Noncontractible loops of symplectic embeddings between convex toric domains

Given two 4-dimensional ellipsoids whose symplectic sizes satisfy a specified inequality, we prove that a certain loop of symplectic embeddings between the two ellipsoids is noncontractible. The statement about symplectic ellipsoids is a particular case of a more general result. Given two convex toric domains whose first and second ECH capacities satisfy a specified inequality, we prove that a certain loop of symplectic embeddings between the two convex toric domains is noncontractible. We show how the constructed loops become contractible if the target domain becomes large enough. The proof involves studying certain moduli spaces of holomorphic cylinders in families of symplectic cobordisms arising from families of symplectic embeddings.Comment: 16 pages, 5 figures. Comments are welcome

The volume growth of complete gradient shrinking Ricci solitons

We prove that any gradient shrinking Ricci soliton has at most Euclidean volume growth. This improves a recent result of H.-D. Cao and D. Zhou by removing a condition on the growth of scalar curvature

Design of boundary stabilizers for the non-autonomous cubic semilinear heat equation, driven by a multiplicative noise

Here we design boundary feedback stabilizers to unbounded trajectories, for semi-linear stochastic heat equation with cubic non-linearity. The feedback controller is linear, given in a simple explicit form and involves only the eigenfunctions of the Laplace operator. It is supported in a given open subset of the boundary of the domain. Via a rescaling argument, we transform the stochastic equation into a random deterministic one. Then, the simple form of the feedback, we propose here, allows to write the solution, of the random equation, in a mild formulation via a kernel. Appealing to a fixed point argument the existence \& stabilization result is proved

Entropy of AT(n) systems

In this paper we show that any ergodic measure preserving transformation of a standard probability space which is AT$(n)$ for some positive integer $n$ has zero entropy. We show that for every positive integer $n$ any Bernoulli shift is not AT($n$). We also give an example of a transformation which has zero entropy but does not have property AT($n$), for any integer $n\geq 1$

The flow of weights of some factors arising as fixed point algebras

In this paper we study the associated flow of some factors arising as fixed point algebras under product type actions on ITPFI factors. We compute their associated flow and show that under certain conditions these flows are approximately transitive and consequently the corresponding fixed point factors are ITPFI.Comment: 13 page

Strongly connected components-Algorithm for finding the strongly connected components of a graph

A directed graph G (V, E) is strongly connected if and only if, for a pair of vertices X and Y from V, there exists a path from X to Y and a path from Y to X. In Computer Science, the partition of a graph in strongly connected components is represented by the partition of all vertices from the graph, so that for any two vertices, X and Y, from the same partition, there exists a path from X to Y and a path from Y to X and for any two vertices, U and V, from different partition, the property is not met. The algorithm presented below is meant to find the partition of a given graph in strongly connected components in O (numberOfNodes + numberOfEdges * log* (numberOfNodes)), where log* function stands for iterated logarithm.Comment: 7 pages, 5 sequences of cod

Present-day financial statement and the foundation of economic decisions

Nowadays we face a world-scale movement towards the privatization of public enterprise and liberalization of commerce, world-wide investments and monetary policies at a global level. These factors have progressively led to a substantial growth of commerce and international investments. The progress that has been achieved in information technology has determined the availability of financial and non-financial information in different parts of the world, using various means of communicating it.financial statement, economic decision, accounting, information technology

Analysis of weighted Laplacian and applications to Ricci solitons

We study both function theoretic and spectral properties of the weighted Laplacian $\Delta_f$ on complete smooth metric measure space $(M,g,e^{-f}dv)$ with its Bakry-\'{E}mery curvature $Ric_f$ bounded from below by a constant. In particular, we establish a gradient estimate for positive $f-$harmonic functions and a sharp upper bound of the bottom spectrum of $\Delta_f$ in terms of the lower bound of $Ric_{f}$ and the linear growth rate of $f.$ We also address the rigidity issue when the bottom spectrum achieves its optimal upper bound under a slightly stronger assumption that the gradient of $f$ is bounded. Applications to the study of the geometry and topology of gradient Ricci solitons are also considered. Among other things, it is shown that the volume of a noncompact shrinking Ricci soliton must be of at least linear growth. It is also shown that a nontrivial expanding Ricci soliton must be connected at infinity provided its scalar curvature satisfies a suitable lower bound.Comment: Will appear in Comm. Anal. Geo

Geometry of shrinking Ricci solitons

The main purpose of this paper is to investigate the curvature behavior of four dimensional shrinking gradient Ricci solitons. For such soliton $M$ with bounded scalar curvature $S$, it is shown that the curvature operator $\mathrm{Rm}$ of $M$ satisfies the estimate $|\mathrm{Rm}|\le c\,S$ for some constant $c$. Moreover, the curvature operator $\mathrm{Rm}$ is asymptotically nonnegative at infinity and admits a lower bound $\mathrm{Rm}\geq -c\,\left(\ln r\right)^{-1/4},$ where $r$ is the distance function to a fixed point in $M$. As application, we prove that if the scalar curvature converges to zero at infinity, then the manifold must be asymptotically conical. As a separate issue, a diameter upper bound for compact shrinking gradient Ricci solitons of arbitrary dimension is derived in terms of the injectivity radius.Comment: 28 pages, submitted, v2 has a new section about the conical structure of soliton

Gradient estimate for harmonic functions on K\"ahler manifolds

We prove a sharp integral gradient estimate for harmonic functions on noncompact K\"ahler manifolds. As application, we obtain a sharp estimate for the bottom of spectrum of the p-Laplacian and prove a splitting theorem for manifolds achieving this estimate.Comment: 34pages, accepte