It is shown that a separable $C^*$-algebra is inner quasidiagonal if and
only if it has a separating family of quasidiagonal irreducible
representations. As a consequence, a separable $C^*$-algebra is a strong
NF algebra if and only if it is nuclear and has a separating family of
quasidiagonal irreducible representations.
We also obtain some permanence properties of the class of inner
quasidiagonal $C^*$-algebras.

We show that every free semigroup algebra has a (strongly) unique
Banach space predual. We also provide a new simpler proof that a
weak-$*$ closed unital operator algebra containing a weak-$*$
dense subalgebra of compact operators has a unique Banach space
predual.

We prove that every complete family of linearly non-degenerate
rational curves of degree $e > 2$ in $\mathbb{P}^{n}$ has at most $n-1$
moduli. For $e = 2$ we prove that such a family has at most $n$
moduli. The general method involves exhibiting a map from the base of
a family $X$ to the Grassmannian of $e$-planes in $\mathbb{P}^{n}$ and
analyzing the resulting map on cohomology.

We study the transfer of nondegeneracy
between Lie triple systems and their standard Lie algebra envelopes
as well as between Kantor pairs, their associated Lie triple systems,
and their Lie algebra envelopes. We also show that simple Kantor
pairs and Lie triple systems in characteristic $0$ are
nondegenerate.

We comment on domain conditions that regulate when the adjoint of the
sum or product of two unbounded operators is the sum or product of their
adjoints, and related closure issues. The quantum mechanical problem PHP
essentially selfadjoint for unbounded Hamiltonians is addressed, with new
results.

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.

We study infinitesimal deformations of holomorphic maps of
compact, complex, Kähler manifolds. In particular, we describe a
generalization of Bloch's semiregularity map that annihilates
obstructions to deform holomorphic maps with fixed codomain.

In this paper, another relationship between the quasi-ideal adequate transversals
of an abundant semigroup is given. We introduce the concept of a weakly multiplicative
adequate transversal and the classic result that an adequate transversal is multiplicative
if and only if it is weakly multiplicative and a quasi-ideal is obtained.
Also, we give two equivalent conditions for an adequate transversal to be weakly
multiplicative. We then consider the case when $I$ and $\Lambda$ (defined below) are
bands. This is analogous to the inverse transversal if the regularity condition is adjoined.

We prove, under some conditions on the domains, that the adjoint of
the sum of two unbounded operators is the sum of their adjoints in
both Hilbert and Banach space settings. A similar result about the
closure of operators is also proved. Some interesting consequences
and examples "spice up" the paper.

In this paper, we are going to investigate the canonical property of solutions of
systems of differential equations having a singularity and turning
point of even order. First, by a replacement, we transform the system
to the Sturm-Liouville equation with turning point. Using of the
asymptotic estimates provided by Eberhard, Freiling, and Schneider
for a special fundamental system of solutions of the Sturm-Liouville
equation, we study the infinite product representation of solutions of the systems. Then we
transform the Sturm-Liouville equation with
turning point to the
equation with singularity, then we study the asymptotic behavior of its solutions. Such
representations are relevant to the inverse spectral problem.

We correct an oversight in the the paper
Cartan Subalgebras of
$\mathrm{gl}_\infty$, Canad. Math. Bull. 46(2003), no. 4,
597-616.
doi: 10.4153/CMB-2003-056-1

Building on the work of Nogin,
we prove that the braid group $B_4$ acts transitively on full exceptional
collections of vector bundles on Fano threefolds with $b_2=1$ and
$b_3=0$. Equivalently,
this group acts transitively on the set of simple helices (considered
up to a shift in the derived category) on such a Fano threefold. We
also prove that on
threefolds with $b_2=1$ and very ample anticanonical class, every
exceptional coherent
sheaf is locally free.

We establish a discrete-time criteria guaranteeing the existence of an
exponential dichotomy in the continuous-time
behavior of an abstract evolution family. We prove that an evolution
family ${\cal U}=\{U(t,s)\}_{t
\geq s\geq 0}$ acting on a Banach space $X$ is uniformly
exponentially dichotomic (with respect to its continuous-time
behavior) if and only if the
corresponding difference equation with the inhomogeneous term from
a vector-valued Orlicz sequence space $l^\Phi(\mathbb{N}, X)$
admits
a solution in the same $l^\Phi(\mathbb{N},X)$. The technique of
proof effectively eliminates the continuity hypothesis on the
evolution family (i.e., we do not assume that $U(\,\cdot\,,s)x$
or $U(t,\,\cdot\,)x$ is continuous on $[s,\infty)$, and respectively
$[0,t]$). Thus, some known results given by
Coffman and Schaffer, Perron, and Ta Li are extended.

Writing $s = \sigma + it$ for a complex variable, it is proved
that the modulus of the gamma
function, $|\Gamma(s)|$, is strictly monotone increasing with
respect to $\sigma$ whenever
$|t| > 5/4$. It is also shown that this result is false for $|t|
\leq 1$.

In this paper we characterize the positive
definite measures with discrete Fourier transform. As an
application we provide a characterization of pure point
diffraction in locally compact Abelian groups.

We show that there is an infinite family of hyperbolic knots such that
each knot admits a cyclic surgery $m$ whose adjacent surgeries $m-1$
and $m+1$ are toroidal. This gives an affirmative answer to a
question asked by Boyer and Zhang.

In this note we give a brief review of the construction of a toric
variety $\mathcal{V}$ coming from a genus $g \geq 2$ Riemann surface
$\Sigma^g$ equipped with a trinion, or pair of pants, decomposition.
This was outlined by J. Hurtubise and L.~C. Jeffrey.
A. Tyurin used this construction on a certain
collection of trinion decomposed surfaces to produce a variety
$DM_g$, the so-called Delzant model of moduli space, for
each genus $g.$ We conclude this note with some basic facts about
the moment polytopes of the varieties $\mathcal{V}.$ In particular,
we show that the varieties $DM_g$ constructed by Tyurin, and claimed
to be smooth, are in fact singular for $g \geq 3.$

We consider approximation of multivariate functions in Sobolev
spaces by high order Parzen windows in a non-uniform sampling
setting. Sampling points are neither i.i.d. nor regular, but are
noised from regular grids by non-uniform 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.