Measurement uncertainty relations

Measurement uncertainty relations are quantitative bounds on the errors in an approximate joint measurement of two observables. They can be seen as a generalization of the error/disturbance tradeoff first discussed heuristically by Heisenberg. Here we prove such relations for the case of two canonically conjugate observables like position and momentum, and establish a close connection with the more familiar preparation uncertainty relations constraining the sharpness of the distributions of the two observables in the same state. Both sets of relations are generalized to means of order $\alpha$ rather than the usual quadratic means, and we show that the optimal constants are the same for preparation and for measurement uncertainty. The constants are determined numerically and compared with some bounds in the literature. In both cases the near-saturation of the inequalities entails that the state (resp. observable) is uniformly close to a minimizing one.Comment: This version 2 contains minor corrections and reformulation

A method for optimizing the ambiguity function concentration

International audienceIn the context of signal analysis and transformation in the time-frequency (TF) domain, controlling the shape of a waveform in this domain is an important issue. Depending on the application, a notion of optimal function may be defined through the properties of the ambiguity function. We present an iterative method for providing such optimal functions under a general concentration constraint of the ambiguity function. At each iteration, it follows a variational approach which maximizes the ambiguity localization via a user-defined weight function F . Under certain assumptions on this latter function, it converges to a waveform which is optimal according to the localization criterion defined by F

Differential entropy and time

We give a detailed analysis of the Gibbs-type entropy notion and its dynamical behavior in case of time-dependent continuous probability distributions of varied origins: related to classical and quantum systems. The purpose-dependent usage of conditional Kullback-Leibler and Gibbs (Shannon) entropies is explained in case of non-equilibrium Smoluchowski processes. A very different temporal behavior of Gibbs and Kullback entropies is confronted. A specific conceptual niche is addressed, where quantum von Neumann, classical Kullback-Leibler and Gibbs entropies can be consistently introduced as information measures for the same physical system. If the dynamics of probability densities is driven by the Schr\"{o}dinger picture wave-packet evolution, Gibbs-type and related Fisher information functionals appear to quantify nontrivial power transfer processes in the mean. This observation is found to extend to classical dissipative processes and supports the view that the Shannon entropy dynamics provides an insight into physically relevant non-equilibrium phenomena, which are inaccessible in terms of the Kullback-Leibler entropy and typically ignored in the literature.Comment: Final, unabridged version; http://www.mdpi.org/entropy/ Dedicated to Professor Rafael Sorkin on his 60th birthda

Between-Group Pigou-Dalton Transfers

This paper introduces the concept of Pigou-Dalton transfers between populations of income receivers. Gini's mean difference and Dagum's Gini index between populations are axiomatically derived in order to gauge the impact of within- and between-group Pigou-Dalton transfers on Dagum's measure. We show its sensitiveness for any given transfer in the sense that inequality-reducing transfers are captured when transfers occur from higher-income donors to lower-income recipients, which belong to two distinct populations. Accordingly, we point out the implications of between-group transfers on: the ordering of multivariate majorization, the multivariate stochastic dominance in the sense of multivariate Lorenz ordering and zonotope inclusions, the Gini decomposition, and on the use of the generalized entropy index.Between-group Gini ; Between-group Transfers ; Entropy ; Pigou-Dalton ; Within-group Transfers

Consumer demand for variety: intertemporal effects of consumption, product switching and pricing policies

The concept of diminishing marginal utility is a cornerstone of economic theory. The consumption of a good typically creates satiation that diminishes the marginal utility of consuming more. Temporal satiation induces consumers to increase their stimulation level by seeking variety and therefore substitute towards other goods (substitutability across time) or other differentiated versions (products) of the good (substitutability across products). The literature on variety-seeking has developed along two strands, each focusing on only one type of substitutability. I specify a demand model that attempts to link these two strands of the literature. This issue is economically relevant because both types of substitutability are important for retailers and manufacturers in designing intertemporal price discrimination strategies. The consumer demand model specified allows consumption to have an enduring effect and the marginal utility of the different products to vary over consumption occasions. Consumers are assumed to make rational purchase decisions by taking into account, not only current and future satiation levels, but also prices and product choices. I then use the model to evaluate the demand implications of a major pricing policy change from hi-low pricing to an everyday low pricing strategy. I find evidence that consumption has a lasting effect on utility that induces substitutability across time and that the median consumer has a taste for variety in her product decisions. Consumers are found to be forward-looking with respect to the duration since the last purchase, to price expectations and product choices. Pricing policy simulations suggest that retailers may increase revenue by reducing the variance of prices, but that lowering the everyday level of prices may be unprofitable.Variety seeking; Intertemporal effects of consumption; product switching; Pricing; Dynamic demand;

Fourier-muunnoksen epävarmuusperiaatteista

This thesis surveys the vast landscape of uncertainty principles of the Fourier transform. The research of these uncertainty principles began in the mid 1920’s following a seminal lecture by Wiener, where he first gave the remark that condenses the idea of uncertainty principles: "A function and its Fourier transform cannot be simultaneously arbitrarily small". In this thesis we examine some of the most remarkable classical results where different interpretations of smallness is applied. Also more modern results and links to active fields of research are presented.We make great effort to give an extensive list of references to build a good broad understanding of the subject matter.Chapter 2 gives the reader a sufficient basic theory to understand the contents of this thesis. First we talk about Hilbert spaces and the Fourier transform. Since they are very central concepts in this thesis, we try to make sure that the reader can get a proper understanding of these subjects from our description of them. Next, we study Sobolev spaces and especially the regularity properties of Sobolev functions. After briefly looking at tempered distributions we conclude the chapter by presenting the most famous of all uncertainty principles, Heisenberg’s uncertainty principle.In chapter 3 we examine how the rate of decay of a function affects the rate of decay of its Fourier transform. This is the most historically significant form of the uncertainty principle and therefore many classical results are presented, most importantly the ones by Hardy and Beurling. In 2012 Hedenmalm gave a beautiful new proof to the result of Beurling. We present the proof after which we briefly talk about the Gaussian function and how it acts as the extremal case of many of the mentioned results.In chapter 4 we study how the support of a function affects the support and regularity of its Fourier transform. The magnificent result by Benedicks and the results following it work as the focal point of this chapter but we also briefly talk about the Gap problem, a classical problem with recent developments.Chapter 5 links density based uncertainty principle to Fourier quasicrystals, a very active field of re-search. We follow the unpublished work of Kulikov-Nazarov-Sodin where first an uncertainty principle is given, after which a formula for generating Fourier quasicrystals, where a density condition from the uncertainty principle is used, is proved. We end by comparing this formula to other recent formulas generating quasicrystals

On the impact of trade on industrial structures : The role of entry cost heterogeneity

entrepreneurship, trade liberalization, externality, heterogeneity, stability

The Affine Uncertainty Principle, Associated Frames and Applications in Signal Processing

Uncertainty relations play a prominent role in signal processing, stating that a signal can not be simultaneously concentrated in the two related domains of the corresponding phase space. In particular, a new uncertainty principle for the affine group, which is directly related to the wavelet transform has lead to a new minimizing waveform. In this thesis, a frame construction is proposed which leads to approximately tight frames based on this minimizing waveform. Frame properties such as the diagonality of the frame operator as well as lower and upper frame bounds are analyzed. Additionally, three applications of such frame constructions are introduced: inpainting of missing audio data, detection of neuronal spikes in extracellular recorded data and peak detection in MALDI imaging data