Volume 36, Issue 4

The Boidol group is the smallest non- ∗ {\ast} -regular exponential Lie group. It is of dimension 4 and its Lie algebra is an extension of the Heisenberg Lie algebra by the reals with the roots 1 and -1. We describe the C*-algebra of the Boidol group as an algebra of operator fields defined over the spectrum of the group. It is the only connected solvable Lie group of dimension less than or equal to 4 whose group C*-algebra had not yet been determined.

In this article, we study the extent to which an n -dimensional compact flat manifold with the cyclic holonomy group of square-free order may be distinguished by the finite quotients of its fundamental group. In particular, we display a formula for the cardinality of profinite genus of the fundamental group of an n -dimensional compact flat manifold with the cyclic holonomy group of square-free order.

A positive rig is a commutative and unitary semi-ring A such that 1 + x {1+x} is invertible for every x ∈ A {x\in A} . We show that the category of positive rigs shares many properties with that of K -algebras for a (non-algebraically closed) field K . In particular, it is coextensive and, although we do not have an analogue of Hilbert’s basis theorem for positive rigs, we show that every finitely presentable positive rig is a finite direct product of directly indecomposable ones. We also describe free positive rigs as rigs of rational functions with non-negative rational coefficients, and we give a characterization of the positive rigs with a unique maximal ideal.

Let p be an odd prime and let ρ : ℤ / p → GL n ⁡ ( ℤ ) {\rho:\mathbb{Z}/p\rightarrow\operatorname{{GL}}_{n}(\mathbb{Z})} be an action of ℤ / p {\mathbb{Z}/p} on a lattice and let Γ := ℤ n ⋊ ρ ℤ / p {\Gamma:=\mathbb{Z}^{n}\rtimes_{\rho}\mathbb{Z}/p} be the corresponding semidirect product. The torus bundle M := T ρ n × ℤ / p S ℓ {M:=T^{n}_{\rho}\times_{\mathbb{Z}/p}S^{\ell}} over the lens space S ℓ / ℤ / p {S^{\ell}/\mathbb{Z}/p} has fundamental group Γ. When ℤ / p {\mathbb{Z}/p} fixes only the origin of ℤ n {\mathbb{Z}^{n}} , Davis and Lück (2021) compute the L -groups L m 〈 j 〉 ⁢ ( ℤ ⁢ [ Γ ] ) {L^{\langle j\rangle}_{m}(\mathbb{Z}[\Gamma])} and the structure set 𝒮 geo , s ⁢ ( M ) {\mathcal{{S}}^{{\rm geo},s}(M)} . In this paper, we extend these computations to all actions of ℤ / p {\mathbb{Z}/p} on ℤ n {\mathbb{Z}^{n}} . In particular, we compute L m 〈 j 〉 ⁢ ( ℤ ⁢ [ Γ ] ) {L^{\langle j\rangle}_{m}(\mathbb{Z}[\Gamma])} and 𝒮 geo , s ⁢ ( M ) {\mathcal{{S}}^{{\rm geo},s}(M)} in a case where E ¯ ⁢ Γ {\underline{E}\Gamma} has a non-discrete singular set.

In this article, we introduce and explore the notion of topological amenability in the broad setting of (locally compact) semihypergroups. We acquire several stationary, ergodic and Banach algebraic characterizations of the same in terms of convergence of certain probability measures, total variation of convolution with probability measures and translation of certain functionals, as well as the F-algebraic properties of the associated measure algebra. We further investigate the interplay between restriction of convolution product and convolution of restrictions of measures on a sub-semihypergroup. Finally, we discuss and characterize topological amenability of sub-semihypergroups in terms of certain invariance properties attained on the corresponding measure algebra of the parent semihypergroup. This in turn provides us with an affirmative answer to an open question posed by J. Wong in 1980.

Let 𝜅 be an inaccessible cardinal, 𝔘 a universal algebra, and ∼ \sim the equivalence relation on U κ \mathfrak{U}^{\kappa} of eventual equality. From mild assumptions on 𝜅, we give general constructions of E ∈ End ( U κ / ∼ ) \mathcal{E}\in\operatorname{End}(\mathfrak{U}^{\kappa}/{\sim}) satisfying E ∘ E = E \mathcal{E}\circ\mathcal{E}=\mathcal{E} which do not descend from Δ ∈ End ⁡ ( U κ ) \Delta\in\operatorname{End}(\mathfrak{U}^{\kappa}) having small strong supports. As an application, there exists an E ∈ End ( Z κ / ∼ ) \mathcal{E}\in\operatorname{End}(\mathbb{Z}^{\kappa}/{\sim}) which does not come from a Δ ∈ End ⁡ ( Z κ ) \Delta\in\operatorname{End}(\mathbb{Z}^{\kappa}) .

In this paper we characterize Li–Yorke chaotic composition operators on Orlicz spaces. Indeed, some necessary and sufficient conditions are provided for Li–Yorke chaotic composition operator C φ {C_{\varphi}} on the Orlicz space L Φ ⁢ ( μ ) {L^{\Phi}(\mu)} . In some cases we have equivalent conditions for composition operators on Orlicz spaces to be Li–Yorke chaotic. The results of this paper extend similar results in L p {L^{p}} -spaces.

In this article, the post quantum analogue of Sheffer polynomial sequences is introduced using concepts of post quantum calculus. The series representation, recurrence relations, determinant expression and certain other properties of this class are established. Further, the 2D-post quantum-Sheffer polynomials are introduced via generating function and their properties are established. Certain identities and integral representations for the 2D-post quantum-Hermite polynomials, 2D-post quantum-Laguerre polynomials, and 2D-post quantum-Bessel polynomials are also considered.

We prove that the non-separating curve complex of every surface of finite type and genus at least three admits an exhaustion by finite rigid sets.

The moduli space of planar polygons with generic side lengths is a smooth, closed manifold. It is known that these manifolds contain the moduli space of distinct points on the real projective line as an open dense subset. Kapranov showed that the real points of the Deligne–Mumford–Knudson compactification can be obtained from the projective Coxeter complex of type 𝐴 (equivalently, the projective braid arrangement) by iteratively blowing up along the minimal building set. In this paper, we show that these planar polygon spaces can also be obtained from the projective Coxeter complex of type 𝐴 by performing an iterative cellular surgery along a subcollection of the minimal building set. Interestingly, this subcollection is determined by the combinatorial data associated with the length vector called the genetic code.

In this paper we discuss some basic properties of octonionic Bergman and Hardy spaces. In the first part we review some fundamental concepts of the general theory of octonionic Hardy and Bergman spaces together with related reproducing kernel functions in the monogenic setting. We explain how some of the fundamental problems in well-defining a reproducing kernel can be overcome in the non-associative setting by looking at the real part of an appropriately defined para-linear octonion-valued inner product. The presence of a weight factor of norm 1 in the definition of the inner product is an intrinsic new ingredient in the octonionic setting. Then we look at the slice monogenic octonionic setting using the classical complex book structure. We present explicit formulas for the slice monogenic reproducing kernels for the unit ball, the right octonionic half-space and strip domains bounded in the real direction. In the setting of the unit ball we present an explicit sequential characterization which can be obtained by applying the special Taylor series representation of the slice monogenic setting together with particular octonionic calculation rules that reflect the property of octonionic para-linearity.

In this paper, we give some new results on transcendence on algebraic groups. These results extend some previous ones established on commutative or linear algebraic groups to arbitrary algebraic groups in complex and p -adic fields, respectively.

We make explicit Bombieri’s refinement of Gallagher’s log-free “large sieve density estimate near σ = 1 {\sigma=1} ” for Dirichlet L -functions. We use this estimate and recent work of Green to prove that if N ≥ 2 {N\geq 2} is an integer, A ⊆ { 1 , … , N } {A\subseteq\{1,\ldots,N\}} , and for all primes p no two elements in A differ by p - 1 {p-1} , then | A | ≪ N 1 - 10 - 18 {|A|\ll N^{1-10^{-18}}} . This strengthens a theorem of Sárközy.

Given a partial action α of a groupoid G on a ring R , we study the associated partial skew groupoid ring R ⋊ α G {R\rtimes_{\alpha}G} , which carries a natural G -grading. We show that there is a one-to-one correspondence between the G -invariant ideals of R and the graded ideals of the G -graded ring R ⋊ α G {R\rtimes_{\alpha}G} . We provide sufficient conditions for primeness, and necessary and sufficient conditions for simplicity of R ⋊ α G {R\rtimes_{\alpha}G} . We show that every ideal of R ⋊ α G {R\rtimes_{\alpha}G} is graded if and only if α has the residual intersection property. Furthermore, if α is induced by a topological partial action θ, then we prove that minimality of θ is equivalent to G -simplicity of R , topological transitivity of θ is equivalent to G -primeness of R , and topological freeness of θ on every closed invariant subset of the underlying topological space is equivalent to α having the residual intersection property. As an application, we characterize condition (K) for an ultragraph in terms of topological properties of the associated partial action and in terms of algebraic properties of the associated ultragraph algebra.

In this paper, we study the combinatorial relations between the cokernels cok ⁡ ( A n + p ⁢ x i ⁢ I n ) {\operatorname{cok}(A_{n}+px_{i}I_{n})} ( 1 ≤ i ≤ m {1\leq i\leq m} ), where A n {A_{n}} is an n × n {n\times n} matrix over the ring of p -adic integers ℤ p {\mathbb{Z}_{p}} , I n {I_{n}} is the n × n {n\times n} identity matrix and x 1 , … , x m {x_{1},\dots,x_{m}} are elements of ℤ p {\mathbb{Z}_{p}} whose reductions modulo p are distinct. For a positive integer m ≤ 4 {m\leq 4} and given x 1 , … , x m ∈ ℤ p {x_{1},\dots,x_{m}\in\mathbb{Z}_{p}} , we determine the set of m -tuples of finitely generated ℤ p {\mathbb{Z}_{p}} -modules ( H 1 , … , H m ) {(H_{1},\dots,H_{m})} for which ( cok ⁡ ( A n + p ⁢ x 1 ⁢ I n ) , … , cok ⁡ ( A n + p ⁢ x m ⁢ I n ) ) = ( H 1 , … , H m ) (\operatorname{cok}(A_{n}+px_{1}I_{n}),\dots,\operatorname{cok}(A_{n}+px_{m}I_% {n}))=(H_{1},\dots,H_{m}) for some matrix A n {A_{n}} . We also prove that if A n {A_{n}} is an n × n {n\times n} Haar random matrix over ℤ p {\mathbb{Z}_{p}} for each positive integer n , then the joint distribution of cok ⁡ ( A n + p ⁢ x i ⁢ I n ) {\operatorname{cok}(A_{n}+px_{i}I_{n})} ( 1 ≤ i ≤ m {1\leq i\leq m} ) converges as n → ∞ {n\rightarrow\infty} .