The polynomial v(x) allows us to determine the points in E(k̄) that lie in the kernel of α. Indeed, we have kerα = {(x0, y0) ∈ E(k̄) : v(x0) = 0} ∪ {0}. If E1 is defined by y 2 = f(x) = x3 + Ax + B, then we get one point in kerα for each root of v that is also a root of f (these are points (x0, 0) of order 2), two points for every other distinct root of v (since α(x0, y0) = 0 implies α(x0,−y0) = ...

At a Simple Branch Point x0 of a parameter-dependent nonlinear system, two distinct solutions branches S1 and S2 intersect. A popular method of branch switching from S1 to S2 at x0 is to use an orthogonal vector to the tangent vector to S1at x0 as an initial guess for the tangent vector to S2 at x0. We describe an improved version of this method using a bisection-like procedure, which is especi...

We prove Lp boundedness results, p>2, for local maximal averaging operators over a smooth 2D hypersurface S with either C1 density function or singularity that grows as |(x,y)|−β β<2. Suppose one is in coordinates such the surface localized near some (x0,y0,z0) at which (0,0,1) normal to surface, and suppose represented graph of z0+s(x−x0,y−y0) (x0,y0), s(0,0)=0. It shown long Taylor series Hes...

Consider a controlled evolutionary game (CEG), where nn players, xxii , ii = 1,⋯ ,nn, are considered as machines and mm players, uuii , ii = 1, ... ,mm, are considered as human players. Assume the machines strategy updating rules are known, and their payoffs yyii, ii = 1, ... ,nn, are also known. Then the dynamic model can be described as � xx1(tt + 1) = ff1(xx1,⋯ , xxnn,uu1,⋯ , uumm) ⋮ xxnn(tt...

The karyotypes of five species of crickets Phalangopsinae were studied: E. onthophagus shows 2n = 19, X0♂, pairs 1, 7 and 8 metacentric, 2, 3, 6 and 9 submetacentric, 4 and 5 acrocentric; E. itatibensis with 2n = 19, X0♂, pairs 1, 7 and 8 metacentric, 2, 3, 4, 6 and 9 submetacentric, pair 5 acrocentric; E. cavernicolus with 2n = 21, X0♂, pairs 7, 8 and 9 metacentric, pairs 1, 2, 3, 4, 5, 6 and ...

Polynomizing is a term that intends to describe the uses of polynomiallike representations as a reasoning strategy and as a tool for scientific heuristics. I show how proof-theory and semantics for classical and several non-classical logics can be approached from this perspective, and discuss the assessment of this prospect, in particular to recover certain ideas of George Boole in unifying log...

In 1945, R. Fox introduced the so-called Fox torus homotopy groups in which the usual homotopy groups are embedded and their Whitehead products are expressed as commutators. A modern treatment of Fox torus homotopy groups and their generalization were given by the authors in an earlier work. In this note, we further explore these groups and their properties. We discuss co-multiplications on Fox...

In this paper we prove that the Poincaré map associated to a Lorenz like flow has exponential decay of correlations with respect to Lipschitz observables. This implies that the hitting time associated to the flow satisfies a logarithm law. The hitting time τr(x, x0) is the time needed for the orbit of a point x to enter for the first time in a ball Br(x0) centered at x0, with small radius r. As...