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O presente trabalho apresenta uma análise da aplicação da ferramenta DSM Design Structure Matrix ao planejamento do processo de projeto de edificações. O objetivo principal desta pesquisa é a análise dos Manuais de Escopo de Projetos e Serviços para a Indústria Imobiliária do ponto de vista do planejamento do processo de projeto, utilizando a ferramenta DSM. Para desenvolvimento da pesquisa primeiramente foi feita uma revisão bibliográfica, onde selecionou-se a metodologia ADePT para o desenvolvimento do planejamento do processo de projeto. Em seguida foram aplicadas as duas primeiras etapas da metodologia ADePT definição do processo e otimização do processo, que forneceram material para uma análise da aplicação da ferramenta DSM ao planejamento do processo de projeto e uma análise crítica sobre os Manuais de Escopo de Projetos e Serviços para a Indústria Imobiliária. O cumprimento das duas primeiras etapas da metodologia ADePT forneceu material com informações que permitiram a observação de algumas incoerências e permitiu a apresentação de críticas e sugestões de alterações em seu conteúdo, como proposta de melhoria à ferramenta-guia que estes se propõem a ser. Por fim, a partir da validação da DSM como ferramenta para o desenvolvimento do planejamento do processo de projeto e com base nas alterações sugeridas para os manuais, foi proposto um novo modelo de processo de projeto. / This study presents an analysis of application of DSM Design Structure Matrix to the planning process of building design. The main objective this study is the analysis of manuals Scope of Projects and Services for the Real State Industry in terms of planning the design process, using the DSM. For development of the study was first done a literature review, where we selected the ADEPT methodology to development planning this process. When we applied the first two stages of ADEPT methodology process definition and optimization of the process, which provided material for an analysis of the implementation of DSM planning in the design process and a critical analysis of the Manuals Scope of Projects and Services for Real State Industry. Compliance with the first two stages of ADEPT methodology provided material with information that allowed the observation of some inconsistencies and allowed the presentation of comments and suggestions for changes in content, as proposed improvements to the tool guide that they purport to be. Finally, from the validation of the DSM as a tool for development planning and design process based on the suggested changes to the manuals, we proposed a new model of the design process.

Cell-derived (CDM) matrix scaffolds provide a 3-dimensional (3D) matrix material that recapitulates a native, human extracellular matrix (ECM) microenvironment. CDMs are a heterogeneous source of ECM proteins with a composition dependent on the cell source and its phenotype. CDMs have several applications, such as for development of cell culture substrates to study stromal cell propagation and differentiation, as well as cell or drug delivery vehicles, or for regenerative biomaterial applications. Although CDMs are versatile and exhibit advantageous structure and activity, their use has been hindered due to the prolonged culture time required for ECM deposition and maturation in vitro. Macromolecular crowding (MMC) has been shown to increase ECM deposition and organization by limiting the diffusion of ECM precursor proteins and allowing the accumulation of matrix at the cell layer. A commonly used crowder that has been shown to increase ECM deposition in vitro is Ficoll, and was used in this study as a positive control to assess matrix deposition. Hyaluronic acid (HA), a natural crowding macromolecule expressed at high levels during fetal development, has been shown to play a role in ECM production, organization, and assembly in vivo. HA has not been investigated as a crowding molecule for matrix deposition or development of CDMs in vitro. This dissertation focused on 2 aims supporting the development of a functional, human dermal fibroblast-derived ECM material for the delivery deliver an antimicrobial peptide, cCBD-LL37, and for potentially promoting a pro-angiogenic environment. The goal of this thesis was to evaluate the effects of high molecular weight (HMW) HA as a macromolecular crowding agent on in vitro deposition of ECM proteins important for tissue regeneration and angiogenesis. A pilot proteomics study supported the use of HA as a crowder, as it preliminarily showed increases in ECM proteins and increased retention of ECM precursor proteins at the cell layer; thus supporting the use of HA as a crowder molecule. In the presence of HA, human dermal fibroblasts demonstrated an increase in ECM deposition comparable to the effects of Ficoll 70/400 at day 3 using Raman microspectroscopy. It was hypothesized that HA promotes matrix deposition through changes on ECM gene expression. However, qRT-PCR results indicate that HA and Ficoll 70/400 did not have a direct effect on collagen gene expression, but differences in matrix crosslinking and proteinase genes were observed. Decellularized CDMs were then used to assess CDM stiffness and endothelial sprouting, which indicated differences in structural organization of collagen, and preliminarily suggests that there are differences in endothelial cell migration depending on the crowder agent used in culture. Finally, the collagen retained in the decellularized CDM matrix prepared under MMC supported the binding of cCBD-LL37 with retention of antimicrobial activity when tested against E.coli. Overall, the differences in matrix deposition profiles in HA versus Ficoll crowded cultures may be attributed to crowder molecule-mediated differences in matrix crosslinking, turnover, and organization as indicated by differences in collagen deposition, matrix metalloproteinase expression, and matrix stiffness. MMC is a valuable tool for increasing matrix deposition, and can be combined with other techniques, such as low oxygen and bioreactor cultures, to promote development of a biomanufactured CDM-ECM biomaterial. Successful development of scalable CDM materials that stimulate angiogenesis and support antimicrobial peptide delivery would fill an important unmet need in the treatment of non-healing, chronic, infected wounds.

This thesis is divided into two parts, each part exploring a different topic within the general area of nonlinear algebra. In the first part, we study several applications of tensors. First, we study tensor networks, and more specifically: uniform matrix product states. We use methods from nonlinear algebra and algebraic geometry to answer questions about topology, defining equations, and identifiability of uniform matrix product states. By an interplay of theorems from algebra, geometry, and quantum physics we answer several questions and conjectures posed by Critch, Morton and Hackbusch. In addition, we prove a tensor version of the so-called quantum Wielandt inequality, solving an open problem regarding the higher-dimensional version of matrix product states. Second, we present new contributions to the study of fast matrix multiplication. Motivated by the symmetric version of matrix multiplication we study the plethysm S^k(sl_n) of the adjoint representation sl_n of the Lie group SL_n . Moreover, we discuss two algebraic approaches for constructing new tensors which could potentially be used to prove new upper bounds on the complexity of matrix multiplication. One approach is based on the highest weight vectors of the aforementioned plethysm. The other approach uses smoothable finite-dimensional algebras. Finally, we study the Hessian discriminant of a cubic surface, a recently introduced invariant defined in terms of the Waring rank. We express the Hessian discriminant in terms of fundamental invariants. This answers Question 15 of the 27 questions on the cubic surface posed by Bernd Sturmfels. In the second part of this thesis, we apply algebro-geometric methods to study matroids and flag matroids. We review a geometric interpretation of the Tutte polynomial in terms of the equivariant K-theory of the Grassmannian. By generalizing Grassmannians to partial flag varieties, we obtain a new invariant of flag matroids: the flag-geometric Tutte polynomial. We study this invariant in detail, and prove several interesting combinatorial properties.

Der humanen dermalen Wundheilung liegt ein komplexes Zusammenspiel verschiedener Faktoren zugrunde. Die Bedeutung dieses fein regulierten Gleichgewichts wird deutlich, wenn es durch Fehlregulationen oder Störungen zu chronischen Wundheilungsstörungen oder lokaler Fibrose mit überschießender Narbenbildung kommt. Eine der möglichen Methoden zur Prävention und Behandlung ist die Deckung der Wunde mit einem Hautersatz. Dabei werden zunehmend sogenannte Biomaterialien aus natürlichen Substanzen mit hoher Biokompatibilität und der Möglichkeit zur Interaktion mit dem nativen Gewebe verwendet. In Studien wurde gezeigt, dass vor allem sulfatierte Glykosaminoglykan-Derivate durch die Interaktion ihrer negativ geladenen Sulfatgruppen mit Zytokinen, Wachstumsfaktoren und dermalen Zellen einen positiven Einfluss auf den Wundheilungsprozess haben können. In der vorliegenden Arbeit wurden daher kollagenbasierte artifizielle extrazelluläre Matrizes mit unsulfatierter oder sulfatierter Hyaluronsäure hinsichtlich ihres Einflusses auf humane dermale Fibroblasten als Komponenten der Wundheilung untersucht. Dermale Fibroblasten spielen im Ablauf der Wundheilung eine tragende Rolle und interagieren eng mit der umgebenden Matrix. Anhand ihrer Proteinexpression lassen sich Rückschlüsse auf wichtige Funktionen wie Adhäsion, Proliferation, Differenzierung und Matrixsynthese ziehen. In den durchgeführten Experimenten zeigte sich, dass sulfatierte Matrix in der Kultur mit dermalen Fibroblasten kein entzündliches Milieu förderte. Die Proliferation, Differenzierung und Migration der Fibroblasten schienen gesteigert, während sich die Matrix-Synthese und ihr Remodeling weder pathologisch gehemmt noch überschießend zeigten. Daher wäre die weitere Untersuchung dieses Biomaterials ein vielversprechender Ansatz, um langfristig dem Risiko von Wundheilungsstörungen wie chronischen Wunden oder fibroproliferativen Wundheilungsstörungen effektiv entgegenzuwirken.

Die Mikrosystemtechnik entwickelt sehr kleine Sensoren und Aktuatoren, deren Größe wie der Name schon sagt in Mikrometern gemessen werden kann. Die meist aus Silizium gefertigten Bauteile werden durch Dotierung elektrisch leitfähig. Die so erzeugten Elektroden können nun mittels elektrostatischer Kräfte bewegt werden. Für die numerische Simulation dieser System ist die Kenntnis der Kapazität dieser Elektrodenanordnungen notwendig. In den folgenden Kapiteln wird eine Möglichkeit der Bestimmung der Kapazitätsmatrix für solche Elektrodenanordnungen aufgezeigt. Dazu werden wir zunächst im Kapitel 2 einige Begriffe der Elektrostatik definieren und ihre Zusammenhänge erläutern. Danach werden wir im Kapitel 3 eine Randintegralgleichung herleiten mit deren Hilfe eine Bestimmung der Kapazitätsmatrix möglich ist. Um diese Gleichung zu Lösen werden wir sie im Kapitel 4 diskretisieren. Diese Diskretisierung wird zu einem vollbesetzten Gleichungssystem führen. Das Lösen dieses Gleichungssystems ist relativ teuer, daher wird in den Kapiteln 5 und 6 eine Approximation erläutert, die den Speicherbedarf und Rechenaufwand reduziert. Im Kapitel 7 werden wir die Fehler, welche durch die Diskretisierung und die Approximation entstehen, näher untersuchen. Abschließend werden wir im Kapitel 8 die Kapazitätsmatrizen einiger Beispiele berechnen und mit früheren Berechnungsergebnissen vergleichen.

This thesis is on the numerical computation of eigenvalues of symmetric hierarchical matrices. The numerical algorithms used for this computation are derivations of the LR Cholesky algorithm, the preconditioned inverse iteration, and a bisection method based on LDLT factorizations. The investigation of QR decompositions for H-matrices leads to a new QR decomposition. It has some properties that are superior to the existing ones, which is shown by experiments using the HQR decompositions to build a QR (eigenvalue) algorithm for H-matrices does not progress to a more efficient algorithm than the LR Cholesky algorithm. The implementation of the LR Cholesky algorithm for hierarchical matrices together with deflation and shift strategies yields an algorithm that require O(n) iterations to find all eigenvalues. Unfortunately, the local ranks of the iterates show a strong growth in the first steps. These H-fill-ins makes the computation expensive, so that O(n³) flops and O(n²) storage are required. Theorem 4.3.1 explains this behavior and shows that the LR Cholesky algorithm is efficient for the simple structured Hl-matrices. There is an exact LDLT factorization for Hl-matrices and an approximate LDLT factorization for H-matrices in linear-polylogarithmic complexity. This factorizations can be used to compute the inertia of an H-matrix. With the knowledge of the inertia for arbitrary shifts, one can compute an eigenvalue by bisectioning. The slicing the spectrum algorithm can compute all eigenvalues of an Hl-matrix in linear-polylogarithmic complexity. A single eigenvalue can be computed in O(k²n log^4 n). Since the LDLT factorization for general H-matrices is only approximative, the accuracy of the LDLT slicing algorithm is limited. The local ranks of the LDLT factorization for indefinite matrices are generally unknown, so that there is no statement on the complexity of the algorithm besides the numerical results in Table 5.7. The preconditioned inverse iteration computes the smallest eigenvalue and the corresponding eigenvector. This method is efficient, since the number of iterations is independent of the matrix dimension. If other eigenvalues than the smallest are searched, then preconditioned inverse iteration can not be simply applied to the shifted matrix, since positive definiteness is necessary. The squared and shifted matrix (M-mu I)² is positive definite. Inner eigenvalues can be computed by the combination of folded spectrum method and PINVIT. Numerical experiments show that the approximate inversion of (M-mu I)² is more expensive than the approximate inversion of M, so that the computation of the inner eigenvalues is more expensive. We compare the different eigenvalue algorithms. The preconditioned inverse iteration for hierarchical matrices is better than the LDLT slicing algorithm for the computation of the smallest eigenvalues, especially if the inverse is already available. The computation of inner eigenvalues with the folded spectrum method and preconditioned inverse iteration is more expensive. The LDLT slicing algorithm is competitive to H-PINVIT for the computation of inner eigenvalues. In the case of large, sparse matrices, specially tailored algorithms for sparse matrices, like the MATLAB function eigs, are more efficient. If one wants to compute all eigenvalues, then the LDLT slicing algorithm seems to be better than the LR Cholesky algorithm. If the matrix is small enough to be handled in dense arithmetic (and is not an Hl(1)-matrix), then dense eigensolvers, like the LAPACK function dsyev, are superior. The H-PINVIT and the LDLT slicing algorithm require only an almost linear amount of storage. They can handle larger matrices than eigenvalue algorithms for dense matrices. For Hl-matrices of local rank 1, the LDLT slicing algorithm and the LR Cholesky algorithm need almost the same time for the computation of all eigenvalues. For large matrices, both algorithms are faster than the dense LAPACK function dsyev.:List of Figures xi List of Tables xiii List of Algorithms xv List of Acronyms xvii List of Symbols xix Publications xxi 1 Introduction 1 1.1 Notation 2 1.2 Structure of this Thesis 3 2 Basics 5 2.1 Linear Algebra and Eigenvalues 6 2.1.1 The Eigenvalue Problem 7 2.1.2 Dense Matrix Algorithms 9 2.2 Integral Operators and Integral Equations 14 2.2.1 Definitions 14 2.2.2 Example - BEM 16 2.3 Introduction to Hierarchical Arithmetic 17 2.3.1 Main Idea 17 2.3.2 Definitions 19 2.3.3 Hierarchical Arithmetic 24 2.3.4 Simple Hierarchical Matrices (Hl-Matrices) 30 2.4 Examples 33 2.4.1 FEM Example 33 2.4.2 BEM Example 36 2.4.3 Randomly Generated Examples 37 2.4.4 Application Based Examples 38 2.4.5 One-Dimensional Integral Equation 38 2.5 Related Matrix Formats 39 2.5.1 H2-Matrices 40 2.5.2 Diagonal plus Semiseparable Matrices 40 2.5.3 Hierarchically Semiseparable Matrices 42 2.6 Review of Existing Eigenvalue Algorithms 44 2.6.1 Projection Method 44 2.6.2 Divide-and-Conquer for Hl(1)-Matrices 45 2.6.3 Transforming Hierarchical into Semiseparable Matrices 46 2.7 Compute Cluster Otto 47 3 QR Decomposition of Hierarchical Matrices 49 3.1 Introduction 49 3.2 Review of Known QR Decompositions for H-Matrices 50 3.2.1 Lintner’s H-QR Decomposition 50 3.2.2 Bebendorf’s H-QR Decomposition 52 3.3 A new Method for Computing the H-QR Decomposition 54 3.3.1 Leaf Block-Column 54 3.3.2 Non-Leaf Block Column 56 3.3.3 Complexity 57 3.3.4 Orthogonality 60 3.3.5 Comparison to QR Decompositions for Sparse Matrices 61 3.4 Numerical Results 62 3.4.1 Lintner’s H-QR decomposition 62 3.4.2 Bebendorf’s H-QR decomposition 66 3.4.3 The new H-QR decomposition 66 3.5 Conclusions 67 4 QR-like Algorithms for Hierarchical Matrices 69 4.1 Introduction 70 4.1.1 LR Cholesky Algorithm 70 4.1.2 QR Algorithm 70 4.1.3 Complexity 71 4.2 LR Cholesky Algorithm for Hierarchical Matrices 72 4.2.1 Algorithm 72 4.2.2 Shift Strategy 72 4.2.3 Deflation 73 4.2.4 Numerical Results 73 4.3 LR Cholesky Algorithm for Diagonal plus Semiseparable Matrices 75 4.3.1 Theorem 75 4.3.2 Application to Tridiagonal and Band Matrices 79 4.3.3 Application to Matrices with Rank Structure 79 4.3.4 Application to H-Matrices 80 4.3.5 Application to Hl-Matrices 82 4.3.6 Application to H2-Matrices 83 4.4 Numerical Examples 84 4.5 The Unsymmetric Case 84 4.6 Conclusions 88 5 Slicing the Spectrum of Hierarchical Matrices 89 5.1 Introduction 89 5.2 Slicing the Spectrum by LDLT Factorization 91 5.2.1 The Function nu(M − µI) 91 5.2.2 LDLT Factorization of Hl-Matrices 92 5.2.3 Start-Interval [a, b] 96 5.2.4 Complexity 96 5.3 Numerical Results 97 5.4 Possible Extensions 100 5.4.1 LDLT Slicing Algorithm for HSS Matrices 103 5.4.2 LDLT Slicing Algorithm for H-Matrices 103 5.4.3 Parallelization 105 5.4.4 Eigenvectors 107 5.5 Conclusions 107 6 Computing Eigenvalues by Vector Iterations 109 6.1 Power Iteration 109 6.1.1 Power Iteration for Hierarchical Matrices 110 6.1.2 Inverse Iteration 111 6.2 Preconditioned Inverse Iteration for Hierarchical Matrices 111 6.2.1 Preconditioned Inverse Iteration 113 6.2.2 The Approximate Inverse of an H-Matrix 115 6.2.3 The Approximate Cholesky Decomposition of an H-Matrix 116 6.2.4 PINVIT for H-Matrices 117 6.2.5 The Interior of the Spectrum 120 6.2.6 Numerical Results 123 6.2.7 Conclusions 130 7 Comparison of the Algorithms and Numerical Results 133 7.1 Theoretical Comparison 133 7.2 Numerical Comparison 135 8 Conclusions 141 Theses 143 Bibliography 145 Index 153

The process of aging causes the major changes in nervous tissue such as changes in the size of brain, architecture of glial cells and extracellular matrix. The size of brain is on the decrease as consequence of aging and there is a change of molecules as well as morphology at all levels. Extracellular space (ECS) is interstitium important especially in communication between cells mediated by diffusion. The limit of diffusion in extracellular space is given by size of ECS, which is discribed by volume fraction and tortuosity, that reflect amount of diffusion barriers. The changes of ECS diffusion parameters during aging were measured by real-time iontophoretic method in four parts of brain (cortex - Cx, hippocampus - Hp, inferior colliculus - IC and corpus trapezoideum - TB). Further, we studied influence of deficiency of Bral2 link protein at differences of ECS diffusion parameters and importance of Bral2 protein at aging and regulation mechanisms of cytotoxic brain edema. Our results show, that aging leads to decreasing of ECS volume v Cx and Hp, but it was not observed in IC and TB, where the intact perineuronal nets act like protecting shield against the degenerative disease induced by aging. However, small differences in composition of perineuronal nets, deficiency of Bral2 link protein, may...

This thesis deals with the generalized version of the Krein-Milman theorem, as it was stated in the work of Webster-Winkler. We introduce basic definitions, extending convexity notions in the classical sense to the setting of matrix convex sets. Further on, we study important theorems which are needed to prove the main result, for example, a representation result, which states that any compact matrix convex set is matrix affinely homeomorphic to the matricial version of the state space on some operator system. In the final part, we provide a proof of the matrix Krein-Milman theorem. 1

An n × n matrix factorization of a polynomial f is a pair of n × n matrices (P, Q) such that PQ = f In, where In is the n × n identity matrix. In this dissertation, we study matrix factorizations of an arbitrary element in a given unital ring. This study is motivated on the one hand by the construction of the unit object in the bicategory LGK of Landau-Ginzburg models (of great utility in quantum physics) whose 1−cells are matrix factorizations of polynomials over a commutative ring K, and on the other hand by the existing tensor product of matrix factorizations b⊗. We observe that the pair of n × n matrices that appear in the matrix factorization of an element in a unital ring is not unique. Next, we propose a new operation on matrix factorizations denoted e⊗ which is such that if X is a matrix factorization of an element f in a unital ring (e.g. the power series ring K[[x1, ..., xr]] f) and Y is a matrix factorization of an element g in a unital ring (e.g. g ∈ K[[y1, ..., ys]]), then Xe⊗Y is a matrix factorization of f g in a certain unital ring (e.g. in case f ∈ K[[x1, ..., xr]] and g ∈ K[[y1, ..., ys]], then f g ∈ K[[x1, ..., xr , y1, ..., ys]]). e⊗ is called the multiplicative tensor product of X and Y. After proving that this product is bifunctorial, many of its properties are also stated and proved. Furthermore, if MF(1) denotes the category of matrix factorizations of the constant power series 1, we define the concept of one-step connected category and prove that there is a one-step connected subcategory of (MF(1),e⊗) which is semi-unital semi-monoidal. We also define the concept of right pseudo-monoidal category which generalizes the notion of monoidal category and we prove that (MF(1),e⊗) is an example of this concept. Furthermore, we define a summand-reducible polynomial to be one that can be written in the form f = t1 + · · · + ts + g11 · · · g1m1 + · · · + gl1 · · · glml under some specified conditions where each tk is a monomial and each gji is a sum of monomials. We then use b⊗ and e⊗ to improve the standard method for matrix factorization of polynomials on this class and we prove that if pji is the number of monomials in gji, then there is an improved version of the standard method for factoring f which produces factorizations of size 2 Qm1 i=1 p1i+···+ Qml i=1 pli−( Pm1 i=1 p1i+···+ Pml i=1 pli) times smaller than the size one would normally obtain with the standard method. Moreover, details are given to elucidate the intricate construction of the unit object of LGK. Thereafter, a proof of the naturality of the right and left unit maps of LGK with respect to 2−morphisms is presented. We also prove that there is no direct inverse for these (right and left) unit maps, thereby justifying the fact that their inverses are found only up to homotopy. Finally, some properties of matrix factorizations are exploited to state and prove a necessary condition to obtain a Morita context in LGK.