Volume 32, Issue 5

The twin group Tn{T_{n}} is a right-angled Coxeter group generated by n-1{n-1} involutions, and the pure twin group PTn{\mathrm{PT}_{n}} is the kernel of the natural surjection from Tn{T_{n}} onto the symmetric group on n symbols. In this paper, we investigate some structural aspects of these groups. We derive a formula for the number of conjugacy classes of involutions in Tn{T_{n}}, which, quite interestingly, is related to the well-known Fibonacci sequence. We also derive a recursive formula for the number of z -classes of involutions in Tn{T_{n}}. We give a new proof of the structure of Aut⁡(Tn){\operatorname{Aut}(T_{n})} for n≥3{n\geq 3}, and show that Tn{T_{n}} is isomorphic to a subgroup of Aut⁡(PTn){\operatorname{Aut}(\mathrm{PT}_{n})} for n≥4{n\geq 4}. Finally, we construct a representation of Tn{T_{n}} to Aut⁡(Fn){\operatorname{Aut}(F_{n})} for n≥2{n\geq 2}.

We study decompositions of length functions on integral domains as sums of length functions constructed from overrings. We find a standard representation when the integral domain admits a Jaffard family, when it is Noetherian and when it is a Prüfer domains such that every ideal has only finitely many minimal primes. We also show that there is a natural bijective correspondence between singular length functions and localizing systems.

In this paper we discuss coefficient problems for functions in the class 𝒞0⁢(k){{\mathcal{C}}_{0}(k)}. This family is a subset of 𝒞{{\mathcal{C}}}, the class of close-to-convex functions, consisting of functions which are convex in the positive direction of the real axis. Our main aim is to find some bounds of the difference of successive coefficients depending on the fixed second coefficient. Under this assumption we also estimate |an+1|-|an|{|a_{n+1}|-|a_{n}|} and |an|{|a_{n}|}. Moreover, it is proved that Re⁡{an}≥0{\operatorname{Re}\{a_{n}\}\geq 0} for all f∈𝒞0⁢(k){f\in{\mathcal{C}}_{0}(k)}.

We prove a uniqueness type theorem for (weak, total) integrals on a Frobenius cowreath in a monoidal category. When the cowreath is, moreover, pre-Galois, we construct a Morita context relating the subalgebra of coinvariants and a certain wreath algebra. Then we see that the strictness of the Morita context is related to the Galois property of the cowreath and the existence of a weak total integral on it. We apply our results to quasi-Hopf algebras.

Solutions of Rough Differential Equations (RDE) may be defined as paths whose increments are close to an approximation of the associated flow. They are constructed through a discrete scheme using a non-linear sewing lemma. In this article, we show that such solutions also solve a fixed point problem by exhibiting a suitable functional. Convergence then follows from consistency and stability, two notions that are adapted to our framework. In addition, we show that uniqueness and convergence of discrete approximations is a generic property, meaning that it holds excepted for a set of vector fields and starting points which is of Baire first category. At last, we show that Brownian flows are almost surely unique solutions to RDE associated to Lipschitz flows. The later property yields almost sure convergence of Milstein schemes.

Classical Castelnuovo Lemma shows that the number of linearly independent quadratic equations of a nondegenerate irreducible projective variety of codimension c is at most (c+12){{{c+1}\choose{2}}} and the equality is attained if and only if the variety is of minimal degree. Also G. Fano’s generalization of Castelnuovo Lemma implies that the next case occurs if and only if the variety is a del Pezzo variety. Recently, these results are extended to the next case in [E. Park, On hypersurfaces containing projective varieties, Forum Math. 27 2015, 2, 843–875]. This paper is intended to complete the classification of varieties satisfying at least (c+12)-3{{{c+1}\choose{2}}-3} linearly independent quadratic equations. Also we investigate the zero set of those quadratic equations and apply our results to projective varieties of degree ≥2⁢c+1{\geq 2c+1}.

We make the polynomial dependence on the fixed representation π in our previous subconvex bound of L⁢(12,π⊗χ){L(\frac{1}{2},\pi\otimes\chi)} for GL2×GL1{\mathrm{GL}_{2}\times\mathrm{GL}_{1}} explicit, especially in terms of the usual conductor 𝐂⁢(πfin){\mathbf{C}(\pi_{\mathrm{fin}})}. There is no clue that the original choice, due to Michel and Venkatesh, of the test function at the infinite places should be the optimal one. Hence we also investigate a possible variant of such local choices in some special situations.

A generalised Postnikov tower for a space X is a tower of principal fibrations with fibres generalised Eilenberg–MacLane spaces, whose inverse limit is weakly homotopy equivalent to X . In this paper we give a characterisation of a polyhedral product ZK⁢(X,A){Z_{K}(X,A)} whose universal cover either admits a generalised Postnikov tower of finite length, or is a homotopy retract of a space admitting such a tower. We also include p -local and rational versions of the theorem. We end with a group theoretic application.

Throughout this paper we show that under certain conditions the method for describing solvable Leibniz (resp. Lie) algebras with given nilradical by means of non-nilpotent outer derivations of the nilradical is also applicable to the case Leibniz (resp. Lie) superalgebras. Moreover, after having established the general method for Lie and Leibniz superalgebras, we classify all the solvable superalgebras on a very important class of each of them, that is, those with nilradical of maximal nilindex. Note that for (n+m){(n+m)}-dimensional superalgebras this maximal nilindex is n+m-1{n+m-1} in the Lie case and n+m{n+m} in Leibniz.

We generalize the Alvis–Curtis duality to the abstract representations of reductive groups with Frobenius maps. Similar to the case of representations of finite reductive groups, we show that the Alvis–Curtis duality of infinite type, which we define in this paper, also interchanges the irreducible representations in the principal representation category.

We introduce the concept of a partial abstract kernel associated to a group G and a semilattice of groups A and relate the partial cohomology group H3⁢(G,C⁢(A)){H^{3}(G,C(A))} with the obstructions to the existence of admissible extensions of A by G which realize the given abstract kernel. We also show that if such extensions exist, then they are classified by H2⁢(G,C⁢(A)){H^{2}(G,C(A))}.

We deal with instanton bundles on the product ℙ1×ℙ2{\mathbb{P}^{1}\times\mathbb{P}^{2}} and the blow up of ℙ3{\mathbb{P}^{3}} along a line. We give an explicit construction leading to instanton bundles. Moreover, we also show that they correspond to smooth points of a unique irreducible component of their moduli space.

Let ℒ=-Δ𝔾+V{\mathcal{L}=-{\Delta}_{\mathbb{G}}+V} be a Schrödinger operator on the stratified Lie group 𝔾{\mathbb{G}}, where Δ𝔾{{\Delta}_{\mathbb{G}}} is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class Bq0{B_{q_{0}}} with q0>𝒬/2{q_{0}>\mathcal{Q}/2} and 𝒬{\mathcal{Q}} is the homogeneous dimension of 𝔾{\mathbb{G}}. In this article, by Campanato type spaces Λℒα⁢(𝔾){\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{G})}, we introduce Hardy type spaces associated with ℒ{\mathcal{L}} denoted by Hℒp⁢(𝔾){H^{{p}}_{\vphantom{\varepsilon}{\mathcal{L}}}(\mathbb{G})} and prove the atomic characterization of Hℒp⁢(𝔾){H^{{p}}_{\vphantom{\varepsilon}{\mathcal{L}}}(\mathbb{G})}. Further, we obtain the following duality relation: Λℒ𝒬⁢(1/p-1)⁢(𝔾)=(Hℒp⁢(𝔾))∗,𝒬/(𝒬+δ)<p<1 for⁢δ=min⁡{1,2-𝒬/q0}.\Lambda_{\mathcal{L}}^{\mathcal{Q}(1/p-1)}(\mathbb{G})=(H^{{p}}_{\vphantom{% \varepsilon}{\mathcal{L}}}(\mathbb{G}))^{\ast},\quad\mathcal{Q}/(\mathcal{Q}+% \delta)<p<1\quad\text{for}\ \delta=\min\{1,2-\mathcal{Q}/q_{0}\}. The above relation enables us to characterize Λℒα⁢(𝔾){\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{G})} via two families of Carleson measures generated by the heat semigroup and the Poisson semigroup, respectively. Also, we obtain two classes of perturbation formulas associated with the semigroups related to ℒ{\mathcal{L}}. As applications, we obtain the boundedness of the Littlewood–Paley function and the Lusin area function on Λℒα⁢(𝔾){\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{G})}.