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## q-Fractional calculus for Rubin’s q-difference operator

### Advances in Difference Equations (2013-09-23) 2013: 1-27 , September 23, 2013

In this paper we introduce a fractional *q*-integral operator and derivative as a generalization of Rubin’s *q*-difference operator. We also reformulate the definition of the
-Fourier transform and the *q*-analogue of the Fourier multiplier introduced by Rubin in (J. Math. Anal. Appl. 212(2):571-582, 1997; Proc. Am. Math. Soc. 135(3):777-785, 2007). As applications, we give summation formulas for
finite series, we also use the
-Fourier transform and Hahn *q*-Laplace transform to solve a fractional *q*-diffusion equation.

*MSC:*39A12, 33D15, 42A38, 35R11.

## Learning by Doing and Technology Sharing in Asymmetric Duopolies

### Advances in Dynamic Games and Applications (1994-01-01) 1: 395-418 , January 01, 1994

The existence of an experience curve for the firm has theoretical foundation in Arrow’s pioneering work (Arrow, 1962). The study of the learning-curve effect was initiated, however, almost thirty years earlier by Wright (1936). It was further developed by Alchian (1959), Hirschleifer (1962), Hirschmann (1964), and by an important work by the Boston Consulting Group (1972). The learning-curve phenomenon appears to be related, in the first place, to the introduction of new products. It has been observed that a doubling in accumulated production of a new product can result in a decline of unit costs by a factor of 10 to 50 percent of their initial level (Teng and Thompson, 1983). Empirical documentation concerning the learning curve can be found in several papers that appeared in the period from the early sixties to more recent years (e.g., Lundberg, 1961; Alchian, 1963; Hollander 1965; Zimmerman, 1982; Lieberman, 1984; Alder and Clark, 1991). This literature considers cost reduction as an outcome of experience and cumulative output is the variable commonly chosen to represent that experience. The revival of interest in the study of the learning curve can be attributed to some important recent contributions that suggest the existence of a link between learning by firms and incremental innovations in both production processes and products (see in particular Rosenberg, 1976; Nelson and Winter, 1982).

## Universality of Coproducts in Categories of Lax Algebras

### Applied Categorical Structures (2006-06-01) 14: 243-249 , June 01, 2006

Categories of lax $(T,V\,)$ -algebras are shown to have pullback-stable coproducts if $T$ preserves inverse images. The general result not only gives a common proof of this property in many topological categories but also shows that important topological categories, like the category of uniform spaces, are not presentable as a category of lax $(T,V\,)$ -algebras, with $T$ preserving inverse images. Moreover, we show that any such category of $(T,V\,)$ -algebras has a concrete, coproduct preserving functor into the category of topological spaces.

## 25 Years of Fourier Integral Operators

### Mathematics Past and Present Fourier Integral Operators (1994-01-01): 1-21 , January 01, 1994

I am grateful to Lars Hörmander and to my collaborator, Hans Duistermaat, for allowing us to reprint the two classical articles on Fourier integral operators: F.I.O., I (Acta Mathematica, Vol. 127, pp. 77–183) and II (Acta Mathematica, Vol. 128, pp. 184–269). Much of the material in these articles is now available in book form. In particular many would regard the *definitive* treatment of Fourier integral operators as being Volumes Ill and IV of Hörmander’s book [26]. Nevertheless I feel there are two very good reasons for going to the trouble of reprinting these articles as an ensemble. One is that they provide a relatively concise introduction to this subject. For someone who is trying to get a sense of the lay of the land they are probably a better place to start than Hörmander’s book or any other standard text book. Though these two papers (particularly F.I.O. I) would probably pose problems for someone who had never seen pseudodifferential operators before, they are essentially self-contained, and, considering the fact that they cover a lot of ground very quickly, are quite readable as well.

## Sparse spectral discretizations for some parabolic PDEs

### Analysis and Mathematical Physics (2012-12-01) 2: 439-446 , December 01, 2012

A method how to look for sparse spectral discretizations of parabolic PDEs is proposed. Due to sparse structure such discretizations have lower computational costs.

## Some projection-like methods for the generalized Nash equilibria

### Computational Optimization and Applications (2010-01-01) 45: 89-109 , January 01, 2010

A generalized Nash game is an *m*-person noncooperative game in which each player’s strategy depends on the rivals’ strategies. Based on a quasi-variational inequality formulation for the generalized Nash game, we present two projection-like methods for solving the generalized Nash equilibria in this paper. It is shown that under certain assumptions, these methods are globally convergent. Preliminary computational experience is also reported.

## Variance Groups and the Structure of Mixed Polytopes

### Rigidity and Symmetry (2014-01-01) 70: 97-116 , January 01, 2014

The natural mixing construction for abstract polytopes provides a way to build a minimal common cover of two regular or chiral polytopes. With the help of the chirality group of a polytope, it is often possible to determine when the mix of two chiral polytopes is still chiral. By generalizing the chirality group to a whole family of variance groups, we can explicitly describe the structure of the mix of two polytopes. We are also able to determine when the mix of two polytopes is invariant under other external symmetries, such as duality and Petrie duality.

## The MDL Principle

### Information and Complexity in Statistical Modeling (2007-01-01): 97-102 , January 01, 2007

We have mentioned the *MDL* principle on several occasions somewhat loosely as the principle that calls for finding the model and model class with which the data together with the model and model class, respectively, can be encoded with the shortest code length. Actually to apply the principle we must distinguish between two types of models — those for data compression and others for general statistical purposes such as prediction. In data compression, we apply the models to the same data from which the models are determined. Hence these models need not have any predictive power; and, in fact, to get the shortest code length we do not even need to fit models in the class considered, say,
$$
\mathcal{M}_\gamma
$$
_{γ}. This is because the universal *NML* model gives a code length, which we called the stochastic complexity and which we consider to be the shortest for all intents and purposes.