D-optimal designs for Poisson regression with synergetic interaction effect

We characterize D-optimal designs in the two-dimensional Poisson regression model with synergetic interaction and provide an explicit proof. The proof is based on the idea of reparameterization of the design region in terms of contours of constant intensity. This approach leads to a substantial reduction in complexity as properties of the sensitivity can be treated along and across the contours separately. Furthermore, some extensions of this result to higher dimensions are presented.

On the quantum equivalence of an antisymmetric tensor field with spontaneous Lorentz violation

We present an explicit proof that a minimal model of rank-2 antisymmetric field with spontaneous Lorentz violation and a classically equivalent vector field model are also quantum equivalent by calculating quantum effective actions of both theories. We comment on the issues encountered while checking quantum equivalence in curved spacetime.

The exponential Lie series for continuous semimartingales

We consider stochastic differential systems driven by continuous semimartingales and governed by non-commuting vector fields. We prove that the logarithm of the flowmap is an exponential Lie series. This relies on a natural change of basis to vector fields for the associated quadratic covariation processes, analogous to Stratonovich corrections. The flowmap can then be expanded as a series in compositional powers of vector fields and the logarithm of the flowmap can thus be expanded in the Lie algebra of vector fields. Further, we give a direct explicit proof of the corresponding Chen–Strichartz formula which provides an explicit formula for the Lie series coefficients. Such exponential Lie series are important in the development of strong Lie group integration schemes that ensure approximate solutions themselves lie in any homogeneous manifold on which the solution evolves.

Family of Smale–Williams Solenoid Attractors as Solutions of Differential Equations: Exact Formula and Conjugacy

We show that the family of the Smale–Williams solenoid attractors parameterized by its contraction rate can be characterized as solutions of a set of differential equations. The exact formula describing the attractor can be obtained by solving the differential equations subject to explicitly given initial conditions. Using the formula, we present in this note a simple and explicit proof of the result that the dynamics on the solenoid is topologically conjugate to the shift on the inverse limit space of the expanding map t ↦ mt mod 1 for some integer m ≥ 2 and to a suspension over the adding machine.

On the normal-mode frequency spectrum of kinetic magnetohydrodynamics

This paper presents an explicit proof that, in the kinetic magnetohydrodynamics framework, the squared frequencies of normal-mode perturbations about a static equilibrium are real. This proof is based on a quadratic form for the square-integrable normal-mode eigenfunctions and does not rely on demonstrating operator self-adjointness. The analysis is consistent with the quasineutrality condition without involving any subsidiary constraint to enforce it, and does not require the assumption that all particle orbits be periodic. It applies to Maxwellian equilibria, spatially bounded by either a rigid conducting wall or by a plasma-vacuum interface where the density goes continuously to zero.

Invariant coupling of determinantal measures on sofic groups

To any positive contraction $Q$ on $\ell ^{2}(W)$, there is associated a determinantal probability measure $\mathbf{P}^{Q}$ on $2^{W}$, where $W$ is a denumerable set. Let ${\rm\Gamma}$ be a countable sofic finitely generated group and $G=({\rm\Gamma},\mathsf{E})$ be a Cayley graph of ${\rm\Gamma}$. We show that if $Q_{1}$ and $Q_{2}$ are two ${\rm\Gamma}$-equivariant positive contractions on $\ell ^{2}({\rm\Gamma})$ or on $\ell ^{2}(\mathsf{E})$ with $Q_{1}\leq Q_{2}$, then there exists a ${\rm\Gamma}$-invariant monotone coupling of the corresponding determinantal probability measures witnessing the stochastic domination $\mathbf{P}^{Q_{1}}\preccurlyeq \mathbf{P}^{Q_{2}}$. In particular, this applies to the wired and free uniform spanning forests, which was known before only when ${\rm\Gamma}$ is residually amenable. In the case of spanning forests, we also give a second more explicit proof, which has the advantage of showing an explicit way to create the free uniform spanning forest as a limit over a sofic approximation. Another consequence of our main result is to prove that all determinantal probability measures $\mathbf{P}^{Q}$ as above are $\bar{d}$-limits of finitely dependent processes. Thus, when ${\rm\Gamma}$ is amenable, $\mathbf{P}^{Q}$ is isomorphic to a Bernoulli shift, which was known before only when ${\rm\Gamma}$ is abelian. We also prove analogous results for sofic unimodular random rooted graphs.