Differentiability of fractal curves

While self-similar sets have no tangents at any single point, self-affine curves can be smooth. We consider plane self-affine curves without double points and with two pieces. There is an open subset of parameter space for which the curve is differentiable at all points except for a countable set. For a parameter set of codimension one, the curve is continuously differentiable. However, there are no twice differentiable self-affine curves in the plane, except for parabolic arcs

Optimal Capacity in the Banking Sector and Economic Growth.

The paper investigates, from the welfare and growth point of view, the determination of the optimal capacity of the banking system. For that purpose, we consider an overlapping generation model with endogenous growth. There is horizontal differentiation and imperfect competition in the banking sector. Macroeconomic shocks affect the return on capital and, together with the expectations of depositors, condition the stability of the banking sector. We specify to what extent deposit insurance may reduce instability and increase the number of deposits, welfare and growth. We also characterize the conditions under which excess banking capacities may appear and how their reduction may improve welfare.Deposit insurance ; imperfect ; competition ; growth, banking.

Boundaries of Disk-like Self-affine Tiles

Let $T:= T(A, {\mathcal D})$ be a disk-like self-affine tile generated by an integral expanding matrix $A$ and a consecutive collinear digit set ${\mathcal D}$, and let $f(x)=x^{2}+px+q$ be the characteristic polynomial of $A$. In the paper, we identify the boundary $\partial T$ with a sofic system by constructing a neighbor graph and derive equivalent conditions for the pair $(A,{\mathcal D})$ to be a number system. Moreover, by using the graph-directed construction and a device of pseudo-norm $\omega$, we find the generalized Hausdorff dimension $\dim_H^{\omega} (\partial T)=2\log \rho(M)/\log |q|$ where $\rho(M)$ is the spectral radius of certain contact matrix $M$. Especially, when $A$ is a similarity, we obtain the standard Hausdorff dimension $\dim_H (\partial T)=2\log \rho/\log |q|$ where $\rho$ is the largest positive zero of the cubic polynomial $x^{3}-(|p|-1)x^{2}-(|q|-|p|)x-|q|$, which is simpler than the known result.Comment: 26 pages, 11 figure

Canonical Representatives of Morphic Permutations

An infinite permutation can be defined as a linear ordering of the set of natural numbers. In particular, an infinite permutation can be constructed with an aperiodic infinite word over $\{0,\ldots,q-1\}$ as the lexicographic order of the shifts of the word. In this paper, we discuss the question if an infinite permutation defined this way admits a canonical representative, that is, can be defined by a sequence of numbers from [0, 1], such that the frequency of its elements in any interval is equal to the length of that interval. We show that a canonical representative exists if and only if the word is uniquely ergodic, and that is why we use the term ergodic permutations. We also discuss ways to construct the canonical representative of a permutation defined by a morphic word and generalize the construction of Makarov, 2009, for the Thue-Morse permutation to a wider class of infinite words.Comment: Springer. WORDS 2015, Sep 2015, Kiel, Germany. Combinatorics on Words: 10th International Conference. arXiv admin note: text overlap with arXiv:1503.0618

Curvature-direction measures of self-similar sets

We obtain fractal Lipschitz-Killing curvature-direction measures for a large class of self-similar sets F in R^d. Such measures jointly describe the distribution of normal vectors and localize curvature by analogues of the higher order mean curvatures of differentiable submanifolds. They decouple as independent products of the unit Hausdorff measure on F and a self-similar fibre measure on the sphere, which can be computed by an integral formula. The corresponding local density approach uses an ergodic dynamical system formed by extending the code space shift by a subgroup of the orthogonal group. We then give a remarkably simple proof for the resulting measure version under minimal assumptions.Comment: 17 pages, 2 figures. Update for author's name chang

MASCOTTE: Model for AnalySing and foreCasting shOrT TErm developments

MASCOTTE is the new version of the Banque de France's macro-econometric forecasting model. Following the last rebasing of National Accounts (currently at 1995 price), the previous version of the model was simplified, re-specified and re-estimated. The model is essentially used for making macro-economic projections of the French economy over a two-to-three year horizon, which requires an accounting framework as close as possible to the French National Accounts. The main agents are companies, households, general government and the rest of the world. The new version now includes a supply block derived from the explicit optimisation behaviour of companies using a Cobb-Douglas technology under imperfect competition, and a new Wage Setting schedule. Full homogeneity of the nominal side of the model ensures the independence between the nominal equilibrium and the real equilibrium, the latter being only determined in the long run by relative prices. Furthermore, as regards the specification of equations, special attention was paid to the consequences of changes in short-term interest rates.Macro-economic model ; Applied econometrics ; Forecasting ; France

The Sustainability Game::AI Technology as an Intervention for Public Understanding of Cooperative Investment

Cooperative behaviour is a fundamental strategy for survival; it positively affects economies, social relationships, and makes larger societal structures possible. People vary, however, in their willingness to engage in cooperative behaviour in a particular context. Here we examine whether AI can be effectively used to to alter individuals' implicit understanding of cooperative dynamics, and hence increase cooperation and participation in public goods projects. We developed an intervention---the Sustainability Game (SG)---to allow players to experience the consequences of individual investment strategies on a sustainable society. %, when personal well being, communal space, and resources limitations are taken into consideration. Results show that the intervention significantly increases individuals' cooperative behaviour in partially anonymised public goods contexts, but enhances competition one-on-one. This indicates our intervention does improve transparency of the systemic consequences of individual cooperative behaviour

On the "Mandelbrot set" for a pair of linear maps and complex Bernoulli convolutions

We consider the "Mandelbrot set" $M$ for pairs of complex linear maps, introduced by Barnsley and Harrington in 1985 and studied by Bousch, Bandt and others. It is defined as the set of parameters $\lambda$ in the unit disk such that the attractor $A_\lambda$ of the IFS $\{\lambda z-1, \lambda z+1\}$ is connected. We show that a non-trivial portion of $M$ near the imaginary axis is contained in the closure of its interior (it is conjectured that all non-real points of $M$ are in the closure of the set of interior points of $M$). Next we turn to the attractors $A_\lambda$ themselves and to natural measures $\nu_\lambda$ supported on them. These measures are the complex analogs of much-studied infinite Bernoulli convolutions. Extending the results of Erd\"os and Garsia, we demonstrate how certain classes of complex algebraic integers give rise to singular and absolutely continuous measures $\nu_\lambda$. Next we investigate the Hausdorff dimension and measure of $A_\lambda$, for $\lambda$ in the set $M$, for Lebesgue-a.e. $\lambda$. We also obtain partial results on the absolute continuity of $\nu_\lambda$ for a.e. $\lambda$ of modulus greater than $\sqrt{1/2}$.Comment: 22 pages, 5 figure