Extremal Values of the Basic Invariants of Plane Curves

2000 ◽  
Vol 09 (08) ◽  
pp. 1085-1126
Author(s):  
Jianming Yu ◽  
Jianyi Zhou ◽  
Jianzhong Pan
Keyword(s):  
Fixed Number ◽  
Plane Curves ◽  
Explicit Formulas ◽  
Double Points ◽  
Fine Structures ◽  
Extremal Values ◽  
Extremal Value

In [A2] V.I. Arnold introduced three basic invariants St, J+ and J- of plane curves and proposed some interesting conjectures concerning the extremal value of these invariants on a given set of curves. Partial answers have been obtained by O. Viro and A. N. Shumakovich. We give explicit formulas for these extremal values of sets of plane curves with fixed number of double points and of Whitney index and we determine on which curves these extremal values are attained (Theorems 3-6). Our arguments are based on understanding of the fine structures of generic curves and some surgery operations on curves.

2. Applications of the Theorem that Two Closed Plane Curves intersect an even number of times

1878 ◽  
Vol 9 ◽  
pp. 237-246 ◽  
Author(s):  
Tait
Keyword(s):  
Special Class ◽  
Double Point ◽  
Closed Curve ◽  
British Association ◽  
Plane Curves ◽  
Double Points ◽  
The Wire ◽  
Closed Plane

The theorem itself may be considered obvious, and is easily applied, as I showed at the late meeting of the British Association, to prove that in passing from any one double point of a plane closed curve continuously along the curve to the same point again, an even number of intersections must be passed through. Hence, if we suppose the curve to be constructed of cord or wire, and restrict the crossings to double points, we may arrange them throughout so that, in following the wire continuously, it goes alternately over and under each branch it meets. When this is done it is obviously as completely knotted as its scheme (defined below) will admit of, and except in a special class of cases cannot have the number of crossings reduced by any possible deformation.

Parity conditions for realizability of Gauss diagrams

2019 ◽  
Vol 28 (01) ◽  
pp. 1950015
Author(s):  
Oleg N. Biryukov
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Plane Curves ◽  
Double Points ◽  
Gauss Diagram ◽  
Necessary And Sufficient ◽  
Closed Plane

We consider a problem of realizability of Gauss diagrams by closed plane curves where the plane curves have only double points of transversal self-intersection. We formulate the necessary and sufficient conditions for realizability. These conditions are based only on the parity of double and triple intersections of the chords in the Gauss diagram.

Unfolding plane curves with cusps and nodes

2015 ◽  
Vol 145 (1) ◽  
pp. 161-174
Author(s):  
Juan J. Nuño Ballesteros
Keyword(s):  
Finite Codimension ◽  
Plane Curves ◽  
Singular Set ◽  
One Dimensional ◽  
Double Points ◽  
Euler Obstruction ◽  
Type Singularity ◽  
Transverse Slice

Given an irreducible surface germ (X, 0) ⊂ (ℂ3, 0) with a one-dimensional singular set Σ, we denote by δ1 (X, 0) the delta invariant of a transverse slice. We show that δ1 (X, 0) ≥ m0 (Σ, 0), with equality if and only if (X, 0) admits a corank 1 parametrization f :(ℂ2, 0) → (ℂ3, 0) whose only singularities outside the origin are transverse double points and semi-cubic cuspidal edges. We then use the local Euler obstruction Eu(X, 0) in order to characterize those surfaces that have finite codimension with respect to -equivalence or as a frontal-type singularity.

Design of systems with extremal dynamic properties

2013 ◽  
Vol 61 (3) ◽  
pp. 563-567 ◽  
Author(s):  
H. Górecki ◽  
M. Zaczyk
Keyword(s):  
Characteristic Equation ◽  
Chemical Industry ◽  
Dynamic Properties ◽  
Dynamic Error ◽  
Analytical Formulae ◽  
Extremal Values ◽  
Extremal Value ◽  
Zeros And Poles

Abstract In this article the problem of determination of coefficients a1, a2, . . . , an of the characteristic equation which yield required extremal values of the solution x(t) at extremal values τ of time is solved. The extremal values of x(t) and τ are treated as functions of the roots s1, s2, . . . , sn. The analytical formulae enable to design the systems with prescribed dynamic properties. The zeros and poles can be located using the known method. The extremal dynamic error x(t) is the most important property of the behaviour of the system. This extremal value of the dynamic error has fundamental role in the chemical industry where for example overrising temperature or pressure can lead to an explosion. A second very important property is the extremal time τ connected with the extremal value of the error. This property is essential in the electroenergetic system, which can be destroyed by the overvoltages waves.

Plane Curves with Consecutive Double Points

1914 ◽  
Vol 16 (1/4) ◽  
pp. 15 ◽  
Author(s):  
F. R. Sharpe ◽  
C. F. Craig
Keyword(s):  
Plane Curves ◽  
Double Points

scholarly journals Relative Node Polynomials for Plane Curves

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Florian Block
Keyword(s):  
Recent Work ◽  
Plane Curves ◽  
Explicit Formulas ◽  
Severi Varieties ◽  
Severi Variety ◽  
Nous Montrons ◽  
International Audience

International audience We generalize the recent work of Fomin and Mikhalkin on polynomial formulas for Severi degrees. The degree of the Severi variety of plane curves of degree d and δ nodes is given by a polynomial in d, provided δ is fixed and d is large enough. We extend this result to generalized Severi varieties parametrizing plane curves which, in addition, satisfy tangency conditions of given orders with respect to a given line. We show that the degrees of these varieties, appropriately rescaled, are given by a combinatorially defined ``relative node polynomial'' in the tangency orders, provided the latter are large enough. We describe a method to compute these polynomials for arbitrary δ , and use it to present explicit formulas for δ ≤ 6. We also give a threshold for polynomiality, and compute the first few leading terms for any δ . Nous généralisons les travaux récents de Fomin et Mikhalkin sur des formules polynomiales pour les degrés de Severi. Le degré de la variété de Severi des courbes planes de degré d et à δ nœuds est donné par un polynôme en d , pour δ fixé et d assez grand. Nous étendons ce résultat aux variétés de Severi généralisées paramétrant les courbes planes et qui, en outre, satisfont à des conditions de tangence d'ordres donnés avec une droite fixée. Nous montrons que les degrés de ces variétés, rééchelonnés de manière appropriée, sont donnés par un ``polynôme de noeud relatif'', défini combinatoirement, en les ordres de tangence, dès que ceux-ci sont assez grands. Nous décrivons une méthode pour calculer ces polynômes pour delta arbitraire, et l'utilisons pour présenter des formules explicites pour δ ≤ 6 . Nous donnons aussi un seuil pour la polynomialité, et calculons les premiers termes dominants pour tout δ .

scholarly journals Design of the oscillatory systems with the extremal dynamic properties

2014 ◽  
Vol 62 (2) ◽  
pp. 241-253 ◽  
Author(s):  
H. Górecki ◽  
M. Zaczyk
Keyword(s):  
Characteristic Equation ◽  
Practical Problem ◽  
Dynamic Properties ◽  
Oscillatory Systems ◽  
Complex Roots ◽  
Analytical Formulae ◽  
Extremal Values ◽  
Extremal Value

Abstract In this article the problem of determination of such coefficients a1, a2, ..., an and eigenvalues s1, s2, ..., sn of the characteristic equation which yield required extremal values of the solution x(t) at the extremal value τ of time is solved. The extremal values of x(τ ) and τ are treated as functions of the roots s1, s2, ..., sn. The analytical formulae enable us to design the systems with prescribed dynamic properties. For solution of the problem the properties of symmetrical equations are used. The method is illustrated by an example of the equation of 4-th degree. The regions of the different kinds of the roots: real, with one pair of complex and two pairs of complex roots are illustrated. A practical problem is shown.

Ultrastructure of NPC/HK1 cells heterotransplanted in athymic nude mice

1982 ◽  
Vol 40 ◽  
pp. 236-237
Author(s):  
E.C. Chew ◽  
C.L. Li ◽  
D.P. Huang ◽  
H.C. Ho ◽  
L.S. Mak ◽  
...  
Keyword(s):  
Cell Line ◽  
Nude Mice ◽  
Biopsy Specimen ◽  
Epithelial Cell Line ◽  
Present Communication ◽  
Recurrent Tumour ◽  
Athymic Nude Mice ◽  
Cell Line Establishment ◽  
Fine Structures ◽  
Well Differentiated

An epithelial cell line, NPC/HK1, has recently been established from a biopsy specimen of a recurrent tumour of the nasopharynx which was histologically diagnosed as a moderately to well differentiated squamous cell carcinoma. A definite decrease in the amount of tonofilaments and desmosomes in the NPC/HK1 cells during the cell line establishment was observed. The present communication reports on the fine structures of the NPC/HK1 cells heterotraneplanted in athymic nude mice.

Sign in / Sign up

Export Citation Format

Share Document