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We prove the first polynomial bound on the number of monotonic homotopy moves required to tighten a collection of closed curves on any compact orientable surface, where the number of crossings in the curve is not allowed to increase at any time during the process. The best known upper bound before was exponential, which can be obtained by combining the algorithm of De Graaf and Schrijver [ J. Comb. Theory Ser. B , 1997] together with an exponential upper bound on the number of possible surface maps. To obtain the new upper bound, we apply tools from hyperbolic geometry, as well as operations in graph drawing algorithms—the cluster and pipe expansions—to the study of curves on surfaces. As corollaries, we present two efficient algorithms for curves and graphs on surfaces. First, we provide a polynomial-time algorithm to convert any given multicurve on a surface into minimal position. Such an algorithm only existed for single closed curves, and it is known that previous techniques do not generalize to the multicurve case. Second, we provide a polynomial-time algorithm to reduce any k -terminal plane graph (and more generally, surface graph) using degree-1 reductions, series-parallel reductions, and Δ Y -transformations for arbitrary integer k . Previous algorithms only existed in the planar setting when k ≤ 4, and all of them rely on extensive case-by-case analysis based on different values of k . Our algorithm makes use of the connection between electrical transformations and homotopy moves and thus solves the problem in a unified fashion.

The Gardner’s equation is the generalization of Kortweg-de Vries equation and modified Kortweg-de Vries equation having real life applications in different fields of science, such as, hydrodynamics, quantum field theory, etc. Moreover, it has applications in fluid mechanics where it explains surface and internal waves. This manuscript addresses application of the new ϕ 6 -model expansion method to Gardner’s equation for getting its solitary wave solutions. Such new results aid in getting noteworthy advances in various disciplines of science and technology. Periodic, kink, anti-kink, bright, dark, w-shaped and singular soliton solutions of the governing equation have been constructed. The proposed method takes into account a variety of closed form solutions using the Jacobi elliptic functions. The graphical illustration of some of the obtained solutions has been presented for a suitable choice of free parameters to depict the dynamic nature of the obtained wave solutions.

Given a natural number k and an orientable surface S of finite type, define the k-curve graph to be the graph with vertices corresponding to isotopy classes of essential simple closed curves on S and with edges corresponding to pairs of such curves admitting representatives that intersect at most k times. We prove that the automorphism group of the k-curve graph of a surface S is isomorphic to the extended mapping class group for all k sufficiently small with respect to the Euler characteristic of S. We prove the same result for the so-called systolic complex, a variant of the curve graph whose complete subgraphs encode the intersection patterns for any collection of systoles with respect to a hyperbolic metric. This resolves a conjecture of Schmutz Schaller., Comment: 33 pages, 23 figures, 1 table. Incorporated referee comments. To appear in the Michigan Mathematical Journal

One of the most common hypotheses on the theory of non-smooth dynamical systems is a regular surface as switching manifold, at which case there is at least well-defined and established Filippov dynamics. However, systems with singular switching manifolds still lack such well-established dynamics, although present in many relevant models of phenomena where multiple switches or multiple abrupt changes occur. At this work, we leverage a methodology that, through blow-ups and singular perturbation, allows the extension of Filippov dynamics to the singular case. Specifically, tridimensional systems whose switching manifold consists of an algebraic manifold with transversal self-intersection are considered. This configuration, known as double discontinuity, represents systems with two switches and whose singular part consists of a straight line, where ordinary Filippov dynamics is not directly applicable. For the general, non-linear case, beyond defining the so-called fundamental dynamics over the singular part, general theorems on its qualitative behavior are provided. For the affine case, however, theorems fully describing the fundamental dynamics are obtained. Finally, this fine-grained control over the dynamics is leveraged to derive Peixoto like theorems characterizing semi-local structural stability., 46 pages, 16 figures; new title, abstract, introduction and conclusion; improved statements and argumentation

This article manages the development of the traveling wave solutions of time-fractional Zakharov-Kuznetsov (ZK) equation by using the modified auxiliary equation method (MAEM) and analytical approximate solution that is constructed through the residual power series method (RPSM). The MAEM with time fractional beta-derivative is utilized to observe the effects of fractional parameter on the governing model. The RPSM is more useful to retrieve a particular form of wave solution when the initial behavior of the system is known. This method is observed by considering Caputo’s time fractional derivative. Some of the obtained results are graphically illustrated using 3D surface graphs and 2D line plots. The proposed methods are shown to be useful for construction of traveling wave solutions which are hoped to be beneficial in exploring different dynamical behaviors expressed by the equation.

In this paper we shall develop a face-based local tangential lifting method to handle the problem of discrete conservation laws for diffusion equations and solve Cahn-Hilliard equations over curved surfaces. To do so, we need to give a new discrete approximation of the Laplace-Beltrami operators on functions over curved surfaces. Our method is a simple effective numerical method for solving diffusion equations on curved surfaces. The key idea of our face-based local tangential lifting method is to approximate locally the underlying surface and functions with respect to faces in the discrete model of the surface by using the centroid method. Indeed, our algorithm is a generalized finite difference method and an intrinsic geometry method. It gives a natural way to approximate directly the Laplace-Beltrami operators on a triangular mesh. Discrete conservation laws for diffusion equations on triangular meshes are also discussed.

We analyze the structure of the logical operators from a class of quantum codes that generalizes the surface codes. These are the hypergraph product codes, restricted to the vertical sector. By generalizing an argument of Bravyi and K\"onig, we find that transversal gates for these codes must be restricted to the Clifford group.

We give a linear-time algorithm to decide 3-colorability of a triangle-free graph embedded in a fixed surface, and a quadratic-time algorithm to output a 3-coloring in the affirmative case. The algorithms also allow to prescribe the coloring of a bounded number of vertices.

This paper considers the sliding mode control (SMC) problem of a class of uncertain Markovian jump systems, in which there exist randomly occurring parameter uncertainties and random gain variations in the controller. By introducing two independent random variables obeying Bernoulli distribution, the random characteristics of parameter uncertainties and controller gain variations are described. A mode-dependent sliding surface is constructed, and then, the non-fragile SMC scheme is synthesized such that the specified sliding surface is reached in finite time. Furthermore, the stochastic finite-time boundedness over both the reaching and sliding stages are ensured simultaneously under some sufficient conditions. Finally, the developed non-fragile SMC approach is verified by a practical example.

Classifying and identifying surface defects is essential during the production and use of aluminum profiles. Recently, the dual-convolutional neural network(CNN) model fusion framework has shown promising performance for defects classification and recognition. Spurred by this trend, this paper proposes an improved dual-CNN model fusion framework to classify and identify defects in aluminum profiles. Compared with traditional dual-CNN model fusion frameworks, the proposed architecture involves an improved fusion layer, fusion strategy, and classifier block. Specifically, the suggested method extracts the feature map of the aluminum profile RGB image from the pre-trained VGG16 model's pool5 layer and the feature map of the maximum pooling layer of the suggested A4 network, which is added after the Alexnet model. then, weighted bilinear interpolation unsamples the feature maps extracted from the maximum pooling layer of the A4 part. The network layer and upsampling schemes ensure equal feature map dimensions ensuring feature map merging utilizing an improved wavelet transform. Finally, global average pooling is employed in the classifier block instead of dense layers to reduce the model's parameters and avoid overfitting. The fused feature map is then input into the classifier block for classification. The experimental setup involves data augmentation and transfer learning to prevent overfitting due to the small-sized data sets exploited, while the K cross-validation method is employed to evaluate the model's performance during the training process. The experimental results demonstrate that the proposed dual-CNN model fusion framework attains a classification accuracy higher than current techniques, and specifically 4.3% higher than Alexnet, 2.5% for VGG16, 2.9% for Inception v3, 2.2% for VGG19, 3.6% for Resnet50, 3% for Resnet101, and 0.7% and 1.2% than the conventional dual-CNN fusion framework 1 and 2, respectively, proving the effectiveness of the proposed strategy.

We consider hyperbolic links that admit alternating projections on surfaces in compact, irreducible 3-manifolds. We show that, under some mild hypotheses, the volume of the complement of such a link is bounded below in terms of a Kauffman bracket function defined on link diagrams on the surface. In the case that the 3-manifold is a thickened surface, this Kauffman bracket function leads to a Jones-type polynomial that is an isotopy invariant of links. We show that coefficients of this polynomial provide 2-sided linear bounds on the volume of hyperbolic alternating links in the thickened surface. As a corollary of the proof of this result, we deduce that the twist number of a reduced, twist reduced, checkerboard alternating link projection with disk regions, is an invariant of the link., 24 pages, 6 figures; To be published in Fundamenta Mathematicae; V2: Corrected typos, updated references

In this paper, we handle focal surfaces of surface of revolution in Galilean 3-space $\mathbb{G}_{3}$. We define the focal surfaces of surface of revolution and we obtain some results for these types of surfaces to become flat and minimal. Also, by giving some examples to these surfaces, we present the visualizations of each type of focal surface of surface of revolution in $\mathbb{G}_{3}$.

Bimaterial anti-plane interfacial crack problems with arbitrary distributed forces applied on crack surfaces are numerically investigated. The problems are in essence a kind of boundary value problems about Laplace's equations. Previous work of using Symplectic analytical singular element (SASE) with high-order accuracy and other singular finite elements for calculating mode III stress intensity becomes invalid when the distributed forces on each surface are different. In this paper, Jordan-form special solutions (JFSS) are proposed for the first time. We derive the JFSS for present problems and use them to improve displacement modes of SASE. The distributed forces are therefore converted into equivalent nodal forces of the nodes around SASE. The new SASE is found to be directly and conveniently used for cases where the two distributed forces are either equal (even both zero) or not equal. Numerical examples are provided to demonstrate the effectiveness and accuracy of the present method. The necessity and effectiveness of the JFSS is also validated. Moreover, the JFSS can be applied to some other physical problems represented by the Laplace's equation.

An efficient and accurate reliability-based design of the geosynthetic-reinforced slopes (GRS) using the pseudo-static and modified pseudo-dynamic framework is proposed in the present study. Deterministic formulation used in the present study is made robust with the help of nonlinear constrained optimization. The collocation based stochastic response surface method (CSRSM) is used to probabilistically analyze the GRS. The critical modes of failure pertaining to the internal and external stability of the GRS are considered in the formation of the performance functions. The horizontal seismic acceleration coefficient (kh), internal friction angle of soil (φ), soil unit weight (γ), shear wave velocity (Vs), and friction angle at the interface between soil and reinforcement (φb) are chosen as the random variables, owing to their high influence on the stability of the GRS. The influence of correlation on the stability of the reinforced slope is illustrated considering the internal and external stability. System reliability analysis considering the internal and external modes of failure is also performed. An illustrative example is presented showing the steps to design a GRS using the proposed formulation. The results confirm the necessity of performing the system reliability analysis to estimate an accurate value of probability of failure of GRS.

The purpose of this paper is to propose an optimization method for shape design of Auxetic Bending-Active Gridshells (ABAGs). A 2-dimensional auxetic structure has negative Poisson’s ratio for in-plane deformation. Positive Gaussian curvature distributed on a curved surface of ABAG is obtained by out-of-plane deformation of the initial flat grid composed of reentrant structural units. We introduce discrete differential geometry into shape design of the surface of ABAG, which can be easily discretized into triangular meshes in accordance with the discrete member locations. The objective function of the proposed optimization problem is formulated using discrete Gaussian curvature, which is defined as the angle defect at node. In addition, discrete mean curvature vector is used for specifying the direction of convexity. We use particle swarm optimization for solving the problem. It is shown in the numerical examples, that surfaces with specified distribution of convex region on an ABAG can be obtained using the proposed method.

Whether sub-optimal local minima and saddle points exist in the highly non-convex loss landscape of deep neural networks has a great impact on the performance of optimization algorithms. Theoretically, we study in this paper the existence of non-differentiable sub-optimal local minima and saddle points for deep ReLU networks with arbitrary depth. We prove that there always exist non-differentiable saddle points in the loss surface of deep ReLU networks with squared loss or cross-entropy loss under reasonable assumptions. We also prove that deep ReLU networks with cross-entropy loss will have non-differentiable sub-optimal local minima if some outermost samples do not belong to a certain class. Experimental results on real and synthetic datasets verify our theoretical findings.

As the solutions of partial differential equations (PDEs), PDE surfaces provide an effective way for physical-based surface design in surface modeling. The bicubic B-spline surface is a useful tool for surface modeling in computer aided geometric design (CAGD). In this paper, we present a method for generating bicubic B-spline surfaces with the uniform knots and the quasi-uniform knots from the sixth order PDEs. From the given boundary condition, based on the cubic B-spline basis representation and the PDE mask, the resulting bicubic B-spline surface can be generated uniquely. The boundary condition is more flexible and can be applied for curvature-continuous surface design, surface blending and hole filling. Some representative examples show the effectiveness of the presented method.

We explore upper bounds on the covering radius of non-hollow lattice polytopes. In particular, we conjecture a general upper bound of $d/2$ in dimension $d$, achieved by the "standard terminal simplices" and direct sums of them. We prove this conjecture up to dimension three and show it to be equivalent to the conjecture of González-Merino \& Schymura (2017) that the $d$-th covering minimum of the standard terminal $n$-simplex equals $d/2$, for every $n>d$. We also show that these two conjectures would follow from a discrete analog for lattice simplices of Hadwiger's formula bounding the covering radius of a convex body in terms of the ratio of surface area versus volume. To this end, we introduce a new notion of discrete surface area of non-hollow simplices. We prove our discrete analog in dimension two and we give strong evidence for its validity in arbitrary dimension., 44 pages, 7 figures

The main problem in cartography is that it is not possible to map/project/transform a spherical or ellipsoidal surface into a plane without distortions. The distortions of areas, angles, and/or distances are immanent to all maps. It is known that scale changes from point to point, and at certain points, the scale usually depends on the direction. The local linear scale factor c is one of the most important indicators of distortion distribution in the theory of map projections. It is not possible to find out the values of the local linear scale factor c in directions of coordinate axes x and y immediately from the definition of c. To solve this problem, in this paper, we derive new formulae for the calculation of c for a rotational ellipsoid. In addition, we derive the formula for computing c in any direction defined by dx and dy. We also considered the position and magnitude of the extreme values of c and derived new formulae for a rotational ellipsoid.

The $$\mathrm {SU}(r)$$ Vafa–Witten partition function, which virtually counts Higgs pairs on a projective surface S, was mathematically defined by Tanaka–Thomas. On the Langlands dual side, the first-named author recently introduced virtual counts of Higgs pairs on $$\mu _r$$ -gerbes. In this paper, we instead use Yoshioka’s moduli spaces of twisted sheaves. Using Chern character twisted by rational B-field, we give a new mathematical definition of the $$\mathrm {SU}(r) / {\mathbb {Z}}_r$$ Vafa-Witten partition function when r is prime. Our definition uses the period-index theorem of de Jong. S-duality, a concept from physics, predicts that the $$\mathrm {SU}(r)$$ and $$\mathrm {SU}(r) / {\mathbb {Z}}_r$$ partition functions are related by a modular transformation. We turn this into a mathematical conjecture, which we prove for all K3 surfaces and prime numbers r.

Let $\Omega\subset\mathbb R^n$ be a $C^1$ domain, or more generally, a Lipschitz domain with small local Lipschitz constant. In this paper it is shown that if $u$ is a function harmonic in $\Omega$ and continuous in $\overline \Omega$ which vanishes in a relatively open subset $\Sigma\subset\partial\Omega$ and moreover the normal derivative $\partial_\nu u$ vanishes in a subset of $\Sigma$ with positive surface measure, then $u$ is identically $0$., Comment: More detailed explanation in some argument involving integration by parts and in Remark 3.3. An additional appendix with a self-contained proof of Lemma 4.3, whose proof was not included in the paper previously

In this paper, we study second-order and fourth-order elliptic problems which include not only a Poisson equation in the bulk but also an inhomogeneous Laplace--Beltrami equation on the boundary of the domain. The bulk and the surface PDE are coupled by a boundary condition that is either of Dirichlet or Robin type. We point out that both the Dirichlet and the Robin type boundary condition can be handled simultaneously through our formalism without having to change the framework. Moreover, we investigate the eigenvalue problems associated with these second-order and fourth-order elliptic systems. We further discuss the relation between these elliptic problems and certain parabolic problems, especially the Allen--Cahn equation and the Cahn--Hilliard equation with dynamic boundary conditions.

One can realize higher laminations as positive configurations of points in the affine building. The duality pairings of Fock and Goncharov give pairings between higher laminations for two Langlands dual groups $G$ and $G^{\vee}$. These pairings are a generalization of the intersection pairing between measured laminations on a topological surface. We give a geometric interpretation of these intersection pairings in the case that $G=SL_n$. In particular, we show that they can be computed as the length of minimal weighted networks in the building. Thus we relate the intersection pairings to the metric structure of the affine building. This proves several of the conjectures from [LO] The key tools are linearized versions of well-known classical results from combinatorics, like Hall's marriage lemma, Konig's theorem, and the Kuhn-Munkres algorithm., Comment: 15 pages. We prove some conjectures from arXiv:1511.00165

A proof of convergence is given for a novel evolving surface finite element semi-discretization of Willmore flow of closed two-dimensional surfaces, and also of surface diffusion flow. The numerical method proposed and studied here discretizes fourth-order evolution equations for the normal vector and mean curvature, reformulated as a system of second-order equations, and uses these evolving geometric quantities in the velocity law interpolated to the finite element space. This numerical method admits a convergence analysis in the case of continuous finite elements of polynomial degree at least two. The error analysis combines stability estimates and consistency estimates to yield optimal-order $$H^1$$ -norm error bounds for the computed surface position, velocity, normal vector and mean curvature. The stability analysis is based on the matrix–vector formulation of the finite element method and does not use geometric arguments. The geometry enters only into the consistency estimates. Numerical experiments illustrate and complement the theoretical results.

Using a novel formulation based on flow force, we prove the existence of a global continuum of periodic traveling wave solutions to the irrotational water wave problem in finite depth where the flow force is held fixed, instead of fixing the mass flux. We further establish the limiting behaviour of solutions in this continuum where the flow force remains fixed throughout the continuum of the solutions. The global solutions, with a fixed flow force, show the possibility that a weak stagnation point can be approached at the surface, which is a characteristic of large amplitude irrotational water waves. The formulation is suitable for applications to numerical studies on water waves.

To develop lightweight and high-strength curved beetle elytron plates (CBEPs), an algorithm to extract a cross-section point set based on three-dimensional point cloud data measured from a beetle forewing and to reconstruct its outer surface was designed. This approach was used to study three beetle species. The curve fitting method and the curvature distribution characteristics of the corresponding forewings during the reconstruction were investigated. (1) Two algorithms, namely, the median surface separation method and segmented microstrip method, were established. (2) Two types of fitting methods were proposed: judging the consistency of the curvature curve peaks to determine higher-degree polynomials (clusters) and implementing k-fold cross-validation to determine the optimal engineering fitting curve. The optimal fitting curves for the cross-section point sets of the various beetle forewings involved in this paper were discovered to be elliptic equations. (3) The curvature characteristics of the forewing cross-sections vary depending on the type of fitting curve used; accordingly, researchers can choose the most appropriate fitting curve. For the research and development of CBEPs, elliptic equations can be used to establish the required bionic structure model, the important structural parameters of which include not only the long and short axes of the ellipse but also the position and length of the segment. This paper, therefore, lays a foundation for the development of CBEPs.

The geometric properties of differential systems are used to demonstrate how the sinh-poisson equation describes a surface with a constant negative curvature in this paper. The canonical reduction of 4-dimensional self dual Yang Mills theorem is the sinh-poisson equation, which explains pseudo spherical surfaces. We derive the B¨acklund transformations and the travelling wave solution for the sinh-poisson equation in specific. As a result, we discover exact solutions to the self-dual Yang-Mills equations.

To each variety $X$ and a nonnegative integer $m$, there is a space $X_m$ over $X$, called the jet scheme of $X$ of order $m$, parametrizing $m$-th jets on $X$. Its fiber over a singular point of $X$ is called a singular fiber. For a surface with a rational double point, Mourtada gave a one-to-one correspondence between the irreducible components of the singular fiber of $X_m$ and the exceptional curves of the minimal resolution of $X$ for $m \gg 0$. In this paper, for a surface $X$ over complex number with a singularity of $A_n$ or $D_4$-type, we study the intersections of irreducible components of the singular fiber and construct a graph using this information. The vertices of the graph correspond to irreducible components of the singular fiber and two vertices are connected when the intersection of the corresponding components is maximal for the inclusion relation. In the case of $A_n$ or $D_4$-type singularity, we show that this graph is isomorphic to the resolution graph for $m \gg 0$., 18 pages

The creative industries have become the government's attention for contributing to economic accretion. But due the lack of artistic creativity and appeal, the evolution of creative industries craft section is not optimal. So that it was needed a variations of relief items to increase the attractiveness. In general, industrial objects design are still limited to the space geometry objects or a Bezier curve of degree two. Therefore, Bezier curves of degree is selected and modified it into a quartic Bezier forms and then applied to the design of industrial objects (glassware). The purpose of this research is to determine the formula of quartic Bezier from of qubic Bezier modifications and to determine the rotary surface shape of quartic Bezier from cubic Bezier modifications. Then, from some form of revolving surface of modified cubic Bezier the glassware designs are generated. The results of this research are, first, the formula of quartic Bezier result of Bezier cubic modifications. Second, the form of revolving surface of modified cubic Bezier which is influenced by five control points and parameter selection . For further Research it is expected to develop a modification of cubic Bezier into Bezier of degree

In her seminal 2008 paper, Maryam Mirzakhani showed that the ratio that two topological types of curves occur in is a rational number. In this paper we describe the process by which we obtained experimental evidence that separating and non-separating curves on the surface of genus two occur in the ratio 1 : 48.

We propose Gauss–Seidel progressive iterative approximation (GS-PIA) for subdivision surface interpolation by combining the Gauss–Seidel iterative method for linear systems and progressive iterative approximation (PIA) for free-form curve and surface interpolation. We address the details of GS-PIA for Loop and Catmull–Clark surface interpolation and prove that they are convergent. In addition, GS-PIA may also be applied to surface interpolation for other stationary approximating subdivision schemes with explicit limit position formula/masks. GS-PIA inherits many good properties of PIA, such as having intuitive geometric meaning and being easy to implement. Compared with some other existing interpolation methods by approximating subdivision schemes, GS-PIA has three main advantages. First, it has a faster convergence rate than PIA and weighted progressive iterative approximation (W-PIA). Second, GS-PIA does not need to compute optimal weights while W-PIA does. Third, GS-PIA does not modify the mesh topology but some methods with fairness measures do. Numerical examples for Loop and Catmull–Clark subdivision surface interpolation illustrated in this paper show the efficiency and effectiveness of GS-PIA.

For $G = \mathrm {GL}_2, \mathrm {SL}_2, \mathrm {PGL}_2$ we compute the intersection E-polynomials and the intersection Poincaré polynomials of the G-character variety of a compact Riemann surface C and of the moduli space of G-Higgs bundles on C of degree zero. We derive several results concerning the P=W conjectures for these singular moduli spaces.

A graph is locally-connected if the neighborhood of each vertex induces a connected graph. It is well known that a triangulation on a closed surface is locally-connected, and some results for triangulations were generalized to those for connected locally-connected graphs. In this paper, we extend two characterization theorems of triangulations for a complete coloring and a facial complete coloring, which are vertex colorings with constraints on the appearance of color tuples, to those of connected locally-connected graphs. Moreover, we also investigate the relation between the corresponding invariants and the number of independent elements.

A multibranch Rao–Wilton–Glisson (MB-RWG) basis function is proposed in this article, which is composed of one positive triangle with a larger edge length and several negative triangles with smaller edge lengths. All negative triangles are aggregated to replace the negative triangle in the traditional RWG basis function, with exactly the same function expression defined on the triangles. The number of negative triangles can be changed in different mesh structures. Similar to traditional RWG basis functions, the proposed MB-RWG basis functions guarantee normal current continuity across the common edges. No line charges exist and the total charge on one MB-RWG is zero. MB-RWG basis functions can be used very conveniently to connect one surface with a coarse mesh scheme to another with a fine mesh scheme and can be flexibly applied in the domain decomposition method (DDM). Numerical results validate the accuracy and demonstrate the versatility of the proposed basis function in modeling multiscale perfect electrically conducting (PEC) objects.

Models of boundary conditions on the surface of a body for rarefied molecular gas taking into account the rotational energy are investigated. A comparison of the temperature fields for the case of the flow past bodies is performed by the supersonic flow of a gas at incomplete energy accommodation on the surface and at various degrees of exchange of the rotational and translational energies. The influence of the intensity of exchange on the gas flow depending on the rarefaction parameter, the free-stream velocity, and the surface temperature is shown.

The main purpose of this paper is to present a methodology for precisely calculating the tooth thickness of cylindrical worm gear, which is expounded detailedly with the worm gear in the toroidal surface enveloping cylindrical worm pair as an example. This method can compute the tooth thickness in any cross section of cylindrical worm gear, and it is universal for cylindrical worm gears. In an objective cross section, the tooth profile equations of both flanks of a typical cylindrical worm gear tooth are derived. A given radial dimension intersects with both flanks of the tooth profile, and the coordinates of intersection points are obtained by solving the corresponding nonlinear equations. Then, the distance between the two intersection points is tooth thickness. By utilizing the acquired key points, the tooth profile of cylindrical worm gear in its cross section is drawn by interpolation method. Based on the theory brought forward, the tooth thickness characteristics of cylindrical worm gear are analyzed. The change rule of the top thickness of cylindrical worm gear along the tooth width is discovered, and the change curve is plotted. The rule obtained is consistent with the experimental results, and thus the correctness of the current work is verified.

It is of a great importance in modern agriculture to study fast, automatic, inexpensive and accurate method of diagnosing plant diseasesTherefore, timely and accurately diagnosis of the disease in the fields is one of the most important factors in dealing with plant diseases. For this reason, in the present study, the image processing method study, has been examined for diagnosing the two important diseases of rice and tomato, brown spots and leaf blasts. In this study, firstly the data section is treated using improved k-means segmentation, after preprocessing. Secondly, comprehensive features are extracted and the disease areas are demarcated. An improved genetic algorithm is used in the feature selection step. Finally, images are categorized using the k-nearest neighbor’s algorithm (k-NN) classifier. The accuracy of the proposed method for the rice data set is 99.12 and for the tomato data set is 97.29, which shows a very good performance compared to other methods.

This article considers the master-slave Markovian jump complex dynamical networks (CDNs) with multiple time-varying delays by designing a non-fragile sliding mode control mechanism to achieve $${H_\infty }$$ /passive synchronization. First, by introducing a non-fragile sliding surface, master-slave CDNs can achieve $${H_\infty }$$ /passive synchronization, and the state trajectory of CDNs can arrive the sliding surface based on a suitable controller. By applying an appropriate Lyapunov–Krasovskii functional (LKF) that combines triple and four integral terms, using the reciprocal convex combination method and utilizing a few novel integral inequalities, the criterion of $${H_\infty }$$ /passive synchronization is established for master-slave CDNs. Besides, the prospective criterion of $${H_\infty }$$ /passive synchronization can be transformed into the form of linear matrix inequalities. Moreover, reasonability of theoretical results is proved via numerical simulations.

Type-II superconductivity is known to persist close to the sample surface in presence of a strong magnetic field. As a consequence, the ground state energy in the Ginzburg-Landau theory is approximated by an effective one-dimensional model. As shown in [CG2], the presence of corners on the surface affects the energy of the sample with a non-trivial contribution. In [CG2], the two-dimensional model problem providing the corner energy is implicitly identified and, although no explicit dependence of the energy on the corner opening angle is derived, a conjecture about its form is proposed. We study here such a conjecture and confirm it, at least to leading order, for corners with almost flat opening angle., Comment: 22 pages, pdfLaTex

The issues of estimating the error of coordinate measurements of the shape characteristics of geometric elements of products depending on the number of control points, taking into account a given confidence probability, are considered. Analytical models are proposed for estimating the error based on a priori data, similar to estimating uncertainty by type B. The correspondence of model and experimental results is verified by the Monte Carlo method using a specially developed program in VBA and the library functions of the Statistical Analysis package of the Microsoft Excel program. Such a characteristic of the shape of the part as roundness is investigated. The influence of the parameters of regular structures associated with the features of the technological processes of manufacturing the part on the parameters of the distribution function of the coordinates of the control points is revealed. Fourier analysis is used to identify and quantify regular structures on the surface of the part. The sources of error that have a significant impact on the results of measurements of the shape characteristics of geometric elements of products are given. Based on the results of statistical calculations, the dependence of the measurement error of the shape characteristics on the number of control points is analyzed, the scope of application of analytical formulas for estimating the error of single measurements for a given confidence probability is determined. The article is intended for specialists in the field of practical coordinate metrology and related fields.

Most of the scientific and industrial experiments usually involve small number of factors, especially biomedical experiments and computer based experiments. We construct some highly useful designs ...

We use the Borisov-Keum equations of a fake projective plane and the Borisov-Yeung equations of the Cartwright-Steger surface to show the existence of a regular surface with canonical map of degree 36 and of an irregular surface with canonical map of degree 27. As a by-product, we get equations (over a finite field) for the $\mathbb Z/3$-invariant fibres of the Albanese fibration of the Cartwright-Steger surface and show that they are smooth., Ancillary files with Magma code included. Final version, to appear in J Algebraic Geom

More and more simulation tools are being used in the development of gears in order to save development time and costs while improving the gears. BECAL is a comprehensive software tool for the tooth contact analysis (TCA) of bevel, hypoid, beveloid and spur gears. The gear geometry is provided by a manufacturing simulation or a geometry import. To determine the exact contact conditions in the TCA, the discrete flank points are converted into a continuous and differentiable surface representation. At present, it is an approximation by means of Bézier tensor product surfaces. With this surface representation, significant deviations to the target points can occur depending on the tooth geometry. In particular tip, root and end relief, strongly curved tooth root geometries or discontinuous topological measurement data due to e.g. micro-pitting can only be considered insufficiently.Hence, a new method for surface approximation with non-uniform rational b‑spline surfaces (NURBS) is presented. Its application can significantly improve the surface representation compared to the target geometry, leading to more realistic results regarding contact stress, tooth root stress and transmission error. To illustrate the advantages, NURBS-based surfaces are compared with the Bézier tensor product surfaces. Finally, the potential of the new approach regarding the prediction of lifetime and acoustics is demonstrated by application to different gear geometries.

We study the dynamics of {\it topologically Anosov} homeomorphisms of non-compact surfaces. In the case of surfaces of genus zero and finite type, we classify them. We prove that if $f\colon S \to S$, is a Topologically Anosov homeomorphism where $S$ is a non-compact surface of genus zero and finite type, then $S= \mathbb{R}^2$ and $f$ is conjugate to a homothety or reverse homothety (depending on wether $f$ preserves or reverses orientation). A weaker version of this result was conjectured in \cite{cgx}.

The problem of studying the structure of planar graphs with sets of points, which should be critical concerning the distance between cells on the boundaries of which the elements of a given set are located in operations of removing vertices or edges of a graph, is considered. Knowing the structure of these planar graphs, it is possible to construct a finite set of planar graphs with given characteristics required for the construction of obstruction graphs of a given nonorientable genus. The main result is to use the constructed list of plane graphs critical concerning distance 2 to construct obstruction graphs of a given nonorientable genus.

Several kinds of approximation of the gravitational potential of a homogeneous body by truncated spherical harmonics series are in use in physical geodesy. However, only one of them is capable of a representation converging to the true potential in the whole layer between the Brillouin sphere and the Bjerhammar sphere of the body. We aim at providing various majorizations, namely upper bounds, of the error with the double purpose of proving explicitly the convergence in the sense of different norms and of giving computable bounds, that might be used in numerical studies. The first aim is reached for all the norms. For the second, however, it turns out that among the bounds, when applied to the example of the terrain correction of the Earth, only those referring to the mean absolute error and the mean squared error at the level of Brillouin sphere of minimum radius give significant and useful results. In order to make the computation an easy exercise, a simple approximate formula has been developed requiring only the use of the distribution function of the heights of the surface of the body with respect to the Bjerhammar sphere.