Real Critical Points Research Articles

Deformations of Functions Without Real Critical Points

Given a real analytic function f(x, y) with one critical point P 0, we study deformations f t of fsuch that, for any t ≠ 0, the analytic function f t has no critical points in a neighborhood of P 0. We give explicitly a deformation without real critical points for any function which has only one real branch with characteristic exponents (4, 2q, r).

Critical Points of Real Polynomials and Topology of Real Algebraic T-Surfaces

The paper is devoted to a special class of real polynomials, so-called T-polynomials, which arise in the combinatorial version of the Viro theorem. We study the relation between the numbers of real critical points of a given index of a T-polynomial and the combinatorics of lattice triangulations of Newton polytopes. We obtain upper bounds for the numbers of extrema and saddles of generic T-polynomials of a given degree in three variables, and derive from them upper bounds for Betti numbers of real algebraic surfaces in \(\mathbb{R}P^3\)defined by T-polynomials. The latter upper bounds are stronger than the known upper bounds for arbitrary real algebraic surfaces in \(\mathbb{R}P^3\). Another result is the existence of an asymptotically maximal family of real polynomials of degree min three variables with 31m3/36 + O(m2) saddle points.

Rational Functions with Real Critical Points and the B. and M. Shapiro Conjecture in Real Enumerative Geometry

Suppose that 2d - 2 tangent lines to the rational normal curve z → (1: z …: z d ) in d-dimensional complex projective space are given. It was known that the number of codimension 2 subspaces intersecting all these lines is always finite; for a generic configuration it is equal to the d th Catalan number. We prove that for real tangent lines, all these codimension 2 subspaces are also real, thus confirming a special case of a general conjecture of B. and M. Shapiro. This is equivalent to the following result: If all critical points of a rational function lie on a circle in the Riemann sphere (for example, on the real line), then the function maps this circle into a circle.

Salmonella – A Brief Summary

Salmonellosis is the main cause of human bacterial gastroenteritis in most European countries. Infections with Salmonella is usually subclinical, whereas clinical cases show symptoms with a wide range of severity. Infection is most commonly associated with the consumption of meat, especially poultry or pork, and eggs and their products.Salmonella can enter the food chain at any point throughout its length. The principal reservoir of Salmonellae is the gastrointestinal tract of mammals and birds, but Salmonellae are able to survive and even multiply in many external environments.In Norway, Sweden and Finland cost effective prevention methods have been used for several years to prevent and control Salmonellea infections. In addition, competitive exclusion (CE) and vaccination might be relevant as biological methods to prevent colonisation of bird intestines by enteropathogens, especially Salmonella.Antibiotic drug resistance has been a problem since the start of the antibiotic era. The cause for anxiety is that more and more bacteria are becoming resistant, often to a whole range of antibiotics.The debate on the use of antimicrobials in veterinary medicine and animal production dates back almost as long as the use itself. There is a clear evidence to show that antibacterial agents given to animals for growth promotion, prophylactic purposes or treatment induce a rise in the number of antibiotic resistant strains isolated from the animals. These bacteria may be transmitted to humans by several possible routes.There are thus strong arguments for preventive efforts which have to be directed towards identifying real critical control points (HACCP) throughout the whole food chain, which starts from the farm and ends at the consumer's table.

Finding at Least One Point in Each Connected Component of a Real Algebraic Set Defined by a Single Equation

Deciding efficiently the emptiness of a real algebraic set defined by a single equation is a fundamental problem of computational real algebraic geometry. We propose an algorithm for this test. We find, when the algebraic set is non empty, at least one point on each semi-algebraically connected component. The problem is reduced to deciding the existence of real critical points of the distance function and computing them.

Limit cycle containing nine critical points in its interior for a class of cubic systems

Discuss a class of real planar cubic systems with a critical point O(0,0) of nine orders and obtain the conditions for its limit cycle surrounding the origin, and prove that when small pertubations of coefficients are made, the critical point O(0,0) of nine orders is split into nine real simple critical points and the limit cycle surrounding the origin becomes the limit cycle containing nine critical points in its interior.

Carrier Concentration Dependence of the Spin Pseudo-Gap Behaviors in YBa2Cu3Oy

Oxygen content ( y ) dependence of the NMR relaxation rates (1/ T 1 , 1/ T 2G ) of the Cu(2) site of YBa 2 Cu 3 O y has been studied to investigate magnetic and spin pseudo-gap phase diagrams of this system. The characteristic spin gap temperature ( T sg ) defined by the peak position of (1/ T 1 T ) has a plateau as a function of oxygen content ( y ), at y ∼6.6, so-called “60 K” phase where T c also has a plateau. This plateau means that T sg may be a monotonic function of hole concentration. On the other hand, it has been confirmed from the Curie-Weiss T -dependence of T 2G 2 well above T sg that the “Weiss Temperature (Θ)” for χ( Q ) changes its sign from negative to positive at y ∼6.85. This means that, unless the spin gap and the superconductivity occurred, the real ground state would change from the weak antiferromagnetic metallic state to the nearly antiferromagnetic one at y ∼6.85, which is far from the real critical point, y =6.42.

Partition function zeros for the two-dimensional Ising model VII

The density of the complex temperature zeros of the partition function of the two-dimensional Ising model on completely anisotropic triangular lattices is investigated near real and complex critical points. The non-uniform behaviour of the density of zeros at interior and boundary critical points is studied analytically, and numerically for a lattice with interactions in the ratios 3:2:1. The limiting behaviour of the density at complex critical points depends on the direction of approach in the complex plane. In the neighbourhood of interior critical points one finds only a single layer of zeros. But generally there are two layers or superposed sets of zeros with different distributions, and different limiting densities at boundary critical points. On anisotropic quadratic and on partially anisotropic triangular lattices the two layers become identical, by symmetry of the partition function. The divergence of the (two- dimensional) density of zeros at real critical points is discussed briefly in relation to scaling theory.