Singular Part Research Articles

An in-silico comparison of anatomical and substrate based AF ablation using electro and optical flow mapping

Abstract Background The pursuit of improved outcomes for persistent AF may necessitate the utilization of innovative technologies and ablation methods. We hypothesize that combining electrophysiologic phase and optical flow mapping could offer a novel approach to identify ablation sites in persistent AF while minimizing ablation area, thereby enhancing the effectiveness of substrate-based ablation compared to a purely empiric strategy. Objective To compare the efficacy of a novel electro-optic flow-based ablation strategy (EOF) with empiric ablation for persistent AF patients using a virtual cohort of bi-atrial models. Methods An exploratory cohort (n=20) from a CT-derived bi-atrial in silico population was utilized to simulate AF. A pipeline was developed to assess ablation targets based on simulated AF and to automatically connect targets to closest inert boundaries. As part of target assessment, phase singularities (PS) and Horn-Schunck average optical flow maps were computed. An intersection of information assessed through PS and curl of average optical flow was utilized to generate EOF maps. 5 x strategies comprising purely anatomical, PS based and EOF guided ablations were simulated (1. PVI; 2. PVI+mitral/roof lines; 3. PVI+mitral/box line; 4. PS+anatomical; 5. EOF+anatomical). Ablation efficacy was determined by first examining change of rhythm via ablation and second by calculating the amount of tissue area ablated. Results Ablating PVI alone (Method 1) did not result in AF termination in any case. Ablating PVI with mitral or roof lines (Method 2) resulted in moderate outcomes (6 AF term. / 3 conv. to AT) whereas ablating PVI with mitral and box lines (Method 3) resulted in improved outcomes (9 AF term. / 4 conv. to AT). Compared to purely anatomical approaches, ablating PS or EOF targets and combining them with roof, mitral lines (Method 4 and Method 5 respectively) saw marked improvement (16 AF term. / 2 conv. to AT). Mapping based on EOF (Method 5) resulted in smaller ablation targets compared to a PS guided strategy (Method 4) with the same outcome. Conclusion AF termination to ablated tissue area ratio was higher for EOF based ablation targets compared to PS based or empiric ablations. Algorithmic generation of ablation lesion masks based on predicted targets can provide novel insights into ablation outcomes. Combining EOF targets with empiric lesions represents a potential ablation strategy of individualized substrate modification for persistent AF.Study Pipeline

Construction and validation of a risk prediction model for soldiers with frostbite in northeast China: a cross-sectional study

BackgroundOne of the challenges of physical training in extreme condition is frostbite, especially in Northeast China. In this study, we aimed to construct a risk prediction model for frostbite among soldiers in Northeast China, and verify its effect.Methods698 participants were selected via convenience sampling from Northeast China from December 2021 to January 2022 (winter). They were randomly divided into a training set (N = 479) and a testing set (N = 202) in a ratio of 7:3. All participants completed a researcher-made questionnaire on frostbite. The prediction model was constructed through the use of Logistic regression analysis, which was used to predict the independent risk factors for frostbite formation and screen significant indicators. The model’s performance was assessed using the receiver operating characteristic (ROC) curve and decision curve analysis (DCA) to evaluate the prediction efficiency and goodness of fit.ResultsThe incidence of frostbite in the training set was 19.83% (95 people), all of which were first-degree frostbite. Among them, frostbite in multiple parts was the most common (58.95%), followed by singular body parts like hands (24.21%), ears (11.58%) and feet (5.26%). Single factor logistic regression analyses showed that ambient temperature, ambient wind speed, outdoor stationary time, stationary status, and history of frostbite are independent risk factors that affect the occurrence of frostbite. Furthermore, we constructed the frostbite risk prediction model for soldiers in the northeastern region of China. The area under the receiver operating characteristic curve (AUC) for the risk of frostbite in the training set and testing set was 0.816 (95% CI, 0.770 ~ 0.862) and 0.787 (95% CI, 0.713 ~ 0.860), respectively. The Hosmer-Lemeshow test of the model showed χ2 = 11.328 and P = 0.184 (> 0.05). The DCA curve indicated that most of the clinical net benefits of the model are greater than 0, demonstrating good clinical usefulness.ConclusionThe constructed frostbite prediction model can effectively identify soldiers with a higher risk of frostbite. It provided theoretical support for commanders to take preventive measures to reduce the incidence of frostbite among soldiers and was of great clinical guiding significance.

On the Nodal Set of Solutions to a Class of Nonlocal Parabolic Equations

We investigate the local properties, including the nodal set and the nodal properties of solutions to the following parabolic problem of Muckenhoupt-Neumann type: { ∂ t u ¯ − y − a ∇ ⋅ ( y a ∇ u ¯ ) = 0 a m p ; in B 1 + × ( − 1 , 0 ) − ∂ y a u ¯ = q ( x , t ) u a m p ; on B 1 × { 0 } × ( − 1 , 0 ) , \begin{equation*} \begin {cases} \partial _t \overline {u} - y^{-a} \nabla \cdot (y^a \nabla \overline {u}) = 0 \quad &\text { in } \mathbb {B}_1^+ \times (-1,0) \\ -\partial _y^a \overline {u} = q(x,t)u \quad &\text { on } B_1 \times \{0\} \times (-1,0), \end{cases} \end{equation*} where a ∈ ( − 1 , 1 ) a\in (-1,1) is a fixed parameter, B 1 + ⊂ R N + 1 \mathbb {B}_1^+\subset \mathbb {R}^{N+1} is the upper unit half ball and B 1 B_1 is the unit ball in R N \mathbb {R}^N . Our main motivation comes from its relation with a class of nonlocal parabolic equations involving the fractional power of the heat operator H s u ( x , t ) = 1 | Γ ( − s ) | ∫ − ∞ t ∫ R N [ u ( x , t ) − u ( z , τ ) ] G N ( x − z , t − τ ) ( t − τ ) 1 + s d z d τ . \begin{equation*} H^su(x,t) = \frac {1}{|\Gamma (-s)|} \int _{-\infty }^t \int _{\mathbb {R}^N} \left [u(x,t) - u(z,\tau )\right ] \frac {G_N(x-z,t-\tau )}{(t-\tau )^{1+s}} dzd\tau . \end{equation*} We characterise the possible blow-ups and we examine the structure of the nodal set of solutions vanishing with a finite order. More precisely, we prove that the nodal set has at least parabolic Hausdorff codimension one in R N × R \mathbb {R}^N\times \mathbb {R} , and can be written as the union of a locally smooth part and a singular part, which turns out to possess remarkable stratification properties. Moreover, the asymptotic behaviour of general solutions near their nodal points is classified in terms of a class of explicit polynomials of Hermite and Laguerre type, obtained as eigenfunctions to an Ornstein-Uhlenbeck type operator. Our main results are obtained through a fine blow-up analysis which relies on the monotonicity of an Almgren-Poon type quotient and some new Liouville type results for parabolic equations, combined with more classical results including Federer’s reduction principle and the parabolic Whitney’s extension.

New insights on the signature change via the Colombeau framework

Abstract In this paper we propose a reinterpretation of the Mansouri–Nozari signature-change approach to the early Universe, in which the metric signature only occurs for t < 0 within the Euclidean regime, posing a challenge in providing coherent physical interpretations in advance. We ensure that the sign change occurs for t > 0 in the Euclidean regime. Our findings show that the energy-momentum tensor does not vanish on the signature-changing surface, even when we adopt conditions commonly used in the literature. Furthermore, we formulate a modified cosmology within the framework of signature-changing and discuss the modifications that our approach introduces to the cosmological equations. We establish a constraint relating the equation of state parameter and its singular part and we determine an effective equation of state as function of the redshift. Finally, we show that the law of conservation of matter in the modified ΛCDM model remains valid.

Inverse scattering for repulsive potential and strong singular interactions

In a previous work of 2014 on a quantum system governed by the repulsive Hamiltonian, the author proved uniqueness for short-range interactions described by a scattering operator consisting of regular and singular parts. In this paper, the singular part is assumed to have much stronger singularities and the same uniqueness theorem is proved. By applying the time-dependent method invented by Enss and Weder [J. Math. Phys. 36(8), 3902–3921 (1995)], the high-velocity limit for a wider class of the scattering operator with stronger singularities also uniquely determines the interactions of a multi-dimensional system.

Marshall–Olkin Bivariate Weibull Model with Modified Singularity (MOBW-μ): A Study of Its Properties and Correlation Structure

We propose the “Marshall–Olkin Bivariate Weibull Model with Modified Singularity MOBW-μ”, which focuses on bivariate distributions essential for reliability and survival analyses. Distributions such as the Marshall–Olkin bivariate exponential (MOBE) and the Marshall–Olkin bivariate Weibull (MOBW) are discussed. The MOBW-μ model is introduced, which incorporates a lag parameter μ in the singular part, and probabilistic properties such as the joint survival function, marginal density functions, and the bivariate hazard rate function are explored. In addition, aspects such as the correlation structure and survival copulation are addressed and we show that the correlation of the MOBW-μ is always lower than that of its copula, regardless of the parameters. The latter result implies that the MOBW-μ does not have the Lancaster’s phenomenon that explains that any nonlinear transformation of variables decreases the correlation in absolute value. The article concludes by presenting a robust theoretical framework applicable to various disciplines.

On Bivariate Distributions with Singular Part

There are many families of bivariate distributions with given marginals. Most families, such as the Farlie–Gumbel–Morgenstern (FGM) and the Ali–Mikhail–Haq (AMH), are absolutely continuous, with an ordinary probability density. In contrast, there are few families with a singular part or a positive mass on a curve. We define a general condition useful to detect the singular part of a distribution. By continuous extension of the bivariate diagonal expansion, we define and study a wide family containing these singular distributions, obtain the probability density, and find the canonical correlations and functions. The set of canonical correlations is described by a continuous function rather than a countable sequence. An application to statistical inference is given.

Spectral analysis of dispersed multiphase flows in the presence of fluid interfaces

Spectral analysis of dispersed two-phase flows is highly desirable to reveal the interplay of the various flow scales, much larger or much smaller than the size of the dispersed bodies. This is a challenging task as the matching conditions at the body interfaces generate singularities in the fields describing the two-phase mixture. The nature of these singularities and their consequences on the spectra are theoretically analyzed for bubble or droplet flows. Results of direct numerical simulations are reported and spatial spectra of the mixture velocity, the flow forces and their power are examined. The regular part of the spectral densities of energy production, dissipation and transfers between scales are separated from their singular part. The resulting spectral energy balance, free of the footprint of the singularities, is found in agreement with coarse-grained simulations where the interfaces are filtered out before solving the Navier–Stokes equations. These results pave the way for the spectral analysis of more complex turbulent dispersed flows.

On the gradient rearrangement of functions

AbstractIn this paper, we introduce a symmetrization technique for the gradient of a $$\text {BV}$$ BV function, which separates its absolutely continuous part from its singular part (sum of jump and Cantorian part). In particular, we prove a $$\text {L }^{\text {1}}$$ L 1 comparison between the function and the symmetrization just mentioned. Furthermore, we apply this result to obtain Saint-Venant type inequalities for some geometric functionals.

SOME MORE ON LOGARITHMIC SINGULARITY INTEGRATION IN BOUNDARY ELEMENT METOD

The accuracy of calculations of integrals with logarithmic singularities for two methods, namely the method of ignoring singularities and the method of subtraction (consisting in separating the singular part from the remaining non-singular), are presented in this paper. Only two-dimensional problems, like Dirichlet's problems, as well as acoustic problems formulated in the frequency domain are considered. Problems related to the accuracy of calculations are discussed and the influence of frequency, as well as the influence of the geometry of the analysed area on the accuracy of calculations, are indicated. When we talk about the influence of geometry, we mean not only discretization, but also the configuration of the area, such as the sharp edges of the boundary line, assuming the use of the classic, without any modifications, Boundary Element Method.

Fate of Chiral Symmetries in the Quark-Gluon Plasma from an Instanton-Based Random Matrix Model of QCD.

We propose a new way of understanding how chiral symmetry is realized in the high temperature phase of QCD. Based on the finding that a simple free instanton gas precisely describes the details of the lowest part of the spectrum of the lattice overlap Dirac operator, we propose an instanton-based random matrix model of QCD with dynamical quarks. Simulations of this model reveal that even for small quark mass the Dirac spectral density has a singularity at the origin, caused by a dilute gas of free instantons. Even though the interaction, mediated by light dynamical quarks, creates small instanton-anti-instanton molecules, those do not influence the singular part of the spectrum, and this singular part is shown to dominate Banks-Casher type sums in the chiral limit. By generalizing the Banks-Casher formula for the singular spectrum, we show that in the chiral limit the chiral condensate vanishes if there are at least two massless flavors. Our model also indicates a possible way of resolving a long-standing debate, as it suggests that for two massless quark flavors the U(1)_{A} symmetry is likely to remain broken up to arbitrarily high finite temperatures.

On the structure of Nevanlinna measures

AbstractIn this paper, we study the structural properties of Nevanlinna measures, that is, Borel measures that arise in the integral representation of Herglotz–Nevanlinna functions. In particular, we give a characterization of these measures in terms of their Fourier transform, characterize measures supported on hyperplanes including extremal measures, describe the structure of the singular part of the measures when some variable are set to a fixed value, and provide estimates for the measure of expanding and shrinking cubes. Corresponding results are stated also in the setting of the polydisc where applicable, and some of our proofs are actually performed via the polydisc.

On the non-uniqueness of the kernel of the Zakharov equation in intermediate and shallow water: the connection with the Davey–Stewartson equation

The Zakharov equation describes the evolution of weakly nonlinear surface gravity waves for arbitrary spectral shape. For deep-water waves, results from the Zakharov equation are well established. However, for two-dimensional propagation, in intermediate and shallow water, there are problems related to the treatment of apparent singularities in the contribution of the wave-induced set-up to the evolution of the surface gravity waves. More specifically, the kernel in the integral term is characterized by a regular and an apparent singular contribution. Here, we show that the Davey–Stewartson equation can be directly derived from the Zakharov equation, also in the shallow water limit. This result provides guidance on how to treat the singular contribution to the evolution of the action variable. A relevant result that is obtained is that the growth rate obtained from the stability analysis of a plane wave in shallow water does not depend on the singular part of the kernel of the Zakharov equation.

Codimension One Foliation and the Prime Spectrum of a Ring

Let F be a transversally oriented codimension-one foliation of class Cr, r ≥ 0, on a closed manifold M. A leaf class of a leaf F is the union of all leaves having the same closure as F . Let X be the leaf classes space and X0 be the union of all open subsets of X homeomorphic to R or S1. In [2, Theorem 3.15] it is shown that if a codimension one foliation has a finite height, then the singular part of the space of leaf classes is homeomorphic to the prime spectrum (or simply the spectrum) of unitary commutative ring. In this paper we prove that the singular part of the space of leaf classes is homeomorphic to the spectrum of unitary commutative ring if and only if every family of totaly ordered leaves is bounded below.

Linear inverse problems with nonnegativity constraints: Singularity of optimisers

We look at continuum solutions in optimisation problems associated to linear inverse problems $ y = Ax $ with non-negativity constraint $ x \geq 0 $. We focus on the case where the noise model leads to maximum likelihood estimation through general divergences, which cover a wide range of common noise statistics such as Gaussian and Poisson. Considering $ x $ as a Radon measure over the domain on which the reconstruction is taking place, we show a general singularity result. In the high noise regime corresponding to $ y \notin\{A x \mid x \geq 0\} $ and under a key assumption on the divergence as well as on the operator $ A $, any optimiser has a singular part with respect to the Lebesgue measure. We hence provide an explanation as to why any possible algorithm successfully solving the optimisation problem will lead to undesirably spiky-looking images when the image resolution gets finer, a phenomenon well documented in the literature. We illustrate these results with several numerical examples inspired by medical imaging.

Nucleon Self-Energy Including Two-Loop Contributions

The nucleon self-energy is calculated in SU(2) covariant chiral perturbation theory to study the pion mass dependence of the nucleon mass up to chiral order O(q6), i.e., including two-loop diagrams. The contributions of the diagrams are expressed by a small set of (scalar) master integrals. The extended-on-mass-shell (EOMS) renormalization scheme is applied by systematically subtracting infrared regular parts of the integrals (besides the divergences) to ensure that the renormalized results are consistent with the power counting. The master integrals are evaluated in two ways: Firstly, they are calculated by means of the chiral expansion in d dimensions, using the strategy of regions to differentiate between the infrared singular and regular part. This yields the physical nucleon mass in a 1/m-expansion and is in agreement with the infrared-renormalized result. Secondly, the master integrals are solved numerically without relying on the 1/m-expansion using the sector decomposition method.

Local estimates for vectorial Rudin–Osher–Fatemi type problems in one dimension

We consider the Rudin–Osher–Fatemi variational denoising model with general regularizing term and L2 fidelity in one-dimensional, vector-valued setting. We obtain local estimates on the singular part of the variation measure of the minimizer in terms of the singular part of the variation measure of the datum. In the case of homogeneous regularizer, we prove local estimates on the whole variation measure of the minimizer and deduce an analogous result for the gradient flow of the regularizer. We also discuss the question of extending our results to other fidelities.

Simultaneous Effect of EBR on LLDPE and PDMS Rubber Blends and Its Nanocomposites for Cable Applications.

The impact of electron beam radiation on the blend of linear low-density polyethylene (LLDPE) and polydimethylsiloxane (PDMS) rubber at different doses from 50 to 300 kGy has been investigated. The irradiated sheets were examined for their morphology, gel content, thermal stability, melt behavior, and electrical and dielectric properties. The radiation treatment has reduced both the melting point and crystallinity of base polymers and their blends because of chain scission. As observed, 100 kGy doses of irradiated blend and 3 wt % of loaded nanosilica composite showed comparatively good thermal stability. The phase morphology of the LLDPE: PDMS rubber blend showed a honeycomb-like design before irradiation because of two-stage morphology, which prominently changed into a solitary stage after electron beam irradiation. This is because of intermolecular cross-link arrangement inside the singular parts, just like cross-linking development at the interface. From the AQFESEM study, it is observed that the stacking of nanosilica particles within the blend matrix is greatly reduced after electron beam irradiation. The addition of nanosilica within the blend increased the electrical conductivity and dielectric permittivity. The dielectric breakdown strength has been observed to be the highest for 3 wt % loaded nanocomposite and its irradiated sample. The result indicates that the nanocomposite can be utilized for high-voltage cable applications in indoor and outdoor fields.

Phase field modeling of dynamic fracture in elastoplastic composites with interfacial debonding

In this work, we extend a phase field formulation for dynamic ductile fracture to consider interfacial debonding in elastoplastic composites. The interfacial weak zone is created through a regularization of the sharp interfaces, and the singular strain part along the interfaces is approximated by using a Taylor expansion. Then, a strain density depending on the displacement jump related to matrix/inclusions decohesion is added to the total energy to take into account interfacial debonding. The coupling problems (displacement, plasticity and damage problems) are derived within the variational framework and a staggered iterative algorithmic procedure is described to solve the coupling problems. Numerical examples demonstrate that this method can handle the initiation, propagation, and interaction between bulk dynamic fracture and interface cracks, as well as the anisotropic behavior in the complex microstructure of elastic–plastic composite materials. It also indicates that this model is convergent in terms of mesh refinement.

Smoothing properties of a linearization of the three-wave collision operator in the bosonic Boltzmann–Nordheim equation

We consider the kinetic theory of a three-dimensional fluid of weakly interacting bosons in a non-equilibrium state which includes both normal fluid and a condensate. More precisely, we look at the previously postulated nonlinear Boltzmann–Nordheim equations for such systems, in a spatially homogeneous state which has an isotropic momentum distribution, and we linearize the equation around an equilibrium state which has a condensate. We study the most singular part of the linearized operator coming from the three-wave collision operator for supercritical initial data. The operator has two types of singularities, one of which is similar to the marginally smoothing operator defined by the symbol ln⁡(1+p2). Our main result in this context is that for initial data in a certain Banach space of functions satisfying a Hölder type condition, at least for some finite time, evolution determined by the linearized operator improves the Hölder regularity. The main difficulty in this problem arises from the combination of a point singularity and a line singularity present in the linear operator, and we have to use certain fine-tuned function spaces in order to carry out our analysis.