1. CMB Online first
 Steinberg, Benjamin; van Gool, Samuel J.

Merge decompositions, twosided KrohnRhodes, and aperiodic pointlikes
This paper provides short proofs of two fundamental theorems
of finite semigroup theory whose previous proofs were significantly
longer, namely the twosided KrohnRhodes decomposition theorem
and Henckell's aperiodic pointlike theorem, using a new algebraic
technique that we call the merge decomposition. A prototypical
application of this technique decomposes a semigroup $T$ into
a twosided semidirect product whose components are built from
two subsemigroups $T_1,T_2$, which together generate $T$, and
the subsemigroup generated by their setwise product $T_1T_2$.
In this sense we decompose $T$ by merging the subsemigroups
$T_1$ and $T_2$. More generally, our technique merges semigroup
homomorphisms from free semigroups.
Keywords:KrohnRhodes theorem, aperiodic pointlikes Categories:20M07, 20M35, 68Q70 

2. CMB 2018 (vol 61 pp. 252)
 Dewar, Megan; Pike, David; Proos, John

Connectivity in Hypergraphs
In this paper we consider two natural notions of connectivity
for hypergraphs: weak and strong. We prove that the strong
vertex connectivity of a connected hypergraph is bounded by its
weak edge connectivity, thereby extending a theorem of Whitney
from graphs to hypergraphs. We find that while determining a
minimum weak vertex cut can be done in polynomial time and is
equivalent to finding a minimum vertex cut in the 2section of
the hypergraph in question, determining a minimum strong vertex
cut is NPhard for general hypergraphs. Moreover, the problem
of finding minimum strong vertex cuts remains NPhard when restricted
to hypergraphs with maximum edge size at most 3. We also discuss
the relationship between strong vertex connectivity and the
minimum
transversal problem for hypergraphs, showing that there are
classes
of hypergraphs for which one of the problems is NPhard while
the other can be solved in polynomial time.
Keywords:hypergraph, connectivity, computational complexity, transversal Categories:05C65, 05C40, 68Q17 

3. CMB 2017 (vol 61 pp. 40)
 Chen, Wengu; Ge, Huanmin

A sharp bound on RIC in generalized orthogonal matching pursuit
Generalized orthogonal matching pursuit (gOMP) algorithm has
received much attention in recent years as a natural extension
of
orthogonal matching pursuit (OMP). It is used to recover sparse
signals in compressive sensing. In this paper, a new bound is
obtained for the exact reconstruction of every $K$sparse signal
via
the gOMP algorithm in the noiseless case. That is, if the restricted
isometry constant (RIC) $\delta_{NK+1}$ of the sensing matrix
$A$
satisfies $ \delta_{NK+1}\lt \frac{1}{\sqrt{\frac{K}{N}+1}}$, then
the
gOMP can perfectly recover every $K$sparse signal $x$ from $y=Ax$.
Furthermore, the bound is proved to be sharp.
In the noisy case, the above bound on RIC combining with an
extra condition on the minimum
magnitude of the nonzero components of $K$sparse signals can
guarantee
that the gOMP selects all of support indices of the $K$sparse
signals.
Keywords:sensing matrix, generalized orthogonal matching pursuit, restricted isometry constant, sparse signal Categories:65D15, 65J22, 68W40 

4. CMB 2011 (vol 54 pp. 566)
 Zhou, XiangJun; Shi, Lei; Zhou, DingXuan

Nonuniform Randomized Sampling for Multivariate Approximation by High Order Parzen Windows
We consider approximation of multivariate functions in Sobolev
spaces by high order Parzen windows in a nonuniform sampling
setting. Sampling points are neither i.i.d. nor regular, but are
noised from regular grids by nonuniform shifts of a probability
density function. Sample function values at sampling points are
drawn according to probability measures with expected values being
values of the approximated function. The approximation orders are
estimated by means of regularity of the approximated function, the
density function, and the order of the Parzen windows, under
suitable choices of the scaling parameter.
Keywords:multivariate approximation, Sobolev spaces, nonuniform randomized sampling, high order Parzen windows, convergence rates Categories:68T05, 62J02 

5. CMB 2011 (vol 54 pp. 288)
 Jacobs, David P.; Rayes, Mohamed O.; Trevisan, Vilmar

The Resultant of Chebyshev Polynomials
Let $T_{n}$ denote the $n$th
Chebyshev polynomial of the first kind,
and let $U_{n}$ denote the $n$th
Chebyshev polynomial of the second kind.
We give an explicit formula for the resultant
$\operatorname{res}( T_{m}, T_{n} )$.
Similarly, we give a formula for
$\operatorname{res}( U_{m}, U_{n} )$.
Keywords:resultant, Chebyshev polynomial Categories:11Y11, 68W20 

6. CMB 2004 (vol 47 pp. 343)
 Drensky, Vesselin; Hammoudi, Lakhdar

Combinatorics of Words and Semigroup Algebras Which Are Sums of Locally Nilpotent Subalgebras
We construct new examples of nonnil algebras with any number of
generators, which are direct sums of two
locally nilpotent subalgebras. Like all previously known examples, our examples
are contracted semigroup algebras and the underlying semigroups are unions
of locally nilpotent subsemigroups.
In our constructions we make more
transparent
than in the past the close relationship between the considered problem
and combinatorics of words.
Keywords:locally nilpotent rings,, nil rings, locally nilpotent semigroups,, semigroup algebras, monomial algebras, infinite words Categories:16N40, 16S15, 20M05, 20M25, 68R15 
