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1. CMB Online first
On an Exponential Functional Inequality and its Distributional Version Let $G$ be a group and $\mathbb K=\mathbb C$ or $\mathbb
R$.
In this article, as a generalization of the result of Albert
and Baker,
we investigate the behavior of bounded
and unbounded functions $f\colon G\to \mathbb K$ satisfying the inequality
$
\Bigl|f
\Bigl(\sum_{k=1}^n x_k
\Bigr)-\prod_{k=1}^n f(x_k)
\Bigr|\le \phi(x_2, \dots, x_n),\quad \forall\, x_1, \dots,
x_n\in G,
$
where $\phi\colon G^{n-1}\to [0, \infty)$. Also, as a distributional
version of the above inequality we consider the stability of
the functional equation
\begin{equation*}
u\circ S - \overbrace{u\otimes \cdots \otimes u}^{n-\text {times}}=0,
\end{equation*}
where $u$ is a Schwartz distribution or Gelfand hyperfunction,
$\circ$ and $\otimes$ are the pullback and tensor product of
distributions, respectively, and $S(x_1, \dots, x_n)=x_1+ \dots
+x_n$.
Keywords:distribution, exponential functional equation, Gelfand hyperfunction, stability Categories:46F99, 39B82 |
2. CMB 2011 (vol 56 pp. 265)
Embedding Distributions of Generalized Fan Graphs Total embedding distributions have been known for a few classes of graphs.
Chen, Gross, and Rieper
computed it for necklaces, close-end ladders and cobblestone
paths. Kwak and Shim computed it for bouquets of circles and
dipoles. In this paper, a splitting theorem is generalized
and the embedding distributions of
generalized fan graphs are obtained.
Keywords:total embedding distribution, splitting theorem, generalized fan graphs Category:05C10 |
3. CMB 2011 (vol 55 pp. 673)
Multiplicity Free Jacquet Modules Let $F$ be a non-Archimedean local field or a finite field.
Let $n$ be a natural number and $k$ be $1$ or $2$.
Consider $G:=\operatorname{GL}_{n+k}(F)$ and let
$M:=\operatorname{GL}_n(F) \times \operatorname{GL}_k(F)\lt G$ be a maximal Levi subgroup.
Let $U\lt G$ be the corresponding unipotent subgroup and let $P=MU$ be the corresponding parabolic subgroup.
Let $J:=J_M^G: \mathcal{M}(G) \to \mathcal{M}(M)$ be the Jacquet functor, i.e., the functor of coinvariants with respect to $U$.
In this paper we prove that $J$ is a multiplicity free functor, i.e.,
$\dim \operatorname{Hom}_M(J(\pi),\rho)\leq 1$,
for any irreducible representations $\pi$ of $G$ and $\rho$ of $M$.
We adapt the classical method of Gelfand and Kazhdan, which proves the ``multiplicity free" property of certain representations to prove the ``multiplicity free" property of certain functors.
At the end we discuss whether other Jacquet functors are multiplicity free.
Keywords:multiplicity one, Gelfand pair, invariant distribution, finite group Categories:20G05, 20C30, 20C33, 46F10, 47A67 |
4. CMB 2011 (vol 54 pp. 464)
A Characterization of the Compound-Exponential Type Distributions In this paper, a fixed point equation of the
compound-exponential type distributions is derived, and under some
regular conditions,
both the existence and uniqueness of
this fixed point equation are investigated.
A question posed by Pitman and Yor
can be partially answered by using our approach.
Keywords:fixed point equation, compound-exponential type distributions Categories:62E10, 60G50 |
5. CMB 2009 (vol 53 pp. 206)
Semi-Slant Submanifolds of an Almost Paracontact Metric Manifold In this paper, we define and study the geometry of semi-slant submanifolds of an almost paracontact metric manifold. We give some characterizations for a submanifold to be semi-slant submanifold to be semi-slant product and obtain integrability conditions for the distributions involved in the definition of a semi-slant submanifold.
Keywords:paracontact metric manifold, slant distribution, semi-slant submanifold, semi-slant product Categories:53C15, 53C25, 53C40 |
6. CMB 2007 (vol 50 pp. 447)
Generalizations of Frobenius' Theorem on Manifolds and Subcartesian Spaces Let $\mathcal{F}$ be a family of vector fields on a manifold or a
subcartesian space spanning a distribution $D$. We prove that an orbit $O$
of $\mathcal{F}$ is an integral manifold of $D$ if $D$ is involutive on $O$
and it has constant rank on $O$. This result implies Frobenius' theorem, and
its various generalizations, on manifolds as well as on subcartesian spaces.
Keywords:differential spaces, generalized distributions, orbits, Frobenius' theorem, Sussmann's theorem Categories:58A30, 58A40 |