Expected volume and Euler characteristic of random submanifolds

In a closed manifold of positive dimension $n$, we estimate the expected volume and Euler characteristic for random submanifolds of codimension $r\in \{1,...,n\}$ in two different settings. On one hand, we consider a closed Riemannian manifold and some positive $\lambda$. Then we take $r$ independent random functions in the direct sum of the eigenspaces of the Laplace-Beltrami operator associated to eigenvalues less than $\lambda$ and consider the random submanifold defined as the common zero set of these $r$ functions. We compute asymptotics for the mean volume and Euler characteristic of this random submanifold as $\lambda$ goes to infinity. On the other hand, we consider a complex projective manifold defined over the reals, equipped with an ample line bundle $\mathcal{L}$ and a rank $r$ holomorphic vector bundle $\mathcal{E}$ that are also defined over the reals. Then we get asymptotics for the expected volume and Euler characteristic of the real vanishing locus of a random real holomorphic section of $\mathcal{E}\otimes\mathcal{L}^d$ as $d$ goes to infinity. The same techniques apply to both settings.Comment: Final version, accepted for publication in J. Funct. Anal., 50 pages.A change in notational convention impacts the statement of the main theorems and most formula

Variance of the volume of random real algebraic submanifolds II

Let $\mathcal{X}$ be a complex projective manifold of dimension $n$ defined over the reals and let $M$ be its real locus. We study the vanishing locus $Z\_{s\_d}$ in $M$ of a random real holomorphic section $s\_d$ of $\mathcal{E} \otimes \mathcal{L}^d$, where $\mathcal{L} \to \mathcal{X}$ is an ample line bundle and $\mathcal{E} \to \mathcal{X}$ is a rank $r$ Hermitian bundle, $r \in \{1,\dots, n\}$. We establish the asymptotic of the variance of the linear statistics associated with $Z\_{s\_d}$, as $d$ goes to infinity. This asymptotic is of order $d^{r-\frac{n}{2}}$. As a special case, we get the asymptotic variance of the volume of $Z\_{s\_d}$. The present paper extends the results of [20], by the first-named author, in essentially two ways. First, our main theorem covers the case of maximal codimension ($r = n$), which was left out in [20]. And second, we show that the leading constant in our asymptotic is positive. This last result is proved by studying the Wiener--It{\=o} expansion of the linear statistics associated with the common zero set in $\mathbb{RP}^n$ of $r$ independent Kostlan--Shub--Smale polynomials.Comment: Final version, published in Indiana Math. Univ.

Linear Approximation Methods and International Real Business Cycles with Incomplete Asset Markets

Most quantitative studies of international real business cycle (IRBC) models require the use of approximate solution methods. We solve an IRBC model with incomplete asset markets using King, Plosser and Rebelo's (1988) linear approximation method. We quantify the additional approximation error brought about by the existence of a unit root in the linear dynamic system and demonstrate that the symmetry of the model helps reduce this approximation error. A central finding is that the parametrizations which address the cross-country consumption correlation puzzle are precisely those where solutions may be least accurate.

Precautionary saving and portfolio allocation: DP by GMM

There is much research on consumption-savings problems with risky labor income and a constant interest rate and also on portfolio allocation with risky returns but nonstochastic labor income. Less is known quantitatively about the interaction between the two forms of risk. Under CRRA utility, undiversifiable income risk should be reflected in both savings rates and portfolio allocations. To quantify these effects in a model of consumption and portfolio choice, we adopt a semi-parametric projection method for solving dynamic programmes, based on generalized method of moments estimation of the parameters of approximate decision rules. We find that background income risk does affect optimal portfolios but that this effect may be difficult to detect empirically.portfolio theory, precautionary saving

Inventories, Sticky Prices and the Propogation of Nominal Shocks

Post-war business cycle fluctuations of output and inflation are remarkably persistent. Many recent sticky-price monetary business cycle models, however, grossly underpredict this persistence. We assess whether adding inventories to a standard sticky-price model raises the persistence of output and inflation. For this addition, we consider three different frameworks: a linear-quadratic inventory model, a factor of production model, and a shopping-cost model. We find that adding inventories increases the persistence of output and inflation, but that the increase is smaller for inflation. Overall, the shopping-cost model best explains the persistence of output and inflation.

What do “residuals” from first-order conditions reveal about DGE models?

The first-order condition (FOC) associated with labour in many dynamic general equilibrium models involves only current period variables. Residuals constructed from this FOC are inconsistent with aggregate US data in that they are very large and highly persistent. The persistence suggests that models which introduce dynamic terms in the labour FOC may be more consistent with the data. Three such models (one with learning by doing, one with habit formation, and one with labour adjustment costs) confirm that they can reduce the persistence in the residuals making the models more consistent with the joint dynamics of consumption, output and hours.dynamic general equilibrium models, real business cycles, first-order conditions.

Development of an ATCA IPMI controller mezzanine board to be used in the ATCA developments for the ATLAS Liquid Argon upgrade

International audienceIn the context of the LHC upgrades, a new Read-Out Driver (ROD) board for the ATLAS LAr calorimeter is being developed. xTCA (Advanced/Micro Telecom Computing Architecture) is becoming a standard in high energy physics and is a serious candidate for future readout systems. We will present our current developments to master ATCA and to integrate a large number of very high speed links (96 links/8.5 Gbps) on a ROD Evaluator ATCA board. To manage our ROD Evaluator, we have developed a versatile ATCA IPMI controller for ATCA boards which is FPGA Mezzanine Card (FMC) compliant

Organizational Capital and the International Co-movement of Investment

A productivity shock leads to a large international transfer of capital and negative co-movement of investment in the typical two-country real business cycle model. Most recent models that attempt to reduce or remove this transfer produce unrealistically low investment volatility. We show that adding organizational capital to the technological environment of a relatively standard international business cycle model can ameliorate this problem. In addition we show that GHH preferences along with the above modification are sufficient to deliver positive cross-country correlations of consumption, hours, output and investment.International RBC; learning by doing; organizational capital; cross-country correlations; investment