Maximum principle, mean value operators and quasi boundedness in non-euclidean settings

This work deals with some classes of linear second order partial differential operators with non-negative characteristic form and underlying non- Euclidean structures. These structures are determined by families of locally Lipschitz-continuous vector fields in RN, generating metric spaces of Carnot- Carath´eodory type. The Carnot-Carath´eodory metric related to a family {Xj}j=1,...,m is the control distance obtained by minimizing the time needed to go from two points along piecewise trajectories of vector fields. We are mainly interested in the causes in which a Sobolev-type inequality holds with respect to the X-gradient, and/or the X-control distance is Doubling with respect to the Lebesgue measure in RN. This study is divided into three parts (each corresponding to a chapter), and the subject of each one is a class of operators that includes the class of the subsequent one. In the first chapter, after recalling “X-ellipticity” and related concepts introduced by Kogoj and Lanconelli in [KL00], we show a Maximum Principle for linear second order differential operators for which we only assume a Sobolev-type inequality together with a lower terms summability. Adding some crucial hypotheses on measure and on vector fields (Doubling property and Poincar´e inequality), we will be able to obtain some Liouville-type results. This chapter is based on the paper [GL03] by Guti´errez and Lanconelli. In the second chapter we treat some ultraparabolic equations on Lie groups. In this case RN is the support of a Lie group, and moreover we require that vector fields satisfy left invariance. After recalling some results of Cinti [Cin07] about this class of operators and associated potential theory, we prove a scalar convexity for mean-value operators of L-subharmonic functions, where L is our differential operator. In the third chapter we prove a necessary and sufficient condition of regularity, for boundary points, for Dirichlet problem on an open subset of RN related to sub-Laplacian. On a Carnot group we give the essential background for this type of operator, and introduce the notion of “quasi-boundedness”. Then we show the strict relationship between this notion, the fundamental solution of the given operator, and the regularity of the boundary points.

From local to global in quasiconformal structures.

We exhibit a large class of metric spaces whose infinitesimal quasiconformal structure is strong enough to capture the global quasiconformal structure. A sufficient condition for this to happen is described in terms of a Poincaré-type inequality.

Very special divisors on 4-gonal real algebraic curves

International audience

Some Mean-Value Theorems of the Cauchy Type

2000 Mathematics Subject Classification: Primary 26A24, 26D15; Secondary 41A05

Tax evasion, inequality, and progressive taxes : a political economy perspective

The standard approach to tax compliance applies the economics-of-crime methodology pioneered by Becker (1968): in its first application, due to Allingham and Sandmo (1972) it models the behaviour of agents as a decision involving a choice of the extent of their income to report to tax authorities, given a certain institutional environment, represented by parameters such as the probability of detection and penalties in the event the agent is caught. While this basic framework yields important insights on tax compliance behavior, it has some critical limitations. Specifically, it indicates a level of compliance that is significantly below what is observed in the data. This thesis revisits the original framework with a view towards addressing this issue, and examining the political economy implications of tax evasion for progressivity in the tax structure. The approach followed involves building a macroeconomic, dynamic equilibrium model for the purpose of examining these issues, by using a step-wise model building procedure starting with some very simple variations of the basic Allingham and Sandmo construct, which are eventually integrated to a dynamic general equilibrium overlapping generations framework with heterogeneous agents. One of the variations involves incorporating the Allingham and Sandmo construct into a two-period model of a small open economy of the type originally attributed to Fisher (1930). A further variation of this simple construct involves allowing agents to initially decide whether to evade taxes or not. In the event they decide to evade, the agents then have to decide the extent of income or wealth they wish to under-report. We find that the ‘evade or not’ assumption has strikingly different and more realistic implications for the extent of evasion, and demonstrate that it is a more appropriate modeling strategy in the context of macroeconomic models, which are essentially dynamic in nature, and involve consumption smoothing across time and across various states of nature. Specifically, since deciding to undertake tax evasion impacts on the consumption smoothing ability of the agent by creating two states of nature in which the agent is ‘caught’ or ‘not caught’, there is a possibility that their utility under certainty, when they choose not to evade, is higher than the expected utility obtained when they choose to evade. Furthermore, the simple two-period model incorporating an ‘evade or not’ choice can be used to demonstrate some strikingly different political economy implications relative to its Allingham and Sandmo counterpart. In variations of the two models that allow for voting on the tax parameter, we find that agents typically choose to vote for a high degree of progressivity by choosing the highest available tax rate from the menu of choices available to them. There is, however, a small range of inequality levels for which agents in the ‘evade or not’ model vote for a relatively low value of the tax rate. The final steps in the model building procedure involve grafting the two-period models with a political economy choice into a dynamic overlapping generations setting with more general, non-linear tax schedules and a ‘cost-of evasion’ function that is increasing in the extent of evasion. Results based on numerical simulations of these models show further improvement in the model’s ability to match empirically plausible levels of tax evasion. In addition, the differences between the political economy implications of the ‘evade or not’ version of the model and its Allingham and Sandmo counterpart are now very striking; there is now a large range of values of the inequality parameter for which agents in the ‘evade or not’ model vote for a low degree of progressivity. This is because, in the ‘evade or not’ version of the model, low values of the tax rate encourages a large number of agents to choose the ‘not-evade’ option, so that the redistributive mechanism is more ‘efficient’ relative to the situations in which tax rates are high. Some further implications of the models of this thesis relate to whether variations in the level of inequality, and parameters such as the probability of detection and penalties for tax evasion matter for the political economy results. We find that (i) the political economy outcomes for the tax rate are quite insensitive to changes in inequality, and (ii) the voting outcomes change in non-monotonic ways in response to changes in the probability of detection and penalty rates. Specifically, the model suggests that changes in inequality should not matter, although the political outcome for the tax rate for a given level of inequality is conditional on whether there is a large or small or large extent of evasion in the economy. We conclude that further theoretical research into macroeconomic models of tax evasion is required to identify the structural relationships underpinning the link between inequality and redistribution in the presence of tax evasion. The models of this thesis provide a necessary first step in that direction.

Parental leave regulations and the persistence of the male breadwinner model: Using fuzzy-set ideal type analysis to assess gender equality in an enlarged Europe

his paper uses fuzzy-set ideal type analysis to assess the conformity of European leave regulations to four theoretical ideal typical divisions of labour: male breadwinner, caregiver parity, universal breadwinner and universal caregiver. In contrast to the majority of previous studies, the focus of this analysis is on the extent to which leave regulations promote gender equality in the family and the transformation of traditional gender roles. The results of this analysis demonstrate that European countries cluster into five models that only partly coincide with countries’ geographical proximity. Second, none of the countries considered constitutes a universal caregiver model, while the male breadwinner ideal continues to provide the normative reference point for parental leave regulations in a large number of European states. Finally, we witness a growing emphasis at the national and EU levels concerning the universal breadwinner ideal, which leaves gender inequality in unpaid work unproblematized.

Certain problems concerning polynomials and transcendental entire functions of exponential type

Soit $\displaystyle P(z):=\sum_{\nu=0}^na_\nu z^{\nu}$ un polynôme de degré $n$ et $\displaystyle M:=\sup_{|z|=1}|P(z)|.$ Sans aucne restriction suplémentaire, on sait que $|P'(z)|\leq Mn$ pour $|z|\leq 1$ (inégalité de Bernstein). Si nous supposons maintenant que les zéros du polynôme $P$ sont à l'extérieur du cercle $|z|=k,$ quelle amélioration peut-on apporter à l'inégalité de Bernstein? Il est déjà connu [{\bf \ref{Mal1}}] que dans le cas où $k\geq 1$ on a $$(*) \qquad |P'(z)|\leq \frac{n}{1+k}M \qquad (|z|\leq 1),$$ qu'en est-il pour le cas où $k < 1$? Quelle est l'inégalité analogue à $(*)$ pour une fonction entière de type exponentiel $\tau ?$ D'autre part, si on suppose que $P$ a tous ses zéros dans $|z|\geq k \, \, (k\geq 1),$ quelle est l'estimation de $|P'(z)|$ sur le cercle unité, en terme des quatre premiers termes de son développement en série entière autour de l'origine. Cette thèse constitue une contribution à la théorie analytique des polynômes à la lumière de ces questions.

Social Inequality and Social Policy outside the OECD

Almost all Latin American countries are still marked by extreme forms of social inequality – and to an extent, this seems to be the case regardless of national differences in the economic development model or the strength of democracy and the welfare state. Recent research highlights the fact that the heterogeneous labour markets in the region are a key source of inequality. At the same time, there is a strengthening of ‘exclusive’ social policy, which is located at the fault lines of the labour market and is constantly (re-)producing market-mediated disparities. In the last three decades, this type of social policy has even enjoyed democratic legitimacy. These dynamics challenge many of the assumptions guiding social policy and democratic theory, which often attempt to account for the specificities of the region by highlighting the purported flaws of certain policies. We suggest taking a different perspective: social policy in Latin American should not be grasped as a deficient or flawed type of social policy, but as a very successful relation of political domination. ‘Relational social analysis’ locates social policy in the ‘tension zone’ constituted by the requirements of economic reproduction, demands for democratic legitimacy and the relative autonomy of the state. From this vantage point, we will make the relation of domination in question accessible for empirical research. It seems particularly useful for this purpose to examine the recent shifts in the Latin American labour markets, which have undergone numerous reforms. We will examine which mechanisms, institutions and constellations of actors block or activate the potentials of redistribution inherent in such processes of political reform. This will enable us to explore the socio-political field of forces that has been perpetuating the social inequalities in Latin America for generations.

Digital Inequality

Unpicking and understanding if and how the web is linked to inequality means: Recognising that the access divide is not over, Thinking beyond hardware, Thinking beyond demographic variables, Developing a conceptual and theoretical toolkit, Beyond technological determinism, Co-constitution, Intersectionality, Technical capital

Inequality of opportunity in educational achievement in Latin America: evidence from PISA 2006-2009

We assess inequality of opportunity in educational achievement in six Latin American countries, employing two waves of PISA data (2006 and 2009). By means of a non-parametric approach using a decomposable inequality index, GE(0), we rank countries according to their degree of inequality of opportunity. We work with alternative characterizations of types: school type (public or private), gender, parental education, and combinations of those variables. We calculate incremental contributions of each set of circumstances to inequality. We provide rankings of countries based on unconditional inequalities (using conventional indices) and on conditional inequalities (EOp indices), and the two sets of rankings do not always coincide. Inequality of opportunities range from less than 1% to up to 27%, with substantial heterogeneity according to the year, the country, the subject and the specificication of circumstances. Robustness checks based on bootstrap and the use of an alternative index confirm most of the initial results.

Inequality, intergenerational mobility and redistributive policies under endogenous information

This paper presents a dynamic model to study how different levels of information about the root determinants of wealth (luck versus effort) can impact inequality and intergenerational mobility through societal beliefs, individual choices and redistributive policies. To my knowledge, the model presented is the first dynamicmodel in which skills are stochastic and both beliefs and voted redistribution are determined endogenously. The model is able to explain a number of empirical facts. Large empirical evidence shows that the difference in the political support for redistribution appears to reflect differences in the social perceptions regarding the determinants of individual wealth and the underlying sources of income inequality. Moreover the beliefs about the determinants of wealth impact individual choices of effort and therefore the beliefs about the determinants of wealth impact inequality and mobility both through choices of effort and redistributive policies. The model generates multiple equilibria (US versus Europe-type) which may account for the observed features not only in terms of societal beliefs and redistribution but also in terms of perceived versus real mobility and inequality.

Coordination, inclusiveness and wage inequality between median and bottom income workers

What explains cross-national variation in wage inequality? Research in comparative political economy stresses the importance of the welfare state and wage coordination in reducing not only disposable income inequality but also gross earnings inequality. However, the cross-national variation in gross earnings inequality between median and low income workers is at odds with this conventional wisdom: the German coordinated market economy is now more unequal in this type of inequality than the UK, a liberal market economy. To solve this puzzle, I argue that non-inclusive coordination benefits median but not bottom income workers and is as a result associated with higher – rather than lower - wage inequality. I find support for this argument using a large N quantitative analysis of wage inequality in a panel of Western European countries. Results are robust to the inclusion of numerous controls, country fixed effects, and also hold with a sample of OECD countries. Taken together these findings force us to reconsider the relationship between coordination and wage inequality at the bottom of the income distribution.

Landau and Kolmogoroff type polynomial inequalities

Let 0type inequalitiesparallel to f((f))parallel to(2) less than or equal to A parallel to f(m)parallel to + B parallel to f parallel to/ A theta(k) + B mu(k),with certain constants theta(k)e mu(k), which hold for every f is an element of pi(n) (pi(n) denotes the space of real algebraic polynomials of degree not exceeding n).For the particular case j=1 and m=2, we provide a complete characterisation of the positive constants A and B, for which the corresponding Landau type polynomial inequalities parallel to f'parallel to less than or equal toA parallel to f parallel to + B parallel to f parallel to/ A theta(k) + B mu(k)hold. In each case we determine the corresponding extremal polynomials for which equalities are attained.

Landau and Kolmogoroff type polynomial inequalities II

Let 0 < j < m ≤ n. Kolmogoroff type inequalities of the form ∥f(j)∥2 ≤ A∥f(m)∥ 2 + B∥f∥2 which hold for algebraic polynomials of degree n are established. Here the norm is defined by ∫ f2(x)dμ(x), where dμ(x) is any distribution associated with the Jacobi, Laguerre or Bessel orthogonal polynomials. In particular we characterize completely the positive constants A and B, for which the Landau weighted polynomial inequalities ∥f′∥ 2 ≤ A∥f″∥2 + B∥f∥ 2 hold. © Dynamic Publishers, Inc.

Changes and inequality in Latin American families

Includes bibliography