Volume 27, Issue 1

Let 𝔽 q be a field of characteristic p with q elements. It is known that the degrees of the irreducible characters of the Sylow p -subgroup of GL(𝔽 q ) are powers of q . On the other hand Sangroniz (2003) showed that this is true for a Sylow p -subgroup of a classical group defined over 𝔽 q if and only if p is odd. For the classical groups of Lie type B , C and D the only bad prime is 2. For the exceptional groups there are others. In this paper we construct irreducible characters for the Sylow p -subgroups of the Chevalley groups D 4 ( q ) with q = 2 f of degree q 3 /2. Then we use an analogous construction for E 6 ( q ) with q = 3 f to obtain characters of degree q 7 /3, and for E 8 ( q ) with q = 5 f to obtain characters of degree q 16 /5. This helps to explain why the primes 2, 3 and 5 are bad for the Chevalley groups of type E in terms of the representation theory of the Sylow p -subgroup.

What sets A ⊂ ℤ n can be written in the form ( K - K ) ∩ ℤ n , where K is a compact subset of ℝ n such that K +ℤ n = ℝ n ? Such sets A are called achievable, and it is known that if A is achievable, then 〈 A 〉 = ℤ n . This condition completely characterizes achievable sets for n = 1, but not much is known for n ≥ 2. We attempt to characterize achievable sets further by showing that with any finite, symmetric set A ⊂ ℤ n containing zero, we may associate a graph 𝒢( A ). Then if A is achievable, we show that the set associated to some connected component of 𝒢( A ) is achievable. In two dimensions, we can strengthen this theorem. Further generalizations and open questions are discussed. Throughout, the language and formalism of algebraic topology are useful.

This paper gives a criterion on the weighted norm estimates of the oscillatory and variation operators for the commutators of Calderón–Zygmund singular integrals in dimension 1. As applications, the weighted L p -boundedness of the oscillation operators and the ρ-variation operators for commutators of the Hilbert transform and the Hermitian Riesz transform are obtained. In addition, the corresponding results for the λ-jump operators and the numbers of up-crossings associated with these operators are also given.

Kropholler and Mislin conjectured that groups acting admissibly on a finite-dimensional G -CW-complex with finite stabilisers admit a finite-dimensional model for E 𝔉 G , the classifying space for proper actions. This conjecture is known to hold for groups with bounded torsion. In this note we consider a large class of groups 𝔘 containing the above and many known examples with unbounded torsion. We show that the conjecture holds for a large subclass of 𝔘.

We prove that iterated Whitehead products of length ( n + 1) vanish in any value of an n -excisive functor in the sense of Goodwillie. We compare then different notions of homotopy nilpotency, from the Berstein–Ganea definition to the Biedermann–Dwyer one. The latter is strongly related to Goodwillie calculus and we analyze the vanishing of iterated Whitehead products in such objects.

We present an infinite series formula based on the Karoubi–Hamida integral, for the universal Borel class evaluated on H 2 n +1 (GL(ℂ)). For a cyclotomic field F we define a canonical set of elements in K 3 ( F ) and present a novel approach (based on a free differential calculus) to constructing them. Indeed, we are able to explicitly construct their images in H 3 (GL(ℂ)) under the Hurewicz map. Applying our formula to these images yields a value V 1 ( F ), which coincides with the Borel regulator R 1 ( F ) when our set is a basis of K 3 ( F ) modulo torsion. For F = ℚ( e 2π i /3 ) a computation of V 1 ( F ) has been made based on our techniques.

Let { R n } n ≥0 be a binary recurrence defined by R n +2 = A R n +1 + B R n ( n ≥ 0), where A , B , R 0 , R 1 are integers and Δ = A 2 + 4 B > 0 is not a perfect square. We show a necessary and sufficient condition for the transcendency of the sum ∑k≥0,Rrk≠-bkakRrk+bk,$ {\sum _{k\ge 0, {R_{r^k}\ne -b_k}}\frac{a_k}{R_{r^k}+b_k}}, $ where r ≥ 2 is an integer and { a k } k ≥0 and { b k } k ≥0 are sequences of algebraic numbers satisfying certain growth conditions. For the proof, we apply the transcendence criterion for Mahler type functions of several variables which gives equivalence between the rationality of the function and the algebraicity of its values at algebraic points.

We consider the following quasi-linear parabolic system of backward partial differential equations on a Banach space E : (∂ t + L ) u + f (·,·, u , A 1/2 ∇ u ) = 0 on [0, T ] × E , u T = φ, where L is a possibly degenerate second-order differential operator with merely measurable coefficients. We solve this system in the framework of generalized Dirichlet forms and employ the stochastic calculus associated to the Markov process with generator L to obtain a probabilistic representation of the solution u by solving the corresponding backward stochastic differential equation. The solution satisfies the corresponding mild equation which is equivalent to being a generalized solution to the PDE. A further main result is the generalization of the martingale representation theorem in infinite dimension using the stochastic calculus associated to the generalized Dirichlet form given by L . The non-linear term f satisfies a monotonicity condition with respect to u and a Lipschitz condition with respect to ∇ u .

For locally convex, nilpotent Lie algebras we construct faithful representations by nilpotent operators on a suitable locally convex space. In the special case of nilpotent Banach–Lie algebras we get norm continuous representations by bounded operators on Banach spaces.

If X is a quasi-compact and quasi-separated scheme, the category Qcoh( X ) of quasi-coherent sheaves on X is locally finitely presented. Therefore categorical flat quasi-coherent sheaves in the sense of Stenström (1968) naturally arise. But there is also the standard definition of flatness in Qcoh( X ) from the stalks. So it makes sense to wonder the relationship (if any) between these two notions. In this paper we show that there are plenty of locally finitely presented categories having no other categorical flats than the zero object. As particular instance, we show that Qcoh(𝐏 n ( R )) has no other categorical flat objects than zero, where R is any commutative ring.

We present a general fixed point theorem which can be seen as the quintessence of the principles of proof for Banach's fixed point theorem, ultrametric and certain topological fixed point theorems. It works in a minimal setting, not involving any metrics. We demonstrate its applications to the metric, ultrametric and topological cases, and to ordered abelian groups and fields.

Corrections to http://dx.doi.org/10.1515/forum-2012-0060.

We study a variant of Erdős–Rényi graph over finite rings. We then use this graph to obtain a Szemerédi–Trotter type theorem a sum-product estimate in finite rings.

We consider a number of linear and non-linear boundary value problems involving generalized Schrödinger equations. The model case is -Δ u = V u for u ∈ W 0 1,2 ( D ) with D a bounded domain in ℝ n . We use the Sobolev embedding theorem, and in some cases the Moser–Trudinger inequality and the Hardy–Sobolev inequality, to 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.

By Li, Miao and Zhang (2009) and Miao, Xu and Zhao (2009), the dynamics of the solutions for the focusing energy-critical Hartree equation have been classified when E ( u 0 ) < E ( W ), where W is the ground state. In this paper, we continue the study on the dynamics of the radial solutions with the threshold energy under the assumption of the nondegeneracy of the linearized operator near ground state. Our arguments closely follow those of Duyckaerts and Merle (2008, 2009), Duyckaerts and Roudenko (2010) and Li and Zhang (2009, 2011). The new ingredient is that we show that the positive solution of the nonlocal elliptic equation in L 2 d /( d -2) (ℝ d ) is regular and unique by the moving plane method in its global form, which plays an important role in the spectral theory of the linearized operator and the dynamics behavior of the threshold solution. Moreover, the nondegeneracy property of the linearized operator can be shown in the Newtonian's potential case (that is, d = 6).

We establish a fixed point property for a certain class of locally compact groups, including almost connected Lie groups and compact groups of finite abelian width, which act by simplicial isometries on finite rank buildings with measurable stabilisers of points. As an application, we deduce amongst other things that all topological one-parameter subgroups of a real or complex Kac–Moody group are obtained by exponentiating ad-locally finite elements of the corresponding Lie algebra.

We address the problem of identifying a Hecke–Maass cusp form f of full level from the central values of the Rankin–Selberg L -functions L (1/2, f ⊗ h ) where h runs through the set of Hecke–Maass eigenforms of full level. We prove a quantitative result in this direction.

We present and discuss methods for embedding a left-ordered group G into the multiplicative group of a skew field D . Among others, we are interested in the case in which any group automorphism of G extends to a ring automorphism of D and we will be able to apply Dubrovin's embedding theorem to a special class of left-ordered groups which we call locally complete.

We establish a conservativeness criterion for symmetric jump-diffusion processes generated by regular Dirichlet forms. Using this criterion, we characterize the conservation property in terms of the volume growth of the underlying measure and the growth of the canonical coefficients.

We construct some irreducible representations of the Leavitt path algebra of an arbitrary quiver. The constructed representations are associated to certain algebraic branching systems. For a row-finite quiver, we classify algebraic branching systems, to which irreducible representations of the Leavitt path algebra are associated. For a certain quiver, we obtain a faithful completely reducible representation of the Leavitt path algebra. The twisted representations of the constructed ones under the scaling action are studied.

This paper is an attempt to find out which properties of a finite group G can be expressed in terms of commutators of elements of coprime orders. A criterion of solubility of G in terms of such commutators is obtained. We also conjecture that every element of a nonabelian simple group is a commutator of elements of coprime orders and we confirm this conjecture for the alternating groups.

We prove Jiang's conjecture (1996) to show the existence of an L-function of a cuspidal representation of GSp(4,𝔸) × GSp(4,𝔸) which has a pole of order 2 at s = 1, even for globally generic representations. Moreover, we show that if a cuspidal representation π of GSp(4,𝔸) comes from GSO(4,𝔸), then π is the Weil transfer of Π 1 ⊗ Π 2 realized as a representation of GSO(4,𝔸) where the Π i are irreducible automorphic cuspidal representations of GL(2,𝔸). This agrees with the Langlands functoriality principle as GSO(4,𝔸) is an endoscopic group for GSp(4,𝔸).

We characterize, in terms of its idempotents, the Leavitt path algebras of an arbitrary graph that satisfies Condition (L) or Condition (NE). In the latter case, we also provide the structure of such algebras. Dual graph techniques are considered and demonstrated to be useful in the approach of the study of Leavitt path algebras of arbitrary graphs. A refining of the so-called Reduction Theorem is achieved and is used to prove that I ( P c ( E )), the ideal of the vertices which are base of cycles without exits of the graph E , a construction with a clear parallelism to the socle, is a ring isomorphism invariant for arbitrary Leavitt path algebras. We also determine its structure in any case.

Given E ⊂ ℝ d , define the volume set of E , 𝒱( E ) = {det( x 1 , x 2 ,..., x d ) : x j ∈ E }. In ℝ 3 , we prove that 𝒱( E ) has positive Lebesgue measure if either the Hausdorff dimension of E ⊂ ℝ 3 is greater than 13/5, or E is a product set of the form E = B 1 × B 2 × B 3 with B j ⊂ ℝ, dim ℋ ( B j ) > 2/3, j = 1,2,3. We show that the same conclusion holds for 𝒱( E ) of Salem subsets E ⊂ ℝ d with dim ℋ ( E ) > d - 1, and give applications to discrete combinatorial geometry.