Curves C1 Research Articles

Addition-deletion results for the minimal degree of a Jacobian syzygy of a union of two curves

Let C:f=0 be a reduced curve in the complex projective plane. The minimal degree mdr(f) of a Jacobian syzygy for f, which is the same as the minimal degree of a derivation killing f, is an important invariant of the curve C, for instance it can be used to determine whether C is free or nearly free. In this note we study the relations of this invariant mdr(f) with a decomposition of C as a union of two curves C1 and C2, without common irreducible components. When all the singularities that occur are quasihomogeneous, a result by Schenck, Terao and Yoshinaga yields finer information on this invariant in this setting. Using this, we give some geometrical criteria, the first ones of this type in the existing literature as far as we know, for a line to be a jumping line for the rank 2 vector bundle of logarithmic vector fields along a reduced curve C.

Cantor sets of low density and Lipschitz functions on C1 curves

We characterize the functions f:(0,1]⟶[0,1] for which there exists a measurable set C⊆[0,1] of positive measure satisfying |C∩I||I|<f(|I|) for any nontrivial interval I⊆[0,1]. As an application, we prove that on any injective C1 curve it is possible to construct a Lipschitz function that cannot be approximated by Lipschitz functions attaining their Lipschitz constant. Finally, we extend this result to more general C1 curves.

Numerical Analysis of Natural Convection Driven Flow of a Non-Newtonian Power-Law Fluid in a Trapezoidal Enclosure with a U-Shaped Constructal

Placement of fins in enclosures has promising utilization in advanced technological processes due to their role as heat reducing/generating elements such as in conventional furnaces, economizers, gas turbines, heat exchangers, superconductive heaters and so forth. The advancement in technologies in power engineering and microelectronics requires the development of effective cooling systems. This evolution involves the utilization of fins of significantly variable geometries enclosed in cavities to increase the heat elimination from heat-generating mechanisms. Since fins are considered to play an effective role in the escalation of heat transmission, the current study is conducted to examine the transfer of heat in cavities embedding fins, as well as the effect of a range of several parameters upon the transmission of energy. The following research is supplemented with the interpretation of the thermo-physical aspects of a power-law liquid enclosed in a trapezoidal cavity embedding a U-shaped fin. The Boussinesq approximation is utilized to generate the mathematical attributes of factors describing natural convection, which are then used in the momentum equation. Furthermore, the Fourier law is applied to formulate the streaming heat inside the fluid flow region. The formulated system describing the problem is non-dimensionalized using similarity transformations. The geometry of the problem comprises a trapezoidal cavity with a non-uniformly heated U-shaped fin introduced at the center of the base of the enclosure. The boundaries of the cavity are at no-slip conditions. Non-uniform heating is provided at the walls (l1 and l2), curves (c1,c2 and c3) and surfaces (s1 and s2) of the fin; the upper wall is insulated whereas the base and sidewalls of the enclosure are kept cold. The solution of the non-dimensionalized equations is procured by the Galerkin finite element procedure. To acquire information regarding the change in displacement w.r.t time and temperature, supplementary quadratic interpolating functions are also observed. An amalgam meshing is constructed to elaborate the triangular and quadrilateral elements of the trapezoidal domain. Observation of significant variation in the flow configurations for a specified range of parameters is taken into consideration i.e., 0.5≤n≤1.5 and 104≤Ra≤106. Furthermore, flow structures in the form of velocity profiles, streamlines, and temperature contours are interpreted for the parameters taken into account. It is deduced from the study that ascending magnitude of (Ra) elevates level of kinetic energy and magnitude of heat flux; however, a contrary configuration is encapsulated for the power-law index. Navier–Stokes equations constituting the phenomenon are written with the help of non-dimensionalized stream function, temperature profiles, and vortices, and the solutions are acquired using the finite element method. Furthermore, the attained outcomes are accessible through velocity and temperature profiles. It is worth highlighting the fact that the following analysis enumerates the pseudo-plastic, viscous and dilatant behavior of the fluid for different values of (n). This study highlights that the momentum profile and the heat transportation increase by increasing (Ra) and decline as the viscosity of the fluid increases. Overall, it can be seen from the current study that heat transportation increases with the insertion of a fin in the cavity. The current communication signifies the phenomenon of a power-law fluid flow filling a trapezoidal cavity enclosing a U-shaped fin. Previously, researchers have studied such phenomena mostly in Newtonian fluids, hence the present effort presents novelty regarding consideration of a power-law liquid in a trapezoidal enclosure by the placement of a U-shaped fin.

Gap Structure of 12442-Type KCa2(Fe1−x Co x )4As4F2 (x = 0, 0.1) Revealed by Temperature Dependence of Lower Critical FieldSupported by the Youth Innovation Promotion Association of the Chinese Academy of Sciences (Grant No. 2015187), the “Strategic Priority Research Program (B)” of the Chinese Academy of Sciences (Grant No. XDB30000000), and the National Natural Science Foundation of China (Grant Nos.

We report an in-depth investigation on the out-of-plane lower critical field H c1 of the KCa2(Fe1−x Co x )4As4F2 (12442-type, x = 0, 0.1). The multi-gap feature is revealed by the kink in the temperature-dependent H c1(T) curve for the two samples with different doping levels. Based on a simplified two-gap model, the magnitudes of the two gaps are determined to be Δ 1 = 1.2 meV and Δ 2 = 5.0 meV for the sample with x = 0, Δ 1 = 0.86 meV and Δ 2 = 2.8 meV for that with x = 0.1. With the cobalt doping, the ratio of energy gap to critical transition temperature (Δ/k B T c) remains almost unchanged for the smaller gap and is suppressed by 20% for the larger gap. For the undoped KCa2Fe4As4F2, the obtained gap sizes are generally consistent with the results of angle-resolved photoemission spectroscopy experiments.

Second order asymptotics for Krein indefinite multipliers with multiplicity two

We consider linear Hamiltonian equations in R2n of the following typedγdt(t)=J2nA(t)γ(t),γ(0)∈Sp(2n,R), where J=J2n=def[0Idn−Idn0] and A:t↦A(t) is a C1 curve in the space of 2n×2n real matrices which are symmetric. Then, t↦γ(t) is a C2 curve in the space of 2n×2n (real) symplectic matrices. We obtain second order asymptotics for the eigenvalues bifurcated from non-real Krein indefinite eigenvalues with algebraic multiplicity two and geometric multiplicity one. As a corollary, we obtain a simple formula about the derivative of the sum of the bifurcated eigenvalues at time t=0. In the end, we discuss possible potential applications for the linear stability of the elliptic Lagrangian solutions of the planar three-body problem.

Two-phase fluids in collision of incompressible inviscid fluids effluxing from two nozzles

This paper is devoted to the mathematical theory of two-dimensional injection of incompressible, irrotational, and inviscid fluids issuing from two infinitely long nozzles into a free stream. In general, there is a free interface with constant jump of the Bernoulli constant on it, which is different and more difficult than the previous related works. Physically, it is called the collision fluid. The main result in this paper is that for given two co-axis symmetric infinitely long nozzles, imposing the incoming mass fluxes in the two nozzles, there exists a unique piecewise smooth collision fluid, such that the free interface of the collision fluid is a C1 curve, and the pressure is continuous across the interface. As byproducts, the asymptotic behaviors, the positivity of the vertical velocity, monotone relationship between the location of the interface and the incoming mass fluxes are also established.

On the topological type of a set of plane valuations with symmetries

AbstractLet be a set of irreducible plane curve singularities. For an action of a finite group G, let be the Alexander polynomial in variables of the algebraic link and let with identical variables in each group. (If , is the monodromy zeta function of the function germ , where is an equation defining the curve C1.) We prove that determines the topological type of the link L. We prove an analogous statement for plane divisorial valuations formulated in terms of the Poincaré series of a set of valuations.

Recognizing projections of algebraic curves

Given two irreducible, algebraic space curves C1 and C2, where C2 is contained in some plane Π, we provide algorithms to check whether or not there exist perspective or parallel projections mapping C1 onto C2, i.e. to recognize C2 as the projection of C1. In the affirmative case, the algorithms provide the eye point(s) of the perspective transformation(s), or the direction(s) of the parallel projection(s). Although the problem is mainly discussed for rational curves, an algorithm for implicit curves is also given. The algorithms presented are mostly symbolic; nevertheless, we include an approximate algorithm for rational curves too.

Cutting convex curves

We show that for any two convex curves C1 and C2 in Rd parametrized by [0,1] with opposite orientations, there exists a hyperplane H with the following property: For any t∈[0,1] the points C1(t) and C2(t) are never in the same open half space bounded by H. This will be deduced from a more general result on equipartitions of ordered point sets by hyperplanes.

CF-Pursuit: A Pursuit Method with a Clothoid Fitting and a Fuzzy Controller for Autonomous Vehicles

Simple and efficient geometric controllers, like Pure-Pursuit, have been widely used in various types of autonomous vehicles to solve tracking problems. In this paper, we have developed a new pursuit method, named CF-Pursuit, which has been based on Pure-Pursuit but with certain differences. In CF-Pursuit, in order to reduce fitting errors, we used a clothoid C1 curve to replace the circle employed in Pure-Pursuit. This improvement to the fitting method helps the Pursuit method to decrease tracking errors. As regards the selection of look-ahead distance, we employed a fuzzy system to directly consider the path's curvature. There are three input variables in this fuzzy system, 6 mcurvature, 9 mcurvature and 12 mcurvature, calculated from the clothoid fit with the current position and the goal position on the defined path. A Sugeno fuzzy model was adapted to output a reasonable look-ahead distance using the experiences of human drivers as well as our own tests. Compared with some other geometric controllers, CF-Pursuit performs better in robustness, cross track errors and stability. The results from field tests have proven the CF-Pursuit is a practical and efficient geometric method for the path tracking problems of autonomous vehicles.

The Darboux Trihedrons of Regular Curves on a Regular Time-Like Surface

In this work, we study Differential Geometry in the Minkowski 3-space R13 of the curves on a time-like regular surface by using parameter curves which are not perpendicular to each other. The aim of this study is to investigate the formulae between the Darboux Vectors of the time-like curve (c) , the time-like parameter curve (c1) and the space-like parameter curve (c2) which are not intersecting perpendicularly.

Proper orthogonal decomposition for parameter estimation in oscillating biological networks

Proper orthogonal decomposition (POD) is frequently applied to estimate parameters of partial differential equations. This study examines the application of the POD method in estimating the parameters of an Ordinary Differential Equation (ODE) model of stable oscillating biological networks. The mathematical model used to simulate molecular interactions in these oscillating networks is related to the Gause–Lotka–Volterra equations. The findings reveal that POD generates accurate estimates of the parameters even in the presence of experimental noise; furthermore, extrapolating biologically measured data points to a number of oscillations improves the curve fits, C1 approximations, and parameter estimations.

Non-homeomorphic Galois conjugate Beauville structures on [formula omitted

Cataneseʼs rigidity results for surfaces isogenous to a product of curves indicate that Beauville surfaces should provide a fertile source of examples of Galois conjugate varieties that are not homeomorphic, a phenomenon discovered by J.P. Serre in the sixties.In this paper, we construct Beauville surfaces S=(C1×C2)/G with group G=PSL(2,p) for p⩾7, and curves C1, C2 such that the orbit of S under the action of the absolute Galois group Gal(Q¯/Q) contains non-homeomorphic conjugate surfaces. When p=7 the orbit consists exactly of two surfaces that have non-isomorphic fundamental groups, and the curves C1, C2 have genera 8 and 49, which is shown to be the minimum for which there is a pair of non-homeomorphic Galois conjugate Beauville surfaces. As p grows the orbits contain an arbitrarily large number of non-homeomorphic surfaces.Along the way we prove a metric rigidity theorem for Beauville surfaces which provides an elementary proof of the part of Cataneseʼs theory needed to prove our results.

A Method to Obtain the Position of Space Object Based on Monocular Vision and Laser Ring

When the cylindrical laser shines on the target object, a spot can be obtained, which the edge is a closed curve, marked as C1. The imaging of the curve C1 on the image surface of CCD is a closed curve C2 too. Coordinate system is established to describe the position relationship among camera, image and light source, and to analyze the principle for monocular vision and laser ring to get the information about the object depth. In order to solve the problem and make the above principle clear, the key is to work out the expression for the curve C2 on the image surface of CCD. In order to calculate the closed curve C2 expression, the curve C2 will firstly be divided into two parts, the upper curve and the lower one. According to least-square polynomial, discrete points on the curves of two parts are drawn out, constraints are established and the curve equations are fitted. Then, to verify practicality of this method, a virtual model scene will be created, through which relevant data describing edge of virtual CCD image and that of a virtual spot when the virtual light source alights on the virtual object will be obtained. At last, closed curve equation will be fitted in accordance with data describing edge of virtual image; the position of space object will be fixed by making use of light source equation and closed curve equation; and a contrast will be made between the calculated value and data of the spot edge to prove whether a method to obtain the position of space objects based on monocular vision and laser ring is feasible.

TENSOR PRODUCT SURFACES WITH POINTWISE 1-TYPE GAUSS MAP

Tensor product immersions of a given Riemannian manifold was initiated by B.-Y. Chen. In the present article we study the tensor product surfaces of two Euclidean plane curves. We show that a tensor product surface M of a plane circle c1 centered at origin with an Euclidean planar curve c2 has harmonic Gauss map if and only if M is a part of a plane. Further, we give necessary and sufficient conditions for a tensor product surface M of a plane circle c1 centered at origin with an Euclidean planar curve c2 to have pointwise 1-type Gauss map.

Initial Experience Using a Radiofrequency Powered Transseptal Needle

The purpose of this study was to determine the safety and efficacy of using a novel radiofrequency (RF) powered transseptal needle to perform transseptal puncture (TSP). TSP was performed in 35 consecutive patients undergoing left-sided catheter ablation (mean age = 51 years; male = 71%) using a RF powered transseptal needle (NRG, Adult Large and Standard Curve C1, 71 cm, Baylis Medical Company, Inc.). Prior TSP had been performed in 34% of patients. The transseptal apparatus was positioned with the tip of the dilator engaged in the fossa ovalis. RF energy was delivered to the tip of the transseptal needle using a proprietary RF generator at 10 W for 2 seconds as gentle pressure was applied to the needle. In 5 of the 41 TSPs, the needle crossed into the left atrium before RF energy was delivered. In 35 of the remaining 36 punctures, the needle was successfully advanced into the left atrium after application of RF current. In 1 patient, the TSP with the powered needle was unsuccessful but was accomplished using a standard needle. The only complication was a transient right atrial thrombus, which occurred in 2 patients. A radiofrequency powered transseptal needle can be used to perform TSP safely and successfully without the need for significant mechanical force, even in patients who have undergone TSP previously. Additional studies are needed to determine whether a powered transseptal needle should be used routinely.

Distance Trisector of a Segment and a Point

Motivated by the work of Asano et al. [1], we consider the distance trisector problem and zone diagram considering segments in the plane as the input geometric objects. As the most basic case, we first consider the pair of curves (distance trisector curves) trisecting the distance between a point and a line, as shown in Figure 1. This is a natural extension of the bisector curve (that is a parabola) of a point and a line. In this paper, we show that these trisector curves C1 and C2 exist and are unique.

Venn Symmetry and Prime Numbers: A Seductive Proof Revisited

An n-Venn diagram is a Venn diagram on n sets, which is defined to be a collection of n simple closed curves (Jordan curves) C1,C2, . . . ,Cn in the plane such that any two intersect in finitely many points and each of the 2n sets of the form ∩C i i is nonempty and connected, where i is one of “interior” or “exterior.” Thus the Venn regions are all bounded except for the region exterior to all curves; each bounded region is the interior of a Jordan curve. See [6] for much more information on Venn diagrams. An n-Venn diagram is symmetric if each curve Ci is ρ i (C1), where ρ is a rotation of order n about some center (we use O for the fixed point of rotation ρ). We use Boolean notation for combinations of sets, with the 0-1 string e1e2 . . . en representing ∩C i i , where i is interior (respectively, exterior) if ei = 1 (respectively, 0). Thus 111 . . . 1 represents F , the full intersection of all the interiors, 000 . . . 0 is the intersection of all the exteriors (the unbounded region), and 100 . . . 0 represents the set of points interior to C1 and exterior to the others. In a symmetric Venn diagram, rotation of a region by ρ corresponds to a rightward cyclic shift of the Boolean string. The universally familiar three-circle Venn diagram is symmetric, as is the one on two sets using two circles. For about 40 years a major open question was whether symmetric n-Venn diagrams exist for all prime n. Henderson found one for n = 5 and also (unpublished) for n = 7. Much later, Hamburger [3] settled the case of 11, which was quite complicated, and then in 2004 Griggs, Killian, and Savage [1] found an approach that works for all primes. So we now have the strikingly beautiful theorem that a symmetric n-Venn diagram exists if and only if n is prime. But there is a small problem: Henderson’s proof, which appears to be very simple, has a gap. Here is the proof from [4]. Suppose 1 ≤ k ≤ n − 1. Since a symmetric n-Venn diagram is symmetric with respect to a rotation of 2π/n, the regions corresponding to the Boolean strings with k 1s must come in groups of size n, each group consisting of one such region and its images under repeated rotation by 2π/n. Therefore n divides (n k ) . This concludes the proof because the only n for which this is true for the specified k-values are the primes (an easy-to-prove fact of number theory; see [5]). This is a very seductive argument. The primeness arises in such a cute way that one wants it to be true. Thus the proof has been repeated in many papers in the decades since it was first published. Yet there are problems. The proof does not call upon the connectedness of the Venn regions. Without connectedness the result is false; see Figure 1 (due to Grunbaum [2]), which shows a diagram satisfying all of the conditions

AFFINE RULINGS OF NORMAL RATIONAL SURFACES

where Xs denotes the smooth locus of X, consider: Problem 1. Find all affine rulings of X. Problem 2. Find all pairs of curves C1, C2 on X such that X n (C1 [ C2) is isomorphic to P2 minus two lines. Problem 3. Find all curves C in X such that (Xs n C) = 1. This paper investigates Problem 1 for an arbitrary X satisfying (†). We define (Definition 1.14) the notion of a “basic” affine ruling of X and our main results describe how to construct all affine rulings of X, assuming that the basic ones are known. In the case where X is a weighted projective plane, the basic affine rulings of X are given in [6]; the present paper and [6] therefore constitute a solution to Problem 1 in that case. Problem 3 (with X = P2) has been considered by several authors ([8], [9], [18], [19], [14]). In his review of [14] (see MR 82k:14013), M. H. Gizatullin mentions some unpublished examples found by V. I. Danilov and himself, and which seem to correspond to the list of basic affine rulings of P2. The case X = P2 was finally solved in [10]. Our generalization to weighted projective planes seems to be new, as well as our method—valid for any X satisfying (†)—which reduces the general problem to the determination of the basic affine rulings. Let us briefly indicate how problems 1–3 are related to each other. Consider the stronger condition (‡) on a surface X:

Some Remarks on the Fučı́k Spectrum of the p-Laplacian and Critical Groups

Let be a bounded domain in n n ≥ 1. For 1 < p < ∞, the Fucik spectrum of the p-Laplacian on W 1 p 0 is defined as the set p of those points a b ∈ 2 such that − pu = a u+ p−1 − b u− p−1 u ∈ W 1 p 0 (1.1) has a nontrivial solution. Here pu = div ( ∇u p−2∇u) and u± = max ±u 0 . It is known that the first eigenvalue λ1 of − p on W 1 p 0 is positive, simple, and admits a positive eigenfunction φ1 ∈ W 1 p 0 ∩C1 (see Lindqvist [8]), so p contains the two lines λ1 × and × λ1. A first nontrivial curve C2 in p through λ2 λ2 , where λ2 is the second eigenvalue of − p, was recently constructed and variationally characterized by a