Complete systems of surfaces in 3-manifolds

1997 ◽  
Vol 122 (1) ◽  
pp. 185-191 ◽  
Author(s):  
FENGCHUN LEI
Keyword(s):  
Finite Number ◽  
Boundary Component ◽  
Pairwise Disjoint ◽  
Complete System ◽  
Closed Surface ◽  
The Other ◽  
Closed Curves ◽  
Orientable Surfaces

A complete system (CS) [Jscr ]={J1, ..., Jn} on a connected closed surface F is a collection of pairwise disjoint simple closed curves on F such that the surface obtained by cutting F open along [Jscr ] is a 2-sphere with 2n-holes. Two CSs on F are equivalent if each can be obtained from the other via finite number of slides (defined in Section 1) and isotopies. Let M be a 3-manifold and F a boundary component of M of genus n. A CS of surfaces for M is a CS on F which bounds n pairwise disjoint incompressible orientable surfaces in M. When [Jscr ] is a CS of discs on the boundary of a handlebody V, it is well known that any CS on F which is equivalent to [Jscr ] is also a CS of discs for V. Our first result says that the same thing happens for a CS of surfaces for M, that is, if [Jscr ] is a CS of surfaces for M, then any CS equivalent to [Jscr ] is also a CS of surfaces for M. The following theorem is our main result on CS of surfaces in 3-manifolds:

Degree and holomorphic extendibility

2013 ◽  
Vol 143 (1) ◽  
pp. 129-139
Author(s):  
Darja Govekar Leban
Keyword(s):  
Continuous Function ◽  
Finite Number ◽  
Bounded Domain ◽  
Pairwise Disjoint ◽  
Continuous Functions ◽  
Closed Curves ◽  
Preceding Theorem ◽  
Holomorphic Extendibility ◽  
Class Of Functions

Recently it was shown that if D is a bounded domain in ℂ whose boundary consists of a finite number of pairwise disjoint simple closed curves, then a continuous function f on bD extends holomorphically through D if and only if, for each g ∈ A(D) such that f + g has no zero on bD, the degree of f + g is non-negative (which, for these special domains, is equivalent to the fact that the change of argument of f + g along bD is non-negative). Here A(D) is the algebra of all continuous functions on D which are holomorphic on D. This fails to hold for general domains, and generalizing to more general domains presents a major problem that often requires a much larger class of functions g. It is shown that the preceding theorem still holds in the case when D is a bounded domain in ℂ such that D is finitely connected and such that D is equal to the interior of D.

scholarly journals Topology of optimal flows with collective dynamics on closed orientable surfaces

2020 ◽  
Vol 13 (2) ◽  
pp. 50-67
Author(s):  
Alexandr Olegovich Prishlyak ◽  
Mariya Viktorovna Loseva
Keyword(s):  
Morse Function ◽  
Hamiltonian Dynamics ◽  
Closed Surface ◽  
Topological Invariant ◽  
The Other ◽  
Heteroclinic Cycles ◽  
Collective Dynamics ◽  
Gradient Dynamics ◽  
Orientable Surfaces

We consider flows on a closed surface with one or more heteroclinic cycles that divide the surface into two regions. One of the region has gradient dynamics, like Morse fields. The other region has Hamiltonian dynamics generated by the field of the skew gradient of the simple Morse function. We construct the complete topological invariant of the flow using the Reeb and Oshemkov-Shark graphs and study its properties. We describe all possible structures of optimal flows with collective dynamics on oriented surfaces of genus no more than 2, both for flows containing a center and for flows without it.

On the Dynamic Isotropy of Mechanisms With Two Degrees of Freedom

2005 ◽  
Author(s):  
Raffaele Di Gregorio ◽  
Alessandro Cammarata ◽  
Rosario Sinatra
Keyword(s):  
Finite Number ◽  
Degrees Of Freedom ◽  
Point Of View ◽  
Closed Curves ◽  
Special Cases ◽  
Dynamic Performances ◽  
Definition Of ◽  
Geometric Locus

The comparison of mechanisms with different topology or with different geometry, but with the same topology, is a necessary operation during the design of a machine sized for a given task. Therefore, tools that evaluate the dynamic performances of a mechanism are welcomed. This paper deals with the dynamic isotropy of 2-dof mechanisms starting from the definition introduced in a previous paper. In particular, starting from the condition that identifies the dynamically isotropic configurations, it shows that, provided some special cases are not considered, 2-dof mechanisms have at most a finite number of isotropic configurations. Moreover, it shows that, provided the dynamically isotropic configurations are excluded, the geometric locus of the configuration space that collects the points associated to configurations with the same dynamic isotropy is constituted by closed curves. This results will allow the classification of 2-dof mechanisms from the dynamic-isotropy point of view, and the definition of some methodologies for the characterization of the dynamic isotropy of these mechanisms. Finally, examples of applications of the obtained results will be given.

Minimal Size of Counters for (Real-Time) Multicounter Automata

2021 ◽  
Vol 181 (2-3) ◽  
pp. 99-127
Author(s):  
Viliam Geffert ◽  
Zuzana Bednárová
Keyword(s):  
Finite Number ◽  
Real Time ◽  
The Other ◽  
Binary Representation ◽  
Minimal Size ◽  
Nondeterministic Automaton ◽  
Other Hand ◽  
Weak Space ◽  
Fixed Constant

We show that, for automata using a finite number of counters, the minimal space that is required for accepting a nonregular language is (log n)ɛ. This is required for weak space bounds on the size of their counters, for real-time and one-way, and for nondeterministic and alternating versions of these automata. The same holds for two-way automata, independent of whether they work with strong or weak space bounds, and of whether they are deterministic, nondeterministic, or alternating. (Here ɛ denotes an arbitrarily small—but fixed—constant; the “space” refers to the values stored in the counters, rather than to the lengths of their binary representation.) On the other hand, we show that the minimal space required for accepting a nonregular language is nɛ for multicounter automata with strong space bounds, both for real-time and one-way versions, independent of whether they are deterministic, nondeterministic, or alternating, and also for real-time and one-way deterministic multicounter automata with weak space bounds. All these bounds are optimal both for unary and general nonregular languages. However, for automata equipped with only one counter, it was known that one-way nondeterministic automata cannot recognize any unary nonregular languages at all, even if the size of the counter is not restricted, while, with weak space bound log n, we present a real-time nondeterministic automaton recognizing a binary nonregular language here.

Appendix, containing the Investigation of a Formula for the Rectification of any Arch of an Ellipse

1805 ◽  
Vol 5 (2) ◽  
pp. 271-293
Keyword(s):  
Finite Number ◽  
The Other ◽  
Algebraic Equations ◽  
Conic Sections ◽  
Straight Line ◽  
Curve Line ◽  
Straight Lines

It is now generally understood, that by the rectification of a curve line, is meant, not only the method of finding a straight line exactly equal to it, but also the method of expressing it by certain functions of the other lines, whether straight lines or circles, by which the nature of the curve is defined. It is evidently in the latter sense that we must understand the term rectification, when applied to the arches of conic sections, seeing that it has hitherto been found impossible, either to exhibit straight lines equal to them, or to express their relation to their co-ordinates, by algebraic equations, consisting of a finite number of terms.

scholarly journals A Stochastic Differential Equation Driven by Poisson Random Measure and Its Application in a Duopoly Market

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Tong Wang ◽  
Hao Liang
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Finite Number ◽  
Random Measure ◽  
Approximate Solutions ◽  
The Other ◽  
Poisson Random Measure ◽  
Verification Theorem ◽  
Hamilton Jacobi Bellman ◽  
Duopoly Market

We investigate a stochastic differential equation driven by Poisson random measure and its application in a duopoly market for a finite number of consumers with two unknown preferences. The scopes of pricing for two monopolistic vendors are illustrated when the prices of items are determined by the number of buyers in the market. The quantity of buyers is proved to obey a stochastic differential equation (SDE) driven by Poisson random measure which exists a unique solution. We derive the Hamilton-Jacobi-Bellman (HJB) about vendors’ profits and provide a verification theorem about the problem. When all consumers believe a vendor’s guidance about their preferences, the conditions that the other vendor’s profit is zero are obtained. We give an example of this problem and acquire approximate solutions about the profits of the two vendors.

Connecting Tort and Crime: Comparative Legal History in England and Spain since 1850

2009 ◽  
Vol 11 ◽  
pp. 247-288
Author(s):  
Matthew Dyson
Keyword(s):  
Criminal Law ◽  
Legal History ◽  
Complete System ◽  
Civil Law ◽  
Tort Law ◽  
The Other ◽  
Criminal Courts ◽  
English System ◽  
History Of ◽  
The Relationship

Abstract This chapter explores the relationship between tort law and criminal law. In particular it tracks one line of developments in the procedural co-ordination of criminal and civil law: the ability of criminal courts to award compensation for harm. It is a study of legal change or development: how and why law has evolved from the middle of the nineteenth century through to the present day. The chapter is also comparative, looking at the English and Spanish legal systems. The history of powers to compensate has highlighted two fundamentally different ways to resolve claims based on a concurrently tortious and criminal wrong. The English system has slowly moved from disparate and piecemeal provisions to a general if under-theorised system. On the other hand, Spain created a novel and complete system of liability to be administered by the criminal courts. This chapter seeks to trace and explain this development with a view to understanding how much civil and criminal law can perform the same function: compensation.

scholarly journals P-harmonic dimensions on ends

1993 ◽  
Vol 132 ◽  
pp. 131-139
Author(s):  
Michihiko Kawamura ◽  
Shigeo Segawa
Keyword(s):  
Riemann Surface ◽  
Boundary Component ◽  
Harmonic Functions ◽  
Ideal Boundary ◽  
Boundary Values ◽  
Hölder Continuous ◽  
Closed Curves ◽  
Open Riemann Surface ◽  
Relative Boundary ◽  
Bounded Harmonic Functions

Consider an end Ω in the sense of Heins (cf. Heins [3]): Ω is a relatively non-compact subregion of an open Riemann surface such that the relative boundary ∂Ω consists of finitely many analytic Jordan closed curves, there exist no non-constant bounded harmonic functions with vanishing boundary values on ∂Ω and Ω has a single ideal boundary component. A density P = P(z)dxdy (z = x + iy) is a 2-form on Ω∩∂Ω with nonnegative locally Holder continuous coefficient P(z).

scholarly journals The effect of vertical crustal fractures on the rifting process

1989 ◽  
Vol 20 (2) ◽  
pp. 175
Author(s):  
V. Anfiloff
Keyword(s):  
Finite Number ◽  
Continental Crust ◽  
Gravity Data ◽  
The Other ◽  
Fracture System ◽  
The Past ◽  
Fundamental Process ◽  
Basic Volcanics ◽  
Transfer Faults

In the past, rifts have mainly been identified in terms of sediment troughs. They account for many of the elongate gravity lows distributed in a coherent rectilinear manner over the continent. Other gravity lows can be attributed to granites intruding rift compartments, and some gravity highs can be attributed to basic volcanics in compartments. The total number of rifts which can be thus inferred from gravity and magnetics is very large, and suggests rifting is pervasive over the whole continent and controlled by a systematically distributed "Cardinal" system of ancient vertical crustal fractures.The extensional concept of rifting is based on a finite number of rifts, all of which have "failed" to split the continent. When a far greater number of rifts is recognised, it becomes difficult to accept that all these rifts have "failed" to reach full opening by extensional processes. In view of the known horizontal compressive forces acting in the crust, it is more probable that rifting is caused by compression. The compartmentalization of rifts, clearly observed in gravity data, also implies compression.Closely spaced rectilinear dyke systems in shield areas may also represent the pervasive "Cardinal" fracture system. In general, this system of orthogonal fractures poses problems for the detatchment rifting concept which assumes that transfer faults are formed at the time a rift forms, whereas they in all probability predate the rift, and owe their existence to a fundamental process operating when continental crust first formed.Two types of compressive rift models are discussed. One is associated with shear couples between widely spaced parallel fractures. The other is based on the concept of a crust cut by closely spaced fractures in which compression is propagated along a network of linked blocks. In both cases the development of basement ridges is a key issue.

The No-Three-In-Line Problem

1968 ◽  
Vol 11 (4) ◽  
pp. 527-531 ◽  
Author(s):  
Richard K. Guy ◽  
Patrick A. Kelly
Keyword(s):  
Finite Number ◽  
The Other ◽  
Probabilistic Argument ◽  
Number Of Solutions ◽  
Other Hand ◽  
Image Position

Let Sn be the set of n2 points with integer coordinates n (x, y), 1 ≤ x, y <n. Let fn be the maximum cardinal of a subset T of Sn such that no three points of T are collinear. Clearly fn < 2n.For 2 ≤ n ≤ 10 it is known ([2], [3] for n = 8, [ 1] for n = 10, also [4], [6]) that fn = 2n, and that this bound is attained in 1, 1, 4, 5, 11, 22, 57, 51 and 156 distinct configurations for these nine values of n. On the other hand, P. Erdös [7] has pointed out that if n is prime, fn ≥ n, since the n points (x, x2) reduced modulo n have no three collinear. We give a probabilistic argument to support the conjecture that there is only a finite number of solutions to the no-three-in-line problem. More specifically, we conjecture that

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