Monge–Ampère of Pac-Man

We show that the Monge–Ampère density of the extremal function $$V_P$$ for a non-convex Pac-Man set $$P\subset {{\mathbb {R}}}^2$$ tends to a finite limit as we approach the vertex p of P along lines but with a value that may vary with the line. On the other hand, along a tangential approach to p, the Monge–Ampère density becomes unbounded. This partially mimics the behavior of the Monge–Ampère density of the union of two quarter disks S of Sigurdsson and Snaebjarnarson (Ann Pol Math 123:481–504, 2019). We also recover their formula for $$V_S$$ by elementary methods.     

A Nash–Moser approach for the Euler–Arnold equations

We study the local well-posedness in the smooth category for a class of Euler equations. A Nash–Moser approach is used to extend, for the case of an invertible elliptic pseudo-differential operator, some results obtained by Escher and Kolev, with the help of some geometric arguments.     

On the persistence and blow up for the generalized two-component Dullin–Gottwald–Holm system

Considered herein is the persistence property of the solutions to the generalized two-component integrable Dullin–Gottwald–Holm system, which was derived from the Euler equation with nonzero constant vorticity in shallow water waves moving over a linear shear flow. Firstly, the persistence properties of the system are investigated in weighted $$L^p$$-spaces for a large class of moderate weights. Then, we establish the new local-in-space blow-up results simplifying and extending earlier blow-up criterion for this system.     

On the Littlewood–Paley Spectrum for Passive Scalar Transport Equations

Journal of Nonlinear Science - We derive time-averaged $$L^1$$ estimates on Littlewood–Paley decompositions for linear advection–diffusion equations. For wave numbers close to the...     

A local Hahn–Banach theorem and its applications

An important consequence of the Hahn–Banach theorem says that on any locally convex Hausdorff topological space X, there are sufficiently many continuous linear functionals to separate points of X. In the paper, we establish a “local” version of this theorem. The result is applied to study the uo-dual of a Banach lattice that was recently introduced in Gao et al. (Positivity 22(3):711–725, 2018). We also provide a simplified approach to the measure-free characterization of uniform integrability established in Kardaras (J Funct Anal 266:1913–1927, 2014).     

Kurdyka–Łojasiewicz Exponent via Inf-projection

Foundations of Computational Mathematics - Kurdyka–Łojasiewicz (KL) exponent plays an important role in estimating the convergence rate of many contemporary first-order methods. In...     

On the topology of metric f – K -contact manifolds

We observe that the class of metric f–K-contact manifolds, which naturally contains that of K-contact manifolds, is closed under forming mapping tori of automorphisms of the structure. We show that the de Rham cohomology of compact metric f–K-contact manifolds naturally splits off an exterior algebra, and relate the closed leaves of the characteristic foliation to its basic cohomology.     

Global existence for the periodic dispersive Hunter–Saxton equation

Monatshefte für Mathematik - In this paper, we study an integrable dispersive Hunter–Saxton equation in periodic domain. Firstly, we establish the local well-posedness of the Cauchy...     

Stability of the wave equations on a tree with local Kelvin–Voigt damping

In this paper we study the stability problem of a tree of elastic strings with local Kelvin–Voigt damping on some of the edges. Under the compatibility condition of displacement and strain and continuity condition of damping coefficients at the vertices of the tree, exponential/polynomial stability are proved. Our results generalize the case of single elastic string with local Kelvin–Voigt damping in Liu and Rao (Z. Angew Math Phys 56:630–644, 2005), Liu and Liu (Z. Angew Math Phys 53:265–280, 2002).     

Note on generalized Archimedes–Borchardt algorithm

Aequationes mathematicae - We investigate convergence and invariance properties of the generalized Archimedes–Borchardt algorithm. The main tool is reducing the problem to an appropriate...     

An improvement of the Diffie–Hellman noncommutative protocol

Designs, Codes and Cryptography - The main purpose of this paper is to propose a new version of the Diffie–Hellman noncommutative key exchange protocol invented in 2000 by Ko, Lee, Cheon,...     

Remarks on Rawnsley’s $$\varvec{\varepsilon }$$ε -function on the Fock–Bargmann–Hartogs domains

In this paper, we mainly study a family of unbounded non-hyperbolic domains in $$\mathbb {C}^{n+m}$$, called Fock–Bargmann–Hartogs domains $$D_{n,m}(\mu )$$ ($$\mu >0$$) which are defined as a Hartogs type domains with the fiber over each $$z\in \mathbb {C}^{n}$$ being a ball of radius $$e^{-\frac{\mu }{2} {\Vert z\Vert }^{2}}$$. The purpose of this paper is twofold. Firstly, we obtain necessary and sufficient conditions for Rawnsley’s $$\varepsilon $$-function $$\varepsilon _{(\alpha ,g)}(\widetilde{w})$$ of $$\big (D_{n,m}(\mu ), g(\mu ;\nu )\big )$$ to be a polynomial in $$\Vert \widetilde{w}\Vert ^2$$, where $$g(\mu ;\nu )$$ is a Kähler metric associated with the Kähler potential $$\nu \mu {\Vert z\Vert }^{2} -\ln (e^{-\mu {\Vert z\Vert }^{2}}-\Vert w\Vert ^2)$$. Secondly, using above results, we study the Berezin quantization on $$D_{n,m}(\mu )$$ with the metric $$\beta g(\mu ;\nu )$$$$(\beta >0)$$.     

Global Algebraic Poincaré–Bendixson Annulus for the van der Pol System

Differential Equations - We construct analytically two algebraic closed curves forming a Poincaré–Bendixson annulus for the van der Pol system for all values of its parameter. The inner...     

Banach spaces in which weakly p -Dunford–Pettis sets are relatively compact

In this paper we introduce p-Dunford–Pettis completely continuous operators and study Banach spaces with the wp-Dunford–Pettis relative compact property (wp-DPrcP). We study the behaviour of p-Dunford–Pettis completely operators on spaces with this property. We give sufficient conditions for spaces of operators to have the wp-DPrcP.     

Compact Difference Schemes for the Multidimensional Klein–Gordon Equation

Differential Equations - Stable compact difference schemes of $$4+2$$ and $$4+4 $$ approximation orders are considered and studied on standard stencils for the multidimensional Klein–Gordon...     

Bregman primal–dual first-order method and application to sparse semidefinite programming

Computational Optimization and Applications - We present a new variant of the Chambolle–Pock primal–dual algorithm with Bregman distances, analyze its convergence, and apply it to the...     

Abelian Categories from Triangulated Categories via Nakaoka–Palu’s Localization

Applied Categorical Structures - The aim of this paper is to provide an expansion of Abe–Nakaoka’s heart construction to the following two different realizations of the module category...     

A note on the Assmus–Mattson theorem for some binary codes

Designs, Codes and Cryptography - We previously proposed the first nontrivial examples of a code having support t-designs for all weights obtained from the Assmus–Mattson theorem and having...     

The Banach–Mazur game and domain theory

Archiv der Mathematik - We prove that player $$\alpha $$ has a winning strategy in the Banach–Mazur game on a space X if and only if X is F-Y countably $$\pi $$-domain representable. We show...     

Markoff–Rosenberger triples and generalized Lucas sequences

Periodica Mathematica Hungarica - We consider the Markoff–Rosenberger equation $$\begin{aligned} ax^2+by^2+cz^2=dxyz \end{aligned}$$ with $$(x,y,z)=(U_i,U_j,U_k)$$ , where $$U_i$$ denotes the...