Configuration Of Singularities Research Articles

Reconstruction of phase maps from the configuration of phase singularities in two-dimensional manifolds

Phase singularity analysis provides a quantitative description of spiral wave patterns observed in chemical or biological excitable media. The configuration of phase singularities (locations and directions of rotation) is easily derived from phase maps in two-dimensional manifolds. The question arises whether one can construct a phase map with a given configuration of phase singularities. The existence of such a phase map is guaranteed provided that the phase singularity configuration satisfies a certain constraint associated with the topology of the supporting medium. This paper presents a constructive mathematical approach to numerically solve this problem in the plane and on the sphere as well as in more general geometries relevant to atrial anatomy including holes and a septal wall. This tool can notably be used to create initial conditions with a controllable spiral wave configuration for cardiac propagation models and thus help in the design of computer experiments in atrial electrophysiology.

A study of singularities on rational curves via syzygies

Consider a rational projective curve C of degree d over an algebraically closed field k. There are n homogeneous forms g1;:::;g n of degree d in B = kk(x;y) which parameterize C in a birational, base point free, manner. We study the singularities of C by studying a Hilbert-Burch matrix ' for the row vector (g1;:::;g n). In the we use the generalized row ideals of ' to identify the singular points on C, their multiplicities, the number of branches at each singular point, and the multiplicity of each branch. Let p be a singular point on the parameterized planar curve C which corresponds to a general- ized zero of '. In the we give a matrix ' 0 whose maximal minors parameterize the closure, in P 2 , of the blow-up at p of C in a neighborhood of p. We apply the General Lemma to ' 0 in order to learn about the singularities of C in the first neighborhood of p. If C has even degree d = 2c and the multiplicity of C at p is equal to c, then we apply the Triple Lemma again to learn about the singularities of C in the second neighborhood of p. Consider rational plane curves C of even degree d = 2c. We classify curves according to the configuration of multiplicity c singularities on or infinitely near C. There are 7 possible configurations of such singularities. We classify the Hilbert-Burch matrix which corresponds to each configuration. The study of multiplicity c singularities on, or infinitely near, a fixed rational plane curve C of degree 2c is equivalent to the study of the scheme of generalized zeros of the fixed balanced Hilbert-Burch matrix ' for a parameterization of C. Let

GLOBAL CLASSIFICATION OF A CLASS OF CUBIC VECTOR FIELDS WHOSE CANONICAL REGIONS ARE PERIOD ANNULI

We study cubic vector fields with inverse radial symmetry, i.e. of the form ẋ = δx - y + ax2+ bxy + cy2+ σ(dx - y)(x2+ y2), ẏ = x + δy + ex2+ fxy + gy2+ σ(x + dy) (x2+ y2), having a center at the origin and at infinity; we shortly call them cubic irs-systems. These systems are known to be Hamiltonian or reversible. Here we provide an improvement of the algorithm that characterizes these systems and we give a new normal form.Our main result is the systematic classification of the global phase portraits of the cubic Hamiltonian irs-systems respecting time (i.e. σ = 1) up to topological and diffeomorphic equivalence. In particular, there are 22 (resp. 14) topologically different global phase portraits for the Hamiltonian (resp. reversible Hamiltonian) irs-systems on the Poincaré disc.Finally we illustrate how to generalize our results to polynomial irs-systems of arbitrary degree. In particular, we study the bifurcation diagram of a 1-parameter subfamily of quintic Hamiltonian irs-systems. Moreover, we indicate how to construct a concrete reversible irs-system with a given configuration of singularities respecting their topological type and separatrix connections.

On the geometry of certain irreducible non-torus plane sextics

An irreducible non-torus plane sextic with simple singularities is said to be special if its fundamental group factors to a dihedral group. There exist (exactly) ten configurations of simple singularities that are realizable by such curves. Among them, six are realizable by non-special sextics as well. We conjecture that for each of these six configurations there always exists a non-special curve whose fundamental group is abelian, and we prove this conjecture for three configurations (another one has already been treated in one of our previous papers). As a corollary, we obtain new explicit examples of Alexander-equivalent Zariski pairs of irreducible sextics.

Topological Configurations of Optical Phase Singularities

Geometric and topological properties of phase singularity lines in three-dimensional complex scalar wavefields are discussed. In particular, their role as the intersections of the zero contour surfaces of the real and imaginary parts of the field gives numerous insights into 3D vortex topology. In addition, complex scalar wavefields (i.e. solutions of the three-dimensional Helmholtz and paraxial equations) are compared to more general complex scalar fields, including those arising naturally from algebraic geometry.

$\pi_1$-Equivalent Weak Zariski Pairs

Consider a moduli space ${{\cal M}(\Sigma,d)}$ of reduced curves in ${\textbf{CP}^2}$ with a given degree ${d}$ and having a prescribed configuration of singularities ${\Sigma}$. Let ${C,C'\in {\cal M}(\Sigma,d)}$. The pair of curves ${(C,C')}$ is called a weak Zariski pair if the pairs of spaces ${(\textbf{CP}^2,C)}$ and ${(\textbf{CP}^2,C')}$ are not homeomorphic. There exists two classical ways to detect weak Zariski pairs: (i) showing that the generic Alexander polynomials ${\Delta_C(t)}$ and ${\Delta_{C'}(t)}$ of~${C}$ and ${C'}$ are different; (ii) showing that the fundamental groups ${\pi_1(\textbf{CP}^2-C)}$ and ${\pi_1(\textbf{CP}^2-C')}$ are not isomorphic. In this paper, we give the first example of a weak Zariski pair ${(C,C')}$ such that ${\pi_1(\textbf{CP}^2-C)}$ and ${\pi_1(\textbf{CP}^2-C')}$ are isomorphic (notice that such an isomorphism automatically implies ${\Delta_C(t)} = {\Delta_{C'}(t)}$). We shall call such a pair a ${\pi_1}$-equivalent weak Zariski pair. The members ${C}$ and ${C'}$ of our pair are reducible sextics with the following configuration of singularities ${\{D_{10}+A_5+A_4\}}$. By the way, we determine the fundamental group ${\pi_1(\textbf{CP}^2-D)}$ for any curve~${D}$ in the moduli space ${{\cal M}(\{D_{10}+A_5+A_4\},6)}$. As an application, we give a new weak Zariski ${4}$-ple (we recall that a ${4}$-ple ${(D_1,D_2,D_3,D_4)}$ of curves in ${{\cal M}(\Sigma,d)}$ is called a weak Zariski ${4}$-ple if for any ${1\leq i<j\leq 4}$ the pairs of spaces ${(\textbf{CP}^2,D_i)}$ and ${(\textbf{CP}^2,D_j)}$ are not homeomorphic).

Fundamental groups of the complements of certain plane non-tame torus sextics

The moduli space of torus sextics with the configuration of singularities { A 2 + A 5 + 2 E 6 } has two connected components. We compute the fundamental groups π 1 ( CP 2 − C ) for sextics C in both components and study their differences.

On topological types of reduced sextics

We present a computational method for obtaining generic forms of sextics with a given configuration of local singularities. Using this method, we first complete the classification of topological types of local singularities appearing on reduced sextics. Next we give the list of possible configurations of singularities containing at least one non-simple singularity and satisfying $\rho(5)\ge 7$, and show that such a sextic is of torus type, which has been conjectured by the third author.

Geometry of Reduced Sextics of Torus Type

In [7], we gave a classification of the configurations of singularities of irreducible sextic of torus type. In this paper, we give a classification of the configurations of singularities on reducible sextics of torus type. We determine the component types and the geometry of the components for each configuration.

Feedback stability of charge amplifiers with continuous reset through forward-biased diode junctions

A low noise charge sensitive amplifier that does not require a feedback resistor nor an additional reset device has recently been proposed and successfully tested in several spectroscopy systems. The discharge of the leakage and signal current from the detector is performed continuously through the forward-biased gate to channel junction of the input JFET or, in general, through a diode shunting the preamplifier input. Up to now this forward-biased FET charge amplifier (FBFA) has been used only with detectors having currents limited to a few tens of pA. Recent applications of the FBPA for larger area detectors operating at room temperature with leakage current up to few nA reveals that the original feedback configuration of the FBFA is no longer stable and can show low frequency oscillations. In this work the double feedback loop of the FBFA and its stability is analyzed in detail, and a solution to the mentioned problem is found. The acceptable limit of the input current value in the original configuration of the amplifier is analytically determined, and a new configuration of the singularities of the loop gain, which guarantees stability of the FBFA response at higher input current, is proposed. Experimental tests are presented that show the critical condition of the original FBFA configuration and verify the feasibly of the new proposed solution.

On the Ginzburg-Landau energy with weight

This paper gives a solution to an open problem raised by Bethuel, Brezis and Hélein. We study the Ginzburg-Landau energy with weight. We find the expression of the renormalized energy and we show that the finite configuration of singularities of the limit is a minimum point of this functional. We find a vanishing gradient type property and then we obtain the renormalized energy by Bethuel, Brezis and Hélein’s shrinking holes method.

Analytic Structure of the Triple-Regge Vertex

We study the triple-Regge, helicity-pole and related asymptotic behaviors of the six-line amplitude using the dual-resonance model (DRM) as a guide. We show that the triple-Regge vertex in the (DRM) can be expressed as a sum of four terms, which we interpret as exhibiting the allowed tree-like configurations of singularities in the asymptotic channel invariants. These four terms have counterparts in the helicity-pole limit where there exist several distinct terms with differing powers of the asymptotic variable, i.e., several different helicity poles. As an application we investigate the interesting property of the (DRM) that the triple-Regge or helicity-pole contribution to single-particle production at zero momentum transfer vanishes for trajectory intercepts near unity. We trace this effect to a nonsense zero. This suggests a mechanism for fulfilling one of the unitarity requirements for general six-line amplitudes. We discuss the possible generality of our results.