We generalize known results about Hilbertian fields to Hilbertian rings. For example, let R be a Hilbertian ring (e.g. R is the ring of integers of a number field) with quotient field K and let A be an abelian variety over K. Then, for every extension M of K in K(Ator(Ksep)), the integral closure RM of R in M is Hilbertian.

Let M be a smooth compact manifold with boundary. Under some geometric conditions on M, a homotopical model for the pair (M, ∂M) can be recovered from the configuration category of M \ ∂M. The grouplike monoid of derived homotopy automorphisms of the configuration category of M \ ∂M then acts on the homotopical model of (M, ∂M). That action is compatible with a better known homotopical action of the homeomorphism group of M \ ∂M on (M, ∂M).

For a multiplier Hopf algebra pairing 〈A,B〉, we construct a class of group-cograded multiplier Hopf algebras D(A,B), generalizing the classical construction of finite dimensional Hopf algebras introduced by Panaite and Staic Mihai [Isr. J. Math. 158 (2007), 349–365]. Furthermore, if the multiplier Hopf algebra pairing admits a canonical multiplier in M(B⊗A) we show the existence of quasitriangular structure on D(A,B). As an application, some special cases and examples are provided.

We generalize the notion of summable Szlenk index from a Banach space to an arbitrary weak*-compact set. We prove that a weak*-compact set has summable Szlenk index if and only if its weak*-closed, absolutely convex hull does. As a consequence, we offer a new, short proof of a result from Draga and Kochanek [J. Funct. Anal. 271 (2016), 642–671] regarding the behavior of summability of the Szlenk index under c0 direct sums. We also use this result to prove that the injective tensor product of two Banach spaces has summable Szlenk index if both spaces do, which answers a question from Draga and Kochanek [Proc. Amer. Math. Soc. 145 (2017), 1685–1698]. As a final consequence of this result, we prove that a separable Banach space has summable Szlenk index if and only if it embeds into a Banach space with an asymptotic c0 finite dimensional decomposition, which generalizes a result from Odell et al. [Q. J. Math. 59, (2008), 85–122]. We also introduce an ideal norm $\mathfrak{s}$ on the class $\mathfrak{S}$ of operators with summable Szlenk index and prove that $(\mathfrak{S}, \mathfrak{s})$ is a Banach ideal. For 1 ⩽ p ⩽ ∞, we prove precise results regarding the summability of the Szlenk index of an ℓp direct sum of a collection of operators.

In this paper, we study the existence of positive solutions to a semilinear nonlocal elliptic problem with the fractional α-Laplacian on Rn, 0 < α < n. We show that the problem has infinitely many positive solutions in $ {C^\tau}({R^n})\bigcap H_{loc}^{\alpha /2}({R^n}) $. Moreover, each of these solutions tends to some positive constant limit at infinity. We can extend our previous result about sub-elliptic problem to the nonlocal problem on Rn. We also show for α ∊ (0, 2) that in some cases, by the use of Hardy’s inequality, there is a nontrivial non-negative $ H_{loc}^{\alpha /2}({R^n}) $ weak solution to the problem

$$ {( - \Delta )^{\alpha /2}}u(x) = K(x){u^p} \quad {\rm{ in}} \ {R^n}, $$

Le diagrams and Grassmann necklaces both index the collection of positroids in the nonnegative Grassmannian Gr≥0(k, n), but they excel at very different tasks: for example, the dimension of a positroid is easily extracted from its Le diagram, while the list of bases of a positroid is far more easily obtained from its Grassmann necklace. Explicit bijections between the two are, therefore, desirable. An algorithm for turning a Le diagram into a Grassmann necklace already exists; in this note, we give the reverse algorithm.

We show that for any n and q, the number of real conjugacy classes in $ \rm{PGL}(\it{n},\mathbb{F}_q) $ is equal to the number of real conjugacy classes of $ \rm{GL}(\it{n},\mathbb{F}_q) $ which are contained in $ \rm{SL}(\it{n},\mathbb{F}_q) $, refining a result of Lehrer [J. Algebra36(2) (1975), 278–286] and extending the result of Gill and Singh [J. Group Theory14(3) (2011), 461–489] that this holds when n is odd or q is even. Further, we show that this quantity is equal to the number of real conjugacy classes in $ \rm{PGU}(\it{n},\mathbb{F}_q) $, and equal to the number of real conjugacy classes of $ \rm{U}(\it{n},\mathbb{F}_q) $ which are contained in $ \rm{SU}(\it{n},\mathbb{F}_q) $, refining results of Gow [Linear Algebra Appl.41 (1981), 175–181] and Macdonald [Bull. Austral. Math. Soc.23(1) (1981), 23–48]. We also give a generating function for this common quantity.

We investigate questions on the existence of nontrivial solution for a class of the critical Kirchhoff-type problems in Hyperbolic space. By the use of the stereographic projection the problem becomes a singular problem on the boundary of the open ball $B_1(0)\subset \mathbb{R}^n$ Combining a version of the Hardy inequality, due to Brezis–Marcus, with the mountain pass theorem due to Ambrosetti–Rabinowitz are used to obtain the nontrivial solution. One of the difficulties is to find a range where the Palais Smale converges, because our equation involves a nonlocal term coming from the Kirchhoff term.

We give a bound on the H-constants of configurations of smooth curves having transversal intersection points only on an algebraic surface of non-negative Kodaira dimension. We also study in detail configurations of lines on smooth complete intersections $X \subset \mathbb{P}_{\mathbb{C}}^{n + 2}$ of multi-degree d = (d1, …, dn), and we provide a sharp and uniform bound on their H-constants, which only depends on d.

In this paper, Berger spheres are regarded as geodesic spheres with sufficiently big radii in a complex projective space. We characterize such real hypersurfaces by investigating their geodesics and contact structures from the viewpoint of submanifold theory.

The purpose of this paper is to understand lattices of certain subcategories in module categories of representation-finite gentle algebras called tiling algebras, as introduced by Coelho Simões and Parsons. We present combinatorial models for torsion pairs and wide subcategories in the module category of tiling algebras. Our models use the oriented flip graphs and noncrossing tree partitions, previously introduced by the authors, and a description of the extension spaces between indecomposable modules over tiling algebras. In addition, we classify two-term simple-minded collections in bounded derived categories of tiling algebras. As a consequence, we obtain a characterization of c-matrices for any quiver mutation-equivalent to a type A Dynkin quiver.

In Ersoy et al. [J. Algebra481 (2017), 1–11], we have proved that if G is a locally finite group with an elementary abelian p-subgroup A of order strictly greater than p2 such that CG(A) is Chernikov and for every non-identity α ∈ A the centralizer CG(α) does not involve an infinite simple group, then G is almost locally soluble. This result is a consequence of another result proved in Ersoy et al. [J. Algebra481 (2017), 1–11], namely: if G is a simple locally finite group with an elementary abelian group A of automorphisms acting on it such that the order of A is greater than p2, the centralizer CG(A) is Chernikov and for every non-identity α ∈ A the set of fixed points CG(α) does not involve an infinite simple groups then G is finite. In this paper, we improve this result about simple locally finite groups: Indeed, suppose that G is a simple locally finite group, consider a finite non-abelian subgroup P of automorphisms of exponent p such that the centralizer CG(P) is Chernikov and for every non-identity α ∈ P the set of fixed points CG(α) does not involve an infinite simple group. We prove that if Aut(G) has such a subgroup, then G ≅PSLp(k) where char k ≠ p and P has a subgroup Q of order p2 such that CG(P) = Q.

An oft-cited result of Peter Shiu bounds the mean value of a nonnegative multiplicative function over a coprime arithmetic progression. We prove a variant where the arithmetic progression is replaced by a sifted set. As an application, we show that the normalized square roots of −1 (mod m) are equidistributed (mod 1) as m runs through the shifted primes q − 1.

The semigroups of unital extensions of separable C*-algebras come in two flavours: a strong and a weak version. By the unital Ext-groups, we mean the groups of invertible elements in these semigroups. We use the unital Ext-groups to obtain K-theoretic classification of both unital and non-unital extensions of C*-algebras, and in particular we obtain a complete K-theoretic classification of full extensions of UCT Kirchberg algebras by stable AF algebras.

We introduce the class of partially invertible modules and show that it is an inverse category which we call the Picard inverse category. We use this category to generalize the classical construction of crossed products to, what we call, generalized epsilon-crossed products and show that these coincide with the class of epsilon-strongly groupoid-graded rings. We then use generalized epsilon-crossed groupoid products to obtain a generalization, from the group-graded situation to the groupoid-graded case, of the bijection from a certain second cohomology group, defined by the grading and the functor from the groupoid in question to the Picard inverse category, to the collection of equivalence classes of rings epsilon-strongly graded by the groupoid.