Triangular algebras, and maximal triangular algebras in particular, have been objects of interest for over 50 years. Rich families of examples have been studied in the context of many w*- and C*-algebras, but there remains a dearth of concrete examples in $B({\cal H})$. In previous work, we described a family of maximal triangular algebras of finite multiplicity. Here, we investigate a related family of maximal triangular algebras with infinite multiplicity, and unearth a new asymptotic structure exhibited by these algebras.

The algebra of all Dirichlet series that are uniformly convergent in the half-plane of complex numbers with positive real part is investigated. When it is endowed with its natural locally convex topology, it is a non-nuclear Fréchet Schwartz space with basis. Moreover, it is a locally multiplicative algebra but not a Q-algebra. Composition operators on this space are also studied.

We consider a nonlinear Robin problem driven by a non-homogeneous differential operator plus an indefinite potential term. The reaction function is Carathéodory with arbitrary growth near±∞. We assume that it is odd and exhibits a concave term near zero. Using a variant of the symmetric mountain pass theorem, we establish the existence of a sequence of distinct nodal solutions which converge to zero.

We introduce full diffeomorphism-invariant Colombeau algebras with added ε-dependence in the basic space. This unites the full and special settings of the theory into one single framework. Using locality conditions we find the appropriate definition of point values in full Colombeau algebras and show that special generalized points suffice to characterize elements of full Colombeau algebras. Moreover, we specify sufficient conditions for the sheaf property to hold and give a definition of the sharp topology in this framework.

The Weierstrass function σ(u) associated with an elliptic curve can be generalized in a natural way to an entire function associated with a higher genus algebraic curve. This generalized multivariate sigma function has been investigated since the pioneering work of Felix Klein. The present paper shows Hurwitz integrality of the coefficients of the power series expansion around the origin of the higher genus sigma function associated with a certain plane curve, which is called an (n, s)-curve or a plane telescopic curve. For the prime (2), the expansion of the sigma function is not Hurwitz integral, but its square is. This paper clarifies the precise structure of this phenomenon. In Appendix A, computational examples for the trigonal genus 3 curve ((3, 4)-curve) y3 + (μ1x + μ4)y2 + (μ2x2 + μ5x + μ8)y = x4 + μ3x3 + μ6x2 + μ9x + μ12 (where μj are constants) are given.

We are concerned with the following Kirchhoff-type equation

$$ - \varepsilon ^2M\left( {\varepsilon ^{2 - N}\int_{{\open R}^N} {\vert \nabla u \vert^2{\rm d}x} } \right)\Delta u + V(x)u = f(u),\quad x \in {{\open R}^N},\quad N{\rm \ges }2,$$

We consider abstract Sobolev spaces of Bessel-type associated with an operator. In this work, we pursue the study of algebra properties of such functional spaces through the corresponding semigroup. As a follow-up to our previous work, we show that by making use of the property of a ‘carré du champ’ identity, this algebra property holds in a wider range than previously shown.

In the following note, we focus on the problem of existence of continuous solutions vanishing at infinity to the equation div v = f for f ∈ Ln(ℝn) and satisfying an estimate of the type ||v||∞ ⩽ C||f||n for any f ∈ Ln(ℝn), where C > 0 is related to the constant appearing in the Sobolev–Gagliardo–Nirenberg inequality for functions with bounded variation (BV functions).

In this paper we define B-Fredholm elements in a Banach algebra A modulo an ideal J of A. When a trace function is given on the ideal J, it generates an index for B-Fredholm elements. In the case of a B-Fredholm operator T acting on a Banach space, we prove that its usual index ind(T) is equal to the trace of the commutator [T, T0], where T0 is a Drazin inverse of T modulo the ideal of finite rank operators, extending Fedosov's trace formula for Fredholm operators (see Böttcher and Silbermann [Analysis of Toeplitz operators, 2nd edn (Springer, 2006)]. In the case of a primitive Banach algebra, we prove a punctured neighbourhood theorem for the index.

An A1−A∞ estimate, improving on a previous result for [b, TΩ] with $\Omega \in L^{infty}({\open S}^{n - 1})$ and b∈BMO is obtained. A new result in terms of the A∞ constant and the one supremum Aq−A∞exp constant is also proved, providing a counterpart for commutators of the result obtained by Li. Both of the preceding results rely upon a sparse domination result in terms of bilinear forms, which is established using techniques from Lerner.

This work is motivated by the question of whether there are spaces X for which the Farber–Grant symmetric topological complexity TCS(X) differs from the Basabe–González–Rudyak–Tamaki symmetric topological complexity TCΣ(X). For a projective space ${\open R}\hbox{P}^m$, it is known that $\hbox{TC}^S ({\open R}\hbox{P}^{m})$ captures, with a few potential exceptional cases, the Euclidean embedding dimension of ${\open R}\hbox{P}^{m}$. We now show that, for all m≥1, $\hbox{TC}^{\Sigma}({\open R}\hbox{P}^{m})$ is characterized as the smallest positive integer n for which there is a symmetric ${\open Z}_{2}$-biequivariant map Sm×Sm→Sn with a ‘monoidal’ behaviour on the diagonal. This result thus lies at the core of the efforts in the 1970s to characterize the embedding dimension of real projective spaces in terms of the existence of symmetric axial maps. Together with Nakaoka's description of the cohomology ring of symmetric squares, this allows us to compute both TC numbers in the case of ${\open R}\hbox{P}^{2^{e}}$ for e≥1. In particular, this leaves the torus S1×S1 as the only closed surface whose symmetric (symmetrized) TCS (TCΣ) invariant is currently unknown.

A module is called unit-endoregular if its endomorphism ring is unit-regular. In this paper, we continue the research in unit-endoregular modules. More characterizations of unit-endoregular modules are obtained. As a special case, we show that for an abelian group G, Endℤ(G) is a unit-regular Baer ring if and only if Endℤ(G) is a two-sided extending regular ring. While the class of unit-endoregular modules is not closed under direct sums, we provide a characterization when there are direct sums of two or more unit-endoregular modules also unit-endoregular under certain conditions. In particular, we investigate unit-endoregular modules which are direct sums of indecomposable modules.

We consider a family of nonlinear rational recurrences of odd order which was introduced by Heideman and Hogan, and recently rediscovered in the theory of Laurent phenomenon algebras (a generalization of cluster algebras). All of these recurrences have the Laurent property, implying that for a particular choice of initial data (all initial values set to 1) they generate an integer sequence. For these particular sequences, Heideman and Hogan gave a direct proof of integrality by showing that the terms of the sequence also satisfy a linear recurrence relation with constant coefficients. Here we present an analogous result for the general solution of each of these recurrences.

For a closed subgroup of a locally compact group the Rieffel induction process gives rise to a C*-correspondence over the C*-algebra of the subgroup. We study the associated Cuntz–Pimsner algebra and show that, by varying the subgroup to be open, compact, or discrete, there are connections with the Exel–Pardo correspondence arising from a cocycle, and also with graph algebras.

For a finite quiver Q without sources, we consider the corresponding radical square zero algebra A. We construct an explicit compact generator for the homotopy category of acyclic complexes of projective A-modules. We call such a generator the projective Leavitt complex of Q. This terminology is justified by the following result: the opposite differential graded endomorphism algebra of the projective Leavitt complex of Q is quasi-isomorphic to the Leavitt path algebra of Qop. Here, Qop is the opposite quiver of Q, and the Leavitt path algebra of Qop is naturally ${\open Z}$-graded and viewed as a differential graded algebra with trivial differential.

We decompose the topological stability (in the sense of P. Walters) into the corresponding notion for points. Indeed, we define a topologically stable point of a homeomorphism f as a point x such that for any C0-perturbation g of f there is a continuous semiconjugation defined on the g-orbit closure of x which tends to the identity as g tends to f. We obtain some properties of the topologically stable points, including preservation under conjugacy, vanishing for minimal homeomorphisms on compact manifolds, the fact that topologically stable chain recurrent points belong to the periodic point closure, and that the chain recurrent set coincides with the closure of the periodic points when all points are topologically stable. Next, we show that the topologically stable points of an expansive homeomorphism of a compact manifold are precisely the shadowable ones. Moreover, an expansive homeomorphism of a compact manifold is topologically stable if and only if every point is topologically stable. Afterwards, we prove that a pointwise recurrent homeomorphism of a compact manifold has no topologically stable points. Finally, we prove that every chain transitive homeomorphism with a topologically stable point of a compact manifold has the pseudo-orbit tracing property. Therefore, a chain transitive expansive homeomorphism of a compact manifold is topologically stable if and only if it has a topologically stable point.

We construct a random model to study the distribution of class numbers in special families of real quadratic fields ${\open Q}(\sqrt d )$ arising from continued fractions. These families are obtained by considering continued fraction expansions of the form $\sqrt {D(n)} = [f(n),\overline {u_1,u_2, \ldots ,u_{s-1} ,2f(n)]} $ with fixed coefficients u1, …, us−1 and generalize well-known families such as Chowla's 4n2 + 1, for which analogous results were recently proved by Dahl and Lamzouri [‘The distribution of class numbers in a special family of real quadratic fields’, Trans. Amer. Math. Soc. (2018), 6331–6356].