Point X0 Research Articles

Various notions of shadowing in triangular system and its component systems

In this article, we investigate various forms of shadowing for a general triangular system. In particular, we relate various notions of shadowing for a triangular system with various notions of shadowing in the component systems. We prove that if the base f for T is transitive then shadowing in the base map and the non-autonomous system generated by a transitive point ensures shadowing of the triangular system. We prove that if the base map for T is expansive then shadowing in the triangular system ensures shadowing in the component systems. We prove that if the non-autonomous component systems form a synchronized family and the base map possesses a globally attracting fixed point x0 then eventual shadowing in system generated by x0 ensures eventual shadowing in each of the non-autonomous component systems. We also investigate chain transitivity and chain mixing for a general triangular system.

Local-in-space wave breaking criteria for a generalized rod equation

A generalized rod equation including the celebrated Camassa–Holm shallow water model is considered. The blow-up features of the equation are discussed on the line R. Using the method to construct the Lyapunov functions, local-in-space wave breaking criteria to the equation are established. Our wave breaking results are only involved in the initial values V0(x0) and V0′(x0) at a single local point x0∈R.

Poincaré inequalities and Ap weights on bow-ties

A metric space X is called a bow-tie if it can be written as ▪, where ▪ and ▪ are closed subsets of X. We show that a doubling measure μ on X supports a (q,p)-Poincaré inequality on X if and only if X satisfies a quasiconvexity-type condition, μ supports a (q,p)-Poincaré inequality on both ▪ and ▪, and a variational ▪-capacity condition holds. This capacity condition is in turn characterized by a sharp measure decay condition at the point x0. In particular, we study the bow-tie XRn consisting of the positive and negative hyperquadrants in Rn equipped with a radial doubling weight and characterize the validity of the ▪-Poincaré inequality on XRn in several ways. For such weights, we also give a general formula for the capacity of annuli around the origin.

On topological entropy (chaos) realized far from fixed point

In many papers, the concept of chaos is associated with positive topological entropy. The points around which chaos is concentrated are also often examined. In this paper we distinguish places of chaos generation and its realization. Moreover, we will consider the possibility of approximation of continuous function by functions having full entropy point x0 (there is chaos generated by any neighbourhood of it) but simultaneously x0 is not a focal entropy point (chaos is realized at some distance from the considered point).

Research on inspection route of hanging environmental robot based on computational fluid dynamics

The environment of a closed piggery is commonly characterized by spatial unevenness, and there are currently no specific standards for installation points of various environmental monitoring sensors. Therefore, the project team used the hanging track inspection robot (HTIR) as an environmental mon-itoring platform to seek the environmental monitoring points and ensure the scientific layout of moni-toring points. Ansys-CFD software was used to study the change rules of environmental parameters at 1.6 m (α plane), 0.7 m (β plane), and 0.4 m (γ plane) above the ground. The 300 monitoring points ((x1~x30) ×(y1~y10)) in each plane were analyzed to determine the most suitable monitoring points and inspection routes for HTIR. The results showed that: (1) All monitoring points could be arranged directly below the y3 track. (2) Monitoring points (x1, y3), (x10, y3) and (x30, y3) were environmental feature points. At (x1, y3), the maximum relative humidity and NH3 concentration on the α plane could be detected, and the maximum wind speed, maximum temperature, and maximum NH3 concentration on other planes could also be detected; At (x10, y3), the minimum temperature and maximum relative humidity of the β and γ planes could be detected; At (x30, y3), the maximum NH3 concentration in the α plane and the minimum relative humidity in all planes could be detected. This study scientifically arranged the inspection track and monitoring points for HTIR, improved the accuracy of environmental monitoring, and put forward suggestions for reducing NH3 concentration in closed piggeries, laying the foundation for the next step.

On a continuation of quaternionic and octonionic logarithm along curves and the winding number

This paper deals with the problem of finding a continuous extension of the hypercomplex (quaternionic or octonionic) logarithm along (quaternionic or octonionic) paths which avoid the origin. The main difficulty depends upon this fact: while a branch of the complex logarithm can be defined in a small open neighbourhood of a strictly negative real point, no continuous branch of the hypercomplex logarithm can be defined in any open set which contains a strictly negative real point. To overcome this difficulty, we use the logarithmic manifold introduced in [4]: in general, the existence of a lift of a path to this manifold is not guaranteed and, indeed, the problem of lifting a path to the logarithmic manifold is completely equivalent to the problem of finding a continuation of the hypercomplex logarithm along this path.The second part of the paper scrutinizes the existence of a notion of winding number (with respect to the origin) for hypercomplex loops that avoid the origin, even though it is known that the definition of winding number for such loops is not natural in Rn when n is greater than 2. The surprise is that, in the hypercomplex setting, the new definition of winding number introduced in this paper can be given and has full meaning for a large class of hypercomplex loops (untwisted loops with companion that avoid the origin).Finally an original but rather natural notion of homotopy for these hypercomplex loops (the c-homotopy) is presented and it is proved to be suitable to comply with the intrinsic geometrical meaning of the winding number for this class of loops, namely, two such hypercomplex loops are c-homotopic if, and only if, they have the same winding number.

On the Numerical Solutions Based On Exponential Finite Difference Method for Kuramoto-Sivashinsky Equation and Numerical Stability Analysis

In this paper, we solve the Kuramoto-Sivashinsky Equation numerically by finite-difference methods, using two different schemes which are the Fully Implicit scheme and Exponential finite difference scheme, because of the existence of the fourth derivative in the equation we suggested a treatment for the numerical solution of the two previous scheme by parting the mesh grid into five regions, the first region represents the first boundary condition, the second at the grid point x1, while the third represents the grid points x2,x3,…xn-2, the fourth represents the grid point xn-1 and the fifth is the second boundary condition. We also, study the numerical stability by Fourier (Von-Neumann) method for the two scheme which used in the solution on all mesh points to ensure the stability of the point which had been treated in the suggested style, we using two interval with two initial condition and the numerical results obtained by using these schemes are compare with Exact Solution of Equation Excellent approximate is found between the Exact Solution and numerical Solutions of these methods.

РЕЗУЛЬТАТЫ ПОИСКА ОПТИМАЛЬНЫХ ПАРАМЕТРОВ ПАССИВНОГО ИЗМЕЛЬЧИТЕЛЯ ПРИ ПРИГОТОВЛЕНИИ ПШЕНИЧНОЙ ПАТОКИ

In order to grain molasses preparing process studying, a laboratory installation has been developed. This installation’s working process’s research results at wheat molasses preparation are presented. Information on the passive shreddering device, one of the feature of which is under either specific grain crop reconfiguration possibility is given. The spring wheat (sort of Moskovskaya 35, it had grown in the Nizhny Novgorod region in 2022) characteristic in specific energy consumption process experiments using is given. According to the search experiments’ results it was revealed that a nozzle with a hole with diameter of 25 mm using leads to specific energy consumption in 18% more increasing then with one of 40 mm. The small distance from the nozzle to the grill also a negative effect has had. In the «Statistica» statistical analysis’s frame, a desirability function with a value of 0,809 for the criterion Y (average values of specific energy consumption E, W/h/kg) at selected factors X1 (grid’s inclination angle is α, deg.) and X2 (grid’s hole diameter is d, mm) optimization was constructed. In order to the optimal values of the studied factors and their mutual influence on this installation’s workflow determining, a matrix of a complete experiment factor 32 was implemented. After that, an adequate response’s function at a given significant level α = 0,05 was obtained. At the same time, it is noted that the X1 factor is more important than X2 one. According to these response function’s extremum studies results, the extreme point X1 = 0,2, so as X2 = 0,19 are revealed, at which the the function’s extremum is Y min = E = 43,27 W/h/kg. Based on technological considerations, the passive shreddering’s optimal tuning parameters at wheat molasses preparation are α = 30° and d = 6 mm, at that the specific energy consumption will be 41,5 W/h/ kg.

A point-picking game

In this article we study a version of the point-picking game defined by Berner and Juhász. Given a space X, the closed game CG(X) on X between Player O and Player P is played as follows: Player O chooses a non-empty open set U1⊂X, then Player P chooses a point x1∈U1, then Player O chooses a non-empty open set U2⊂X, then Player P chooses a point x2∈U2, and so on. An infinite sequence w=(U1,x1,U2,x2,…) such that Un is a nonempty open set and xn∈Un for every n∈N is called a play inCG(X). We will say that PlayerPwins w if {xn:n∈N} is closed in X, otherwise, PlayerOwins w. We prove that if Player O does not have a Markov winning strategy in CG(X) then X is selectively closed. We show that if X is the σ-product {f∈{0,1}ω1:f−1(1) is finite }, then Player O has a winning strategy in CG(X), yet X is selectively closed and selectively discrete. We also construct a selectively discrete space with a stationary winning strategy for Player O in CG(X).

Theoretical foundations of point cloud coordinate system transformation

Purpose. To provide theoretical foundations and develop mathematical models for the efficient transformation of coordinate systems for point clouds in geophysical research; the scientific analysis is aimed at developing algorithms and establishing necessary dependencies for the reliable integration of data obtained at different time points into a unified coordinate system, opening up prospects for further study and analysis of processes in geophysical research. The methods.The calculation is carried out using the following steps. Determination of known coordinates of four points (x1', y1', z1'; x2', y2', z2'; x3', y3', z3'; x4', y4', z4') in a hypothetical coordinate system (X', Y', Z') and the coordinates of the same points (x1, y1, z1; x2, y2, z2; x3, y3, z3; x4, y4, z4) in the coordinate system (X, Y, Z) to which the point clouds need to be transformed. Determination of constants a1, a2, a3, d, b1, b2, b3, e, c1, c2, c3, f through a system of equations. After determining the constants, the coordinates of points (x', y', z') in the hypothetical coordinate system (X', Y', Z') are calculated using equations where each equation expresses the coordinates of points (x', y', z') in terms of coordinates of points (x, y, z) in the coordinate system (X, Y, Z) and the determined constants. After performing the calculations, point clouds can be merged into a single coordinate system using the computed coordinates (x', y', z'). This methodology allows for the successful transformation of coordinate systems for point clouds in geophysical research. Findings. Analytical regularities have been established based on known coordinates of four points in both coordinate systems, allowing for the efficient transformation of a point cloud from one coordinate system to another. The originality. For the first time, precise analytical dependencies have been established that enable the efficient transformation of point clouds from one coordinate system to another using known coordinates of four points in both systems. Practical implementation. The obtained dependencies enable the efficient transformation of point clouds from one coordinate system to another using known coordinates of four points in both systems.

Neural-Network-Assisted Finite Difference Discretization for Numerical Solution of Partial Differential Equations

A neural-network-assisted numerical method is proposed for the solution of Laplace and Poisson problems. Finite differences are applied to approximate the spatial Laplacian operator on nonuniform grids. For this, a neural network is trained to compute the corresponding coefficients for general quadrilateral meshes. Depending on the position of a given grid point x0 and its neighbors, we face with a nonlinear optimization problem to obtain the finite difference coefficients in x0. This computing step is executed with an artificial neural network. In this way, for any geometric setup of the neighboring grid points, we immediately obtain the corresponding coefficients. The construction of an appropriate training data set is also discussed, which is based on the solution of overdetermined linear systems. The method was experimentally validated on a number of numerical tests. As expected, it delivers a fast and reliable algorithm for solving Poisson problems.

Separation of near-boiling mixtures (acetic acid and water) enhanced by ionic liquid

Acetic acid aqueous solution is a common and typical near-boiling mixture in the petroleum, chemical, polymer, pharmaceutical and food industries. Because of the large number of ideal plates for separating acetic acid and water by ordinary distillation, the energy consumption and production cost are high. Designable ionic liquids (ILs) and deep eutectic solvents (DESs) are environmentally-friendly entrainers for extractive distillation. Many azeotropic systems enhanced by adding ILs or DESs have been studied, but the near-boiling water/acetic acid system is rarely discussed.In this study, an IL 1,2,3-trimethylimidazolium dimethylphosphate ([TMIM]+[Me2PO4]−) with easy synthesis, low production cost and good thermal stability was employed as the rectification entrainer for water/acetic acid system, while the DESs were easy to react in acetic acid aqueous solution and was abandoned. First, the relative volatility of water (1) and acetic acid (2) at the more difficult separation point (x1′ = 0.9) was measured using a modified Othmer still at 101.32 kPa. When the mole fraction of entrainers was 0.071, these relative volatilities were 2.80 (n,n-dimethylformamide), 2.88 (1-methyl-2-pyrrolidinone), 4.82 ([TMIM]+[Me2PO4]−) respectively. The separation performance of [TMIM]+[Me2PO4]− was 67% higher than that of 1-methyl-2-pyrrolidinone. Then, the effect of [TMIM]+[Me2PO4]− concentration on the relative volatility was investigated. The relative volatility was dramatically increased from 1.60 to 6.85 with 0–0.103 mol fraction of [TMIM]+[Me2PO4]− and was 4.28 times that in an IL-free system. After that, isobaric vapor–liquid equilibrium (VLE) data of water + acetic acid, water + [TMIM]+[Me2PO4]− and water + acetic acid + [TMIM]+[Me2PO4]− ([TMIM]+[Me2PO4]− at ∼ 5, 15, and 25 mass%) systems were determined at 101.32 kPa. Also, Based on the chemical theory and Hayden–O’Connell method, the binary energy parameters of the NRTL (Non-Random-Two-Liquid) model for these systems were regressed and the predicted values were found to fit the experimental VLE data quite well..For the first time, we have found that [TMIM]+[Me2PO4]− as an entrainer is superior to conventional volatile solvents for water/acetic acid distillation in terms of performance, which provides more choices and imagination for industry. At the same time, it promotes understanding of the influence of IL structure on water/acetic acid separation performance, and is also an important inspiration for the distillation of other near-boiling systems.

Temperature on rods with Robin boundary conditions

We consider solutions uf to the one-dimensional Robin problem with the heat source f∈L1[−π,π] and Robin parameter α>0. For given m, M, and s, 0≤m<s<M, we identify the heat sources f0, such that uf0 maximizes the temperature gap max[−π,π]⁡uf−min[−π,π]⁡uf over all heat sources f such that m≤f≤M and ‖f‖L1=2πs. In particular, this answers a question raised by J. J. Langford and P. McDonald in [5]. We also identify heat sources, which maximize/minimize uf at a given point x0∈[−π,π] over the same class of heat sources as above and discuss a few related questions.

Generalized Equilibrium Problems

If X is a convex subset of a topological vector space and f is a real bifunction defined on X×X, the problem of finding a point x0∈X such that f(x0,y)≥0 for all y∈X, is called an equilibrium problem. When the bifunction f is defined on the cartesian product of two distinct sets X and Y we will call it a generalized equilibrium problem. In this paper, we study the existence of the solutions, first for generalized equilibrium problems and then for equilibrium problems. In the obtained results, apart from the bifunction f, another bifunction is introduced, the two being linked by a certain compatibility condition. The particularity of the equilibrium theorems established in the last section consists of the fact that the classical equilibrium condition (f(x,x)=0, for all x∈X) is missing. The given applications refer to the Minty variational inequality problem and quasi-equilibrium problems.

One-dimensional Dynamics for a Discontinuous Singular Map and the Routes to Chaos

We visit a previously proposed discontinuous, two-parameter generalization of the continuous, one-parameter logistic map and present exhaustive numerical studies of the behavior for different values of the two parameters and initial points x0. In particular, routes to chaos exist that do not exhibit period-doubling whereas period-doubling is the sole route to chaos in the logistic map. Aperiodic maps are found that lead to cobwebs with x = ±∞ as accumulation points, where every neighborhood contains infinitely many points generated by the map.

Maintaining the Union of Unit Discs under Insertions with Near-Optimal Overhead

We present efficient dynamic data structures for maintaining the union of unit discs and the lower envelope of pseudo-lines in the plane. More precisely, we present three main results in this paper: (i) We present a linear-size data structure to maintain the union of a set of unit discs under insertions. It can insert a disc and update the union in O (( k +1)log 2 n ) time, where n is the current number of unit discs and k is the combinatorial complexity of the structural change in the union due to the insertion of the new disc. It can also compute, within the same time bound, the area of the union after the insertion of each disc. (ii) We propose a linear-size data structure for maintaining the lower envelope of a set of x -monotone pseudo-lines. It can handle insertion/deletion of a pseudo-line in O (log 2 n ) time; for a query point x 0 ∈ ℝ, it can report, in O (log n ) time, the point on the lower envelope with x -coordinate x 0 ; and for a query point q ∈ ℝ 2 , it can return all k pseudo-lines lying below q in time O (log n + k log 2 n ). (iii) We present a linear-size data structure for storing a set of circular arcs of unit radius (not necessarily on the boundary of the union of the corresponding discs), so that for a query unit disc D , all input arcs intersecting D can be reported in O ( n 1/2+ɛ + k ) time, where k is the output size and ɛ &gt; 0 is an arbitrarily small constant. A unit-circle arc can be inserted or deleted in O (log 2 n ) time.

Математическая модель принятия решений на нечетком множестве данных в сфере логистики

Insufficiently effective management of transportation process is associated with the cases of irregularity and inconsistency of the work of transport various modes among themselves, insignificant development of processing capacities for cargo transshipment between transport modes as well as of insufficient number of terminals serving to «smooth» work inconsistency in time in transport flows. This article describes mathematical model for choosing a rational variant of cargo delivery from the position of a transportation client. The main advantage of the described mathematical model is the minimization of production and technological costs on the platform of fuzzy set theory with an “eye” on exact sets (topology on the set of parameters of real numbers R1). Sufficiently smooth function f ( x), x ∈ R1, is adopted. For it, the Taylor representation takes place locally at the point x0. Presented in the paper integrated approach allows to evaluate the rationality of routes and the choice of the best among them which takes into account wide range of criteria, limitations of transport parameters and cargo owner requirements. It is worth to note that the rationality of choice is sometimes formed not only from the best result, but also from emerging alternatives and factors affecting this or that choice and process. A client chooses a competitive route option according to the variation of claimed by him demands on logistics projects. The paper also presents existing method analysis on reveal of the rationalization of transport constituent in logistics field.

On the robustness of the topological derivative for Helmholtz problems and applications

Abstract We consider Helmholtz problems in two and three dimensions. The topological sensitivity of a given cost function J(u ∈) with respect to a small hole B ∈ around a given point x 0 ∈ B ∈ ⊂ Ω depends on various parameters, like the frequency k chosen or certain material parameters or even the shape parameters of the hole B ∈. These parameters are either deliberately chosen in a certain range, as, e.g., the frequencies, or are known only up to some bounds. The problem arises as to whether one can obtain a uniform design using the topological gradient. We show that for 2-d and 3-d Helmholtz problems such a robust design is achievable.

Critical points of solutions to a kind of linear elliptic equations in multiply connected domains

In this paper, we mainly study the critical points and critical zero points of solutions u to a kind of linear elliptic equations with nonhomogeneous Dirichlet boundary conditions in a multiply connected domain Ω in ℝ2. Based on the delicate analysis about the distributions of connected components of the super-level sets {x ∈ Ω: u(x) > t} and sub-level sets {x ∈ Ω: u(x) < t} for some t, we obtain the geometric structure of interior critical point sets of u. Precisely, let Ω be a multiply connected domain with the interior boundary γI and the external boundary γE, where u∣γI = ψ1(x), u∣γE = ψ2(x). When ψ1(x) and ψ2(x) have N1 and N2 local maximal points on γI and γE respectively, we deduce that \(\sum\nolimits_{i = 1}^k {{m_i} \le {N_1} + {N_2}} \), where m1,…,mk are the respective multiplicities of interior critical points x1,…,xk of u. In addition, when \({\min _{{\gamma _E}}}{\psi _2}\left( x \right) \ge {\max _{{\gamma _I}}}{\psi _1}\left( x \right)\) and u has only N1 and N2 equal local maxima relative to \(\overline \Omega \) on γI and γe respectively, we develop a new method to show that one of the following three results holds \(\sum\nolimits_{i = 1}^k {{m_i} = {N_1} + {N_2}} \) or \(\sum\nolimits_{i = 1}^k {{m_i} + 1 = {N_1} + {N_2}} \) or \(\sum\nolimits_{i = 1}^k {{m_i} + 2 = {N_1} + {N_2}} \). Moreover, we investigate the geometric structure of interior critical zero points of u. We obtain that the sum of multiplicities of the interior critical zero points of u is less than or equal to the half of the number of its isolated zero points on the boundaries.

An algebra of distributions related to a star product with separation of variables

Given a star product with separation of variables ⋆ on a pseudo-Kähler manifold M and a point x0∈M, we construct an associative algebra of formal distributions supported at x0. We use this algebra to express the formal oscillatory exponents of a family of formal oscillatory integrals related to the star product ⋆.