Non-Smooth Stochastic Lyapunov Functions With Weak Extension of Viscosity Solutions

This paper proposes a notion of viscosity weak supersolutions to build a bridge between stochastic Lyapunov stability theory and viscosity solution theory. Different from ordinary differential equations, stochastic differential equations can have the origins being stable despite having no smooth stochastic Lyapunov functions (SLFs). The feature naturally requires that the related Lyapunov equations are illustrated via viscosity solution theory, which deals with non-smooth solutions to partial differential equations. This paper claims that stochastic Lyapunov stability theory needs a weak extension of viscosity supersolutions, and the proposed viscosity weak supersolutions describe non-smooth SLFs ensuring a large class of the origins being noisily (asymptotically) stable and (asymptotically) stable in probability. The contribution of the non-smooth SLFs are confirmed by a few examples; especially, they ensure that all the linear-quadratic-Gaussian (LQG) controlled systems have the origins being noisily asymptotically stable for any additive noises

Stabilization by Unbounded-Variation Noises

In this paper, we claim the availability of deterministic noises for stabilization of the origins of dynamical systems, provided that the noises have unbounded variations. To achieve the result, we first consider the system representations based on rough path analysis; then, we provide the notion of asymptotic stability in roughness to analyze the stability for the systems. In the procedure, we also confirm that the system representations include stochastic differential equations; we also found that asymptotic stability in roughness is the same property as uniform almost sure asymptotic stability provided by Bardi and Cesaroni. After the discussion, we confirm that there is a case that deterministic noises are capable of making the origin become asymptotically stable in roughness while stochastic noises do not achieve the same stabilization results.Comment: 22 pages, 5 figure

Merlin-Arthur with efficient quantum Merlin and quantum supremacy for the second level of the Fourier hierarchy

We introduce a simple sub-universal quantum computing model, which we call the Hadamard-classical circuit with one-qubit (HC1Q) model. It consists of a classical reversible circuit sandwiched by two layers of Hadamard gates, and therefore it is in the second level of the Fourier hierarchy. We show that output probability distributions of the HC1Q model cannot be classically efficiently sampled within a multiplicative error unless the polynomial-time hierarchy collapses to the second level. The proof technique is different from those used for previous sub-universal models, such as IQP, Boson Sampling, and DQC1, and therefore the technique itself might be useful for finding other sub-universal models that are hard to classically simulate. We also study the classical verification of quantum computing in the second level of the Fourier hierarchy. To this end, we define a promise problem, which we call the probability distribution distinguishability with maximum norm (PDD-Max). It is a promise problem to decide whether output probability distributions of two quantum circuits are far apart or close. We show that PDD-Max is BQP-complete, but if the two circuits are restricted to some types in the second level of the Fourier hierarchy, such as the HC1Q model or the IQP model, PDD-Max has a Merlin-Arthur system with quantum polynomial-time Merlin and classical probabilistic polynomial-time Arthur.Comment: 30 pages, 4 figure

Stochastic Asymptotic Stabilizers for Deterministic Input-Affine Systems based on Stochastic Control Lyapunov Functions

In this paper, a stochastic asymptotic stabilization method is proposed for deterministic input-affine control systems, which are randomized by including Gaussian white noises in control inputs. The sufficient condition is derived for the diffucion coefficients so that there exist stochastic control Lyapunov functions for the systems. To illustrate the usefulness of the sufficient condition, the authors propose the stochastic continuous feedback law, which makes the origin of the Brockett integrator become globally asymptotically stable in probability.Comment: A preliminary version of this paper appeared in the Proceedings of the 48th Annual IEEE Conference on Decision and Control [14

Unveiling hidden topological phases of a one-dimensional Hadamard quantum walk

Quantum walks, whose dynamics is prescribed by alternating unitary coin and shift operators, possess topological phases akin to those of Floquet topological insulators, driven by a time-periodic field. While there is ample theoretical work on topological phases of quantum walks where the coin operators are spin rotations, in experiments a different coin, the Hadamard operator is often used instead. This was the case in a recent photonic quantum walk experiment, where protected edge states were observed between two bulks whose topological invariants, as calculated by the standard theory, were the same. This hints at a hidden topological invariant in the Hadamard quantum walk. We establish a relation between the Hadamard and the spin rotation operator, which allows us to apply the recently developed theory of topological phases of quantum walks to the one-dimensional Hadamard quantum walk. The topological invariants we derive account for the edge state observed in the experiment, we thus reveal the hidden topological invariant of the one-dimensional Hadamard quantum walk.Comment: 11 pages, 4 figure

Estimated pretreatment hemodynamic prognostic factors of aneurysm recurrence after endovascular embolization.

BACKGROUND:Hemodynamic factors play important roles in aneurysm recurrence after endovascular treatment. OBJECTIVE:Predicting the risk of recurrence by hemodynamic analysis using an untreated aneurysm model is important because such prediction is required before treatment. METHODS:We retrospectively analyzed hemodynamic factors associated with aneurysm recurrence from pretreatment models of five recurrent and five stable posterior communicating artery (Pcom) aneurysms with no significant differences in aneurysm volume, coil packing density, or sizes of the dome, neck, or Pcom. Hemodynamic factors of velocity ratio, flow rate, pressure ratio, and wall shear stress were investigated. RESULTS:Among the hemodynamic factors investigated, velocity ratio and flow rate of the Pcom showed significant differences between the recurrence group and stable group (0.630 ± 0.062 and 0.926 ± 0.051, P= 0.016; 56.4 ± 8.9 and 121.6 ± 6.7, P= 0.008, respectively). CONCLUSIONS:Our results suggest that hemodynamic factors may be associated with aneurysm recurrence among Pcom aneurysms. Velocity and flow rate in the Pcom may be a pretreatment prognostic factor for aneurysm recurrence after endovascular treatment

Combination of high-resolution cone beam computed tomography and metal artefact reduction software: a new image fusion technique for evaluating intracranial stent apposition after aneurysm treatment.

We introduce a new imaging technique to improve visualisation of stent apposition after endovascular treatment of brain aneurysms employing high-resolution cone beam CT and three-dimensional digital subtraction angiography. After performing a stent-assisted coil embolisation of brain aneurysm, the image datasets were processed with a metal artefact reduction software followed by the automated image fusion programmes. Two patients who underwent aneurysm coiling using a Neuroform stent were evaluated. The reconstructed 3D images showed a detailed structure of the stent struts and identified malappositions of the deployed stents. Case 1 showed good apposition on the outer curvature side of the carotid siphon, while the inner curvature side showed prominent malapposition. Case 2, with multiple aneurysms, showed good apposition on both outer and inner curvature sides, although inward prolapse of the struts was observed. This new imaging technique may help evaluate stent apposition after the endovascular aneurysm treatment

Accuracy of Length of Virtual Stents in Treatment of Intracranial Wide-Necked Aneurysms.

Background and purposePrecise stent deployment is important for successful treatment of intracranial aneurysms by stent-assisted coiling (SAC). We evaluated the accuracy of virtual stents generated using commercial stent planning software by comparing the length of virtual and actually deployed intracranial laser cut stents on three-dimensional digital subtraction angiography (3D-DSA) images.MethodsWe retrospectively analyzed the data of 75 consecutive cases of intracranial wide-necked aneurysms treated with the SAC technique using laser cut stents. Based on 3D-DSA images acquired by C-arm CT, stent sizing and placement were intraoperatively simulated by a commercial software application. The difference in length of the stents was estimated by measuring proximal discrepancies between the end points of the virtual and actually deployed stents on fused pre-procedural and post-procedural 3D-DSA images. Discrepancies between distal stent end points were manually minimized. The Kruskal-Wallis test was applied to test whether stent location, type, and length had an effect on difference in length between virtual and real stent.ResultsThe median difference in length between virtual and real stents was 1.58 mm with interquartile range 1.12-2.12 mm. There was no evidence for an effect of stent location (p = 0.23), stent type (p = 0.33), or stent length (p = 0.53) on difference in length between virtual and real stents.ConclusionsStent planning software allows 3D simulation of laser cut stents overlain on 3D-DSA images of vessels and may thus be useful for stent selection and deployment of laser cut stents during stent-assisted coiling of intracranial aneurysms