Search

Every irreducible germ of singular complex plane analytic curve at the origin can be described in two ways in a given coordinate system: by a parametrization x = T n , y = ∑ i = 1 ∞ a i T i or by an equation f (x , y) = 0 where f = ∑ c i j X i Y j is a complex power series in two variables. We show that every polynomial condition on the coefficients of a parametrization is, under some natural invariance assumptions, equivalent with a polynomial condition on the coefficients of f. [ABSTRACT FROM AUTHOR]

We characterise, in terms of Dixmier–Ohno invariants, the types of singularities that a plane quartic curve can have. We then use these results to obtain new criteria for determining the stable reduction types of non-hyperelliptic curves of genus 3. [ABSTRACT FROM AUTHOR]

In the study of plane curves, one of the problems is to classify the embedded topology of plane curves in the complex projective plane that have a given fixed combinatorial type, where the combinatorial type of a plane curve is data equivalent to the embedded topology in its tubular neighborhood. A pair of plane curves with the same combinatorial type but distinct embedded topology is called a Zariski pair. In this paper, we consider Zariski pairs consisting of conic-line arrangements that arise from Poncelet's closure theorem. We study unramified double covers of the union of two conics that are induced by a $2m$ -sided Poncelet transverse. As an application, we show the existence of families of Zariski pairs of degree $2m+6$ for $m\geq 2$ that consist of reducible curves having two conics and $2m+2$ lines as irreducible components. [ABSTRACT FROM AUTHOR]

The aim of this paper is an exposition of fields of multiply periodic, or Kleinian, ℘-functions. Such a field arises on the Jacobian variety of an algebraic curve, providing natural algebraic models for the Jacobian and Kummer varieties, possessing the addition law, and accommodating dynamical equations with solutions. All of this will be explained in detail for plane algebraic curves in their canonical forms. Examples of hyperelliptic and non-hyperelliptic curves are presented. [ABSTRACT FROM AUTHOR]

This paper considers the construction of the fundamental function and Abelian differentials of the third kind on a plane algebraic curve over the field of complex numbers that has no singular points. The algorithm for constructing differentials of the third kind was described in Weierstrass's lectures. The paper discusses its implementation in the Sage computer algebra system. The specifics of this algorithm, as well as the very concept of the differential of the third kind, implies the use of both rational numbers and algebraic numbers, even when the equation of a curve has integer coefficients. Sage has a built-in tool for computations in algebraic number fields, which allows the Weierstrass algorithm to be implemented almost literally. The simplest example of an elliptic curve shows that it requires too many resources, far beyond the capabilities of an office computer. A symmetrization of the method is proposed and implemented, which makes it possible to solve the problem while saving a significant amount of computational resources. [ABSTRACT FROM AUTHOR]

In this paper, we explore how the typical second year undergraduate topic of matrix diagonalisation can be applied to solve a geometric problem which arises frequently in Vector Calculus: to find the curve of intersection of a plane and a surface. We use GeoGebra, which offers a cost-free and dynamic interface, to explore a specific variant of this problem, namely the parametrisation of the intersection of a plane and sphere. GeoGebra not only allows students to confirm their calculations, but it can also provide motivation to create connections between prior learning experiences and new knowledge. [ABSTRACT FROM AUTHOR]

In this paper, we construct the first examples of what we call fake ES-irreducible components; Definition 2.8. In our way to do so, we classify the automorphism groups of smooth plane sextics that only have automorphisms of order ≤ 3 ; Theorems 2.1, 2.4 and 2.5, Corollaries 2.9 and 2.11. [ABSTRACT FROM AUTHOR]

In this work, it is taken into account that in nature, due to pressure and temperature, water drops in general are either spherical or ellipsoidal. Thus, starting from a more general structure, a 3D elliptical surface (oblate spheroid) is constructed, which, by means of parameters, can be turned into a spherical shape. Hexagons are built on a rectangular horizontal plane, then this plane is passed through an elliptical surface at height h, which is determined by a parameter θ. As a result of the cutting of these surfaces, a curve and a plane are obtained, both horizontal ellipsoidal; if these hexagons are within the perimeter of the horizontal ellipse obtained as a function of θ , they are marked with an N, and if they are outside the perimeter, they are marked with an E. Several frozen nucleation centers are established, either in the same layer or in different planes, marking them with an F and their first eight neighbors with a B. The calculations based on a modified snowflake model are carried out tile by tile and layer by layer, governed by the thermodynamic factors α , β , and γ , leading to results that depend on the position of the nucleator, which can be symmetrical or asymmetrical for a snowflake with more than one nucleation center and an external surface formed by water vapor that functions as a membrane. [ABSTRACT FROM AUTHOR]

We define a family of special functions (the CSI ones), which can be used to write any parameterized plane curve with polynomial curvature explicitly. These special functions generalize the Fresnel integrals, and may have an interest in their own right. We prove that any plane curve with polynomial curvature is asymptotically a pseudo-spiral. Using the CSI functions, we can approximate, locally, any plane curve; this approach provides a useful criterion for a (local) classification of plane curves. In addition, we present a new algorithm for finding an arc-length parametrization for any curve, within a prescribed degree of approximation. [ABSTRACT FROM AUTHOR]

Given an algebraic germ of a plane curve at the origin, in terms of a bivariate polynomial, we analyze the complexity of computing an irreducible decomposition up to any given truncation order. With a suitable representation of the irreducible components, and whenever the characteristic of the ground field is zero or larger than the degree of the germ, we design a new algorithm that involves a nearly linear number of arithmetic operations in the ground field plus a small amount of irreducible univariate polynomial factorizations. [ABSTRACT FROM AUTHOR]

Anguilliform locomotion, an efficient aquatic locomotion mode where the whole body is engaged in fluid–body interaction, contains sophisticated physics. We hypothesized that data-driven modeling techniques may extract models or patterns of the swimmers' dynamics without implicitly measuring the hydrodynamic variables. This work proposes empirical kinematic control and data-driven modeling of a soft swimming robot. The robot comprises six serially connected segments that can individually bend with the segmental pneumatic artificial muscles. Kinematic equations and relations are proposed to measure the desired actuation to mimic anguilliform locomotion kinematics. The robot was tested experimentally and the position and velocities of spatially digitized points were collected using QualiSys® Tracking Manager (QTM) 1.6.0.1. The collected data were analyzed offline, proposing a new complex variable delay-embedding dynamic mode decomposition (CDE DMD) algorithm that combines complex state filtering and time embedding to extract a linear approximate model. While the experimental results exhibited exotic curves in phase plane and time series, the analysis results showed that the proposed algorithm extracts linear and chaotic modes contributing to the data. It is concluded that the robot dynamics can be described by the linearized model interrupted by chaotic modes. The technique successfully extracts coherent modes from limited measurements and linearizes the system dynamics. [ABSTRACT FROM AUTHOR]

In the present article we construct new families of free and nearly free curves starting from a plane cubic curve |$C$| and adding some of its hyperosculating conics. We present results that involve nodal cubic curves and the Fermat cubic. In addition, we provide new insight into the geometry of the |$27$| hyperosculating conics of the Fermat cubic curve using well-chosen group actions. [ABSTRACT FROM AUTHOR]

An examination was previously derived to conclude the understanding of the response of a cantilever beam with a tip mass (CBTM) that is stimulated by a parameter to undergo small changes in flexibility (stiffness) and tip mass. The study of this problem is essential in structural and mechanical engineering, particularly for evaluating dynamic performance and maintaining stability in engineering systems. The existing work aims to study the same problem but in different situations. He's frequency formula (HFF) is utilized with the non-perturbative approach (NPA) to transform the nonlinear governing ordinary differential equation (ODE) into a linear form. Mathematica Software 12.0.0.0 (MS) is employed to confirm the high accuracy between the nonlinear and the linear ODE. Actually, the NPA is completely distinct from any traditional perturbation technique. It simply inspects the stability criteria in both the theoretical and numerical calculations. Temporal histories of the obtained results, in addition to the corresponding phase plane curves, are graphed to explore the influence of various parameters on the examined system's behavior. It is found that the NPA is simple, attractive, promising, and powerful; it can be adopted for the highly nonlinear ODEs in different classes in dynamical systems in addition to fluid mechanics. Bifurcation diagrams, phase portraits, and Poincaré maps are used to study the chaotic behavior of the model, revealing various types of motion, including periodic and chaotic behavior. [ABSTRACT FROM AUTHOR]

This paper delves into the study of curvilinear Hilbert schemes associated with a singular variety |$(X,0)$| and the relationship between their motivic classes and the motivic measure on the arc scheme |$X_\infty $| of |$X$| introduced by Denef and Loeser. We introduce an Igusa zeta function specifically tailored for curvilinear Hilbert schemes for which we provide an explicit formulation in terms of an embedded resolution of the singularity, and we consequently obtain a recursive formula to compute the motivic classes of curvilinear Hilbert schemes in terms of the resolution. In addition, the paper explores and analyzes the geometry and combinatorics of curvilinear Hilbert schemes in the context of plane curve singularities and their topological invariants. [ABSTRACT FROM AUTHOR]

In 2011 Hefez and Hernandes completed Zariski's analytic classification of plane branches belonging to a given equisingularity class by creating "very short" parameterizations over the complex numbers. Their results were used by Mehmood and Pfister to classify unimodal plane branches in characteristic 0 by giving lists of normal forms. The aim of this paper is to give a complete classification of unimodal plane branches over an algebraically closed field of positive characteristic. Since the methods of Hefez and Hernandes cannot be used in positive characteristic, we use a different approach and, for some sporadic singularities in small characteristic, computations with Singular. Our methods are characteristic-independent and provide a different proof of the classification in characteristic 0 showing at the same time that this classification holds also in large characteristic. The main theoretical ingredients are the semicontinuity of the semigroup and of the modality, which we prove and which may be of independent interest. [ABSTRACT FROM AUTHOR]

In the initial part of this paper, we survey (in arbitrary spacetime dimension) the general FLRW cosmologies with non-interacting perfect fluids and with a canonical or phantom scalar field, minimally coupled to gravity and possibly self-interacting; after integrating the evolution equations for the fluids, any model of this kind can be described as a Lagrangian system with two degrees of freedom, where the Lagrange equations determine the evolution of the scale factor and the scalar field as functions of the cosmic time. We analyze specific solvable models, paying special attention to cases with a phantom scalar; the latter favors the emergence of nonsingular cosmologies in which the Big Bang is replaced, e.g., with a Big Bounce or a periodic behavior. As a first example, we consider the case with dust (i.e., pressureless matter), radiation, and a scalar field with a constant self-interaction potential (this is equivalent to a model with dust, radiation, a free scalar field and a cosmological constant in the Einstein equations). In the phantom subcase (say, with nonpositive spatial curvature), this yields a Big Bounce cosmology, which is a non-absurd alternative to the standard (Λ CDM) Big Bang cosmology; this Big Bounce model is analyzed in detail, even from a quantitative viewpoint. We subsequently consider a class of cosmological models with dust and a phantom scalar, whose self-potential has a special trigonometric form. The Lagrange equations for these models are decoupled passing to suitable coordinates (x , y) , which can be interpreted geometrically as Cartesian coordinates in a Euclidean plane: in this description, the scale factor is a power of the radius r = x 2 + y 2 . Each one of the coordinates x , y evolves like a harmonic repulsor, a harmonic oscillator, or a free particle (depending on the signs of certain constants in the self-interaction potential of the phantom scalar). In particular, in the case of two harmonic oscillators, the curves in the plane described by the point (x , y) as a function of time are the Lissajous curves, well known in other settings but not so popular in cosmology. A general comparison is performed between the contents of the present work and the previous literature on FLRW cosmological models with scalar fields, to the best of our knowledge. [ABSTRACT FROM AUTHOR]

The article "Heron Triangles and the Hunt for Unicorns" delves into the search for Pythagorean triples and Heron triangles with rational sides and medians. It explores the concept of perfect triangles with integer sides, medians, and area, as well as the mathematical beauty of Somos-5 sequences and their connection to integrable maps. The text discusses the work of Buchholz and Rathbun in finding Heron triangles with two rational medians using specific algorithms, presenting tables and equations to illustrate the relationships between the parameters of these triangles. It also mentions the existence of an infinite family of such triangles and sporadic cases that do not conform to this family, highlighting the intricate mathematical patterns and challenges in studying these triangles. [Extracted from the article]

The article explores geodesics on polyhedra, particularly focusing on simple closed geodesics and their properties. It discusses the classification of geodesics on regular polyhedra, the relationship between geodesics and billiards, and the uniqueness of disphenoids in having arbitrarily long geodesics. The text also addresses the existence of simple closed geodesics on tetrahedra and the properties of geodesics on nonconvex polyhedra, including the construction of long closed geodesics using seven cubes. Open questions are posed regarding geodesics on nonconvex polyhedra in different spaces and the minimal number of vertices needed for long geodesics. The author acknowledges the contributions of an anonymous referee for their feedback. [Extracted from the article]

In this study first, natural logarithm function f(x) = lnx with base e has been examined as polynomial function of 5th, 6th, 7th order Bézier curve. By modelling matrix representation of 5th, 6th, 7th order Bézier curve we have found the control points in plane. Further, Mercator series for the curves In(1+x) and ln(1-x) have been written too as the polynomial functions as 5th, 6th, 7th order Bézier curve in plane based on the control points with matrix form in E². Finally, the curve In(1-x²) has been expressed as 5th, 6th, 7th order Bézier curve, examined the control points and given matrix forms. [ABSTRACT FROM AUTHOR]

Copyright of Machine Tool & Hydraulics is the property of Guangzhou Mechanical Engineering Research Institute (GMERI) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We construct and analyze families of periodic delay orbits for a class of delay differential equations in two dimensions depending on two real-valued functions. These families are parametrized by the delay parameter. It is possible to represent the dependency of these periodic delay orbits on the delay parameter by a curve in the plane, without loss of information. It turns out that the singularities of these curves necessarily are cusps in the non-degenerate case. After discussing degenerate situations in general, we explain how to glue different families of periodic delay orbits at degeneracies in the delay parameter. [ABSTRACT FROM AUTHOR]

The document provides a list of references to scientific studies on fluid mechanics, droplet coalescence, and the Leidenfrost effect. Topics include thin film drainage measurements, droplet behavior on heated liquid pools, and the levitation of drops on liquid nitrogen. The studies offer insights into phenomena like hydrodynamic slippage and the motion of bubbles in capillaries, contributing to our understanding of fluid dynamics and interfacial phenomena. [Extracted from the article]

Copyright of Journal of Advanced Mathematical Modeling (JAMM) is the property of Shahid Chamran University of Ahvaz and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The article "A singularly altered streamline topology allows faster transport from deformed drops" delves into the impact of drop deformation on scalar transport rate in planar linear flows. It reveals that deformed drops exhibit enhanced transport rates due to altered streamline topologies, leading to faster transport compared to spherical drops. The study emphasizes the significance of streamline topology in convective enhancement and its implications for various suspended elastic microstructures. These findings contribute valuable insights for research on heat and mass transport in multiphase scenarios, showcasing the complexity of transport phenomena in deformed drops. [Extracted from the article]

The present study aims to examine the movement of a simple pendulum that is connected by a lightweight spring and connected with a rotating wheel. The motivation behind this topic is to gain a comprehensive understanding of intricate dynamic systems that involve consistent mechanical components and response with time delay. This system is not only theoretically attractive but also practically appropriate in domains such as robotics, engineering, and control systems. As well-known, all classical perturbation methods exploit Taylor expansion to simplify the practicality of restoring forces. In contrast, the non-perturbative approach, as a novel methodology, transforms any nonlinear ordinary differential equation into a linear one. It scrutinizes the restoring forces, away from employing Taylor expansion; hence it eliminates the previous weakness. The concept of the non-perturbative approach is based mainly on the He's frequency formula. The confidence of the non-perturbative approach comes from the numerical compatibility between the nonlinear and linear ordinary differential equation via the Mathematica Software. Therefore, instead of handling the nonlinear ordinary differential equation, we investigate the linear one. The achieved response is plotted over time to show the impact of the acted parameters during a specified time interval. Moreover, the phase plane curves that correspond to the plotted solution are presented and examined. The stability criteria of the analogous linear ordinary differential equation are provided and drawn to explore the stability/instability zones. The performance is applicable in engineering and other fields due to its ease of adaptation to different nonlinear systems. Therefore, the non-perturbative approach can be regarded as substantial, successful, and interesting and can extended to be applied in further categories within the field of couples dynamical systems. [ABSTRACT FROM AUTHOR]

The Fréchet distance is a popular measure of dissimilarity for polygonal curves. It is defined as a min–max formulation that considers all orientation-preserving bijective mappings between the two curves. Because of its susceptibility to noise, Driemel and Har-Peled introduced the shortcut Fréchet distance in 2012, where one is allowed to take shortcuts along one of the curves, similar to the edit distance for sequences. We analyse the parameterised version of this problem, where the number of shortcuts is bounded by a parameter \(k\). The corresponding decision problem can be stated as follows: Given two polygonal curves \(T\) and \(B\) of at most \(n\) vertices, a parameter \(k\) and a distance threshold \(\delta\) , is it possible to introduce \(k\) shortcuts along \(B\) such that the Fréchet distance of the resulting curve and the curve \(T\) is at most \(\delta\) ? We study this problem for polygonal curves in the plane. We provide a complexity analysis for this problem with the following results: (1) there exists a decision algorithm with running time in \(\mathcal{O}(kn^{2k+2}\log n)\) ; (2) assuming the exponential-time hypothesis (ETH), there exists no algorithm with running time bounded by \(n^{o(k)}\). In contrast, we also show that efficient approximate decider algorithms are possible, even when \(k\) is large. We present a \((3+\varepsilon)\) -approximate decider algorithm with running time in \(\mathcal{O}(kn^{2}\log^{2}n)\) for fixed \(\varepsilon\). In addition, we can show that, if \(k\) is a constant and the two curves are \(c\) -packed for some constant \(c\) , then the approximate decider algorithm runs in near-linear time. [ABSTRACT FROM AUTHOR]

We consider a reaction-diffusion predator-prey model with predator-taxis, in which both the cost and the benefit induced by fear of predators are incorporated. In particular, the benefit of fear is reflected in the Beddington-DeAngelis functional response that can be derived in view of avoidance behaviours of prey. Critical conditions for Turing instability are determined with the help of the first Turing bifurcation curve in the two-parameter plane composed by the random diffusion rate of prey and the predator-taxis rate. For predator-taxis-induced bifurcation from simple eigenvalues, the existence and stability of non-homogeneous positive steady state solutions are established. Especially, we use decomposition in space and apply the implicit function theorem to obtain the bifurcation theorem with double eigenvalues. Theoretical and numerical results show that low predator-taxis sensitivity may cause the occurrence of spatial patterns when the random diffusion rate of prey is slow. However, the presence of predator-taxis can not lead to pattern formation when the random diffusion rate of prey is fast enough. In addition, increasing the level of fear may stabilize the system at the expense of the density of prey or predators. [ABSTRACT FROM AUTHOR]

Planar point sets with many triple lines (which contain at least three distinct points of the set) have been studied for 180 years, started with Jackson and followed by Sylvester. Green and Tao (Discret Comput Geom 50(2):409–468, 2013) have shown that the maximum possible number of triple lines for an n-element set is ⌊ n (n - 3) / 6 ⌋ + 1 . Here we address the related problem of describing the structure of the asymptotically near-optimal configurations, i.e., of those for which the number of straight lines which go through three or more points has a quadratic (i.e., best possible) order of magnitude. We pose the problem whether such point sets must always be related to cubic curves. To support this conjecture we settle various special cases; some of them are also related to the four-in-a-line problem of Erdős. [ABSTRACT FROM AUTHOR]

Let F \in \mathbb {Z}[x_0, \ldots, x_n] be homogeneous of degree d and assume that F is not a 'nullform', i.e., there is an invariant I of forms of degree d in n+1 variables such that I(F) \neq 0. Equivalently, F is semistable in the sense of Geometric Invariant Theory. Minimizing F at a prime p means to produce T \in Mat(n+1, \mathbb {Z}) \cap GL(n+1, \mathbb {Q}) and e \in \mathbb {Z}_{\ge 0} such that F_1 = p^{-e} F([x_0, \ldots, x_n] \cdot T) has integral coefficients and v_p(I(F_1)) is minimal among all such F_1. Following Kollár [Electron. Res. Announc. Amer. Math. Soc. 3 (1997), pp. 17–27], the minimization process can be described in terms of applying weight vectors w \in \mathbb {Z}_{\ge 0}^{n+1} to F. We show that for any dimension n and degree d, there is a complete set of weight vectors consisting of [0,w_1,w_2,\dots,w_n] with 0 \le w_1 \le w_2 \le \dots \le w_n \le 2 n d^{n-1}. When n = 2, we improve the bound to d. This answers a question raised by Kollár. These results are valid in a more general context, replacing \mathbb {Z} and p by a PID R and a prime element of R. Based on this result and a further study of the minimization process in the planar case n = 2, we devise an efficient minimization algorithm for ternary forms (equivalently, plane curves) of arbitrary degree d. We also describe a similar algorithm that allows to minimize (and reduce) cubic surfaces. These algorithms are available in the computer algebra system Magma. [ABSTRACT FROM AUTHOR]

Neural implicit surfaces are a promising tool for geometry processing that represent a solid object as the zero level set of a neural network. Usually trained to approximate a signed distance function of the considered object, these methods exhibit great visual fidelity and quality near the surface, yet their properties tend to degrade with distance, making geometrical queries hard to perform without the help of complex range analysis techniques. Based on recent advancements in Lipschitz neural networks, we introduce a new method for approximating the signed distance function of a given object. As our neural function is made 1‐Lipschitz by construction, it cannot overestimate the distance, which guarantees robustness even far from the surface. Moreover, the 1‐Lipschitz constraint allows us to use a different loss function, called the hinge‐Kantorovitch‐Rubinstein loss, which pushes the gradient as close to unit‐norm as possible, thus reducing computation costs in iterative queries. As this loss function only needs a rough estimate of occupancy to be optimized, this means that the true distance function need not to be known. We are therefore able to compute neural implicit representations of even bad quality geometry such as noisy point clouds or triangle soups. We demonstrate that our methods is able to approximate the distance function of any closed or open surfaces or curves in the plane or in space, while still allowing sphere tracing or closest point projections to be performed robustly. [ABSTRACT FROM AUTHOR]

In this paper, we define novel topological invariants of doubly periodic tangles (DP tangles). DP tangles are embeddings of curves in the thickened plane with translational symmetries in two independent directions. We first organize the components of a DP tangle into different interlinked compounds, which are invariants of a DP tangle. The notion of an interlinked compound leads to the classification of DP tangles according to their directional type. We then prove that the directional type is an invariant of DP tangles using the concept of axis-motif, which can be viewed as the blueprint of a DP tangle. [ABSTRACT FROM AUTHOR]

In this paper, we study the bivariate truncated moment problem (TMP) on curves of the form $ y=q(x) $ y = q (x) , $ q(x)\in \mathbb R[x] $ q (x) ∈ R [ x ] , $ \deg q\geq ~3 $ deg ⁡ q ≥ 3 and $ yx^\ell =1 $ y x ℓ = 1 , $ \ell \in \mathbb N\setminus \{1\} $ ℓ ∈ N ∖ { 1 }. For even degree sequences, the solution based on the size of moment matrix extensions was first given by Fialkow [Fialkow L. Solution of the truncated moment problem with variety $ y=x^3 $ y = x 3 . Trans Amer Math Soc. 2011;363:3133–3165.] using the truncated Riesz–Haviland theorem [Curto R, Fialkow L. An analogue of the Riesz–Haviland theorem for the truncated moment problem. J Funct Anal. 2008;255:2709–2731.] and a sum-of-squares representations for polynomials, strictly positive on such curves [Fialkow L. Solution of the truncated moment problem with variety $ y=x^3 $ y = x 3 . Trans Amer Math Soc. 2011;363:3133–3165.; Stochel J. Solving the truncated moment problem solves the moment problem. Glasgow J Math. 2001;43:335–341.]. Namely, the upper bound on this size is quadratic in the degrees of the sequence and the polynomial determining a curve. We use a reduction to the univariate setting technique, introduced in [Zalar A. The truncated Hamburger moment problem with gaps in the index set. Integral Equ Oper Theory. 2021;93:36.doi: .; Zalar A. The truncated moment problem on the union of parallel lines. Linear Algebra Appl. 2022;649:186–239. .; Zalar A. The strong truncated Hamburger moment problem with and without gaps. J Math Anal Appl. 2022;516:126563. doi: .], and improve Fialkow's bound to $ \deg q-1 $ deg ⁡ q − 1 (resp. $ \ell +1 $ ℓ + 1) for curves $ y=q(x) $ y = q (x) (resp. $ yx^\ell =1 $ y x ℓ = 1). This in turn gives analogous improvements of the degrees in the sum-of-squares representations referred to above. Moreover, we get the upper bounds on the number of atoms in the minimal representing measure, which are $ k\deg q $ kdeg ⁡ q (resp. $ k(\ell +1) $ k (ℓ + 1)) for curves $ y=q(x) $ y = q (x) (resp. $ yx^\ell =1 $ y x ℓ = 1) for even degree sequences, while for odd ones they are $ k\deg q-\big \lceil \frac {\deg q}{2} \big \rceil $ kdeg ⁡ q − ⌈ deg ⁡ q 2 ⌉ (resp. $ k(\ell +1)-\big \lfloor \frac {\ell }{2} \big \rfloor +1 $ k (ℓ + 1) − ⌊ ℓ 2 ⌋ + 1) for curves $ y=q(x) $ y = q (x) (resp. $ yx^\ell =1 $ y x ℓ = 1). In the even case, these are counterparts to the result by Riener and Schweighofer [Riener C, Schweighofer M. Optimization approaches to quadrature:a new characterization of Gaussian quadrature on the line and quadrature with few nodes on plane algebraic curves, on the plane and in higher dimensions. J Complex. 2018;45:22–54., Corollary 7.8], which gives the same bound for odd degree sequences on all plane curves. In the odd case, their bound is slightly improved on the curves we study. Further on, we give another solution to the TMP on the curves studied based on the feasibility of a linear matrix inequality, corresponding to the univariate sequence obtained, and finally we solve concretely odd degree cases to the TMP on curves $ y=x^\ell $ y = x ℓ , $ \ell =2,3 $ ℓ = 2 , 3 , and add a new solvability condition to the even degree case on the curve $ y=x^2 $ y = x 2 . [ABSTRACT FROM AUTHOR]

We introduce generalized Darboux frames along a null Cartan curve lying on a timelike surface in Minkowski space 피13 and define null Cartan normal isophotic and normal silhouette curves in terms of the vector field that lies in the normal plane of the curve and belongs to its generalized Darboux frame of the first kind. We investigate null Cartan normal isophotic and normal silhouette curves with constant geodesic curvature kg$$ {k}_g $$ and constant geodesic torsion τg$$ {\tau}_g $$. We obtain the parameter equations of their axes and prove that such curves are the null Cartan helices or the null Cartan cubics. In particular, we show that null Cartan normal isophotic curves with a non‐zero constant curvatures kg$$ {k}_g $$ and τg$$ {\tau}_g $$ have a remarkable property that they are general helices, relatively normal‐slant helices and isophotic curves with respect to the same axis. We prove that null Cartan cubics lying on a timelike surface are normal isophotic curves with a spacelike axis and normal silhouette curves with a lightlike axis. We obtain the relation between Minkowski Pythagorean hodograph cubic curves and null Cartan normal isophotic and normal silhouette curves. Finally, we give numerical examples of null Cartan normal isophotic and normal silhouette curves obtained by integrating the system of two the first order differential equations under the initial conditions. [ABSTRACT FROM AUTHOR]

A novel method for separating the matrix spectrum by a straight line based on a fractional linear transformation is proposed. This method has a number of advantages over the approaches based on an exponential transformation; more precisely, the range of its application is wider and the number of iterations needed for its convergence is much lower. The proposed method is used to study flutter problems for an infinite strip under various edge fastening conditions, which, after suitable discretization of differential operators, are reduced to spectral problems for linear operators. The study of stability regions by the method of spectrum dichotomy by the imaginary axis makes it possible to construct neutral curves in the plane of parameters of the flutter problem. [ABSTRACT FROM AUTHOR]

In this study, we consider the problem of how to characterize a surface family from a given common asymptotic curve in a three-dimensional Galilean space 3 . We obtain a parametrization of the surface such that the surface tangent plane is coincident with the curve osculating plane. The extension to ruled surfaces is also outlined. We illustrate this method by presenting some examples. [ABSTRACT FROM AUTHOR]

In this paper, the motion of strictly convex closed plane curves under dissipative hyperbolic mean curvature flow is studied. The hyperbolic Monge–Amp è$$ \overset{\grave }{e} $$re equation is derived by using the support function. The short‐time existence of the flow is proved, and some evolution equations are derived. Furthermore, according to different initial velocities, we discuss the expansion and contraction of the dissipative hyperbolic curvature flow; that is, if the initial velocity v˜0<−α2ω0(θ)$$ {\tilde{v}}_0<-\frac{\alpha }{2}{\omega}_0\left(\theta \right) $$, the flow will converge to the limit curve at a finite time; if the initial velocity v˜0=−α2ω0(θ)$$ {\tilde{v}}_0&#x0003D;-\frac{\alpha }{2}{\omega}_0\left(\theta \right) $$, the flow will converge to a point t→∞$$ t\to \infty $$; if the initial velocity −−α2ω0(θ)0$$ {\tilde{v}}_0\left(\theta \right)>0 $$, the flow will expand to a limit curve as t→∞$$ t\to \infty $$. [ABSTRACT FROM AUTHOR]

In this paper, we explore the thermodynamic and phase transition properties of asymptotically AdS black holes within Einstein–Gauss–Bonnet gravity, focusing on Joule–Thomson expansion. Thermodynamics is studied in the extended phase space, where the cosmological constant serves as thermodynamic pressure. We observe that the black hole undergoes a phase transition similar to that of a van der Waals system. We analyze charged and neutral cases separately to distinguish the effect of charge and Gauss–Bonnet parameter on critical behavior and examine the phase structure. We find that the Gauss–Bonnet coupling parameter behaves similarly to black hole charge or spin, guiding the phase structure. To understand the underlying phase structure determined by the Gauss–Bonnet coefficient α , we introduce a new order parameter. We discover that the change in the conjugate variable to the Gauss–Bonnet parameter acts as an order parameter, demonstrating a critical exponent of 1 ∕ 2 in the vicinity of the critical point. Since the phase structure is analogous to that of a van der Waals fluid, we investigate the Joule–Thomson expansion of the black hole. We analytically study the Joule–Thomson expansion, focusing on three key characteristics: the Joule–Thomson coefficient, inversion curves and isenthalpic curves. We obtain isenthalpic curves in the T–P plane and illustrate the cooling–heating regions. [ABSTRACT FROM AUTHOR]

Frenet and Darboux trihedrons are the objects of research. At the current point of the direction curve of the Frenet trihedron, three mutually perpendicular unit orthogonal vectors can be uniquely constructed. The orthogonal vector of the tangent is directed along the tangent to the curve at the current point. The orthogonal vector of the main normal is located in the plane, which is formed by three points of the curve on different sides from the current one when they are maximally close to the current point. It is directed to the center of the curvature of the curve. The orthogonal vector of the binormal is perpendicular to the two previous orthogonal vectors and has a direction according to the rule of the right coordinate system. Thus, the movement of the Frenet trihedron along the base curve, as a solid body, is determined. The Darboux trihedron is also a right-hand coordinate system that moves along the base curve lying on the surface. Its orthogonal vector of the tangent is directed identically to the Frenet trihedron, and other orthogonal vectors in pairs form a certain angle ε with the orthogonal vectors of the Frenet trihedron. This is because one of the orthogonal vectors of the Darboux trihedron is normal to the surface and forms a certain angle ε with the binormal. Accordingly, the third orthogonal vector of the Darboux trihedron forms an angle ε with the orthogonal vector of the normal of the Frenet trihedron. This orthogonal vector and orthogonal vector of the tangent form the tangent plane to the surface at the current point of the curve, and the corresponding orthogonal vectors of the tangent and the normal of the Frenet trihedron form the tangent plane of the curve at the same point. Thus, when the Frenet and Darboux trihedrons move along a curve with combined vertices, there is a rotation around the common orthogonal vector point of the tangent at an angle ε between the osculating plane of the Frenet trihedron and the tangent plane to the surface of the Darboux trihedron. These trihedrons coincide in a separate case (for a flat curve) (ε = 0). The connection between Frenet and Darboux trihedrons – finding the expression for the angle ε, is considered in the article. The inverse problem – the determination of the movement of the Darboux trihedron at a given regularity of the change of the angle ε, is also considered. A partial case is considered and it is shown that for a flat base curve at ε = const, the set of positions of the orthogonal vector of normal forms a developable surface of the same angle of inclination of the generators. In addition, the inverse problem of finding the regularity of the change of the angle ε between the corresponding orthogonal vectors of the trihedrons allows constructing a ruled surface for the gravitational descent of the load, conventionally assumed to be a particle. At the same time, the balance of forces in the projections on the orthogonal vectors of the trihedron in the common normal plane of the trajectory is considered. This balance depends on the angle ε [ABSTRACT FROM AUTHOR]

Let R=K[x,y,z]$R=\mathbb {K}[x,y,z]$. A reduced plane curve C=V(f)⊂P2$C=V(f)\subset \mathbb {P}^2$ is free if its associated module of tangent derivations Der(f)$\mathrm{Der}(f)$ is a free R$R$‐module, or equivalently if the corresponding sheaf TP2(−logC)$T_ {\mathbb {P}^2 }(-\log C)$ of vector fields tangent to C$C$ splits as a direct sum of line bundles on P2$\mathbb {P}^2$. In general, free curves are difficult to find, and in this paper, we describe a new method for constructing free curves in P2$\mathbb {P}^2$. The key tools in our approach are eigenschemes and pencils of curves, combined with an interpretation of Saito's criterion in this context. Previous constructions typically applied only to curves with quasihomogeneous singularities, which is not necessary in our approach. We illustrate our method by constructing large families of free curves. [ABSTRACT FROM AUTHOR]

For every n, we construct two arcs in the plane that intersect at least n times and do not form spirals. The construction is in three stages: we first exhibit two closed curves on the torus that do not form double spirals, then two arcs on the torus that do not form spirals, and finally two arcs in the plane that do not form spirals. The planar arcs provide a counterexample to a proof of Pach and Tóth concerning string graphs. [ABSTRACT FROM AUTHOR]

In this paper, we present the unsolvability of |$\overline \partial $| -equation with weighted |$L^{2}$| estimates involved curvature terms on any singular normal complex space in general. Moreover, in the non-normal case, we also give a complete description on |$L^{2}$| -solvability of the |$\overline \partial $| -equation with weighted |$L^{2}$| estimates for plane curve singularities and their variants in the higher dimension. [ABSTRACT FROM AUTHOR]

Bounding the number of rational points of height at most |$H$| on irreducible algebraic plane curves of degree |$d$| has been an intense topic of investigation since the work by Bombieri and Pila. In this paper we establish optimal dependence on |$d$| by showing the upper bound |$C d^{2} H^{2/d} (\log H)^{\kappa }$| with some absolute constants |$C$| and |$\kappa $|⁠. This bound is optimal with respect to both |$d$| and |$H$|⁠ , except for the constants |$C$| and |$\kappa $|⁠. This answers a question raised by Salberger, leading to a simplified proof of his results on the uniform dimension growth conjectures of Heath-Brown and Serre, and where at the same time we replace the |$H^{\varepsilon }$| factor by a power of |$\log H$|⁠. The main strength of our approach comes from the combination of a new, efficient form of smooth parametrizations of algebraic curves with a century-old criterion of Pólya, which allows us to save one extra power of |$d$| compared with the standard approach using Bézout's theorem. [ABSTRACT FROM AUTHOR]

Motivated by the classical Rodrigues' formula, we study below the root asymptotic of the polynomial sequence R [ α n ] , n , P (z) = d [ α n ] P n (z) d z [ α n ] , n = 0 , 1 , ⋯ where P(z) is a fixed univariate polynomial, α is a fixed positive number smaller than deg P , and [ α n ] stands for the integer part of α n . Our description of this asymptotic is expressed in terms of an explicit harmonic function uniquely determined by the plane rational curve emerging from the application of the saddle point method to the integral representation of the latter polynomials using Cauchy's formula for higher derivatives. As a consequence of our method, we conclude that this curve is birationally equivalent to the zero locus of the bivariate algebraic equation satisfied by the Cauchy transform of the asymptotic root-counting measure for the latter polynomial sequence. We show that this harmonic function is also associated with an abelian differential having only purely imaginary periods and the latter plane curve belongs to the class of Boutroux curves initially introduced in Bertola (Anal Math Phys 1: 167–211, 2011), Bertola and Mo (Adv Math 220(1): 154–218, 2009). As an additional relevant piece of information, we derive a linear ordinary differential equation satisfied by { R [ α n ] , n , P (z) } as well as higher derivatives of powers of more general functions. [ABSTRACT FROM AUTHOR]

Geometric shape models often help to extract specific contours in digital images (the segmentation process) with major precision. Motivated by this idea, we introduce two models for the representation of prostate shape in the axial plane of magnetic resonance images. In more detail, the models are two parametric closed curves of the plane. The analytic study of the models includes the geometric role of the parameters describing the curves, symmetries, invariants, special cases, elliptic Fourier descriptors, conditions for simple curves and area of the enclosed surfaces. The models were validated for prostate shapes by fitting the curves to prostate contours delineated by a radiologist and measuring the errors with the mean distance, the Hausdorff distance and the Dice similarity coefficient. Validation was also conducted by comparing our models with the deformed superellipse model used in literature. Our models are equivalent in fitting metrics to the deformed superellipse model; however, they have the advantage of a more straightforward formulation and they depend on fewer parameters, implying a reduced computational time for the fitting process. Due to the validation, our models may be applied for developing innovative and performing segmentation methods or improving existing ones. [ABSTRACT FROM AUTHOR]

This document is an author index for Volume 109 of the Bulletin of the Australian Mathematical Society. It lists the authors and titles of articles published in the journal. Some of the topics covered include birth-and-death processes, convex functions, plane curves, solvable groups, Diophantine equations, meromorphic functions, shape analysis, numerical semigroups, Brownian motion, and nonlinear Schrödinger equations. The index provides a comprehensive overview of the mathematical research published in the journal. [Extracted from the article]