It is shown that the quotients of Kähler surfaces under free anti-holomorphic involutions have vanishing Seiberg-Witten invariants. Various vanishing theorems have played important roles in gauge theory. The first among them, due to S. K. Donaldson [2], states that the Donaldson invariants vanish for any smooth closed oriented 4-manifold X which decomposes to a connected sum X1#X2 with b2 (X1) ...

We examine the behavior of accelerated gradient methods in smooth nonconvex unconstrained optimization. Each of these methods is typically a linear combination of a gradient direction and the previous step. We show by means of the stable manifold theorem that the heavyball method method does not converge to critical points that do not satisfy second-order necessary conditions. We then examine t...

We show that the number of stabilizations needed to interchange the handlebodies of a Heegaard splitting of a closed 3manifold by an isotopy is bounded below by the smaller of twice its genus or half its Hempel distance. This is a combinatorial version of a proof by Hass, Thompson and Thurston of a similar theorem, but with an explicit bound in terms of distance. We also show that in a 3-manifo...

We show that the R/Z part of the analytically defined η invariant of Atiyah-PatodiSinger for a Dirac operator on an odd dimensional closed spin manifold can be expressed purely geometrically through a stable Chern-Simons current on a higher dimensional sphere. As a preliminary application, we discuss the relation with the Atiyah-PatodiSinger R/Z index theorem for unitary flat vector bundles, an...

In this paper two aspects of numerical dynamics are used for an artificial neural network (ANN) analysis. It is shown that topological conjugacy of gradient dynamical systems and both the shadowing and inverse shadowing properties have nontrivial implications in the analysis of a perceptron learning process. The main result is that, generically, any such process is stable under numerics and rob...

In General Relativity, asymptotically flat manifolds naturally arise as spacelike slices in space-times describing isolated gravitational systems. Due to their self-evident physical relevance, their geometry has been extensively studied and a variety of fundamental results have been obtained. Among these, the mean-curvature proof of the Positive Mass Theorem proposed by Schoen and Yau [SY79] ha...

In this paper we show the existence of stable nonconstant equilibrium (patterns) for reaction-diffusion equations with nonlinear boundary conditions on small domains connected by thin channels. We prove the convergence of eigenvalues and eigenfunctions of the Laplace operator is such domains. This information is used to show that the asymptotic dynamic of the heat equations in this domain is eq...

We prove that the family of measured dynamical systems which can be realised as uniquely ergodic minimal homeomorphisms on a given manifold (of dimension at least two) is stable under measured extension. As a corollary, any ergodic system with an irrational eigenvalue is isomorphic to a uniquely ergodic minimal homeomorphism on the two-torus. The proof uses the following improvement of Weiss re...

In this paper we analyze the global asymptotic behavior of the asymmetric May– Leonard model of three competing species: dxi dt = xi(1−xi−βixi−1−αixi+1), xi(0) > 0, i = 1, 2, 3 with x0 = x3, x4 = x1 under the assumption 0 < αi < 1 < βi, i = 1, 2, 3. Let Ai = 1−αi and Bi = βi − 1, i = 1, 2, 3. The linear stability analysis shows that the interior equilibrium P = (p1, p2, p3) is asymptotically st...