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The problem of the spatial motion of a passively gravitating body during an to the central body of a perturbing body – a test star – is considered. Using the exact expression of the force function, an integral invariant relationship – a quasi-integral – was found. Due to the quasi-integral, the regions of possible motion of the passively gravitating body, the surfaces of minimal energy (a generalization of the zero velocity surfaces), and the singular points of these surfaces were determined. The stability of planetary motion according to Hill during the approach of a test star to the Solar System was investigated. Criteria for the possibility, as well as the impossibility of capturing the passively gravitating body by the test star, were established. According to the Hill stability criteria, critical values of the orbital parameters of the test star were established, at which the planets of the Solar System either become satellites of the test star or leave the bounds of the Solar System.

The paper studies a nonlinear equation including both fractional and ordinary derivatives. The singular Cauchy problem is considered. The theorem of the existence of uniqueness of a solution in the neighborhood of a fixed singular point of algebraic type is proved. An analytical approximate solution is built, an a priori estimate is obtained. A formula for calculating the area where the proven theorem works is obtained. The theoretical results are confirmed by a numerical experiment in both digital and graphical versions. The technology of optimizing an a priori error using an a posteriori error is demonstrated. [ABSTRACT FROM AUTHOR]

The propagation of slit shear waves in a quasi-symmetric structure of piezoelectrics of the 4mm symmetry class has been theoretically studied. It has been shown that taking into account the unequal level of loss and gain in piezoelectrics results in either intersection, touching, or convergence of two modes at the point of their degeneracy (singular point) in the shear wave spectrum. It has been found that the intersection of the mode spectra occurs only in the case of equal loss and gain magnitudes (a PT-symmetric structure). Based on this, it is concluded that using the pattern of the spectra near a singular point, it is possible to determine the level of imbalance of gain and loss in piezoelectric waveguides. As in the case of a purely PT-symmetric structure, the frequency dependence of the amplitude at an exceptional point of a quasi PT-symmetric structure (with a rather small difference in loss and gain levels) exhibits a very narrow peak, which opens up the possibility of creating hypersensitive sensors based on them. Thus, it has been shown that even at unequal loss and gain levels in piezoelectrics (a quasi PT-symmetric structure), it is possible to obtain a structure with all the characteristics of a PT-symmetric structure. [ABSTRACT FROM AUTHOR]

The classical and the new Euler-Jacobi formulae for simple and double points provide an algebraic relation between the singular points of a polynomial vector field and their topological indices. Using these formulae we obtain the geometrical configuration of the singular points together with their topological indices for several classes of polynomial differential systems in R n when these differential systems, having the maximum number of singular points, either all their singular points are simple, or at most one singular point is double (i.e. it has multiplicity two). [ABSTRACT FROM AUTHOR]

The purpose of this paper is to develop a practical strategy to accelerate Newton's method in the vicinity of singular points. We present an adaptive safeguarding scheme with a tunable parameter, which we call adaptive γ-safeguarding, that one can use in tandem with Anderson acceleration to improve the performance of Newton's method when solving problems at or near singular points. The key features of adaptive γ-safeguarding are that it converges locally for singular problems, and it can detect nonsingular problems automatically, in which case the Newton-Anderson iterates are scaled towards a standard Newton step. The result is a flexible algorithm that performs well for singular and nonsingular problems, and can recover convergence from both standard Newton and Newton-Anderson with the right parameter choice. This leads to faster local convergence compared to both Newton's method, and Newton-Anderson without safeguarding, with effectively no additional computational cost. We demonstrate three strategies one can use when implementing Newton-Anderson and γ-safeguarded Newton-Anderson to solve parameterdependent problems near singular points. For our benchmark problems, we take two parameter-dependent incompressible flow systems: flow in a channel and Rayleigh-Bénard convection. [ABSTRACT FROM AUTHOR]

Background. Visual non-destructive testing of the inner surface of pipes is an important aspect in their production and operation. A defect detected and corrected in a timely manner can significantly reduce the number of defects in production and prevent various emergency incidents during operation. Formation of a complete panoramic image of the inner surface of pipes suitable for quality analysis is an urgent and sought-after task that can be solved using computer vision systems. Aim. This work is the research and development of television methods for forming a complete panoramic image of the inner surface of a pipe, which can be analyzed to search for defects. Methods. To form a cylindrical panoramic image, mathematical models for the formation of an equidistant projection of spherical images obtained using a fisheye lens were used. For high-quality stitching of the resulting frames, digital image processing methods were used, including brightness and contrast transformations, and searching for special points using the MSER algorithm. Theoretical results are verified by full-scale modeling. Results. The result of this work is an algorithm for stitching frames of a video sequence generated by a television camera with a fisheye optical system, uniformly moved along the longitudinal axis of the pipe, into a single panoramic image of the internal surface. Conclusion. The algorithm ensures the formation of a high-quality image of a full panorama of the inner surface of the pipes with the absence of brightness artifacts.

The problem of the spatial motion of a passively gravitating body during an to the central body of a perturbing body – a test star – is considered. Using the exact expression of the force function, an integral invariant relationship – a quasi-integral – was found. Due to the quasi-integral, the regions of possible motion of the passively gravitating body, the surfaces of minimal energy (a generalization of the zero velocity surfaces), and the singular points of these surfaces were determined. The stability of planetary motion according to Hill during the approach of a test star to the Solar System was investigated. Criteria for the possibility, as well as the impossibility of capturing the passively gravitating body by the test star, were established. According to the Hill stability criteria, critical values of the orbital parameters of the test star were established, at which the planets of the Solar System either become satellites of the test star or leave the bounds of the Solar System. [ABSTRACT FROM AUTHOR]

In this paper we construct several arrangements of lines and/or conics that are derived from the geometry of the Klein arrangement of 21 lines in the complex projective plane. [ABSTRACT FROM AUTHOR]

Let Σ (f) be the singular points of a polynomial f ∈ K [ x , y ] in the plane K 2 , where K is R or C . Our goal is to study the singular point map S d , it sends polynomials f of degree d to their singular points Σ (f) . Very roughly speaking, a polynomial f is essentially determined when any other g sharing the singular points of f satisfies that f = λ g ; here both are polynomials of degree d, λ ∈ K ∗ . In order to describe the degree d essentially determined polynomials, a computation of the required number of isolated singular points δ (d) is provided. A dichotomy appears for the values of δ (d) ; depending on a certain parity, the space of essentially determined polynomials is an open or closed Zariski set. We compute the map S 3 , describing under what conditions a configuration of 4 points leads to a degree 3 essentially determined polynomial. Furthermore, we describe explicitly configurations supporting degree 3 non essential determined polynomials. The quotient space of essentially determined polynomials of degree 3 up to the action of the affine group Aff (K 2) determines a singular K -analytic surface. [ABSTRACT FROM AUTHOR]

Interpolation of a point series is a necessary step in solving such problems as building graphs de-scribing phenomena or processes, as well as modelling based on a set of reference points of the line frames defining the surface. To obtain an adequate model, the following conditions are imposed upon the interpolating curve: a minimum number of singular points (kinking points, inflection points or points of extreme curvature) and a regular curvature change along the curve. The aim of the work is to develop the algorithm for assigning characteristics (position of normals and curvature value) to the interpolating curve at reference points, at which the curve complies with the specified conditions. The characteristics of the curve are assigned within the area of their possible location. The possibilities of the proposed algorithm are investigated by interpolating the point series assigned to the branches of the parabola. In solving the test example, deviations of the normals and curvature radii from the corresponding characteristics of the original curve have been determined. The values obtained confirm the correctness of the solutions proposed in the paper.

The generalized CR equation u z ¯ = a u + b u ¯ + f is studied when the coefficients a and b have a finite number of singular points inside the domain. Solutions are constructed via the study of an associated integral operator and the existence of nontrivial solutions of the associated homogeneous equation is established. [ABSTRACT FROM AUTHOR]

In this study, the ruled developable surfaces with pointwise 1-type Gauss map of Frenet-type framed base (Ftfb) curve are introduced in Euclidean 3-space. The tangent developable surfaces, focal developable surfaces, and rectifying developable surfaces with singular points are considered. Then the conditions for the Gauss map of these surfaces to be pointwise 1-type are obtained separately. In order to form a basis for the study, first, the basic concepts related to the Ftfb curve and the Gauss map of a surface are recalled. Later, the necessary and sufficient conditions are found for these surfaces to be of the pointwise 1-type of the Gauss map. Finally, examples for each type of these surfaces are given, and their graphics are illustrated.

Given an L ∞ -algebra V and an L ∞ -subalgebra W, we give sufficient conditions for all small Maurer–Cartan elements of V to be equivalent to Maurer–Cartan elements lying in W. As an application, we obtain a stability criterion for fixed points of a Dirac structure in an arbitrary Courant algebroid of split signature (for instance a twisted Poisson structure), i.e., points where the corresponding leaf is zero-dimensional. The criterion guarantees that any nearby Dirac structure also has a fixed point. [ABSTRACT FROM AUTHOR]

Resolving equations describing stationary flows of an incompressible viscoelastic polymer fluid through a circular pipe are derived on the basis of the rheological mesoscopic Pokrovskii–Vinogradov model. Exact solutions of the equations are obtained, and constraints on the values of rheological parameters that ensure their existence are formulated. These results enable one to constructively describe the breakdown of Poiseuille-type laminar flow. The size and orientation of polymer fluid macromolecules play a key role in the mechanics of this process. The mathematical description of the process uses singular points of the solutions. [ABSTRACT FROM AUTHOR]

In this paper, we consider the singular points of suitable weak solutions to the 3D co-rotational Beris–Edwards system modeling the hydrodynamical motion of nematic liquid crystal flows, which is a coupled system with the Navier–Stokes equations for the fluid and a parabolic system of Q-tensor for the liquid average orientation. We prove that if (u , Q) defined on R 3 × (0 , T) is a suitable weak solution to the 3D co-rotational Beris–Edwards system, and satisfies ‖ (u , ∇ Q) ‖ L q , ∞ (0 , T ; L p (R 3)) < ∞ with 3 < p < ∞ and 2 q + 3 p = 1 , then for a given open subset Ω ⊆ R 3 and for a given moment of time t 0 ∈ (0 , T) , the number of points of the set Σ (t 0) ∩ Ω is finite, where Σ (t 0) ≡ { (x , t 0) ∈ Σ } and Σ is the set of singular points for (u , Q) . Moreover, if T 1 ∈ (0 , T) is the first time for singularity appears, and if (u , Q) satisfies ‖ (u , ∇ Q) (· , t) ‖ L p (R 3) ≤ c 0 (T 1 - t) p - 3 2 p for all 0 < t < T 1 , with 3 < p ≤ ∞ and c 0 is a postive constant, then we show that (u , Q) preserves the energy equality on the closed interval [ 0 , T 1 ] including the first blow-up time T 1 . [ABSTRACT FROM AUTHOR]

The phase equilibrium in an acetonitrile + water + cyclohexene + chloroform system was studied at 101.3 kPa. A prediction regarding the internal structure of the composition tetrahedron (presence/absence of one/two internal singular points) was made using thermodynamic modeling in AspenPlus V.10.0. The existence of two internal quaternary azeotropes (of node type with a minimum boiling point and of saddle type) was confirmed as a result of a full-scale experiment. Thermodynamic-topological analysis of the structure of phase equilibrium diagrams was carried out to confirm the correctness of the diagram construction. [ABSTRACT FROM AUTHOR]

Mechanical fields over thin elastic surfaces can develop singularities at isolated points and curves in response to constrained deformations (e.g., crumpling and folding of paper), singular body forces and couples, distributions of isolated defects (e.g., dislocations and disclinations), and singular metric anomaly fields (e.g., growth and thermal strains). With such concerns as our motivation, we model thin elastic surfaces as von Kármán plates and generalize the classical von Kármán equations, which are restricted to smooth fields, to fields which are piecewise smooth, and can possibly concentrate at singular curves, in addition to being singular at isolated points. The inhomogeneous sources to the von Kármán equations, given in terms of plastic strains, defect induced incompatibility, and body forces, are likewise allowed to be singular at isolated points and curves in the domain. The generalized framework is used to discuss the singular nature of deformation and stress arising due to conical deformations, folds, and folds terminating at a singular point. [ABSTRACT FROM AUTHOR]

The paper consideres linear systems of ordinary differential equations of arbitrary order with a matrix identically degenerate in the domain of definition at the highest derivative of the desired vector function and with loads in the form of Volterra and Fredholm integral operators. The initial value problems are formulated using projections onto admissible sets of initial vectors. Special attention is paid to systems having singular points on the interval of integration. The concept of a singular point is formalized. Their classification in the case of differential equations is given. A number of examples illustrating the theoretical results are presented. [ABSTRACT FROM AUTHOR]

This study presents an advanced computational Levenberg–Marquardt backpropagation (LMB) neural network for the novel third order (NTO) pantograph Emden–Fowler system (PEFS), i.e., (NTO-PEFS) together with its two forms. The designed novel NTO-PEFS is achieved using the pantograph system and standard form of the Emden–Fowler system. The detail of each form of the NTO-PEFS based on the singular points, pantographs and shape factors is also provided. The numerical performance using the LMB neural network is tested for three different variants of the model and obtained results will be compared through the designed dataset based exact solutions. To assess the approximate solutions of the NTO-PEFS for both forms of each example, the process of testing, authentication and training are implemented to reduce the mean square error (MSE) based on the LMB. One can find the values based absolute error are close to 10–04 to 10–08 for each problem to solve the NTO-PEFS using the stochastic computing paradigms. The relative studies and performance investigations for the error histograms, regression, correlation and MSE enhance the effectiveness as well as the exactness of the designed LMB neural network scheme. [ABSTRACT FROM AUTHOR]

The article presents the construction and study of a mathematical model of small oscillations of an aerodynamic pendulum in a flow of a moving medium. The model of the impact of the medium on the body is taken as the model of the quasi-static flow around the plate by the medium. According to this hypothesis, the aerodynamic forces acting on the body are applied at the center of pressure. In our task, the pressure centre is movable relative to the plate. The equations of motion for the considered body are obtained. A transition to new dimensionless variables is carried out. Uniqueness is shown to be impaired when determining the angle of attack at points where the air velocity of the pressure center is near zero. Some areas of ambiguity are constructed using multiple solutions of algebraic nonlinear equations obtained from kinematic relations. This method of creating areas of ambiguity is rather laborious and does not allow it to be done within a wide range of phase variables. An easier way of dealing with ambiguity areas is to define the coordinates of return points and draw the boundaries of ambiguity areas as two envelopes. The algorithm for obtaining these envelopes is described in detail. In the mathematical package MATLAB 18, a program is written that builds the upper and lower boundaries of the ambiguity regions. With its help, a parametric analysis of folds was carried out for various values of phase variables. It is shown that the sizes of the obtained areas of ambiguity are small. It is specified that the change of the angle of the attack during integration must be continuous, otherwise there is a failure in the search for a numerical solution. A set of programs has also been created, which makes it possible to build areas of ambiguity by repeatedly solving a nonlinear equation. Thus, a mathematical model of plate vibrations has been developed and a geometric analysis of surfaces and folds has been carried out using a set of programs based on a specialized computer mathematics system MATLAB 18. In this way, a very interesting problem in which there is a multi-valued angle of attack is considered, which is more understandable if we consider a mechanical interpretation, but non-trivial from the point of view of the general theory of solving differential equations.

Last decades have been characterized by increasing global climate change that is associated with an increase in the content of greenhouse gases in the Earth's atmosphere. The main greenhouse gases include water vapor, carbon dioxide, ozone, nitrogen oxides and freons. Carbon biogeochemical cycles researches are of particular relevance. Wetland ecosystems are one of the biogeochemical cycles significant reservoirs. It is known that wetland contain huge reserves of carbon in the form of weakly decomposed organic matter and they are active sources of methane and a sink of carbon dioxide from the atmosphere. A dynamic model of biogeochemical carbon cycles in swamp ecosystems proposed. Qualitative research of ordinary differential equations system was carried out, the types of singular points were investigated, integral curves and phase portraits were constructed. [ABSTRACT FROM AUTHOR]

We consider linear systems of ordinary differential equations with a singular matrix that multiplies a higher derivative higher derivative of the desired vector-function. Initial value problems have projectors on admissible sets of initial vectors. Special attention is paid to systems with singular points in the domain. We formalize and classify the notion of singular points for systems under study. The results are illustrated by specially constructed examples. [ABSTRACT FROM AUTHOR]

In this paper, we study the evolution of framed curves, which may have singular points. We express flows of framed curves in Euclidean 3‐space. We obtain necessary and sufficient conditions as partial differential equations involving the framed curvatures for the inextensible flow of framed curve. Also, we give examples for the evolution equation of framed curvatures. [ABSTRACT FROM AUTHOR]

The new Euler-Jacobi formula for double points provides an algebraic relation between the singular points of a polynomial vector field and their topological indices. Using this formula we obtain the configuration of the singular points together with their topological indices for the planar quadratic–quartic polynomial differential systems when these systems have seven finite singular points being one of them with multiplicity two. The case with eight finite singular points has already been solved using the classical Euler-Jacobi formula. [ABSTRACT FROM AUTHOR]

We study the behaviour of solutions of quasi-linear differential equations of the second order at their singular points, where the coefficient of the second-order derivative vanishes. We consider solutions entering a singular point either with definite tangential direction (proper solutions) or without definite tangential direction (oscillating solutions). Equations of this type appear in many problems arising in analysis, geometry, dynamical systems theory, and physics. First, we prove that a generic equation of the considered type has no oscillating solutions. Then we concentrate on proper solutions, which can enter a singular point in admissible tangential directions only. Great attention is paid to second-order differential equations, whose right-hand sides are cubic polynomials by the first-order derivative. We obtain local representations for solutions of such equations in a form similar to Newton--Puiseux series -- series with fractional exponents (and, in a special case, with logarithmic terms). [ABSTRACT FROM AUTHOR]

The new Euler–Jacobi formula for points with multiplicity two provides an algebraic relation between the singular points of a polynomial vector field and their topological indices. Using this formula we obtain the configuration of the singular points together with their topological indices for the planar cubic polynomial differential systems when these systems have eight finite singular points, being one of them with multiplicity two. The case with nine finite singular points has already been solved using the classical Euler–Jacobi formula. [ABSTRACT FROM AUTHOR]

The new Euler–Jacobi formula for points with multiplicity two provides an algebraic relation between the singular points of a polynomial vector field and their topological indices. Using this formula we obtain the configuration of the singular points together with their topological indices for the planar cubic polynomial differential systems when these systems have eight finite singular points, being one of them with multiplicity two. The case with nine finite singular points has already been solved using the classical Euler–Jacobi formula. [ABSTRACT FROM AUTHOR]

We give three different proofs of the log-epiperimetric inequality at singular points for the obstacle problem. In the first, direct proof, we write the competitor explicitly; the second proof is also constructive, but this time the competitor is given through the solution of an evolution problem on the sphere. We compare the competitors obtained in the different proofs and their relation to other similar results that appeared recently. Finally, in the appendix, we give a general theorem, which can be applied also in other contexts and in which the construction of the competitor is reduced to finding a flow satisfying two differential inequalities.

Mechanical fields over thin elastic surfaces can develop singularities at isolated points and curves in response to constrained deformations (e.g., crumpling and folding of paper), singular body forces and couples, distributions of isolated defects (e.g., dislocations and disclinations), and singular metric anomaly fields (e.g., growth and thermal strains). With such concerns as our motivation, we model thin elastic surfaces as von Kármán plates and generalize the classical von Kármán equations, which are restricted to smooth fields, to fields which are piecewise smooth, and can possibly concentrate at singular curves, in addition to being singular at isolated points. The inhomogeneous sources to the von Kármán equations, given in terms of plastic strains, defect induced incompatibility, and body forces, are likewise allowed to be singular at isolated points and curves in the domain. The generalized framework is used to discuss the singular nature of deformation and stress arising due to conical deformations, folds, and folds terminating at a singular point. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of obtaining the asymptotics of solutions of differential operators in a neighborhood of an irregular singular point. More precisely, we construct uniform asymptotics for solutions of linear differential equations with second-order meromorphic coefficients in a neighborhood of a singular point and apply the results obtained to the equations of mathematical physics. The main results related to the construction of uniform asymptotics are obtained using resurgent analysis methods applied to differential equations with irregular singularities. These results allow us to construct asymptotics for any second-order equations with meromorphic coefficients—that is, with an arbitrary order of degeneracy. This also allows one to determine the type of a singular point and highlight the cases where the point is non-singular or regular. [ABSTRACT FROM AUTHOR]

Computer vision technology is used to obtain the necessary information from images. The two-dimensional image was transformed into its corresponding three-dimensional structure based on the reconstruction process and singular points. A review of software tools designed for scanning objects and measuring their dimensions is performed. A mathematical model and algorithm for constructing three-dimensional structures of objects in images are presented. Based on the design pattern, the architecture of the software application is developed. It consists of an image processing module, a module for calculating the size of objects, and a module for storing the received data. The fastest algorithm for finding singular points and their descriptors is used. In this paper, both homogeneous and Euclidean coordinates of characteristic points are calculated. The method of using the mobile application in automatic and manual modes is described. The results of scanning objects with the camera of a mobile device and determining their size in real time are presented.

We show that the classical result on the stability of the origin in a quadratic planar system of ODEs can be formulated using either matrix theory or via its associated real and complex Marcus algebra. A generalization to a three-dimensional case is considered and some counterexamples provided. [ABSTRACT FROM AUTHOR]

Three-dimensional (axisymmetric) flows of an ideal incompressible fluid are considered in multiply connected regions. In a coordinate system fitted to the velocity potential and the stream function the analogs of the powers of a complex variable are constructed for quasiconformal axisymmetric mappings; they are similar with those used for conformal mappings in the two-dimensional case. Quasianalytical polynomials of an arbitrary degree are considered and the functions conjugate to them are chosen. The solutions satisfying the Laplace equation in the cylindrical coordinate system are determined in the form of formal powers in the "operator" form. The solutions of problems of flow past bodies of finite dimensions, in semi-infinite domains, and in axysimmetric channels of complicated shape in the presence of an internal body are presented. [ABSTRACT FROM AUTHOR]

In this paper we study polynomial vector fields on C 2 of degree larger than 2 with n 2 isolated singularities. More precisely, we show that if two polynomial vector fields share n 2 - 1 singularities with the same spectra (trace and determinant) and from these singularities n 2 - 2 have the same positions, then both vector fields have identical position and spectra at all the singularities. Moreover we also show that if two polynomial vector fields share n 2 - 1 singularities with the same positions and from these singularities n 2 - 2 have the same spectra, then both vector fields have identical position and spectra at all the singularities. Moreover we also prove that generic vector fields of degree n > 2 have no twins and that for any n > 2 there exist two uniparametric families of twin vector fields, i.e. two different families of vector fields having exactly the same singular points and for each singular point both vector fields have the same spectrum. [ABSTRACT FROM AUTHOR]

The evolutes of regular curves in the Euclidean plane are given by the caustics of regular curves. In this paper, we define the generalized evolutes of planar curves which are spatial curves, and the projection of generalized evolutes along a fixed direction are the evolutes. We also prove that the generalized evolutes are the locus of centers of slant circles of the curvature of planar curves. Moreover, we define the generalized parallels of planar curves and show that the singular points of generalized parallels sweep out the generalized evolute. In general, we cannot define the generalized evolutes at the singular points of planar curves, but we can define the generalized evolutes of fronts by using moving frames along fronts and curvatures of the Legendre immersion. Then we study the behaviors of generalized evolutes at the singular points of fronts. Finally, we give some examples to show the generalized evolutes. [ABSTRACT FROM AUTHOR]

Spatially varied flows with increasing discharge can be encountered in several hydraulic systems. The analysis of water surface profiles in such flow conditions is useful for the verification and design of these systems. In this paper, a comprehensive analysis of the possible flow profiles in non-prismatic trapezoidal channels is performed using the method of singular points, by taking into account bed slope and friction. Different cases are identified depending on whether singular points occur or not along the collecting channel. It is verified that a singular point can be of the saddle, nodal, or spiral type, and that two singular points may occur in special circumstances. Furthermore, water depth profiles may be non-monotonic, showing a minimum or a maximum. The method is applied to both a theoretical example and three experimental tests taken from the literature, and to a real-field case concerning the side spillway channel of a dam. [ABSTRACT FROM AUTHOR]

We study the partial regularity problem for a three dimensional simplified Ericksen–Leslie system, which consists of the Navier–Stokes equations for the fluid velocity coupled with a convective Ginzburg-Landau type equations for the molecule orientation, modelling the incompressible nematic liquid crystal flows. Base on the recent studies on the Navier–Stokes equations, we first prove some new local energy bounds and an εε-regularity criterion for suitable weak solutions to the simplified Ericksen-Leslie system, i.e., for σ ∈[0,1], there exists a ε > 0 such that if (u,d,P) is a suitable weak solution in Qr(z0) with 0 < r ≤ 1 and z0 = (x0,t0), and satisfies r −3/2−σ ∫t0t0−r2(∥|u|2∥2/2−σ/H−σ(Br(x0)) + ∥|∇d|2∥2/2−σ/H−σ(Br(x0)) + ∥P∥2/2−σ/H−σ(Br(x0))) dt ≤ ε, then (u,d) is regular at z0. Here, H−σ(Br(x)) is the dual space of Hσ0(Br(x)), the space of functions ƒ in the homogeneous Sobolev space Hσ(R3) such that supp ƒ ⊂ Br(x). Inspired by this ε-regularity criterion, we then improve the known upper Minkowski dimension of the possible interior singular points for suitable weak solutions from 95/63(≈1.50794) given by [24] (Nonlinear Anal. RWA, 44 (2018), 246–259.) to 835/613(≈1.36215). [ABSTRACT FROM AUTHOR]

A study is conducted on the asymptotic stability of the rest points of the fractional oscillator Van der Pol-Duffing. The fractional van der Pol-Duffing oscillator is an oscillatory system of two differential equations with fractional order derivatives in the sense of Gerasimov-Caputo. The orders of fractional derivatives characterize the properties of the medium (memory effects) in which the oscillatory process takes place and can be the same (commensurate) or different (incommensurable). Using theorems for commensurable and incommensurable systems, the asymptotic stability of the rest points of the fractional van der Pol-Duffing oscillator is investigated with concrete examples. The results of the studies were confirmed by constructing the appropriate waveforms and phase trajectories.

We consider linear systems of ordinary differential equations (ODE) with rectangular matrices of coefficients, including the case when the matrix before the derivative of the desired vector function is not full rank for all argument values from the domain. Systems of this type are usually called differential-algebraic equations (DAEs). We obtained criteria for the existence of nonsingular transformations splitting the system into subsystems, whose solution can be written down analytically using generalized inverse matrices. The resulting solution formula is called a generalized split form of a DAE and can be viewed as a certain analogue of the Weierstrass-Kronecker canonical form. In particular, it is shown that arbitrary DAEs with rectangular coefficient matrices are locally reducible to a generalized split form. The structure of these forms (if it is defined on the integration segment) completely determines the structure of general solutions to the systems. DAEs are commonly characterizes by an integer number called index, as well as by the solution space dimension. The dimension of the solution space determines arbitrariness of the the general solution manifold. The index determines how many times we should differentiate the entries on which the solution to the problem depends. We show the ways of calculating these main characteristics.

This paper develops a new active contour (AC) model capable of multiple complex objects segmentation in the presence of heavy noise. The model segments images in the framework of two types of partial differential equations (PDEs): the Euler–Lagrange and Poisson PDEs. The former is used to build an evolution algorithm, while the Poisson solution gradient vector field (PGVF) directs the evolution toward the boundaries of all image objects. The AC halts on boundaries and PGVF separatrices, splits on the latter, and leaves at least one segment (called label) on every boundary. Each label tracks its boundary until the corresponding object is enveloped. The advantages of the new method are validated on a number of skin lesions, road, and aircraft images of varying sizes and in the presence of Gaussian noise. The obtained results are compared against results by contemporary and established active contours and neural networks. [ABSTRACT FROM AUTHOR]

The results of a theoretical analysis with an assessment of technological indicators that provide the possibility of real implementation of a continuous biotechnological process for obtaining lactic acid are presented. The analysis was carried out using mathematical modeling, including accounting for the formation of a product, biomass, and by-product. The main substrate consumed by microorganisms and the component that forms the main substrate in the synthesis process are considered as raw materials. In the analysis, two problems were solved: determining the range of values of the input indicators that ensure the real implementation of the technology for a given value of productivity for the target product (lactic acid) and assessing the multiplicity of input indicators for a given value of one of them. Analytical relationships for calculating the multiplicity indicators for points limiting the region of real implementation of the process are presented. Examples of numerical calculations that implement theoretical conclusions are given. The results of numerical calculations have shown significant user capabilities in the choice of the initial process indicators for the implementation of the technology with the same value of productivity. [ABSTRACT FROM AUTHOR]