KAM-tori near an analytic elliptic fixed point

We study the accumulation of an elliptic fixed point of a real analytic Hamiltonian by quasi-periodic invariant tori. We show that a fixed point with Diophantine frequency vector \o_0 is always accumulated by invariant complex analytic KAM-tori. Indeed, the following alternative holds: If the Birkhoff normal form of the Hamiltonian at the invariant point satisfies a R\"ussmann transversality condition, the fixed point is accumulated by real analytic KAM-tori which cover positive Lebesgue measure in the phase space (in this part it suffices to assume that \o_0 has rationally independent coordinates). If the Birkhoff normal form is degenerate, there exists an analytic subvariety of complex dimension at least $d+1$ passing through 0 that is foliated by complex analytic KAM-tori with frequency $\omega_0$. This is an extension of previous results obtained in \cite{EFK} to the case of an elliptic fixed point

Network Mutual Information and Synchronization under Time Transformations

We investigate the effect of general time transformations on the phase synchronization (PS) phenomenon and the mutual information rate (MIR) between pairs of nodes in dynamical networks. We demonstrate two important results concerning the invariance of both PS and the MIR. Under time transformations PS can neither be introduced nor destroyed and the MIR cannot be raised from zero. On the other hand, for proper time transformations the timing between the cycles of the coupled oscillators can be largely improved. Finally, we discuss the relevance of our findings for communication in dynamical networks.Comment: 15 p

Towards a method for rigorous development of generic requirements patterns

We present work in progress on a method for the engineering, validation and verification of generic requirements using domain engineering and formal methods. The need to develop a generic requirement set for subsequent system instantiation is complicated by the addition of the high levels of verification demanded by safety-critical domains such as avionics. Our chosen application domain is the failure detection and management function for engine control systems: here generic requirements drive a software product line of target systems. A pilot formal specification and design exercise is undertaken on a small (twosensor) system element. This exercise has a number of aims: to support the domain analysis, to gain a view of appropriate design abstractions, for a B novice to gain experience in the B method and tools, and to evaluate the usability and utility of that method.We also present a prototype method for the production and verification of a generic requirement set in our UML-based formal notation, UML-B, and tooling developed in support. The formal verification both of the structural generic requirement set, and of a particular application, is achieved via translation to the formal specification language, B, using our U2B and ProB tools

Nonstandard smooth realizations of Liouville rotations

Abstract. We augment the C ∞ conjugation approximation method with explicit estimates on the conjugacy map. This allows us to construct ergodic volume-preserving diffeomorphisms measure-theoretically isomorphic to any a priori given Liouville rotation on a variety of manifolds. In the special case of tori the maps can be made uniquely ergodic. Introduction We call a diffeomorphism f of a compact manifold M that preserves a smooth measure µ a smooth realization of an abstract system (X, T, ν) if they are measure-theoretically isomorphic. A diffeomorphism of a compact manifold has finite entropy with respect to any Borel measure. The natural question therefore becomes whether every finite-entropy automorphism of a Lebesgue space has a smooth realization. This problem remains stubbornly intractable and there remain abstract examples that have no known smooth realizations. We seek to find smooth realizations of one of the simplest types of automorphisms: aperiodic automorphisms with pure point spectrum with a group of eigenvalues with a single generator. Such automorphisms are measure-theoretically isomorphic to irrational rotations of the circle. They therefore have a natural smooth realization. We seek smooth realizations on manifolds other than T. Such realizations are called non-standard smooth realizations. We extend the conjugation approximation method of Anosov and Katok [1] to construct non-standard smooth realizations of a given Liouville rotation on T on a variety o

Infinite ergodic theory and Non-extensive entropies

We bring into account a series of result in the infinite ergodic theory that we believe that they are relevant to the theory of non-extensive entropie

Cerebral Venous Sinus Thrombosis (CVST): Long-Term Single-Center Experience

CVST is a rare location of thrombosis involving Dural/ cerebral venous sinuses. It affects around 5-10 people per million population annually. It is an uncommon but life-threatening form of stroke affecting younger individuals. Therefore, identifying and treating in a timely manner is critical. Rarer thrombotic disorders like paroxysmal nocturnal hemoglobinuria (PNH) or Janus Kinase 2 (JAK2) mutation positive myeloproliferative neoplasms (MPN) can rarely present with CVST. It can also present during pregnancy for the first time. Diagnosis is often established by Computed Tomography (CT) or Magnetic Resonance Imaging (MRI). Infections, certain medication use (asparaginase or birth control pills) could lead to CVST. Patients often present with headaches, seizures or neurological deficits. Management is often with systemic anticoagulation despite intraparenchymal hemorrhage. Reducing intracranial pressure by invasive approaches is sometimes needed.https://digitalcommons.unmc.edu/surp2022/1024/thumbnail.jp

Holomorphic linearization of commuting germs of holomorphic maps

Let $f_1, ..., f_h$ be $h\ge 2$ germs of biholomorphisms of \C^n fixing the origin. We investigate the shape a (formal) simultaneous linearization of the given germs can have, and we prove that if $f_1, ..., f_h$ commute and their linear parts are almost simultaneously Jordanizable then they are simultaneously formally linearizable. We next introduce a simultaneous Brjuno-type condition and prove that, in case the linear terms of the germs are diagonalizable, if the germs commutes and our Brjuno-type condition holds, then they are holomorphically simultaneously linerizable. This answers to a multi-dimensional version of a problem raised by Moser.Comment: 24 pages; final version with erratum (My original paper failed to cite the work of L. Stolovitch [ArXiv:math/0506052v2]); J. Geom. Anal. 201

(Non)Invariance of dynamical quantities for orbit equivalent flows

We study how dynamical quantities such as Lyapunov exponents, metric entropy, topological pressure, recurrence rates, and dimension-like characteristics change under a time reparameterization of a dynamical system. These quantities are shown to either remain invariant, transform according to a multiplicative factor or transform through a convoluted dependence that may take the form of an integral over the initial local values. We discuss the significance of these results for the apparent non-invariance of chaos in general relativity and explore applications to the synchronization of equilibrium states and the elimination of expansions

GAMER MRI: Gated-attention mechanism ranking of multi-contrast MRI in brain pathology.

During the last decade, a multitude of novel quantitative and semiquantitative MRI techniques have provided new information about the pathophysiology of neurological diseases. Yet, selection of the most relevant contrasts for a given pathology remains challenging. In this work, we developed and validated a method, Gated-Attention MEchanism Ranking of multi-contrast MRI in brain pathology (GAMER MRI), to rank the relative importance of MR measures in the classification of well understood ischemic stroke lesions. Subsequently, we applied this method to the classification of multiple sclerosis (MS) lesions, where the relative importance of MR measures is less understood. GAMER MRI was developed based on the gated attention mechanism, which computes attention weights (AWs) as proxies of importance of hidden features in the classification. In the first two experiments, we used Trace-weighted (Trace), apparent diffusion coefficient (ADC), Fluid-Attenuated Inversion Recovery (FLAIR), and T1-weighted (T1w) images acquired in 904 acute/subacute ischemic stroke patients and in 6,230 healthy controls and patients with other brain pathologies to assess if GAMER MRI could produce clinically meaningful importance orders in two different classification scenarios. In the first experiment, GAMER MRI with a pretrained convolutional neural network (CNN) was used in conjunction with Trace, ADC, and FLAIR to distinguish patients with ischemic stroke from those with other pathologies and healthy controls. In the second experiment, GAMER MRI with a patch-based CNN used Trace, ADC and T1w to differentiate acute ischemic stroke lesions from healthy tissue. The last experiment explored the performance of patch-based CNN with GAMER MRI in ranking the importance of quantitative MRI measures to distinguish two groups of lesions with different pathological characteristics and unknown quantitative MR features. Specifically, GAMER MRI was applied to assess the relative importance of the myelin water fraction (MWF), quantitative susceptibility mapping (QSM), T1 relaxometry map (qT1), and neurite density index (NDI) in distinguishing 750 juxtacortical lesions from 242 periventricular lesions in 47 MS patients. Pair-wise permutation t-tests were used to evaluate the differences between the AWs obtained for each quantitative measure. In the first experiment, we achieved a mean test AUC of 0.881 and the obtained AWs of FLAIR and the sum of AWs of Trace and ADC were 0.11 and 0.89, respectively, as expected based on previous knowledge. In the second experiment, we achieved a mean test F1 score of 0.895 and a mean AW of Trace = 0.49, of ADC = 0.28, and of T1w = 0.23, thereby confirming the findings of the first experiment. In the third experiment, MS lesion classification achieved test balanced accuracy = 0.777, sensitivity = 0.739, and specificity = 0.814. The mean AWs of T1map, MWF, NDI, and QSM were 0.29, 0.26, 0.24, and 0.22 (p < 0.001), respectively. This work demonstrates that the proposed GAMER MRI might be a useful method to assess the relative importance of MRI measures in neurological diseases with focal pathology. Moreover, the obtained AWs may in fact help to choose the best combination of MR contrasts for a specific classification problem

In Vivo Mapping of Vascular Inflammation Using Multimodal Imaging

Plaque vulnerability to rupture has emerged as a critical correlate to risk of adverse coronary events but there is as yet no clinical method to assess plaque stability in vivo. In the search to identify biomarkers of vulnerable plaques an association has been found between macrophages and plaque stability--the density and pattern of macrophage localization in lesions is indicative of probability to rupture. In very unstable plaques, macrophages are found in high densities and concentrated in the plaque shoulders. Therefore, the ability to map macrophages in plaques could allow noninvasive assessment of plaque stability. We use a multimodality imaging approach to noninvasively map the distribution of macrophages in vivo. The use of multiple modalities allows us to combine the complementary strengths of each modality to better visualize features of interest. Our combined use of Positron Emission Tomography and Magnetic Resonance Imaging (PET/MRI) allows high sensitivity PET screening to identify putative lesions in a whole body view, and high resolution MRI for detailed mapping of biomarker expression in the lesions.Macromolecular and nanoparticle contrast agents targeted to macrophages were developed and tested in three different mouse and rat models of atherosclerosis in which inflamed vascular plaques form spontaneously and/or are induced by injury. For multimodal detection, the probes were designed to contain gadolinium (T1 MRI) or iron oxide (T2 MRI), and Cu-64 (PET). PET imaging was utilized to identify regions of macrophage accumulation; these regions were further probed by MRI to visualize macrophage distribution at high resolution. In both PET and MR images the probes enhanced contrast at sites of vascular inflammation, but not in normal vessel walls. MRI was able to identify discrete sites of inflammation that were blurred together at the low resolution of PET. Macrophage content in the lesions was confirmed by histology.The multimodal imaging approach allowed high-sensitivity and high-resolution mapping of biomarker distribution and may lead to a clinical method to predict plaque probability to rupture