Characterizations of Lojasiewicz inequalities and applications

The classical Lojasiewicz inequality and its extensions for partial differential equation problems (Simon) and to o-minimal structures (Kurdyka) have a considerable impact on the analysis of gradient-like methods and related problems: minimization methods, complexity theory, asymptotic analysis of dissipative partial differential equations, tame geometry. This paper provides alternative characterizations of this type of inequalities for nonsmooth lower semicontinuous functions defined on a metric or a real Hilbert space. In a metric context, we show that a generalized form of the Lojasiewicz inequality (hereby called the Kurdyka- Lojasiewicz inequality) relates to metric regularity and to the Lipschitz continuity of the sublevel mapping, yielding applications to discrete methods (strong convergence of the proximal algorithm). In a Hilbert setting we further establish that asymptotic properties of the semiflow generated by -∂f are strongly linked to this inequality. This is done by introducing the notion of a piecewise subgradient curve: such curves have uniformly bounded lengths if and only if the Kurdyka- Lojasiewicz inequality is satisfied. Further characterizations in terms of talweg lines -a concept linked to the location of the less steepest points at the level sets of f- and integrability conditions are given. In the convex case these results are significantly reinforced, allowing in particular to establish the asymptotic equivalence of discrete gradient methods and continuous gradient curves. On the other hand, a counterexample of a convex C2 function in R2 is constructed to illustrate the fact that, contrary to our intuition, and unless a specific growth condition is satisfied, convex functions may fail to fulfill the Kurdyka- Lojasiewicz inequality.

Análise geoambiental da região de marília, SP: Suscetibilidade a processos erosivos frente ao histórico de ocupação da área

The study area comprises the western portion of the Marília Plateau, State of São Paulo, southeastern Brazil. The geological substrate encompasses Cretaceous sedimentary rocks of the Bauru Group and local Neocenozoic units like colluvium aprons, fluvial terrace deposits, in situ regoliths, and modern alluvial deposits. In a geomorphologic sense, the study area might be characterized as showing two main pediplanes, viz. P1 Surface (upper) and P2 Surface (lower), both surfaces are presently under dissecation processes. After the 1920's the expansion of the railroad system fostered the removal of the natural vegetation, which in turn was followed by the introduction of coffee, peanut, corn and cotton crops. This intense exploitation was conducted without respect to the soil carrying capacity and its natural susceptibility to erosional processes, including an aggressive form known as voçoroca. As a result, the study shows that the most susceptible material includes colluvium aprons, in situ regoliths, and colluvium-alluvium or alluvium deposits. The most critical situation is in the P2 Surface context, near the bottom of the Marília Plateau scarpment, where surface runoff can be very strong. Another point of active erosion is represented by the exposed walls of gullies and voçorocas, mainly in fluvial reaches subjected to talweg lowering. In a general sense, this study shows current evidence of erosional stability due to the ìntroduction of pastures as a predominant type of land occupation and to a series of erosion control procedures. Among these actions are terrace implantation, construction of small pits for runoff control, natural or induced reforestation by land owners. Despite these efforts, some erosion points remain chiefly in steeper country roads and trail stretches, in areas of concentration of cattle tracks (e.g., near cattle ponds), gullies or voçorocas exposed walls, and badly planned urbanization. The permanence of these erosion points demonstrates the necessity of a continuous monitoring of surface dynamics as well as a rapid and effective intervining measures of erosion and/or silting control.

Recensement cartographique des anciens cours d’eau de l’île de Montréal et tracé des creux et des crêtes

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Carte des creux et crêtes et des voiries de l’île de Montréal

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Carte des creux et crêtes et de l'altimétrie de l’île de Montréal

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Cartographie des anciens cours d’eau, lignes de creux et des bassins versants de l'île de Montréal

Guide et ses annexes à destination des utilisateurs des cartes relatives à la recherche "Cartographie des anciens cours d’eau, lignes de creux et des bassins versants de l'île de Montréal". Aide à la compréhension du contexte, des définitions, de la méthodologie et des hypothèses relatives à la réalisation des documents cartographiques suivants: ■ Carte des anciens cours d'eau de l'île de Montréal, Mahaut V., 2016 ■ Carte des creux et crêtes et des voiries de l'île de Montréal, 1:20.000, Mahaut V., 2016 [http://hdl.handle.net/1866/16313] ■ Carte des creux et crêtes et de l'altimétrie de l'île de Montréal, 1:20.000, Mahaut V.,2016 [http://hdl.handle.net/1866/16312] ■ Recensement cartographique des anciens cours d’eau de l’île de Montréal et tracé des creux et des crêtes, Carte index 1:50.000, Cartes A4, B3, B4, C3, C4, D1, D2, D3, D4, E1, E2, E3, E4, 1:10.000, Mahaut V., 2016 [http://hdl.handle.net/1866/16311]

Recensement cartographique des anciens cours d’eau de l’île de Montréal et tracé des creux et des crêtes

Pdf en couleur au format 119 x 84 cm, échelle 1:10.000 et 1:50.000

Carte des creux et crêtes et des voiries de l’île de Montréal

Pdf en couleur au format 252 x 119 cm, échelle 1:20.000

Carte des creux et crêtes et de l'altimétrie de l’île de Montréal

Pdf en couleur au format 252 x 119 cm, échelle 1:20.000

Cartographie des anciens cours d’eau, lignes de creux et des bassins versants de l'île de Montréal

Guide et ses annexes à destination des utilisateurs des cartes relatives à la recherche "Cartographie des anciens cours d’eau, lignes de creux et des bassins versants de l'île de Montréal". Aide à la compréhension du contexte, des définitions, de la méthodologie et des hypothèses relatives à la réalisation des documents cartographiques suivants: ■ Carte des anciens cours d'eau de l'île de Montréal, Mahaut V., 2016 ■ Carte des creux et crêtes et des voiries de l'île de Montréal, 1:20.000, Mahaut V., 2016 [http://hdl.handle.net/1866/16313] ■ Carte des creux et crêtes et de l'altimétrie de l'île de Montréal, 1:20.000, Mahaut V.,2016 [http://hdl.handle.net/1866/16312] ■ Recensement cartographique des anciens cours d’eau de l’île de Montréal et tracé des creux et des crêtes, Carte index 1:50.000, Cartes A4, B3, B4, C3, C4, D1, D2, D3, D4, E1, E2, E3, E4, 1:10.000, Mahaut V., 2016 [http://hdl.handle.net/1866/16311]