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In this article, we study the growth of solutions of the homogeneous complex linear differential equation f(k) + Ak-1(z)f(k-1) + ··· + A1(z)f' + A0(z)f = 0, where the coefficients Aj(z) (j = 0, 1,..., k-1) are analytic or meromorphic functions in C\{z0}. Under the sufficient condition that there exists one dominant coefficient by its logarithmic lower order or by its logarithmic lower type, we extend some precedent results due to Liu, Long, and Zeng and others. [ABSTRACT FROM AUTHOR]

We present a new method to determine the optimal regularity of positive solutions u ∈ C4(Ω\{0}) ∩ C0(Ω̄) of the Hénon-Hardy equation, i.e., (0.1) where Ω ⊂ ℝN (N ⩾ 4) is a bounded smooth domain with 0 ∈ Ω, α > −4, and p ∈ ℝ. It is clear that 0 is an isolated singular point of solutions of (0.1) and the optimal regularity of u in Ω relies on the parameter α. It is also important to see that the regularity of u at x = 0 determines the regularity of u in Ω. We first establish asymptotic expansions up to arbitrary orders at x = 0 of prescribed positive solutions u ∈ C4(Ω\{0}) ∩ C0(Ω̄) of (0.1). Then we show that the regularity at x = 0 of each positive solution u of (0.1) can be determined by some terms in asymptotic expansions of the related positive radial solution of the equation (0.1) with Ω = B, where B is the unit ball of ℝN. The main idea works for more general equations with singular weights. [ABSTRACT FROM AUTHOR]

We investigate vertices for plane curves with singular points. As plane curves with singular points, we consider Legendre curves (respectively, Legendre immersions) in the unit tangent bundle over the Euclidean plane and frontals (respectively, fronts) in the Euclidean plane. We define a vertex using evolutes of frontals. After that we define a vertex of a frontal in the general case. It is also known that the four vertex theorem does not hold for simple closed fronts. We give conditions under which a frontal has a vertex and the four vertex theorem holds for closed frontals. We also give examples and counter examples of the four vertex theorem. [ABSTRACT FROM AUTHOR]

Self-intersections and local singular points of the curves play an important role in algebraic geometry and many other areas. In the present paper, we study the self-intersection and local singular points of the n-member chains. For this purpose, we derive and use several new results on trigonometric formulas. A unified approach for calculating self-intersection and local singular points for a wide class of curves is presented. An application to the spectral theory of integro-differential operators with difference kernels is given as well. [ABSTRACT FROM AUTHOR]

A function I(x, y, t) constant on the solutions of a differential system in R 2 is called an invariant. We classify all planar quadratic differential systems having invariants of the form I (x , y , t) = a x 2 + b x y + c y 2 + d x + e y + c 1 t with c 1 ≠ 0 . There are 13 different families of quadratic systems having invariants of this form. As far as we know this is the first time that quadratic differential systems having an invariant different from a Darboux invariant have been classified [ABSTRACT FROM AUTHOR]

This paper presents a new computational envelope solution for profiling the helical gear disc tool. It uses the normal projection of the disc tool axis onto the helical gear surface to generate the characteristic curve and then automatically computes the geometric data of the characteristic curve to create the disc tool in 3D solid form. As a popular profile, the ISO heliacal gear was a typical proper example to verify and clarify the proposed solution. The solution can quickly create the helical gear disc tool in 3D solid form with high accuracy. The 3D comparison average error of the helical gear disc tool surfaces generated by the proposed solution and the Boolean method is 0.004 mm, and the RMS error is 0.009 mm. [ABSTRACT FROM AUTHOR]

We have introduced framed surfaces as smooth surfaces with singular points. The framed surface is a surface with a moving frame based on the unit normal vector of the surface. Thus, the notion of framed surfaces (respectively, framed base surfaces) is locally equivalent to the notion of Legendre surfaces (respectively, frontals). A more general notion of singular surfaces, called generalised framed surfaces, is introduced in this paper. The notion of generalised framed surfaces includes not only the notion of framed surfaces, but also the notion of one-parameter families of framed curves. It also includes surfaces with corank one singularities. We investigate the properties of generalised framed surfaces. [ABSTRACT FROM AUTHOR]

This study shows, by means of numerical analysis, that the characteristics of discrete dynamical systems, in which chaos and catastrophe coexist, are closely related to the geometric statistics in Finsler geometry. The two geometric statistics introduced are nonlinear connections information, denoted as N I , and the mean deviation curvature, denoted as P ¯. The quantity N I can be used to determine the occurrence of chaos in terms of nonequilibrium stability. The resulting chaos is characterized by P ¯ in terms of the trajectory's robustness, which is related to the localization or globalization of chaos. The characteristics of catastrophe-induced chaos are clearly visualized through the contour topography of N I , in which an abrupt change is represented by cliff topography (i.e. a line of critical points); initial dependence is reflected in the reversibility of topographic patterns. On overlaying the contour topography with the singularity pattern, it is evident that chaos does not arise around the singular point. Furthermore, the extensive development of cusp and butterfly chaos demands information on the nonlinear connections within the singularity pattern. The asymmetry in swallowtail chaos is less distinguishable in an equilibrated state, but becomes more evident when the system is in a state of nonequilibrium. In many analyses, chaos and catastrophe are examined separately. However, these results demonstrate that when both are present, the two have a complex relationship constrained by the singularity. [ABSTRACT FROM AUTHOR]

This article analyzes the inhomogeneous Hilbert boundary value problem for an upper half-plane with the finite index and boundary condition on the real axis for one generalized Cauchy–Riemann equation with a singular point on the real axis. A structural formula was obtained for the general solution of this equation under restrictions leading to an infinite index of the logarithmic order of the accompanying Hilbert boundary value problem for analytic functions. This formula and the solvability results of the Hilbert problem in the theory of analytic functions were applied to solve the set boundary value problem.

We carry out qualitative analysis of a fourth-order tumor growth control model using ordinary differential equations. We show that the system has one positive equilibrium point, and its stability is independent of the feedback gain. Using a Lyapunov function method, we prove that there exist realistic parameter values for which the systems admit limit cycle oscillations due to a supercritical Hopf bifurcation. The time evolution of the state variables is also represented. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a wide family of dynamical systems whose characteristic feature is a polynomial right-hand side containing coprime forms of the phase variables of the system. One of the equations of the system contains a third-degree polynomial (cubic form), the other equation contains a quadratic form. We consider the problem of constructing all possible phase portraits in the Poincaré disk for systems from the family considered and establish criteria for the implementation of each portrait that are close to coefficient criteria. This problem is solved by using the central and orthogonal Poincaré methods of sequential mappings and a number of other methods developed by the authors for the purposes of this study. We obtained rigorous qualitative and quantitative results. More than 250 topologically distinct phase portraits of various systems were constructed. The absence of limit cycles of systems of this family is proved. Methods developed can be useful for the further study of systems with polynomial right-hand sides of other forms. [ABSTRACT FROM AUTHOR]

The theory of calculating the curvature interference limit line of the offset involute cylindrical worm drive based on the singular point condition on the enveloping surface is established. The tooth surface equations and the meshing function of the worm drive are derived. Based on the singular point condition that the normal vector of the enveloping surface is zero, the equation of the curvature interference limit line is obtained. By employing the resultant elimination method and the geometric construction, the existence of the solutions of the curvature interference limit line is determined, and the reasonable initial values for the iteration program are afforded. The numerical outcomes show that there is one meaningful curvature interference limit line on each flank of single tooth of the worm gear, which usually does not enter the worm gear tooth surface. Both flanks of the worm gear are all settled in the side of the limit line, the curvature interference does not happen on the worm gear tooth surface. The undercutting of the cutting engagement process generally does not happen. The results also reveal that the curvature interference limit line is closer to the addendum and heel of the face worm gear on the i flank, which has the greatest potential venture inflict the undercutting. [ABSTRACT FROM AUTHOR]

Applying Geometric Invariant Theory (GIT), we study the stability of foliations of degree 3 on P 2 with a unique singular point of multiplicity 1, 2, or 3 and Milnor number 13. In particular, we characterize those foliations for multiplicity 2 in three cases: stable, strictly semistable, and unstable. [ABSTRACT FROM AUTHOR]

For a nonlightlike nondegenerate regular curve, we have the arc-length parameter and the Frenet-Serret type formula by using a moving frame like a regular space curve in the Euclidean space. If a point of the curve moves between spacelike and timelike regions, then there is a lightlike point. In this paper, we consider mixed types of not only regular curves but also curves with singular points. In order to consider mixed type of curves with singular points, we introduce a frame, so-called the lightcone frame, and lightcone framed curves. We investigate differential geometric properties of lightcone framed curves. [ABSTRACT FROM AUTHOR]

In this paper, we focus on using the low-rank factorization approach to solve the SDP relaxation of a graph equipartition problem, which involves an additional spectral upper bound over the traditional linear SDP. We discuss the equivalence between the decomposed problem and the original SDP problem. We also derive a sufficient condition, under which a second order stationary point of the non-convex problem is also a global minimum. Moreover, the constraints of the non-convex problem involve an algebraic variety with conducive geometric properties which we analyse. We also develop a method to escape from a non-optimal singular point on this variety. This allows us to use Riemannian optimization techniques to solve the SDP problem very efficiently with certified global optimality. [ABSTRACT FROM AUTHOR]

A singular nonlinear differential equation zσdwdz=aw+zwf(z,w),{z}^{\sigma }\frac{{\rm{d}}w}{{\rm{d}}z}=aw+zwf\left(z,w), where σ>1\sigma \gt 1, is considered in a neighbourhood of the point z=0z=0 located either in the complex plane C{\mathbb{C}} if σ\sigma is a natural number, in a Riemann surface of a rational function if σ\sigma is a rational number, or in the Riemann surface of logarithmic function if σ\sigma is an irrational number. It is assumed that w=w(z)w=w\left(z), a∈C⧹{0}a\in {\mathbb{C}}\setminus \left\{0\right\}, and that the function ff is analytic in a neighbourhood of the origin in C×C{\mathbb{C}}\times {\mathbb{C}}. Considering σ\sigma to be an integer, a rational, or an irrational number, for each of the above-mentioned cases, the existence is proved of analytic solutions w=w(z)w=w\left(z) in a domain that is part of a neighbourhood of the point z=0z=0 in C{\mathbb{C}} or in the Riemann surface of either a rational or a logarithmic function. Within this domain, the property limz→0w(z)=0{\mathrm{lim}}_{z\to 0}w\left(z)=0 is proved and an asymptotic behaviour of w(z)w\left(z) is established. Several examples and figures illustrate the results derived. The blow-up phenomenon is discussed as well.

The paper is a continuation and completion of the paper Bruno, A.D.; Azimov, A.A. Parametric Expansions of an Algebraic Variety Near Its Singularities. Axioms 2023, 5, 469, where we calculated parametric expansions of the three-dimensional algebraic manifold Ω , which appeared in theoretical physics, near its 3 singular points and near its one line of singular points. For that we used algorithms of Nonlinear Analysis: extraction of truncated polynomials, using the Newton polyhedron, their power transformations and Formal Generalized Implicit Function Theorem. Here we calculate parametric expansions of the manifold Ω near its one more singular point, near two curves of singular points and near infinity. Here we use 3 new things: (1) computation in algebraic extension of the field of rational numbers, (2) expansions near a curve of singular points and (3) calculation of branches near infinity. [ABSTRACT FROM AUTHOR]

We propose the method for numerical solution of four-wave kinetic equations that arise in the wave turbulence (weak turbulence) theory when describing a homogeneous isotropic interaction of waves. To calculate the collision integral in the right-hand side of equation, the cubature formulas of high rate of convergence are developed, which allow for adaptation of the algorithm to the singularities of the solutions and of the integral kernels. The convergence tests in the problems of integration arising from real applications are done. To take into account the multi-scale nature of turbulence problems in our algorithm, rational approximations of the solutions and a new time marching scheme are implemented and tested. The efficiency of the developed algorithm is demonstrated by modelling the inverse cascade of Bose gas particles during the formation of a Bose–Einstein condensate. [ABSTRACT FROM AUTHOR]

We study Nijenhuis operators, that is, .1; 1/-tensors with vanishing Nijenhuis torsion under the additional assumption that they are gl-regular, i.e., every eigenvalue has geometric multiplicity one. We prove the existence of a coordinate system in which the operator takes first or second companion form, and give a local description of such operators. We apply this local description to study singular points. In particular, we obtain normal forms of gl-regular Nijenhuis operators near singular points in dimension two and discover topological restrictions for the existence of gl-regular Nijenhuis operators on closed surfaces. [ABSTRACT FROM AUTHOR]

It is proved that if an anisotropic medium has an open set of singular directions, then this medium has two slowness surfaces that completely coincide. The coinciding slowness surfaces form one double singular slowness surface. The corresponding anisotropic medium is an elliptical orthorhombic (ORT) medium with equal stiffness coefficients c44=c55=c66 rotated to an arbitrary coordinate system. Based on the representation of the Christoffel matrix as a uniaxial tensor and considering that the elements of the Christoffel matrix are quadratic forms in the components of the slowness vector, a system of homogeneous polynomial equations was derived. Then, the identical equalities between homogeneous polynomials are replaced by the equalities between their coefficients. As a result, a new system of equations is obtained, the solution of which is the values of the reduced (density normalized) stiffness coefficients in a medium with a singular surface. Conditions for the positive definite of the obtained stiffness matrix are studied. For the defined medium, the Christoffel equations and equations of group velocity surfaces are derived. The orthogonal rotation matrix that transforms the medium with a singular surface into an elliptic ORT medium in the canonical coordinate system is determined. In the canonical coordinate system, the slowness surfaces S1 and S2 waves coincide and are given by a sphere with a radius . The slowness surface of qP waves in the canonical coordinate system is an ellipsoid with semi-axes , , . The polarization vectors of S1 and S2 waves can be arbitrarily selected in the plane orthogonal to the polarization vector of the qP wave. However, the qP wave polarization vector can be significantly different from the wave vector. This feature should be taken into account in the joint processing and modelling of S and qP waves. The results are illustrated in one example of an elliptical ORT medium.

In the paper, series of exponential monomials are considered. We study the problem of the distribution of singular points of the sum of a series on the boundary of its domain of convergence. We study the conditions under which, for any sequence of coefficients of the series with a chosen domain of convergence, the domain of existence of the sum of this series coincides with the given domain of convergence. We consider sequences of exponents having an angular density (measurable) and the zero condensation index. Various criteria related to the distribution of singular points of the sum of a series of exponential monomials on the boundary of its convergence domain are obtained. In particular, in the class of the indicated sequences, a criterion is obtained that all boundary points of a chosen convex domain are special for any sum of a series with a given domain of convergence. The criteria are formulated using simple geometric characteristics of the sequence of exponents and a convex domain (the angular density and the length of the boundary arc). It is also shown that the condition that the condensation index is equal to zero is essential. [ABSTRACT FROM AUTHOR]

Analysis of traffic flow signals plays an important role in traffic prediction and management. As an intrinsic property, the singular point of a traffic flow signal labels a new nonsteady status. Therefore, detecting the singular point is an effective approach to determine the moment of traffic flow prediction. In this paper, an improved wavelet transform is proposed to detect singular points of real-time traffic flow signals. The number of detected singular points is output via the heuristic selection of multiple scales. Then, a weighted similarity measurement of historical traffic flow signals is utilized to predict the next singular point. The position of the next singular point decides the duration of prediction adaptively. The detected and predicted singular points are applied to dynamically update the unequal interval prediction of traffic flow. Furthermore, a Vasicek model is used to predict the traffic flow by minimizing the sum of the relative mean standard error (RMSE) between the traffic flow increment in the predicted interval and the sampled increments of previous intervals. A decomposition method is used to solve the unequal matrix problem. Based on the scenario and traffic flow imported from the real-world map, the simulation results show that the proposed algorithm outperforms existing approaches with high prediction accuracy and much lower computing cost. [ABSTRACT FROM AUTHOR]

Presently, there is a method based on Power Geometry that allows one to find asymptotic forms and asymptotic expansions of solutions to different kinds of non-linear equations near their singularities. The method contains three algorithms: (1) Reducing the equation to its normal form, (2) separating truncated equations, and (3) power transformations of coordinates. Here, we describe the method for the simplest case, a single algebraic equation, and apply it to an algebraic variety, as described by an algebraic equation of order 12 in three variables. The variety was considered in study of Einstein's metrics and has several singular points and singular curves. Near some of them, we compute a local parametric expansion of the variety. [ABSTRACT FROM AUTHOR]

Singularity representation at the crack tip and/or smaller region will be considered using six scale transitional physical parameters: three assigned for the nano/micro range (µ* na/mi, σ* na/mi, d* na/mi) and three assigned for the micro/macro range µ* mi/ma, σ* mi/ma, d* mi/ma). The subscripts nano, micro, and macro are self-evident. Only the ratio of two successive scale sensitive parameters are needed. Although time dependent physical parameters at the lower scale can be found analytical, they regarded as fictitious, mainly because they are not conducible to measurements. The transitional character of multiscale changes according to nano→micro→macro with the respective singularity strength of λ are given by 1.00/0.75/0.50. Since λ=05 corresponds to the inverse square root r-0.5, where r is the distance from the macro singular point. The micro and nano singular point possess the singularities r-0.75 and r-1.00, respectively. For example, a critical device component may be designed to operate at the nano/micro/macro scale with a life distribution of 2.5+ / 3.5+ / 5.5+ and total life of 11.5+ years. Progressive changes are assumed to occur in the direction of nano→micr→macro. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlinear q-Sturm-Liouville problem on the whole real axis in which the limit-circle case holds for q-Sturm-Liouville expression at infinity. We established the existence and uniqueness of solutions for this problem. [ABSTRACT FROM AUTHOR]

The paper is a continuation and completion of the paper Bruno, A.D.; Azimov, A.A. Parametric Expansions of an Algebraic Variety Near Its Singularities. Axioms 2023, 5, 469, where we calculated parametric expansions of the three-dimensional algebraic manifold Ω, which appeared in theoretical physics, near its 3 singular points and near its one line of singular points. For that we used algorithms of Nonlinear Analysis: extraction of truncated polynomials, using the Newton polyhedron, their power transformations and Formal Generalized Implicit Function Theorem. Here we calculate parametric expansions of the manifold Ω near its one more singular point, near two curves of singular points and near infinity. Here we use 3 new things: (1) computation in algebraic extension of the field of rational numbers, (2) expansions near a curve of singular points and (3) calculation of branches near infinity.

The position of singular lines for orthorhombic (ORT) media with fixed diagonal elements of the elasticity matrix cij, i=1…6 is studied under the condition that c11, c22, c33>c66>c44>c55. In this case, the off-diagonal coefficients of the elasticity matrix c12, c13, c23 are chosen so that some of the values of d12=c12+c66, d13=c13+c55, d23=c23+c44 are zero. For orthorhombic medium, where the only one of d12, d13, d23 is zero, contains only singular points in the planes of symmetry. If two or all three dij are zero, then the ORT medium contains singular lines and discrete singular points. We call such media pathological. A degenerate ORT medium with positive d12, d13, d23 has at most two singular lines, which are the intersection of a quadratic cone with a sphere. The pathological media may have up to 6 singular lines on the surface of the slowness. Singular lines for pathological media are described by more complex equations than conventional degenerate ORT models. The article proposes to using squares x, y, z of the components of the slowness vector in the equations. In a new coordinate system, equations defining singular lines for pathological media become linear or quadratic. Intersecting with the plane x+y+z =1, they define the straight lines, ellipses, or hyperbolas. If non-zero values d12, d13, d23 increase, the singular lines pass through four fixed points on the plane x+y+z =1, which makes it possible to describe the evolution of their change. Conditions are derived under which the singular curves of pathological ORT models are limiting the singular curves for degenerate ORT models with positive values of d12, d13, d23. Formulas are derived for transforming surfaces of slowness and singular lines of pathological media into the region of group velocities. The results are demonstrated with examples of pathological models obtained from the standard model of the ORT medium by changing the elasticity coefficients c12, c13, c23 so that some of the values d12, d13, d23 are zero

Using the method of different scales, formulas for the hydrodynamic fields of acoustic-gravity waves (AGWs) with vertical wavelengths that are small compared to the scales of changes in the background temperature and wind fields are derived. These formulas are equivalent to the conventional WKB approximation, but explicitly include the vertical gradients of the background fields. The conditions for the applicability of the formulas for describing the propagation of AGWs from the troposphere to the thermosphere are formulated and analyzed. The absence of singular points (critical levels) in the equations for wave modes in the analyzed height range is one of the conditions for the applicability of approximate formulas. For the wind from the empirical HWM model, singular points are often located below 200 km and are typical for internal gravity waves (IGWs) with lengths on the order of 10 km. As the wavelength increases, the number of singular points decreases. For IGWs with scales on the order of 300 km or more, there are usually no singular points. It is shown that IGWs with periods of less than 20 min propagating upward from tropospheric heights usually have one turning point in the altitude range from 100 to 130 km. The formulas are useful, in particular, for parametrizing the effects of AGWs in numerical models of atmospheric dynamics and energy. [ABSTRACT FROM AUTHOR]

In this paper, we have described the B-Lift curve in Euclidean space as a curve obtained by combining the endpoints of the binormal vector of a unit speed curve. Subsequently, we have explored the Frenet frames of the B-Lift curves. Moreover, we have introduced the tangent, normal and binormal surfaces of the B-Lift curve and examined the geometric invariants of these surfaces. Finally, we have investigated the singularities of these surface and visualized the surfaces with MATLAB program. [ABSTRACT FROM AUTHOR]

In this article, we investigate the resolvent of dynamic Dirac operators on unbounded time scales. For the resolvent of these operators, integral representations are obtained. Finally, a formula for the Titchmarsh-Weyl function is given. [ABSTRACT FROM AUTHOR]

Monte Carlo method and envelope method are used to analyze the cooperative space and singular points of multi manipulator cooperative system model. The standard D-H parameter method is used to establish the coordinate system， the kinematics model of the robot is established in Matlab， and the forward kinematics equation of the robot is obtained. The workspace of each manipulator is obtained by Monte Carlo method， the common area points are extracted by envelope method， and the cooperation space of the multi manipulator cooperation system and the joint angle range of each manipulator in the cooperation space are obtained. An algorithm to calculate the operability is presented. According to the algorithm， the singularity distribution of multi manipulator in the cooperative space is calculated and analyzed. The Matlab/Robotics toolbox is used to build the simulation platform， and the algorithm is compiled for simulation analysis， which verifies the correctness and rationality of the kinematics model analysis of the multi robot cooperative system， the feasibility of the proposed algorithm is proved and the foundation for the subsequent coordinated operation and trajectory planning to avoid singular points is laid.

We consider φ ⁎ T P 3 , the pull-back of T P 3 (the tangent bundle to P 3) via a generically one-to-one parametrization φ of a rational curve D of degree d D ≥ 3 and spanning P 3. We study the splitting of φ ⁎ T P 3 as direct sum of line bundles in terms of ancillary curve(s) of degree strictly smaller than d D and spanning P r with r ≤ 3. The degrees of these line bundles are determined by (i) the largest multiplicity of D at a point (if the largest multiplicity is greater than or equal to ⌊ d D / 2 ⌋) and (ii) the largest multiplicities at a non-coplanar quadruple of points (if all the multiplicities of D are strictly smaller than ⌊ d D / 2 ⌋). [ABSTRACT FROM AUTHOR]

The authors analyze the special properties of two classes of quadratic and twisted Edwards curves over a prime field, which take into account their non-cyclic structure and the incompleteness of the point addition law. Both classes of curves contain singular points of 2nd and 4th orders with respect to one infinite coordinate, which generate points with uncertainty 0/0 in one of the coordinates of the sum, called fuzzy points. Five theorems are formulated and proved, which allow resolving these uncertainties and establishing the conditions whereby the point addition law in these classes of curves is complete. [ABSTRACT FROM AUTHOR]

The paper is devoted to the investigation of the second order differential equation of the Riemann-type with three and many singular points. A three-pointed differential operator is introduced, and with the help of this operator, a second-order operator-differential equation is constructed. Further, it is shown the equivalence of the constructed operator-differential equation to the second-order Riemann equation with three singular points, but only with the distinction that the Fuchs condition is replaced by a Fuchs-type condition with zero right-hand side. It is also obtained the explicit solutions of the second order operator-differential equation with many singular points. Some explicit elementary solution of the Riemann–Gilbert problem for the second order differential equation with three singular points is obtained. The solution of the constructed equations in depending of the roots of the characteristic equation is found in two cases: when the roots are different and when they are multiple. [ABSTRACT FROM AUTHOR]

The results of experimental study of the excessive hypersound absorption in an aqueous acetone solution are discussed based on the conclusions of the theory of high-frequency sound scattering near the critical point (developed by Chaban) and the Landau theory. These results are described within the framework of the Landau and Chaban theories and explained by the existence of two different states with minimum thermodynamic stability in the solution. The main conclusions of the theory are presented, and a method for measuring the absorption coefficient of hypersound near the temperature of the unstable thermodynamic state is described. [ABSTRACT FROM AUTHOR]

Background. Searching of numbers of Poincare limit cycles of polynomial dynamic systems belongs to second part of the 16th Gilbert problem, which is not solved in general. The purpose of this work is generalization of earlier results for the generalized Kukles system and new estimation of numbers of limit cycles the Kukles system 10 degree is got. Materials and methods. The methods of qualitative theory of dynamic systems and averaging theory were applied. Results. Singular points were researched of the generalized Kukles polynomial differential system and classification of phase portrait in the Poincare disc was showed. In addition, the program, which accelerated researching of numbers of limit cycles, was written using average theory. For the first time numbers of limit cycles for the Kukles system 10 degree depending on average degree are got. Conclusions. The classification of global phase portrait in the Poincare disc finishes a question about probable trajectory the generalized polynomial Kukles system. There is a potential for the future researching to get accurate assessment of numbers of limit cycles in respect to degree of the system without using of the program. In the future we are going to get analytic dependence numbers of limit cycles on system and average degrees.

Analysis of traffic flow signals plays an important role in traffic prediction and management. As an intrinsic property, the singular point of a traffic flow signal labels a new nonsteady status. Therefore, detecting the singular point is an effective approach to determine the moment of traffic flow prediction. In this paper, an improved wavelet transform is proposed to detect singular points of real-time traffic flow signals. The number of detected singular points is output via the heuristic selection of multiple scales. Then, a weighted similarity measurement of historical traffic flow signals is utilized to predict the next singular point. The position of the next singular point decides the duration of prediction adaptively. The detected and predicted singular points are applied to dynamically update the unequal interval prediction of traffic flow. Furthermore, a Vasicek model is used to predict the traffic flow by minimizing the sum of the relative mean standard error (RMSE) between the traffic flow increment in the predicted interval and the sampled increments of previous intervals. A decomposition method is used to solve the unequal matrix problem. Based on the scenario and traffic flow imported from the real-world map, the simulation results show that the proposed algorithm outperforms existing approaches with high prediction accuracy and much lower computing cost.

Presently, there is a method based on Power Geometry that allows one to find asymptotic forms and asymptotic expansions of solutions to different kinds of non-linear equations near their singularities. The method contains three algorithms: (1) Reducing the equation to its normal form, (2) separating truncated equations, and (3) power transformations of coordinates. Here, we describe the method for the simplest case, a single algebraic equation, and apply it to an algebraic variety, as described by an algebraic equation of order 12 in three variables. The variety was considered in study of Einstein’s metrics and has several singular points and singular curves. Near some of them, we compute a local parametric expansion of the variety.

This paper analyzes the properties of the nonequilibrium singular point in one-dimensional elementary catastrophe. For this analysis, the Kosambi–Cartan–Chern (KCC) theory is applied to characterize the dynamical system based on differential geometrical quantities. When both the nonlinear connection and deviation curvature are zero, that is, when the geometric stability of the KCC theory is neutral, two bifurcation curves are obtained: one is the known curve with an equilibrium singular point, and the other is a new curve with a nonequilibrium singular point. The two singular points are distinguished based on the vanishing condition of the Berwald connection. Applied to the ecosystem described by the Hill function, the absolute value of the cuspidal curvature of the nonequilibrium singular point is larger than that of the equilibrium singular point. The ecological interpretation of this result is that the range of bistability of the ecosystem in the nonequilibrium state is greater than that in the equilibrium state. The type of singular points in equilibrium and nonequilibrium bifurcation curves are not necessarily the same. For instance, there is a combination in which even if the former has one cusp, the latter may show various types, depending on the parametric space. These results demonstrate that there are cases where simply shifting the system from the equilibrium to nonequilibrium state expands the range of bistability and changes the type of singularity. Although singularity analysis is often performed near the equilibrium point, nonequilibrium analysis, i.e. analysis based on the KCC theory, provides a useful perspective for analyzing singularity theory according to the bifurcation phenomenon. [ABSTRACT FROM AUTHOR]

In this work, we consider one dimensional Dirac operators on unbounded time scales. We construct a spectral function of such an operator. Using this function, we establish a Parseval equality and an expansion formula in eigenfunctions for the Dirac operator on unbounded time scales. [ABSTRACT FROM AUTHOR]

A nonlinear Hahn–Sturm–Liouville problem on (−∞,∞) is studied. The existence and uniqueness of the solutions for such equations are proved. [ABSTRACT FROM AUTHOR]

Detecting the singular point accurately and efficiently is one of the most important tasks for fingerprint recognition. In recent years, deep learning has been gradually used in the fingerprint singular point detection. However, the existing deep learning-based singular point detection methods are either two-stage or multi-stage, which makes them time-consuming. More importantly, their detection accuracy is yet unsatisfactory, especially for the low-quality fingerprint. In this paper, we make a Real One-Stage Effort to detect fingerprint singular points more accurately and efficiently, and therefore, we name the proposed algorithm ROSE for short, in which the multi-scale spatial attention, the Gaussian heatmap and the variant of focal loss are integrated together to achieve a higher detection rate. Experimental results on the datasets FVC2002 DB1 and NIST SD4 show that our ROSE outperforms the state-of-the-art algorithms in terms of detection rate, false alarm rate and detection speed. [ABSTRACT FROM AUTHOR]