It is shown that the Pinney equation, Ermakov systems, and their higher-order generalizations describeself-similar solutions of plane curve motions in centro-affine and affine geometries.
Group-invariant solutions to certain plane curve motions in Euclidean and centro-affine geometries areobtained. The behavior of some solutions is also presented.
We introduce a new algebraic approach dealing with the problem of computing the topology of an arrangement of a finite set of real algebraic plane curves presented implicitly. The main achievement of the presented met...We introduce a new algebraic approach dealing with the problem of computing the topology of an arrangement of a finite set of real algebraic plane curves presented implicitly. The main achievement of the presented method is a complete avoidance of irrational numbers that appear when using the sweeping method in the classical way for solving the problem at hand. Therefore, it is worth mentioning that the efficiency of the proposed method is only assured for low-degree curves.展开更多
For the plane curves Γ,the maximal operator associated to it is defined by Mf(x)=sup|∫f(x-Γ(t))(r^(-1)t)r^(-1)dt| where is a Schwartz function.For a certain class of curves in R^2,M is shown to bounded on (H(R^2)...For the plane curves Γ,the maximal operator associated to it is defined by Mf(x)=sup|∫f(x-Γ(t))(r^(-1)t)r^(-1)dt| where is a Schwartz function.For a certain class of curves in R^2,M is shown to bounded on (H(R^2),Weak L^1(R^2).This extends the theorem of Stein & Wainger and the theo- rem of Weinberg.展开更多
We define the notion of evolutes of curves in a hyperbolic plane and establish the relation-ships between singularities of these subjects and geometric invariants of curves under the action of theLorentz group.We also...We define the notion of evolutes of curves in a hyperbolic plane and establish the relation-ships between singularities of these subjects and geometric invariants of curves under the action of theLorentz group.We also describe how we can draw the picture of an evolute of a hyperbolic plane curvein the Poincaré disk.展开更多
The author gives another linear-algebraic proof of the famous result of Zariski, Delorme,Briancon- Granger- Maisonobe about the moduli number of plane curve singularities with the same topological type as Xa + Yb = 0 ...The author gives another linear-algebraic proof of the famous result of Zariski, Delorme,Briancon- Granger- Maisonobe about the moduli number of plane curve singularities with the same topological type as Xa + Yb = 0 (i.e.,with one characteristic pair). Since the original proof depends very much on the division theorem of Briancon, it cannot be generalized to higher dimensions. It is hopeful that the proof here will be applied to the higher dimensional cases.展开更多
We describe the evolution of certain multiplicities and intersection numbers of plane curve singularities under the iterated action of an analytic morphism.
Verifiable secret sharing is a special kind of secret sharing. In this paper, A secure and efficient threshold secret sharing scheme is proposed by using the plane parametric curve on the basis of the principle of sec...Verifiable secret sharing is a special kind of secret sharing. In this paper, A secure and efficient threshold secret sharing scheme is proposed by using the plane parametric curve on the basis of the principle of secret sharing. And the performance of this threshold scheme is analyzed. The results reveal that the threshold scheme has its own advantage of one-parameter representation for a master key, and it is a perfect ideal secret sharing scheme. It can easily detect cheaters by single operation in the participants so that the probability of valid cheating is less than 1/<em>p</em> (where <em>p</em> is a large prime).展开更多
For an almost complex structure J on U R4 pseudo-holomorphically fibered over C a J-holomorphic curve C U can be described by a Weierstrass polynomial. The J-holomorphicity equation descends to a perturbed 8-operator ...For an almost complex structure J on U R4 pseudo-holomorphically fibered over C a J-holomorphic curve C U can be described by a Weierstrass polynomial. The J-holomorphicity equation descends to a perturbed 8-operator on the coefficients; the operator is typically (0, 2/m)-Holder continuous if m is the local degree of C over C. This sheds some light on the problem of parametrizing pseudo-holomorphic deformations of J-holomorphic curve singu-larities.展开更多
A new simple Lagrangian method with favorable stability and efficiencyproperties for computing general plane curve evolutions is presented. The methodis based on the flowing finite volume discretization of the intrins...A new simple Lagrangian method with favorable stability and efficiencyproperties for computing general plane curve evolutions is presented. The methodis based on the flowing finite volume discretization of the intrinsic partial differentialequation for updating the position vector of evolving family of plane curves. A curvecan be evolved in the normal direction by a combination of fourth order terms relatedto the intrinsic Laplacian of the curvature, second order terms related to the curva-ture, first order terms related to anisotropy and by a given external velocity field. Theevolution is numerically stabilized by an asymptotically uniform tangential redistri-bution of grid points yielding the first order intrinsic advective terms in the governingsystem of equations. By using a semi-implicit in time discretization it can be numer-ically approximated by a solution to linear penta-diagonal systems of equations (inpresence of the fourth order terms) or tri-diagonal systems (in the case of the secondorder terms). Various numerical experiments of plane curve evolutions, including, inparticular, nonlinear, anisotropic and regularized backward curvature flows, surfacediffusion and Willmore flows, are presented and discussed.展开更多
A Riemann surface S having field of moduli M,but not a field of definition,is called pseudo-real.This means that S has anticonformal automorphisms,but none of them is an involution.A Riemann surface is said to be plan...A Riemann surface S having field of moduli M,but not a field of definition,is called pseudo-real.This means that S has anticonformal automorphisms,but none of them is an involution.A Riemann surface is said to be plane if it can be described by a smooth plane model of some degree d≥4 in P^2/C.We characterize pseudo-real-plane Riemann surfaces»S,whose conformal automorphism group Aut+(S)is PGL3(C)-conjugate to a finite non-trivial group that leaves invariant infinitely many points of P^2/C.In particular,we show that such pseudo-real-plane Riemann surfaces exist only if Aut+(S)is cyclic of even order n dividing the degree d.Explicit families of pseudo-reai-plane Riemann surfaces are given for any degree d=2pm with m>1 odd,p prime and n=d/p.展开更多
Compared with conventional planar optical image sensors,a curved focal plane array can simplify the lens design and improve the field of view.In this paper,we introduce the design and implementation of a segmented,hem...Compared with conventional planar optical image sensors,a curved focal plane array can simplify the lens design and improve the field of view.In this paper,we introduce the design and implementation of a segmented,hemispherical,CMOS-compatible silicon image plane for a 10-mm diameter spherical monocentric lens.To conform to the hemispherical focal plane of the lens,we use flexible gores that consist of arrays of spring-connected silicon hexagons.Mechanical functionality is demonstrated by assembling the 20-μm-thick silicon gores into a hemispherical test fixture.We have also fabricated and tested a photodiode array on a siliconon-insulator substrate for use with the curved imager.Optical testing shows that the fabricated photodiodes achieve good performance;the hemispherical imager enables a compact 160°field of view camera with >80% fill factor using a single spherical lens.展开更多
文摘It is shown that the Pinney equation, Ermakov systems, and their higher-order generalizations describeself-similar solutions of plane curve motions in centro-affine and affine geometries.
文摘Group-invariant solutions to certain plane curve motions in Euclidean and centro-affine geometries areobtained. The behavior of some solutions is also presented.
基金Project (No. MTM2005-08690-C02-02) partially supported by the Spanish Ministry of Science and Innovation Grant
文摘We introduce a new algebraic approach dealing with the problem of computing the topology of an arrangement of a finite set of real algebraic plane curves presented implicitly. The main achievement of the presented method is a complete avoidance of irrational numbers that appear when using the sweeping method in the classical way for solving the problem at hand. Therefore, it is worth mentioning that the efficiency of the proposed method is only assured for low-degree curves.
文摘For the plane curves Γ,the maximal operator associated to it is defined by Mf(x)=sup|∫f(x-Γ(t))(r^(-1)t)r^(-1)dt| where is a Schwartz function.For a certain class of curves in R^2,M is shown to bounded on (H(R^2),Weak L^1(R^2).This extends the theorem of Stein & Wainger and the theo- rem of Weinberg.
文摘We define the notion of evolutes of curves in a hyperbolic plane and establish the relation-ships between singularities of these subjects and geometric invariants of curves under the action of theLorentz group.We also describe how we can draw the picture of an evolute of a hyperbolic plane curvein the Poincaré disk.
文摘The author gives another linear-algebraic proof of the famous result of Zariski, Delorme,Briancon- Granger- Maisonobe about the moduli number of plane curve singularities with the same topological type as Xa + Yb = 0 (i.e.,with one characteristic pair). Since the original proof depends very much on the division theorem of Briancon, it cannot be generalized to higher dimensions. It is hopeful that the proof here will be applied to the higher dimensional cases.
文摘We describe the evolution of certain multiplicities and intersection numbers of plane curve singularities under the iterated action of an analytic morphism.
文摘Verifiable secret sharing is a special kind of secret sharing. In this paper, A secure and efficient threshold secret sharing scheme is proposed by using the plane parametric curve on the basis of the principle of secret sharing. And the performance of this threshold scheme is analyzed. The results reveal that the threshold scheme has its own advantage of one-parameter representation for a master key, and it is a perfect ideal secret sharing scheme. It can easily detect cheaters by single operation in the participants so that the probability of valid cheating is less than 1/<em>p</em> (where <em>p</em> is a large prime).
文摘For an almost complex structure J on U R4 pseudo-holomorphically fibered over C a J-holomorphic curve C U can be described by a Weierstrass polynomial. The J-holomorphicity equation descends to a perturbed 8-operator on the coefficients; the operator is typically (0, 2/m)-Holder continuous if m is the local degree of C over C. This sheds some light on the problem of parametrizing pseudo-holomorphic deformations of J-holomorphic curve singu-larities.
基金This work was supported by grants:VEGA 1/0269/09,APVV-0351-07,APVV-RPEU-0004-07(K.Mikula and M.Balazovjech)and APVV-0247-06(D.Sevcovic).
文摘A new simple Lagrangian method with favorable stability and efficiencyproperties for computing general plane curve evolutions is presented. The methodis based on the flowing finite volume discretization of the intrinsic partial differentialequation for updating the position vector of evolving family of plane curves. A curvecan be evolved in the normal direction by a combination of fourth order terms relatedto the intrinsic Laplacian of the curvature, second order terms related to the curva-ture, first order terms related to anisotropy and by a given external velocity field. Theevolution is numerically stabilized by an asymptotically uniform tangential redistri-bution of grid points yielding the first order intrinsic advective terms in the governingsystem of equations. By using a semi-implicit in time discretization it can be numer-ically approximated by a solution to linear penta-diagonal systems of equations (inpresence of the fourth order terms) or tri-diagonal systems (in the case of the secondorder terms). Various numerical experiments of plane curve evolutions, including, inparticular, nonlinear, anisotropic and regularized backward curvature flows, surfacediffusion and Willmore flows, are presented and discussed.
文摘A Riemann surface S having field of moduli M,but not a field of definition,is called pseudo-real.This means that S has anticonformal automorphisms,but none of them is an involution.A Riemann surface is said to be plane if it can be described by a smooth plane model of some degree d≥4 in P^2/C.We characterize pseudo-real-plane Riemann surfaces»S,whose conformal automorphism group Aut+(S)is PGL3(C)-conjugate to a finite non-trivial group that leaves invariant infinitely many points of P^2/C.In particular,we show that such pseudo-real-plane Riemann surfaces exist only if Aut+(S)is cyclic of even order n dividing the degree d.Explicit families of pseudo-reai-plane Riemann surfaces are given for any degree d=2pm with m>1 odd,p prime and n=d/p.
文摘Compared with conventional planar optical image sensors,a curved focal plane array can simplify the lens design and improve the field of view.In this paper,we introduce the design and implementation of a segmented,hemispherical,CMOS-compatible silicon image plane for a 10-mm diameter spherical monocentric lens.To conform to the hemispherical focal plane of the lens,we use flexible gores that consist of arrays of spring-connected silicon hexagons.Mechanical functionality is demonstrated by assembling the 20-μm-thick silicon gores into a hemispherical test fixture.We have also fabricated and tested a photodiode array on a siliconon-insulator substrate for use with the curved imager.Optical testing shows that the fabricated photodiodes achieve good performance;the hemispherical imager enables a compact 160°field of view camera with >80% fill factor using a single spherical lens.
