If $(X,d)$ is a metric space then the map $f\colon X\to X$ is defined to be a
weak contraction if $d(f(x),f(y))<d(x,y)$ for all $x,y\in X$, $x\neq y$. We
determine the simplest non-closed sets $X\subseteq\RR^n$ in the sense of
descriptive set theoretic complexity such that every weak contraction $f\colon
X\to X$ is constant. In order to do so, we prove that there exists a non-closed
$F_{\sigma}$ set $F\subseteq \RR$ such that every weak contraction $f\colon
F\to F$ is constant.

Let D be a planar Lipschitz domain and consider the Beurling transform of the
characteristic function of D, B(1_D). Let 1<p<\infty and 0<a<1 with ap>1. In
this paper we show that if the outward unit normal N on bD, the boundary of D,
belongs to the Besov space B_{p,p}^{a-1/p}(bD), then the Beurling transform of
1_D is in the Sobolev space W^{a,p}(D). This result is sharp. Further, together
with recent results by Cruz, Mateu and Orobitg, this implies that the Beurling
transform is bounded in W^{a,p}(D) if N belongs to B_{p,p}^{a-1/p}(bD),
assuming that ap>2.

We study an extension to Fourier transforms of the old problem on absolute
convergence of the re-expansion in the sine (cosine) Fourier series of an
absolutely convergent cosine (sine) Fourier series. The results are obtained by
revealing certain relations between the Fourier transforms and their Hilbert
transforms.

In this paper, we consider some cubic near-Hamiltonian systems obtained from
perturbing the symmetric cubic Hamiltonian system with two symmetric singular
points by cubic polynomials. First, following Han [2012] we develop a method to
study the analytical property of the Melnikov function near the origin for
near-Hamiltonian system having the origin as its elementary center or nilpotent
center.

There exists a remarkable connection between the quantum mechanical
Landau-Zener problem and purely classical-mechanical problem of a ball rolling
on a Cornu spiral. This correspondence allows us to calculate a complicated
multiple integral, a kind of multi-dimensional generalization of Fresnel
integrals. A direct method of calculation is also considered but found to be
successful only in some low-dimensional cases. As a byproduct of this direct
method, an interesting new integral representation for $\zeta(2)$ is obtained.

We study boundedness properties of a class of multiparameter paraproducts on
the dual space of the dyadic Hardy space H_d^1(T^N), the dyadic product BMO
space BMO_d(T^N). For this, we introduce a notion of logarithmic mean
oscillation on the polydisc. We also obtain a result on the boundedness of
iterated commutators on BMO([0,1]^2).

We find a formula that relates the Fourier transform of a radial function on
$\mathbf{R}^n$ with the Fourier transform of the same function defined on
$\mathbf{R}^{n+2}$. This formula enables one to explicitly calculate the
Fourier transform of any radial function $f(r)$ in any dimension, provided one
knows the Fourier transform of the one-dimensional function $t\to f(|t|)$ and
the two-dimensional function $(x_1,x_2)\to f(|(x_1,x_2)|)$. We prove analogous
results for radial tempered distributions.

In 1986 J. Nuttall published in Constructive Approximation the paper
"Asymptotics of generalized Jacobi polynomials", where with his usual insight
he studied the behavior of the denominators ("generalized Jacobi polynomials")
and the remainders of the Pade approximants to a special class of algebraic
functions with 3 branch points. 25 years later we try to look at this problem
from a modern perspective. On one hand, the generalized Jacobi polynomials
constitute an instance of the so-called Heine-Stieltjes polynomials, i.e. they
are solutions of linear ODE with polynomial coefficients.

Porosity and dimension are two useful, but different, concepts that quantify
the size of fractal sets and measures. An active area of research concerns
understanding the relationship between these two concepts. In this article we
will survey the various notions of porosity of sets and measures that have been
proposed, and how they relate to dimension. Along the way, we will introduce
the idea of local entropy averages, which arose in a different context, and was
then applied to obtain a bound for the dimension of mean porous measures.

Turan's method, as expressed in his books, is a careful study of
trigonometric polynomials from different points of view. The present article
starts from a problem asked by Turan: how to construct a sequence of real
numbers x(j) (j= 1,2,...n) such that the almost periodic polynomial whose
frequencies are the x(j) and the coefficients are 1 are small (say, their
absolute values are less than n d, d< given) for all integral values of the
variable m between 1 and M= M(n,d) ? The best known answer is a random choice
of the x(j) modulo 1.

In this paper, we study the necessary conditions and sufficient conditions
for the twisted angles of the central configurations formed by two twisted
regular polygons, specially, we prove that for the 2N-body problems, the
twisted angles must be$\theta=0 {or} \theta=\pi/N$.

We investigate the Pompeiu property for subsets of the real line, under no
assumption of connectedness. In particular we focus our study on finite unions
of bounded (disjoint) intervals, and we emphasize the different results
corresponding to the cases where the function in question is supposed to have
constant integral on all isometric images, or just on all the
translation-images of the domain.

We derive simple linear, inhomogeneous recurrences for the variance of the
index by utilising the fact that the generating function for the distribution
of the number of positive eigenvalues of a Gaussian unitary ensemble is a
$\tau$-function of the fourth Painlev\'e equation. From this we deduce a simple
summation formula, several integral representations and finally an exact
hypergeometric function evaluation for the variance.

In this paper we extend the results of the research started by the first
author, in which Karlin-McGregor diagonalization of certain reversible Markov
chains over countably infinite general state spaces by orthogonal polynomials
was used to estimate the rate of convergence to a stationary distribution.

We prove a single sum formula for the linearization coefficients of the
Bessel polynomials. In two special cases we show that our formula reduces
indeed to Berg and Vignat's formulas in their proof of the positivity results
about these coefficients (Constructive Approximation, 27(2008), 15-32). As a
bonus we also obtain a generalization of an integral formula of Boros and Moll
(J. Comput. Appl. Math. 106 (1999), 361-368).

It it known that extremizers for the $L^2$ to $L^6$ adjoint Fourier
restriction inequality on the cone in $\mathbbm R^3$ exist. Here we show that
nonnegative extremizing sequences are precompact, after the application of
symmetries of the cone. If we use the knowledge of the exact form of the
extremizers, as found by Carneiro, then we can show that nonnegative
extremizing sequences converge, after the application of symmetries.

We obtain existence and uniqueness for odd second order oscillators in the
space of odd functions without topological assumptions on the nonlinear part.

We use the recent findings of Cohl [arXiv:1105.2735] and evaluate the
principal and the residual values of the integral
$\int_{-1}^{1}\mathrm{d}t\:(1-t^{2})^{\lambda-1/2}(x-t)^{-\kappa-1/2}C_{n}^{\lambda}(t)$,
with $\textrm{Re}(\lambda)>-1/2$, $\kappa\in\mathbb{C}$, $-1<x<1$, where
$C_{n}^{\lambda}(t)$ is the Gegenbauer polynomial.

In this article, the study of the orthogonality properties of $q$-polynomials
of the Hahn class started in the initial article by R. \'{A}lvarez-Nodarse, R.
Sevinik-Ad\i g{\"{u}}zel, and H. Ta\c{s}eli, \textit{On the orthogonality of
$q$-classical polynomials of the Hahn class I} is proceeded.

Relaxation theorems for multiple integrals on W^{1,p}(\Omega;\RR^m), where
p\in]1,\infty[, are proved under general conditions on the integrand
L:\MM\to[0,\infty] which is Borel measurable and not necessarily finite. We
involve a localization principle that we previously used to prove a general
lower semicontinuity result.

We isolate a general condition on L:\MM\to[0,\infty], assumed to be
continuous, under which W^{1,q}-quasiconvexity with q\in[1,\infty] is a
sufficient condition for I(u)=\int_\Omega L(\nabla u(x))dx to be sequentially
weakly lower semicontinuous on W^{1,p}(\Omega;\RR^m) with p\in]1,\infty[.

We study the notion of regular singularities for parameterized complex
ordinary linear differential systems, prove an analogue of the Schlesinger
theorem for systems with regular singularities and solve both a parameterized
version of the weak Riemann-Hilbert Problem and a special case of the inverse
problem in parameterized Picard-Vessiot theory.

On a metric measure space satisfying the doubling property, we establish
several optimal characterizations of Besov and Triebel-Lizorkin spaces,
including a pointwise characterization. Moreover, we discuss their
(non)triviality under a Poincar\'e inequality.

We study the properties of the logarithm of the derivative operator and show
that its action on a constant is not zero, but yields the sum of the
logarithmic function and the Euler-Mascheroni constant. We discuss more general
aspects concerning the logarithm of an operator for the study of the properties
of the Bessel functions.

We show that the coefficients of the three-term recurrence relation for
orthogonal polynomials with respect to a semi-classical extension of the
Laguerre weight satisfy the fourth Painlev\'e equation when viewed as functions
of one of the parameters in the weight. We compare different approaches to
derive this result, namely, the ladder operators approach, the isomonodromy
deformations approach and combining the Toda system for the recurrence
coefficients with a discrete equation.

In this paper a new integral for the remainder of $\pi(x)$ is obtained. It is
proved that there is an infinite set of the formulae containing miscellaneous
parts of this integral.

In this paper we obtain the $L^p$-boundedness of Riesz transforms for Dunkl
transform for all $1<p<\infty$.

We obtain simple proofs of certain inequalites for bivariate means.

We analyze the Synchrosqueezing transform, a consistent and invertible
time-frequency analysis tool that can identify and extract oscillating
components (of time-varying frequency and amplitude) from regularly sampled
time series. We first describe a fast algorithm implementing the transform.
Second, we show Synchrosqueezing is robust to bounded perturbations of the
signal. This stability property extends the applicability of Synchrosqueezing
to the analysis of nonuniformly sampled and noisy time series, which are
ubiquitous in engineering and the natural sciences.

In this paper, we discuss foliations by real curves. We investigate
differential equations which are modifications of du/dx = v along leaves. Our
focus is on having a solution operator so that u is continuous if v is
continuous.

We provide evaluations of several recently studied higher and multiple Mahler
measures using log-sine integrals. This is complemented with an analysis of
generating functions and identities for log-sine integrals which allows the
evaluations to be expressed in terms of zeta values or more general
polylogarithmic terms. The machinery developed is then applied to evaluation of
further families of multiple Mahler measures.

We introduce a new class of Hardy spaces $H^{\phi(\cdot,\cdot)}(\mathbb
R^n)$, called Hardy spaces of Musielak-Orlicz type, which generalize the
Hardy-Orlicz spaces of Janson and the weighted Hardy spaces of Garc\'ia-Cuerva,
Str\"omberg, and Torchinsky. Here, $\phi: \mathbb R^n\times [0,\infty)\to
[0,\infty)$ is a function such that $\phi(x,\cdot)$ is an Orlicz function and
$\phi(\cdot,t)$ is a Muckenhoupt $A_\infty$ weight. A function $f$ belongs to
$H^{\phi(\cdot,\cdot)}(\mathbb R^n)$ if and only if its maximal function $f^*$
is so that $x\mapsto \phi(x,|f^*(x)|)$ is integrable.

We continue the analysis of higher and multiple Mahler measures using
log-sine integrals as started in "Log-sine evaluations of Mahler measures" and
"Special values of generalized log-sine integrals" by two of the authors. This
motivates a detailed study of various multiple polylogarithms and worked
examples are given. Our techniques enable the reduction of several multiple
Mahler measures, and supply an easy proof of two conjectures by Boyd.

In this paper we present a theory of vessels, defined originally by M. S.
Livsic and studied in the Phd thesis of the author. A vessel is a collection of
two Hilbert spaces and operators acting between them with certain properties.
Although we present vessels in a full generality at the beginning, we consider
as an application a special case, corresponding to one dimensional Schrodinger
equation (or Sturm Liouville equation) with a spectral parameter s:
$-\frac{d^2}{dx^2} y(x) + q(x) y(x) = s^2 y(x)$.

We prove that if $\phi: {\Bbb R}^d \times {\Bbb R}^d \to {\Bbb R}$, $d \ge
2$, is a homogeneous function, smooth away from the origin and having non-zero
Monge-Ampere determinant away from the origin, then $$ R^{-d} # \{(n,m) \in
{\Bbb Z}^d \times {\Bbb Z}^d: |n|, |m| \leq CR; R \leq \phi(n,m) \leq R+\delta
\} \lesssim \max \{R^{d-2+\frac{2}{d+1}}, R^{d-1} \delta \}.$$

We consider a multi-parameter family of canonical coordinates and mirror maps
o\ riginally introduced by Zudilin [Math. Notes 71 (2002), 604-616]. This
family includes many of the known one-variable mirror maps as special cases, in
particular many of modular origin and the celebrated example of Candelas, de la
Ossa, Green and\

Let F be a differential field of characteristic zero. In this article, we
construct Picard-Vessiot extensions of F whose differential Galois group is
isomorphic to the full unipotent subgroup of the upper triangular group defined
over the field of constants of F. We will also give a procedure to compute
linear differential operators for our Picard-Vessiot extensions. We do not
require the condition that the field of constants be algebraically closed.

We consider measures supported on the bi-circle and review the recurrence
relations satisfied by the orthogonal polynomials associated with these
measures constructed using the lexicographical or reverse lexicographical
ordering. New relations are derived among these recurrence coefficients. We
extend the results of [8] on a parameterization for Bernstein-Szego measures
supported on the bi-circle.

In the paper, we establish an inequality involving the gamma and digamma
functions and use it to prove the negativity and monotonicity of a function
involving the gamma and digamma functions.

In 1969, Vic Klee asked whether a convex body is uniquely determined (up to
translation and reflection in the origin) by its inner section function, the
function giving for each direction the maximal area of sections of the body by
hyperplanes orthogonal to that direction. We answer this question in the
negative by constructing two infinitely smooth convex bodies of revolution
about the $x_n$-axis in $\R^n$, $n\ge 3$, one origin symmetric and the other
not centrally symmetric, with the same inner section function. Moreover, the
pair of bodies can be arbitrarily close to the unit ball.

We consider the moments of products of complete elliptic integrals of the
first and second kinds. In particular, we derive new results using elementary
means, aided by computer experimentation and a theorem of W. Zudilin. Diverse
related evaluations, and two conjectures, are also given.

We apply the Bennett-Carbery-Tao multilinear restriction estimate in order to
bound restriction operators and more general oscillatory integral operators. We
get improved L^p estimates in the Stein restriction problem for dimension at
least 5 and a small improvement in dimension 3. We also get improved estimates
on Hormander-type oscillatory integral operators when the dimension is even or
when the quadratic term in the phase function is positive definite. The
oscillatory estimates are related to improved bounds on the dimensions of
curved Kakeya sets in even dimensions.

The Marcinkiewicz integral is essentially a Littlewood-Paley $g$-function,
which plays a important role in harmonic analysis. In this article, by using
the atomic decomposition theory of weighted Hardy spaces and homogeneous
weighted Herz-type Hardy spaces, we will obtain some weighted weak type
estimates for Marcinkiewicz integrals on these spaces.

By systematically applying ten well-known and inequivalent two-part relations
between hypergeometric sums 3F2(...|1) to the published database of all such
sums, 62 new sums are obtained. The existing literature is summarized, and many
purportedly novel results extracted from that literature are shown to be
special cases of these new sums. The general problem of finding elements
contiguous to Watson's, Dixon's and Whipple's theorems is reduced to a simple
algorithm suitable for machine computation. Several errors in the literature
are corrected or noted.

We give equivalent forms of Askey-Wilson (AW) polynomials expressing them
with a help of Al-Salam-Chihara polynomials. After restricting parameters of AW
polynomials to complex conjugate pairs we give probabilistic interpretation of
AW weight function and expand it in the series similar to Poisson-Mehler
expansion formula and give its probabilistic interpretation. On the way (by
setting certain parameter q to to 0) we get some formulae useful in rapidly
developing so called 'free probability'.

We investigate $g$-functions and Lusin's area type integrals related to
certain multi-dimensional Dunkl and Laguerre settings. We prove that the
considered square functions are bounded on weighted $L^p$, $1<p<\infty$, and
from $L^1$ into weak $L^1$.

The paper studies solutions of ODEs killed on the domain boundary and their
first exit times from the domain. Some regularity is obtained in the form of
estimates of L_2-norm for functionals on solutions and the first exit times.
Some regularity properties for the density of solutions killed on the boundary
is also studied for the case of random initial conditions.

Main purpose of this paper is to reconstruct generating function of the
Bernstein type polynomials. Some properties this generating functions are
given. By applying this generating function, not only derivative of these
polynomials but also recurrence relations of these polynomials are found.
Interpolation function of these polynomials is also constructed via Mellin
Transformation. This function interpolates these polynomials at negative
integers which are given explicitly.

This note contains suficient conditions for the probability density function
of an arbitrary continuous univariate distribution such that the corresponding
Mills ratio to be reciprocally convex (concave). To illustrate the applications
of the main results, the Mills ratio of some common continuous univariate
distributions, like gamma, log-normal and Student's t distributions, are
discussed in details. The application to monopoly theory is also summarized.

Let $P\in\Z[n]$ with $P(0)=0$ and $\VE>0$. We show, using Fourier analytic
techniques, that if $N\geq \exp\exp(C\VE^{-1}\log\VE^{-1})$ and
$A\subseteq\{1,\...,N\}$, then there must exist $n\in\N$ such that
\[\frac{|A\cap (A+P(n))|}{N}>\left(\frac{|A|}{N}\right)^2-\VE.\]

This paper investigates positive harmonic functions on a domain which
contains an infinite cylinder, and whose boundary is contained in the union of
parallel hyperplanes. (In the plane its boundary consists of two sets of
vertical semi-infinite lines.) It characterizes, in terms of the spacing
between the hyperplanes, those domains for which there exist minimal harmonic
functions with a certain exponential growth.

We consider the linear span S of the functions tak (with some ak > 0) in
weighted L2 spaces, with rather general weights. We give one necessary and one
sufficient condition for S to be dense. Some comparisons are also made between
the new results and those that can be deduced from older ones in the
literature.

We study some properties of smooth sets in the sense defined by Hungerford.
We prove a sharp form of Hungerford's Theorem on the Hausdorff dimension of
their boundaries on Euclidean spaces and show the invariance of the definition
under a class of automorphisms of the ambient space.

We present a new criterion for the boundedness in weighted $L^p$ spaces of
multiplier operators for Laguerre and Hermite expansions that arise from a
Laplace-Stieltjes transform. As a special case, we recover known results on
weighted estimates for Laguerre and Hermite fractional integrals with a unified
and simpler approach.

This paper solves the point mass problem on the real line when the recurrence
coefficients are asymptotically periodic. First, we give formulae for the
perturbed orthogonal polynomials and the perturbed recurrence coefficients when
a point mass is added to any non-trivial measure supported on the real line.
Then we analyze the recurrence relation and prove new asymptotic formulae for
the orthogonal polynomials associated to a measure with asymptotically periodic
coefficients.

We present the construction of a probability measure with compact support on
R such that adding a discrete pure point results in changes in the recursion
coefficients without exponential decay.

In this paper, interpolation by scaled multi-integer translates of Gaussian
kernels is studied. The main result establishes $L_p$ Sobolev error estimates
and shows that the error is controlled by the $L_p$ multiplier norm of a
Fourier multiplier closely related to the cardinal interpolant, and comparable
to the Hilbert transform. Consequently, its multiplier norm is bounded
independent of the grid spacing when $1<p<\infty$, and involves a logarithmic
term when $p=1$ or $\infty$.

The notions of higher-order weighted multilinear Poincar\'e and Sobolev
inequalities in Carnot groups are introduced. As an application, weighted
Leibnitz-type rules in Campanato-Morrey spaces are established.

Let $X$ be a Banach space and suppose $Y$ is a Banach subspace compactly
embedded into $X$, and $(a_k)$ is a bounded weakly null sequence of functionals
in $X^*$. Then there exists a sequence $\{\varepsilon_n\} \searrow 0$ such that
$|a_n(y)| \leq \varepsilon_n \|y\|_Y$ for every $n\in\mathbb{N}$ and every
$y\in Y$. We prove this result and we use it for the study of fast decay of
Fourier coefficients in $L^p(\mathbb{T})$ and frame coefficients in the Hilbert
setting.

We consider a family of orthogonal polynomials extensively studied by Hoare
and Rahman in a probability context and from an algebraic point of view by
Aomoto, Gelfand, Mizukawa and Tanaka. Our approch is different from theirs.

We investigate how the integrability of the derivatives of Orlicz-Sobolev
mappings defined on open subsets of $\mathbb{R}^n$ affect the sizes of the
images of sets of Hausdorff dimension less than $n$. We measure the sizes of
the image sets in terms of generalized Hausdorff measures.

The construction of needlet-type wavelets on sections of the spin line
bundles over the sphere has been recently addressed in Geller and Marinucci
(2008), and Geller et al. (2008,2009). Here we focus on an alternative proposal
for needlets on this spin line bundle, in which needlet coefficients arise from
the usual, rather than the spin, spherical harmonics, as in the previous
constructions.

We prove weighted strong inequalities for the multilinear potential operator
${\cal T}_{\phi}$ and its commutator, where the kernel $\phi$ satisfies certain
growth condition. For these operators we also obtain Fefferman-Stein type
inequalities and Coifman type estimates. On the other hand we prove weighted
weak type inequalities for the multilinear maximal operator
$\mathcal{M}_{\varphi,B}$ associated to a essentially nondecreasing function
$\varphi$ and to a submultiplicative Young function $B$.

The polar representation theorem for the n-dimensional time-dependent linear
Hamiltonian system with continuous coefficients, states that, given two
isotropic solutions (Q1, P1) and (Q2, P2), with the identity matrix as
Wronskian,the formula Q2 = rcos(f), Q1 = rsin(f), holds, where r and f are
continuous matrices, r is non-singular and f is symmetric. In this article we
use the monotonicity properties of the matrix f eigenvalues in order to obtain
results on the Sturm-Liouville problem.

Let $\mathcal{E}_1(M)^+$ be the local ring of germs at $0$ of functions
belonging to a given Denjoy-Carleman quasianalytic class in a neighborhood of
$0$ in $[0,+\infty[$. We show that the ring $\mathcal{E}_1(M)^+$ contains
elements that cannot be extended quasianalytically in a neighborhood of $0$ in
$\mathbb{R}$, unless it coincides with the ring of real-analytic germs.

We present a formal derivation of a simplified version of Compressible
Primitive Equations (CPEs) for atmosphere modeling. They are obtained from
$3$-D compressible Navier-Stokes equations with an \emph{anisotropic viscous
stress tensor} where viscosity depends on the density. We then study the
stability of the weak solutions of this model by using an intermediate model,
called model problem, which is more simple and practical, to achieve the main
result.

We introduce a new class of fully nonlinear integro-differential operators
with possible nonsymmetric kernels, which includes the ones that arise from
stochastic control problems with purely jump L\`evy processes. If the index of
the operator $\sigma$ is in $ (1,2)$ (subcritical case), then we obtain a
comparison principle, a nonlocal version of the Alexandroff-Backelman-Pucci
estimate, a Harnack inequality, a H\"older regularity, and an interior $\rm
C^{1,\alpha}$-regularity for fully nonlinear integro-differential equations
associated with such a class.

A modification of the well-known step-by-step process for solving
Nevanlinna-Pick problems in the class of $\bR_0$-functions gives rise to a
linear pencil $H-\lambda J$, where $H$ and $J$ are Hermitian tridiagonal
matrices. First, we show that $J$ is a positive operator. Then it is proved
that the corresponding Nevanlinna-Pick problem has a unique solution iff the
densely defined symmetric operator $J^{-1/2}HJ^{-1/2}$ is self-adjoint and some
criteria for this operator to be self-adjoint are presented.

We generalise the Bernoulli numbers to include the case where the index may
be a continuous variable.

In this paper we give a precise estimate of the discrepancy of a class of
uniformly distributed sequences of partitions. Among them we found a large
class having low discrepancy (which means of order 1/N. One of them is the
Kakutani-Fibonacci sequence.

It is proved in this paper that there is a nonlocal asymptotic splitting (in
the integral sense) of the function $Z^4(t)$ into two factors. The
corresponding formula cannot be obtained in the known theories of
Balasubramanian, Heath-Brown and Ivic.

In this paper, we extend the Montogomery identities for the Riemann-Liouville
fractional integrals. We also use this Montogomery identities to establish some
new integral inequalities for convex functions.

In this paper, we establish new an inequality of weighted Ostrowski-type for
double integrals involving functions of two independent variables by using
fairly elementary analysis.

In this paper, we establish several new inequalities for some twice
differantiable mappings that are connected with the celebrated Hermite-Hadamard
integral inequality. Some applications for special means of real numbers are
also provided.

We prove that the holonomy group at infinity of the Painleve VI equation is
virtually commutative.

Iterated commutators of multilinear Calderon-Zygmund operators and pointwise
multiplication with functions in $BMO$ are studied in products of Lebesgue
spaces. Both strong type and weak end-point estimates are obtained, including
weighted results involving the vectors weights of the multilinear
Calderon-Zygmund theory recently introduced in the literature. Some better than
expected estimates for certain multilinear operators are presented too.

In this paper we study the generalized Erdos-Falconer distance problems in
the finite field setting. The generalized distances are defined in terms of
polynomials, and various formulas for sizes of distance sets are obtained. In
particular, we develop a simple formula for estimating the cardinality of
distance sets determined by diagonal polynomials. As a result, we generalize
the spherical distance problems due to Iosevich and Rudnev and the cubic
distance problems due to Iosevich and Koh. Moreover, our results are of higher
dimensional version for Vu's work on two dimension.

Multiple orthogonal polynomials are a generalization of orthogonal
polynomials in which the orthogonality is distributed among a number of
orthogonality weights. They appear in random matrix theory in the form of
special determinantal point processes that are called multiple orthogonal
polynomial (MOP) ensembles. The correlation kernel in such an ensemble is
expressed in terms of the solution of a Riemann-Hilbert problem, that is of
size (r+1) x (r+1) in the case of r weights.

We show that the solutions of first order nonlinear ODEs can be controlled
globally in the complex domain, using a finite set of constants of motion
defined in regions of $\CC$. These constants of motion enable us to obtain
quantitative behaviors of the solutions far away from the origin, as well as to
determine the position of singularities of the solution.

The present paper considers two concepts of nonuniform exponential dichotomy
(in the sense of Barreira-Valls) for evolution operators in Banach spaces. Some
examples clarify the relations between these concepts. A variant for the case
of nonuniform exponential dichotomy of a well-known theorem due to Datko is
obtained. We also prove a sufficient condition for the existence of exponential
dichotomy of evolution operators in terms of the existence of a Lyapunov
function in the general case of Banach spaces.

We tackle the Riemann-Hilbert problem on the Riemann sphere as stalk-wise
logarithmic modifications of the classical R\"ohrl-Deligne vector bundle. We
show that the solutions of the Riemann-Hilbert problem are in bijection with
some families of local filtrations which are stable under the prescribed
monodromy maps. We introduce the notion of Birkhoff-Grothendieck
trivialisation, and show that its computation corresponds to geodesic paths in
some local affine Bruhat-Tits building.

We discuss factorization of the hypergeometric-type difference equations on
the uniform lattices and show how one can construct a dynamical algebra, which
corresponds to each of these equations. Some examples are exhibited, in
particular, we show that several models of discrete harmonic oscillators,
previously considered in a number of publications, can be treated in a unified
form.

We argue that one can factorize the difference equation of hypergeometric
type on the nonuniform lattices in general case. It is shown that in the most
cases of q-linear spectrum of the eigenvalues this directly leads to the
dynamical symmetry algebra $su_q(1,1)$, whose generators are explicitly
constructed in terms of the difference operators, obtained in the process of
factorization. Thus all models with the $q$-linear spectrum (some of them, but
not all, previously considered in a number of publications) can be treated in a
unified form.

We construct a $\Lambda(4)$ set which is not a finite union of $B_2[G]$ sets,
answering a question of Rudin. Our construction is an interesting combinatorial
object in its own right, and provides a counterexample to a weaker
characterization of $\Lambda(4)$ sets than stated in Rudin's original question.
It also serves as a counterexample to several natural conjectures in the
pursuit of an "anti-Freiman" theory in additive combinatorics. In particular,
we answer a question along these lines posed by O'Bryant.

We say that the polynomial sequence $(Q^{(\lambda)}_n)$ is a semiclassical
Sobolev polynomial sequence when it is orthogonal with respect to the inner
product $$ <p, r>_S=<{{\bf u}} ,{p\, r}> +\lambda <{{\bf u}}, {{\mathscr D}p
\,{\mathscr D}r}>, $$ where ${\bf u}$ is a semiclassical linear functional,
${\mathscr D}$ is the differential, the difference or the $q$--difference
operator, and $\lambda$ is a positive constant.

Jacques Peyri\`ere investigated Riesz products associated with a given set of
frequencies and the corresponding coefficients : mutual singularity or absolute
continuity of the measures defined by two such products, Hausdorff dimensions
of the sets carrying such measures, multifractal analysis. The present
expository paper starts from Peyri\`ere's results and give some new ways to
investigate the same questions. Since Riesz products are also an essential tool
for Sidon sets, the characterisations of Sidon sets by Pisier and Bourgain,
using quasi-independent sets, are given and commented.

We analyze families of non-autonomous systems of first-order ordinary
differential equations admitting a common time-dependent superposition rule,
i.e., a time-dependent map expressing any solution of each of these systems in
terms of a generic set of particular solutions of the system and some
constants. We next study relations of these families, called Lie families, with
the theory of Lie and quasi-Lie systems and apply our theory to provide common
time-dependent superposition rules for certain Lie families.

We formulate and discuss a conjecture which would extend a classical
inequality of Bernstein.

We consider a Cauchy problem for some family of q-difference-differential
equations with Fuchsian and irregular singularities, that admit a unique formal
power series solution in two variables t and z for given formal power series
initial conditions. Under suitable conditions and by the application of certain
q-Borel and Laplace transforms (introduced by J.-P. Ramis and C.

For functions that take values in the Clifford algebra, we study the
Clifford-Fourier transform on $R^m$ defined with a kernel function $K(x,y) :=
e^{\frac{i \pi}{2} \Gamma_{y}}e^{-i <x,y>}$, replacing the kernel $e^{i <x,y>}$
of the ordinary Fourier transform, where $\Gamma_{y} := - \sum_{j<k} e_{j}e_{k}
(y_{j} \partial_{y_{k}} - y_{k}\partial_{y_{j}})$. An explicit formula of
$K(x,y)$ is derived, which can be further simplified to a finite sum of Bessel
functions when $m$ is even.

Let $S$ be the Lie group $\mathrm{R}^n\ltimes \mathrm{R}^+$ endowed with the
left-invariant Riemannian symmetric space structure and the right Haar measure
$\rho$, which is a Lie group of exponential growth. Hebisch and Steger in
[Math. Z. 245(2003), 37--61] proved that any integrable function on $(S,\rho)$
admits a Calder\'on--Zygmund decomposition which involves a particular family
of sets, called Calder\'on--Zygmund sets. In this paper, we first show the
existence of a dyadic grid in the group $S$, which has {nice} properties
similar to the classical Euclidean dyadic cubes.

In this paper we prove that the Hankel multipliers of Laplace transform type
on $(0,1)^n$ are of weak type (1,1). Also we analyze Lp-boundedness properties
for the imaginary powers of Bessel operator on $(0,1)^n$.

A theorem on the existence of exactly $N$ limit cycles around a critical
point for the Lienard system $\ddot{x}+f(x) \dot{x}+g(x) =0$ is proved. An
alogrithm on the determination of a desired number of limit cycles for this
system has been considered which might become relevant for a Lienard system
with incomplete data.

In this paper we study the orthogonality conditions satisfied by the
classical q-orthogonal polynomials that are located at the top of the q-Hahn
tableau (big q-jacobi polynomials (bqJ)) and the Nikiforov-Uvarov tableau
(Askey-Wilson polynomials (AW)) for almost any complex value of the parameters
and for all non-negative integers degrees. We state the degenerate version of
Favard's theorem, which is one of the keys of the paper, that allow us to
extend the orthogonality properties valid up to some integer degree N to
Sobolev type orthogonality properties.

In this paper we prove the "Tiling implies Spectral" part of Fuglede's paper
for the case of three intervals. Then we prove the "Spectral implies Tiling"
part of the conjecture for the case of three equal intervals as also when the
intervals have lengths 1/2, 1/4, 1/4. For the general case we change our
approach to get information on the structure of the spectrum for the n-interval
case. Finally, we use symbolic computations on Mathematica, and prove this part
of the conjecture with an additional assumption on the spectrum.

In the paper, we first survey some results on inequalities for bounding
harmonic numbers or Euler-Mascheroni constant, and then we establish a new
sharp double inequality for bounding harmonic numbers as follows: For
$n\in\mathbb{N}$, the double inequality
-\frac{1}{12n^2+{2(7-12\gamma)}/{(2\gamma-1)}}\le H(n)-\ln
n-\frac1{2n}-\gamma<-\frac{1}{12n^2+6/5} is valid, with equality in the
left-hand side only when $n=1$, where the scalars
$\frac{2(7-12\gamma)}{2\gamma-1}$ and $\frac65$ are the best possible.

A class of ultrametric Cantor sets $(C, d_{u})$ introduced recently in
literature (Raut, S and Datta, D P (2009), Fractals, 17, 45-52) is shown to
enjoy some novel properties. The ultrametric $d_{u}$ is defined using the
concept of {\em relative infinitesimals} and an {\em inversion} rule. The
associated (infinitesimal) valuation which turns out to be both scale and
reparametrisation invariant, is identified with the Cantor function associated
with a Cantor set $\tilde C$ where the relative infinitesimals are supposed to
live in.

We prove some sufficient conditions implying $l^p$ inequalities of the form
$||x||_p \leq ||y||_p$ for vectors $ x, y \in [0,\infty)^n$ and for $p$ in
certain positive real intervals. Our sufficient conditions are strictly weaker
than the usual majorization relation. The conditions are expressed in terms of
certain homogeneous symmetric polynomials in the entries of the vectors. These
polynomials include the elementary symmetric polynomials as a special case. We
also give a characterization of the majorization relation by means of symmetric
polynomials.

The Cauchy functional equation f(x+y)=f(x)+f(y) has been investigated by many
authors, under various "regularity" conditions. We present a new method for
solving this equation assuming only local integrability of the unknown
function, or, more generally, that a complex exponent of the function is
locally measurable. The (rather simple) proof can be generalized, e.g., to
higher dimensions. A key idea is to consider the equation as an initial value
problem. As a by-product of this approach we arrive to a class of wild
functions.

We construct a continuous Lagrangian, strictly convex and superlinear in the
third variable, such that the associated variational problem has a Lipschitz
minimizer which is non-differentiable on a dense set. More precisely, the upper
and lower Dini derivatives of the minimizer differ by a constant on a dense
(hence second category) set. In particular, we show that mere continuity is an
insufficient smoothness assumption for Tonelli's partial regularity theorem.

We give an alternative definition of integral at the generality of the Perron
integral and propose an exposition of the foundations of integral theory
starting from this new definition. Both definition and proofs needed for the
development are unexpectedly simple. We show how to adapt the definition to
cover the multidimensional and Stieltjes case and prove that our integral is
equivalent to the Henstock-Kurzweil(-Stieltjes) integral.

Let T be Takagi's continuous but nowhere-differentiable function. Using a
representation in terms of Rademacher series due to N. Kono, we give a complete
characterization of those points where T has a left-sided, right-sided, or
two-sided infinite derivative. This characterization is illustrated by several
examples. A consequence of the main result is that the sets of points where
T'(x) is infinite have Hausdorff dimension one. As a byproduct of the method of
proof, some exact results concerning the modulus of continuity of T are also
obtained.

In this paper we give some necessary and sufficient characterizations for
weak exponential instability of evolution operators. Variants for the classical
results due to Datko and Lyapunov are obtained.

Burkholder obtained a sharp estimate of $\E|W|^p$ via $\E|Z|^p$, where $W$ is
a martingale transform of $Z$, or, in other words, for martingales $W$
differentially subordinated to martingales $Z$. His result is that $\E|W|^p\le
(p^*-1)^p\E|Z|^p$, where $p^* =\max (p, \frac{p}{p-1})$. What happens if the
martingales have an extra property of being orthogonal martingales? This
property is an analog (for martingales) of the Cauchy-Riemann equation for
functions, and it naturally appears from a problem on singular integrals (see
the references at the end of Section~1).

We consider the problem of finding a best uniform approximation to the
standard monomial on the unit ball in $\bbC^2$ by polynomials of lower degree
with complex coefficients. We reduce the problem to a one-dimensional weighted
minimization problem on an interval. In a sense, the corresponding extremal
polynomials are uniform counterparts of the classical orthogonal Jacobi
polynomials. They can be represented by means of special conformal mappings on
the so-called comb-like domains.

A multivariable version of the strong maximal function is introduced and a
sharp distributional estimate for this operator in the spirit of the Jessen,
Marcinkiewicz, and Zygmund theorem is obtained. Conditions that characterize
the boundedness of this multivariable operator on products of weighted Lebesgue
spaces equipped with multiple weights are obtained. Results for other
multi(sub)linear maximal functions associated with bases of open sets are
studied too.

We prove that all the meromorphic solutions of the nonlinear differential
equation c0 u"' + 6 u^4 + c1 u" + c2 u u' + c3 u^3 + c4 u'+ c5 u^2 + c6 u +c7=0
are elliptic or degenerate elliptic, and we build them explicitly.

The aim of this paper is to emphasize various concepts of dichotomies for
evolution equations in Banach spaces, due to the important role they play in
the approach of stable, instable and central manifolds. The asymptotic
properties of the solutions of the evolution equations are studied by means of
the asymptotic behaviors for skew-evolution semiflows.

We unify the known three distinct inequality sequences [Abramowitz 9.5.2] of
real zeros of Bessel functions into a single, generalized one. This result is
triggered by a uniqueness proof concerning a particular inverse scattering
problem.

We prove well-posedness of global solutions for a class of coagulation
equations which exhibit the gelation phase transition. Considering the
generating functions, we solve an associated partial differential equation
before and after the phase transition. Applications include the classical
Smoluchowski and Flory equations with multiplicative coagulation rate and the
recently introduced symmetric model with limited aggregations.

Using some resolution of singularities and oscillatory integral methods in
conjunction with appropriate damping and interpolation techniques, L^p
boundedness theorems for p > 2 are obtained for maximal operators over a wide
range of hypersurfaces. These estimates are sharp in many situations, including
the convex hypersurfaces of finite line type considered by Iosevich, Sawyer,
and others.

A nonlinear inequality is formulated in the paper. An estimate of the rate of
growth/decay of solutions to this inequality is obtained. This inequality is of
interest in a study of dynamical systems and nonlinear evolution equations. It
can be applied to a study of global existence of solutions to nonlinear PDE.

We complement a recent result of S. Furuichi, by showing that the differences
$\sum_{i=1}^n \alpha_i x_i - \prod_{i=1}^n x_i^{\alpha_i}$ associated to
distinct sequences of weights are comparable, with constants that depend on the
smallest and largest weights.

Given a positive definite kernel in a locally compact space, we study a
minimal energy problem in the presence of an external field over the class of
all nonnegative Radon measures that are supported by a given closed noncompact
set, satisfy certain normalizing assumptions, and do not exceed a fixed measure
serving as a constraint. Under general assumptions, we establish the existence
of a minimizing measure and analyze its continuity properties in the weak* and
strong topologies when the set and the constraint are both varied.

In this study, we are concerned with spectral problems of second-order vector
dynamic equations with two-point boundary value conditions and mixed
derivatives, where the matrix-valued coefficient of the leading term may be
singular, and the domain is non-uniform but finite. A concept of
self-adjointness of the boundary conditions is introduced. The self-adjointness
of the corresponding dynamic operator is discussed on a suitable admissible
function space, and fundamental spectral results are obtained. The dual
orthogonality of eigenfunctions is shown in a special case.

The two weight inequality for the Hilbert transform is characterized in terms
of (1) a Poisson A_2 condition on the weights (2) A forward testing condition,
in which the two weight inequality is tested on intervals (3) and a backwards
testing condition, dual to (2). A critical new concept in the proof is an
Energy Condition, which incorporates information about the distribution of the
weights in question inside intervals. This condition is a consequence of the
three conditions above.

We study non-self-adjoint Hamiltonian systems on Sturmian time scales,
defining Weyl-Sims sets, which replace the classical Weyl circles, and a
matrix-valued $M-$function on suitable cone-shaped domains in the complex
plane. Furthermore, we characterize realizations of the corresponding dynamic
operator and its adjoint, and construct their resolvents. Even-order scalar
equations and the Orr-Sommerfeld equation on time scales are given as examples
illustrating the theory, which are new even for difference equations.

We look at the time-frequency localisation of generators of lattice Gabor
systems. For a generator of a Riesz basis, this localisation is described by
the classical Balian-Low theorem.

We investigate g-functions based on semigroups related to multi-dimensional
Laguerre function expansions of convolution type. We prove that these operators
can be viewed as Calderon-Zygmund operators in the sense of the underlying
space of homogeneous type, hence their mapping properties follow from the
general theory.

We investigate the dependence on parameters for the discrete boundary value
problem connected with the Emden-Fowler equation. A variational method is used
in order to obtain a general scheme allowing for investigation the dependence
on paramaters of discrete boundary value problems.

We consider a class of linear ODEs of second order with variable coefficients
and construct its Lie algebra of Lie group of equivalence transformations.
Further we find invariants and differential invariants of this Lie algebra and
by using them we formulate criteria of equivalence of the equations under
consideration. These criteria enable us to characterize some classes integrable
in quadratures.

This paper describes a search algorithm to locate values of t where the real
part of the Riemann zeta function, zeta(sigma+it), is negative for sigma>1. The
run-time to execute the search is much less than a brute-force approach and
relies on certain symmetries of congruence equations related to the zeta
function. Initial results show the smallest value of t where this begins to
occur is much nearer to the real axis than conservative estimates would
suggest.

We investigate connections between radial Fourier multipliers on $R^d$ and
certain conical Fourier multipliers on $R^{d+1}$. As an application we obtain a
new weak type endpoint bound for the Bochner-Riesz multipliers associated to
the light cone in $R^{d+1}$, where $d\ge 4$, and results on characterizations
of $L^p\to L^{p,\nu}$ inequalities for convolutions with radial kernels.

We give a computationally efficient method for constructing the linear
differential operator with polynomial coefficients whose space of holomorphic
solutions is spanned by all the branches of a function defined by a generic
algebraic curve. The proposed method does not require solving the algebraic
equation and can be applied in the case when its Galois group is not solvable.

We study the possible analogous of the Div-Curl Lemma in classical harmonic
analysis and partial differential equations, but from the point of view of the
multi-parameter setting. In this context we see two possible Div-Curl lemmas
that arise. Extensions to differential forms are also given.

A new modulus of smoothness based on the Euler angles is introduced on the
unit sphere and is shown to satisfy all the usual characteristic properties of
moduli of smoothness, including direct and inverse theorem for the best
approximation by polynomials and its equivalence to a $K$-functional, defined
via partial derivatives in Euler angles. The set of results on the moduli on
the sphere serves as a basis for defining new moduli of smoothness and their
corresponding $K$-functionals on the unit ball, which are used to characterize
the best approximation by polynomials on the ball.

It is proved in this paper that there is a fine correlation between the
values of $|\zeta(1/2+i\vp_2(t))|^4$ and $|\zeta(1/2+it)|^4$ where $\vp_2(t)$
stands for the Jacob's ladder of the second order. This new asymptotic formula
cannot be obtained in known theories of Balasubramanian, Heath-Brown and Ivic.

We prove optimal bounds in L^2(R^2) for the maximal oper- ator obtained by
taking a singular integral along N arbitrary directions in the plane. We also
give a new proof for the optimal L^2 bound for the single scale Kakeya maximal
function.

The existence of higher derivative discontinuous solutions to a first order
ordinary differential equation is shown to reveal a nonlinear SL(2,R) structure
of analysis in the sense that a real variable $t$ can now accomplish changes
not only by linear translations $t \to t + h$ but also by inversions $t \to
1/t$. We show that the real number set has the structure of a positive Lebesgue
measure Cantor set. We also present an extension of the Picard's theorem in
this new light.

We elegantly refine a double inequality for the gamma function and improve
some known results for bounding the gamma function.

We discuss the existence of solutions with oblique asymptotes to a class of
second order nonlinear ordinary differential equations by means of Lyapunov
functions. The approach is new in this field and allows for simpler proofs of
general results regarding Emden-Fowler like equations.