Conjecture Let and are monovalued, entirely defined funcoids with . Then there exists a pointfree funcoid such that (for every filter on ) (The join operation is taken on the lattice of filters with reversed order.)
A positive solution of this problem may open a way to prove that some funcoids-related categories are cartesian closed.
Conjecture There is a finite upper bound on the multiplicities of entries in Pascal's triangle, other than the number .
The number appears once in Pascal's triangle, appears twice, appears three times, and appears times. There are infinite families of numbers known to appear times. The only number known to appear times is . It is not known whether any number appears more than times. The conjectured upper bound could be ; Singmaster thought it might be or . See Singmaster's conjecture.
Conjecture Every complete geometric graph with an even number of vertices has a partition of its edge set into plane (i.e. non-crossing) spanning trees.
Conjecture For every , there exists an integer such that if is a digraph whose arcs are colored with colors, then has a set which is the union of stables sets so that every vertex has a monochromatic path to some vertex in .
Problem Find a constant such that for any there is a sequence of tautologies of depth that have polynomial (or quasi-polynomial) size proofs in depth Frege system but requires exponential size proofs.
Conjecture Let be the space of Diffeomorphisms on the connected , compact and boundaryles manifold M and the space of vector fields. There is a dense set ( ) such that exhibit a finite number of attractor whose basins cover Lebesgue almost all ambient space
This is a very Deep and Hard problem in Dynamical Systems . It present the dream of the dynamicist mathematicians .
Given integers , the 2-stage Shuffle-Exchange graph/network, denoted , is the simple -regular bipartite graph with the ordered pair of linearly labeled parts and , where , such that vertices and are adjacent if and only if (see Fig.1).
Given integers , the -stage Shuffle-Exchange graph/network, denoted , is the proper (i.e., respecting all the orders) concatenation of identical copies of (see Fig.1).
Let be the smallest integer such that the graph is rearrangeable.
Conjecture For which values of and are there bi-colored graphs on vertices and different colors with the property that all the monochromatic colorings have unit weight, and every other coloring cancels out?
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is a 
-chromatic graph on at most 
vertices, then 
. 
be a digraph all of whose strong components are nontrivial. Then 
. ![$ p \in [0,1] $](/files/tex/1d076f7332523eb59bd7deae19667f83f0a3b6e0.png)
, we let 
denote a subgraph of 
.
so that 
? 
-spheres bound (rational) homology 
-balls?
-flow 
on 
from the edge set of 
, for all 
, and 
.
-regular graph 
for every 
with 
odd.
be an integer. If 
. 
so that the power series 
is bounded for all 
. 
and 
are monovalued, entirely defined funcoids with 
. Then there exists a pointfree funcoid 
such that (for every filter 
on 
) 
(The join operation is taken on the lattice of filters with reversed order.) 
exist a 
so that both 
are prime.
are simple finite graphs, then 
. 
is the 
.
. 
appears once in Pascal's triangle, 
appears three times, and 
appears 
times is 
. It is not known whether any number appears more than 
. See 
such that if 
set which is the union of 
, then 
there is a sequence of tautologies of depth 
Frege system 
but requires exponential size 
proofs. 
be the space of 
Diffeomorphisms on the connected , compact and boundaryles manifold M and 
the space of 
(
) such that 
exhibit a finite number of attractor whose basins cover Lebesgue almost all ambient space 
, the 2-stage Shuffle-Exchange graph/network, denoted 
, is the simple 
of linearly labeled parts 
and 
, where 
, such that vertices 
and 
are adjacent if and only if 
(see Fig.1).
, the 
, is the proper (i.e., respecting all the orders) concatenation of 
identical copies of 
be the smallest integer 
such that the graph 
. 
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