Yes / Age estimation methods are often based on the age-related morphological changes of the auricular surface and the pubic bone. In this study, a mathematical approach to quantify these changes has been tested analyzing the curvature variation on 3D models from CT and laser scans. The sample consisted of the 24 Suchey–Brooks (SB) pubic bone casts, 19 auricular surfaces from the Buckberry and Chamberlain (BC) “recording kit” and 98 pelvic bones from the Terry Collection (Smithsonian Institution). Strong and moderate correlations between phases and curvature were found in SB casts (ρ 0.60–0.93) and BC “recording kit” (ρ 0.47–0.75), moderate and weak correlations in the Terry Collection bones (pubic bones: ρ 0.29–0.51, auricular surfaces: ρ 0.33–0.50) but associated with large individual variability and overlap of curvature values between adjacent decades. The new procedure, requiring no expert judgment from the operator, achieved similar correlations that can be found in the classic methods.

Yes / Recent studies have reported that quantifying symphyseal and auricular surfaces curvature changes on 3D models acquired by laser scanners have a potential for age estimation. However, no tests have been carried out to evaluate the repeatability of the results between different laser scanners. 3D models of the two pelvic joints were generated using three laser scanners (Custom, Faro, Minolta). The surface curvature, the surface area and the distance between co-registered meshes were investigated. Close results were found for surface areas (differences between 0.3% and 2.4%) and for distance deviations (average < 20 μm, SD < 200 μm). The curvature values were found to be systematically biased between different laser scanners, but still showing similar trends with increasing phases / scores. Applying a smoothing factor to the 3D models, it was possible to separate anatomy from the measurement error of each instrument, so that similar curvature values could be obtained (p < 0.05) independent of the specific laser scanner. / The full text was made available at the end of the publisher's embargo: 31st March 2016

Yes / Humans manipulate objects chiefly within their lower visual field, a consequence of upright posture and the anatomical position of hands and arms.This study tested the hypothesis of enhanced sensitivity to a range of stimuli within the lower visual field. Following current models of hierarchical processing within the ventral steam, discrimination sensitivity was measured for orientation, curvature, shape (radial frequency patterns), and faces at various para-central locations (horizontal, vertical, and main diagonal meridians) and eccentricities (5° and 10°). Peripheral sensitivity was isotropic for orientation and curvature. By contrast, observers were significantly better at discriminating shapes throughout the lower visual field compared to elsewhere. For faces, however, peak sensitivity was found in the left visual field, corresponding to the right hemispheric localization of human face processing. Presenting head outlines without any internal features (e.g., eyes, mouth) recovered the lower visual field advantage found for simple shapes. A lower visual field preference for the shape of an object, which is absent for more localized information (orientation and curvature) but also for more complex objects (faces), is inconsistent with a strictly feed-forward model and poses a challenge for multistage models of object perception. The distinct lower visual field preference for contour shapes is, however, consistent with an asymmetry at intermediate stages of visual processing, which may play a key role in representing object characteristics that are particularly relevant to visually guided actions.

The interaction between the study of geometric and analytic aspects of Riemannian manifolds and that of graphs is a very amazing subject. The study of synthetic curvature notions on graphs adds new contributions to this topic. In this thesis, we mainly study two kinds of synthetic curvature notions: the Ollivier-Ricci cuvature on locally finite graphs and the combinatorial curvature on infinite semiplanar graphs. In the first part, we study the Ollivier-Ricci curvature. As known in Riemannian geometry, a lower Ricci curvature bound prevents geodesics from diverging too fast on average. We translate this Riemannian idea into a combinatorial setting using the Olliver-Ricci curvature notion. Note that on a graph, the analogue of geodesics starting in different directions, but eventually approaching each other again, would be a triangle. We derive lower and upper Ollivier-Ricci curvature bounds on graphs in terms of number of triangles, which is sharp for instance for complete graphs. We then describe the relation between Ollivier-Ricci curvature and the local clustering coefficient, which is an important concept in network analysis introduced by Watts-Strogatz. Furthermore, positive lower boundedness of Ollivier-Ricci curvature for neighboring vertices imply the existence of at least one triangle. It turns out that the existence of triangles can also improve Lin-Yau\'s curvature dimension inequality on graphs and then produce an implication from Ollivier-Ricci curvature lower boundedness to the curvature dimension inequality. The existence of triangles prevents a graph from being bipartite. A finite graph is bipartite if and only if its largest eigenvalue equals 2. Therefore it is natural that Ollivier-Ricci curvature is closely related to the largest eigenvalue estimates. We combine Ollivier-Ricci curvature notion with the neighborhood graph method developed by Bauer-Jost to study the spectrum estimates of a finite graph. We can always obtain nontrivial estimates on a non-bipartite graph even if its curvature is nonpositive. This answers one of Ollivier\'s open problem in the finite graph setting. In the second part of this thesis, we study systematically infinite semiplanar graphs with nonnegative combinatorial curvature. Unlike the previous Gauss-Bonnet formula approach, we explore an Alexandrov approach based on the observation that the nonnegative combinatorial curvature on a semiplanar graph is equivalent to nonnegative Alexandrov curvature on the surface obtained by replacing each face by a regular polygon of side length one with the same facial degree and gluing the polygons along common edges. Applying Cheeger-Gromoll splitting theorem on the surface, we give a metric classification of infinite semiplanar graphs with nonnegative curvature. We also construct the graphs embedded into the projective plane minus one point. Those constructions answer a question proposed by Chen. We further prove the volume doubling property and Poincare inequality which make the running of Nash-Moser iteration possible. We in particular explore the volume growth behavior on Archimedean tilings on a plane and prove that they satisfy a weak version of relative volume comparison with constant 1. With the above two basic inequalities in hand, we study the geometric function theory of infinite semiplanar graphs with nonnegative curvature. We obtain the Liouville type theorem for positive harmonic functions, the parabolicity. We also prove a dimension estimate for polynomial growth harmonic functions, which is an extension of the solution of Colding-Minicozzi of a conjecture of Yau in Riemannian geometry.

The interaction between the study of geometric and analytic aspects of Riemannian manifolds and that of graphs is a very amazing subject. The study of synthetic curvature notions on graphs adds new contributions to this topic. In this thesis, we mainly study two kinds of synthetic curvature notions: the Ollivier-Ricci cuvature on locally finite graphs and the combinatorial curvature on infinite semiplanar graphs. In the first part, we study the Ollivier-Ricci curvature. As known in Riemannian geometry, a lower Ricci curvature bound prevents geodesics from diverging too fast on average. We translate this Riemannian idea into a combinatorial setting using the Olliver-Ricci curvature notion. Note that on a graph, the analogue of geodesics starting in different directions, but eventually approaching each other again, would be a triangle. We derive lower and upper Ollivier-Ricci curvature bounds on graphs in terms of number of triangles, which is sharp for instance for complete graphs. We then describe the relation between Ollivier-Ricci curvature and the local clustering coefficient, which is an important concept in network analysis introduced by Watts-Strogatz. Furthermore, positive lower boundedness of Ollivier-Ricci curvature for neighboring vertices imply the existence of at least one triangle. It turns out that the existence of triangles can also improve Lin-Yau\''s curvature dimension inequality on graphs and then produce an implication from Ollivier-Ricci curvature lower boundedness to the curvature dimension inequality. The existence of triangles prevents a graph from being bipartite. A finite graph is bipartite if and only if its largest eigenvalue equals 2. Therefore it is natural that Ollivier-Ricci curvature is closely related to the largest eigenvalue estimates. We combine Ollivier-Ricci curvature notion with the neighborhood graph method developed by Bauer-Jost to study the spectrum estimates of a finite graph. We can always obtain nontrivial estimates on a non-bipartite graph even if its curvature is nonpositive. This answers one of Ollivier\''s open problem in the finite graph setting. In the second part of this thesis, we study systematically infinite semiplanar graphs with nonnegative combinatorial curvature. Unlike the previous Gauss-Bonnet formula approach, we explore an Alexandrov approach based on the observation that the nonnegative combinatorial curvature on a semiplanar graph is equivalent to nonnegative Alexandrov curvature on the surface obtained by replacing each face by a regular polygon of side length one with the same facial degree and gluing the polygons along common edges. Applying Cheeger-Gromoll splitting theorem on the surface, we give a metric classification of infinite semiplanar graphs with nonnegative curvature. We also construct the graphs embedded into the projective plane minus one point. Those constructions answer a question proposed by Chen. We further prove the volume doubling property and Poincare inequality which make the running of Nash-Moser iteration possible. We in particular explore the volume growth behavior on Archimedean tilings on a plane and prove that they satisfy a weak version of relative volume comparison with constant 1. With the above two basic inequalities in hand, we study the geometric function theory of infinite semiplanar graphs with nonnegative curvature. We obtain the Liouville type theorem for positive harmonic functions, the parabolicity. We also prove a dimension estimate for polynomial growth harmonic functions, which is an extension of the solution of Colding-Minicozzi of a conjecture of Yau in Riemannian geometry.

Tesis por compendio / [ES] El desarrollo tecnológico en la creación de trayectorias que permitan navegación libre de colisiones de Vehículos Autónomos (AVs) ha sido un objetivo constante de estudio debido a su fuerte interés científico y tecnológico en las últimas tres décadas. Las diferentes clases de AVs, ya sean Vehículos Aéreos no Tripulados (UAVs), Vehículos Terrestres no Tripulados (UGVs) o Vehículos Submarinos no Tripulados (UUVs), fomentan el desarrollo e implementación de trayectorias en el espacio tridimensional (3D). Un grupo especial de tecnología UAV está caracterizado por su ala fija, lo cual destaca características particulares en los AVs, debido a las restricciones no-holonómicas (un sistema que se describe mediante un conjunto de parámetros sujetos a restricciones diferenciales que no permiten que un vehículo se mueva de forma instantánea en cualquier dirección). En este sentido, las trayectorias navegables para estos UAVs no deben ser construidas como un conjunto de líneas rectas y círculos como en la gran mayoría de planificadores basados en primitivas, ya que no se garantiza una continuidad en su curvatura. Por lo tanto, las trayectorias construidas para esta rama tecnológica deben ser resueltas considerando las diferentes restricciones de maniobrabilidad del UAV, además de criterios de continuidad de curvas (el problema de continuidad se refiere principalmente a la continuidad geométrica, en términos de continuidad tangencial o de curvatura), suavidad en las curvas (una curva es suave si sus derivadas son continuas en el intervalo definido) y la seguridad en el vuelo (el control de seguridad garantiza que una trayectoria suave esté suficientemente lejos de los obstáculos). Finalmente, la cinemática del movimiento de los vehículos es otro factor que debe ser considerado mientras se suavizan las trayectorias. El presente trabajo está enfocado en la creación de trayectorias navegables en el espacio 3D, para UAVs de características no-holonómicas. La principal dificultad al solventar este problema se debe a la movilidad de esta clase de UAVs, pues se ven obligados a avanzar sin la posibilidad de detenerse a través de trayectorias 3D, realizando curvas con curvaturas limitadas (una máxima capacidad de giro a una velocidad definida). En consecuencia, se han desarrollado las herramientas necesarias para proporcionar una completa caracterización de trayectorias óptimas (con un radio de giro limitado) para UAVs que se mueven en el espacio 3D a una velocidad constante. Esta tesis se centra en la generación de caminos con trayectorias navegables en el espacio Euclídeo 3D, que contenga curvas con curvatura continua, considerando de esta manera las restricciones cinemáticas de los UAVs. Por tal motivo el objetivo principal es el desarrollo de la matemática necesaria para definir curvas clotoides en el espacio tridimensional, de modo que puedan ser utilizadas como primitivas en la generación de trayectorias. Finalmente, culminado el desarrollo de esta herramienta básica, y en función de los obstáculos del entorno, se puede completar una planificación y replanificación activa de movimientos. Para complementar la investigación, la verificación de las herramientas de planificación de trayectorias y del sistema, se han realizado simulaciones con la ayuda del entorno de desarrollo integrado (IDE) Matlab. De la misma forma, se ha preparado una plataforma de simulación de vuelo, tomando las virtudes del simulador de vuelo FlightGear 2018, y el modelo dinámico del avión de ala fija con restricciones no-holonómicas (Kadett 2400 ). En cuanto a la generación de trayectorias 3D, se han desarrollado simulaciones off-line, donde las acciones de control que debe ejecutar el avión para que siga la trayectoria calculada son definidas por: acceleración, brusquedad de curvatura y brusquedad de torsión. Por último, el enfoque de revisión bibliográfica presente en este documento se ha centrado en trabajos realizados que buscan cumplir con las tareas de planificación. / [CA] El desenvolupament tecnològic en la creació de trajectòries que permeten navegació lliure de col·lisions de Vehicles Autònoms (AVs) ha estat un objectiu constant d'estudi a causa del seu fort interés científic i tecnològic en les últimes tres dècades. Les diferents classes d'AVs, ja siguen Vehicles Aeris no Tripulats (UAVs), Vehicles Terrestres no Tripulats (UGVs) o Vehicles Submarins no Tripulats (UUVs), fomenten el desenvolupament i la implementació de trajectòries a l'espai tridimensional (3D). Un grup especial de tecnologia UAV està caracteritzat per la seua ala fixa, cosa que destaca característiques particulars en els AVs, a causa de les restriccions no-holonòmiques (un sistema que es descriu mitjançant un conjunt de paràmetres subjectes a restriccions diferencials que no permeten que un vehicle es menege de forma instantània en qualsevol direcció). En aquest sentit, les trajectòries navegables per a aquests UAVs no han de ser construïdes com un conjunt de línies rectes i cercles com a la gran majoria de planificadors basats en primitives, ja que no es garanteix una continuïtat en la seua curvatura. Per tant, les trajectòries construïdes per a aquesta branca tecnològica han de ser resoltes considerant les diferents restriccions de maniobrabilitat de l'UAV, a més de criteris de continuïtat de corbes (el problema de continuïtat es refereix principalment a la continuïtat geomètrica, en termes de continuïtat tangencial o de curvatura), suavitat a les corbes (una corba és suau si les seves derivades són contínues en l'interval definit) i la seguretat en el vol (el control de seguretat garanteix que una trajectòria suau estiga prou lluny dels obstacles). Finalment, la cinemàtica del moviment dels vehicles és un altre factor que cal considerar mentre se suavitzen les trajectòries. Aquest treball està enfocat a la creació de trajectòries navegables a l'espai 3D, per a UAVs de característiques no-holonòmiques. La principal dificultat en solucionar aquest problema es deu a la mobilitat d'aquesta classe de UAVs, ja que es veuen obligats a avançar sense la possibilitat d'aturarse a través de trajectòries 3D, fent corbes amb curvatures limitades (una màxima capacitat de gir a una velocitat definida). En conseqüència, s'han desenvolupat les ferramentes necessàries per proporcionar una completa caracterització de trajectòries òptimes (amb un radi de gir limitat) per a UAVs que es mouen al pla 3D a una velocitat constant. Aquesta tesi se centra en la generació de camins amb trajectòries navegables a l'espai Euclidià 3D, que continguen corbes amb curvatura contínua, considerant així les restriccions cinemàtiques dels UAVs. Per aquest motiu, l'objectiu principal és el desenvolupament de la matemàtica necessària per definir corbes clotoides a l'espai tridimensional, de manera que puguen ser utilitzades com a primitives en la generació de trajectòries. Finalment, culminat el desenvolupament d'aquesta ferramenta bàsica, i en funció dels obstacles de l'entorn, es pot completar una planificació i una replanificació activa de moviments. Per complementar la investigació, la verificació de les ferramentes de planificació de trajectòries i del sistema, s'han fet simulacions amb l'ajuda de l'entorn de desenvolupament integrat (IDE) Matlab. De la mateixa manera, s'ha preparat una plataforma de simulació de vol, prenent les virtuts del simulador de vol FlightGear 2018 i el model dinàmic de l'avió d'ala fixa amb restriccions no-holonòmiques (Kadett 2400). Pel que fa a la generació de trajectòries 3D, s'han desenvolupat simulacions off-line, on les accions de control que ha d'executar l'avió perquè seguisca la trajectòria calculada són definides per: acceleració, brusquedat de curvatura i brusquedat de torsió. Finalment, l'enfocament de revisió bibliogràfica present en aquest document s'ha centrat en treballs realitzats que busquen complir les tasques de planificació de trajectòria, planificació de moviment i construcció de corbes suaus per a AVs. / [EN] The technological development in the creation of trajectories that allow collision-free navigation of Autonomous Vehicles (AVs) has been a continuous target of study due to its strong scientific and technological interest in the last three decades. Different classes of AVs, whether, Unmanned Aerial Vehicles (UAVs), Unmanned Ground Vehicles (UGVs) or Unmanned Underwater Vehicles (UUVs), encourage the development and implementation of paths in three-dimensional (3D) space. A special group of UAV technology is characterized by its fixed wing, which emphasizes particular characteristics in UAVs, due to non-holonomic constraints (a system that is described by a set of parameters subject to differential constraints that do not allow a vehicle to move instantaneously in any direction). In this sense, navigable paths for these UAVs should not be built as a set of straight lines and circles as in the vast majority of primitive-based planners, since no continuity in their curvature is guaranteed. Therefore, the paths built for this technology branch must be solved considering the different maneuverability constraints of the UAV, in addition to curve continuity criteria (the continuity problem refers mainly to geometric continuity, in terms of tangential or curvature continuity), curve smoothness (a curve is smooth if its derivatives are continuous in the defined interval) and flight safety (safety control ensures that a smooth path is sufficiently far away from obstacles). Finally, the kinematics of vehicle motion is another factor to be considered while smoothing paths. This thesis work is focused on the creation of navigable paths in 3D space for UAVs with non-holonomic characteristics. The main difficulty in solving this problem is due to the mobility of this kind of UAVs, since they are forced to move without the possibility of stopping through 3D paths, performing curves with limited curvatures (a maximum turning capacity at a defined speed). Consequently, the needed tools have been developed to provide a complete characterization of optimal paths (with a limited turning radius) for UAVs moving in the 3D plane at a constant velocity. This thesis focuses on the generation of paths with navigable trajectories in 3D Euclidean space, containing curves with continuous curvature, thus considering the kinematic constraints of UAVs. Therefore, the main aim is the development of the necessary mathematics to define clothoid curves in the three-dimensional space, so that they can be used as primitives in the generation of paths. Finally, once the development of this basic tool has been completed, and depending on the obstacles in the environment, an active planning and replanning of movements can be completed. To complement the research, the verification of the path planning tools and the system, simulations have been performed with the help of the integrated development environment (IDE) Matlab. In the same way, a flight simulation platform has been prepared, taking the virtues of the FlightGear 2018 flight simulator, and the dynamic model of the fixed-wing aircraft with non-holonomic constraints (Kadett 2400 ). Regarding the generation of 3D paths, off-line simulations have been developed, where the control actions to be executed by the aircraft to follow the calculated path are defined by: acceleration, curvature sharpness and torsion sharpness. Finally, the literature review approach presented in this document has focused on works that address the tasks of path planning, motion planning and construction of smooth curves for AVs. Special care has been taken in the methodologies used, the variety of techniques, in addition to the advantages and disadvantages presented throughout the literature review. / The authors are grateful to the financial support of Spanish Ministry of Economy and European Union, grant DPI2016-81002-R (AEI/FEDER, UE). This work was also supported by the postdoctoral fellowship “APOSTD/2017/055” and the local administration “GV/2017/029” (Generalitat Valenciana, Conse- lleria d’Educació) Valencia - Spain. / Vanegas Zabala, GI. (2024). Generación de maniobras suaves en el espacio 3D [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/203122 / Compendio

This study performed detailed measurements on and around scaled up conducting and adiabatic airfoils with and without film cooling. The conducting vane was a matched Bi airfoil, which accurately scaled the convective heat transfer and conduction through the solid, in order to produce non-dimensional surface temperatures and thermal boundary layers that were representative of an actual engine. Measurements made on all vane models included surface temperature measurements and thermal profiles above the walls. Separate measurements on non-film cooled and film cooled conducting models allowed for the individual contributions of the internal convective cooling and external film cooling to the overall cooling scheme to be quantified. Surface temperature and thermal field measurements above the wall were also performed on a film cooled adiabatic model. For the conducting model with internal cooling only, strong streamwise temperature variations were seen. The surface temperature variations were highly dependent on the local external and internal heat transfer coefficients. Spanwise temperature variations also existed, but were modest in comparison to streamwise variations. Comparing the thermal fields above the film cooled adiabatic and conducting walls allowed for the assumption that the conducting wall would not significantly affect the thermal field in the film cooling jet to be tested. Near the edge of the film cooling jet the developing thermal boundary layer had a clear effect on the overlying gas temperature, suggesting that the common assumption that the adiabatic wall temperature is the appropriate driving temperature for heat transfer to a film cooled wall was invalid. On the jet centerline thermal boundary layer effects were less influential, due to the development of a new, thin boundary layer. This suggested that the adiabatic wall temperature as driving temperature for heat transfer was a reasonable assumption on the jet centerline for most cases tested. As film cooling momentum flux ratio increase, thermal boundary layer effects became more influential on the jet centerline. Additionally, the high resolution surface temperature measurements and thermal field measurements above the wall presented in the current study represent a significant improvement in the data available for validation of computational simulations of conducting turbine airfoils. / text

The existence of pathwise stationary solutions of this stochastic partial differential equation (SPDE, for abbreviation) is demonstrated. In Part II, a connection between certain kind of state constrained controlled Forward-Backward Stochastic Differential Equations (FBSDEs) and Hamilton-Jacobi-Bellman equations (HJB equations) are demonstrated. The special case provides a probabilistic representation of some geometric flows, including the mean curvature flows. Part II includes also a probabilistic proof of the finite time existence of the mean curvature flows.

No description available.

L'objectif de ce travail est d'étudier le comportement en temps long d'un modèle de particules avec une interaction de type branchement. Plus précisément, les particules se déplacent indépendamment suivant une dynamique markovienne jusqu'au temps de branchement, où elles donnent naissance à de nouvelles particules dont la position dépend de celle de leur mère et de son nombre d'enfants. Dans la première partie de ce mémoire nous omettons le branchement et nous étudions le comportement d'une seule lignée. Celle-ci est modélisée via un processus de Markov qui peut admettre des sauts, des parties diffusives ou déterministes par morceaux. Nous quantifions la convergence de ce processus hybride à l'aide de la courbure de Wasserstein, aussi nommée courbure grossière de Ricci. Cette notion de courbure, introduite récemment par Joulin, Ollivier, et Sammer correspond mieux à l'étude des processus avec sauts. Nous établissons une expression du gradient du semigroupe des processus de Markov stochastiquement monotone, qui nous permet d'expliciter facilement leur courbure. D'autres bornes fines de convergence en distance de Wasserstein et en variation totale sont aussi établies. Dans le même contexte, nous démontrons qu'un processus de Markov, qui change de dynamique suivant un processus discret, converge rapidement vers un équilibre, lorsque la moyenne des courbures des dynamiques sous-jacentes est strictement positive. Dans la deuxième partie de ce mémoire, nous étudions le comportement de toute la population de particules. Celui-ci se déduit du comportement d'une seule lignée grâce à une formule many-to-one, c'est-à-dire un changement de mesure de type Girsanov. Via cette transformation, nous démontrons une loi des grands nombres et établissons une limite macroscopique, pour comparer nos résultats aux résultats déjà connus en théorie des équations aux dérivées partielles. Nos résultats sont appliqués sur divers modèles ayant des applications en biologie et en informatique. Parmi ces modèles, nous étudierons le comportement en temps long de la plus grande particule dans un modèle simple de population structurée en taille / The aim of this work is to study the long time behavior of a branching particle model. More precisely, the particles move independently from each other following a Markov dynamics until the branching event. When one of these events occurs, the particle produces some random number of individuals whose position depends on the position of its mother and her number of offspring. In the first part of this thesis, we only study one particle line and we ignore the branching mechanism. So we are interested by the study of a Markov process which can jump, diffuse or be piecewise deterministic. The long time behavior of these hybrid processes is described with the notion of Wasserstein or coarse Ricci curvature. This notion of curvature, introduced by Joulin, Ollivier and Sammer, is more appropriate for the study of processes with jumps. We establish an expression of the gradient of the Markov semigroup of stochastically monotone processes which gives the curvature of these processes. Others sharp bounds of convergence, in Wasserstein distance and total variation distance, are also established. In the same way, we prove that if a Markov process evolves according to one of finitely many underlying Markovian dynamics, with a choice of dynamics that changes at the jump times of a second Markov process, then it is exponentially ergodic, under the assumption that the mean of the curvature of the underlying dynamics is positive. In the second part of the work, we study all the population. Its behaviour can be deduced to the study of the first part using a Girsavov-type transform which is called a many-to-one formula. Using this relation, we establish a law of large numbers and a macroscopic limit, in order to compare our results to the well know results on deterministic setting. Several examples, based on biology and computer science problems, illustrate our results, including the study of the largest individual in a size-structured population model