Volume 27, Issue 2

A compact p -group G ( p prime) is called near abelian if it contains an abelian normal subgroup A such that G / A has a dense cyclic subgroup and that every closed subgroup of A is normal in G . We relate near abelian groups to a class of compact groups, which are rich in permuting subgroups. A compact group is called quasihamiltonian (or modular ) if every pair of compact subgroups commutes setwise. We show that for p ≠ 2 a compact p -group G is near abelian if and only if it is quasihamiltonian. The case p = 2 is discussed separately.

We study the geometry of the birational map between an intersection of a web of quadrics in ℙ 7 that contain a plane and the double octic branched along the discriminant of the web.

We provide new formulas for the coefficients in the partial fraction decomposition of the restricted partition generating function. These techniques allow us to partially resolve a recent conjecture of Sills and Zeilberger. We also describe upcoming work, giving a resolution to Rademacher's conjecture on the asymptotics of these coefficients.

Recently, we developed a method for the study of holonomy properties of non-Riemannian Finsler manifolds and obtained that the holonomy group cannot be a compact Lie group if the Finsler manifold of dimension >2 has non-zero constant flag curvature. The purpose of this paper is to move further, exploring the holonomy properties of projectively flat Finsler manifolds of non-zero constant flag curvature. We prove in particular that projectively flat Randers and Bryant–Shen manifolds of non-zero constant flag curvature have infinite dimensional holonomy group.

In this paper, we first establish some results about weighted Carleson measure, and then apply them to multilinear Fourier multiplier. By using a different method from the one used by Fujita and Tomita (2012), we obtain some weighted result of multilinear multiplier operators by considering the missing parts of their results in the end-point cases such as from L 2 (ω) × L ∞ (ω) × L ∞ (ω) to L 2 (ω), using weighted Carleson measure theory and employing the multilinear interpolation theory.

We present a purely category-theoretic characterization of retracts of Fraïssé limits. For this aim, we consider a natural version of injectivity with respect to a pair of categories (a category and its subcategory). It turns out that retracts of Fraïssé limits are precisely the objects that are injective relatively to such a pair. One of the applications is a characterization of non-expansive retracts of Urysohn's universal metric space.

Classical Castelnuovo's lemma shows that the number of linearly independent quadratic equations of a nondegenerate irreducible projective variety of codimension c is at most c+12${{\binom{c+1}{2}}}$ and the equality is attained if and only if the variety is of minimal degree. Also a generalization of Castelnuovo's lemma by G. Fano implies that the next case occurs if and only if the variety is a del Pezzo variety. For curve case, these results are extended to equations of arbitrary degree respectively by J. Harris and S. L'vovsky. This paper is intended to extend these results to arbitrary dimensional varieties and to the next cases.

In this paper we prove a form of the Zilber's trichotomy conjecture for reducts of algebraically closed valued fields of characteristic 0 which are expansions of the valued vector space structure. We prove first that a non-modular reduct of a nontrivially valued algebraically closed field containing the valued vector space structure defines a non-semilinear curve. Then we show that the expansion of such a reduct by a non-semilinear curve defines multiplication on a nonempty open set.

Fix g a self-dual Hecke–Maass form for SL 3 (ℤ). Let f be a holomorphic newform of prime level q and fixed weight. Conditional on a lower bound for a short sum of squares of Fourier coefficients of f , we prove a subconvexity bound in the q aspect for L ( s , g × f ) at the central point.

We prove an algebraic property of the elements defining Hecke operators on period polynomials associated with modular forms, which implies that the pairing on period polynomials corresponding to the Petersson scalar product of modular forms is Hecke equivariant. As a consequence of this proof, we derive two indefinite theta series identities which can be seen as analogues of Jacobi's formula for the theta series associated with the sum of four squares.

Let s1,⋯,sd$s_1,\dots ,s_d$ be d positive integers and define the multiple t -values of depth d by t(s1,⋯,sd)=∑n1>⋯>nd≥11(2n1-1)s1⋯(2nd-1)sd,$ t(s_1,\dots ,s_d)=\sum _{n_1>\cdots >n_d\ge 1} \frac{1}{(2n_1-1)^{s_1}\cdots (2n_d-1)^{s_d}}, $ which is equal to the multiple Hurwitz-zeta value 2-wζ(s1,⋯,sd;-12,⋯,-12)$2^{-w}\zeta (s_1,\dots ,s_d;-\frac{1}{2},\dots ,-\frac{1}{2})$ where w=s1+⋯+sd$w=s_1+\cdots +s_d$ is called the weight. For d≤n$d\le n$, let T(2n,d)$T(2n,d)$ be the sum of all multiple t -values with even arguments whose weight is 2n$2n$ and whose depth is d . In 2011, Shen and Cai gave formulas for T(2n,d)$T(2n,d)$ for d≤5$d\le 5$ in terms of t(2n)$t(2n)$, t(2)t(2n-2)$t(2)t(2n-2)$ and t(4)t(2n-4)$t(4)t(2n-4)$. In this short note we generalize their results to arbitrary depth by using the theory of symmetric functions established by Hoffman (2012).

We give a complete classification of isomorphism classes of homogeneous holomorphic hermitian principal bundles over an irreducible hermitian symmetric space of noncompact type. This generalizes the results of our earlier work [Kyoto J. Math. 50 (2010), 325–363] where homogeneous holomorphic hermitian principal bundles over the upper half-plane were considered.

In a recent paper, Bruinier and Ono prove that the coefficients of certain weight -1/2$-1/2$ harmonic Maass forms are traces of singular moduli for weak Maass forms. In particular, for the partition function p ( n ), they prove that p(n)=124n-1·∑Pp(αQ),$ p(n)=\frac{1}{24n-1}\cdot \sum P_p(\alpha _Q), $ where P p is a weak Maass form and αQ$\alpha _Q$ ranges over a finite set of discriminant -24n+1$-24n+1$ CM points. Moreover, they show that 6·(24n-1)·Pp(αQ)$6\cdot (24n-1)\cdot P_p(\alpha _Q)$ is always an algebraic integer, and they conjecture that (24n-1)·Pp(αQ)$(24n-1)\cdot P_p(\alpha _Q)$ is always an algebraic integer. Here we prove a general theorem which implies this conjecture as a corollary.

Let L be an elliptic differential operator on a complete connected Riemannian manifold M such that the associated heat kernel has two-sided Gaussian bounds as well as a Gaussian type gradient estimate. Let L(α)${L^{(\alpha )}}$ be the α-stable subordination of L for α∈(1,2)${\alpha \in (1,2)}$. We found some classes 𝕂αγ,β(β,γ∈[0,α))${\mathbb {K}_{\alpha }^{\gamma ,\beta } (\beta ,\gamma \in [0,\alpha ))}$ of time-space functions containing the Kato class, such that for any measurable functions b:[0,∞)×M→TM${b: [0,\infty )\times M\rightarrow TM}$ and c:[0,∞)×M→M${c: [0,\infty )\times M\rightarrow M}$ with |b|,c∈𝕂α1,1${|b|, c\in \mathbb {K}_{\alpha }^{1,1}}$, the operator Lb,c(α)(t,x):=L(α)(x)+〈b(t,x),∇·〉+c(t,x),(t,x)∈[0,∞)×M,$ L_{b,c}^{(\alpha )}(t,x):= L^{(\alpha )}(x)+ \langle b(t,x),\nabla \cdot \,\rangle +c(t,x),\quad (t,x)\in [0,\infty )\times M,\vspace*{-1.42262pt} $ has a unique heat kernel pb,c(α)(t,x;s,y)${p_{b,c}^{(\alpha )}(t,x;s,y)}$, 0≤s<t${0\le s&lt;t}$, x,y∈M${x,y\in M}$, which is jointly continuous and satisfies t-sC{ρ(x,y)∨(t-s)1α}d+α≤pb,c(α)(t,x;s,y)≤C(t-s){ρ(x,y)∨(t-s)1α}d+α$ \frac{t-s}{C\lbrace \rho (x,y)\vee (t-s)^{\frac{1}{\alpha }}\rbrace ^{d+\alpha }}\le p_{b,c}^{(\alpha )}(t,x;s,y)\le \frac{C(t-s)}{\lbrace \rho (x,y)\vee (t-s)^{\frac{1}{\alpha }}\rbrace ^{d+\alpha }}\vspace*{-1.42262pt} $ and |∇xpb,c(α)(t,x;s,y)|≤C(t-s)α-1α{ρ(x,y)∨(t-s)1α}d+α,0≤s<t,x,y∈M,$ \big |\nabla _x p_{b,c}^{(\alpha )}(t,x; s,y)\big |\le \frac{C(t-s)^{\frac{\alpha -1}{\alpha }}}{\lbrace \rho (x,y)\vee (t-s)^{\frac{1}{\alpha }}\rbrace ^{d+\alpha }},\quad 0\le s&lt;t,\; x,y\in M,\vspace*{-1.42262pt} $ for some constant C > 1, where ρ is the Riemannian distance. The estimate of ∇ypb,c(α)${\nabla _yp^{(\alpha )}_{b,c}}$ and the Hölder continuity of ∇xpb,c(α)${\nabla _x p_{b,c}^{(\alpha )}}$ are also considered. The resulting estimates of the gradient and its Hölder continuity are new even in the standard case where L=Δ${L=\Delta }$ on ℝ d and b , c are time-independent.

In this paper, we first introduce the new class of multiple weights Ap→∞${A^\infty _{\vec{p}}}$ which is larger than the class of multiple weights in [Adv. Math. 220 (2009), 1222–1264]. Then, using this class of weights, we study the weighted norm inequalities for certain classes of multilinear operators and their commutators with new BMO functions introduced in [J. Fourier Anal. Appl. 17 (2011), 115–134]. Finally, we show that some multilinear pseudodifferential operators fall within the scope of the theory obtained in this paper.

The invariant balanced Hermitian geometry of nilmanifolds of dimension 6 is described. We prove that the (restricted) holonomy group of the associated Bismut connection reduces to a proper subgroup of SU(3) if and only if the complex structure is abelian. As an application we show that if J is abelian, then any invariant balanced J -Hermitian structure provides solutions of the Strominger system.

Let k be an algebraically closed field. Given an extension A : B of finite-dimensional k -algebras, we establish criteria ensuring that the representation-theoretic notion of polynomial growth is preserved under ascent and descent. These results are then used to show that principal blocks of finite group schemes of odd characteristic are of polynomial growth if and only if they are Morita equivalent to trivial extensions of radical square zero tame hereditary algebras. In that case, the blocks are of domestic representation type and the underlying group schemes are closely related to binary polyhedral group schemes.

We prove a Hörmander type multiplier theorem for multilinear Fourier multipliers with multiple weights. We also give weighted estimates for their commutators with vector BMO functions.

This paper shows a new phenomenon in higher cluster tilting theory. For each positive integer d , we exhibit a triangulated category 𝖢 with the following properties. On the one hand, the d -cluster tilting subcategories of 𝖢 have very simple mutation behaviour: Each indecomposable object has exactly d mutations. On the other hand, the weakly d -cluster tilting subcategories of 𝖢 which lack functorial finiteness can have much more complicated mutation behaviour: For each 0 ≤ ℓ ≤ d - 1, we show a weakly d -cluster tilting subcategory 𝖳ℓ${\mathsf {T}_{\ell }}$ which has an indecomposable object with precisely ℓ mutations. The category 𝖢 is the algebraic triangulated category generated by a ( d + 1)-spherical object and can be thought of as a higher cluster category of Dynkin type A ∞ .

For a fixed ring, different classes of ring epimorphisms and localisation maps are compared. In fact, we provide sufficient conditions for a ring epimorphism to be a universal localisation. Furthermore, we consider recollements induced by some homological ring epimorphisms and investigate whether they yield recollements of derived module categories.

For each n ≥ 1, we define an algebra having many properties that one might expect to hold for a Brauer algebra of type Bn${\rm B}_{n}$. It is defined by means of a presentation by generators and relations. We show that this algebra is a subalgebra of the Brauer algebra of type Dn+1${\rm D}_{n+1}$ and point out a cellular structure in it. This work is a natural sequel to the introduction of Brauer algebras of type Cn${\rm C}_n$, which are subalgebras of classical Brauer algebras of type A2n-1${\rm A}_{2n-1}$ and differ from the current ones for n > 2. A novel feature is the failure of admissible root sets to describe the tops and bottoms of the diagrams corresponding to monomials in the Brauer algebra of type Bn${\rm B}_{n}$; instead of these sets we use extended admissible sets in order to find normal forms for monomials in the algebra.

We consider a number of boundary value problems involving the p -Laplacian. The model case is -Δpu=V|u|p-2u${-\Delta _p u = V|u|^{p-2}u}$ for u∈W01,p(D)${u\in W_0^{1,p}(D)}$ with D a bounded domain in ℝ n . We derive necessary conditions for the existence of nontrivial solutions. These conditions usually involve a lower bound for a product of powers of the norm of V , the measure of D , and a sharp Sobolev constant. In most cases, these inequalities are best possible. Applications to non-linear eigenvalue problems are also discussed.

We study mapping properties of the averaging operator related to the variety V={x∈𝔽qd:Q(x)=0}$ V=\lbrace x\in \mathbb {F}_q^d: Q(x)=0\rbrace $, where Q ( x ) is a nondegenerate quadratic polynomial over a finite field 𝔽 q with q elements. This paper is devoted to eliminating the logarithmic bound proved by Koh and Shen (to appear in Proc. Edinb. Math. Soc. ). As a consequence, we settle down the averaging problems over the quadratic surfaces V in the case when the dimensions d ≥ 4 are even and V contains a d /2-dimensional subspace.

In this paper we study the action of the Fourier–Mukai transform on the Lawson homology of abelian varieties and a Beauville-type eigenspace decomposition of Lawson homology with rational coefficients.