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3  Automorphismes naturels de l'espace de Douady de points sur une surface Boissière, Samuel
On établit quelques résultats généraux relatifs à la taille
du groupe d'automorphismes de l'espace de Douady de points sur une
surface, puis on étudie quelques propriétés des automorphismes
provenant d'un automorphisme de la surface, en particulier leur action
sur la cohomologie et la classification de leurs points fixes.


24  Lower Order Terms of the Discrete Minimal Riesz Energy on Smooth Closed Curves Borodachov, S. V.
We consider the problem of minimizing the energy of $N$ points
repelling each other on curves in $\mathbb{R}^d$ with the potential
$xy^{s}$, $s\geq 1$, where $\, \cdot\, $ is
the Euclidean norm. For a sufficiently smooth, simple, closed,
regular curve, we find the next order term in the asymptotics of the
minimal $s$energy. On our way, we also prove that at
least for $s\geq 2$, the minimal pairwise distance in optimal configurations
asymptotically equals $L/N$, $N\to\infty$, where $L$ is the length
of the curve.


44  Surfaces of Rotation with Constant Mean Curvature in the Direction of a Unitary Normal Vector Field in a Randers Space Carvalho, T. M. M.; Moreira, H. N.; Tenenblat, K.
We consider the Randers space $(V^n,F_b)$ obtained by perturbing the Euclidean metric by a translation, $F_b=\alpha+\beta$, where $\alpha$ is the Euclidean metric and $\beta$ is a $1$form with norm $b$, $0\leq b\lt 1$. We introduce the concept of a hypersurface with constant mean curvature in the direction of a unitary normal vector field. We obtain the ordinary differential equation that characterizes the rotational surfaces $(V^3,F_b)$ of constant mean curvature (cmc) in the direction of a unitary normal vector field. These equations reduce to the classical equation of the rotational cmc surfaces in Euclidean space, when $b=0$. It also reduces to the equation that characterizes the minimal rotational surfaces in $(V^3,F_b)$ when $H=0$, obtained by M. Souza and K. Tenenblat. Although the differential equation depends on the choice of the normal direction, we show that both equations determine the same rotational surface, up to a reflection. We also show that the round cylinders are cmc surfaces in the direction of the unitary normal field. They are generated by the constant solution of the differential equation. By considering the equation as a nonlinear dynamical system, we provide a qualitative analysis, for $0\lt b\lt \frac{\sqrt{3}}{3}$. Using the concept of stability and considering the linearization around the single equilibrium point (the constant solution), we verify that the solutions are locally asymptotically stable spirals. This is proved by constructing a Lyapunov function for the dynamical system and by determining the basin of stability of the equilibrium point. The surfaces of rotation generated by such solutions tend asymptotically to one end of the cylinder.


81  Pseudoprime Reductions of Elliptic Curves David, C.; Wu, J.
Let $E$ be an elliptic curve over $\mathbb Q$ without complex multiplication,
and for each prime
$p$ of good reduction, let $n_E(p) =  E(\mathbb F_p) $. For any integer
$b$, we consider elliptic pseudoprimes to the base
$b$. More precisely, let $Q_{E,b}(x)$ be the number of primes $p \leq
x$ such that $b^{n_E(p)} \equiv b\,({\rm mod}\,n_E(p))$, and let $\pi_{E,
b}^{\operatorname{pseu}}(x)$ be the number of compositive $n_E(p)$ such
that $b^{n_E(p)} \equiv b\,({\rm mod}\,n_E(p))$ (also called
elliptic curve pseudoprimes). Motivated by cryptography applications,
we address the problem of finding upper bounds for
$Q_{E,b}(x)$ and $\pi_{E, b}^{\operatorname{pseu}}(x)$,
generalising some of the literature for the classical pseudoprimes
to this new setting.


102  Quandle Cocycle Invariants for Spatial Graphs and Knotted Handlebodies Ishii, Atsushi; Iwakiri, Masahide
We introduce a flow of a spatial graph and see how invariants for
spatial graphs and handlebodylinks are derived from those for flowed
spatial graphs.
We define a new quandle (co)homology by introducing a subcomplex of the
rack chain complex.
Then we define quandle colorings and quandle cocycle invariants for
spatial graphs and handlebodylinks.


123  Gosset Polytopes in Picard Groups of del Pezzo Surfaces Lee, JaeHyouk
In this article, we study the correspondence between the geometry of
del Pezzo surfaces $S_{r}$ and the geometry of the $r$dimensional Gosset
polytopes $(r4)_{21}$. We construct Gosset polytopes $(r4)_{21}$ in
$\operatorname{Pic} S_{r}\otimes\mathbb{Q}$ whose vertices are lines, and we identify
divisor classes in $\operatorname{Pic} S_{r}$ corresponding to $(a1)$simplexes ($a\leq
r$), $(r1)$simplexes and $(r1)$crosspolytopes of the polytope $(r4)_{21}$.
Then we explain how these classes correspond to skew $a$lines($a\leq r$),
exceptional systems, and rulings, respectively.


151  Moments of the Rank of Elliptic Curves Miller, Steven J.; Wong, Siman
Fix an elliptic curve $E/\mathbb{Q}$ and assume the Riemann Hypothesis
for the $L$function $L(E_D, s)$ for every quadratic twist $E_D$ of
$E$ by $D\in\mathbb{Z}$. We combine Weil's
explicit formula with techniques of HeathBrown to derive an asymptotic
upper bound for the weighted moments of the analytic rank of $E_D$. We
derive from this an upper bound for the density of lowlying zeros of
$L(E_D, s)$ that is compatible with the random matrix models of Katz and
Sarnak. We also show that for any unbounded increasing function $f$ on $\mathbb{R}$,
the analytic rank and (assuming in addition the Birch and SwinnertonDyer
conjecture)
the number of integral points of $E_D$ are less than $f(D)$
for almost all $D$.


183  Negative Powers of Laguerre Operators Nowak, Adam; Stempak, Krzysztof
We study negative powers of Laguerre differential operators in $\mathbb{R}^d$, $d\ge1$.
For these operators we prove twoweight $L^pL^q$ estimates with ranges of $q$ depending
on $p$. The case of the harmonic oscillator (Hermite operator) has recently
been treated by Bongioanni and Torrea by using a straightforward
approach of kernel estimates. Here these results are applied in certain Laguerre settings.
The procedure is fairly direct for Laguerre function expansions of
Hermite type,
due to some monotonicity properties of the kernels involved.
The case of Laguerre function expansions of convolution type is less straightforward.
For halfinteger type indices $\alpha$ we transfer the desired results from the Hermite setting
and then apply an interpolation argument based on a device we call the
<it>
convexity principle
</it>
to cover the continuous range of $\alpha\in[1/2,\infty)^d$. Finally, we investigate negative powers
of the Dunkl harmonic oscillator in the context of a finite reflection group acting on $\mathbb{R}^d$ and
isomorphic to $\mathbb Z^d_2$. The two weight $L^pL^q$ estimates we obtain in this setting are essentially
consequences of those for Laguerre function expansions of convolution type.


217  $W_\omega^{2,p}$Solvability of the CauchyDirichlet Problem for Nondivergence Parabolic Equations with BMO Coefficients Tang, Lin
In this paper, we establish
the regularity of strong solutions to
nondivergence parabolic equations with BMO coefficients in nondoubling weighted spaces.


241  Triangles of BaumslagSolitar Groups Allcock, Daniel
Our main result is that many triangles of BaumslagSolitar groups
collapse to finite groups, generalizing a famous example of Hirsch and
other examples due to several authors. A triangle of BaumslagSolitar
groups means a group with three generators, cyclically ordered, with
each generator conjugating some power of the previous one to another
power. There are six parameters, occurring in pairs, and we show that
the triangle fails to be developable whenever one of the parameters
divides its partner, except for a few special cases. Furthermore,
under fairly general conditions, the group turns out to be finite and
solvable of derived length $\leq3$. We obtain a lot of information about
finite quotients, even when we cannot determine developability.


254  Corrigendum to ``On $\mathbb{Z}$modules of Algebraic Integers'' Bell, Jason P.; Hare, Kevin G.
We fix a mistake in the proof of Theorem 1.6 in the paper in the title.


257  Compactness of Commutators for Singular Integrals on Morrey Spaces Chen, Yanping; Ding, Yong; Wang, Xinxia
In this paper we characterize the
compactness of the commutator $[b,T]$ for the singular integral
operator on the Morrey spaces $L^{p,\lambda}(\mathbb R^n)$. More
precisely, we prove that if
$b\in \operatorname{VMO}(\mathbb R^n)$, the $\operatorname {BMO}
(\mathbb R^n)$closure of $C_c^\infty(\mathbb R^n)$,
then $[b,T]$ is a compact operator on the
Morrey spaces $L^{p,\lambda}(\mathbb R^n)$ for $1\lt p\lt \infty$ and
$0\lt \lambda\lt n$. Conversely, if $b\in \operatorname{BMO}(\mathbb R^n)$ and
$[b,T]$ is a compact operator on the $L^{p,\,\lambda}(\mathbb R^n)$
for some $p\ (1\lt p\lt \infty)$, then $b\in \operatorname {VMO}(\mathbb R^n)$.
Moreover, the boundedness of a rough singular integral operator $T$
and its commutator $[b,T]$ on $L^{p,\,\lambda}(\mathbb R^n)$ are also
given. We obtain a sufficient condition for a
subset in Morrey space to be a strongly precompact set,
which has interest in its own right.


282  Level Lowering Modulo Prime Powers and Twisted Fermat Equations Dahmen, Sander R.; Yazdani, Soroosh
We discuss a clean level lowering theorem modulo prime powers
for weight $2$ cusp forms.
Furthermore, we illustrate how this can be used to completely
solve certain twisted Fermat equations
$ax^n+by^n+cz^n=0$.


301  Hermite's Constant for Function Fields Hurlburt, Chris; Thunder, Jeffrey Lin
We formulate an analog of Hermite's constant for function fields over a finite field and
state a conjectural value for this analog. We prove our conjecture in many cases, and
prove slightly weaker results in all other cases.


318  Cubic Polynomials with Periodic Cycles of a Specified Multiplier Ingram, Patrick
We consider cubic polynomials $f(z)=z^3+az+b$ defined over
$\mathbb{C}(\lambda)$, with a marked point of period $N$ and multiplier
$\lambda$. In the case $N=1$, there are infinitely many such objects,
and in the case $N\geq 3$, only finitely many (subject to a mild
assumption). The case $N=2$ has particularly rich structure, and we
are able to describe all such cubic polynomials defined over the field
$\bigcup_{n\geq 1}\mathbb{C}(\lambda^{1/n})$.


345  Salem Numbers and Pisot Numbers via Interlacing McKee, James; Smyth, Chris
We present a general construction of Salem numbers via rational
functions whose zeros and poles mostly lie on the unit circle and
satisfy an interlacing condition. This extends and unifies earlier
work. We then consider the ``obvious'' limit points of the set of Salem
numbers produced by our theorems and show that these are all Pisot
numbers, in support of a conjecture of Boyd. We then show that all
Pisot numbers arise in this way. Combining this with a theorem of
Boyd, we produce all Salem numbers via an interlacing construction.


368  C$^*$Algebras over Topological Spaces: Filtrated KTheory Meyer, Ralf; Nest, Ryszard
We define the filtrated Ktheory of a $\mathrm{C}^*$algebra over a finite topological space \(X\)
and explain how to construct a spectral sequence that computes the bivariant Kasparov theory over \(X\)
in terms of filtrated Ktheory.


409  Lifting Quasianalytic Mappings over Invariants Rainer, Armin
Let $\rho \colon G \to \operatorname{GL}(V)$ be a rational finite dimensional complex representation of a reductive linear
algebraic group $G$, and let $\sigma_1,\dots,\sigma_n$ be a system of generators of the algebra of
invariant polynomials $\mathbb C[V]^G$.
We study the problem of lifting mappings $f\colon \mathbb R^q \supseteq U \to \sigma(V) \subseteq \mathbb C^n$
over the mapping of invariants
$\sigma=(\sigma_1,\dots,\sigma_n) \colon V \to \sigma(V)$. Note that $\sigma(V)$ can be identified with the categorical quotient $V /\!\!/ G$
and its points correspond bijectively to the closed orbits in $V$. We prove that if $f$ belongs to a quasianalytic subclass
$\mathcal C \subseteq C^\infty$ satisfying some mild closedness properties that guarantee resolution of singularities in
$\mathcal C$,
e.g., the real analytic class, then $f$ admits a lift of the
same class $\mathcal C$ after desingularization by local blowups and local power substitutions.
As a consequence we show that $f$ itself allows for a lift
that belongs to $\operatorname{SBV}_{\operatorname{loc}}$, i.e., special functions of bounded variation.
If $\rho$ is a real representation of a compact Lie group, we obtain stronger versions.


429  Holomorphic Mappings between Domains in $\mathbb C^2$ Shafikov, Rasul; Verma, Kaushal
An extension theorem for holomorphic mappings between two domains in
$\mathbb C^2$ is proved under purely local hypotheses.


455  On Cardinal Invariants and Generators for von Neumann Algebras Sherman, David
We demonstrate how most common cardinal invariants associated with a von
Neumann algebra $\mathcal M$ can be computed from the decomposability number,
$\operatorname{dens}(\mathcal M)$, and the minimal cardinality of a generating
set, $\operatorname{gen}(\mathcal M)$.
Applications include the equivalence of the wellknown generator
problem, ``Is every separablyacting von Neumann algebra
singlygenerated?", with the formally stronger questions, ``Is every
countablygenerated von Neumann algebra singlygenerated?" and ``Is
the $\operatorname{gen}$ invariant monotone?" Modulo the generator problem, we
determine the range of the invariant $\bigl( \operatorname{gen}(\mathcal M),
\operatorname{dens}(\mathcal M) \bigr)$,
which is mostly governed by the inequality $\operatorname{dens}(\mathcal M) \leq
\mathfrak C^{\operatorname{gen}(\mathcal M)}$.


481  Some Functional Inequalities on Polynomial Volume Growth Lie Groups Chamorro, Diego
In this article we study some Sobolevtype inequalities on polynomial volume growth Lie groups.
We show in particular that improved Sobolev inequalities can be extended to this general framework
without the use of the LittlewoodPaley decomposition.


497  Le lemme fondamental pondéré pour le groupe métaplectique Li, WenWei
Dans cet article, on énonce une variante du lemme fondamental
pondéré d'Arthur pour le groupe métaplectique de Weil, qui sera un
ingrédient indispensable de la stabilisation de la formule des
traces. Pour un corps de caractéristique résiduelle suffisamment
grande, on en donne une démonstration à l'aide de la méthode de
descente, qui est conditionnelle: on admet le lemme fondamental
pondéré non standard sur les algèbres de Lie. Vu les travaux de
Chaudouard et Laumon, on s'attend à ce que cette condition soit
ultérieurement vérifiée.


544  On the Simple Inductive Limits of Splitting Interval Algebras with Dimension Drops Li, Zhiqiang
A Ktheoretic classification is given of the simple inductive limits
of finite direct sums of the
type I $C^*$algebras known as splitting interval algebras with
dimension drops. (These are the subhomogeneous $C^*$algebras, each
having spectrum a finite union
of points and an open interval, and torsion $\textrm{K}_1$group.)


573  Fundamental Group of Simple $C^*$algebras with Unique Trace III Nawata, Norio
We introduce the fundamental group ${\mathcal F}(A)$ of
a simple $\sigma$unital $C^*$algebra $A$ with unique (up to scalar multiple)
densely defined lower semicontinuous trace.
This is a generalization of ``Fundamental Group of Simple
$C^*$algebras with Unique Trace I and II'' by Nawata and Watatani.
Our definition in this paper makes sense for stably projectionless $C^*$algebras.
We show that there exist separable stably projectionless $C^*$algebras such that
their fundamental groups are equal to $\mathbb{R}_+^\times$
by using the classification theorem of Razak and Tsang.
This is a contrast to the unital case in Nawata and Watatani.
This study is motivated by the work of Kishimoto and Kumjian.


588  Level Raising and Anticyclotomic Selmer Groups for Hilbert Modular Forms of Weight Two Nekovář, Jan
In this article we refine the method of Bertolini and Darmon
and prove several finiteness results for
anticyclotomic Selmer groups of Hilbert modular forms of parallel
weight two.


669  The Genuine Omegaregular Unitary Dual of the Metaplectic Group Pantano, Alessandra; Paul, Annegret; SalamancaRiba, Susana A.
We classify all genuine unitary representations of the metaplectic group whose
infinitesimal character is real and at least as regular as that of the
oscillator representation. In a previous paper we exhibited a certain family
of representations satisfying these conditions, obtained by cohomological
induction from the tensor product of a onedimensional representation and an
oscillator representation. Our main theorem asserts that this family exhausts
the genuine omegaregular unitary dual of the metaplectic group.


705  Pure Infiniteness of the Crossed Product of an AHAlgebra by an Endomorphism Thomsen, Klaus
It is shown that simplicity of the crossed product of
a unital AHalgebra with slow dimension growth by an endomorphism
implies that the algebra is also purely infinite, provided only that
the endomorphism leaves no trace state invariant and takes the unit
to a full projection.


721  Analysis of the BrylinskiKostant Model for Spherical Minimal Representations Achab, Dehbia; Faraut, Jacques
We revisit with another view point the construction by R. Brylinski
and B. Kostant of minimal representations of simple Lie groups. We
start from a pair $(V,Q)$, where $V$ is a complex vector space and $Q$
a homogeneous polynomial of degree 4 on $V$.
The manifold $\Xi $ is an orbit of a covering of ${\rm Conf}(V,Q)$,
the conformal group of the pair $(V,Q)$, in a finite dimensional
representation space.
By a generalized KantorKoecherTits construction we obtain a complex
simple Lie algebra $\mathfrak g$, and furthermore a real
form ${\mathfrak g}_{\mathbb R}$. The connected and simply connected Lie
group $G_{\mathbb R}$ with ${\rm Lie}(G_{\mathbb R})={\mathfrak
g}_{\mathbb R}$ acts unitarily on a Hilbert space of holomorphic
functions defined on the manifold $\Xi $.


755  Homotopy Classification of Projections in the Corona Algebra of a Nonsimple $C^*$algebra Brown, Lawrence G.; Lee, Hyun Ho
We study projections in the corona algebra of $C(X)\otimes K$, where K
is the $C^*$algebra of compact operators on a separable infinite
dimensional Hilbert space and $X=[0,1],[0,\infty),(\infty,\infty)$,
or $[0,1]/\{ 0,1 \}$. Using BDF's essential codimension, we determine
conditions for a projection in the corona algebra to be liftable to a
projection in the multiplier algebra. We also determine the
conditions for two projections to be equal in $K_0$, Murrayvon
Neumann equivalent, unitarily equivalent, or homotopic. In light of
these characterizations, we construct examples showing that the
equivalence notions above are all distinct.


778  Ricci Solitons and Geometry of Fourdimensional Nonreductive Homogeneous Spaces Calvaruso, Giovanni; Fino, Anna
We study the geometry of nonreductive $4$dimensional homogeneous
spaces. In particular, after describing their LeviCivita connection
and curvature properties, we classify homogeneous Ricci solitons on
these spaces, proving the existence of shrinking, expanding and steady
examples. For all the nontrivial examples we find, the Ricci operator
is diagonalizable.


805  Quantum Random Walks and Minors of Hermitian Brownian Motion Chapon, François; Defosseux, Manon
Considering quantum random walks, we construct discretetime
approximations of the eigenvalues processes of minors of Hermitian
Brownian motion. It has been recently proved by Adler, Nordenstam, and
van Moerbeke that the process of eigenvalues of
two consecutive minors of a Hermitian Brownian motion is a Markov
process; whereas, if one considers more than two consecutive minors,
the Markov property fails. We show that there are analog results in
the noncommutative counterpart and establish the Markov property of
eigenvalues of some particular submatrices of Hermitian Brownian
motion.


822  A Compositional Shuffle Conjecture Specifying Touch Points of the Dyck Path Haglund, J.; Morse, J.; Zabrocki, M.
We introduce a $q,t$enumeration of Dyck paths that are forced to touch the main diagonal
at specific points and forbidden to touch elsewhere
and conjecture that it describes the action of
the Macdonald theory $\nabla$ operator applied to a HallLittlewood
polynomial. Our conjecture refines several earlier conjectures concerning
the space of diagonal harmonics including the ``shuffle conjecture"
(Duke J. Math. $\mathbf {126}$ ($2005$), 195232) for $\nabla e_n[X]$.
We bring to light that certain generalized HallLittlewood polynomials
indexed by compositions are the building blocks for the algebraic
combinatorial theory of $q,t$Catalan sequences, and we prove a number of
identities involving these functions.


845  Monodromy Filtrations and the Topology of Tropical Varieties Helm, David; Katz, Eric
We study the topology of tropical varieties that arise from a certain
natural class of varieties. We use the theory of tropical
degenerations to construct a natural, ``multiplicityfree''
parameterization of $\operatorname{Trop}(X)$ by a topological space
$\Gamma_X$ and give a geometric interpretation of the cohomology of
$\Gamma_X$ in terms of the action of a monodromy operator on the
cohomology of $X$. This gives bounds on the Betti numbers of
$\Gamma_X$ in terms of the Betti numbers of $X$ which constrain the
topology of $\operatorname{Trop}(X)$. We also obtain a description of
the top power of the monodromy operator acting on middle cohomology of
$X$ in terms of the volume pairing on $\Gamma_X$.


869  Balayage of SemiDirichlet Forms Hu, ZeChun; Sun, Wei
In this paper we study the balayage of semiDirichlet forms. We
present new results on balayaged functions and balayaged measures
of semiDirichlet
forms. Some of the results are new even in the Dirichlet forms setting.


892  Boundedness of CalderónZygmund Operators on Nonhomogeneous Metric Measure Spaces Hytönen, Tuomas; Liu, Suile; Yang, Dachun; Yang, Dongyong
Let $({\mathcal X}, d, \mu)$ be a
separable metric measure space satisfying the known upper
doubling condition, the geometrical doubling condition, and the
nonatomic condition that $\mu(\{x\})=0$ for all $x\in{\mathcal X}$.
In this paper, we show that the boundedness of a CalderónZygmund
operator $T$ on $L^2(\mu)$ is equivalent to that of $T$ on
$L^p(\mu)$ for some $p\in (1, \infty)$, and that of $T$ from $L^1(\mu)$
to $L^{1,\,\infty}(\mu).$ As an application, we prove that if $T$ is a
CalderónZygmund operator bounded on $L^2(\mu)$,
then its maximal operator is bounded on $L^p(\mu)$
for all $p\in (1, \infty)$ and from
the space of all complexvalued Borel measures on
${\mathcal X}$ to $L^{1,\,\infty}(\mu)$.
All these results generalize the corresponding results of Nazarov et al.
on metric spaces with
measures satisfying the socalled polynomial growth condition.


924  Rectifiability of Optimal Transportation Plans McCann, Robert J.; Pass, Brendan; Warren, Micah
The regularity of solutions to optimal transportation problems has become
a hot topic in current research. It is well known by now that the optimal measure
may not be concentrated on the graph of a continuous mapping unless both the transportation
cost and the masses transported satisfy very restrictive hypotheses (including sign conditions
on the mixed fourthorder derivatives of the cost function).
The purpose of this note is to show that in spite of this,
the optimal measure is supported on a Lipschitz manifold, provided only
that the cost is $C^{2}$ with nonsingular mixed second derivative.
We use this result to provide a simple proof that solutions to Monge's
optimal transportation problem satisfy a change of variables equation
almost everywhere.


935  The H and K Families of Mock Theta Functions McIntosh, Richard J.
In his last letter to Hardy, Ramanujan
defined 17 functions $F(q)$, $q\lt 1$, which he called mock $\theta$functions.
He observed that as $q$ radially approaches any root of unity $\zeta$ at which
$F(q)$ has an exponential singularity, there is a $\theta$function
$T_\zeta(q)$ with $F(q)T_\zeta(q)=O(1)$. Since then, other functions have
been found that possess this property. These functions are related to
a function $H(x,q)$, where $x$ is usually $q^r$ or $e^{2\pi i r}$ for some
rational number $r$. For this reason we refer to $H$ as a ``universal'' mock
$\theta$function. Modular transformations of $H$ give rise to the functions
$K$, $K_1$, $K_2$. The functions $K$ and $K_1$ appear in Ramanujan's lost
notebook. We prove various linear relations between these functions using
AppellLerch sums (also called generalized Lambert series). Some relations
(mock theta ``conjectures'') involving mock $\theta$functions
of even order and $H$ are listed.


961  Densities of Short Uniform Random Walks Borwein, Jonathan M.; Straub, Armin; Wan, James; Zudilin, Wadim
We study the densities of uniform random walks in the plane. A special focus
is on the case of short walks with three or four steps and less completely
those with five steps. As one of the main results, we obtain a hypergeometric
representation of the density for four steps, which complements the classical
elliptic representation in the case of three steps. It appears unrealistic
to expect similar results for more than five steps. New results are also
presented concerning the moments of uniform random walks and, in particular,
their derivatives. Relations with Mahler measures are discussed.


991  Poisson Brackets with Prescribed Casimirs Damianou, Pantelis A.; Petalidou, Fani
We consider the problem of constructing Poisson brackets on smooth
manifolds $M$ with prescribed Casimir functions. If $M$ is of even
dimension, we achieve our construction by considering a suitable
almost symplectic structure on $M$, while, in the case where $M$ is
of odd dimension, our objective is achieved by using a convenient
almost cosymplectic structure. Several examples and applications are
presented.


1019  On a Theorem of Bombieri, Friedlander, and Iwaniec Fiorilli, Daniel
In this article, we show to which extent one can improve a theorem of Bombieri, Friedlander and Iwaniec by using Hooley's variant of the divisor switching technique. We also give an application of the theorem in question, which is a BombieriVinogradov type theorem for the Tichmarsh divisor problem in arithmetic progressions.


1036  Harmonic Analysis Related to Homogeneous Varieties in Three Dimensional Vector Spaces over Finite Fields Koh, Doowon; Shen, ChunYen
In this paper we study the extension problem, the
averaging problem, and the generalized ErdősFalconer distance
problem associated with arbitrary homogeneous varieties in three
dimensional vector spaces over finite fields. In the case when the
varieties do not contain any plane passing through the origin, we
obtain the best possible results on the aforementioned three problems. In
particular, our result on the extension problem modestly generalizes
the result by Mockenhaupt and Tao who studied the particular conical
extension problem. In addition, investigating the Fourier decay on
homogeneous varieties enables us to give complete mapping properties
of averaging operators. Moreover, we improve the size condition on a
set such that the cardinality of its distance set is nontrivial.


1058  Optimal Roughening of Convex Bodies Plakhov, Alexander
A body moves in a rarefied medium composed of point particles at
rest. The particles make elastic reflections when colliding with the
body surface, and do not interact with each other. We consider a
generalization of Newton's minimal resistance problem: given two
bounded convex bodies $C_1$ and $C_2$ such that $C_1 \subset C_2
\subset \mathbb{R}^3$ and $\partial C_1 \cap \partial C_2 = \emptyset$, minimize the
resistance in the class of connected bodies $B$ such that $C_1 \subset
B \subset C_2$. We prove that the infimum of resistance is zero; that
is, there exist "almost perfectly streamlined" bodies.


1075  A Stochastic Difference Equation with Stationary Noise on Groups Raja, Chandiraraj Robinson Edward
We consider the stochastic difference equation $$\eta _k = \xi _k
\phi (\eta _{k1}), \quad k \in \mathbb Z $$ on a locally compact group $G$
where $\phi$ is an automorphism of $G$, $\xi _k$ are given $G$valued
random variables and $\eta _k$ are unknown $G$valued random variables.
This equation was considered by Tsirelson and Yor on
onedimensional torus. We consider the case when $\xi _k$ have a
common law $\mu$ and prove that if $G$ is a distal group and $\phi$
is a distal automorphism of $G$ and if the equation has a solution,
then extremal solutions of the equation are in oneone
correspondence with points on the coset space $K\backslash G$ for
some compact subgroup $K$ of $G$ such that $\mu$ is supported on
$Kz= z\phi (K)$ for any $z$ in the support of $\mu$. We also provide
a necessary and sufficient condition for the existence of solutions
to the equation.


1090  Classic and Mirabolic RobinsonSchenstedKnuth Correspondence for Partial Flags Rosso, Daniele
In this paper we first generalize to the case of
partial flags a result proved both by Spaltenstein and by Steinberg
that relates the relative position of two complete flags and the
irreducible components of the flag variety in which they lie, using
the RobinsonSchenstedKnuth correspondence. Then we use this result
to generalize the mirabolic RobinsonSchenstedKnuth correspondence
defined by Travkin, to the case of two partial flags and a line.


1122  $p$adic $L$functions and the Rationality of Darmon Cycles Seveso, Marco Adamo
Darmon cycles are a higher weight analogue of StarkHeegner points. They
yield local cohomology classes in the Deligne representation associated with a
cuspidal form on $\Gamma _{0}( N) $ of even weight $k_{0}\geq 2$.
They are conjectured to be the restriction of global cohomology classes in
the BlochKato Selmer group defined over narrow ring class fields attached
to a real quadratic field. We show that suitable linear combinations of them
obtained by genus characters satisfy these conjectures. We also prove $p$adic GrossZagier type formulas, relating the derivatives of $p$adic $L$functions of the weight variable attached to imaginary (resp. real)
quadratic fields to Heegner cycles (resp. Darmon cycles). Finally we express
the second derivative of the MazurKitagawa $p$adic $L$function of the
weight variable in terms of a global cycle defined over a quadratic
extension of $\mathbb{Q}$.


1182  PFA$(S)[S]$: More Mutually Consistent Topological Consequences of $PFA$ and $V=L$ Tall, Franklin D.
Extending the work of Larson and Todorcevic,
we show there
is a model of set theory in which normal spaces are collectionwise
Hausdorff if they are either first countable or locally compact, and
yet there are no first countable $L$spaces or compact
$S$spaces. The model is one of the form PFA$(S)[S]$, where $S$
is a coherent Souslin tree.


1201  The Central Limit Theorem for Subsequences in Probabilistic Number Theory Aistleitner, Christoph; Elsholtz, Christian
Let $(n_k)_{k \geq 1}$ be an increasing sequence of positive integers, and let $f(x)$ be a real function satisfying
\begin{equation}
\tag{1}
f(x+1)=f(x), \qquad \int_0^1 f(x) ~dx=0,\qquad
\operatorname{Var_{[0,1]}}
f \lt \infty.
\end{equation}
If $\lim_{k \to \infty} \frac{n_{k+1}}{n_k} = \infty$
the distribution of
\begin{equation}
\tag{2}
\frac{\sum_{k=1}^N f(n_k x)}{\sqrt{N}}
\end{equation}
converges to a Gaussian distribution. In the case
$$
1 \lt \liminf_{k \to \infty} \frac{n_{k+1}}{n_k}, \qquad \limsup_{k \to \infty} \frac{n_{k+1}}{n_k} \lt \infty
$$
there is a complex interplay between the analytic properties of the
function $f$, the numbertheoretic properties of $(n_k)_{k \geq 1}$,
and the limit distribution of (2).


1222  Normality of Maximal Orbit Closures for Euclidean Quivers Bobiński, Grzegorz
Let $\Delta$ be an Euclidean quiver. We prove that the closures of
the maximal orbits in the varieties of representations of $\Delta$
are normal and CohenMacaulay (even complete intersections).
Moreover, we give a generalization of this result for the tame
concealedcanonical algebras.


1248  Darmon's Points and Quaternionic Shimura Varieties Gärtner, Jérôme
In this paper, we generalize a conjecture due to Darmon and Logan in
an adelic setting. We study the relation between our construction and
Kudla's works on cycles on orthogonal Shimura varieties. This relation
allows us to conjecture a GrossKohnenZagier theorem for Darmon's
points.


1289  Systems of Weakly Coupled HamiltonJacobi Equations with Implicit Obstacles Gomes, Diogo; Serra, António
In this paper we study systems of weakly coupled HamiltonJacobi equations
with implicit obstacles that arise in optimal switching problems.
Inspired by methods from the theory of viscosity solutions and
weak KAM theory, we
extend the notion of Aubry set for these
systems. This enables us
to prove a new result on existence and uniqueness of
solutions for the Dirichlet problem, answering a question
of F. Camilli, P. Loreti and N. Yamada.


1310  Uniquely $D$colourable Digraphs with Large Girth Harutyunyan, Ararat; Kayll, P. Mark; Mohar, Bojan; Rafferty, Liam
Let $C$ and $D$ be digraphs. A mapping $f\colon V(D)\to V(C)$ is a
$C$colouring if for every arc $uv$ of $D$, either $f(u)f(v)$
is an arc of $C$ or $f(u)=f(v)$, and the preimage of every
vertex of $C$ induces an acyclic subdigraph in $D$. We say
that $D$ is $C$colourable if it admits a $C$colouring and
that $D$ is uniquely $C$colourable if it is surjectively
$C$colourable and any two $C$colourings of $D$ differ by an
automorphism of $C$. We prove that if a digraph $D$ is not
$C$colourable, then there exist digraphs of arbitrarily large
girth that are $D$colourable but not
$C$colourable. Moreover, for every digraph $D$ that is
uniquely $D$colourable, there exists a uniquely
$D$colourable digraph of arbitrarily large girth. In
particular, this implies that for every rational number $r\geq
1$, there are uniquely circularly $r$colourable digraphs with
arbitrarily large girth.


1329  Composition Operators Induced by Analytic Maps to the Polydisk Izuchi, Kei Ji; Nguyen, Quang Dieu; Ohno, Shûichi
We study properties of composition operators
induced by symbols acting from the unit disk to the polydisk.
This result will be involved in the investigation
of weighted composition operators on the Hardy space on the unit disk
and moreover be concerned with composition operators acting
from the Bergman space to the Hardy space on the unit disk.


1341  Bowen Measure From Heteroclinic Points Killough, D. B.; Putnam, I. F.
We present a new construction of the entropymaximizing, invariant
probability measure on a Smale space (the Bowen measure). Our
construction is based on points that are unstably equivalent to one
given point, and stably equivalent to another: heteroclinic points.
The spirit of the construction is similar to Bowen's construction from
periodic points, though the techniques are very different. We also
prove results about the growth rate of certain sets of heteroclinic
points, and about the stable and unstable components of the Bowen
measure. The approach we take is to prove results through direct
computation for the case of a Shift of Finite type, and then use
resolving factor maps to extend the results to more general Smale
spaces.


1359  Note on Cubature Formulae and Designs Obtained from Group Orbits Nozaki, Hiroshi; Sawa, Masanori
In 1960,
Sobolev proved that for a finite reflection group $G$,
a $G$invariant cubature formula is of degree $t$ if and only if
it is exact for all $G$invariant polynomials of degree at most $t$.
In this paper,
we find some observations on invariant cubature formulas and Euclidean designs
in connection with the Sobolev theorem.
First, we give an alternative proof of
theorems by Xu (1998) on necessary and sufficient conditions
for the existence of cubature formulas with some strong symmetry.
The new proof is shorter and simpler compared to the original one by Xu, and
moreover gives a general interpretation of
the analyticallywritten conditions of Xu's theorems.
Second,
we extend a theorem by Neumaier and Seidel (1988) on
Euclidean designs to invariant Euclidean designs, and thereby
classify tight Euclidean designs obtained from
unions of the orbits of the corner vectors.
This result generalizes a theorem of Bajnok (2007) which classifies
tight Euclidean designs invariant under the Weyl group of type $B$
to other finite reflection groups.


1378  On Weakly Tight Families Raghavan, Dilip; Steprāns, Juris
Using ideas from Shelah's recent proof that a completely
separable maximal almost disjoint family exists when
$\mathfrak{c} \lt {\aleph}_{\omega}$, we construct a weakly tight family
under the hypothesis $\mathfrak{s} \leq \mathfrak{b} \lt
{\aleph}_{\omega}$.
The case when $\mathfrak{s} \lt \mathfrak{b}$
is handled in $\mathrm{ZFC}$ and does not require $\mathfrak{b} \lt {\aleph}_{\omega}$,
while an additional PCF type hypothesis, which holds when $\mathfrak{b} \lt
{\aleph}_{\omega}$ is used to treat the case $\mathfrak{s} = \mathfrak{b}$. The notion of
a weakly tight family is a natural weakening of the well studied
notion of a Cohen indestructible maximal almost disjoint family. It
was introduced by Hrušák and García
Ferreira, who applied it to the Katétov order on almost
disjoint families.


1395  Existence of Weak Solutions of Linear Subelliptic Dirichlet Problems With Rough Coefficients Rodney, Scott
This article gives an existence theory for weak solutions of second order nonelliptic linear Dirichlet problems of the form
\begin{align*}
\nabla'P(x)\nabla u +{\bf HR}u+{\bf S'G}u +Fu &= f+{\bf T'g} \text{ in }\Theta
\\
u&=\varphi\text{ on }\partial \Theta.
\end{align*}
The principal part $\xi'P(x)\xi$ of the above equation is assumed to
be comparable to a quadratic form ${\mathcal Q}(x,\xi) = \xi'Q(x)\xi$ that
may vanish for nonzero $\xi\in\mathbb{R}^n$. This is achieved using
techniques of functional analysis applied to the degenerate Sobolev
spaces $QH^1(\Theta)=W^{1,2}(\Theta,Q)$ and
$QH^1_0(\Theta)=W^{1,2}_0(\Theta,Q)$ as defined in
previous works.
Sawyer and Wheeden give a regularity theory
for a subset of the class of equations dealt with here.


1415  Global WellPosedness and Convergence Results for 3DRegularized Boussinesq System Selmi, Ridha
Analytical study to the regularization of the Boussinesq system is
performed in frequency space using Fourier theory. Existence and
uniqueness of weak solution with minimum regularity requirement are
proved. Convergence results of the unique weak solution of the
regularized Boussinesq system to a weak LerayHopf solution of the
Boussinesq system are established as the regularizing parameter
$\alpha$ vanishes. The proofs are done in the frequency space and use
energy methods, ArselàAscoli compactness theorem and a Friedrichs
like approximation scheme.

