1. CJM 2013 (vol 67 pp. 450)
 Santoprete, Manuele; Scheurle, Jürgen; Walcher, Sebastian

Motion in a Symmetric Potential on the Hyperbolic Plane
We study the motion of a particle in the hyperbolic plane (embedded in Minkowski space), under the action of a potential that depends only on one variable. This problem is the analogous to the spherical pendulum in a unidirectional force field. However, for the discussion of the hyperbolic plane one has to distinguish three inequivalent cases, depending on the direction of the force field. Symmetry reduction, with respect to groups that are not necessarily compact or even reductive, is carried out by way of Poisson varieties and Hilbert maps. For each case the dynamics is discussed, with special attention to linear potentials.
Keywords:Hamiltonian systems with symmetry, symmetries, noncompact symmetry groups, singular reduction Categories:37J15, 70H33, 70F99, 37C80, 34C14, , 20G20 

2. CJM 2012 (vol 65 pp. 927)
 Wang, Liping; Zhao, Chunyi

Infinitely Many Solutions for the Prescribed Boundary Mean Curvature Problem in $\mathbb B^N$
We consider the following prescribed boundary mean curvature problem
in $ \mathbb B^N$ with the Euclidean metric:
\[
\begin{cases}
\displaystyle \Delta u =0,\quad u\gt 0 &\text{in }\mathbb B^N,
\\[2ex]
\displaystyle \frac{\partial u}{\partial\nu} + \frac{N2}{2} u =\frac{N2}{2} \widetilde K(x) u^{2^\#1} \quad & \text{on }\mathbb S^{N1},
\end{cases}
\]
where $\widetilde K(x)$ is positive and rotationally symmetric on $\mathbb
S^{N1}, 2^\#=\frac{2(N1)}{N2}$.
We show that if $\widetilde K(x)$ has a local maximum point,
then the above problem has infinitely many positive solutions
that are not rotationally symmetric on $\mathbb S^{N1}$.
Keywords:infinitely many solutions, prescribed boundary mean curvature, variational reduction Categories:35J25, 35J65, 35J67 

3. CJM 2012 (vol 65 pp. 553)
4. CJM 2008 (vol 60 pp. 391)
 Migliore, Juan C.

The Geometry of the Weak Lefschetz Property and Level Sets of Points
In a recent paper, F. Zanello showed that level Artinian algebras in 3
variables can fail to have the Weak Lefschetz Property (WLP), and can
even fail to have unimodal Hilbert function. We show that the same is
true for the Artinian reduction of reduced, level sets of points in
projective 3space. Our main goal is to begin an understanding of how
the geometry of a set of points can prevent its Artinian reduction
from having WLP, which in itself is a very algebraic notion. More
precisely, we produce level sets of points whose Artinian reductions
have socle types 3 and 4 and arbitrary socle degree $\geq 12$ (in the
worst case), but fail to have WLP. We also produce a level set of
points whose Artinian reduction fails to have unimodal Hilbert
function; our example is based on Zanello's example. Finally, we show
that a level set of points can have Artinian reduction that has WLP
but fails to have the Strong Lefschetz Property. While our
constructions are all based on basic double Glinkage, the
implementations use very different methods.
Keywords:Weak Lefschetz Property, Strong Lefschetz Property, basic double Glinkage, level, arithmetically Gorenstein, arithmetically CohenMacaulay, socle type, socle degree, Artinian reduction Categories:13D40, 13D02, 14C20, 13C40, 13C13, 14M05 

5. CJM 2007 (vol 59 pp. 109)
 Jayanthan, A. V.; Puthenpurakal, Tony J.; Verma, J. K.

On Fiber Cones of $\m$Primary Ideals
Two formulas for the multiplicity of the fiber cone
$F(I)=\bigoplus_{n=0}^{\infty} I^n/\m I^n$ of an $\m$primary ideal of
a $d$dimensional CohenMacaulay local ring $(R,\m)$ are derived in
terms of the mixed multiplicity $e_{d1}(\m  I)$, the multiplicity
$e(I)$, and superficial elements. As a consequence, the
CohenMacaulay property of $F(I)$ when $I$ has minimal mixed
multiplicity or almost minimal mixed multiplicity is characterized
in terms of the reduction number of $I$ and lengths of certain ideals.
We also characterize the CohenMacaulay and Gorenstein properties of
fiber cones of $\m$primary ideals with a $d$generated minimal
reduction $J$ satisfying $\ell(I^2/JI)=1$ or
$\ell(I\m/J\m)=1.$
Keywords:fiber cones, mixed multiplicities, joint reductions, CohenMacaulay fiber cones, Gorenstein fiber cones, ideals having minimal and almost minimal mixed multiplicities Categories:13H10, 13H15, 13A30, 13C15, 13A02 

6. CJM 2004 (vol 56 pp. 638)
 Śniatycki, Jędrzej

Multisymplectic Reduction for Proper Actions
We consider symmetries of the Dedonder equation arising from
variational problems with partial derivatives. Assuming a proper
action of the symmetry group, we identify a set of reduced equations
on an open dense subset of the domain of definition of the fields
under consideration. By continuity, the Dedonder equation is
satisfied whenever the reduced equations are satisfied.
Keywords:Dedonder equation, multisymplectic structure, reduction,, symmetries, variational problems Categories:58J70, 35A30 

7. CJM 2001 (vol 53 pp. 715)
 Cushman, Richard; Śniatycki, Jędrzej

Differential Structure of Orbit Spaces
We present a new approach to singular reduction of Hamiltonian systems
with symmetries. The tools we use are the category of differential
spaces of Sikorski and the StefanSussmann theorem. The former is
applied to analyze the differential structure of the spaces involved
and the latter is used to prove that some of these spaces are smooth
manifolds.
Our main result is the identification of accessible sets of the
generalized distribution spanned by the Hamiltonian vector fields of
invariant functions with singular reduced spaces. We are also able
to describe the differential structure of a singular reduced space
corresponding to a coadjoint orbit which need not be locally closed.
Keywords:accessible sets, differential space, Poisson algebra, proper action, singular reduction, symplectic manifolds Categories:37J15, 58A40, 58D19, 70H33 

8. CJM 1999 (vol 51 pp. 26)
 Fabian, Marián; Mordukhovich, Boris S.

Separable Reduction and Supporting Properties of FrÃ©chetLike Normals in Banach Spaces
We develop a method of separable reduction for Fr\'{e}chetlike
normals and $\epsilon$normals to arbitrary sets in general Banach
spaces. This method allows us to reduce certain problems involving
such normals in nonseparable spaces to the separable case. It is
particularly helpful in Asplund spaces where every separable subspace
admits a Fr\'{e}chet smooth renorm. As an applicaton of the separable
reduction method in Asplund spaces, we provide a new direct proof of a
nonconvex extension of the celebrated BishopPhelps density theorem.
Moreover, in this way we establish new characterizations of Asplund
spaces in terms of $\epsilon$normals.
Keywords:nonsmooth analysis, Banach spaces, separable reduction, FrÃ©chetlike normals and subdifferentials, supporting properties, Asplund spaces Categories:49J52, 58C20, 46B20 

9. CJM 1997 (vol 49 pp. 675)
 de Cataldo, Mark Andrea A.

Some adjunctiontheoretic properties of codimension two nonsingular subvarities of quadrics
We make precise the structure of the first two reduction morphisms
associated with codimension two nonsingular subvarieties
of nonsingular quadrics $\Q^n$, $n\geq 5$.
We give a coarse classification of the same class of subvarieties
when they are assumed not to be of loggeneraltype.}
Keywords:Adjunction Theory, classification, codimension two, conic bundles,, low codimension, non loggeneraltype, quadric, reduction, special, variety. Categories:14C05, 14E05, 14E25, 14E30, 14E35, 14J10 
