Question I've been working on this for a long time and I'm getting nowhere. Could you help me or at least tell me where to look for help. Suppose D is an m-by-m diagonal matrix with integer elements all . Suppose X is an m-by-n integer matrix . Consider the partitioned matrix M = [D X]. Obviously M has full row rank so it has a right inverse of rational numbers. The question is, under what conditions does it have an integer right inverse? My guess, which I can't prove, is that the integers in each row need to be relatively prime.
The zeroes of the Riemann zeta function that are inside the Critical Strip (i.e. the vertical strip of the complex plane where the real part of the complex variable is in ]0;1[), are actually located on the Critical line ( the vertical line of the complex plane with real part equal to 1/2)
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 all positive integers and , there exists an integer such that every graph of average degree at least contains a subgraph of average degree at least and girth greater than .
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?
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?
Problem Consider the set of all topologically inequivalent polyhedra with edges. Define a form parameter for a polyhedron as where is the number of vertices. What is the distribution of for ?
Conjecture The following statements are equivalent for every endofuncoid and a set : \item is connected regarding . \item For every there exists a totally ordered set such that , , and for every partion of into two sets , such that , we have .
Conjecture Let if is odd and if is even. Let . Assume we start with some number and repeatedly take the of the current number. Prove that no matter what the initial number is we eventually reach .
To divide a given 2D convex region C into a specified number n of convex pieces all of equal area (perimeters could be different) such that the total perimeter of pieces is (1) maximized (2) minimized.
Remark: It appears maximizing the total perimeter is the easier problem.
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-norm of a path partition on a directed graph 
is no more than the maximal size of an induced 
be an integer and let 
be a minor minimal 
. Then either 
minor for some 
or 
. Suppose X is an m-by-n integer matrix 
. Consider the partitioned matrix M = [D X]. Obviously M has full row rank so it has a right inverse of rational numbers. The question is, under what conditions does it have an integer right inverse? My guess, which I can't prove, is that the integers in each row need to be relatively prime.
be a planar graph of maximum degree 
. The chromatic number of its square is
if 
, \item at most 
if 
, \item at most 
if 
. 
is called a Diophantine 
-tuple if 
is a perfect square for all 
. 
is a Diophantine quadruple and 
, then 
be an abelian group. Is the 
, 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 
-stage Shuffle-Exchange graph/network, denoted 
, is the proper (i.e., respecting all the orders) concatenation of 
identical copies of 
be the smallest integer 
such that the graph 
. 
of 
hyperplanes in dimension 
is not greater than 
and ![$ p \in {\mathbb Q}[x] $](/files/tex/a72fcb0a006c3c2afb3ba69722ccdb2599b83e90.png)
rationally derived if all roots of 
and the nonzero derivatives of 
are invertible 
matrices with entries in 
for a prime 
submatrix 
of ![$ [B_1 B_2 \ldots B_p] $](/files/tex/86661dc2948aeca789b4392c2e2a9cbf7d96f735.png)
so that 
(first time when the players occupy the same point) which can be achieved assuming the two players must adopt the same strategy? 
, 
does not divide 
where 
is the Legendre symbol. 
, we write 
if 
satisfies 
. The problem of finding such an integer 
(with 
) is the Discrete Log Problem.
in the hypercube. 
-neighbour bootstrap process in the hypercube. 
for reloid 
. 
for every funcoid 
(see below mentioned online article).
, it is impossible to cover a square of side greater than 
unit squares. 
where 
is the number of vertices. What is the distribution of 
for 
? 
be a 
.
and a set 
:
there exists a totally ordered set 
such that 
, 
, and for every partion 
of 
into two sets 
, 
such that 
, we have ![$ X \mathrel{[ \mu]^{\ast}} Y $](/files/tex/0ef560be389646efd1fdde5ebc9afc9ac98ee64e.png)
. 
if 
if 
. Assume we start with some number 
. 
always rational? 
to be the maximum degree, 
to be the size of the largest 
to be the chromatic number of 
for every graph 
-connected cubic graph on 
vertices admit a partition into 
? 
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