We introduce and study a remarkable family of real probability
measures $\pi_{st}$ that we call free Bessel laws. These are related
to the free Poisson law $\pi$ via the formulae
$\pi_{s1}=\pi^{\boxtimes s}$ and ${\pi_{1t}=\pi^{\boxplus t}}$. Our
study includes definition and basic properties, analytic aspects
(supports, atoms, densities), combinatorial aspects (functional
transforms, moments, partitions), and a discussion of the relation
with random matrices and quantum groups.

We investigate the number of integral solutions possessed by a
pair of diagonal cubic equations in a large box. Provided that the number of
variables in the system is at least fourteen, and in addition the number of
variables in any non-trivial linear combination of the underlying forms is at
least eight, we obtain an asymptotic formula for the number of integral
solutions consistent with the product of local densities associated with the
system.

Perelman established a differential Li--Yau--Hamilton
(LYH) type inequality for fundamental solutions of the conjugate
heat equation corresponding to the Ricci flow on compact manifolds.
As an application of the LYH inequality,
Perelman proved a pseudolocality result for the Ricci flow on
compact manifolds. In this article we provide the details for the
proofs of these results in the case of a complete noncompact
Riemannian manifold. Using these results we prove that under
certain conditions, a finite time singularity of the Ricci flow
must form within a compact set. The conditions are satisfied by
asymptotically flat manifolds. We also prove a long time existence
result for the K\"ahler--Ricci flow on complete nonnegatively curved K\"ahler
manifolds.

Reversibility of the Fleming--Viot process with mutation, selection,
and recombination is well understood. In this paper, we study the
reversibility of a system of Fleming--Viot processes that live on a
countable number of colonies interacting with each other through
migrations between the colonies. It is shown that reversibility
fails when both migration and mutation are non-trivial.

Let $A_p(G)$ be the Figa-Talamanca,
Herz Banach Algebra on $G$; thus $A_2(G)$
is the Fourier algebra. Strong Ditkin (SD) and
Extremely Strong Ditkin (ESD) sets for the Banach algebras
$A_p^r(G)$ are investigated for abelian and nonabelian
locally compact groups $G$. It is shown that SD and ESD sets
for $A_p(G)$ remain SD and ESD sets for $A_p^r(G)$,
with strict inclusion for ESD sets. The case for the strict
inclusion of SD sets is left open.

A result on the weak sequential completeness of $A_2(F)$
for ESD sets $F$ is proved and used to show that Varopoulos,
Helson, and Sidon sets are not ESD sets for $A_2(G)$, yet they
are such for $A_2^r(G)$ for discrete groups $G$, for
any $1\le r\le 2$.

A result is given on the equivalence of the sequential and the net
definitions of SD or ESD sets for $\sigma$-compact groups.

In this paper, we study the non-vanishing and transcendence of
special values of a varying class of $L$-functions and their derivatives.
This allows us to investigate the transcendence of Petersson norms
of certain weight one modular forms.

We establish the asymptotic behaviour of the partition function, the
heat content, the integrated eigenvalue counting function, and, for
certain points, the on-diagonal heat kernel of generalized
Sierpinski carpets. For all these functions the leading term is of
the form $x^{\gamma}\phi(\log x)$ for a suitable exponent $\gamma$
and $\phi$ a periodic function. We also discuss similar results for
the heat content of affine nested fractals.

We characterize the continuous $q$-ultraspherical polynomials in
terms of the special form of the coefficients in the expansion
$\mathcal{D}_q P_n(x)$ in the basis $\{P_n(x)\}$, $\mathcal{D}_q$
being the Askey--Wilson divided difference operator. The polynomials
are assumed to be symmetric, and the connection coefficients
are multiples of the reciprocal of the square of the $L^2$ norm of
the polynomials. A similar characterization is given for the discrete
$q$-ultraspherical polynomials. A new proof of the evaluation of
the connection coefficients for big $q$-Jacobi polynomials is given.

Using standard transformation and summation formulas for basic
hypergeometric series we obtain an explicit polynomial form of the
$q$-analogue of the 9-$j$ symbols, introduced by the author in a
recent publication. We also consider a limiting case in which the
9-$j$ symbol factors into two Hahn polynomials. The same
factorization occurs in another limit case of the corresponding
$q$-analogue.

In this paper we determine the limiting distributional behavior for
sums of infinitesimal conditionally free random variables. We show that the weak
convergence of classical convolution and that of conditionally free convolution
are equivalent for measures in an infinitesimal triangular array,
where the measures may have unbounded support. Moreover, we use these
limit theorems to study the conditionally free infinite divisibility. These results
are obtained by complex analytic methods without reference to the
combinatorics of c-free convolution.