Splitting Type Research Articles

Efficient simulation of complex Ginzburg–Landau equations using high-order exponential-type methods

In this paper, we consider the task of efficiently computing the numerical solution of evolutionary complex Ginzburg–Landau equations on Cartesian product domains with homogeneous Dirichlet/Neumann or periodic boundary conditions. To this aim, we employ for the time integration high-order exponential methods of splitting and Lawson type with constant time step size. These schemes enjoy favorable stability properties and, in particular, do not show restrictions on the time step size due to the underlying stiffness of the models. The needed actions of matrix exponentials are efficiently realized by using a tensor-oriented approach that suitably employs the so-called μ-mode product (when the semidiscretization in space is performed with finite differences) or with pointwise operations in Fourier space (when the model is considered with periodic boundary conditions). The overall effectiveness of the approach is demonstrated by running simulations on a variety of two- and three-dimensional (systems of) complex Ginzburg–Landau equations with cubic or cubic-quintic nonlinearities, which are widely considered in literature to model relevant physical phenomena. In fact, we show that high-order exponential-type schemes may outperform standard techniques to integrate in time the models under consideration, i.e., the well-known second-order split-step method and the explicit fourth-order Runge–Kutta integrator, for stringent accuracies.

Dynamic mechanical behaviours of frozen rock under sub-zero temperatures and dynamic loads

Dynamic mechanical behaviours of frozen rocks are essential for dealing with the design and stability of various engineering structures in cold regions. A dynamic-cryogenic testing system integrated with high-speed Digital Image Correlation (DIC) is adopted to investigate the dynamic mechanical behaviours of rock subjected to ambient room and sub-zero temperatures, along with high-rate loading. Dynamic compression, tensile and Mode I dynamic fracture tests were performed at 20, -20, −50 and −70 °C using dry and saturated sandstone from Yulong Copper Mine at high-altitude cold region. A DIC-based approach using pixel-based virtual extensometers is proposed to accurately capture the real-time cracking speed with high precision. The proportion of ice and unfrozen water in rock pores across various temperatures is quantified using the nuclear magnetic resonance (NMR) method. The results indicate significant temperature and rate dependencies in dynamic fracture toughness, crack propagation velocity, dynamic tensile strength, dynamic compressive strength, and failure modes; These parameters exhibit an increase with decreasing ambient temperature as unfrozen water transitions into ice, demonstrating the frozen effect of ice filling pores, supporting the rock skeleton, and bonding ice-rock interfaces. With decreasing temperature, crack branching becomes more pronounced in dynamic notched semi-circular bending (NSCB) tests. Additionally, dynamically compressed frozen rocks tend to transition from pulverization (Type II) to splitting (Type I) with multiple elongations, attributed to the strengthening effect. The findings elucidate the failure mechanisms observed in a higher proportion of larger blast blocks and hold valuable implications for blasting designs and dynamic disaster prevention in cold regions.

Investigation on failure mechanism and time-delay fracturing behavior of hard-rock tunnel under extremely high geostress state

To investigate the failure mechanism and time-delay fracturing behavior of hard-rock tunnel under extremely high geostress state, a series of conventional triaxial compression tests and time-delay loading–unloading mechanical tests on rocks were conducted. Combining testing results and on-site data of deep-buried tunnels, the generation mechanism and failure process of time-delay rockbursts were analyzed. The results indicate that with the increase of confining pressure, the failure mode of hard-rock exhibits a transition from axial splitting type and shear-slip type to special squeezing-expansion type. The progressive damage of rocks under extremely high stress state mainly manifests as aging lateral and volumetric expansion. Under time-delay compression tests, the long-term strength of hard-rock is lower than the threshold stress of σeb. In extremely high stress state, the fracture and dissipated energy evolution of rocks has significant time-dependent characteristics. Under extremely high stress conditions, rocks can store a large amount of releasable energy internally and gradually generate numerous cracks without the need for external work, ultimately leading to sudden unstable failure. Affected by the superposition of time-delay pressure and loading–unloading effects, the hard-rock also exhibits the same time-dependent fracturing behavior as soft-rock under high stress state, but its deformation mechanism is dominated by lateral expansion and crack volumetric dilatancy. When the stress level is far from reaching the ultimate strength required for rockmass failure, it is sufficient to cause the propagation of internal microcracks. Under the action of time-delay loads, the spalling and peeling failure of surrounding rock is significantly reduced, but the scale of unstable blocks is larger. Based on the failure process and microseismic monitoring date of rockburst in hard-rock tunnels, the entire evolution process of time-delay rockbursts can be divided into the following six stages: before excavation, rockmss excavation, aging cracking, squeezing-expansion, instability and spalling, time-delay rockburst. Time-delay rockburst has an obvious acoustic emission calm period before its occurrence, making its incubation process has strong suddenness and randomness.

Restricting the splitting types of a positive density set of places in number field extensions

We prove necessary and sufficient conditions for a finite group G G with an ordering of G G -extensions to satisfy the following property: for every positive density set of places A A of a number field K K and every splitting type given by a conjugacy class c c in G G , 0 0% of G G -extensions avoid this splitting type for each p ∈ A p\in A .

Searching for Dark Matter Axions via Atomic Excitations

Axions can be considered as good dark matter candidates. The detection of such light particles can be achieved by observing axion-induced atomic excitations. The target is in a magnetic field so that the m-degeneracy is removed, and the energy levels can be suitably adjusted. Using an axion-electron coupling indicated by the limit obtained by the Borexino experiment, which is quite stringent, reasonable axion absorption rates have been obtained for various atomic targets The obtained results depend, of course, on the atom considered through the parameters ϵ (the spin−orbit splitting) as well as δ ( the energy splitting due to the magnetic moment interaction). This assumption allows axion masses in the tens of μeV if the transition occurs between members of the same multiplet, i.e., |J1,M1=−J1⟩→|J1,M1=−J+1⟩,J1≠0, and axion masses in the range 1 meV–1 eV for transitions of the spin−orbit splitting type |J1,M=−J1⟩→|J2,M2=−J1+q⟩,q=−1,0,1, i.e., three types of transition. The axion mass that can be detected is very close to the excitation energy involved, which can vary by adjusting the magnetic field. Furthermore, since the axion is absorbed by the atom, the calculated cross-section exhibits the behavior of a resonance, which can be exploited by experiments to minimize any background events.

Splitting type results for pseudoconvex domains and remarks on their Nebenhülle

We give a very general splitting type theorem for biholomorphic maps close to identity in the context of smoothly bounded pseudoconvex domains (Theorem 1.4). As a particular case, in the context of worm domains, we essentially reprove the splitting type result (Theorem 1.3) from Bracci et al. (Math Z 292:879–893, 2019) (by a different method). We also discuss some properties of the Nebenhülle of worm domains.

Normal bundle of monomial curves: an application to rational curves

In this note, we give an application to the study of general rational curves in \mathbb{P}^s(\mathbb{C}) of the calculation of the splitting type of the normal bundle of any smooth monomial rational curve (i.e., embedded by monomial functions).

NUMERICAL SIMULATION OF FRACTIONAL POWER DIFFUSION BIOSENSORS

The main aim of this paper is to propose new mathematical models for simulation of biosensors and to construct and investigate discrete methods for the efficient solution of the obtained systems of nonlinear PDEs. The classical linear diffusion operators are substituted with nonlocal fractional powers of elliptic operators. The splitting type finite volume scheme is used as a basic template for the introduction of new mathematical models. Therefore the accuracy of the splitting scheme is investigated and compared with the symmetric Crank-Nicolson scheme. The dependence of the approximation error on the regularity of the solution is investigated. Results of computational experiments for different values of fractional parameters are presented and analysed.

Special lines on contact manifolds

In a series of two articles Kebekus studied deformation theory of minimal rational curves on contact Fano manifolds. Such curves are called contact lines. Kebekus proved that a contact line through a general point is necessarily smooth and has a fixed standard splitting type of the restricted tangent bundle. In this paper we study singular contact lines and those with special splitting type. We provide restrictions on the families of such lines, and on contact Fano varieties which have reducible varieties of minimal rational tangents. We also show that the results about singular lines naturally generalise to complex contact manifolds, which are not necessarily Fano, for instance, quasiprojective contact manifolds or contact manifolds of Fujiki class C. In particular, in many cases the dimension of a family of singular lines is at most 2 less than the dimension of the contact manifold.

A General Alternating-Direction Implicit Framework with Gaussian Process Regression Parameter Prediction for Large Sparse Linear Systems

This paper proposes an efficient general alternating-direction implicit (GADI) framework for solving large sparse linear systems. The convergence property of the GADI framework is discussed. Most of the existing ADI methods can be viewed as particular schemes of the developed framework. Meanwhile the GADI framework can derive new ADI methods. Moreover, as the algorithm efficiency is sensitive to the splitting parameters, we offer a data-driven approach, the Gaussian process regression (GPR) method based on the Bayesian inference, to predict the GADI framework's relatively optimal parameters. The GPR method requires a small training data set to learn the regression prediction mapping, which has sufficient accuracy and high generalization capability. It allows us to efficiently solve linear systems with a one-shot computation, and does not require any repeated computations to obtain relatively optimal splitting parameters. Finally, we use the three-dimensional convection-diffusion equation and continuous Sylvester matrix equation to examine the performance of our proposed methods. Numerical results demonstrate that the proposed framework is faster tens to thousands of times than the existing ADI methods, such as (inexact) Hermitian and skew-Hermitian splitting type methods in which the consumption of obtaining relatively optimal splitting parameters is ignored. Due to the efficiency of the developed methods, we can solve much larger linear systems which these existing ADI methods have been not reached.

Ayurvedic management of Hirayama disease

Hirayama disease (HD), also known as monomelic amyotrophy, is rare cervical myelopathy due to spinal cord compression by forward displacement of the posterior dural sac during neck flexion that manifests itself as asymmetrical, slowly progressive atrophic weakness of the forearms and hands predominantly in young males. A 23-year-old man came to the outpatient department with a complaint of weakness of the right upper limb for the past 3 years which started in the right hand and forearm along with tremors in both hands. Considering the clinical history, physical examination, and electromyography, the condition was diagnosed as HD by an allopathic consultant two years ago. The symptoms can be included under Vata vyadhi. Some of the symptoms such as Asthibheda (~splitting type of pain in the bones), Sandhishoola (~pain in joints), Mamsa kshaya (~diminution of muscle mass), and Bala kshaya (~diminished strength) come under Asthimajjagata vata. Following the symptomatology and its resemblance with Asthimajjagata vata, the patient was treated with Snehapana (~internal oleation) with Ashwagandha ghrita followed by Virechana karma (~purgation therapy). Anuvasana basti with Panchatikta kshira along with Shalishastika pinda svedana was done after Virechana. Rasayana kalpa ksheera chikitsa was given after completion of Basti procedure. After 40 days of treatment, the patient showed significant improvement in muscle bulk. The magnetic resonance imaging shows no neural compression. This case shows that Ayurveda treatment may be helpful in pathologies like HD. Further, better results may be obtained if Ayurvedic therapies start at an earlier stage and continued for a longer duration.

Assessing the spectral characteristics of band splitting type II radio bursts observed by CALLISTO spectrometers

Metric type II radio bursts are usually early indicators of CME-driven shocks and other space weather phenomena in the solar corona. This paper presents a detailed investigation of the spectral properties of band splitting type II radio bursts and their association with sunspot number. Using type II radio bursts in a frequency range [20 MHz–200 MHz] observed by CALLISTO during 2010–2017, it was discovered that the analyzed type II shock height, magnetic field strength, CME/shock speed, and Alfvén speed synchronize with the trajectory of the solar cycle 24. Also, the study revealed that the onset of the declining phase of solar cycle 24 has the highest electron density, whiles the frequency of type II bursts depicts a bimodal distribution during the study period, with peaks in 2012 and 2015. Further, a good correlation (with correlation factor R = 0.87) was obtained between the estimated CME/shock speeds from the dynamic spectra and the associated CME speeds from SOHO/LASCO. Moreover, the study confirmed a significant correlation (R= ∼0.8) between the absolute drift rates and the plasma frequency. Additionally, the study explored that ∼ 60% of the type II radio bursts considered in this study emanated from the western longitudes. Hence, the findings emphasize that the temporal dynamics of the physical conditions of band splitting type II radio bursts are essential parameters for space weather monitoring and forecasting.

Pull-through failure of crest-fixed steel claddings under combined wind and bushfire conditions

Cold-formed steel (CFS) roof and wall cladding systems are commonly used in Australia and around the world. The use of CFS claddings in bushfire prone areas is becoming more popular due to their non-combustibility and is recommended in the Australian standard AS3959 for construction in bushfire areas. However, severe fire events occur more frequently and in recent bushfire events, strong winds and fire–wind interactions have been observed. Thus, the behaviour of CFS claddings exposed to a combined action of fire and wind needs to be considered for the safe design of buildings in bushfire prone areas. In this study, a small-scale test set-up was developed and used to simulate the wind suction loading on crest-fixed CFS roof and wall claddings while also being exposed to elevated temperatures. Experiments were conducted on three commonly used 0.42 mm high strength (G550) CFS cladding profiles at ambient and elevated temperatures and their critical localised pull-through failures of the crests under the screw heads were investigated at increasing temperatures. The results showed that the failure mode of trapezoidal cladding profiles changed from a transverse splitting type to a localised dimpling type pull-through failure. They also showed different rates of reduction in the pull-through capacity with increasing temperatures and the type of cladding profile. This paper has proposed a suitable equation for the pull-through capacity reduction with increasing temperatures and made suitable recommendations in relation to the use of CFS cladding systems in bushfire prone areas.

Tropical methods in Hurwitz-Brill-Noether theory

Splitting type loci are the natural generalizations of Brill-Noether varieties for curves with a distinguished map to the projective line. We give a tropical proof of a theorem of H. Larson, showing that splitting type loci have the expected dimension for general elements of the Hurwitz space. Our proof uses an explicit description of splitting type loci on a certain family of tropical curves. We further show that these tropical splitting type loci are connected in codimension one, and describe an algorithm for computing their cardinality when they are zero-dimensional. We provide a conjecture for the numerical class of splitting type loci, which we confirm in a number of cases.

Mesoscale modelling of bond performance between deformed steel bar and concrete subjected to dynamic loads

The good bond property is the base for rebar and concrete to work together, influencing the mechanical behaviour of reinforced concrete structures. To investigate the bond-slip performance between deformed steel bars and concrete under dynamic loadings, a three-dimensional meso-scale numerical model was established, considering the inherent heterogeneity of concrete and explicitly modelling the surface characteristics of the rebar. The interaction between the rebar and concrete was implemented by the surface-to-surface contact strategy. The strain rate effect of materials was also taken into account. The rationality and accuracy of the proposed model were verified by the good consistency between the simulation and available test results in terms of failure pattern and bond stress-slip curves. With the help of the meso-scale model, the bond-slip behaviour under various loading rates was simulated and analysed. Finally, a simplified model to describe the dynamic bond stress-slip relationship was developed. It has been found that due to the propagation of stress waves, the cracking process of the specimen varies with the strain rate while the final failure pattern is similar, exhibiting splitting type failure. Thus, a considerable part of pull-out loading was resisted by the inertia of the system at the initial stage. With the increase of strain rate, the shape of the dynamic bond stress-slip curves is similar with a significantly increased bond strength and reduced slip, especially at high strain rates. The ratio between the residual and peak bond stress is approximately constant, irrespective of strain rate. The proposed simplified model is able to effectively characterize the dynamic bond stress-slip relationship between the deformed rebar and concrete.

Fully-discrete Spectral-Galerkin scheme with second-order time-accuracy and unconditionally energy stability for the volume-conserved phase-field lipid vesicle model

In this work, for the phase-field model of lipid vesicles with the property of accurate volume conservation, we construct an effective fully-discrete numerical scheme, in which, the time marching method is based on a novel splitting type technique, and space is discretized by using the Spectral-Galerkin method. The advantage of this scheme is its high efficiency and ease of implementation. Specifically, although the model is highly nonlinear, just by solving two independent linear biharmonic equations with constant coefficients at each time step, the scheme can achieve the second-order accuracy in time, spectral accuracy in space, and unconditional energy stability. The essence of the scheme is to introduce several additional auxiliary variables and use the specially designed ordinary differential equations to reformulate the system. In this way, energy stability can be obtained unconditionally, while avoiding the calculation of variable-coefficient systems. We strictly prove that the energy stability in the fully-discrete form that the scheme holds and give a detailed implementation process. Numerical experiments in 2D and 3D are further carried out to verify the convergence rate, energy stability, and effectiveness of the developed algorithm.

On the splitting type of holomorphic vector bundles induced from regular systems of differential equation

Abstract We calculate the splitting type of holomorphic vector bundles on the Riemann sphere induced by a Fuchsian system of differential equations. Using this technique, we indicate the relationship between Hölder continuous matrix functions and a moduli space of vector bundles on the Riemann sphere. For second order systems with three singular points we give a complete characterization of the corresponding vector bundles by the invariants of Fuchsian system.

Refined Brill–Noether theory for all trigonal curves

Trigonal curves provide an example of Brill–Noether special curves. Theorem 1.3 of Larson (Invent Math 224(3):767–790, 2021) characterizes the Brill–Noether theory of general trigonal curves and the refined stratification by Brill–Noether splitting loci, which parametrize line bundles whose push-forward to \(\mathbb {P}^1\) has a specified splitting type. This note describes the refined stratification for all trigonal curves. Given the Maroni invariant of a trigonal curve, we determine the dimensions of all Brill–Noether splitting loci and describe their irreducible components. When the dimension is positive, these loci are connected, and if furthermore the Maroni invariant is 0 or 1, they are irreducible.

Syzygy bundles and the weak Lefschetz property of monomial almost complete intersections

Deciding the presence of the weak Lefschetz property often is a challenging problem. Continuing studies of Brenner and Kaid (2007), Cook II and Nagel (2011) and Migliore, Miró-Roig, Murai and Nagel (2013) we carry out an in-depth study of Artinian monomial ideals with four generators in three variables. We use a connection to lozenge tilings to describe semistability of the syzygy bundle of such an ideal, to determine its generic splitting type, and to decide the presence of the weak Lefschetz property. We provide results in both characteristic zero and positive characteristic.

On the loci of morphisms from [formula omitted] to G(r,n) with fixed splitting type of the restricted universal sub-bundle or quotient bundle

Let n≥4, 2≤r≤n−2 and e≥1. We show that the intersection of the locus of degree e morphisms from P1 to G(r,n) with the restricted universal sub-bundles having a given splitting type and the locus of degree e morphisms with the restricted universal quotient-bundle having a given splitting type is non-empty and generically transverse along at least one component of the intersection. As a consequence, we get that the locus of degree e morphisms from P1 to G(r,n) with the restricted tangent bundle having a given splitting type need not always be irreducible.