Conjecture Define a array of positive integers where the first row consists of some distinct positive integers arranged in increasing order, and the second row consists of any positive integers in any order. Create a new array where the first row consists of all the integers that occur in the first array, arranged in increasing order, and the second row consists of their multiplicities. Repeat the process. For example, starting with the array , the sequence is: -> -> -> -> -> -> -> -> -> -> -> , and we now have a fixed point (loop of one array).
The process always results in a loop of 1, 2, or 3 arrays.
Conjecture A Fermat prime is a Fermat number that is prime. The only known Fermat primes are F_0 =3,F_1=5,F_2=17,F_3 =257 ,F_4=65537 It is unknown if other fermat primes exist.
Problem Determine a computable set of invariants that allow one to determine, given a compact boundaryless 3-manifold, whether or not it embeds smoothly in the 4-sphere. This should include a constructive procedure to find an embedding if the manifold is embeddable.
Problem Two players start at a distance of 2 on an (undirected) line (so, neither player knows the direction of the other) and both move at a maximum speed of 1. What is the infimum expected meeting time (first time when the players occupy the same point) which can be achieved assuming the two players must adopt the same strategy?
For and positive integers, the (mixed) van der Waerden number is the least positive integer such that every (red-blue)-coloring of admits either a -term red arithmetic progression or an -term blue arithmetic progression.
Conjecture There exists an integer such that every -arc-strong digraph with specified vertices and contains an out-branching rooted at and an in-branching rooted at which are arc-disjoint.
Conjecture Let be a simple graph with vertices and list chromatic number . Suppose that and each vertex of is assigned a list of colors. Then at least vertices of can be colored from these lists.
Setup Fix a tree and for every vertex a non-negative integer which we think of as the amount of gold at .
2-Player game Players alternate turns. On each turn, a player chooses a leaf vertex of the tree, takes the gold at this vertex, and then deletes . The game ends when the tree is empty, and the winner is the player who has accumulated the most gold.
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array of positive integers where the first row consists of some distinct positive integers arranged in increasing order, and the second row consists of any positive integers in any order. Create a new array where the first row consists of all the integers that occur in the first array, arranged in increasing order, and the second row consists of their multiplicities. Repeat the process. For example, starting with the array ![$ [1; 1] $](/files/tex/73f8649de444361674c157a2fe98e0c5783f1c46.png)
, the sequence is: ![$ [1; 2] $](/files/tex/83c3d9d7589f716ed6b0f05d26d36dabe8ba47aa.png)
-> ![$ [1, 2; 1, 1] $](/files/tex/a6a696aec4e84df6bc046cf6d30a4df80e156a14.png)
-> ![$ [1, 2; 3, 1] $](/files/tex/98b2f3e4134422c8a286f0326fc2f57ca9be2ab7.png)
-> ![$ [1, 2, 3; 2, 1, 1] $](/files/tex/19a9d24510f4551fa45b950aed32efed0636b355.png)
-> ![$ [1, 2, 3; 3, 2, 1] $](/files/tex/b00d573110ba8c2472820409103dd4cbd7bff7cc.png)
-> ![$ [1, 2, 3; 2, 2, 2] $](/files/tex/2aae90b65ae3f116402a1c0143127f80d86acdf0.png)
-> ![$ [1, 2, 3; 1, 4, 1] $](/files/tex/d614399c4078a70fbffb24eb99816a0974423175.png)
-> ![$ [1, 2, 3, 4; 3, 1, 1, 1] $](/files/tex/bf4dd82eaf95bd71a88cb78d3367efa2c7e4c942.png)
-> ![$ [1, 2, 3, 4; 4, 1, 2, 1] $](/files/tex/a4ef5900a9dde9b32311afe7dee2b143f42c405f.png)
-> ![$ [1, 2, 3, 4; 3, 2, 1, 2] $](/files/tex/ff367376ed7bb74fc6207273d7157062b83a1be6.png)
-> ![$ [1, 2, 3, 4; 2, 3, 2, 1] $](/files/tex/81ba2b3608ddba2fc95d32b74b70279f3f5adc5b.png)
, and we now have a fixed point (loop of one array).
, 
so that 
does not contain an odd edge-cut. 
be positive semidefinite, by Jensen's inequality, it is easy to see ![$ [tr(A^s+B^s)]^{\frac{1}{s}}\leq [tr(A^r+B^r)]^{\frac{1}{r}} $](/files/tex/f5fa90ccf8c1dde9b9bf1f21399b3b5428dab81d.png)
, whenever 
.
, is it still valid?
-graph if it has a bipartition 
so that every vertex in 
has degree 
and every vertex in 
has degree 
. 
-connected cubic planar graph can be decomposed into two Hamilton cycles. ![\[ F_n = 2^{2^n } + 1 \]](/files/tex/0da5a50010e4e5df91c0d58080245ece34ec9ca6.png)
that is prime. The only known Fermat primes are F_0 =3,F_1=5,F_2=17,F_3 =257 ,F_4=65537 It is unknown if other fermat primes exist.
for some filter objects 
, 
. Particularly, is this formula true for 
? 
for some filter objects 
(first time when the players occupy the same point) which can be achieved assuming the two players must adopt the same strategy? 
rectangular 
-dimensional boxes each of which has side lengths 
inside an 
? 
and 
positive integers, the (mixed) van der Waerden number 
is the least positive integer ![$ [1,n] $](/files/tex/661d0acc09fbc5b62f49645a71cf7d831a26563f.png)
admits either a 
, 
. 
be integers. Set 
and 
for 
. Eventually we have 
; put 
.
, since 
, 
, 
, 
, 
, 
, 
, 
.
.
with specified vertices 
and 
contains an out-branching rooted at 
be a simple graph with 
. Suppose that 
and each vertex of 
colors. Then at least 
vertices of 
and graph 
. 
and for every vertex 
a non-negative integer 
which we think of as the amount of gold at 
be an abelian group. Is the 
, for 
a large prime, always contain a rainbow 
if each of the color classes is of size of either 
or 
? 
be a sequence of nonnegative integers which strictly decreases until 
.
with 
crossings, but not with less crossings. 
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