Limit cycle problem for quadratic differential system x = -y + lx^2 + mxy, y =x(1 +ax +by)

The maximal number of limit cycles for a particular type Ⅲ system x = -y + lx2 + mxy, y =x(1 + ax + by) is studied and some errors that appeared in the paper by Suo Mingxia and Yue Xiting (Annals of Differential Equations, 2003,19(3):397-401) are corrected. By translating the system to be considered into the Lienard type and by using some related properties, we obtain several theorems with suitable conditions coefficients of the system, under which we prove that the system has at most two limit cycles. The conclusions improve the results given in Suo and Yue's paper mentioned above.

LIMIT CYCLE PROBLEM OF QUADRATIC SYSTEM OF TYPE （ Ⅲ ）m=0, （ Ⅲ ）

To continue the discussion in （Ⅰ ） and （ Ⅱ ）,and finish the study of the limit cycle problem for quadratic system （ Ⅲ ）m=0 in this paper. Since there is at most one limit cycle that may be created from critical point O by Hopf bifurcation,the number of limit cycles depends on the different situations of separatrix cycle to be formed around O. If it is a homoclinic cycle passing through saddle S1 on 1 ＋ax-y = 0,which has the same stability with the limit cycle created by Hopf bifurcation,then the uniqueness of limit cycles in such cases can be proved. If it is a homoclinic cycle passing through saddle N on x= 0,which has the different stability from the limit cycle created by Hopf bifurcation,then it will be a case of two limit cycles. For the case when the separatrix cycle is a heteroclinic cycle passing through two saddles at infinity,the discussion of the paper shows that the number of limit cycles will change from one to two depending on the different values of parameters of system.

ON NUMBER OF LIMIT CYCLES FOR THE QUADRATIC SYSTEMS WITH A WEAK FOCUS AND A STRONG FOCUS

It is proved that the quadratic system with a weak focus and a strong focus has at most one limit cycle around the strong focus, and as the weak focus is a 2nd order(or 3rd order) weak focus the quadratic system has at most two(one) limit cycles which have (1,1) distribution ((0,1) distribution).

On the(1,3)Distributions of Limit Cycles of Plane Quadratic Systems

In this paper, (a) we rerise Theorem 2 of Ref [1] omit the condition V_7>0 .(b) we discuss the relative positions of six curves M(s￣2, r)=0, J( s￣2, r)=0, L(s￣2,r)=0, T(s￣2,r)=0, Under the condition of the (1.3) distri-butions of limit cycles, we expand the variable regions of parameters ( s , r) and clearly. show them in figure, (c) we study the (1, 3) distributions of limit cycles of one kind quadratic systems with two singular points at the infinite: and (d) we give a generalmethod to discuss the ( 1 ,3) distibutions`of limit cycles of system (1.1) whatever there isone, two or three singular points at the infinite.

BIFURCATION OF LIMIT CYCLE FOR QUADRATIC SYSTEM WITH INVARIANT PARABOLA

In a previous paper, we have proved that a planar quadratic system with invariant parabola r has at most one limit cycle. In this paper, we use geometric characteristics to give necessary and sufficient conditions un'der which a PQSp with three non-degenerate singular points can be transformed into twO different definite forms. In this wayl we obtain all the bifurcations of such a system.

Lateral earth pressure of granular backfills on retaining walls with expanded polystyrene geofoam inclusions under limited surcharge loading

Existing studies have focused on the behavior of the retaining wall equipped with expanded polystyrene(EPS)geofoam inclusions under semi-infinite surcharge loading rather than limited surcharge loading.In this paper,the failure mode and the earth pressure acting on the rigid retaining wall with EPS geofoam inclusions and granular backfills(henceforth referred to as EPS-wall),under limited surcharge loading are investigated through two-and three-dimensional model tests.The testing results show that different from the sliding of almost all the backfill in the EPS-wall under semi-infinite surcharge loading,only an approximately triangular backfill slides in the wall under limited surcharge loading.The distribution of the lateral earth pressure on the EPS-wall under limited surcharge loading is non-linear,and the distribution changes from the increase of the wall depth to the decrease with the increase of the limited surcharge loading.An approach based on the force equilibrium of a differential element is developed to predict the lateral earth pressure behind the EPS-wall subjected to limited surcharge loading,and its performance was fully validated by the three-dimensional model tests.

Performance Limits and Advancements in Single 2D Transition Metal Dichalcogenide Transistor

Two-dimensional(2D)transition metal dichalcogenides(TMDs)allow for atomic-scale manipulation,challenging the conventional limitations of semiconductor materials.This capability may overcome the short-channel effect,sparking significant advancements in electronic devices that utilize 2D TMDs.Exploring the dimension and performance limits of transistors based on 2D TMDs has gained substantial importance.This review provides a comprehensive investigation into these limits of the single 2D-TMD transistor.It delves into the impacts of miniaturization,including the reduction of channel length,gate length,source/drain contact length,and dielectric thickness on transistor operation and performance.In addition,this review provides a detailed analysis of performance parameters such as source/drain contact resistance,subthreshold swing,hysteresis loop,carrier mobility,on/off ratio,and the development of p-type and single logic transistors.This review details the two logical expressions of the single 2D-TMD logic transistor,including current and voltage.It also emphasizes the role of 2D TMD-based transistors as memory devices,focusing on enhancing memory operation speed,endurance,data retention,and extinction ratio,as well as reducing energy consumption in memory devices functioning as artificial synapses.This review demonstrates the two calculating methods for dynamic energy consumption of 2D synaptic devices.This review not only summarizes the current state of the art in this field but also highlights potential future research directions and applications.It underscores the anticipated challenges,opportunities,and potential solutions in navigating the dimension and performance boundaries of 2D transistors.

Dynamic Event-Triggered Quadratic Nonfragile Filtering for Non-Gaussian Systems:Tackling Multiplicative Noises and Missing Measurements

This paper focuses on the quadratic nonfragile filtering problem for linear non-Gaussian systems under multiplicative noises,multiple missing measurements as well as the dynamic event-triggered transmission scheme.The multiple missing measurements are characterized through random variables that obey some given probability distributions,and thresholds of the dynamic event-triggered scheme can be adjusted dynamically via an auxiliary variable.Our attention is concentrated on designing a dynamic event-triggered quadratic nonfragile filter in the well-known minimum-variance sense.To this end,the original system is first augmented by stacking its state/measurement vectors together with second-order Kronecker powers,thus the original design issue is reformulated as that of the augmented system.Subsequently,we analyze statistical properties of augmented noises as well as high-order moments of certain random parameters.With the aid of two well-defined matrix difference equations,we not only obtain upper bounds on filtering error covariances,but also minimize those bounds via carefully designing gain parameters.Finally,an example is presented to explain the effectiveness of this newly established quadratic filtering algorithm.

A Stroke-Limitation AMD Control System with Variable Gain and Limited Area for High-Rise Buildings

Collisions between a moving mass and an anti-collision device increase structural responses and threaten structural safety.An active mass damper(AMD)with stroke limitations is often used to avoid collisions.However,a strokelimited AMD control system with a fixed limited area shortens the available AMD stroke and leads to significant control power.To solve this problem,the design approach with variable gain and limited area(VGLA)is proposed in this study.First,the boundary of variable-limited areas is calculated based on the real-time status of the moving mass.The variable gain(VG)expression at the variable limited area is deduced by considering the saturation of AMD stroke.Then,numerical simulations of a stroke-limited AMD control system with VGLA are conducted on a high-rise building structure.These numerical simulations show that the proposed approach has superior strokelimitation performance compared with a stroke-limited AMD control system with a fixed limited area.Finally,the proposed approach is validated through experiments on a four-story steel frame.

Supplementary Information to“Balancing the quantum speed limit and instantaneous energy cost in adiabatic quantum evolution”

I.SUPPLEMENTARY NOTE 1:THEORETICAL MATERIALS.The quantum speed limit(QSL)is essential for quantum computing and quantum communication,referring to the minimum time required for a quantum system to evolve from one state to another.Two well-known forms of the QSL are the Mandelstam-Tamm(MT)relation TqsL≥πh/2△E[S1]and the Margolus-Levitin(ML)relation TqsL≥πh/2(E)[S2]where Tqst is denoted as the QSL time,h is the reduced Planck's constant,△E is the energy uncertainty(standard deviation)of the system,and(E)is the average energy of the system above its ground state.Both of relations provide a lower bound on the evolution time.

Representation of an Integer by a Quadratic Form through the Cornacchia Algorithm

Cornachia’s algorithm can be adapted to the case of the equation x2+dy2=nand even to the case of ax2+bxy+cy2=n. For the sake of completeness, we have given modalities without proofs (the proof in the case of the equation x2+y2=n). Starting from a quadratic form with two variables f(x,y)=ax2+bxy+cy2and n an integer. We have shown that a primitive positive solution (u,v)of the equation f(x,y)=nis admissible if it is obtained in the following way: we take α modulo n such that f(α,1)≡0modn, u is the first of the remainders of Euclid’s algorithm associated with n and α that is less than 4cn/| D |) (possibly α itself) and the equation f(x,y)=n. has an integer solution u in y. At the end of our work, it also appears that the Cornacchia algorithm is good for the form n=ax2+bxy+cy2if all the primitive positive integer solutions of the equation f(x,y)=nare admissible, i.e. computable by the algorithmic process.

Cosmology-Independent Photon Mass Limits from Localized Fast Radio Bursts by Using Artificial Neural Networks

A hypothetical photon mass m_(γ) can produce a frequency-dependent vacuum dispersion of light, which leads to an additional time delay between photons with different frequencies when they propagate through a fixed distance. The dispersion measure and redshift measurements of fast radio bursts(FRBs) have been widely used to constrain the rest mass of the photon. However, all current studies analyzed the effect of the frequency-dependent dispersion for massive photons in the standard ΛCDM cosmological context. In order to alleviate the circularity problem induced by the presumption of a specific cosmological model based on the fundamental postulate of the masslessness of photons, here we employ a new model-independent smoothing technique, artificial neural network(ANN), to reconstruct the Hubble parameter H(z) function from 34 cosmic-chronometer measurements.By combining observations of 32 well-localized FRBs and the H(z) function reconstructed by ANN, we obtain an upper limit of m_(γ) ≤ 3.5 × 10^(-51)kg, or equivalently m_(γ) ≤ 2.0 × 10^(-15)eV/c^(2)(m_(γ) ≤ 6.5 × 10^(-51)kg, or equivalently m_(γ) ≤ 3.6 × 10^(-15)eV/c_(2)) at the 1σ(2σ) confidence level. This is the first cosmology-independent photon mass limit derived from extragalactic sources.

Parity-Time symmetry helps breaking a new limit

Parity-Time(PT)symmetry is an emerging concept in quantum mechanics where non-Hermitian Hamiltonians can exhibit real eigenvalues.Now,PT symmetric optical microresonators have been demonstrated to break the bandwidth-efficiency limit for nonlinear optical signal processing.

Speed limit effect during lane change in a two-lane lattice model under V2X environment

Speed limit measures are ubiquitous due to the complexity of the road environment,which can be supplied with the help of vehicle to everything(V2X)communication technology.Therefore,the influence of speed limit on traffic system will be investigated to construct a two-lane lattice model accounting for the speed limit effect during the lane change process under V2X environment.Accordingly,the stability condition and the mKdV equation are closely associated with the speed limit effect through theory analysis.Moreover,the evolution of density and hysteresis loop is simulated to demonstrate the positive role of the speed limit effect on traffic stability in the cases of strong reaction intensity and high limited speed.

Proteomic response of Phaeocystis globosa to nitrogen limitation

Phaeocystis globosa is an important unicellular eukaryotic alga that can also form colonies.P.globosa can cause massive harmful algal blooms and plays an important role in the global carbon or sulfur cycling.Thus far,the ecophysiology of P.globosa has been investigated by numerous studies.However,the proteomic response of P.globosa to nitrogen depletion remains largely unknown.We compared four protein preparation methods of P.globosa for two-dimensional electrophoresis(2-DE)(Urea/Triton X-100 with trichloroacetic acid(TCA)/acetone precipitation;TCA/acetone precipitation;Radio Immuno Precipitation Assay(RIPA)with TCA/acetone precipitation;and Tris buffer).Results show that the combination of RIPA with TCA/acetone precipitation had a clear gel background and showed the best protein spot separation effect,based on which the proteomic response to nitrogen depletion was studied using 2-DE.In addition,we identified six differentially expressed proteins whose relative abundance increased or decreased more than 1.5-fold(P<0.05).Most proteins could not be identified,which might be attributed to the lack of genomic sequences of P.globosa.Under nitrogen limitation,replication protein-like,RNA ligase,and sn-glycerol-3-phosphate dehydrogenase were reduced,which may decrease the DNA replication level and ATP production in P.globosa cells.The increase of endonucleaseⅢand transcriptional regulator enzyme may affect the metabolic and antioxidant function of P.globosa cells and induce cell apoptosis.These findings provide a basis for further proteomic study of P.globosa and the optimization of protein preparation methods of marine microalgae.

Seismic-induced surficial failure of cohesive slopes using three-dimensional limit analysis:A case study of the Wangjiayan landslide in Beichuan, China

A seismic-induced landslide is a common geological catastrophe that occurs in nature.The Wangjiayan landslide,which was triggered by the Wenchuan earthquake,is a typical case in point.The Wanjiayan landslide caused many casualties and resulted in enormous property loss.This study constructs a simple surficial failure model based on the upper bound approach of three-dimensional(3D)limit analysis to evaluate the slope stability of the Wangjiayan case,while a traditional two-dimensional(2D)analysis is also conducted as a reference for comparison with the results of the 3D analysis.A quasi-static calculation is used to study the effect of the earthquake in terms of horizontal ground acceleration,while a parametric study is conducted to evaluate the critical cohesion of slopes.Rather than employing a 3D analysis,using the 2D analysis yields an underestimation regarding the safety factor.In the Wangjiayan landslide,the difference in the factors of safety between the 3D and 2D analyses can reach 20%.The sliding surface morphology as determined by the 3D method is similar to actual morphology,and the parameters of both are also compared to analyze the reliability of the proposed 3D method.

Numerical Methods for a Class of Quadratic Matrix Equations

Quadratic matrix equations arise in many elds of scienti c computing and engineering applications.In this paper,we consider a class of quadratic matrix equations.Under a certain condition,we rst prove the existence of minimal nonnegative solution for this quadratic matrix equation,and then propose some numerical methods for solving it.Convergence analysis and numerical examples are given to verify the theories and the numerical methods of this paper.

THE LIMIT CYCLE BIFURCATIONS OF A WHIRLING PENDULUM WITH PIECEWISE SMOOTH PERTURBATIONS

This paper deals with the problem of limit cycles for the whirling pendulum equation x=y,y=sin x(cosx-r)under piecewise smooth perturbations of polynomials of cos x,sin x and y of degree n with the switching line x=0.The upper bounds of the number of limit cycles in both the oscillatory and the rotary regions are obtained using the Picard-Fuchs equations,which the generating functions of the associated first order Melnikov functions satisfy.Furthermore,the exact bound of a special case is given using the Chebyshev system.At the end,some numerical simulations are given to illustrate the existence of limit cycles.

Relaxed Stability Criteria for Time-Delay Systems:A Novel Quadratic Function Convex Approximation Approach

This paper develops a quadratic function convex approximation approach to deal with the negative definite problem of the quadratic function induced by stability analysis of linear systems with time-varying delays.By introducing two adjustable parameters and two free variables,a novel convex function greater than or equal to the quadratic function is constructed,regardless of the sign of the coefficient in the quadratic term.The developed lemma can also be degenerated into the existing quadratic function negative-determination(QFND)lemma and relaxed QFND lemma respectively,by setting two adjustable parameters and two free variables as some particular values.Moreover,for a linear system with time-varying delays,a relaxed stability criterion is established via our developed lemma,together with the quivalent reciprocal combination technique and the Bessel-Legendre inequality.As a result,the conservatism can be reduced via the proposed approach in the context of constructing Lyapunov-Krasovskii functionals for the stability analysis of linear time-varying delay systems.Finally,the superiority of our results is illustrated through three numerical examples.

The Limits of Etiological Diagnosis of Convulsions in Children at the Bangui Pediatric Hospital

Introduction: Convulsions are a frequent cause of admission to paediatric wards in countries with limited resources, and a major cause of neurological and cognitive sequelae. In sub-Saharan Africa, the aetiology of febrile convulsions is dominated by infections of the central nervous system. A detailed clinical examination and laboratory and imaging tests are carried out to identify the cause of the disease. Computed tomography is reserved for emergency situations or as a second line of defence, after Magnetic Resonance Imaging (MRI) to look for calcifications. Imaging thus helps to establish the nature of the epileptogenic lesion, specify its extent and guide its therapeutic management. The aim of this study is to clarify the contribution of MRI in the etiological investigation of convulsions in children at the paediatric university hospital, in addition to the usual means of exploration. Patient and Method: This was a cross-sectional, descriptive study from January 2022 to December 2023. We carried out an exhaustive sampling of all children aged between 1 and 59 months admitted for convulsion with fever or not who had undergone complementary examinations for aetiological purposes in relation to our technical platform during the study period. Results: Thirty patients were recruited. Children aged 12 to 59 months were the most represented (69.7%). The sex ratio was 1.30. The mean age was 36 months (range 30 days to 59 months). Of the 30 patients, 21.8% had a history of febrile convulsions. Most of our patients were from Bangui (73.3%). On admission, the majority of patients presented with an elevated temperature ranging from 38.5 degrees Celsius to 39.4 degrees Celsius in 33.1% of cases. The dominant clinical manifestations were tonic-clonic convulsions (53.2%), tonic convulsions (35.3%) and clonic convulsions (11.5%). According to their characteristics, convulsions were complex in 53.7% of cases and simple in 46.3%. Examinations for infectious diseases, tumors or malformation has been ruled out. The anticonvulsants most commonly used were diazepam, phenobarbital and phenytoin. We recorded a 73.3% cure rate and 4 (13.3%) cases of death. Conclusion: The investigation of seizures in search of aetiology outside the usual means in our context still presents difficulties. MRI is the examination of choice for exploring the posterior fossa and midline lesions. Combined with CT, it is better for exploring hemispheric tumours.