We characterize all bounded Hankel operators $\unicode[STIX]{x1D6E4}$ such that $\unicode[STIX]{x1D6E4}^{\ast }\unicode[STIX]{x1D6E4}$ has finite spectrum. We identify spectral data corresponding to such operators and construct inverse spectral theory including the characterization of these spectral data.

The goal of this article is to study the coaugmented curved A∞-coalgebra structure of the Koszul codual of a filtered dg algebra over a field k. More precisely, we first extend one result of B. Keller that allowed to compute the A∞-coalgebra structure of the Koszul codual of a nonnegatively graded connected algebra to the case of any unitary dg algebra provided with a nonnegative increasing filtration whose zeroth term is k. We then show how to compute the coaugmented curved A∞-coalgebra structure of the Koszul codual of a Poincaré-Birkhoff-Witt (PBW) deformation of an N-Koszul algebra.

We consider a meromorphic family of endomorphisms of degree at least 2 of a complex projective space that is parameterized by the unit disk. We prove that the measure of maximal entropy of these endomorphisms converges to the equilibrium measure of the associated non-Archimedean dynamical system when the system degenerates. The convergence holds in the hybrid space constructed by Berkovich and further studied by Boucksom and Jonsson. We also infer from our analysis an estimate for the blow-up of the Lyapunov exponent near a pole in one-dimensional families of endomorphisms.

Some congruences on conjectures of van Hamme are established. These results give partial answers to some of Swisher's conjectures.

We study a class of parabolic equations which can be viewed as a generalized mean curvature flow acting on cylindrically symmetric surfaces with a Dirichlet condition on the boundary. We prove the existence of a unique solution by means of an approximation scheme. We also develop the theory of asymptotic stability for solutions of general parabolic problems.

We prove that one-parameter families of real germs of conformal diffeomorphisms tangent to the involution x ↦−x are rigid in the parameter. We establish a connection between the dynamics in the Poincaré and Siegel domains. Although repeatedly employed in the literature, the dynamics in the Siegel domain does not explain the intrinsic real properties of these germs. Rather, these properties are fully elucidated in the Poincaré domain, where the fixed points are linearizable. However, a detailed study of the dynamics in the Siegel domain is of crucial importance. We relate both points of view on the intersection of the Siegel normalization domains.

We prove an inverse theorem for the Gowers $U^{2}$-norm for maps $G\rightarrow {\mathcal{M}}$ from a countable, discrete, amenable group $G$ into a von Neumann algebra ${\mathcal{M}}$ equipped with an ultraweakly lower semi-continuous, unitarily invariant (semi-)norm $\Vert \cdot \Vert$. We use this result to prove a stability result for unitary-valued $\unicode[STIX]{x1D700}$-representations $G\rightarrow {\mathcal{U}}({\mathcal{M}})$ with respect to $\Vert \cdot \Vert$.

We are interested in the Korteweg–de Vries (KdV), Burgers and Whitham limits for a spatially periodic Boussinesq model with non-small contrast. We prove estimates of the relations between the KdV, Burgers and Whitham approximations and the true solutions of the original system that guarantee these amplitude equations make correct predictions about the dynamics of the spatially periodic Boussinesq model over their natural timescales. The proof is based on Bloch wave analysis and energy estimates and is the first justification result of the KdV, Burgers and Whitham approximations for a dispersive partial differential equation posed in a spatially periodic medium of non-small contrast.

In this paper we are concerned with the well-posedness and the exponential stabilization of the generalized Korteweg–de Vries–Burgers equation, posed on the whole real line, under the effect of a damping term. Both problems are investigated when the exponent p in the nonlinear term ranges over the interval [1, 5). We first prove the global well-posedness in Hs(ℝ) for 0 ≤ s ≤ 3 and 1 ≤ p < 2, and in H3(ℝ) when p ≥ 2. For 2 ≤ p < 5, we prove the existence of global solutions in the L2-setting. Then, by using multiplier techniques and interpolation theory, the exponential stabilization is obtained with an indefinite damping term and 1 ≤ p < 2. Under the effect of a localized damping term the result is obtained when 2 ≤ p < 5. Combining multiplier techniques and compactness arguments, we show that the problem of exponential decay is reduced to proving the unique continuation property of weak solutions. Here, the unique continuation is obtained via the usual Carleman estimate.

We show that under the Eikonal abrasion model, prescribing uniform normal speed in the direction of the inward surface normal, the isoperimetric quotient of a convex shape is decreasing monotonically.

Several Lebesgue-type decomposition theorems in analysis have a strong relation to the operation called the parallel sum. The aim of this paper is to investigate this relation from a new point of view. Namely, using a natural generalization of Arlinskii's approach (which identifies the singular part as a fixed point of a single-variable map) we prove the existence of a Lebesgue-type decomposition for non-negative sesquilinear forms. As applications, we also show how this approach can be used to derive analogous results for representable functionals, non-negative finitely additive measures, and positive definite operator functions. The focus is on the fact that each theorem can be proved with the same completely elementary method.

A new criterion for local invertibility of slice regular quaternionic functions is obtained. This paper is motivated by the need to find a geometrical interpretation for analytic conditions on the real Jacobian associated with a slice regular function f. The criterion involves spherical and Cullen derivatives of f and gives rise to several geometric implications, including an application to related conformality properties.

Given a smooth compact surface without focal points and of higher genus, it is shown that its geodesic flow is semi-conjugate to a continuous expansive flow with a local product structure such that the semi-conjugation preserves time parametrization. It is concluded that the geodesic flow has a unique measure of maximal entropy.

Let π: X → ℙn be the d-cyclic covering branched along a smooth hypersurface Y ⊂ ℙn of degree d, 3 ≤ d ≤ n. We identify the minimal rational curves on X with d-tangent lines of Y and describe the scheme structure of the variety of minimal rational tangents 𝒞x ⊂ ℙTx(X) at a general point x ∈ X. We also show that the projective isomorphism type of 𝒞x varies in a maximal way as x moves over general points of X.

This note is a report on the observation that the Fano–Enriques threefolds with terminal cyclic quotient singularities admit Calabi–Yau threefolds as their double coverings. We calculate the invariants of those Calabi–Yau threefolds when the Picard number is one. It turns out that all of them are new examples.

We study almost prime solutions of systems of Diophantine equations in the Birch setting. Previous work shows that there exist integer solutions of size $B$ with each component having no prime divisors below $B^{1/u}$, where $u$ equals $c_{0}n^{3/2}$, $n$ is the number of variables and $c_{0}$ is a constant depending on the degree and the number of equations. We improve the polynomial growth $n^{3/2}$ to the logarithmic $(\log n)(\log \log n)^{-1}$. Our main new ingredients are the generalization of the Brüdern–Fouvry vector sieve in any dimension and the incorporation of smooth weights into the Davenport–Birch version of the circle method.

Assuming a conjecture on distinct zeros of Dirichlet $L$-functions we get asymptotic results on the average number of representations of an integer as the sum of two primes in arithmetic progression. On the other hand the existence of good error terms gives information on the location of zeros of $L$-functions. Similar results are obtained for an integer in a congruence class expressed as the sum of two primes.

This issue is dedicated to nonsmooth dynamics, particularly the widening applications where dynamics is modelled by nonsmooth differential equations. The modeling of electrical or mechanical switches as nonsmooth can be traced throughout the (at least) 90-year history in which nondifferentiable terms, such as sign or step functions, have been turning up in differential equations. In recent decades, nonsmooth models have found increasing use in areas like contact mechanics, climate modeling, and the life sciences, among others, with a wealth of new theory and novel dynamical phenomena discovered along the way. Our aim here is to give just a partial snapshot of the current landscape of research topics in the field. We open with Paul Glendinning's article extending a classic phenomenon of nonlinear dynamics — a form of chaos introduced by Leonid Pavlovich Shilnikov — to nonsmooth systems. The scenario introduces a fundamental notion of nonsmooth dyamics, that of trajectories (in this case the crucial homoclinic orbit) that can ‘slide’ along a discontinuity. Shilnikov's scenario is moreover shown to occur naturally as an equilibrium hits a discontinuity, helping with a fundamental yet complex problem, namely that of extending the notion of boundary equilibrium bifurcations beyond systems of two variables.

We study the decompositions of Hilbert schemes induced by the Schubert cell decomposition of the Grassmannian variety and show that Hilbert schemes admit a stratification into locally closed subschemes along which the generic initial ideals remain the same. We give two applications. First, we give completely geometric proofs of the existence of the generic initial ideals and of their Borel fixed properties. Second, we prove that when a Hilbert scheme of non-constant Hilbert polynomial is embedded by the Grothendieck–Plücker embedding of a high enough degree, it must be degenerate.