Analytic Kramer kernels, Lagrange-type interpolation series and de Branges spaces

The classical Kramer sampling theorem provides a method for obtaining orthogonal sampling formulas. In particular, when the involved kernel is analytic in the sampling parameter it can be stated in an abstract setting of reproducing kernel Hilbert spaces of entire functions which includes as a particular case the classical Shannon sampling theory. This abstract setting allows us to obtain a sort of converse result and to characterize when the sampling formula associated with an analytic Kramer kernel can be expressed as a Lagrange-type interpolation series. On the other hand, the de Branges spaces of entire functions satisfy orthogonal sampling formulas which can be written as Lagrange-type interpolation series. In this work some links between all these ideas are established

Enumeration of connected Catalan objects by type

Noncrossing set partitions, nonnesting set partitions, Dyck paths, and rooted plane trees are four classes of Catalan objects which carry a notion of type. There exists a product formula which enumerates these objects according to type. We define a notion of `connectivity' for these objects and prove an analogous product formula which counts connected objects by type. Our proof of this product formula is combinatorial and bijective. We extend this to a product formula which counts objects with a fixed type and number of connected components. We relate our product formulas to symmetric functions arising from parking functions. We close by presenting an alternative proof of our product formulas communicated to us by Christian Krattenthaler which uses generating functions and Lagrange inversion

ON NECESSARY CONDITIONS FOR EFFICIENCY IN DIRECTIONALLY DIFFERENTIABLE OPTIMIZATION PROBLEMS

This paper deals with multiobjective programming problems with in- equality, equality and set constraints involving Dini or Hadamard differentiable func- tions. A theorem of the alternative of Tucker type is established, and from which Kuhn-Tucker necessary conditions for local Pareto minima with positive Lagrange multipliers associated with all the components of objective functions are derived.Theorem of the alternative, Kuhn-Tucker necessary conditions, direc- tionally differentiable functions.

Modelling Conditional and Unconditional Heteroskedasticity with Smoothly Time-Varying Structure

In this paper, we propose two parametric alternatives to the standard GARCH model. They allow the conditional variance to have a smooth time-varying structure of either additive or multiplicative type. The suggested parameterizations describe both nonlinearity and structural change in the conditional and unconditional variances where the transition between regimes over time is smooth. A modelling strategy for these new time-varying parameter GARCH models is developed. It relies on a sequence of Lagrange multiplier tests, and the adequacy of the estimated models is investigated by Lagrange multiplier type misspecification tests. Finite-sample properties of these procedures and tests are examined by simulation. An empirical application to daily stock returns and another one to daily exchange rate returns illustrate the functioning and properties of our modelling strategy in practice. The results show that the long memory type behaviour of the sample autocorrelation functions of the absolute returns can also be explained by deterministic changes in the unconditional variance.Conditional heteroskedasticity; Structural change; Lagrange multiplier test; Misspecification test; Nonlinear time series; Time-varying parameter model.

Almost Kaehler Ricci Flows and Einstein and Lagrange-Finsler Structures on Lie Algebroids

In this work we investigate Ricci flows of almost Kaehler structures on Lie algebroids when the fundamental geometric objects are completely determined by (semi) Riemannian metrics, or effective) regular generating Lagrange/ Finsler, functions. There are constructed canonical almost symplectic connections for which the geometric flows can be represented as gradient ones and characterized by nonholonomic deformations of Grigory Perelman's functionals. The first goal of this paper is to define such thermodynamical type values and derive almost K\"ahler - Ricci geometric evolution equations. The second goal is to study how fixed Lie algebroid, i.e. Ricci soliton, configurations can be constructed for Riemannian manifolds and/or (co) tangent bundles endowed with nonholonomic distributions modelling (generalized) Einstein or Finsler - Cartan spaces. Finally, there are provided some examples of generic off-diagonal solutions for Lie algebroid type Ricci solitons and (effective) Einstein and Lagrange-Finsler algebroids.Comment: This version is accepted by Mediterranian J. Math. and modified following editor/referee's requests. File latex2e 11pt generates 29 page