\ \{V\cap X : v\in V\in T\}\ $ is a maximal filter in $\ X\ $ REMARK The principal maximal filters in $\ X := X_v\ $ correspond 1-1 to maximal topologies which are non-Hausdorff. (The max. filters with the empty intersection correspond to maximal topologies which are Hausdorff).
May
29
comment If $X$ is compact, is $[X]^2$ compact, too?
I'd vote to close this obviously non-research question if it were not for Ramiro's nice answer.
May
28
comment Hausdorff space $X$ with $X\cong [X]^2$
Adam, thank you.
May
27
comment Hausdorff space $X$ with $X\cong [X]^2$
Up-voted, but this is an overkill though :-) (no, not too bad!).
May
26
comment What's the detailed proof of “the composition of planar tangles is well-defined”?
@SébastienPalcoux -- thank you (now I can SEE :-).
May
26
comment What's the detailed proof of “the composition of planar tangles is well-defined”?
To me the wikipedia's definition of planar algebra is unreadable, just horrible.
May
26
answered Graph $G$ with $\omega(G) = 2$ but $\chi(G) \geq \aleph_0$
May
24
comment Combinatorial formula for the number of different words
Mine would be not so much from another world as off the wall. Your q. is harder than I thought at first. Thus I up-voted it. Now I expect the specialists to answer your question, possibly using some Bernoulli numbers or similar--let me see (let them sweat :-).
May
24
revised Connectedness of the complements of the connected subsets
cosmetic
May
24
comment Combinatorial formula for the number of different words
I misread words in the title as worlds. Too bad.
May
24
comment Extremely messy proofs
For starters (to conclude this series of comments), let $\ \mathbf X\ :=\ (X\ T)\ $ and $\ \mathbf X_p\ :=\ (X\ T_p),\ $ and $\ \mathbf Y\ :=\ (Y\ T').\ $ Then THEOREM 3: $\ f:X\rightarrow Y\ $ is continuous with respect to topologies $\ T\ T'\ $ at $\ p\in X\ \Leftrightarrow\ f:\mathbf X_p\rightarrow\mathbf Y\ $ is continuous. Furthermore, THEOREM 4: $\ f:X\rightarrow Y\ $ is continuous with respect to topologies $\ T\ T'\ \Leftrightarrow\ \forall_{p\in X}\,f:\mathbf X_p\rightarrow\mathbf Y\ $ is continuous.
May
24
comment Extremely messy proofs
Let $\ \mathbf X:=(X\ T)\ $ be a topological space. Let $\ p\in X.\ $ Then the induced (generalized!) singular topology $\ T_p\ $ is defined as $\ T_p\ :=\ \{G\subseteq X\,:\,p\notin G\ \ or\ \ G\in T\}.\ $ Topology $\ T_p\ $ is discrete when $\ p\ $ is isolated. Otherwise it is singular. THEOREM 1: $\ T\ =\ \bigcap_{p\in X} T_p.\ $ Furthermore, let $\ \mathbf M\ $ be the set of all maximal topologies in $\ X.\ $ Then THEOREM 2: $\ T\ =\ 2^X\cap \bigcap\,\{M\in\mathbf M\,:\,T\subseteq M\}.\ $ Now filter-like road to topology is open.
May
24
comment Extremely messy proofs
I've develop that approach (privately, so to speak) a very long time, and years later posted it only on some my obscure Internet pages. Thus there is virtually no reference. My method is to develop the whole (general) topology based on the notion of singular topological spaces and maximal singular topological spaces: a topological space $\ (X\ T)\ $ is singular $\ \Leftrightarrow\ $ it has exactly one limit (i.e. non-isolated) point. It is maximal $\ \Leftrightarrow\ $ topology $\ T\ $ is maximal among all non-discrete topologies. Thus topology is developed purely topologically.
May
22
comment Is g( ) rational if it looks that way on a large rational subset?
The M.A., good job!--you turned a nothing q. title into something sensible.
May
21
comment Inverse limit in shape theory
I could write a minimal (brief) exposition of theory of shape for compact spaces and abstract categories in that compact direction. It'd take a separate MO pseudo-answer though. Otherwise, there is my Fund.Math. paper or monograph Shape Theory by J.-M. Cordier and T.Porter.