Volume 30, Issue 4

We prove that if we are given a set of points 𝒫{\mathcal{P}} and set of lines ℒ{\mathcal{L}} in 𝔽q2{\mathbb{F}_{q}^{2}} such that |𝒫|⁢|ℒ|≳q8/3{|\mathcal{P}||\mathcal{L}|\gtrsim q^{8/3}}, then the set of distinct distances between points from 𝒫{\mathcal{P}} and lines from ℒ{\mathcal{L}} contains a positive proportion of all distances. Using the same techniques, we also obtain a generalization of this result in higher dimensional cases.

For a strict Lie 2-group, we develop a notion of Lie 2-algebra-valued differential forms on Lie groupoids, furnishing a differential graded-commutative Lie algebra equipped with an adjoint action of the Lie 2-group and a pullback operation along Morita equivalences between Lie groupoids. Using this notion, we define connections on principal 2-bundles as Lie 2-algebra-valued 1-forms on the total space Lie groupoid of the 2-bundle, satisfying a condition in complete analogy to connections on ordinary principal bundles. We carefully treat various notions of curvature, and prove a classification result by the non-abelian differential cohomology of Breen–Messing. This provides a consistent, global perspective to higher gauge theory.

The integral group ring ℤ⁢G{\mathbb{Z}G} of a group G has only trivial central units if the only central units of ℤ⁢G{\mathbb{Z}G} are ±z{\pm z} for z in the center of G . We show that the order of a finite solvable group G with this property can only be divisible by the primes 2, 3, 5 and 7, by linking this to inverse semi-rational groups and extending one result on this class of groups. We also classify the Frobenius groups whose integral group rings have only trivial central units.

In [9], Migliore, Miró-Roig and Nagel proved that if R=𝕂⁢[x,y,z]{R=\mathbb{K}[x,y,z]}, where 𝕂{\mathbb{K}} is a field of characteristic zero, and I=(L1a1,…,L4a4){I=(L_{1}^{a_{1}},\dots,L_{4}^{a_{4}})} is an ideal generated by powers of four general linear forms, then the multiplication by the square L2{L^{2}} of a general linear form L induces a homomorphism of maximal rank in any graded component of R/I{R/I}. More recently, Migliore and Miró-Roig proved in [7] that the same is true for any number of general linear forms, as long the powers are uniform. In addition, they conjectured that the same holds for arbitrary powers. In this paper, we will prove that this conjecture is true, that is, we will show that if I=(L1a1,…,Lrar){I=(L_{1}^{a_{1}},\dots,L_{r}^{a_{r}})} is an ideal of R generated by arbitrary powers of any set of general linear forms, then the multiplication by the square L2{L^{2}} of a general linear form L induces a homomorphism of maximal rank in any graded component of R/I{R/I}.

We prove that for certain irrational numbers α, there exist quasi-convex null sequences in the subgroup generated by α+ℤ{\alpha+\mathbb{Z}} in the torus.

It is well known that the set of commutators in a group usually does not form a subgroup. A similar phenomenon occurs for the set of autocommutators. There exists a group of order 64 and nilpotency class 2, where the set of autocommutators does not form a subgroup, and this group is of minimal order with this property. However, for finite abelian groups, the set of autocommutators is always a subgroup. We will show in this paper that this is no longer true for infinite abelian groups. We characterize finitely generated infinite abelian groups in which the set of autocommutators does not form a subgroup and show that in an infinite abelian torsion group the set of commutators is a subgroup. Lastly, we investigate torsion-free abelian groups with finite automorphism group and we study whether the set of autocommutators forms a subgroup in those groups.

This is the first in a series of articles where we will study the Iwasawa theory of an elliptic modular form f along the anticyclotomic ℤp{\mathbb{Z}_{p}}-tower of an imaginary quadratic field K where the prime p splits completely. Our goal in this portion is to prove the Iwasawa main conjecture for suitable twists of f assuming that f is p -ordinary, both in the definite and indefinite setups simultaneously, via an analysis of Beilinson–Flach elements.

When Γ is a row-finite digraph, we classify all finite-dimensional modules of the Leavitt path algebra L⁢(Γ){L(\Gamma)} via an explicit Morita equivalence given by an effective combinatorial (reduction) algorithm on the digraph Γ. The category of (unital) L⁢(Γ){L(\Gamma)}-modules is equivalent to a full subcategory of quiver representations of Γ. However, the category of finite-dimensional representations of L⁢(Γ){L(\Gamma)} is tame in contrast to the finite-dimensional quiver representations of Γ, which are almost always wild.

In the paper, our main aim is to generalize the dual affine quermassintegrals to the Orlicz space. Under the framework of Orlicz dual Brunn–Minkowski theory, we introduce a new affine geometric quantity by calculating the first-order variation of the dual affine quermassintegrals, and call it the Orlicz dual affine quermassintegral. The fundamental notions and conclusions of the dual affine quermassintegrals and the Minkoswki and Brunn–Minkowski inequalities for them are extended to an Orlicz setting, and the related concepts and inequalities of Orlicz dual mixed volumes are also included in our conclusions. The new Orlicz–Minkowski and Orlicz–Brunn–Minkowski inequalities in a special case yield the Orlicz dual Minkowski inequality and Orlicz dual Brunn–Minkowski inequality, which also imply the Lp{L_{p}}-dual Minkowski inequality and Brunn–Minkowski inequality for the dual affine quermassintegrals.

In 1959, Philip Hall introduced the locally finite group 𝒰{\mathcal{U}}, today known as Hall’s universal group. This group is countable, universal, simple, and any two finite isomorphic subgroups are conjugate in 𝒰{\mathcal{U}}. It can explicitly be described as a direct limit of finite symmetric groups. It is homogeneous in the model-theoretic sense since it is the Fraïssé limit of the class of all finite groups. Since its introduction Hall’s group and several natural generalisations have been studied widely. In this article we use a generalisation of Fraïssé’s theory to construct a countable, universal, locally finite semigroup 𝒯{\mathcal{T}}, that arises as a direct limit of finite full transformation semigroups, and has the highest possible degree of homogeneity. We prove that it is unique up to isomorphism among semigroups satisfying these properties. We prove an analogous result for inverse semigroups, constructing a maximally homogeneous universal locally finite inverse semigroup ℐ{\mathcal{I}} which is a direct limit of finite symmetric inverse semigroups (semigroups of partial bijections). The semigroups 𝒯{\mathcal{T}} and ℐ{\mathcal{I}} are the natural counterparts of Hall’s universal group for semigroups and inverse semigroups, respectively. While these semigroups are not homogeneous, they still exhibit a great deal of symmetry. We study the structural features of these semigroups and locate several well-known homogeneous structures within them, such as the countable generic semilattice, the countable random bipartite graph, and Hall’s group itself.

It is shown that the projection lattice of a von Neumann algebra, or more generally every orthomodular lattice X , admits a natural embedding into a group G⁢(X){G(X)} with a lattice ordering so that G⁢(X){G(X)} determines X up to isomorphism. The embedding X↪G⁢(X){X\hookrightarrow G(X)} appears to be a universal (non-commutative) group-valued measure on X , while states of X turn into real-valued group homomorphisms on G⁢(X){G(X)}. The existence of completions is characterized by a generalized archimedean property which simultaneously applies to X and G⁢(X){G(X)}. By an extension of Foulis’ coordinatization theorem, the negative cone of G⁢(X){G(X)} is shown to be the initial object among generalized Baer∗{{}^{\ast}}-semigroups. For finite X , the correspondence between X and G⁢(X){G(X)} provides a new class of Garside groups.

Given measurable functions ϕ, ψ on ℝ+{\mathbb{R}^{+}} and a kernel function k⁢(x,y)≥0{k(x,y)\geq 0} satisfying the Oinarov condition, we study the Hardy operator K⁢f⁢(x)=ψ⁢(x)⁢∫0xk⁢(x,y)⁢ϕ⁢(y)⁢f⁢(y)⁢𝑑y,x>0,Kf(x)=\psi(x)\int_{0}^{x}k(x,y)\phi(y)f(y)\,dy,\quad x>0, between Orlicz–Lorentz spaces ΛXG⁢(w){\Lambda_{X}^{G}(w)}, where f is a measurable function on ℝ+{\mathbb{R}^{+}}. We obtain sufficient conditions of boundedness of K:Λu0G0⁢(w0)→Λu1G1⁢(w1){K:\Lambda_{u_{0}}^{G_{0}}(w_{0})\rightarrow\Lambda_{u_{1}}^{G_{1}}(w_{1})} and K:Λu0G0⁢(w0)→Λu1G1,∞⁢(w1){K:\Lambda_{u_{0}}^{G_{0}}(w_{0})\rightarrow\Lambda_{u_{1}}^{G_{1},\infty}(w_{% 1})}. We also look into boundedness and compactness of K:Λu0p0⁢(w0)→Λu1p1,q1⁢(w1){K:\Lambda_{u_{0}}^{p_{0}}(w_{0})\rightarrow\Lambda_{u_{1}}^{p_{1},q_{1}}(w_{1% })} between weighted Lorentz spaces. The function spaces considered here are quasi-Banach spaces rather than Banach spaces. Specializing the weights and the Orlicz functions, we restore the existing results as well as we achieve new results in the new and old settings.

In this paper, we estimate the shifted convolution sum ∑n⩾1λ1⁢(1,n)⁢λ2⁢(n+h)⁢V⁢(nX),\sum_{n\geqslant 1}\lambda_{1}(1,n)\lambda_{2}(n+h)V\Big{(}\frac{n}{X}\Big{)}, where V is a smooth function with support in [1,2]{[1,2]}, 1⩽|h|⩽X{1\leqslant|h|\leqslant X}, and λ1⁢(1,n){\lambda_{1}(1,n)} and λ2⁢(n){\lambda_{2}(n)} are the n -th Fourier coefficients of SL⁢(3,𝐙){\mathrm{SL}(3,\mathbf{Z})} and SL⁢(2,𝐙){\mathrm{SL}(2,\mathbf{Z})} Hecke–Maass cusp forms, respectively. We prove an upper bound O⁢(X2122+ε){O(X^{\frac{21}{22}+\varepsilon})}, updating a recent result of Munshi.

The classical quadratic Gauss sum can be thought of as an exponential sum attached to a quadratic form on a cyclic group. We introduce an equivariant version of Gauss sum for arbitrary finite quadratic forms, which is an exponential sum twisted by the action of the orthogonal group. We prove that simple arithmetic formulas hold for some basic classes of quadratic forms. In applications, such invariant appears in the dimension formula for certain vector-valued modular forms.

In the present paper, right 2-Engel elements, central automorphisms and commuting automorphisms of Lie algebras will be studied. For this purpose, first the structure of the set of all right 2-Engel elements of a Lie algebra will be examined and then, by taking advantage of it, a number of interesting results about central and commuting automorphisms of Lie algebras will be presented. Finally, a characterization of Lie algebras for which the set of central automorphisms is trivial or the set of commuting automorphisms is trivial will be given.

Answering a question of Pálfy and Pyber, we first prove the following extension of the k⁢(G⁢V){k(GV)}-problem: Let G be a finite group and let A be a coprime automorphism group of G . Then the number of conjugacy classes of the semidirect product G⋊A{G\rtimes A} is at most |G|{\lvert G\rvert}. As a consequence, we verify Brauer’s k⁢(B){k(B)}-conjecture for π-blocks of π-separable groups which was proposed by Y. Liu. This generalizes the corresponding result for blocks of p -solvable groups. We also discuss equality in Brauer’s Conjecture. On the other hand, we construct a counterexample to a version of Olsson’s Conjecture for π-blocks which was also introduced by Liu.