Vortex Filament Equation for a Regular Polygon in the Hyperbolic Plane

The aim of this paper is twofold. First, we show the evolution of the vortex filament equation (VFE) for a regular planar polygon in the hyperbolic space. Unlike in the Euclidean space, the planar polygon is open and both of its ends grow up exponentially, which makes the problem more challenging from a numerical point of view. However, using a finite difference scheme in space combined with a fourth-order Runge–Kutta method in time and fixed boundary conditions, we show that the numerical solution is in complete agreement with the one obtained by means of algebraic techniques. Second, as in the Euclidean case, we claim that, at infinitesimal times, the evolution of VFE for a planar polygon as the initial datum can be described as a superposition of several one-corner initial data. As a consequence, not only can we compute the speed of the center of mass of the planar polygon, but the relationship also allows us to compare the time evolution of any of its corners with the evolution in the Euclidean case.

Critical nonlocal Schrödinger-Poisson system on the Heisenberg group

In this paper, we are concerned with the following a new critical nonlocal Schrödinger-Poisson system on the Heisenberg group: − a − b ∫ Ω | ∇ H u | 2 d ξ Δ H u + μ ϕ u = λ | u | q − 2 u + | u | 2 u , in Ω , − Δ H ϕ = u 2 , in Ω , u = ϕ = 0 , on ∂ Ω , $$\begin{equation*}\begin{cases} -\left(a-b\int_{\Omega}|\nabla_{H}u|^{2}d\xi\right)\Delta_{H}u+\mu\phi u=\lambda|u|^{q-2}u+|u|^{2}u,\quad &\mbox{in} \, \Omega,\\ -\Delta_{H}\phi=u^2,\quad &\mbox{in}\, \Omega,\\ u=\phi=0,\quad &\mbox{on}\, \partial\Omega, \end{cases} \end{equation*}$$ where Δ H is the Kohn-Laplacian on the first Heisenberg group H 1 $ \mathbb{H}^1 $ , and Ω ⊂ H 1 $ \Omega\subset \mathbb{H}^1 $ is a smooth bounded domain, a, b > 0, 1 < q < 2 or 2 < q < 4, λ > 0 and μ ∈ R $ \mu\in \mathbb{R} $ are some real parameters. Existence and multiplicity of solutions are obtained by an application of the mountain pass theorem, the Ekeland variational principle, the Krasnoselskii genus theorem and the Clark critical point theorem, respectively. However, there are several difficulties arising in the framework of Heisenberg groups, also due to the presence of the non-local coefficient (a − b∫Ω∣∇ H u∣2 dx) as well as critical nonlinearities. Moreover, our results are new even on the Euclidean case.

Fixed angle inverse scattering in the presence of a Riemannian metric

We consider a fixed angle inverse scattering problem in the presence of a known Riemannian metric. First, assuming a no caustics condition, we study the direct problem by utilizing the progressing wave expansion. Under a symmetry assumption on the metric, we obtain uniqueness and stability results in the inverse scattering problem for a potential with data generated by two incident waves from opposite directions. Further, similar results are given using one measurement provided the potential also satisfies a symmetry assumption. This work extends the results of [Rakesh and M. Salo, Fixed angle inverse scattering for almost symmetric or controlled perturbations, SIAM J. Math. Anal. 52 2020, 6, 5467–5499] and [Rakesh and M. Salo, The fixed angle scattering problem and wave equation inverse problems with two measurements, Inverse Problems 36 2020, 3, Article ID 035005] from the Euclidean case to certain Riemannian metrics.

Improved approximate rips filtrations with shifted integer lattices and cubical complexes

Rips complexes are important structures for analyzing topological features of metric spaces. Unfortunately, generating these complexes is expensive because of a combinatorial explosion in the complex size. For n points in $$\mathbb {R}^d$$ R d , we present a scheme to construct a 2-approximation of the filtration of the Rips complex in the $$L_\infty $$ L ∞ -norm, which extends to a $$2d^{0.25}$$ 2 d 0.25 -approximation in the Euclidean case. The k-skeleton of the resulting approximation has a total size of $$n2^{O(d\log k +d)}$$ n 2 O ( d log k + d ) . The scheme is based on the integer lattice and simplicial complexes based on the barycentric subdivision of the d-cube. We extend our result to use cubical complexes in place of simplicial complexes by introducing cubical maps between complexes. We get the same approximation guarantee as the simplicial case, while reducing the total size of the approximation to only $$n2^{O(d)}$$ n 2 O ( d ) (cubical) cells. There are two novel techniques that we use in this paper. The first is the use of acyclic carriers for proving our approximation result. In our application, these are maps which relate the Rips complex and the approximation in a relatively simple manner and greatly reduce the complexity of showing the approximation guarantee. The second technique is what we refer to as scale balancing, which is a simple trick to improve the approximation ratio under certain conditions.

SPECIAL VINBERG CONES

The paper is devoted to the generalization of the Vinberg theory of homogeneous convex cones. Such a cone is described as the set of “positive definite matrices” in the Vinberg commutative algebra ℋn of Hermitian T-matrices. These algebras are a generalization of Euclidean Jordan algebras and consist of n × n matrices A = (aij), where aii ∈ ℝ, the entry aij for i < j belongs to some Euclidean vector space (Vij ; 𝔤) and $$ {a}_{ji}={a}_{ij}^{\ast }=\mathfrak{g}\left({a}_{ij},\cdot \right)\in {V}_{ij}^{\ast } $$ a ji = a ij ∗ = g a ij ⋅ ∈ V ij ∗ belongs to the dual space $$ {V}_{ij}^{\ast }. $$ V ij ∗ . The multiplication of T-Hermitian matrices is defined by a system of “isometric” bilinear maps Vij × Vjk → Vij ; i < j < k, such that |aij ⋅ ajk| = |aij| ⋅ |aik|, alm ∈ Vlm. For n = 2, the Hermitian T-algebra ℋn= ℋ2 (V) is determined by a Euclidean vector space V and is isomorphic to a Euclidean Jordan algebra called the spin factor algebra and the associated homogeneous convex cone is the Lorentz cone of timelike future directed vectors in the Minkowski vector space ℝ1,1⊕ V . A special Vinberg Hermitian T-algebra is a rank 3 matrix algebra ℋ3(V; S) associated to a Clifford Cl(V )-module S together with an “admissible” Euclidean metric 𝔤S.We generalize the construction of rank 2 Vinberg algebras ℋ2(V ) and special Vinberg algebras ℋ3(V; S) to the pseudo-Euclidean case, when V is a pseudo-Euclidean vector space and S = S0 ⊕ S1 is a ℤ2-graded Clifford Cl(V )-module with an admissible pseudo-Euclidean metric. The associated cone 𝒱 is a homogeneous, but not convex cone in ℋm; m = 2; 3. We calculate the characteristic function of Koszul-Vinberg for this cone and write down the associated cubic polynomial. We extend Baez’ quantum-mechanical interpretation of the Vinberg cone 𝒱2 ⊂ ℋ2(V ) to the special rank 3 case.

The Engulfing Property for Sections of Convex Functions on the Heisenberg Group and the Associated Quasi-distance

In this paper we investigate the property of engulfing for H-convex functions defined on the Heisenberg group $${\mathbb {H}^n}$$ H n . Starting from the horizontal sections introduced by Capogna and Maldonado (Proc Am Math Soc 134:3191–3199, 2006) , we consider a new notion of section, called $${\mathbb {H}^n}$$ H n -section, as well as a new condition of engulfing associated to the $${\mathbb {H}^n}$$ H n -sections, for an H-convex function defined in $$\mathbb {H}^n.$$ H n . These sections, that arise as suitable unions of horizontal sections, are dimensionally larger; as a matter of fact, the $${\mathbb {H}^n}$$ H n -sections, with their engulfing property, will lead to the definition of a quasi-distance in $${\mathbb {H}^n}$$ H n in a way similar to Aimar et al. in the Euclidean case (J Fourier Anal Appl 4:377–381, 1998). A key role is played by the property of round H-sections for an H-convex function, and by its connection with the engulfing properties.

Hyperbolic Center of Mass for a System of Particles in a Two-Dimensional Space with Constant Negative Curvature: An Application to the Curved 2-Body Problem

In this article, a simple expression for the center of mass of a system of material points in a two-dimensional surface of Gaussian constant negative curvature is given. By using the basic techniques of geometry, we obtained an expression in intrinsic coordinates, and we showed how this extends the definition for the Euclidean case. The argument is constructive and serves to define the center of mass of a system of particles on the one-dimensional hyperbolic sphere LR1.

Bra-ket wormholes in gravitationally prepared states

We consider two dimensional CFT states that are produced by a gravitational path integral.As a first case, we consider a state produced by Euclidean AdS2 evolution followed by flat space evolution. We use the fine grained entropy formula to explore the nature of the state. We find that the naive hyperbolic space geometry leads to a paradox. This is solved if we include a geometry that connects the bra with the ket, a bra-ket wormhole. The semiclassical Lorentzian interpretation leads to CFT state entangled with an expanding and collapsing Friedmann cosmology.As a second case, we consider a state produced by Lorentzian dS2 evolution, again followed by flat space evolution. The most naive geometry also leads to a similar paradox. We explore several possible bra-ket wormholes. The most obvious one leads to a badly divergent temperature. The most promising one also leads to a divergent temperature but by making a projection onto low energy states we find that it has features that look similar to the previous Euclidean case. In particular, the maximum entropy of an interval in the future is set by the de Sitter entropy.

Approximation of critical regularity functions on stratified homogeneous groups

Let [Formula: see text] be a stratified homogeneous group with homogeneous dimension [Formula: see text] and whose Lie algebra is generated by the left-invariant vector fields [Formula: see text]. Let [Formula: see text], [Formula: see text] and [Formula: see text]. We prove that for any function [Formula: see text] there exists a function [Formula: see text] such that [Formula: see text] [Formula: see text] where [Formula: see text] is the largest integer smaller than [Formula: see text] and [Formula: see text] is a positive constant depending only on [Formula: see text]. Here, [Formula: see text] is a homogeneous Triebel–Lizorkin type space adapted to [Formula: see text]. This generalizes earlier results of Bourgain, Brezis [New estimates for eliptic equations and Hodge type systems, J. Eur. Math. Soc. 9(2) (2007) 277–315] and of Bousquet, Russ, Wang, Yung [Approximation in fractional Sobolev spaces and Hodge systems, J. Funct. Anal. 276(5) (2019) 1430–1478] in the Euclidean case and answers an open problem in the latter reference.