A simpler characterization of Sheffer polynomial

We characterize the Sheffer sequences by a single convolution identity $$F^{(y)} p_{n}(x) = \sum _{k=0}^{n}\ p_{k}(x)\ p_{n-k}(y)$$ where $F^{(y)}$ is a shift-invariant operator. We then study a generalization of the notion of Sheffer sequences by removing the requirement that $F^{(y)}$ be shift-invariant. All these solutions can then be interpreted as cocommutative coalgebras. We also show the connection with generalized translation operators as introduced by Delsarte. Finally, we apply the same convolution to symmetric functions where we find that the ``Sheffer'' sequences differ from ordinary full divided power sequences by only a constant factor

A Note on Krugman\u27s Liquidity Trap

The 1998 stylized model of Krugman constituted a ground-breaking contribution explaining the long lasting Japanese stagnation as the consequence of a ‘liquidity trap’ situation featuring a negative natural interest rate. Our critique to such a proposal will focus on three aspects. First, we will question the logical structure of the model, providing an alternative interpretation of its closure. Second, we will argue that aggregate demand has no role in the explanation, as the cause for the persistent excess of savings over desired investment is the result of a supply side shock plus a financial rigidity on the nominal interest rate. Finally, we will discuss the restrictive assumptions needed to get a negative natural interest rate, the concept that lies at the foundation of the entire theoretical apparatus. Our conclusion is that the explanation offered within the 1998 contribution does not provide a satisfying rationale for the Japanese stagnation

Maple umbral calculus package

We are developing a Maple package of functions related to Rota's Umbral Calculus. A Mathematica version of this package is being developed in parallel

Sheffer sequences, probability distributions and approximation operators

We present a new method to compute formulas for the action on monomials of a generalization of binomial approximation operators of Popoviciu type, or equivalently moments of associated discrete probability distributions with finite support. These quantities are necessary to check the assumptions of the Korovkin Theorem for approximation operators, or equivalently the Feller Theorem for convergence of the probability distributions. Our method unifies and simplifies computations of well-known special cases. It only requires a few basic facts from Umbral Calculus. We illustrate our method to well-known approximation operators and probability distributions, as well as to some recent q-generalizations of the Bernstein approximation operator introduced by Lewanowicz and Wo´zny, Lupa¸s, and Phillips

A selected survey of umbral calculus

We survey the mathematical literature on umbral calculus (otherwise known as the calculus of finite differences) from its roots in the 19th century (and earlier) as a set of "magic rules" for lowering and raising indices, through its rebirth in the 1970’s as Rota’s school set it on a firm logical foundation using operator methods, to the current state of the art with numerous generalizations and applications. The survey itself is complemented by a fairly complete bibliography (over 500 references) which we expect to update regularly

Proof of a conjecture of Narayana on dominance refinements of the Smirnov two-sample test

We prove the following conjecture of Narayana: there are no dominance refinements of the Smirnov two-sample test if and only if the two sample sizes are relatively prime

Data-driven online monitoring of wind turbines

Condition based maintenance is a modern approach to maintenance which has been successfully used in several industrial sectors. In this paper we present a concrete statistical approach to condition based maintenance for wind turbine by applying ideas from statistical process control. A specific problem in wind turbine maintenance is that failures of a certain part may have causes that originate in other parts a long time ago. This calls for methods that can produce timely warnings by combining sensor data from different sources. Our method improves on existing methods used in wind turbine maintenance by using adaptive alarm thresholds for the monitored parameters that correct for values of other relevant parameters. We illustrate our method with a case study that shows that our method is able to predict upcoming failures much earlier than currently used methods