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3  On the Asymptotic Behavior of Complete Kähler Metrics of Positive Ricci Curvature Anchouche, Boudjemâa
Let $( X,g) $ be a complete noncompact Kähler manifold, of
dimension $n\geq2,$ with positive Ricci curvature and of standard type
(see the definition below). N. Mok proved that $X$ can be
compactified, i.e., $X$ is biholomorphic to a quasiprojective
variety$.$ The aim of this paper is to prove that the $L^{2}$
holomorphic sections of the line bundle $K_{X}^{q}$ and the volume
form of the metric $g$ have no essential singularities near the
divisor at infinity. As a consequence we obtain a comparison between
the volume forms of the Kähler metric $g$ and of the FubiniStudy
metric induced on $X$. In the case of $\dim_{\mathbb{C} }X=2,$ we
establish a relation between the number of components of the divisor
$D$ and the dimension of the groups $H^{i}( \overline{X},
\Omega_{\overline{X}}^{1}( \log D) )$.


19  Solutions for Semilinear Elliptic Systems with Critical Sobolev Exponent and Hardy Potential Bouchekif, Mohammed; Nasri, Yasmina
In this paper we consider an elliptic system with an inverse square potential and
critical Sobolev exponent in a bounded domain of $\mathbb{R}^N$. By variational
methods we study the existence results.


34  Branching Rules for Ramified Principal Series Representations of $\mathrm{GL}(3)$ over a $p$adic Field Campbell, Peter S.; Nevins, Monica
We decompose the restriction of ramified principal series
representations of the $p$adic group $\mathrm{GL}(3,\mathrm{k})$ to its
maximal compact subgroup $K=\mathrm{GL}(3,R)$. Its decomposition is
dependent on the degree of ramification of the inducing characters and
can be characterized in terms of filtrations of the Iwahori subgroup
in $K$. We establish several irreducibility results and illustrate
the decomposition with some examples.


52  An Algebraic Approach to Weakly Symmetric Finsler Spaces Deng, Shaoqiang
In this paper, we introduce a new algebraic notion, weakly symmetric
Lie algebras, to give an algebraic description of an
interesting class of homogeneous RiemannFinsler spaces, weakly symmetric
Finsler spaces. Using this new definition, we are able to give a
classification of weakly symmetric Finsler spaces with dimensions $2$
and $3$. Finally, we show that all the nonRiemannian reversible weakly
symmetric Finsler spaces we find are nonBerwaldian and with vanishing
Scurvature. This means that reversible nonBerwaldian Finsler spaces
with vanishing Scurvature may exist at large. Hence the generalized
volume comparison theorems due to Z. Shen are valid for a rather large
class of Finsler spaces.


74  Projectors on the Generalized Eigenspaces for Neutral Functional Differential Equations in $L^{p}$ Spaces Ducrot, Arnaud; Liu, Zhihua; Magal, Pierre
We present the explicit formulas for the projectors on the generalized
eigenspaces associated with some eigenvalues for linear neutral functional
differential equations (NFDE) in $L^{p}$ spaces by using integrated
semigroup theory. The analysis is based on the main result
established elsewhere by the authors and results by Magal and Ruan
on nondensely defined Cauchy problem.
We formulate the NFDE as a nondensely defined Cauchy problem and obtain
some spectral properties from which we then derive explicit formulas for
the projectors on the generalized eigenspaces associated with some
eigenvalues. Such explicit formulas are important in studying bifurcations
in some semilinear problems.


94  The Langlands Correspondence on the Generic Irreducible Constituents of Principal Series Kuo, Wentang
Let $G$ be a connected semisimple split group over a $p$adic field.
We establish the explicit link between principal nilpotent
orbits and the irreducible constituents of principal series
in terms of $L$group objects.


109  Sum of Hermitian Matrices with Given Eigenvalues: Inertia, Rank, and Multiple Eigenvalues Li, ChiKwong; Poon, YiuTung
Let $A$ and $B$ be $n\times n$ complex Hermitian (or real symmetric) matrices
with eigenvalues $a_1 \ge \dots \ge a_n$ and $b_1 \ge \dots \ge b_n$.
All possible inertia values, ranks, and multiple eigenvalues
of $A + B$ are determined. Extension of the results to the sum of $k$ matrices
with $k > 2$ and connections of the results to other subjects such
as algebraic combinatorics are also discussed.


133  Some Applications of the Perturbation Determinant in Finite von Neumann Algebras Makarov, Konstantin A.; Skripka, Anna
In the finite von Neumann algebra setting, we introduce the concept
of a perturbation determinant associated with a pair of selfadjoint
elements $H_0$ and $H$ in the algebra and relate it to the concept of
the de la HarpeSkandalis homotopy invariant determinant associated
with piecewise $C^1$paths of operators joining $H_0$ and $H$. We
obtain an analog of Krein's formula that relates the perturbation
determinant and the spectral shift function and, based on this
relation, we derive subsequently (i) the BirmanSolomyak formula for
a general nonlinear perturbation, (ii) a universality of a spectral
averaging, and (iii) a generalization of the
DixmierFugledeKadison differentiation formula.


157  Special Values of Class Group $L$Functions for CM Fields Masri, Riad
Let $H$ be the Hilbert class field of a CM number field $K$ with
maximal totally real subfield $F$ of degree $n$ over $\mathbb{Q}$. We
evaluate the second term in the Taylor expansion at $s=0$ of the
Galoisequivariant $L$function $\Theta_{S_{\infty}}(s)$ associated to
the unramified abelian characters of $\operatorname{Gal}(H/K)$. This is an identity
in the group ring $\mathbb{C}[\operatorname{Gal}(H/K)]$ expressing
$\Theta^{(n)}_{S_{\infty}}(0)$ as essentially a linear combination of
logarithms of special values $\{\Psi(z_{\sigma})\}$, where $\Psi\colon
\mathbb{H}^{n} \rightarrow \mathbb{R}$ is a Hilbert modular function for a congruence
subgroup of $SL_{2}(\mathcal{O}_{F})$ and $\{z_{\sigma}: \sigma \in
\operatorname{Gal}(H/K)\}$ are CM points on a universal Hilbert modular variety. We
apply this result to express the relative class number $h_{H}/h_{K}$
as a rational multiple of the determinant of an $(h_{K}1) \times
(h_{K}1)$ matrix of logarithms of ratios of special values
$\Psi(z_{\sigma})$, thus giving rise to candidates for higher analogs
of elliptic units. Finally, we obtain a product formula for
$\Psi(z_{\sigma})$ in terms of exponentials of special values of
$L$functions.


182  Mutually Aposyndetic Decomposition of Homogeneous Continua Prajs, Janusz R.
A new decomposition, the mutually aposyndetic decomposition of
homogeneous continua into closed, homogeneous sets is introduced. This
decomposition is respected by homeomorphisms and topologically
unique. Its quotient is a mutually aposyndetic homogeneous continuum,
and in all known examples, as well as in some general cases, the
members of the decomposition are semiindecomposable continua. As
applications, we show that hereditarily decomposable homogeneous
continua and path connected homogeneous continua are mutually
aposyndetic. A class of new examples of homogeneous continua is
defined. The mutually aposyndetic decomposition of each of these
continua is nontrivial and different from Jones' aposyndetic
decomposition.


202  Interior $h^1$ Estimates for Parabolic Equations with $\operatorname{LMO}$ Coefficients Tang, Lin
In this paper we establish
a priori $h^1$estimates in a bounded domain for parabolic
equations with vanishing $\operatorname{LMO}$ coefficients.


218  The General Definition of the Complex MongeAmpère Operator on Compact Kähler Manifolds Xing, Yang
We introduce a wide subclass ${\mathcal F}(X,\omega)$ of
quasiplurisubharmonic functions in a compact Kähler manifold, on
which the complex MongeAmpère operator is well defined and the
convergence theorem is valid. We also prove that ${\mathcal F}(X,\omega)$
is a convex cone and includes all quasiplurisubharmonic functions
that are in the Cegrell class.


242  A Second Order Smooth Variational Principle on Riemannian Manifolds Azagra, Daniel; Fry, Robb
We establish a second order smooth variational principle valid for functions defined on (possibly infinitedimensional) Riemannian manifolds which are uniformly locally convex and have a strictly positive injectivity radius and bounded sectional curvature.


261  Erratum to: On Value Distribution Theory of Second Order Periodic ODEs, Special Functions and Orthogonal Polynomials Chiang, YikMan; Ismail, Mourad E. H. No abstract.


262  On the Spectrum of the Equivariant Cohomology Ring Goresky, Mark; MacPherson, Robert
If an algebraic torus $T$ acts on a complex projective algebraic
variety $X$, then the affine scheme $\operatorname{Spec}
H^*_T(X;\mathbb C)$ associated with the equivariant cohomology is
often an arrangement of linear subspaces of the vector space
$H_2^T(X;\mathbb C).$ In many situations the ordinary cohomology ring
of $X$ can be described in terms of this arrangement.


284  SelfMaps of Low Rank Lie Groups at Odd Primes Grbić, Jelena; Theriault, Stephen
Let G be a simple, compact, simplyconnected Lie group localized at an odd prime~p. We study the group of homotopy classes of selfmaps $[G,G]$ when the rank of G is low and in certain cases describe the set of homotopy classes of multiplicative selfmaps $H[G,G]$. The low rank condition gives G certain structural properties which make calculations accessible. Several examples and applications are given.


305  Approximation and Similarity Classification of Stably Finitely Strongly Irreducible Decomposable Operators Hua, He; Yunbai, Dong; Xianzhou, Guo
Let $\mathcal H$ be a complex separable Hilbert space and ${\mathcal L}({\mathcal H})$ denote the collection of bounded linear operators on ${\mathcal H}$. In this paper, we show that for any operator $A\in{\mathcal L}({\mathcal H})$, there exists a stably finitely (SI) decomposable operator $A_\epsilon$, such that $\AA_{\epsilon}\<\epsilon$ and ${\mathcal{\mathcal A}'(A_{\epsilon})}/\operatorname{rad} {{\mathcal A}'(A_{\epsilon})}$ is commutative, where $\operatorname{rad}{{\mathcal A}'(A_{\epsilon})}$ is the Jacobson radical of ${{\mathcal A}'(A_{\epsilon})}$. Moreover, we give a similarity classification of the stably finitely decomposable operators that generalizes the result on similarity classification of CowenDouglas operators given by C. L. Jiang.


320  Some Rigidity Results Related to Monge—Ampère Functions Jerrard, Robert L.
The space of MongeAmpère functions, introduced by J. H. G. Fu, is
a space of rather rough functions in which the map $u\mapsto \operatorname{Det} D^2
u$ is well defined and weakly continuous with respect to a natural
notion of weak convergence. We prove a rigidity theorem for
Lagrangian integral currents that allows us to extend the original
definition of MongeAmpère functions. We also
prove that if a MongeAmpère function $u$ on a bounded set
$\Omega\subset\mathcal{R}^2$ satisfies the equation $\operatorname{Det} D^2 u=0$ in a
particular weak sense, then the graph of $u$ is a developable surface,
and moreover $u$ enjoys somewhat better regularity properties than an
arbitrary MongeAmpère function of $2$ variables.


355  Characterisation Results for Steiner Triple Systems and Their Application to EdgeColourings of Cubic Graphs Král', Daniel; Máčajová, Edita; Pór, Attila; Sereni, JeanSébastien
It is known that a Steiner triple system is projective if and only if it does not contain the fourtriple configuration $C_{14}$. We find three configurations such that a Steiner triple system is affine if and only if it does not contain one of these configurations. Similarly, we characterise Hall triple systems using two forbidden configurations. Our characterisations have several interesting corollaries in the area of edgecolourings of graphs. A cubic graph G is Sedgecolourable for a Steiner triple system S if its edges can be coloured with points of S in such a way that the points assigned to three edges sharing a vertex form a triple in S. Among others, we show that all cubic graphs are Sedgecolourable for every nonprojective nonaffine pointtransitive Steiner triple system S.


382  Verma Modules over Quantum Torus Lie Algebras Lü, Rencai; Zhao, Kaiming
Representations of various onedimensional central
extensions of quantum tori (called quantum torus Lie algebras) were
studied by several authors. Now we define a central extension of
quantum tori so that all known representations can be regarded as
representations of the new quantum torus Lie algebras $\mathfrak{L}_q$. The
center of $\mathfrak{L}_q$ now is generally infinite dimensional.


400  On pAdic Properties of Central LValues of Quadratic Twists of an Elliptic Curve Prasanna, Kartik
We study $p$indivisibility of the central values $L(1,E_d)$ of
quadratic twists $E_d$ of a semistable elliptic curve $E$ of
conductor $N$. A consideration of the conjecture of Birch and
SwinnertonDyer shows that the set of quadratic discriminants $d$
splits naturally into several families $\mathcal{F}_S$, indexed by subsets $S$
of the primes dividing $N$. Let $\delta_S= \gcd_{d\in \mathcal{F}_S}
L(1,E_d)^{\operatorname{alg}}$, where $L(1,E_d)^{\operatorname{alg}}$ denotes the algebraic part
of the central $L$value, $L(1,E_d)$. Our main theorem relates the
$p$adic valuations of $\delta_S$ as $S$ varies. As a consequence we
present an application to a refined version of a question of
Kolyvagin. Finally we explain an intriguing (albeit speculative)
relation between Waldspurger packets on $\widetilde{\operatorname{SL}_2}$ and
congruences of modular forms of integral and halfintegral weight. In
this context, we formulate a conjecture on congruences of
halfintegral weight forms and explain its relevance to the problem of
$p$indivisibility of $L$values of quadratic twists.


415  Classification of Reducing Subspaces of a Class of Multiplication Operators on the Bergman Space via the Hardy Space of the Bidisk Sun, Shunhua; Zheng, Dechao; Zhong, Changyong
In this paper we obtain a complete description of nontrivial minimal reducing subspaces of the multiplication operator by a Blaschke product with four zeros on the Bergman space of the unit disk via the Hardy space of the bidisk.


439  On Hankel Forms of Higher Weights: The Case of Hardy Spaces Sundhäll, Marcus; Tchoundja, Edgar
In this paper we study bilinear Hankel forms of higher weights on Hardy spaces in several dimensions. (The Schatten class Hankel forms of higher weights on weighted Bergman spaces have already been studied by Janson and Peetre for one dimension and by Sundhäll for several dimensions). We get a full characterization of Schatten class Hankel forms in terms of conditions for the symbols to be in certain Besov spaces. Also, the Hankel forms are bounded and compact if and only if the symbols satisfy certain Carleson measure criteria and vanishing Carleson measure criteria, respectively.


456  The Chowla—Selberg Formula and The Colmez Conjecture Yang, Tonghai
In this paper, we reinterpret the Colmez conjecture on the Faltings height of CM abelian varieties in terms of Hilbert (and Siegel) modular forms. We construct an elliptic modular form involving the Faltings height of a CM abelian surface and arithmetic intersection numbers, and prove that the Colmez conjecture for CM abelian surfaces is equivalent to the cuspidality of this modular form.


473  Goresky—MacPherson Calculus for the Affine Flag Varieties Yun, Zhiwei
We use the fixed point arrangement technique developed by
Goresky and MacPherson to calculate the part of the
equivariant cohomology of the affine flag variety $\mathcal{F}\ell_G$ generated
by degree 2. We use this result to show that the vertices of the
moment map image of $\mathcal{F}\ell_G$ lie on a paraboloid.


481  Elements of Algebraic Geometry and the Positive Theory of Partially Commutative Groups CasalsRuiz, Montserrat; Kazachkov, Ilya V.
The first main result of the paper is a criterion for a partially commutative group $\mathbb G$ to be a domain. It allows us to reduce the study of algebraic sets over $\mathbb G$ to the study of irreducible algebraic sets, and reduce the elementary theory of $\mathbb G$ (of a coordinate group over $\mathbb G$) to the elementary theories of the direct factors of $\mathbb G$ (to the elementary theory of coordinate groups of irreducible algebraic sets). Then we establish normal forms for quantifierfree formulas over a nonabelian directly indecomposable partially commutative group $\mathbb H$. Analogously to the case of free groups, we introduce the notion of a generalised equation and prove that the positive theory of $\mathbb H$ has quantifier elimination and that arbitrary firstorder formulas lift from $\mathbb H$ to $\mathbb H\ast F$, where $F$ is a free group of finite rank. As a consequence, the positive theory of an arbitrary partially commutative group is decidable.


520  Computing Noncommutative Deformations of Presheaves and Sheaves of Modules Eriksen, Eivind
We describe a noncommutative deformation theory for presheaves and
sheaves of modules that generalizes the commutative deformation
theory of these global algebraic structures and the noncommutative
deformation theory of modules over algebras due to Laudal.


543  More Variations on the Sierpiński Sieve Hare, Kevin G.
This paper answers a question of Broomhead, Montaldi and Sidorov about the existence of gaskets of a particular type related to the Sierpiński sieve. These gaskets are given by iterated function systems that do not satisfy the open set condition. We use the methods of Ngai and Wang to compute the dimension of these gaskets.


563  Whittaker Functions on Real Semisimple Lie Groups of Rank Two Ishii, Taku
We give explicit formulas for Whittaker functions on real semisimple Lie groups of real rank two belonging to the class one principal series representations. By using these formulas we compute certain archimedean zeta integrals.


582  On the Distribution of Pseudopowers Konyagin, Sergei V.; Pomerance, Carl; Shparlinski, Igor E.
An xpseudopower to base g is a positive integer that is not a power of g, yet is so modulo p for all primes $ple x$. We improve an upper bound for the least such number, due to E.~Bach, R.~Lukes, J.~Shallit, and H.~C.~Williams. The method is based on a combination of some bounds of exponential sums with new results about the average behaviour of the multiplicative order of g modulo prime numbers.


595  On Locally Uniformly Rotund Renormings in C(K) Spaces Martínez, J. F.; Moltó, A.; Orihuela, J.; Troyanski, S.
A characterization of the Banach spaces of type $C(K)$ that admit an equivalent locally uniformly rotund norm is obtained, and a method to apply it to concrete spaces is developed. As an application the existence of such renorming is deduced when $K$ is a NamiokaPhelps compact or for some particular class of Rosenthal compacta, results which were originally proved with ad hoc methods.


614  Translation Groupoids and Orbifold Cohomology Pronk, Dorette; Scull, Laura
We show that the bicategory of (representable) orbifolds and good maps is equivalent to the bicategory of orbifold translation groupoids and generalized equivariant maps, giving a mechanism for transferring results from equivariant homotopy theory to the orbifold category. As an application, we use this result to define orbifold versions of a couple of equivariant cohomology theories: Ktheory and Bredon cohomology for certain coefficient diagrams.


646  Reducibility in A_{R}(K), C_{R}(K), and A(K) Rupp, R.; Sasane, A.
Let $K$ denote a compact real symmetric subset of $\mathbb{C}$ and let
$A_{\mathbb R}(K)$ denote the real Banach algebra of all real
symmetric continuous functions on $K$ that are analytic in the
interior $K^\circ$ of $K$, endowed with the supremum norm. We
characterize all unimodular pairs $(f,g)$ in $A_{\mathbb R}(K)^2$
which are reducible.
In addition, for an arbitrary compact $K$ in $\mathbb C$, we give a
new proof (not relying on Banach algebra theory or elementary stable
rank techniques) of the fact that the Bass stable rank of $A(K)$ is
$1$.
Finally, we also characterize all compact real symmetric sets $K$ such
that $A_{\mathbb R}(K)$, respectively $C_{\mathbb R}(K)$, has Bass
stable rank $1$.


668  The Supersingular Locus of the Shimura Variety for GU(1,s) Vollaard, Inken
In this paper we study the supersingular locus of the reduction modulo $p$ of the Shimura variety for $GU(1,s)$ in the case of an inert prime $p$. Using Dieudonné theory we define a stratification of the corresponding moduli space of $p$divisible groups. We describe the incidence relation of this stratification in terms of the BruhatTits building of a unitary group. In the case of $GU(1,2)$, we show that the supersingular locus is equidimensional of dimension 1 and is of complete intersection. We give an explicit description of the irreducible components and their intersection behaviour.


721  Formal Fibers of Unique Factorization Domains Boocher, Adam; Daub, Michael; Johnson, Ryan K.; Lindo, H.; Loepp, S.; Woodard, Paul A.
Let $(T,M)$ be a complete local (Noetherian) ring such that $\dim T\geq 2$ and
$T=T/M$ and let $\{p_i\} _{i \in \mathcal I}$ be a collection of
elements of T indexed by a set $\mathcal I$ so that $\mathcal I  < T$.
For each $i \in \mathcal{I}$, let $C_i$:={$Q_{i1}$,$\dots$,$Q_{in_i}$}
be a set of nonmaximal prime ideals containing $p_i$ such that the $Q_{ij}$
are incomparable and $p_i\in Q_{jk}$ if and only if $i=j$. We provide necessary
and sufficient conditions so that T is the ${\bf m}$adic completion of a local unique
factorization domain $(A, {\bf m})$, and for each $i \in \mathcal I$, there exists a unit
$t_i$ of T so that $p_{i}t_i \in A$ and $C_i$
is the set of prime ideals $Q$ of $T$ that are maximal with respect to the condition
that $Q \cap A = p_{i}t_{i}A$.


737  Approximation by Dilated Averages and KFunctionals Ditzian, Z.; Prymak, A.
For a positive finite measure $d\mu(\mathbf{u})$ on $\mathbb{R}^d$
normalized to satisfy $\int_{\mathbb{R}^d}d\mu(\mathbf{u})=1$, the dilated average of
$f( \mathbf{x})$ is given by \[ A_tf(\mathbf{x})=\int_{\mathbb{R}^d}f(\mathbf{x}t\mathbf{u})d\mu(\mathbf{u}). \] It
will be shown that under some mild assumptions on $d\mu(\mathbf{u})$ one has
the equivalence \[ \A_tff\_B\approx \inf \{
(\fg\_B+t^2 \P(D)g\_B): P(D)g\in B\}\quad\text{for }t>0, \]
where $\varphi(t)\approx \psi(t)$ means
$c^{1}\le\varphi(t)/\psi(t)\le c$, $B$ is a Banach space of functions
for which translations are continuous isometries and $P(D)$ is an
elliptic differential operator induced by $\mu$. Many applications are
given, notable among which is the averaging operator with $d\mu(\mathbf{u})=
\frac{1}{m(S)}\chi_S(\mathbf{u})d\mathbf{u}$, where $S$ is a bounded convex set
in $\mathbb{R}^d$ with an interior point, $m(S)$ is the Lebesgue measure of
$S$, and $\chi_S(\mathbf{u})$ is the characteristic function of $S$. The rate
of approximation by averages on the boundary of a convex set under
more restrictive conditions is also shown to be equivalent to an
appropriate $K$functional.


758  General Preservers of QuasiCommutativity Dolinar, Gregor; Kuzma, Bojan
Let ${ M}_n$ be the algebra of all $n \times n$ matrices over $\mathbb{C}$. We say that $A, B \in { M}_n$ quasicommute if there exists a nonzero $\xi \in \mathbb{C}$ such that $AB = \xi BA$. In the paper we classify bijective not necessarily linear maps $\Phi \colon M_n \to M_n$ which preserve quasicommutativity in both directions.


787  An Explicit Treatment of Cubic Function Fields with Applications Landquist, E.; Rozenhart, P.; Scheidler, R.; Webster, J.; Wu, Q.
We give an explicit treatment of cubic function fields of characteristic at least five. This includes an efficient technique for converting such a field into standard form, formulae for the field discriminant and the genus, simple necessary and sufficient criteria for nonsingularity of the defining curve, and a characterization of all triangular integral bases. Our main result is a description of the signature of any rational place in a cubic extension that involves only the defining curve and the order of the base field. All these quantities only require simple polynomial arithmetic as well as a few squarefree polynomial factorizations and, in some cases, square and cube root extraction modulo an irreducible polynomial. We also illustrate why and how signature computation plays an important role in computing the class number of the function field. This in turn has applications to the study of zeros of zeta functions of function fields.


808  Extrema of Low Eigenvalues of the DirichletNeumann Laplacian on a Disk Legendre, Eveline
We study extrema of the first and the second mixed eigenvalues of the Laplacian on the disk among some families of DirichletNeumann boundary conditions. We show that the minimizer of the second eigenvalue among all mixed boundary conditions lies in a compact $1$parameter family for which an explicit description is given. Moreover, we prove that among all partitions of the boundary with bounded number of parts on which Dirichlet and Neumann conditions are imposed alternately, the first eigenvalue is maximized by the uniformly distributed partition.


827  BMO Functions and Carleson Measures with Values in Uniformly Convex Spaces Ouyang, Caiheng; Xu, Quanhua
This paper studies the relationship between vectorvalued BMO functions and the Carleson measures defined by their gradients. Let $dA$ and $dm$ denote Lebesgue measures on the unit disc $D$ and the unit circle $\mathbf{T}$, respectively. For $1< q<\infty$ and a Banach space $B$, we prove that there exists a positive constant $c$ such that $$\sup_{z_0\in D}\int_{D}(1z)^{q1}\\nabla f(z)\^q P_{z_0}(z) dA(z) \le c^q\sup_{z_0\in D}\int_{\mathbf{T}}\f(z)f(z_0)\^qP_{z_0}(z) dm(z)$$ holds for all trigonometric polynomials $f$ with coefficients in $B$ if and only if $B$ admits an equivalent norm which is $q$uniformly convex, where $$P_{z_0}(z)=\frac{1z_0^2}{1\bar{z_0}z^2} .$$ The validity of the converse inequality is equivalent to the existence of an equivalent $q$uniformly smooth norm.


845  Biflatness and PseudoAmenability of Segal Algebras Samei, Ebrahim; Spronk, Nico; Stokke, Ross
We investigate generalized amenability and biflatness properties of various (operator) Segal algebras in both the group algebra, $L^1(G)$, and the Fourier algebra, $A(G)$, of a locally compact group~$G$.


870  The BrascampLieb Polyhedron Valdimarsson, Stefán Ingi
A set of necessary and sufficient conditions for the BrascampLieb inequality to hold has recently been found by Bennett, Carbery, Christ, and Tao. We present an analysis of these conditions. This analysis allows us to give a concise description of the set where the inequality holds in the case where each of the linear maps involved has corank $1$. This complements the result of Barthe concerning the case where the linear maps all have rank $1$. Pushing our analysis further, we describe the case where the maps have either rank $1$ or rank $2$. A separate but related problem is to give a list of the finite number of conditions necessary and sufficient for the BrascampLieb inequality to hold. We present an algorithm which generates such a list.


889  Singular Integral Operators and Essential Commutativity on the Sphere Xia, Jingbo
Let ${\mathcal T}$ be the $C^\ast $algebra generated by the Toeplitz operators $\{T_\varphi : \varphi \in L^\infty (S,d\sigma )\}$ on the Hardy space $H^2(S)$ of the unit sphere in $\mathbf{C}^n$. It is well known that ${\mathcal T}$ is contained in the essential commutant of $\{T_\varphi : \varphi \in \operatorname{VMO}\cap L^\infty (S,d\sigma )\}$. We show that the essential commutant of $\{T_\varphi : \varphi \in \operatorname{VMO}\cap L^\infty (S,d\sigma )\}$ is strictly larger than ${\mathcal T}$.


914  Reducibility of the Principal Series for Sp^{~}_{2}(F) over a padic Field Zorn, Christian
Let $G_n=\mathrm{Sp}_n(F)$ be the rank $n$ symplectic group with
entries in a nondyadic $p$adic field $F$. We further let $\widetilde{G}_n$ be
the metaplectic extension of $G_n$ by $\mathbb{C}^{1}=\{z\in\mathbb{C}^{\times}
\mid z=1\}$ defined using the Leray cocycle. In this paper, we aim to
demonstrate the complete list of reducibility points of the genuine
principal series of ${\widetilde{G}_2}$. In most cases, we will use
some techniques developed by Tadić that analyze the Jacquet
modules with respect to all of the parabolics containing a fixed
Borel. The exceptional cases involve representations induced from
unitary characters $\chi$ with $\chi^2=1$. Because such
representations $\pi$ are unitary, to show the irreducibility of
$\pi$, it suffices to show that
$\dim_{\mathbb{C}}\mathrm{Hom}_{{\widetilde{G}}}(\pi,\pi)=1$. We will accomplish this
by examining the poles of certain intertwining operators associated to
simple roots. Then some results of Shahidi and Ban decompose arbitrary
intertwining operators into a composition of operators corresponding
to the simple roots of ${\widetilde{G}_2}$. We will then be able to
show that all such operators have poles at principal series
representations induced from quadratic characters and therefore such
operators do not extend to operators in
$\mathrm{Hom}_{{\widetilde{G}_2}}(\pi,\pi)$ for the $\pi$ in question.


961  Multiplicative Isometries and Isometric ZeroDivisors Aleman, Alexandru; Duren, Peter; Martín, María J.; Vukotić, Dragan
For some Banach spaces of analytic functions in the unit disk
(weighted Bergman spaces, Bloch space, Dirichlettype spaces), the
isometric pointwise multipliers are found to be unimodular constants.
As a consequence, it is shown that none of those spaces have isometric
zerodivisors. Isometric coefficient multipliers are also
investigated.


975  Revisiting TietzeNakajima: Local and Global Convexity for Maps Bjorndahl, Christina; Karshon, Yael
A theorem of Tietze and Nakajima, from 1928, asserts that
if a subset $X$ of $\mathbb{R}^n$ is closed, connected, and locally convex,
then it is convex.
We give an analogous ``local to global convexity" theorem
when the inclusion map of $X$ to $\mathbb{R}^n$ is replaced by a map
from a topological space $X$ to $\mathbb{R}^n$ that satisfies
certain local properties.
Our motivation comes from the CondevauxDazordMolino proof
of the AtiyahGuilleminSternberg convexity theorem in symplectic geometry.


994  Curvature Bounds for Surfaces in Hyperbolic 3Manifolds Breslin, William
A triangulation of a hyperbolic $3$manifold is Lthick if each
tetrahedron having all vertices in the thick part of $M$ is
$L$bilipschitz diffeomorphic to the standard Euclidean tetrahedron.
We show that there exists a fixed constant $L$ such that every
complete hyperbolic $3$manifold has an $L$thick geodesic
triangulation. We use this to prove the existence of universal bounds on
the principal curvatures of $\pi_1$injective surfaces and strongly
irreducible Heegaard surfaces in hyperbolic $3$manifolds.


1011  Functoriality of the Canonical Fractional Galois Ideal Buckingham, Paul; Snaith, Victor
The fractional Galois ideal
is a conjectural improvement on the higher Stickelberger
ideals defined at negative integers, and is expected to provide
nontrivial annihilators for higher $K$groups of rings of integers of
number fields. In this article, we extend the definition of the
fractional Galois ideal to arbitrary (possibly infinite and
nonabelian) Galois extensions of number fields under the assumption
of Stark's conjectures and prove naturality properties under
canonical changes of extension. We discuss applications of this to the
construction of ideals in noncommutative Iwasawa algebras.


1037  Riemann Extensions of TorsionFree Connections with Degenerate Ricci Tensor CalviñoLouzao, E.; GarcíaRío, E.; VázquezLorenzo, R.
{Correspondence} between torsionfree connections with {nilpotent skewsymmetric curvature operator} and IP Riemann
extensions is shown. Some consequences are derived in the study of
fourdimensional IP metrics and locally homogeneous affine surfaces.


1058  On a Conjecture of S. Stahl Chen, Yichao; Liu, Yanpei
S. Stahl conjectured that the zeros of genus polynomials are real. In
this note, we disprove this conjecture.


1060  Heegner Points over Towers of Kummer Extensions Darmon, Henri; Tian, Ye
Let $E$ be an elliptic curve, and let $L_n$ be the Kummer extension
generated by a primitive $p^n$th root of unity and a $p^n$th root of
$a$ for a fixed $a\in \mathbb{Q}^\times\{\pm 1\}$. A detailed case study
by Coates, Fukaya, Kato and Sujatha and V. Dokchitser has led these
authors to predict unbounded and strikingly regular growth for the
rank of $E$ over $L_n$ in certain cases. The aim of this note is to
explain how some of these predictions might be accounted for by
Heegner points arising from a varying collection of Shimura curve
parametrisations.


1082  The Fundamental Group of $S^1$manifolds Godinho, Leonor; SousaDias, M. E.
We address the problem of computing the fundamental
group of a symplectic $S^1$manifold for nonHamiltonian actions on
compact manifolds, and for Hamiltonian actions on noncompact
manifolds with a proper moment map. We generalize known results for
compact manifolds equipped with a Hamiltonian $S^1$action. Several
examples are presented to illustrate our main results.


1099  Character Sums to Smooth Moduli are Small Goldmakher, Leo
Recently, Granville and Soundararajan have made
fundamental breakthroughs in the study of character sums. Building
on their work and using estimates on short character sums developed
by GrahamRingrose and Iwaniec, we improve the
PólyaVinogradov inequality for characters with smooth conductor.


1116  Degenerate pLaplacian Operators and Hardy Type Inequalities on
HType Groups Jin, Yongyang; Zhang, Genkai
Let $\mathbb G$ be a steptwo nilpotent group of Htype with Lie algebra $\mathfrak G=V\oplus \mathfrak t$. We define a class of vector fields $X=\{X_j\}$ on $\mathbb G$ depending on a real parameter $k\ge 1$, and we consider the corresponding $p$Laplacian operator $L_{p,k} u= \operatorname{div}_X (\nabla_{X} u^{p2} \nabla_X u)$. For $k=1$ the vector fields $X=\{X_j\}$ are the left invariant vector fields corresponding to an orthonormal basis of $V$; for $\mathbb G$ being the Heisenberg group the vector fields are the Greiner fields. In this paper we obtain the fundamental solution for the operator $L_{p,k}$ and as an application, we get a Hardy type inequality associated with $X$.


1131  Moduli Spaces of Reflexive Sheaves of Rank 2 Kleppe, Jan O.
Let $\mathcal{F}$ be a coherent rank $2$ sheaf on a scheme $Y \subset \mathbb{P}^{n}$ of
dimension at least two and let $X \subset Y$ be the zero set of a section
$\sigma \in H^0(\mathcal{F})$. In this paper, we study the relationship between the
functor that deforms the pair $(\mathcal{F},\sigma)$ and the two functors that deform
$\mathcal{F}$ on $Y$, and $X$ in $Y$, respectively. By imposing some conditions on two
forgetful maps between the functors, we prove that the scheme structure of
e.g., the moduli scheme ${\rm M_Y}(P)$ of stable sheaves on a threefold $Y$
at $(\mathcal{F})$, and the scheme structure at $(X)$ of the Hilbert scheme of curves
on $Y$ become closely related. Using this relationship, we get criteria for the
dimension and smoothness of $ {\rm M_{Y}}(P)$ at $(\mathcal{F})$, without assuming $
{\textrm{Ext}^2}(\mathcal{F} ,\mathcal{F} ) = 0$. For reflexive sheaves on $Y=\mathbb{P}^{3}$ whose
deficiency module $M = H_{*}^1(\mathcal{F})$ satisfies $ {_{0}\! \textrm{Ext}^2}(M ,M ) = 0 $
(e.g., of diameter at most 2),
we get necessary and sufficient conditions of unobstructedness that coincide
in the diameter one case. The conditions are further equivalent to the
vanishing of certain graded Betti numbers of the free graded minimal
resolution of $H_{*}^0(\mathcal{F})$. Moreover, we show that every irreducible
component of ${\rm M}_{\mathbb{P}^{3}}(P)$ containing a reflexive sheaf of diameter
one is reduced (generically smooth) and we compute its dimension. We also
determine a good lower bound for the dimension of any component of ${\rm
M}_{\mathbb{P}^{3}}(P)$ that contains a reflexive stable sheaf with ``small''
deficiency module $M$.


1155  Moments of the Critical Values of Families of Elliptic Curves, with Applications Young, Matthew P.
We make conjectures on the moments of the central values of the family
of all elliptic curves and on the moments of the first derivative of
the central values of a large family of positive rank curves. In both
cases the order of magnitude is the same as that of the moments of the
central values of an orthogonal family of $L$functions. Notably, we
predict that the critical values of all rank $1$ elliptic curves is
logarithmically larger than the rank $1$ curves in the positive rank
family.


1182  A Fractal Function Related to the JohnNirenberg Inequality for $Q_{\alpha}({\mathbb R^n})$ Yue, Hong
A borderline case function $f$ for $ Q_{\alpha}({\mathbb R^n})$ spaces
is defined as a Haar wavelet decomposition, with the coefficients
depending on a fixed parameter $\beta>0$. On its support $I_0=[0,
1]^n$, $f(x)$ can be expressed by the binary expansions of the
coordinates of $x$. In particular, $f=f_{\beta}\in Q_{\alpha}({\mathbb
R^n})$ if and only if $\alpha<\beta<\frac{n}{2}$, while for
$\beta=\alpha$, it was shown by Yue and Dafni that $f$ satisfies a
JohnNirenberg inequality for $ Q_{\alpha}({\mathbb R^n})$. When
$\beta\neq 1$, $f$ is a selfaffine function. It is continuous almost
everywhere and discontinuous at all dyadic points inside $I_0$. In
addition, it is not monotone along any coordinate direction in any
small cube. When the parameter $\beta\in (0, 1)$, $f$ is onto from
$I_0$ to $[\frac{1}{12^{\beta}}, \frac{1}{12^{\beta}}]$, and the
graph of $f$ has a noninteger fractal dimension $n+1\beta$.


1201  Criteria for Very Ampleness of Rank Two Vector Bundles over Ruled Surfaces Alzati, Alberto; Besana, Gian Mario
Very ampleness criteria for rank $2$ vector bundles over smooth, ruled
surfaces over rational and elliptic curves are given. The criteria are then
used to settle open existence questions for some special threefolds of low
degree.


1228  Valuations for Matroid Polytope Subdivisions Ardila, Federico; Fink, Alex; Rincón, Felipe
We prove that the ranks of the subsets and the activities of the bases
of a matroid define valuations for the subdivisions of a matroid
polytope into smaller matroid polytopes.


1246  Quantum Cohomology of Minuscule Homogeneous Spaces III. SemiSimplicity and Consequences Chaput, P. E.; Manivel, L.; Perrin, N.
We prove that the quantum cohomology ring of any minuscule or
cominuscule homogeneous space, specialized at $q=1$, is semisimple.
This implies that complex conjugation defines an algebra automorphism
of the quantum cohomology ring localized at the quantum
parameter. We check that this involution coincides with the strange
duality defined in our previous article. We deduce VafaIntriligator type
formulas for the GromovWitten invariants.


1264  Holomorphic variations of minimal disks with boundary on a Lagrangian surface Chen, Jingyi; Fraser, Ailana
Let $L$ be an oriented Lagrangian submanifold in an $n$dimensional
Kähler manifold~$M$. Let $u \colon D \to M$ be a minimal immersion
from a disk $D$ with $u(\partial D) \subset L$ such that $u(D)$ meets
$L$ orthogonally along $u(\partial D)$. Then the real dimension of
the space of admissible holomorphic variations is at least
$n+\mu(E,F)$, where $\mu(E,F)$ is a boundary Maslov index; the minimal
disk is holomorphic if there exist $n$ admissible holomorphic
variations that are linearly independent over $\mathbb{R}$ at some
point $p \in \partial D$; if $M = \mathbb{C}P^n$ and $u$ intersects
$L$ positively, then $u$ is holomorphic if it is stable, and its
Morse index is at least $n+\mu(E,F)$ if $u$ is unstable.


1276  A Generalized Poisson Transform of an $L^{p}$Function over the Shilov Boundary of the $n$Dimensional Lie Ball El Wassouli, Fouzia
Let $\mathcal{D}$ be the $n$dimensional Lie ball and let
$\mathbf{B}(S)$ be the space of hyperfunctions on the Shilov
boundary $S$ of $\mathcal{D}$.
The aim of this paper is to give a
necessary and sufficient condition on the generalized Poisson
transform $P_{l,\lambda}f$ of an element $f$ in the space
$\mathbf{B}(S)$ for $f$ to be in $ L^{p}(S)$, $1 > p > \infty.$
Namely, if $F$ is the Poisson transform of some $f\in
\mathbf{B}(S)$ (i.e., $F=P_{l,\lambda}f$), then for any
$l\in \mathbb{Z}$ and $\lambda\in \mathbb{C}$ such that
$\mathcal{R}e[i\lambda] > \frac{n}{2}1$, we show that $f\in L^{p}(S)$ if and
only if $f$ satisfies the growth condition
$$
\F\_{\lambda,p}=\sup_{0\leq r
< 1}(1r^{2})^{\mathcal{R}e[i\lambda]\frac{n}{2}+l}\Big[\int_{S}F(ru)^{p}du
\Big]^{\frac{1}{p}} < +\infty.
$$


1293  Canonical Toric Fano Threefolds Kasprzyk, Alexander M.
An inductive approach to classifying all toric Fano varieties is
given. As an application of this technique, we present a
classification of the toric Fano threefolds with at worst canonical
singularities. Up to isomorphism, there are $674,\!688$ such
varieties.


1310  IwahoriHecke Algebras of $SL_2$ over $2$Dimensional Local Fields Lee, KyuHwan
In this paper we construct an analogue of IwahoriHecke algebras of $\operatorname{SL}_2$ over $2$dimensional local fields. After considering coset decompositions of double cosets of a Iwahori subgroup, we define a convolution product on the space of certain functions on $\operatorname{SL}_2$, and prove that the product is welldefined, obtaining a Hecke algebra. Then we investigate the structure of the Hecke algebra. We determine the center of the Hecke algebra and consider IwahoriMatsumoto type relations.


1325  On Some Explicit Constructions of Finsler Metrics with Scalar Flag Curvature Mo, Xiaohuan; Yu, Changtao
We give an explicit construction of polynomial (of
arbitrary degree) $(\alpha,\beta)$metrics with scalar flag curvature
and determine their scalar flag curvature. These Finsler metrics
contain all nontrivial projectively flat $(\alpha,\beta)$metrics of
constant flag curvature.


1340  Holomorphie des opérateurs d'entrelacement normalisés à l'aide des paramètres d'Arthur Mœglin, C.
In this paper we prove holomorphy for certain intertwining operators
arising from the theory of Eisenstein series. To do that we need to
normalize using the LanglandsShahidi's normalization arising from
the twisted endoscopy and the associated representations of the
general linear group.


1387  Homotopy SelfEquivalences of 4manifolds with Free Fundamental Group Pamuk, Mehmetcik
We calculate the group of homotopy classes of homotopy
selfequivalences of $4$manifolds with free fundamental group and
obtain a classification of such $4$manifolds up to $s$cobordism.


1404  Characterizations of Extremals for some Functionals on Convex Bodies Saroglou, Christos
We investigate equality cases in inequalities for Sylvestertype
functionals. Namely, it was proven by Campi, Colesanti, and Gronchi
that the quantity
$$
\int_{x_0\in K}\cdots\int_{x_n\in
K}[V(\textrm{conv}\{x_0,\dots,x_n\})]^pdx_0\cdots dx_n , n\geq d, p\geq
1
$$
is maximized by triangles among all planar convex bodies $K$
(parallelograms in the symmetric case). We show that these are the
only maximizers, a fact proven by Giannopoulos for $p=1$.
Moreover, if $h$: $\mathbb{R}_+\rightarrow \mathbb{R}_+$ is a
strictly increasing function and $W_j$ is the $j$th
quermassintegral in $\mathbb{R}^d$, we prove that the functional
$$
\int_{x_0\in K_0}\cdots\int_{x_n\in
K_n}h(W_j(\textrm{conv}\{x_0,\dots,x_n\}))dx_0\cdots dx_n , n \geq d
$$
is
minimized among the $(n+1)$tuples of convex bodies of fixed
volumes if and only if $K_0,\dots,K_n$ are homothetic ellipsoids
when $j=0$ (extending a result of Groemer) and Euclidean balls
with the same center when $j>0$ (extending a result of Hartzoulaki
and Paouris).


1419  BMOEstimates for Maximal Operators via Approximations of the Identity with NonDoubling Measures Yang, Dachun; Yang, Dongyong
Let $\mu$ be a nonnegative Radon measure
on $\mathbb{R}^d$ that satisfies the growth condition that there exist
constants $C_0>0$ and $n\in(0,d]$ such that for all $x\in\mathbb{R}^d$ and
$r>0$, ${\mu(B(x,\,r))\le C_0r^n}$, where $B(x,r)$ is the open ball
centered at $x$ and having radius $r$. In this paper, the authors prove
that if $f$ belongs to the $\textrm {BMO}$type space $\textrm{RBMO}(\mu)$ of Tolsa, then
the homogeneous maximal function $\dot{\mathcal{M}}_S(f)$ (when $\mathbb{R}^d$ is not an
initial cube) and the inhomogeneous maximal function
$\mathcal{M}_S(f)$ (when $\mathbb{R}^d$ is an initial cube)
associated with a given approximation of the identity $S$ of Tolsa are
either infinite everywhere or finite almost everywhere,
and in the latter case, $\dot{\mathcal{M}}_S$ and $\mathcal{M}_S$ are bounded from
$\textrm{RBMO}(\mu)$ to the $\textrm {BLO}$type
space $\textrm{RBLO}(\mu)$. The authors also prove that the inhomogeneous
maximal operator $\mathcal{M}_S$ is bounded from the local
$\textrm {BMO}$type space $\textrm{rbmo}(\mu)$
to the local $\textrm {BLO}$type space $\textrm{rblo}(\mu)$.

