A formula for the number of branches for one-dimensional semianalytic sets

1992 ◽  
Vol 112 (3) ◽  
pp. 527-534 ◽  
Author(s):  
Zbigniew Szafraniec
Keyword(s):  
Singular Point ◽  
Disjoint Union ◽  
Connected Components ◽  
One Dimensional ◽  
Finite Disjoint Union ◽  
Analytic Curves ◽  
Number Of Branches ◽  
Analytic Mappings

Let F = (F1, …, Fn-1): (ℝn, 0)→(ℝn-1, 0) and G:(ℝn, 0)→(ℝ, 0) be germs of analytic mappings, and let X = F-1(0). Assume that 0 ∈ ℝn is an isolated singular point in X, i.e. 0 ∈ ℝn is isolated in {x ∈ X|rank[DF(x)] < n-1}. Hence a germ of X/{0} at the origin is either void or a finite disjoint union of analytic curves. Let b denote the number of branches, i.e. connected components, of X/{0} and let b+ (resp. b-, b0) denote the number of branches of X/{0} on which G is positive (resp. G is negative, G vanishes). The problem is to calculate the numbers b, b+, b-, b0 in terms of F and G.

scholarly journals Remarks on Stationary and Uniformly-rotating Vortex Sheets: Rigidity Results

2021 ◽  
Author(s):  
Javier Gómez-Serrano ◽  
Jaemin Park ◽  
Jia Shi ◽  
Yao Yao
Keyword(s):  
Angular Velocity ◽  
Disjoint Union ◽  
Vortex Sheet ◽  
Vortex Sheets ◽  
Smooth Curves ◽  
Rigidity Results ◽  
Positive Vorticity ◽  
Constant Vorticity ◽  
Concentric Circles ◽  
Finite Disjoint Union

AbstractIn this paper, we show that the only solution of the vortex sheet equation, either stationary or uniformly rotating with negative angular velocity $$\Omega $$ Ω , such that it has positive vorticity and is concentrated in a finite disjoint union of smooth curves with finite length is the trivial one: constant vorticity amplitude supported on a union of nested, concentric circles. The proof follows a desingularization argument and a calculus of variations flavor.

scholarly journals A REMARK ON ONE-DIMENSIONAL MANY-BODY PROBLEMS WITH POINT INTERACTIONS

2000 ◽  
Vol 14 (07) ◽  
pp. 721-727 ◽  
Author(s):  
SERGIO ALBEVERIO ◽  
LUDWIK DABROWSKI ◽  
SHAO-MING FEI
Keyword(s):  
Singular Point ◽  
Bound States ◽  
Quantum Mechanical ◽  
Contact Interactions ◽  
Many Body ◽  
Point Interactions ◽  
One Dimensional ◽  
Integrable Quantum ◽  
Scattering Matrices ◽  
Many Body Systems

The integrability of one-dimensional quantum mechanical many-body problems with general contact interactions is extensively studied. It is shown that besides the pure (repulsive or attractive) δ-function interaction there is another singular point interactions which gives rise to a new one-parameter family of integrable quantum mechanical many-body systems. The bound states and scattering matrices are calculated for both bosonic and fermionic statistics.

scholarly journals Global Components of Positive Bounded Variation Solutions of a One-Dimensional Indefinite Quasilinear Neumann Problem

2019 ◽  
Vol 19 (3) ◽  
pp. 437-473 ◽  
Author(s):  
Julian López-Gómez ◽  
Pierpaolo Omari
Keyword(s):  
Neumann Problem ◽  
Positive Solutions ◽  
Bounded Variation ◽  
Connected Components ◽  
One Dimensional ◽  
Linearized Problem ◽  
Bounded Variation Functions ◽  
The One ◽  
First Time ◽  
Prime 2

Abstract This paper investigates the topological structure of the set of the positive solutions of the one-dimensional quasilinear indefinite Neumann problem \begin{dcases}-\Bigg{(}\frac{u^{\prime}}{\sqrt{1+{u^{\prime}}^{2}}}\Bigg{)}^{% \prime}=\lambda a(x)f(u)\quad\text{in }(0,1),\\ u^{\prime}(0)=0,\quad u^{\prime}(1)=0,\end{dcases} where {\lambda\in\mathbb{R}} is a parameter, {a\in L^{\infty}(0,1)} changes sign, and {f\in C^{1}(\mathbb{R})} is positive in {(0,+\infty)} . The attention is focused on the case {f(0)=0} and {f^{\prime}(0)=1} , where we can prove, likely for the first time in the literature, a bifurcation result for this problem in the space of bounded variation functions. Namely, the existence of global connected components of the set of the positive solutions, emanating from the line of the trivial solutions at the two principal eigenvalues of the linearized problem around 0, is established. The solutions in these components are regular, as long as they are small, while they may develop jump singularities at the nodes of the weight function a, as they become larger, thus showing the possible coexistence along the same component of regular and singular solutions.

Parity inversion property of the double ring-shaped oscillator in cylindrical coordinates

2015 ◽  
Vol 30 (39) ◽  
pp. 1550200 ◽  
Author(s):  
Dong-Sheng Sun ◽  
Fa-Lin Lu ◽  
Yuan You ◽  
Chang-Yuan Chen ◽  
Shi-Hai Dong
Keyword(s):  
Functional Analysis ◽  
Singular Point ◽  
Harmonic Oscillator ◽  
Two Dimensional ◽  
Spherical Coordinates ◽  
Analysis Method ◽  
Cylindrical Coordinates ◽  
One Dimensional ◽  
Double Ring ◽  
The One

Using the functional analysis method, we present the exact solutions of the double ring-shaped oscillator (DRSO) potential with certain parity in the cylindrical coordinates. Such a quantum system is separated to two differential equations, i.e. a one-dimensional harmonic oscillator plus an inverse square term and a two-dimensional harmonic oscillator plus an inverse square term. The key point is how to find the adapted symmetrical solutions of the one-dimensional harmonic oscillator plus an inverse square term at the singular point [Formula: see text]. The obtained results are compared with those in the spherical coordinates. We also explore intimate connections [Formula: see text] and [Formula: see text] by substituting [Formula: see text] and [Formula: see text].

scholarly journals A connection between blowing-up and gluings in one-dimensional rings

1984 ◽  
Vol 94 ◽  
pp. 75-87 ◽  
Author(s):  
Grazia Tamone
Keyword(s):  
Singular Point ◽  
Singular Surface ◽  
One Dimensional ◽  
Closed Point ◽  
Blowing Up

Let C be an affine curve, contained on a non-singular surface X as a closed 1-dimensional subscheme. If P is a closed point on C, the blowing-up C′ of C with center P (induced by the blowing-up of X with center P) is an affine curve. It is known that there is a sequence:where C is the normalization of C, and each Ci + 1 is the blowing-up of Ci with center a singular point Pt on Ci (i = 0, …, k – 1).

Configuration Bifurcation Characteristics of the Semi-Regular Hexagon Gough-Stewart Parallel Manipulator

2007 ◽  
Author(s):  
Yu-Xin Wang ◽  
Yu-Tong Li ◽  
Yi-Ming Wang ◽  
Xiu-Tian Yan
Keyword(s):  
Singular Point ◽  
Parallel Manipulator ◽  
Turning Point ◽  
Parallel Manipulators ◽  
Unperturbed System ◽  
Type Ii ◽  
Bifurcation Equation ◽  
One Dimensional ◽  
Perturbed System ◽  
Dimensional Equation

Type-II singularities exist in parallel manipulators commonly. At this kind of singularities, the end-effector is locally movable and uncertain even when all the actuate joints are located. In order to explore a possible approach to obtain the concrete output of the mobile platform at the very small vicinity (germ space) of the singular point, in this paper, the configuration bifurcation characteristics at the germ space have been investigated. At first, the type-II singularity has been identified with Golubitsky-Schaeffer normal form. The result shows that the type-II singular points belong to the turning points. Then, the configuration bifurcation equation is reduced into one dimensional form. Based on this one dimensional equation, the unperturbed and perturbed configuration bifurcation behaviors at the germ space of the turning point have been analyzed. It is found that all configuration branches converged in the same singular point in the unperturbed system can be separated in the perturbed system. This discovery has presented a possible approach to control the parallel manipulator passing through the singular point with a desired configuration.

Border collision bifurcations of chaotic attractors in one-dimensional maps with multiple discontinuities

2021 ◽  
Vol 477 (2253) ◽  
pp. 20210432
Author(s):  
Viktor Avrutin ◽  
Anastasiia Panchuk ◽  
Iryna Sushko
Keyword(s):  
Geometrical Structure ◽  
Sudden Change ◽  
Piecewise Smooth ◽  
Connected Components ◽  
Chaotic Attractors ◽  
One Dimensional ◽  
Smooth Maps ◽  
Border Collision Bifurcations ◽  
Piecewise Smooth Maps ◽  
One Dimensional Maps

In one-dimensional piecewise smooth maps with multiple borders, chaotic attractors may undergo border collision bifurcations, leading to a sudden change in their structure. We describe two types of such border collision bifurcations and explain the mechanisms causing the changes in the geometrical structure of the attractors, in particular, in the number of their bands (connected components).

scholarly journals A Bound on Partitioning Clusters

10.37236/6797 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Daniel Kane ◽  
Terence Tao
Keyword(s):  
Disjoint Union ◽  
Phylogenetic Reconstruction ◽  
Finite Collection ◽  
Time Analysis ◽  
One Dimensional ◽  
Elementary Calculus ◽  
Run Time Analysis ◽  
Additive Energy ◽  
Cluster A ◽  
Similar Vein

Let $X$ be a finite collection of sets (or "clusters"). We consider the problem of counting the number of ways a cluster $A \in X$ can be partitioned into two disjoint clusters $A_1, A_2 \in X$, thus $A = A_1 \uplus A_2$ is the disjoint union of $A_1$ and $A_2$; this problem arises in the run time analysis of the ASTRAL algorithm in phylogenetic reconstruction. We obtain the bound$$ | \{ (A_1,A_2,A) \in X \times X \times X: A = A_1 \uplus A_2 \} | \leq |X|^{3/p} $$where $|X|$ denotes the cardinality of $X$, and $p := \log_3 \frac{27}{4} = 1.73814\dots$, so that $\frac{3}{p} = 1.72598\dots$. Furthermore, the exponent $p$ cannot be replaced by any larger quantity. This improves upon the trivial bound of $|X|^2$. The argument relies on establishing a one-dimensional convolution inequality that can be established by elementary calculus combined with some numerical verification.In a similar vein, we show that for any subset $A$ of a discrete cube $\{0,1\}^n$, the additive energy of $A$ (the number of quadruples $(a_1,a_2,a_3,a_4)$ in $A^4$ with $a_1+a_2=a_3+a_4$) is at most $|A|^{\log_2 6}$, and that this exponent is best possible.

scholarly journals ONE DIMENSIONAL T.T.T STRUCTURES

2014 ◽  
Vol 79 (3) ◽  
pp. 748-775
Author(s):  
DANIEL LOWENGRUB
Keyword(s):  
Topological Group ◽  
Finite Number ◽  
Group Structure ◽  
Connected Components ◽  
Elementary Extension ◽  
Connected Component ◽  
Dimensional Simplex ◽  
One Dimensional ◽  
The Relationship ◽  
Uniformly Bounded

AbstractIn this paper we analyze the relationship between o-minimal structures and the notion of ω-saturated one-dimensional t.t.t structures. We prove that if removing any point from such a structure splits it into more than one definably connected component then it must be a one-dimensional simplex of a finite number of o-minimal structures. In addition, we show that even if removing points doesn’t split the structure, additional topological assumptions ensure that the structure is locally o-minimal. As a corollary we obtain the result that if an ω-saturated one-dimensional t.t.t structure admits a topological group structure then it is locally o-minimal. We also prove that the number of connected components in a definable family is uniformly bounded, which implies that an elementary extension of an ω-saturated one-dimensional t.t.t structure is t.t.t as well.

Sign in / Sign up

Export Citation Format

Share Document