Complete Ricci Solitons via Estimates on the Soliton Potential

In this paper, a growth estimate on the soliton potential is shown for a large class of cohomogeneity one manifolds. This is used to construct continuous families of complete steady and expanding Ricci solitons in the setups of Lü–Page–Pope [ 24] and Dancer–Wang [ 17]. It also provides a different approach to the two summands system [ 30] that applies to all known geometric examples.

Recovery will depend on Greek cooperation with lenders

Subject Greece’s relationship with its creditors. Significance The first annual GDP growth estimate for 2019 has fallen short of expectations. This is partly counterbalanced by the positive assessment of reform progress given in the European Commission’s fifth Enhanced Surveillance Report. Yet investor trust remains fragile, hinging on the Greek government’s ability to meet economic and financial targets consistently and to maintain a constructive dialogue with its international creditors. Impacts In 2020, investors’ flight to safety could undermine demand for Greek bonds. Cheaper borrowing has improved debt sustainability, supporting government attempts to renegotiate primary surplus targets with creditors. Under-execution of planned public investment could restrict economic growth in 2020.

A CR Analogue of Yau’s Conjecture on Pseudoharmonic Functions of Polynomial Growth

In this paper, we first derive the CR volume doubling property, CR Sobolev inequality, and the mean value inequality. We then apply them to prove the CR analogue of Yau’s conjecture on the space consisting of all pseudoharmonic functions of polynomial growth of degree at most $d$ in a complete noncompact pseudohermitian $(2n+1)$-manifold. As a by-product, we obtain the CR analogue of the volume growth estimate and the Gromov precompactness theorem.

CODIMENSIONS GROWTH ESTIMATE OF THE VARIETIES OF DIALGEBRAS

The estimates connecting codimensions of varieties of non-associative algebras and corresponding varieties of dialgebras are obtained.

New normal for trade will shape global economy

Subject The slowdown in global trade. Significance The WTO forecast for world trade has fallen below GDP growth for the first time since the global financial crisis. The forecast hinges on a substantial recovery in the second half of 2016 to meet its meagre trade growth estimate of 1.7% for 2016 as a whole. The first quarter of 2016 actually saw a decline (-1.1%) and the second quarter was barely positive (0.3%). Impacts Economies dependent on goods exports will remain constrained by the weakness in both volumes and prices. Liberalisation of non-tariff barriers may prove difficult politically, particularly health, environment and labour protections. Technology will continue to lower transportation and communication costs, supporting goods trade and facilitating stronger services trade.

Orbit growth for algebraic flip systems

An algebraic flip system is an action of the infinite dihedral group by automorphisms of a compact abelian group $X$. In this paper, a fundamental structure theorem is established for irreducible algebraic flip systems, that is, systems for which the only closed invariant subgroups of $X$ are finite. Using irreducible systems as a foundation, for expansive algebraic flip systems, periodic point counting estimates are obtained that lead to the orbit growth estimate $$\begin{eqnarray}Ae^{hN}\leqslant {\it\pi}(N)\leqslant Be^{hN},\end{eqnarray}$$ where ${\it\pi}(N)$ denotes the number of orbits of length at most $N$, $A$ and $B$ are positive constants and $h$ is the topological entropy.

On CR volume growth estimate in a complete pseudohermitian 3-manifold

In this paper, we first derive the CR Reilly's formula and a CR Ricatti equation for sub-Laplacian of the Carnot–Carathéodory distance in a complete pseudohermitian 3-manifold. As a consequence, we obtain the CR volume growth estimate in a complete pseudohermitian 3-manifold under a lower bound of pseudohermitian curvature tensors. This is a generalization of Nagel, Stein and Wainger's volume growth estimate for the Heisenberg ball in the standard Heisenberg group.