Slow Motion

Objects – from natural to domestic – constitute an essential element in various cinematic image motifs of Andrei Tarkovsky, as depicted in his films Solaris (1972), Mirror (1975), Stalker (1979), Nostalgia (1983) and Sacrifice (1986). In his polemic against Sergei Eisenstein's film theory of montage, Tarkovsky emphasises the temporality of objects that precludes the overpowering subjective temporalisation (which the theory of montage would presuppose). Time and the temporality of objects should, therefore, remain intact in the mise-en-scène. Most of these objects in Tarkovsky's image motifs have a characteristic in common; they are leftover objects – left over from the past, from the narrative continuity of the present, from dreams and mysterious incarnations, in ruins and uninhabited and abandoned terrains, from memory and oblivion. Tarkovsky's notion of mise-en-scène as sculpting in time suggests a mode of temporal existence which lingers on the leftover objects. The time that elapses in mise-en-scène is apparent in the creation of a film image, in which the present persists in the objects left behind. Therefore, in the scene compositions of image motifs, the camera does not necessarily follow the narration and its continuity, but often remains fixed on simple objects that are otherwise hardly regarded in the montage. In my talk I attempt to show how Tarkovsky's cinematic image motif, as displayed in slow motion, flight motif (overcoming the gravity), cinematic still life or constellations of painting, poetry and music, is mostly built on leftover objects. The accompanying music and recitation (of poems by his father, Arseniy Tarkovsky) creates a synthesis of verbal, auditory and visual metaphors that essentially characterises Tarkovsky's image motifs. The constant presence and perseverance of the leftover objects also points to an epistemic potential of Tarkovsky's image motifs, that the status of objects as remnants resists the prevailing subjective transcendentalism (to which the montage principle belongs within the framework of film aesthetics) and its reduction of objects to a merely given entity in perception.

To my parents Yusuf Kholmatov and Inobat Kholmatova, my wife Nihola and our beloved son Akobir, for their sincere love and support. vi Acknowledgment vii G reat praises to the Almighty Lord for giving me strength and ability to understand, learn and complete this work and for all other countless gifts offered me. Enormous cordial thanks to the International Centre for Theoretical Physics (ICTP) and Scuola Internazionale Superiore di Studi Avanzati (SISSA) for the support and given opportunity to be introduced the marvels of Mathematics!!! I and my family owe our deepest gratitude also to Trieste for the precious and fascinating years!!! My appreciation also extends to all professors and academic and non academic staff of both entities for the sincere help and support. The special gratitude is towards my supervisor Prof. Giovanni Bellettini, to whom I am highly indebted for the continuous encouragement, advices and attention, for the patience to my failures and laziness.

The contact angle of a liquid drop on a rigid surface is determined by the classical theory of Young-Laplace. For chemically homogeneous surfaces, this angle is a constant. We study the minimal-energy configurations of liquid drops on rough surfaces. Here the actual angle is still constant for homogeneous surfaces, but the apparent angle can fluctuate widely. A limit theorem is introduced for minimal energy configuration, where the rigid surface converges to a smooth one, but the roughness parameter is kept constant. It turns out that the limit of minimal energy configurations correspond to liquid drop on a smooth surface with an appropriately defined effective chemical interaction energy. It turns out that the effective chemical interaction depends linearly on the roughness in a certain range of parameters, corresponding to full wetting. Outside this range the most stable configuration corresponds to a partial wetting and the effective interaction energy depends on the geometry in an essential way. This result partially justifies and extends Wenzel and Cassie's laws and can be used to deduce the actual inclination angle in the most stable state, where the apparent one is known by measurement. This, in turn, may be applied to deduce the roughness parameter if the interfacial energy is known, or visa versa.

This research aims to investigate student-generation and elaboration of visual representations as a tool for promoting understanding of difficult conceptual domains. The paper focuses on students' naturally occurring representational practices as identified in an activity of student-generated representations (SGR). The research is based on observations of pairs of students, ages 10-14, generating representations while trying to understand the phenomenon of the moon phases. The activity involves a few stages. First, each student generates a representation to promote his or her own understanding of the phenomenon. Then, the students negotiate and co-construct representations with their peers. Finally, they design representations for explaining the phenomenon to an external audience. The analysis identifies various representational practices utilized by students for making sense of the phenomenon, developing explanations, and communicating their ideas to their peers. The analysis examines how these practices support students in achieving some cognitive and communicative goals. This research aims to investigate student-generation and elaboration of visual representations, as a tool for promoting understanding of difficult conceptual domains. In particular, it examines students' practices of using drawings and visual representations to advance their own understanding. Generally speaking, the idea of SGR can be motivated by combining two important components that have been shown valuable for learning: First, a wide range of research has shown that learning with visual representations enhance learning and understanding (e.g.

Suppose that Ω is a 2D domain with holes ω 0 , ω 1 , . . . , ω j , j = 1...k. In the perforated domain A = Ω \ ( ∪ k j=0 ω j ) we consider the class J of complex valued maps having degrees 1 and -1 on the boundaries ∂Ω, ∂ω 0 respectively and degree 0 on the boundaries of other holes. We investigate whether the minimum of the Ginzburg-Landau energy E κ is attained in J , as well as the asymptotic behavior of minimizers as the coherency length κ -1 tends to 0. We show that the answer to these questions is determined by the value of the H 1 -capacity cap(A) of the domain. If cap(A) ≥ π (domain A is "thin"), minimizers exist for each κ. Moreover they are vortexless and converge in H 1 (A) (and even better) to a minimizing S 1 -valued harmonic map as κ → ∞. When cap(A) < π (domain A is "thick"), we establish existence of quasi-minimizers (maps with "almost minimal energy"), which exhibit a different qualitative behavior : they have exactly two zeroes (vortices) rapidly converging to ∂A as κ → ∞ . Finally we formulate a conjecture on non-existence of the minimizers in thick domains. Update. This preprint was written at the end 2004. We added (in June 2008) a short final section accounting subsequent developments.

Objects – from natural to domestic – constitute an essential element in various cinematic image motifs of Andrei Tarkovsky, as depicted in his films Solaris (1972), Mirror (1975), Stalker (1979), Nostalgia (1983) and Sacrifice (1986). In his polemic against Sergei Eisenstein's film theory of montage, Tarkovsky emphasises the temporality of objects that precludes the overpowering subjective temporalisation (which the theory of montage would presuppose). Time and the temporality of objects should, therefore, remain intact in the mise-en-scène. Most of these objects in Tarkovsky's image motifs have a characteristic in common; they are leftover objects – left over from the past, from the narrative continuity of the present, from dreams and mysterious incarnations, in ruins and uninhabited and abandoned terrains, from memory and oblivion. Tarkovsky's notion of mise-en-scène as sculpting in time suggests a mode of temporal existence which lingers on the leftover objects. The time that elapses in mise-en-scène is apparent in the creation of a film image, in which the present persists in the objects left behind. Therefore, in the scene compositions of image motifs, the camera does not necessarily follow the narration and its continuity, but often remains fixed on simple objects that are otherwise hardly regarded in the montage. In my paper I attempt to show how Tarkovsky's cinematic image motif, as displayed in slow motion, flight motif (overcoming the gravity), cinematic still-life or constellations of painting, poetry and music, is mostly built on  leftover objects. The accompanying music and recitation (of poems by his father, Arseniy Tarkovsky) creates a synthesis of verbal, auditory and visual metaphors that essentially characterises Tarkovsky's image motifs. The constant presence and perseverance of the leftover objects also points to an epistemic potential of Tarkovsky's image motifs, that the status of objects as remnants resists the prevailing subjective transcendentalism (to which the montage principle belongs within the framework of film aesthetics) and its reduction of objects to a merely given entity in perception.

Along with Fritz Lang, a master of the silent film era, and Stanley Kubrick, Andrei Tarkovsky is one of the few film directors in the classical film history who closely confronted a technologised or machined modernity – with its tragic destruction and ruination. Particularly his films Solaris (1972) and Stalker (1979) – based on the science fictions by the authors Stanislaw Lem and Arkady/Boris Strugatsky – depict the ruins of a triumphantly emerged technological modernity by revealing them in various image motifs. These pictorial motifs are primarily technical objects or machines and buildings, which remain leftover objects in a state of oblivion and constant ruination. Tarkovsky’s image motifs, which characterize and aesthetically enrich his unique mise-en-scène, are known to be the slow motion, flight-/levitation- and fall-motifs, overcoming inertia, the conspicuous presence of fluidity (rain, water surface on decaying swampy ground), ruined interiors and landscapes, etc.

In my paper I try to show how these different image motifs of Tarkovsky are based on a general and omnipotent natural phenomenon, namely gravity, and how it functions as a tacit principle in all transformations or metamorphoses of a technologically modernised world and the humans, who create and inhabit it. Some of the narrative leitmotifs can be derived from these pictorial motifs, which mostly depict and build on the ruination of a machined modernity. While the film Solaris – as a celestial science fiction – displays striking anti-gravitational image motifs and thus suggests narrative leitmotifs of redemption and resurrection, the film Stalker is rich in earthly image motifs such as the pro-gravitational fall-motif, as represented by the gravitational fall, dispersion, disintegration or ruination of machines and buildings, and also by the fall of humans striving for redemption to death and the subsequent decomposition of bodies in a mysterious part of the world, the Zone. In these films, which are based on celestial and terrestrial science fictions, Tarkovsky seems to juxtapose pro- and anti-gravitational image motifs that relate closely to nature and technology as well as to the human self – to its exteriority and interiority. These image motifs are tacitly accompanied by the equally juxtaposed narrative leitmotifs of redemption and immortality – or repeated resurrection in space (Solaris) – as against the earthly fall into death and ruination in the 'Zone' of destiny (Stalker). This explicates a central motif of Tarkovsky’s film aesthetics, namely, the exploration of a transitory nexus between the human self and the world, which includes both the primordial nature and the technologically developed culture or civilisation.

Objects – from natural to domestic – constitute an essential element in various cinematic image motifs of Andrei Tarkovsky, as depicted in his films Solaris (1972), Mirror (1975), Stalker (1979), Nostalgia (1983) and Sacrifice (1986). In his polemic against Sergei Eisenstein's film theory of mon¬tage, Tarkovsky emphasises the temporality of objects that precludes the overpowering subjective temporalisation (which the theory of montage would presuppose). Time and the temporality of objects should, therefore, remain intact in the mise-en-scène. Most of these objects in Tarkovsky's image motifs have a characteristic in common; they are leftover ob-jects – left over from the past, from the narrative continuity of the present, from dreams and mysteri¬ous incarnations, in ruins and uninhabited and abandoned terrains, from memory and obliv¬ion. Tarkovsky's notion of mise-en-scène as sculpting in time suggests a mode of tem-poral existence which lingers on the leftover objects. The time that elapses in mise-en-scène is apparent in the creation of a film image, in which the present persists in the objects left behind. Therefore, in the scene compositions of image motifs, the camera does not necessarily fol¬low the narration and its continuity, but often remains fixed on simple objects that are other-wise hardly regarded in the montage. In my paper I attempt to show how Tarkovsky's cinematic image motif, as displayed in slow motion, flight motif (overcoming the gravity), cinematic still-life or constellations of painting, poetry and music, is mostly built on  leftover objects. The accompany¬ing music and recitation (of poems by his father, Arseniy Tarkovsky) creates a synthe-sis of verbal, auditory and visual metaphors that essentially characterises Tarkovsky's im-age motifs. The constant presence and perseverance of the leftover objects also points to an epis-temic potential of Tarkovsky's image motifs, that the status of objects as remnants resists the prevail¬ing subjective transcendentalism (to which the montage principle belongs within the frame¬work of film aesthetics) and its reduction of objects to a merely given entity in perception.

La marche est généralement évaluée par une observation simple, mais de plus en plus de bilans cliniques sont filmés pour ensuite être analysés. Grâce à l'utilisation de la caméra ultra-rapide Miro Phantom C110, nous avons étudié la marche dans le plan sagittal de 10 sujets sains. L'approche a consisté à détecter des marqueurs placés au niveau de repère anatomique précis sur le membre inférieur, d'en extraire les coordonnées en 2D pour calculer différents paramètres de la marche. Les objectifs de ce projet sont de vérifier l'utilisation d'une caméra ultra-rapide pour déterminer des paramètres cinétiques et cinématiques de la marche ainsi que d'analyser l'intérêt de la haute fréquence d'acquisition sur la détection d'évènement précis au cours du cycle de marche.

International audienceLa marche est généralement évaluée par une observation simple, mais de plus en plus de bilans cliniques sont filmés pour ensuite être analysés. Grâce à l&#39;utilisation de la caméra ultra-rapide Miro Phantom C110, nous avons étudié la marche dans le plan sagittal de 10 sujets sains. L&#39;approche a consisté à détecter des marqueurs placés au niveau de repère anatomique précis sur le membre inférieur, d’en extraire les coordonnées en 2D pour calculer différents paramètres de la marche. Les objectifs de ce projet sont de vérifier l’utilisation d’une caméra ultra-rapide pour déterminer des paramètres cinétiques et cinématiques de la marche ainsi que d’analyser l’intérêt de la haute fréquence d’acquisition sur la détection d’évènement précis au cours du cycle de marche

La marche est généralement évaluée par une observation simple, mais de plus en plus de bilans cliniques sont filmés pour ensuite être analysés. Grâce à l'utilisation de la caméra ultra-rapide Miro Phantom C110, nous avons étudié la marche dans le plan sagittal de 10 sujets sains. L'approche a consisté à détecter des marqueurs placés au niveau de repère anatomique précis sur le membre inférieur, d'en extraire les coordonnées en 2D pour calculer différents paramètres de la marche. Les objectifs de ce projet sont de vérifier l'utilisation d'une caméra ultra-rapide pour déterminer des paramètres cinétiques et cinématiques de la marche ainsi que d'analyser l'intérêt de la haute fréquence d'acquisition sur la détection d'évènement précis au cours du cycle de marche.

International audienceLa marche est généralement évaluée par une observation simple, mais de plus en plus de bilans cliniques sont filmés pour ensuite être analysés. Grâce à l&#39;utilisation de la caméra ultra-rapide Miro Phantom C110, nous avons étudié la marche dans le plan sagittal de 10 sujets sains. L&#39;approche a consisté à détecter des marqueurs placés au niveau de repère anatomique précis sur le membre inférieur, d’en extraire les coordonnées en 2D pour calculer différents paramètres de la marche. Les objectifs de ce projet sont de vérifier l’utilisation d’une caméra ultra-rapide pour déterminer des paramètres cinétiques et cinématiques de la marche ainsi que d’analyser l’intérêt de la haute fréquence d’acquisition sur la détection d’évènement précis au cours du cycle de marche

The contact angle of a liquid drop on a rigid surface is determined by the classical theory of Young-Laplace. For chemically homogeneous surfaces, this angle is a constant. We study the minimal-energy configurations of liquid drops on rough surfaces. Here the actual angle is still constant for homogeneous surfaces, but the apparent angle can fluctuate widely. A limit theorem is introduced for minimal energy configuration, where the rigid surface converges to a smooth one, but the roughness parameter is kept constant. It turns out that the limit of minimal energy configurations correspond to liquid drop on a smooth surface with an appropriately defined effective chemical interaction energy. It turns out that the effective chemical interaction depends linearly on the roughness in a certain range of parameters, corresponding to full wetting. Outside this range the most stable configuration corresponds to a partial wetting and the effective interaction energy depends on the geometry in an essential way. This result partially justifies and extends Wenzel and Cassie's laws and can be used to deduce the actual inclination angle in the most stable state, where the apparent one is known by measurement. This, in turn, may be applied to deduce the roughness parameter if the interfacial energy is known, or visa versa.

In this paper, we obtain an explicit formula for the discrepancy between the limit of the second inner variations of p-Laplace Allen-Cahn energies and the second inner variation of their Γ-limit which is the area functional. Our analysis explains the mysterious discrepancy term found in our previous paper [8] in the case p = 2. The discrepancy term turns out to be related to the convergence of certain 4-tensors which are absent in the usual Allen-Cahn functional. These (hidden) 4-tensors suggest that, in the complex-valued Ginzburg-Landau setting, we should expect a different discrepancy term which we are able to identify. Along the way, we partially answer a question of Kohn and Sternberg [6] by giving a relation between the limit of second variations of the Allen-Cahn functional and the second inner variation of the area functional at local minimizers. Moreover, our analysis reveals an interesting identity connecting second inner variation and Poincaré inequality for area-minimizing surfaces with volume constraint in the work of Sternberg and Zumbrun [16]. Résumé Dans cet article, nous obtenons une formule explicite pour la différence entre la limite des deuxièmes variations internes desénergies du p-Laplacien de Allen-Cahn et la seconde variation interne de leur Γ-limite qui est la fonctionnelle d'aire. Notre analyse explique la différence mystérieuse trouvée dans notre article précédent [8] dans les cas p = 2. Cette différence se révèleêtre en rapport avec la convergence de certains 4-tenseurs qui sont absents dans la fonctionnelle Allen-Cahn habituelle. Ces 4-tenseurs (cachés) suggèrent que, dans le cadre de Ginzburg-Landauà valeurs complexe, nous nous devons attendreà un terme de divergence différent que nous sommes en mesure d'identifier. En particulier, nous répondons en partie une question de Kohn et Sternberg [6] en donnant une relation entre la limite des deuxièmes variations de la fonctionnelle Allen-Cahn et la deuxième variation interne de la fonctionnelle d'aire aux points de minimum locaux. De plus, notre analyse révèle une identité intéressante qui relie les deuxièmes variations internes et l'inégalité de Poincaré pour les surfaces d'aire minimisante avec contrainte de volume dans le travail de Sternberg et Zumbrun [16].

In this paper, we study the relation between the second inner variations of the Allen-Cahn functional and its Gamma-limit, the area functional. Our result implies that the Allen-Cahn functional only approximates well the area functional up to the first order. However, as an application of our result, we prove, assuming the single-multiplicity property of the limiting energy, that the Morse indices of critical points of the Allen-Cahn functional are bounded from below by the Morse index of the limiting minimal hypersurface.

Learners’ physical performances can serve as focal objects for reflection and insight across a variety of contexts and content areas. This session brings together a set of projects that leverage the physical performances of learners, construct concrete and abstract representations of those performances, and investigate how learners reflect on and understand the relationships between their performances and target content—physics, health and fitness, data literacy and navigation, animal foraging, and climate change. The session will share findings and design principles from each of the studies around constructing technological scaffolds for physical performance reflections. The symposium highlights the various ways performance can be used to engage learners, and how different settings and learning goals affect the designs of performance representations.

Learners' physical performances can serve as focal objects for reflection and insight across a variety of contexts and content areas. This session brings together a set of projects that leverage the physical performances of learners, construct concrete and abstract representations of those performances, and investigate how learners reflect on and understand the relationships between their performances and target content-physics, health and fitness, data literacy and navigation, animal foraging, and climate change. The session will share findings and design principles from each of the studies around constructing technological scaffolds for physical performance reflections. The symposium highlights the various ways performance can be used to engage learners, and how different settings and learning goals affect the designs of performance representations.

We describe some aspects of the dynamics of the Cahn-Hilliard equation. In particular we consider the dynamics of spherical interfaces and discuss a result showing that spherical interfaces either persist for ever or until they reach the boundary. We also discuss the dynamics of a small interface attached to the boundary.

Recently, Giorgio Fusco and the author in [1] studied the system ∆u−Wu(u) = 0 for a class of potentials that possess several global minima and are invariant under a general finite reflection group, and established existence of equivariant solutions connecting the minima in certain directions at infinity, together with an estimate. In this paper a new proof is given which, in particular, avoids the introduction of a pointwise constraint in the minimization process.

We consider the sharp interface limit of a diffuse phase field model with prescribed total mass taking into account a spatially inhomogeneous anisotropic interfacial energy and an elastic energy. The main aim is the derivation of a weak formulation of an anisotropic, inhomogeneous, elastically modified Gibbs-Thomson law in the sharp interface limit. To this end we show that one can pass to the limit in the weak formulation of the Euler-Lagrange equation of the diffuse phase field energy.

We consider the functional Ω g(∇u + X *) dL 2n where g is convex and X * (x, y) = 2(−y, x) and we study the minimizers in BV(Ω) of the associated Dirichlet problem. We prove that, under the bounded slope condition on the boundary datum, and suitable conditions on g, there exists a unique minimizer which is also Lipschitz continuous.

Art creates a specific relation to time and especially to the present moment. It opens the experience of the present towards its indeterminacy and emergence. On the contrary, AI does not know the present. It recognizes only the past and future. We could even say that artificial neural networks do not "know" time at all. Instead, they know only logical functions which process patterns of information. Yet, what makes time "time" is genuine transformation, which happens outside of the abstract realm of logic. I support these observations by analysing two works of art: Bill Viola's The Raft (2004) and Hito Steyerl's This Is the Future (2019). While artistic creation opens up the intervals "in-between seconds" for an unpredictable and transformative event to occur (The Raft), AI closes these intervals and fastens the future into predictability calculated on the basis of past data (This Is the Future). Although this machinic operation makes the present even more unpredictable and prone to catastrophes, its potential for transformation seems to be withdrawn.

We present an approach to the problem of the blow up at the triple junction of three fluids in equilibrium. Although many of our results can already be found in the literature, our approach is almost self-contained and uses the theory of sets of finite perimeter without making use of more advanced topics within geometric measure theory. Specifically, using only the calculus of variations we prove two monotonicity formulas at the triple junction for the three-fluid configuration, and show that blow up limits exist and are always cones. We discuss some of the geometric consequences of our results.

We present an approach to the problem of the blow up at the triple junction of three fluids in equilibrium. Although many of our results can already be found in the literature, our approach is almost self-contained and uses the theory of sets of finite perimeter without making use of more advanced topics within geometric measure theory. Specifically, using only the calculus of variations we prove two monotonicity formulas at the triple junction for the three-fluid configuration, and show that blow up limits exist and are always cones. We discuss some of the geometric consequences of our results.

The problem of video inter-frame interpolation is an essential task in the field of image processing. Correctlyincreasing the number of frames in the recording while maintaining smooth movement allows to improve thequality of played video sequence, enables more effective compression and creating a slow-motion recording. Thispaper proposes the FastRIFE algorithm, which is some speed improvement of the RIFE (Real-Time IntermediateFlow Estimation) model. The novel method was examined and compared with other recently published algorithms.All source codes are available at:https://gitlab.com/malwinq/interpolation-of-images-for-slow-motion-videos.

This paper is devoted to the analysis of qualitative properties of flux-saturated type operators in dimension one. Specifically, we study regularity properties and smoothing effects, discontinuous interfaces, the existence of travelling wave profiles, sub-and supersolutions and waiting time features. The aim of the paper is to better understand these phenomena through two prototypic operators: the relativistic heat equation and the porous media flux-limited equation. As an important consequence of our results we deduce that solutions to the one-dimensional relativistic heat equation become smooth inside their support in the long time run.

The aim of this paper is to study a boundary value problem for a linear scalar equation with discontinuous coefficients. This kind of problem appears in the framework of the analysis of the linearized stability of a fluid flow with respect to small perturbations of the boundary data. The linear equation that we are interested in is obtained by linearizing the equations which govern the flow, and involves discontinuous coefficients and nontrivial products. We present a direct approach based on the one introduced by Godlewski and Raviart, which leads to measure solutions, gives a sense of these nontrivial products, and yields simple numerical schemes that give good results.

There are many problems that lead to analysis of dynamical systems in which one can distinguish motions of two types: slow one and fast one. An averaging over fast motion is used for approximate description of the slow motion. First integrals of the averaged system are approximate first integrals of the exact system, i.e. adiabatic invariants. Resonant phenomena in fast motion (capture into resonance, scattering on resonance) lead to inapplicability of averaging, destruction of adiabatic invariance, dynamical chaos and transport in large domains in the phase space. In the paper perturbation theory methods for description of these phenomena are outlined. We also consider as an example the problem of surfatron acceleration of a relativistic charged particle.

In this paper, we will consider an optimal shape problem of heat insulation introduced by [D. Bucur, G. Buttazzo and C. Nitsch, Two optimization problems in thermal insulation, Notices Amer. Math. Soc. 64 (2017), 8, 830–835]. We will establish the existence of optimal shapes in the class of 𝑀-uniform domains. We will also show that balls are stable solutions of the optimal heat insulation problem.

Learners' physical performances can serve as focal objects for reflection and insight across a variety of contexts and content areas. This session brings together a set of projects that leverage the physical performances of learners, construct concrete and abstract representations of those performances, and investigate how learners reflect on and understand the relationships between their performances and target content-physics, health and fitness, data literacy and navigation, animal foraging, and climate change. The session will share findings and design principles from each of the studies around constructing technological scaffolds for physical performance reflections. The symposium highlights the various ways performance can be used to engage learners, and how different settings and learning goals affect the designs of performance representations.

my article "Dialectics of Transcendental and Embodied Self: Andrei Tarkovsky's Image Motif in Context" in the edited volume: "Assured Self, Restive Self. Encounters with Crisis", edited by Prasanta Chakravarty and published by Bloomsbury in July 2023.

A crisis is at once a historically situated phenomenon and a recurring idea of endangerment or a breakdown in creaturely living. By making our choices stark and dif cult, crisis opens up the possibility for genuinely fresh and unexpected beginnings. At the most fundamental level, crisis is the disintegration of relationality among creatures. In fact, crisis is a battle of attrition with and within selfhood. It has the potential to turn into a norm in everyday interaction. It then stops being an exception and becomes the very condition of our living. Through the rubrics of the assured and the restive, the volume addresses how selfhood encounters and negotiates concentric circles of crisis in life and literature. Does the idea of crisis allow us to formulate the idea of self in a particular way? How do certain sources and resources within the self-stoic or heroic, political and creative-come into being during crisis? The book explores the encounter of the self with situations of crisis from diverse disciplinary and cultural perspectives from antiquity to contemporary times. While some essays delve into questions of repose and sensuality by highlighting speci c cases and trajectories from the subcontinent, others deal with questions of mythology, politics and art in a wider sense. One essay directly addresses the core literary question of the uncanny and its relation to selfhood. While speci c concerns illuminate each essay, the volume speaks with a collective, global sense of crisis that faces humanity now and tentatively offers some prospects to deal with it.

This paper is devoted to the analysis of qualitative properties of flux-saturated type operators in dimension one. Specifically, we study regularity properties and smoothing effects, discontinuous interfaces, the existence of travelling wave profiles, sub-and supersolutions and waiting time features. The aim of the paper is to better understand these phenomena through two prototypic operators: the relativistic heat equation and the porous media flux-limited equation. As an important consequence of our results we deduce that solutions to the one-dimensional relativistic heat equation become smooth inside their support in the long time run.

This paper is devoted to the analysis of qualitative properties of flux-saturated type operators in dimension one. Specifically, we study regularity properties and smoothing effects, discontinuous interfaces, the existence of travelling wave profiles, sub-and supersolutions and waiting time features. The aim of the paper is to better understand these phenomena through two prototypic operators: the relativistic heat equation and the porous media flux-limited equation. As an important consequence of our results we deduce that solutions to the one-dimensional relativistic heat equation become smooth inside their support in the long time run.

This paper is devoted to the analysis of qualitative properties of flux-saturated type operators in dimension one. Specifically, we study regularity properties and smoothing effects, discontinuous interfaces, the existence of travelling wave profiles, sub-and supersolutions and waiting time features. The aim of the paper is to better understand these phenomena through two prototypic operators: the relativistic heat equation and the porous media flux-limited equation. As an important consequence of our results we deduce that solutions to the one-dimensional relativistic heat equation become smooth inside their support in the long time run.

We consider two well known variational problems associated with the phenomenon of phase separation: the isoperimetric problem and minimization of the Cahn-Hilliard energy. The two problems are related through a classical result in Γ-convergence and we explore the behavior of global and local minimizers for these problems in the periodic setting. More precisely, we investigate these variational problems for competitors defined on the flat 2-or 3-torus. We view these two problems as prototypes for periodic phase separation. We give a complete analysis of stable critical points of the 2-d periodic isoperimetric problem and also obtain stable solutions to the 2-d and 3-d periodic Cahn-Hilliard problem. We also discuss some intriguing open questions regarding triply periodic constant mean curvature surfaces in 3-d and possible counterparts in the Cahn-Hilliard setting.

We strengthen the classical inequality of C. B. Morrey concerning the optimal Hölder continuity of functions in W 1,p when p > n, by replacing the L p-modulus of the gradient with the sharp Hardy difference involving distance to the boundary. When p = n we do the same strengthening in the integral form of a well known inequality due to F. John and L. Nirenberg.

In a general Banach space we consider gradient-like dynamical systems with the property that there is a manifold along which solutions move slowly compared to attraction in the transverse direction. Conditions are given on the energy (or Lyapunov functional) that ensure solutions starting near the manifold stay near for a long time or even for ever. The abstract results are then applied to a finite-dimensional approximation of a geometrical evolution, and also in infinite-dimensional examples to show the super slow motion of interfaces for the vector Allen-Cahn and Cahn-Morral systems in one space dimension.

Divergence-measure fields in L ∞ over sets of finite perimeter are analyzed. A notion of normal traces over boundaries of sets of finite perimeter is introduced, and the Gauss-Green formula over sets of finite perimeter is established for divergencemeasure fields in L ∞. The normal trace introduced here over a class of surfaces of finite perimeter is shown to be the weak-star limit of the normal traces introduced in Chen & Frid [6] over the Lipschitz deformation surfaces, which implies their consistency. As a corollary, an extension theorem of divergence-measure fields in L ∞ over sets of finite perimeter is also established. Then we apply the theory to the initial-boundary value problem of nonlinear hyperbolic conservation laws over sets of finite perimeter.

In this paper, we establish the convergence of the Ohta-Kawasaki equation to motion by nonlocal Mullins-Sekerka law on any smooth domain in space dimensions N ≤ 3. These equations arise in modeling microphase separation in diblock copolymers. The only assumptions that guarantee our convergence result are (i) well-preparedness of the initial data and (ii) smoothness of the limiting interface. Our method makes use of the "Gamma-convergence" of gradient flows scheme initiated by Sandier and Serfaty and the constancy of multiplicity of the limiting interface due to its smoothness. For the case of radially symmetric initial data without well-preparedness, we give a new and short proof of the result of M. Henry for all space dimensions. Finally, we establish transport estimates for solutions of the Ohta-Kawasaki equation characterizing their transport mechanism.

The isoperimetric problem with respect to the product-type density e − |x| 2 2 dx dy on the Euclidean space R h × R k is studied, with particular emphasis on the case k = 1. A conjecture about the minimality of large cylinders in the case k > 1 is also formulated.

We discuss the local minimality of certain configurations for a nonlocal isoperimetric problem used to model microphase separation in diblock copolymer melts. We show that critical configurations with positive second variation are local minimizers of the nonlocal area functional and, in fact, satisfy a quantitative isoperimetric inequality with respect to sets that are L 1close. The link with local minimizers for the diffuse-interface Ohta-Kawasaki energy is also discussed. As a byproduct of the quantitative estimate, we get new results concerning periodic local minimizers of the area functional and a proof, via second variation, of the sharp quantitative isoperimetric inequality in the standard Euclidean case. As a further application, we address the global and local minimality of certain lamellar configurations.

In this paper, we establish the convergence of the Ohta-Kawasaki equation to motion by nonlocal Mullins-Sekerka law on any smooth domain in space dimensions N ≤ 3. These equations arise in modeling microphase separation in diblock copolymers. The only assumptions that guarantee our convergence result are (i) well-preparedness of the initial data and (ii) smoothness of the limiting interface. Our method makes use of the "Gamma-convergence" of gradient flows scheme initiated by Sandier and Serfaty and the constancy of multiplicity of the limiting interface due to its smoothness. For the case of radially symmetric initial data without well-preparedness, we give a new and short proof of the result of M. Henry for all space dimensions. Finally, we establish transport estimates for solutions of the Ohta-Kawasaki equation characterizing their transport mechanism.

This paper presents the methodology and results of teaching an entire engineering course using challenge-based instruction. The challenges consisted of eight biomechanics multimedia learning modules developed by the authors as part of a broader NSF educational coalition. The biomechanics modules were presented in an undergraduate mechanical engineering course titled "Biomechanics of Human Movement." The class (N 18) was divided into three-member teams. Each team performed the eight computer-based assignments in intervals of one-two weeks per challenge during the semester. Pre-tests, post-tests, affect surveys, a biomechanics topics matrix, and student outcomes surveys were conducted during the course to determine the success of this approach. This paper outlines the challenge-based approaches used and presents assessment methods for a full, semester-long course. Thus, it is directed at faculty who may wish to use this interesting approach in their own engineering courses.

We consider some properties of the solutions of free boundary problems of obstacletype with two phases for a class of heterogeneous quasilinear elliptic operators, including the p-Laplacian operator with 1 &lt; p &lt; ∞. Under a natural non-degeneracy assumption on the interface, when the level set of the change of phase has null Lebesgue measure, we prove a continuous dependence result for the characteristic functions of each phase and we establish sharp estimates on the variation of its Lebesgue measure with respect to the L1-variation of the data, in a rather general framework. For elliptic quasilinear equations which heterogeneities have appropriate integrable derivatives, we show that the characteristic functions of both phases are of bounded variation for general data with bounded variation. This extends recent results for the obstacle problem and is a first result on the regularity of the free boundary of the heterogeneous two phases problem, which is therefore an interface l...

We address small volume-fraction asymptotic properties of a nonlocal isoperimetric functional with a confinement term, derived as the sharp interface limit of a variational model for self-assembly of diblock copolymers under confinement by nanoparticle inclusion. We introduce a small parameter η to represent the size of the domains of the minority phase, and study the resulting droplet regime as η → 0. By considering confinement densities which are spatially variable and attain a unique nondegenerate maximum, we present a two-scale asymptotic analysis wherein a separation of length scales is captured due to competition between the nonlocal repulsive and confining attractive effects in the energy. A key role is played by a parameter M which gives the total volume of the droplets at order η 3 and its relation to existence and non-existence of Gamow's Liquid Drop model on R 3. For large values of M , the minority phase splits into several droplets at an intermediate scale η 1/3 , while for small M minimizers form a single droplet converging to the maximum of the confinement density.