Mathematics Commons^™

Existence Of Positive Definite Noncoercive Sums Of Squares, Gregory C. Verchota

Mathematics - All Scholarship

Positive definite forms

f 2 R[x₁, . . . , x_n] which are sums of squares of forms of R[x₁, . . . , x_n] are constructed to have the additional property that the members of any collection of forms whose squares sum to f must share a nontrivial complex root in Cⁿ.

Rigidity Of Gradient Ricci Solitons, Peter Petersen, William Wylie

Mathematics - All Scholarship

We define a gradient Ricci soliton to be rigid if it is a flat bundle N*_GammaR^k where N is Einstein. It is known that not all gradient solitons are rigid. Here we offer several natural conditions on the curvature that characterize rigid gradient solitons. Other related results on rigidity of Ricci solitons are also explained in the last section.

Domains Of Definition Of Monge-Ampère Operators On Compact Kähler Manifolds, Dan Coman, Vincent Guedj, Ahmed Zeriahi

Mathematics - All Scholarship

Let (X, w) be a compact Kahler manifold. We introduce and study the largest set DMA(X, w) of w-plurisubharmonic (psh) functions on which the complex Monge-Ampere operator is well defined. It is much larger than the corresponding local domain of definition, though still a proper subset of the set PSH(X, w) of all w-psh functions. We prove that certain twisted Monge-Ampere operators are well defined for all w-psh functions. As a consequence, any w-psh function with slightly attenuated singularities has finite weighted Monge-Ampere energy.

Stable Algebras Of Entire Functions, Dan Coman, Evgeny A. Poletsky

Mathematics - All Scholarship

Suppose that h and g belong to the algebra B generated by the rational functions and an entire function f of finite order on Cⁿ and that h/g has algebraic polar variety. We show that either h/g in B or f = q₁e^p +q₂, where p is a polynomial and q¹, q² are rational functions. In the latter case, h/g belongs to the algebra generated by the rational functions, e^p and e^−p.

Complete Shrinking Ricci Solitons Have Finite Fundamental Group, William Wylie

Mathematics - All Scholarship

We show that if a complete Riemannian manifold supports a vector field such that the Ricci tensor plus the Lie derivative of the metric with respect to the vector field has a positive lower bound, then the fundamental group is finite. In particular, it follows that complete shrinking Ricci solitons and complete smooth metric measure spaces with a positive lower bound on the Bakry-Emery tensor have finite fundamental group. The method of proof is to generalize arguments of Garcia-Rio and Fernandez-Lopez in the compact case.

Modelling And Testing For Structural Changes In Panel Cointegration Models With Common And Idiosyncratic Stochastic Trend, Chihwa Kao, Lorenzo Trapani, Giovanni Urga

Center for Policy Research

In this paper, we propose an estimation and testing framework for parameter instability in cointegrated panel regressions with common and idiosyncratic trends. We develop tests for structural change for the slope parameters under the null hypothesis of no structural break against the alternative hypothesis of (at least) one common change point, which is possibly unknown. The limiting distributions of the proposed test statistics are derived. Monte Carlo simulations examine size and power of the proposed tests. We are grateful for discussions with Robert De Jong, Long-Fei Lee, Zongwu Cai, and Yupin Hu. We would also like to thank participants in …

Asymptotic Properties Of Estimators For The Linear Panel Regression Model With Individual Effects And Serially Correlated Errors: The Case Of Stationary And Non-Stationary Regressors And Residuals, Badi H. Baltagi, Chihwa Kao, Long Liu

Center for Policy Research

This paper studies the asymptotic properties of standard panel data estimators in a simple panel regression model with error component disturbances. Both the regressor and the remainder disturbance term are assumed to be autoregressive and possibly non-stationary. Asymptotic distributions are derived for the standard panel data estimators including ordinary least squares, fixed effects, first-difference, and generalized least squares (GLS) estimators when both T and n are large. We show that all the estimators have asymptotic normal distributions and have different convergence rates dependent on the non-stationarity of the regressors and the remainder disturbances. We show using Monte Carlo experiments that …

Consistent Estimation With Weak Instruments In Panel Data, Chihwa Kao, Long Liu

Center for Policy Research

This note analyzes the asymptotic distribution for instrumental variables regression for panel data when the available instruments are weak. We show that consistency can be established in panel data.

Forecasting With Panel Data, Badi H. Baltagi

Center for Policy Research

This paper gives a brief survey of forecasting with panel data. Starting with a simple error component regression model and surveying best linear unbiased prediction under various assumptions of the disturbance term. This includes various ARMA models as well as spatial autoregressive models. The paper also surveys how these forecasts have been used in panel data applications, running horse races between heterogeneous and homogeneous panel data models using out of sample forecasts.