Band 27, Heft 6

Given a finite group G , denote by Γ(G)$\Gamma (G)$ the simple undirected graph whose vertices are the distinct sizes of noncentral conjugacy classes of G , and set two vertices of Γ(G)$\Gamma (G)$ to be adjacent if and only if they are not coprime numbers. In this note we prove that, if Γ(G)$\Gamma (G)$ is a k -regular graph with k ≥ 1, then Γ(G)$\Gamma (G)$ is a complete graph with k+1$k+1$ vertices.

Within crystallization theory, (Matveev's) complexity of a 3-manifold can be estimated by means of the combinatorial notion of GM-complexity . In this paper, we prove that the GM-complexity of any lens space L(p,q)${L(p,q)}$, with p ≥ 3, is bounded by S(p,q)-3${S(p,q) -3}$, where S(p,q)${S(p,q)}$ denotes the sum of all partial quotients in the expansion of qp${\frac{q}{p}}$ as a regular continued fraction. The above upper bound had been already established with regard to complexity; its sharpness was conjectured by Matveev himself and has recently been proved for some infinite families of lens spaces by Jaco, Rubinstein and Tillmann. As a consequence, infinite classes of 3-manifolds turn out to exist, where complexity and GM-complexity coincide. Moreover, we present and briefly analyze results arising from crystallization catalogues up to order 32, which prompt us to conjecture, for any lens space L(p,q)${L(p,q)}$ with p ≥ 3, the following relation: k(L(p,q))=5+2c(L(p,q))${k(L(p,q)) = 5 + 2 c(L(p,q))}$, where c ( M ) denotes the complexity of a 3-manifold M and k(M)+1${k(M) +1}$ is half the minimum order of a crystallization of M .

Recently, Zhu gave an alternating circular summation of theta functions (see [Appl. Anal. Discrete Math. 6 (2012), 114–125]). In this paper, we further extend Zhu's results. Some applications are also shown.

A question of Gillespie concerning the completeness of the induced cotorsion pairs is settled in the general setting of any bicomplete abelian category. That is, given a complete hereditary cotorsion pair (𝒜,ℬ)${(\mathcal {A},\mathcal {B})}$ in a bicomplete abelian category, we show that the cotorsion pairs of chain complexes (𝒜˜,dgℬ˜)${(\mathcal {\protect \widetilde{A}},\operatorname{dg}\mathcal {\protect \widetilde{B}})}$ and (dg𝒜˜,ℬ˜)${(\operatorname{dg}\mathcal {\protect \widetilde{A}},\mathcal {\protect \widetilde{B}})}$ are complete, compatible and hereditary. This immediately puts a model structure on the chain complex category. As an application of this result, we establish homological dimensions of unbounded complexes in a Grothendieck category by means of dg𝒜${\operatorname{dg}\mathcal {A}}$ complexes and dgℬ${\operatorname{dg}\mathcal {B}}$ complexes respectively. Some examples of complete hereditary cotorsion pairs in Grothendieck categories are given.

In this work we introduce the definition of Schwarzenberger bundle on a Grassmannian. Recalling the notion of Steiner bundle, we generalize the concept of jumping pair for a Steiner bundle on a Grassmannian. After studying the jumping locus variety and bounding its dimension, we give a complete classification of Steiner bundles with jumping locus of maximal dimension, which all are Schwarzenberger bundles.

Recurrence criteria and related capacitary inequalities for a class of symmetric pure jump processes are given under suitable upper volume growth conditions for the underlying metric measure space. Recurrence problems for symmetric jump processes on d -sets are also investigated.

We study parabolically induced representations for GSpin m(F)${\mathrm {GSpin}_m(F)}$ with F a p -adic field of characteristic zero. The Knapp–Stein R -groups are described and shown to be elementary two groups. We show the associated cocycle is trivial proving multiplicity one for induced representations. We classify the elliptic tempered spectrum. For GSpin 2n+1(F)${\mathrm {GSpin}_{2n+1}(F)}$, we describe the Arthur (Endoscopic) R -group attached to Langlands parameters, and show these are isomorphic to the corresponding Knapp–Stein R -groups. This relies on a non-trivial equality of certain L -functions, and we give a direct proof of this equality. We also reduce the isomorphism, in the general case of split groups, to the maximal case.

Let k be an arbitrary positive integer and let f(x)∈ℤ[x]${f(x)\in {\mathbb {Z}}[x]}$ be a quadratic polynomial with a and D as its leading coefficient and discriminant, respectively. Associated to the least common multiple lcm0≤i≤k{f(n+i)}${\operatorname{lcm}_{0\le i\le k}\lbrace f(n+i)\rbrace }$ of any k+1${k+1}$ consecutive terms in the quadratic progression {f(n)}n∈ℕ*${\lbrace f(n)\rbrace _{n\in \mathbb {N}^*}}$, we define the function gk,f(n):=∏i=0k|f(n+i)|1lcm0≤i≤k{f(n+i)}$ {g_{k, f}(n):= \Biggl (\prod _{i=0}^{k}|f(n+i)|\Biggr ) \frac{1}{\operatorname{lcm}_{0\le i\le k}\lbrace f(n+i)\rbrace }} $ for all n∈ℕ*∖Zk,f${n\in \mathbb {N}^*\setminus Z_{k, f}}$, where Zk,f:=⋃i=0k{n∈ℕ*:f(n+i)=0}${Z_{k,f}:= \bigcup _{i=0}^k\lbrace n\in \mathbb {N}^*: f(n+i)=0\rbrace }$. In this paper, we first show that g k , f is eventually periodic if and only if D≠a2i2${D\ne a^2i^2}$ for all integers i with 1≤i≤k${1\le i\le k}$. Consequently, we develop a detailed p -adic analysis of g k , f and determine its smallest period. Finally, we obtain asymptotic formulas of loglcm0≤i≤k{f(n+i)}${\log \operatorname{lcm}_{0\le i\le k}\lbrace f(n+i)\rbrace }$ for all quadratic polynomials f as n goes to infinity.

Let D<0${D &lt; 0}$ be an odd fundamental discriminant and q be a prime number which splits in ℚ(D)${{\mathbb {Q}}(\sqrt{D})}$. Given a suitable smooth function f supported on [X,2X]${[X, 2X]}$ for X ≥ 1, we establish a uniform bound in X,D${X, D}$ and q for ∑c≡0(modq)Wh(D;c)f(c),$ {\sum _{c \equiv 0 \hspace*{1.42262pt}(\operatorname{mod} ~q)}W_h(D;c)f(c)}, $ where Wh(D;c):=∑b(mod2c),b2≡D(mod4c)ehb2c,h∈ℤ,e(z):=e2πiz,$ {W_h(D;c) := \sum _{b \hspace*{1.42262pt}(\operatorname{mod} ~2c),\, b^2 \equiv D \hspace*{1.42262pt}(\operatorname{mod} ~4c)}e\biggl (\frac{hb}{2c} \biggr )}, \quad h \in {\mathbb {Z}}, \quad e(z):= e^{2 \pi i z}, $ is the Weyl sum for roots of the quadratic congruence x2≡D(mod4c)${x^2 \equiv D~(\text{mod}~{4c})}$. We use this result to study several problems of arithmetic interest, including level-aspect versions of equidistribution of quadratic roots, and the asymptotic distribution of traces of CM values of weakly holomorphic modular functions of level q . By work of Zagier and Bruinier–Funke, the generating functions for these traces are weight 3/2 weakly holomorphic modular forms of level 4q${4q}$ satisfying Kohnen's plus space condition.

Let ρ:G↪ GL (n,𝔽)${\rho : G \hookrightarrow \mathrm {GL}(n, \mathbb {F})}$ be a representation of a finite group G over the field 𝔽 and let H≤G${H \le G}$ be a subgroup of G . Form the algebra of polynomial functions 𝔽[V]${\mathbb {F}[V]}$ on the representation space V=𝔽n${V = \mathbb {F}^n}$ of ρ. The action of G on V extends to 𝔽[V]${\mathbb {F}[V]}$ and we denote by 𝔽[V]G${\mathbb {F}[V]^G}$ the invariant algebra ; namely, the subalgebra of G -invariant forms. The coinvariant algebra of ρ is the quotient algebra 𝔽[V]G=𝔽[V]/𝔥(G)${\mathbb {F}[V]_G = \mathbb {F}[V] / \mathfrak {h}(G) }$ where the Hilbert ideal 𝔥(G)${\mathfrak {h}(G)}$ of G (or better put, of ρ) is the ideal of 𝔽[V]${\mathbb {F}[V]}$ generated by all the homogeneous G -invariant forms of strictly positive degree. The group H acts on 𝔽[V]G${\mathbb {F}[V]_G}$ and the subject of this manuscript is the algebra (𝔽[V]G)H${(\mathbb {F}[V]_G)^H}$ of H -invariants of 𝔽[V]G${\mathbb {F}[V]_G}$.

In this paper, we give a criterion of boundedness for the operators of convolution type on Triebel–Lizorkin spaces. As applications, under rather weak size conditions, the bounds for the singular integrals along certain compound surfaces on the above function spaces are obtained. Some previous results are improved and extended.

In a previous paper, Broussous and the author prove that for symmetric spaces of the form G ( E )/ G ( F ), where G is a reductive group defined and split over a p -adic field F and E is an unramified quadratic extension of F , when the residual field of F is large enough, the Steinberg representation is ε-distinguished for a unique quadratic character ε of G ( F ). This proves a conjecture stated by Dipendra Prasad in 2001, under the assumption that ε is the same character as the character χ used in the statement of that conjecture. This article completes the proof of Prasad's conjecture in the unramified case assuming that G is split over F , by proving that we actually have ε=χ${\varepsilon =\chi }$, and also extends the aforementioned distinguishedness result by removing the condition on the residual field.

Perturbations of super Poincaré and weak Poincaré inequalities for Lévy type Dirichlet forms are studied. When the range of jumps is finite, our results are natural extensions to the corresponding ones derived earlier for diffusion processes; and we show that the study for the situation with infinite range of jumps is essentially different. Some examples are presented to illustrate the optimality of our results.

For any field K and for a completely arbitrary graph E , we characterize the Leavitt path algebras L K ( E ) that are indecomposable (as a direct sum of two-sided ideals) in terms of the underlying graph. When the algebra decomposes, it actually does so as a direct sum of Leavitt path algebras for some suitable graphs. Under certain finiteness conditions, a unique indecomposable decomposition exists.

Let R be a commutative ring of characteristic zero and G be an arbitrary group. In the present paper we classify the groups G for which the set of symmetric elements with respect to the classical involution of the group ring RG is Lie metabelian.

On a metric measure space ( M ,ρ,μ) we consider a family of Lipschitz–Besov spaces Λp,qs${\Lambda _{p,q}^{s}}$ that is defined only using the metric ρ and measure μ, and a family of Besov spaces Bp,qs${B_{p,q}^{s}}$ that is defined using an auxiliary self-adjoint operator L and the associated heat semigroup. Under certain assumptions about the heat kernel of L , we prove the identity of the two families of the function spaces.

We prove that there is at most one algebraic model for modules over the K(1)${K(1)}$-local sphere at odd primes that retains some monoidal information.

We develop a systematic study of the Schur tensor product both in the category of operator spaces and in that of C * -algebras.

In this paper, we establish the least upper bound of the product B i ( K ) B i ( K * ) for width-integrals of index i of a convex body K . Further, the least upper bound of the product A p , i ( K ) A q , i ( K * ) for mixed width-integrals of index p ( q ) is given.

For groups with a uniform bound on the length of chains of finite subgroups, we study the relationship between the Bredon cohomological dimension for proper actions and the notions of cohomological dimension one obtains by restricting the coefficients of Bredon cohomology to (cohomological) Mackey functors or fixed point functors. We also investigate the closure properties of the class of groups with finite Bredon cohomological dimension for Mackey functors.

The Cremona group of birational transformations of ℙ ℂ 2 acts on the space 𝔽(2) of holomorphic foliations on the complex projective plane. Since this action is not compatible with the natural graduation of 𝔽(2) by the degree, its description is complicated. The fixed points of the action are essentially described by Cantat and Favre in [J. Reine Angew. Math. 561 (2003), 199–235]. In that article we are interested in problems of “aberration of the degree”, that is, pairs (φ,ℱ)$(\phi ,\mathcal {F})$ from Bir (ℙℂ2)×𝔽(2)${\mathrm {Bir}(\mathbb {P}^2_\mathbb {C})\times \mathbb {F}(2)}$ for which degφ*ℱ=degℱ<(degℱ+1)degφ+degφ-2$\deg \phi ^*\mathcal {F}=\deg \mathcal {F}&lt;(\deg \mathcal {F}+1)\deg \phi +\deg \phi -2$, the generic degree of such a pull-back. We introduce the notion of numerical invariance (degφ*ℱ=degℱ${\deg \phi ^*\mathcal {F}=\deg \mathcal {F}}$) and relate it in small degrees to the existence of transversal structure for the considered foliations.

As a dual of the Auslander transpose of modules, we introduce and study the cotranspose of modules with respect to a semidualizing module C . Then using it we introduce n - C -cotorsionfree modules, and show that n - C -cotorsionfree modules possess many dual properties of n -torsionfree modules. In particular, we show that n - C -cotorsionfree modules are useful in characterizing the Bass class and investigating the approximation theory for modules. Moreover, we study n -cotorsionfree modules over Artin algebras and answer negatively an open question of Huang and Huang posed in 2012.

Let G be a finite group and c be an element of ℤ+∪{∞}${\mathbb {Z}^+\cup \lbrace \infty \rbrace }$. A subgroup H of G is said to be c -nilpotent if it is nilpotent and has nilpotency class at most c . A subset X of G is said to be non- c -nilpotent if it contains no two elements x and y such that the subgroup 〈x,y〉${\langle x,y\rangle }$ is c -nilpotent. In this paper we study the quantity ωc(G)${\omega _c({G})}$, defined to be the size of the largest non- c -nilpotent subset of L . In the case that L is a finite group of Lie type, we identify covers of L by c -nilpotent subgroups, and we use these covers to construct large non- c -nilpotent sets in the group L . We prove that for groups L of fixed rank r , there exist constants D r and E r such that DrN≤ω∞(L)≤ErN${D_r N \le \omega _\infty (L) \le E_r N}$, where N is the number of maximal tori in L . In the case of groups L with twisted rank 1, we provide exact formulae for ωc(L)${\omega _c({L})}$ for all c∈ℤ+∪{∞}${c\in \mathbb {Z}^+\cup \lbrace \infty \rbrace }$. If we write q for the level of the Frobenius endomorphism associated with L and assume that q > 5, then ω∞(L)${\omega _\infty ({L})}$ may be expressed as a polynomial in q with coefficients in {0,1}$\lbrace 0,1\rbrace $.