Given a nonnegative integer $m$ and a finite collection $\mathcal A$ of
linear forms on $\mathcal Q^d$, the arrangement of affine hyperplanes in
$\mathcal Q^d$ defined by the equations $\alpha(x) = k$ for $\alpha
\in \mathcal A$
and integers $k \in [-m, m]$ is denoted by $\mathcal A^m$. It is proved that
the coefficients of the characteristic polynomial of $\mathcal A^m$ are
quasi-polynomials in $m$ and that they satisfy a simple combinatorial
reciprocity law.

Let $f\colon \mathbb R^n\to \mathbb R$ be $C^\infty$ and let $h\colon
\mathbb R^n\to\mathbb R$ be positive
and continuous. For any unbounded nondecreasing sequence $\{c_k\}$
of nonnegative real numbers and for any sequence without
accumulation points $\{x_m\}$ in $\mathbb R^n$, there exists an entire
function $g\colon\mathbb C^n\to\mathbb C$ taking real values on $\mathbb R^n$ such that
\begin{align*}
&|g^{(\alpha)}(x)-f^{(\alpha)}(x)|\lt h(x), \quad |x|\ge c_k, |\alpha|\le k,
k=0,1,2,\dots,
\\
&g^{(\alpha)}(x_m)=f^{(\alpha)}(x_m), \quad |x_m|\ge c_k, |\alpha|\le k,
m,k=0,1,2,\dots.
\end{align*}
This is a version for functions of several variables of the
case $n=1$ due to L. Hoischen.

In this paper, the boundedness from below of multiplication
operators between $\alpha$-Bloch spaces $\mathcal B^\alpha$, $\alpha\gt 0$, on the
unit disk $D$ is studied completely. For a bounded multiplication
operator $M_u\colon \mathcal B^\alpha\to\mathcal B^\beta$, defined by $M_uf=uf$ for
$f\in\mathcal B^\alpha$, we prove the following result:

(i) If $0\lt \beta\lt \alpha$, or
$0\lt \alpha\le1$ and $\alpha\lt \beta$, $M_u$ is not bounded below;

(ii) if $0\lt \alpha=\beta\le1$, $M_u$ is bounded below if and only if
$\liminf_{z\to\partial D}|u(z)|\gt 0$;

(iii) if $1\lt \alpha\le\beta$, $M_u$ is
bounded below if and only if there exist a $\delta\gt 0$ and a positive
$r\lt 1$ such that for every point $z\in D$ there is a point $z'\in
D$ with the property $d(z',z)\lt r$ and
$(1-|z'|^2)^{\beta-\alpha}|u(z')|\ge\delta$, where $d(\cdot,\cdot)$ denotes
the pseudo-distance on $D$.

We describe the representation theory of $C^*$-crossed-products of a unital $C^*$-algebra A by the cyclic group of order~2. We prove that there are two main types of irreducible representations for the crossed-product: those whose restriction to A is irreducible and those who are the sum of two unitarily unequivalent representations of~A. We characterize each class in term of the restriction of the representations to the fixed point $C^*$-subalgebra of~A. We apply our results to compute the K-theory of several crossed-products of the free group on two generators.

We show that if the general real method $(\cdot ,\cdot )_\Gamma$
preserves the Banach-algebra structure, then a bilinear
interpolation theorem holds for $(\cdot ,\cdot )_\Gamma$.

In this paper, we prove that the unboundedness of ranks in families of Jacobian varieties of twisted Fermat curves is equivalent to the divergence of certain infinite series.

We show that for every separable Banach space $X$,
either $\mathrm{SP_w}(X)$ (the set of all spreading models
of $X$ generated by weakly-null sequences in $X$, modulo
equivalence) is countable, or $\mathrm{SP_w}(X)$ contains an
antichain of the size of the continuum. This answers
a question of S.~J. Dilworth, E. Odell, and B. Sari.

An example is given of a UFD which has an infinitely generated Derksen invariant. The ring is "almost rigid" meaning that the Derksen invariant is equal to the Makar-Limanov invariant. Techniques to show that a ring is (almost) rigid are discussed, among which is a generalization of Mason's abc-theorem.

We prove that the group of rational points of a non-isotrivial elliptic curve defined over the perfect closure of a function field in positive characteristic is finitely generated.

Let $\phi$ be a Drinfeld module of generic characteristic, and let X be a sufficiently generic affine subvariety of $\mathbb{G_a^g}$. We show that the intersection of X with a finite rank $\phi$-submodule of $\mathbb{G_a^g}$ is finite.

We show that the sequence of integers which have nearly the typical number of distinct prime factors forms a Poisson process. More precisely, for $\delta$ arbitrarily small and positive, the nearest neighbor spacings between integers n with $|\omega(n) - log log n| < (log log n)^{\delta}$ obey the Poisson distribution law.

Classical results of Kalton and techniques of Feder are used to study the complementation of the space $W(X, Y)$ of weakly compact operators and the space $K(X,Y)$ of compact operators in the space $L(X,Y)$ of all bounded linear maps from X to Y.

In this paper, we find a class of $(\alpha,\beta)$ metrics which have a bounded Cartan torsion. This class contains all Randers metrics. Furthermore, we give some applications and obtain two corollaries about curvature of this metrics.

We give a further decay estimate for the Dziubański-Hernández wavelets that are band-limited and have subexponential decay. This is done by constructing an appropriate bell function and using the Paley-Wiener theorem for ultradifferentiable functions.

A Pisot number is a real algebraic integer greater than 1, all of whose conjugates lie strictly inside the open unit disk; a Salem number is a real algebraic integer greater than 1, all of whose conjugate roots are inside the closed unit disk, with at least one of them of modulus exactly 1. Pisot numbers have been studied extensively, and an algorithm to generate them is well known. Our main result characterises all Pisot numbers whose minimal polynomial is derived from a Newman polynomial -- one with $\{0,1\}$-coefficients -- and shows that they form a strictly increasing sequence with limit $(1+\sqrt{5}) / 2$. It has long been known that every Pisot number is a limit point, from both sides, of sequences of Salem numbers. We show that this remains true, from at least one side, for the restricted sets of Pisot and Salem numbers that are generated by Newman polynomials.

In this paper, we establish several weighted $L^p (1\lt p \lt \infty)$
Hardy type inequalities related to the generalized Greiner operator
by improving the method of Kombe. Then the best
constants in inequalities are discussed by introducing new polar
coordinates.

We establish variants of stability estimates in norms
somewhat stronger than the $H^1$-norm under Arnold's stability hypotheses on
steady solutions to the Euler equations for fluid flow on planar
domains.

Multiplicity-free algebraic geometry is the study of subvarieties
$Y\subseteq X$ with the ``smallest invariants'' as witnessed by a
multiplicity-free Chow ring decomposition of
$[Y]\in A^{\star}(X)$ into a predetermined
linear basis.

This paper concerns the case of Richardson subvarieties of the Grassmannian
in terms of the Schubert basis. We give a nonrecursive combinatorial
classification of multiplicity-free Richardson varieties, i.e.,
we classify multiplicity-free products of Schubert classes. This answers
a question of W. Fulton.