We consider an inhomogeneous XX spin chain which interpolates between the Krawtchouk one with perfect state transfer and the homogeneous XX chain. This model can be used to perform qubit state transfer with sufficiently high fidelity. The advantage of this model with respect to the Krawtchouk chain is that while maintaining high transfer fidelity, the disparity between the largest and the smallest coupling strengths grows much more slowly as the number of sites increases. The construction is fully analytic and is based on spectral surgery transformations of the homogeneous chain. An explicit lower bound on the transfer amplitude is derived; it provides a readout time in closed form and quantifies the trade-off between fidelity and coupling strengths.
We introduce and analyze a two-dimensional variant of Newton’s aerodynamic problem of minimal resistance, where the uniform parallel velocity field is replaced by a non-uniform radial flow emanating from a central source. Under this radial framework, we derive the corresponding resistance functional and characterize its smooth extremals, which are globally expressed as radial graphs and explicitly parametrized by the angle of attack. Furthermore, we investigate the well-posedness of the minimization problem under geometric constraints, analyze the non-existence of non-smooth minimizers via the Weierstrass–Erdmann corner conditions, and prove that the regular extremal curves constitute strong global minimizers within a well-defined domain using classical Weierstrass field theory.
A Poisson line process is a random set of straight lines contained in the plane, as the image of the map (x,v)↦(x+vt)t∈R, for each point (x, v) of a Poisson process in the space-velocity plane. By associating a step with each line of the process, a random surface called multi-time walk field is obtained. The diffusive rescaling of the surface converges to the multi-time Brownian field, a classical Gaussian field also called Lévy-Chentsov field. A cut of the multi-time fields with a perpendicular plane reveals a one dimensional continuous-time random walk and a Brownian motion, respectively. A hard rod is an interval contained in R that travels ballistically until it collides with another hard rod, at which point they interchange positions. By associating each line with the ballistic displacement of a hard rod and associating surface steps with hard rod jumps, we obtain the hydrodynamic limits of the hard rods in the Euler and diffusive scalings. The main tools are law of large numbers and central limit theorems for Poisson processes. When rod sizes are zero we have the dynamics of an ideal gas. We describe the relation between ideal gas and hard-rod invariant measures.
We introduce and develop the notion of directional entropy dimension for measure preserving Z2-actions. We establish a partial variational principle relating the measure theoretic and topological directional entropy dimensions. Using deterministic walks in random sceneries and product constructions, we construct examples showing that every value in the interval [0, 1] can be realized as the directional metric entropy dimension.
We formulate a non-Hermitian extension of the Dirac equation in the conical spacetime of a cosmic string by introducing a non-PT-symmetric scalar interaction. We show that the competition between topology and complex deformation induces a nontrivial spectral structure with geometry-controlled shifts and a radial gain/loss profile. A central result is the emergence of a spatially localized exceptional ring at a finite radius, generated by the competition between conical geometry and non-Hermitian effects. In contrast to conventional exceptional points in parameter space, this structure is intrinsically realized in real space and marks a transition between distinct dynamical regimes. The radial Dirac system is shown to admit an exact decoupling into Schrödinger type equations associated with biconfluent Heun structures. These results establish a direct link between spacetime topology and non-Hermitian spectral properties.
