We first provide a detailed background of a geometric projection methodology developed by Professor Roswitha Marz at Humboldt University in Berlin for showing uniqueness and existence of solutions for ordinary differential-algebraic equations (DAEs). Because of the geometric and operator-theoretic aspects of this particular method, it can be extended to the case of infinite-dimensional abstract DAEs. For example, partial differential equations (PDEs) are often formulated as abstract Cauchy or evolution problems which we label abstract ordinary differential equations or AODE. Using this abstract formulation, existence and uniqueness of the Cauchy problem has been studied. Similarly, we look at an AODE system with operator constraint equations to formulate an abstract differential-algebraic equation or ADAE problem. Existence and uniqueness of solutions is shown under certain conditions on the operators for both index-1 and index-2 abstract DAEs. These existence and uniqueness results are then applied to some index-1 DAEs in the area of thermodynamic modeling of a chemical vapor deposition reactor and to a structural dynamics problem. The application for the structural dynamics problem, in particular, provides a detailed construction of the model and development of the DAE framework. Existence and uniqueness are primarily demonstrated using a semigroup approach. Finally, an exploration of some issues which arise from discretizing the abstract DAE are discussed. / Ph. D.

This article examines how presence and participation in contemporary installation art is reconfigured in online documentation. Considering documentation as an essential component of the art experience, it discusses its ramifications from an artistic as well as an institutional perspective. Using internationally renowned installation artist Olafur Eliasson as example, the article focuses on the documentation of his large-scale installation works in the ongoing art project “Your Uncertain Archive” (https://olafureliasson.net/uncertain). This online archive gathers Eliasson’s artistic output in one (virtual) place, using various techniques to capture and expand on the original on-site art experience. Special attention is devoted to the video documentation of the recent exhibition Life (Fondation Beyeler, 2021), which uses subjective shots, masking and optical filters in order to make the claims of the exhibition more accessible to online audiences.

[pt] Não raras vezes Soren Kierkegaard foi acusado de negligenciar a vida pública em favor de uma subjetividade encerrada em si mesma. Nada mais equivocado. A presente pesquisa pretende demonstrar que a relação com o outro desempenha um papel de suma importância no pensamento deste autor (e tanto é assim que em lugar de um tratamento indiferente, Kierkegaard chama o outro de próximo). No entanto, também é verdade que o indivíduo é incomensurável com a realidade ao seu redor. Ultrapassa o geral, o coletivo, a multidão a fim de se constituir como um indivíduo, a fim de se tornar quem ele é. Enquanto não entrar nessa relação de oposição com o mundo, não terá alcançado a verdade. Como então articular existencialmente a necessidade de ser um indivíduo acima do geral com a não menos necessária demanda de ir ao encontro do próximo? Como se valer do geral para manifestar a própria singularidade? Tentaremos mostrar ao longo deste trabalho que o indivíduo é justamente este ser que quando ameaçado pelo impessoal, se refugia na sua individualidade; e que quando quer manifestar a sua individualidade, se volta ao geral: é como qualquer outro. / [en] Many times Soren Kierkegaard was accused of neglecting public life in behalf of an inner subjectivity. This could not be more mistaken. This research aims to show that interaction with the other plays a primary role in the author´s thinking. Nevertheless, it is correct to say that the individual is incommensurable with reality. So how can an individual express his need to be in contact with collectivity without losing his individuality? How can he resort to the impersonal to manifest his self? Throughout this paper we will attempt to demonstrate that the individual is a person who seeks refuge in privacy when threatened by the collective, and is communal to manifest his individuality.

[pt] Tem a presente pesquisa o escopo de examinar as relações entre ipseidade e alteridade, tendo como ponto de partida o terreno em que Martin Heidegger e Søren Kierkegaard desenvolvem seus pensamentos. O que se pretende ao longo da dissertação é discutir a possibilidade de acesso à ipseidade a partir de uma noção de existência que compreenda a tensão que descola ipseidade e existência. Tomando como referência a análise que deita suas raízes sobre uma leitura fenomenológica da existência e percorre diferentes traços da constituição ontológica do ente que existe, procuro demonstrar, com escólio nas concepções filosóficas de Heidegger e Kierkegaard, que o acesso à ipseidade se dá na relação de alteridade. / [en] The objective in the present dissertation is to examine the relation between ipseity and alterity, having Martin Heidegger´s and Søren Kierkegaard´s philosophys as a starting point. It intends to discuss on this paper the possibility of finding access to ipseity taking as reference a certain notion of existence which comprehends the tension that sets ipseity apart from existence. In this way the research brings us to the following conclusion: the access to ipseity comes from the relation to alterity.

[pt] Nesta tese de Doutorado estudamos uma classe de equações diferenciais parciais de segunda ordem, uniformemente elípticas, completamente não-lineares na forma não-divergência, com crescimento superlinear no gradiente e coeficientes mensuráveis. Para equações com crescimento quadrático, provamos que ocorre multiplicidade de soluções quando o operador não é coercivo e investigamos o comportamento qualitativo dos contínuos de soluções obtidos para uma família parametrizada de problemas. Para isso, estendemos a regularidade e as estimativas C1, alfa, de Caffarelli-Swiech-Winter para equações com crescimento, no máximo quadrático, no gradiente, mostrando que as soluções são continuamente diferenciáveis até o bordo. Além disso, mostramos estimativas a priori na norma uniforme via técnicas puramente não-lineares na forma não-divergência, entre elas desigualdades do tipo Harnack e o princípio do máximo forte de Vázquez para equações de nosso tipo. / [en] In this Ph.D. thesis we study a class of uniformly elliptic partial differential equations of second order in fully nonlinear nondivergence form with superlinear growth in the gradient and measurable coefficients. For equations with quadratic growth, we prove that multiplicity of solutions occurs when the operator is not coercive. We investigate the qualitative behavior of the continuums of solutions obtained for a parameterized family of problems. For this, we extend the Caffarelli-Swiech-Winter C1, alpha, regularity estimates to equations with at most quadratic gradient growth, showing that the solutions are continuously differentiable up to the boundary. Furthermore, we show a priori estimates in the uniform norm using purely nonlinear techniques in the nondivergence form, such as Harnack type inequalities and a Vázquez’s strong maximum principle for equations of our type.

[pt] A Singularização do ser-aí na obra Ser e Tempo de Heidegger procura traçar os passos do ser-aí desde a compreensão prévia do si-mesmo primordialmente mergulhado no modo do impessoal e sob sua mediana tutela guiado, até o resgate do ser-aí [impessoalmente (impropriamente) decaído de si mesmo] de volta ao [seu] próprio ser através e a partir da perspectiva imprópria essencialmente característica do meio ao qual em sendo o ser-aí já sucumbiu - a medianidade cotidiana que rege o modo impessoal de ser si-mesmo do ser-aí. Satisfeito ontológica e propriamente (existencial e fenomenologicamente) o angustiante recolhimento do ser-aí do anonimato [de volta] ao seu próprio ser [si-mesmo], prosseguimos o percurso rumo à singularização deste ente em sentido ôntico, tendo que para isso lançá-lo uma vez mais ao esquecimento cotidiano de si, agora, porém, enquanto seu próprio estar-lançado ou poder-ser: enquanto eu, propriamente. / [en] The singularizing of the there-being in Heidegger s Being and Time seeks to draw the steps of the there-being from the previous comprehension of the self-being primarily dived into the impersonal mode and guided by its median light, to the rescue of the there-being [impersonally (improperly) fallen from himself] back to his [own] proper being through and from the improper perspective essentially characteristic of the middle to which on being the there-being has already surrendered - the daily median light that rules the there-being s impersonal way of being self-being. Once ontologically and properly (existentially and phenomenologically) satisfied the anguishing withdrawal of the there-being from anonymity [back] to his own proper being [self-being], we go on the path towards the singularizing of this being in ontical sense, in order to which we ll have to throw him once again at his daily self-forgetfulness, although this time as his own proper being-thrown or being-possible: as I, properly.

Dans cette thèse, nous étudions l’équation parabolique non linéaire ∂ t u = ∆u + a |x|⎺⥾ |u|ᵅ u, t > 0, x ∈ Rᴺ \ {0}, N ≥ 1, ⍺ ∈ R, α > 0, 0 < Ƴ < min(2,N) et avec une donnée initiale u(0) = φ. On établit l’existence et l’unicité locale dans Lq(Rᴺ) et dans Cₒ(Rᴺ). En particulier, la valeur q = N ⍺/(2 − γ) joue un rôle critique. Pour ⍺ > (2 − γ)/N, on montre l’existence de solutions auto-similaires globales avec données initiales φ(x) = ω(x) |x|−(2−γ)/⍺, où ω ∈ L∞(Rᴺ) homogène de degré 0 et ||ω||∞ est suffisamment petite. Nous montrons ainsi que si φ(x)∼ω(x) |x| ⎺(²⎺⥾)/⍺ pour |x| grande, alors la solution est globale et asymptotique dans L∞(Rᴺ) à une solution auto-similaire de l’équation non linéaire. Tandis que si φ(x)∼ω(x) |x| (x)|x|−σ pour des |x| grandes avec (2 − γ)/⍺ < σ < N, alors la solution est globale, mais elle est asymptotique dans L∞(Rᴺ) à eᵗ∆(ω(x) |x|−σ). L’équation avec un potentiel plus général, ∂ t u = ∆u + V(x) |u|ᵅ u, V(x) |x |⥾ ∈ L∞(Rᴺ), est également étudiée. En particulier, pour des données initiales φ(x)∼ω(x) |x| ⎺(²⎺⥾)/⍺, |x| grande, nous montrons que le comportement à grand temps est linéaire si V est à support compact au voisinage de l’origine, alors qu’il est non linéaire si V est à support compact au voisinage de l’infini. Nous étudions également le système non linéaire ∂ t u = ∆u + a |x|⎺⥾ |v|ᴾ⎺¹v, ∂ t v = ∆v + b |x|⎺ ᴾ |u|q⎺¹ u, t > 0, x ∈ Rᴺ \ {0}, N ≥ 1, a,b ∈ R, 0 < y < min(2,N)? 0 < p < min(2,N), p,q > 1. Sous des conditions sur les paramètres p, q, γ et ρ nous montrons l’existence et l’unicité de solutions globales avec données initiales petites par rapport à certaines normes. En particulier, on montre l’existence de solutions auto-similaires avec donnée initiale Φ = (φ₁, φ₂), où φ₁, φ₂ sont des données initiales homogènes. Nous montrons également que certaines solutions globales sont asymptotiquement auto-similaires. Comme deuxième objectif, nous considérons l’équation de la chaleur non linéaire ut = ∆u + |u|ᴾ⎺¹u - |u| q⎺¹u, avec t ≥ 0 et x ∈ Ω, la boule unité de Rᴺ, N ≥ 3, avec des conditions aux limites de Dirichlet. Soit h une solution stationnaire à symétrie radiale avec changement de signe de (E). On montre que la solution de (E) avec donnée initiale λh explose en temps fini si |λ − 1| > 0 est suffisamment petit et si 1 < q < p < Ps = N+2/N−2 et p suffisamment proche de Ps. Ceci prouve que l’ensemble des données initiales pour lesquelles la solution est globale n’est pas étoilé au voisinage de 0. / In this thesis, we study the nonlinear parabolic equation ∂ t u = ∆u + a |x|⎺⥾ |u|ᵅ u, t > 0, x ∈ Rᴺ \ {0}, N ≥ 1, ⍺ ∈ R, α > 0, 0 < Ƴ < min(2,N) and with initial value u(0) = φ. We establish local well-posedness in Lq(Rᴺ) and in Cₒ(Rᴺ). In particular, the value q = N ⍺/(2 − γ) plays a critical role.For ⍺ > (2 − γ)/N, we show the existence of global self-similar solutions with initial values φ(x) = ω(x) |x|−(2−γ)/⍺, where ω ∈ L∞(Rᴺ) is homogeneous of degree 0 and ||ω||∞ is sufficiently small. We then prove that if φ(x)∼ω(x) |x| ⎺(²⎺⥾)/⍺ for |x| large, then the solution is global and is asymptotic in the L∞-norm to a self-similar solution of the nonlinear equation. While if φ(x)∼ω(x) |x| (x)|x|−σ for |x| large with (2 − γ)/α < σ < N, then the solution is global but is asymptotic in the L∞-norm toe t(ω(x) |x|−σ). The equation with more general potential, ∂ t u = ∆u + V(x) |u|ᵅ u, V(x) |x |⥾ ∈ L∞(Rᴺ), is also studied. In particular, for initial data φ(x)∼ω(x) |x| ⎺(²⎺⥾)/⍺, |x| large , we show that the large time behavior is linear if V is compactly supported near the origin, while it is nonlinear if V is compactly supported near infinity. we study also the nonlinear parabolic system ∂ t u = ∆u + a |x|⎺⥾ |v|ᴾ⎺¹v, ∂ t v = ∆v + b |x|⎺ ᴾ |u|q⎺¹ u, t > 0, x ∈ Rᴺ \ {0}, N ≥ 1, a,b ∈ R, 0 < y < min(2,N)? 0 < p < min(2,N), p,q > 1. Under conditions on the parameters p, q, γ and ρ we show the existence and uniqueness of global solutions for initial values small with respect of some norms. In particular, we show the existence of self-similar solutions with initial value Φ = (φ₁, φ₂), where φ₁, φ₂ are homogeneous initial data. We also prove that some global solutions are asymptotic for large time to self-similar solutions. As a second objective we consider the nonlinear heat equation ut = ∆u + |u|ᴾ⎺¹u - |u| q⎺¹u, where t ≥ 0 and x ∈ Ω, the unit ball of Rᴺ, N ≥ 3, with Dirichlet boundary conditions. Let h be a radially symmetric, sign-changing stationary solution of (E). We prove that the solution of (E) with initial value λ h blows up in finite time if |λ − 1| > 0 is sufficiently small and if 1 < q < p < Ps = N+2/N−2 and p sufficiently close to Ps. This proves that the set of initial data for which the solution is global is not star-shaped around 0.

EN: Jaspers came to philosophy from medicine and psychology. Unique personal situation, authentic life experience, unrepeatable faithful self-respect turned psychopathologist Jaspers from psychiatric physiology of soul to the thesis of ciphers of transcendence, borderline, edge and horizon. He constituted his own pattern of the thinker not only as an explaining teacher, but also as co-creator of original ethics, based on deep comprehension of the other, importance of encompassing communication between man and man. First section of my work deals with the concept of Border. Our attitudes and picture of the universe and its evaluation are limited by borders. The holistic complex being remains behind the horizon The second part is dedicated to border-line situation as an phenomenon of possibility, as seen from the point of view of Jaspers own existence, as presented in his Philosophy, including commentary based on new translations by Vaclav Nemec Closing part deals with existential communication and holistic transition from possible to new horizons of transcendence. My interpretation of chosen excerpts comes mostly from "Psychologie der Weltanschauungen." Quotations appear in my own translation.

EN: Jaspers came to philosophy from medicine and psychology. Unique personal situation, authentic life experience, unrepeatable faithful self-respect turned psychopathologist Jaspers from psychiatric physiology of soul to the thesis of ciphers of transcendence, borderline, edge and horizon. He constituted his own pattern of the thinker not only as an explaining teacher, but also as co-creator of original ethics, based on deep comprehension of the other, importance of encompassing communication between man and man. First section of my work deals with the concept of Border. Our attitudes and picture of the universe and its evaluation are limited by borders. The holistic complex being remains behind the horizon The second part is dedicated to border-line situation as an phenomenon of possibility, as seen from the point of view of Jaspers own existence, as presented in his Philosophy, including commentary based on new translations by Vaclav Nemec Closing part deals with existential communication and holistic transition from possible to new horizons of transcendence. My interpretation of chosen excerpts comes mostly from "Allgemeine Psychopathologie" "Psychologie der Weltanschauungen." Quotations appear in my own translation. KEY WORDS: Prophetic philosophy, philosophy as treatment of world opinion,...

Dans cette thèse, nous nous intéressons à l'étude de quatre équations d'évolution non-locales. Les solutions de ces quatre équations peuvent exploser en temps fini. Dans la théorie des équations d'évolution non-linéaires, une solution est qualifiée de globale si elle est définie pour tout temps positif. Au contraire, si une solution existe seulement sur un intervalle de temps [0; T) borné, elle est dite locale. Dans ce dernier cas et quand le temps maximal d'existence est relié à une alternative d'explosion, on dit aussi que la solution explose en temps fini. Dans un premier travail, nous considérons l'équation de Schrödinger non-linéaire avec une puissance fractionnaire du laplacien, et nous obtenons l'explosion de la solution en temps fini Tmax > 0 pour toute condition initiale positive et non-triviale dans le cas d'exposant sous-critique. Ensuite, nous étudions une équation des ondes amorties avec un potentiel d'espace-temps et un terme non-linéaire et non-local en temps. Nous obtenons un résultat d'existence locale d'une solution dans l'espace d'énergie sous des conditions restrictives sur les données initiales, la dimension de l'espace et la croissance du terme non-linéaire. De plus, nous obtenons l'explosion de la solution en temps fini pour toute condition initiale de moyenne strictement positive. De plus, nous étudions un problème de Cauchy pour l'équation d'évolution avec un p- Laplacien avec une non linéarité non-locale en temps. Dans ce cadre, nous nous intéressons à l'étude de l'existence locale d'une solution de cette équation ainsi qu'un résultat de non-existence de solution globale. Finalement, nous étudions l'intervalle maximal d'existence des solutions de l'équation des milieux poreux avec un terme non-linéaire non-local en temps. / In this thesis, we study four non-local evolution equations. The solutions of these four equations can blow up in finite time. In the theory of nonlinear evolution equations, a solution is qualified as global if it isdefined for any time. Otherwise, if a solution exists only on a bounded interval [0; T), it is called local solution. In this case and when the maximum time of existence is related to a blow up alternative, we say that the solution blows up in finite time. First, we consider the nonlinear Schröodinger equation with a fractional power of the Laplacien operator, and we get a blow up result in finite time Tmax > 0 for any non-trivial non-negative initial condition in the case of sub-critical exponent. Next, we study a damped wave equation with a space-time potential and a non-local in time non-linear term. We obtain a result of local existence of a solution in the energy space under some restrictions on the initial data, the dimension of the space and the growth of nonlinear term. Additionally, we get a blow up result of the solution in finite time for any initial condition positive on average. In addition, we study a Cauchy problem for the evolution p-Laplacien equation with nonlinear memory. We study the local existence of a solution of this equation as well as a result of non-existence of global solution. Finally, we study the maximum interval of existence of solutions of the porous medium equation with a nonlinear non-local in time term.