26. CMB Online first
 He, Ziyi; Yang, Dachun; Yuan, Wen

LittlewoodPaley Characterizations of SecondOrder Sobolev Spaces via Averages on Balls
In this paper, the authors characterize secondorder Sobolev
spaces $W^{2,p}({\mathbb R}^n)$,
with $p\in [2,\infty)$ and $n\in\mathbb N$ or $p\in (1,2)$ and
$n\in\{1,2,3\}$, via the Lusin area
function and the LittlewoodPaley $g_\lambda^\ast$function in
terms of ball means.
Keywords:Sobolev space, ball means, Lusinarea function, $g_\lambda^*$function Categories:46E35, 42B25, 42B20, 42B35 

27. CMB Online first
 Hanson, Brandon

Character sums over Bohr sets
We prove character sum estimates for additive Bohr subsets modulo
a prime.
These estimates are analogous to classical character sum bounds
of
PÃ³lyaVinogradov and Burgess. These estimates are applied to
obtain results on
recurrence mod $p$ by special elements.
Keywords:character sums, Bohr sets, finite fields Categories:11L40, 11T24, 11T23 

28. CMB Online first
 Chen, Caisheng; Song, Hongxue; Yan, Qinglun

Existence of multiple solutions for a $p$Laplacian system in $\textbf{R}^{N}$ with signchanging weight functions
In this paper, we consider the quasilinear elliptic
problem
\[
\left\{
\begin{aligned}
&
M
\left(\int_{\mathbb{R}^{N}}x^{ap}\nabla u^{p}dx
\right){\rm
div}
\left(x^{ap}\nabla u^{p2}\nabla u
\right)
\\
&
\qquad=\frac{\alpha}{\alpha+\beta}H(x)u^{\alpha2}uv^{\beta}+\lambda
h_{1}(x)u^{q2}u,
\\
&
M
\left(\int_{\mathbb{R}^{N}}x^{ap}\nabla v^{p}dx
\right){\rm
div}
\left(x^{ap}\nabla v^{p2}\nabla v
\right)
\\
&
\qquad=\frac{\beta}{\alpha+\beta}H(x)v^{\beta2}vu^{\alpha}+\mu
h_{2}(x)v^{q2}v,
\\
&u(x)\gt 0,\quad v(x)\gt 0, \quad x\in \mathbb{R}^{N}
\end{aligned}
\right.
\]
where $\lambda, \mu\gt 0$, $1\lt p\lt N$,
$1\lt q\lt p\lt p(\tau+1)\lt \alpha+\beta\lt p^{*}=\frac{Np}{Np}$, $0\leq
a\lt \frac{Np}{p}$, $a\leq b\lt a+1$, $d=a+1b\gt 0$, $M(s)=k+l s^{\tau}$,
$k\gt 0$, $l, \tau\geq0$ and the weight $H(x), h_{1}(x), h_{2}(x)$
are
continuous functions which change sign in $\mathbb{R}^{N}$. We
will prove that the problem has at least two positive solutions
by
using the Nehari manifold and the fibering maps associated with
the Euler functional for this problem.
Keywords:Nehari manifold, quasilinear elliptic system, $p$Laplacian operator, concave and convex nonlinearities Category:35J66 

29. CMB Online first
 Sands, Jonathan W.

$L$functions for Quadratic Characters and Annihilation of Motivic Cohomology Groups
Let $n$ be a positive even integer, and let $F$ be a totally real
number field and $L$ be an abelian Galois extension which is totally
real or CM.
Fix a finite set $S$ of primes of $F$ containing the infinite primes
and all those which ramify in
$L$, and let $S_L$ denote the primes of $L$ lying above those in
$S$. Then $\mathcal{O}_L^S$ denotes the ring of $S_L$integers of $L$.
Suppose that $\psi$ is a quadratic character of the Galois group of
$L$ over $F$. Under the assumption of the motivic Lichtenbaum
conjecture, we obtain a nontrivial annihilator of the motivic
cohomology group
$H_\mathcal{M}^2(\mathcal{O}_L^S,\mathbb{Z}(n))$ from the lead term of the Taylor series for the
$S$modified Artin $L$function $L_{L/F}^S(s,\psi)$ at $s=1n$.
Keywords:motivic cohomology, regulator, Artin Lfunctions Categories:11R42, 11R70, 14F42, 19F27 

30. CMB Online first
 Ongaro, Jared; Shapiro, Boris

A note on planarity stratification of Hurwitz spaces
One can easily show that any meromorphic function
on a complex closed Riemann surface can be represented as a
composition of a birational map of this surface to $\mathbb{CP}^2$ and
a projection of the image curve from an appropriate point
$p\in \mathbb{CP}^2$ to the pencil of lines through $p$. We introduce
a natural stratification of Hurwitz spaces according to the
minimal degree of a plane curve such that a given meromorphic
function can be represented in the above way and calculate the
dimensions of these strata. We observe that they are closely
related to a family of Severi varieties studied earlier by J. Harris,
Z. Ran and I. Tyomkin.
Keywords:Hurwitz spaces, meromorphic functions, Severi varieties 

31. CMB Online first
 Silberman, Lior

Quantum unique ergodicity on locally symmetric spaces: the degenerate lift
Given a measure $\bar\mu_\infty$ on a locally symmetric space $Y=\Gamma\backslash
G/K$,
obtained as a weak{*} limit of probability measures associated
to
eigenfunctions of the ring of invariant differential operators,
we
construct a measure $\bar\mu_\infty$ on the homogeneous space $X=\Gamma\backslash
G$
which lifts $\bar\mu_\infty$ and which is invariant by a connected subgroup
$A_{1}\subset A$ of positive dimension, where $G=NAK$ is an Iwasawa
decomposition. If the functions are, in addition, eigenfunctions
of
the Hecke operators, then $\bar\mu_\infty$ is also the limit of measures
associated
to Hecke eigenfunctions on $X$. This generalizes results of the
author
with A. Venkatesh in the case where the spectral parameters
stay
away from the walls of the Weyl chamber.
Keywords:quantum unique ergodicity, microlocal lift, spherical dual Categories:22E50, 43A85 

32. CMB Online first
 Aulaskari, Rauno; Chen, Huaihui

On classes $Q_p^\#$ for hyperbolic Riemann surfaces
The $Q_p$ spaces of holomorphic functions on
the disk, hyperbolic Riemann surfaces or complex unit ball have
been studied deeply.
Meanwhile, there are a lot of papers devoted to the $Q^\#_p$
classes of meromorphic functions on the disk or hyperbolic Riemann
surfaces. In this paper, we prove the nesting property (inclusion
relations) of $Q^\#_p$ classes on hyperbolic Riemann surfaces.
The same property for $Q_p$ spaces was also established systematically
and precisely in
earlier work
by the authors of this paper.
Keywords:$Q_p^\#$ class, hyperbolic Riemann surface, spherical Dirichlet function, Categories:30D50, 30F35 

33. CMB 2015 (vol 58 pp. 320)
 Llamas, Aurora; MartínezBernal, José

Cover Product and Betti Polynomial of Graphs
For disjoint graphs $G$ and $H$, with fixed
vertex covers
$C(G)$ and $C(H)$, their cover product is the graph $G
\circledast
H$ with vertex set
$V(G)\cup V(H)$ and edge set $E(G)\cup E(H)\cup\{\{i,j\}:i\in
C(G), j\in
C(H)\}$. We describe the graded Betti numbers of $G\circledast
H$ in terms of those of
$G$ and $H$. As applications we obtain: (i) For any positive
integer $k$ there
exists a connected bipartite graph $G$ such that $\operatorname{reg}
R/I(G)=\mu_S(G)+k$, where,
$I(G)$ denotes the edge ideal of $G$, $\operatorname{reg} R/I(G)$
is the CastelnuovoMumford
regularity of $R/I(G)$ and $\mu_S(G)$ is the induced or strong
matching number of
$G$; (ii) The graded Betti numbers of the complement of a tree
only depends upon
its number of vertices; (iii) The $h$vector of $R/I(G\circledast
H)$ is described in
terms of the $h$vectors of $R/I(G)$ and $R/I(H)$. Furthermore,
in a different
direction, we give a recursive formula for the graded Betti numbers
of chordal
bipartite graphs.
Keywords:CastelnuovoMumford regularity, chordal bipartite graph, edge ideal, graded Betti number, induced matching number, monomial ideal Categories:13D02, 05E45 

34. CMB 2015 (vol 58 pp. 402)
 Tikuisis, Aaron Peter; Toms, Andrew

On the Structure of Cuntz Semigroups in (Possibly) Nonunital C*algebras
We examine the ranks of operators in semifinite $\mathrm{C}^*$algebras
as measured by their densely defined lower semicontinuous traces.
We first prove that a unital simple $\mathrm{C}^*$algebra whose
extreme tracial boundary is nonempty and finite contains positive
operators of every possible rank, independent of the property
of strict comparison. We then turn to nonunital simple algebras
and establish criteria that imply that the Cuntz semigroup is
recovered functorially from the Murrayvon Neumann semigroup
and the space of densely defined lower semicontinuous traces.
Finally, we prove that these criteria are satisfied by notnecessarilyunital
approximately subhomogeneous algebras of slow dimension growth.
Combined with results of the firstnamed author, this shows that
slow dimension growth coincides with $\mathcal Z$stability,
for approximately subhomogeneous algebras.
Keywords:nuclear C*algebras, Cuntz semigroup, dimension functions, stably projectionless C*algebras, approximately subhomogeneous C*algebras, slow dimension growth Categories:46L35, 46L05, 46L80, 47L40, 46L85 

35. CMB 2015 (vol 58 pp. 271)
36. CMB 2015 (vol 58 pp. 306)
 Khoshkhah, Kaveh; Zaker, Manouchehr

On the Largest Dynamic Monopolies of Graphs with a Given Average Threshold
Let $G$ be a graph and $\tau$ be an assignment of nonnegative
integer thresholds to the vertices of $G$. A subset of vertices,
$D$ is said to be a $\tau$dynamic monopoly, if $V(G)$ can be
partitioned into subsets $D_0, D_1, \ldots, D_k$ such that $D_0=D$
and for any $i\in \{0, \ldots, k1\}$, each vertex $v$ in $D_{i+1}$
has at least $\tau(v)$ neighbors in $D_0\cup \ldots \cup D_i$.
Denote the size of smallest $\tau$dynamic monopoly by $dyn_{\tau}(G)$
and the average of thresholds in $\tau$ by $\overline{\tau}$.
We show that the values of $dyn_{\tau}(G)$ over all assignments
$\tau$ with the same average threshold is a continuous set of
integers. For any positive number $t$, denote the maximum $dyn_{\tau}(G)$
taken over all threshold assignments $\tau$ with $\overline{\tau}\leq
t$, by $Ldyn_t(G)$. In fact, $Ldyn_t(G)$ shows the worstcase
value of a dynamic monopoly when the average threshold is a given
number $t$. We investigate under what conditions on $t$, there
exists an upper bound for $Ldyn_{t}(G)$ of the form $cG$, where
$c\lt 1$. Next, we show that $Ldyn_t(G)$ is coNPhard for planar
graphs but has polynomialtime solution for forests.
Keywords:spread of influence in graphs, irreversible dynamic monopolies, target set selection Categories:05C69, 05C85 

37. CMB 2015 (vol 58 pp. 285)
 Karpukhin, Mikhail

Spectral Properties of a Family of Minimal Tori of Revolution in Fivedimensional Sphere
The normalized eigenvalues $\Lambda_i(M,g)$ of the LaplaceBeltrami
operator can be considered as functionals on the space of all
Riemannian metrics $g$ on a fixed surface $M$. In recent papers
several explicit examples of extremal metrics were provided.
These metrics are induced by minimal immersions of surfaces in
$\mathbb{S}^3$ or $\mathbb{S}^4$. In the present paper a family
of extremal metrics induced by minimal immersions in $\mathbb{S}^5$
is investigated.
Keywords:extremal metric, minimal surface Category:58J50 

38. CMB 2015 (vol 58 pp. 415)
 Willson, Benjamin

A Fixed Point Theorem and the Existence of a Haar Measure for Hypergroups Satisfying Conditions Related to Amenability
In this paper we present a fixed point property for amenable
hypergroups which is analogous to Rickert's fixed point theorem
for semigroups. It equates the existence of a left invariant
mean on the space of weakly right uniformly continuous functions
to the existence of a fixed point for any action of the hypergroup.
Using this fixed point property, a certain class of hypergroups
are shown to have a left Haar measure.
Keywords:invariant measure, Haar measure, hypergroup, amenability, function translations Categories:43A62, 43A05, 43A07 

39. CMB 2015 (vol 58 pp. 281)
 Kalus, Matthias

On the Relation of Real and Complex Lie Supergroups
A complex Lie supergroup can be described as a real Lie supergroup
with integrable almost complex structure. The necessary and
sufficient conditions on an almost complex structure on a real
Lie supergroup for defining a complex Lie supergroup are deduced.
The classification of real Lie supergroups with such almost
complex
structures yields a new approach to the known classification
of complex Lie supergroups by complex HarishChandra superpairs.
A universal complexification of a real Lie supergroup is
constructed.
Keywords:Lie supergroup, almost complex structure, HarishChandra pair, universal complexification Categories:32C11, 58A50 

40. CMB 2015 (vol 58 pp. 241)
 Botelho, Fernanda

Isometries and Hermitian Operators on Zygmund Spaces
In this paper we characterize the isometries of subspaces of the little Zygmund space. We show that the isometries of these spaces are surjective and represented as integral operators. We also show that all hermitian operators on these settings are bounded.
Keywords:Zygmund spaces, the little Zygmund space, Hermitian operators, surjective linear isometries, generators of oneparameter groups of surjective isometries Categories:46E15, 47B15, 47B38 

41. CMB 2015 (vol 58 pp. 334)
 Medini, Andrea

Countable Dense Homogeneity in Powers of Zerodimensional Definable Spaces
We show that, for a coanalytic subspace $X$ of $2^\omega$, the
countable dense homogeneity of $X^\omega$ is equivalent to $X$
being Polish. This strengthens a result of HruÅ¡Ã¡k and Zamora
AvilÃ©s. Then, inspired by results of HernÃ¡ndezGutiÃ©rrez,
HruÅ¡Ã¡k and van Mill, using a technique of Medvedev, we
construct a nonPolish subspace $X$ of $2^\omega$ such that $X^\omega$
is countable dense homogeneous. This gives the first $\mathsf{ZFC}$ answer
to a question of HruÅ¡Ã¡k and Zamora AvilÃ©s. Furthermore,
since our example is consistently analytic, the equivalence result
mentioned above is sharp. Our results also answer a question
of Medini and Milovich. Finally, we show that if every countable
subset of a zerodimensional separable metrizable space $X$ is
included in a Polish subspace of $X$ then $X^\omega$ is countable
dense homogeneous.
Keywords:countable dense homogeneous, infinite power, coanalytic, Polish, $\lambda'$set Categories:54H05, 54G20, 54E52 

42. CMB 2015 (vol 58 pp. 225)
 Aghigh, Kamal; Nikseresht, Azadeh

Characterizing Distinguished Pairs by Using Liftings of Irreducible Polynomials
Let $v$ be a henselian valuation of any rank of a field
$K$ and $\overline{v}$ be the unique extension of $v$ to a
fixed algebraic closure $\overline{K}$ of $K$. In 2005, it was studied properties
of those pairs $(\theta,\alpha)$ of elements of $\overline{K}$
with $[K(\theta): K]\gt [K(\alpha): K]$ where $\alpha$ is an element
of smallest degree over $K$ such that
$$
\overline{v}(\theta\alpha)=\sup\{\overline{v}(\theta\beta)
\ \beta\in \overline{K}, \ [K(\beta): K]\lt [K(\theta): K]\}.
$$
Such pairs are referred to as distinguished pairs.
We use the concept of liftings of irreducible polynomials to give a
different characterization of distinguished pairs.
Keywords:valued fields, nonArchimedean valued fields, irreducible polynomials Categories:12J10, 12J25, 12E05 

43. CMB Online first
 Deng, Shaoqiang; Hu, Zhiguang; Li, Jifu

Cohomogeneity one Randers metrics
An action of a Lie group $G$ on a smooth manifold $M$ is called
cohomogeneity one if the orbit space $M/G$ is of dimension $1$.
A Finsler metric $F$ on $M$ is called invariant if $F$ is
invariant under the action of $G$. In this paper,
we study invariant
Randers metrics on cohomogeneity one manifolds. We first give a
sufficient and necessary condition for the existence of invariant
Randers metrics on cohomogeneity one manifolds. Then we obtain
some results on invariant Killing vector fields on the
cohomogeneity one manifolds and use that to deduce some
sufficient and necessary condition for a cohomogeneity one
Randers metric to be Einstein.
Keywords:cohomogeneity one actions, normal geodesics, invariant vector fields, Randers metrics Categories:53C30, 53C60 

44. CMB Online first
 Kong, Qingjun; Guo, Xiuyun

On $s$semipermutable or $s$quasinormally embedded subgroups of finite groups
Suppose that $G$ is a
finite group and $H$ is a subgroup of $G$. $H$ is said to be
$s$semipermutable in $G$ if $HG_{p}=G_{p}H$ for any Sylow
$p$subgroup $G_{p}$ of $G$ with $(p,H)=1$; $H$ is said to be
$s$quasinormally embedded in $G$ if for each prime $p$ dividing the
order of $H$, a Sylow $p$subgroup of $H$ is also a Sylow
$p$subgroup of some $s$quasinormal subgroup of $G$. We fix in
every noncyclic Sylow subgroup $P$ of $G$ some subgroup $D$
satisfying $1\lt D\lt P$ and study the structure of $G$ under the
assumption that every subgroup $H$ of $P$ with $H=D$ is either
$s$semipermutable or $s$quasinormally embedded in $G$.
Some recent results are generalized and unified.
Keywords:$s$semipermutable subgroup, $s$quasinormally embedded subgroup, saturated formation. Categories:20D10, 20D20 

45. CMB 2014 (vol 58 pp. 80)
 Harada, Megumi; Horiguchi, Tatsuya; Masuda, Mikiya

The Equivariant Cohomology Rings of Peterson Varieties in All Lie
Types
Let $G$ be a complex semisimple linear algebraic group and let
$Pet$ be the associated Peterson variety in the flag
variety $G/B$.
The main theorem of this note gives an efficient presentation
of the equivariant cohomology ring $H^*_S(Pet)$ of the
Peterson variety as a quotient of a polynomial ring by an ideal
$J$ generated by quadratic polynomials, in the spirit of the
Borel presentation of the cohomology of the flag variety. Here
the group $S \cong \mathbb{C}^*$ is a certain subgroup of a maximal
torus $T$ of $G$.
Our description of the ideal $J$ uses the Cartan matrix and is
uniform across Lie types. In our arguments we use the Monk formula
and Giambelli formula for the equivariant cohomology rings of
Peterson varieties for all Lie types, as obtained in the work
of Drellich. Our result generalizes a previous theorem of FukukawaHaradaMasuda,
which was only for Lie type $A$.
Keywords:equivariant cohomology, Peterson varieties, flag varieties, Monk formula, Giambelli formula Categories:55N91, 14N15 

46. CMB 2014 (vol 58 pp. 432)
 Yang, Dachun; Yang, Sibei

Secondorder Riesz Transforms and Maximal Inequalities Associated with Magnetic SchrÃ¶dinger Operators
Let $A:=(\nablai\vec{a})\cdot(\nablai\vec{a})+V$ be a
magnetic SchrÃ¶dinger operator on $\mathbb{R}^n$,
where $\vec{a}:=(a_1,\dots, a_n)\in L^2_{\mathrm{loc}}(\mathbb{R}^n,\mathbb{R}^n)$
and $0\le V\in L^1_{\mathrm{loc}}(\mathbb{R}^n)$ satisfy some reverse
HÃ¶lder conditions.
Let $\varphi\colon \mathbb{R}^n\times[0,\infty)\to[0,\infty)$ be such that
$\varphi(x,\cdot)$ for any given $x\in\mathbb{R}^n$ is an Orlicz function,
$\varphi(\cdot,t)\in {\mathbb A}_{\infty}(\mathbb{R}^n)$ for all $t\in (0,\infty)$
(the class of uniformly Muckenhoupt weights) and its uniformly critical upper type index
$I(\varphi)\in(0,1]$. In this article, the authors prove that
secondorder Riesz transforms $VA^{1}$ and
$(\nablai\vec{a})^2A^{1}$ are bounded from the
MusielakOrliczHardy space $H_{\varphi,\,A}(\mathbb{R}^n)$, associated with $A$,
to the MusielakOrlicz space $L^{\varphi}(\mathbb{R}^n)$. Moreover, the authors
establish the boundedness of $VA^{1}$ on $H_{\varphi, A}(\mathbb{R}^n)$. As applications, some
maximal inequalities associated with $A$ in the scale of $H_{\varphi,
A}(\mathbb{R}^n)$ are obtained.
Keywords:MusielakOrliczHardy space, magnetic SchrÃ¶dinger operator, atom, secondorder Riesz transform, maximal inequality Categories:42B30, 42B35, 42B25, 35J10, 42B37, 46E30 

47. CMB Online first
 MartinezMaure, Yves

Plane Lorentzian and Fuchsian Hedgehogs
Parts of the BrunnMinkowski theory can be extended to hedgehogs, which are
envelopes of families of affine hyperplanes parametrized by their Gauss map.
F. Fillastre introduced Fuchsian convex bodies, which are the
closed convex sets of LorentzMinkowski space that are globally invariant
under the action of a Fuchsian group. In this paper, we undertake a study of
plane Lorentzian and Fuchsian hedgehogs. In particular, we prove the
Fuchsian analogues of classical geometrical inequalities (analogues which
are reversed as compared to classical ones).
Keywords:Fuchsian and Lorentzian hedgehogs, evolute, duality, convolution, reversed isoperimetric inequality, reversed Bonnesen inequality Categories:52A40, 52A55, 53A04, 53B30 

48. CMB 2014 (vol 58 pp. 51)
 De Nitties, Giuseppe; SchulzBaldes, Hermann

Spectral Flows of Dilations of Fredholm Operators
Given an essentially unitary contraction and an arbitrary unitary
dilation of it, there is a naturally associated spectral flow which is
shown to be equal to the index of the operator. This result is
interpreted in terms of the $K$theory of an associated mapping
cone. It is then extended to connect $\mathbb{Z}_2$ indices of odd symmetric
Fredholm operators to a $\mathbb{Z}_2$valued spectral flow.
Keywords:spectral flow, Fredholm operators, Z2 indices Categories:19K56, 46L80 

49. CMB 2014 (vol 58 pp. 182)
50. CMB 2014 (vol 58 pp. 196)