Este libro es una guía teórico-práctica que permite al estudiante de administración y carreras afines entender los conceptos sobre los que se fundamenta cada tema y aplicarlos a sus análisis administrativos. Para ello, además de una breve explicación teórica, en cada tema se presentan ejemplos resueltos y luego por resolver para fijar el aprendizaje. Finalmente, se cierra cada unidad con una serie de ejercicios aplicados. Esta obra es producto de la experiencia obtenida a lo largo de varios años en la coordinación y dictado de los cursos de la Universidad Peruana de Ciencias Aplicadas como Nivelación de Matemáticas, Lógica Matemática y, principalmente, Matemática Básica para Administradores. Asimismo, está complementado con los aportes y problemas propuestos por la mayoría de los profesores de estos cursos. Se trata entonces de una publicación útil y práctica para administradores, estudiantes y profesores.

Dans la première partie de cette thèse, on formule deux méthodes pour le calcul d'une décomposition polyadique canonique avec facteurs matriciels linéairement structurés (tels que des facteurs de Toeplitz ou en bande): un algorithme de moindres carrés alternés contraint (CALS) et une solution algébrique dans le cas où tous les facteurs sont circulants. Des versions exacte et approchée de la première méthode sont étudiées. La deuxième méthode fait appel à la transformée de Fourier multidimensionnelle du tenseur considéré, ce qui conduit à la résolution d'un système d'équations monomiales homogènes. Nos simulations montrent que la combinaison de ces approches fournit un estimateur statistiquement efficace, ce qui reste vrai pour d'autres combinaisons de CALS dans des scénarios impliquant des facteurs non-circulants. La seconde partie de la thèse porte sur la récupération de tenseurs de rang faible et, en particulier, sur le problème de reconstruction tensorielle (TC). On propose un algorithme efficace, noté SeMPIHT, qui emploie des projections séquentiellement optimales par mode comme opérateur de seuillage dur. Une borne de performance est dérivée sous des conditions d'isométrie restreinte habituelles, ce qui fournit des bornes d'échantillonnage sous-optimales. Cependant, nos simulations suggèrent que SeMPIHT obéit à des bornes optimales pour des mesures Gaussiennes. Des heuristiques de sélection du pas et d'augmentation graduelle du rang sont aussi élaborées dans le but d'améliorer sa performance. On propose aussi un schéma d'imputation pour TC basé sur un seuillage doux du coeur du modèle de Tucker et son utilité est illustrée avec des données réelles de trafic routier / In the first part of this thesis, we formulate two methods for computing a canonical polyadic decomposition having linearly structured matrix factors (such as, e.g., Toeplitz or banded factors): a general constrained alternating least squares (CALS) algorithm and an algebraic solution for the case where all factors are circulant. Exact and approximate versions of the former method are studied. The latter method relies on a multidimensional discrete-time Fourier transform of the target tensor, which leads to a system of homogeneous monomial equations whose resolution provides the desired circulant factors. Our simulations show that combining these approaches yields a statistically efficient estimator, which is also true for other combinations of CALS in scenarios involving non-circulant factors. The second part of the thesis concerns low-rank tensor recovery (LRTR) and, in particular, the tensor completion (TC) problem. We propose an efficient algorithm, called SeMPIHT, employing sequentially optimal modal projections as its hard thresholding operator. Then, a performance bound is derived under usual restricted isometry conditions, which however yield suboptimal sampling bounds. Yet, our simulations suggest SeMPIHT obeys optimal sampling bounds for Gaussian measurements. Step size selection and gradual rank increase heuristics are also elaborated in order to improve performance. We also devise an imputation scheme for TC based on soft thresholding of a Tucker model core and illustrate its utility in completing real-world road traffic data acquired by an intelligent transportation

The paper describes a storage scheme for sparse symmetric or nonsymmetric matrices which has been developed and used for many years at the Technical University of Chemnitz. An overview of existing library subroutines using such matrices is included.

Three-dimensional fibrillar networks reconstituted from collagen I are widely used as biomimetic scaffolds for in vitro and in vivo cell studies. Various physicochemical parameters of buffer conditions for in vitro fibril formation are well known, including pH-value, ion concentrations and temperature. However, there is a lack of a detailed understanding of reconstituting well-defined 3D network topologies, which is required to mimic specific properties of the native extracellular matrix. We screened a wide range of relevant physicochemical buffer conditions and characterized the topology of the reconstituted 3D networks in terms of mean pore size and fibril diameter. A congruent analysis of fibril formation kinetics by turbidimetry revealed the adjustment of the lateral growth phase of fibrils by buffer conditions to be key in the determination of pore size and fibril diameter of the networks. Although the kinetics of nucleation and linear growth phase were affected by buffer conditions as well, network topology was independent of those two growth phases. Overall, the results of our study provide necessary insights into how to engineer 3D collagen matrices with an independent control over topology parameters, in order to mimic in vivo tissues in in vitro experiments and tissue engineering applications.

This thesis presents an analysis of several smoothness energies (also called smoothing energies) in geometry processing, and introduces new methods as well as a mathematical proof of correctness and convergence for a well-established method. Geometry processing deals with the acquisition, modification, and output (be it on a screen, in virtual reality, or via fabrication and manufacturing) of complex geometric objects and data. It is closely related to computer graphics, but is also used by many other fields that employ applied mathematics in the context of geometry. The popular Laplacian energy is a smoothness energy that quantifies smoothness and that is closely related to the biharmonic equation (which gives it desirable properties). Minimizers of the Laplacian energy solve the biharmonic equation. This thesis provides a proof of correctness and convergence for a very popular discretization method for the biharmonic equation with zero Dirichlet and Neumann boundary conditions, the piecewise linear Lagrangian mixed finite element method. The same approach also discretizes the Laplacian energy. Such a proof has existed for flat surfaces for a long time, but there exists no such proof for the curved surfaces that are needed to represent the complicated geometries used in geometry processing. This proof will improve the usefulness of this discretization for the Laplacian energy. In this thesis, the novel Hessian energy for curved surfaces is introduced, which also quantifies the smoothness of a functions, and whose minimizers solve the biharmonic equation. This Hessian energy has natural boundary conditions that allow the construction of functions that are not significantly biased by the geometry and presence of boundaries in the domain (unlike the Laplacian energy with zero Neumann boundary conditions), while still conforming to constraints informed by the application. This is useful in any situation where the boundary of the domain is not an integral part of the problem itself, but just an artifact of data representation---be it, because of artifacts created by an imprecise scan of the surface, because information is missing outside of a certain region, or because the application simply demands a result that should not depend on the geometry of the boundary. Novel discretizations of this energy are also introduced and analyzed. This thesis also presents the new developability energy, which quantifies a different kind of smoothness than the Laplacian and Hessian energies: how easy is it to unfold a surface so that it lies flat on the plane without any distortion (surfaces for which this is possible are called developable surfaces). Developable surfaces are interesting, as they can be easily constructed from cheap material such as paper and plywood, or manufactured with methods such as 5-axis CNC milling. A novel definition of developability for discrete triangle meshes, as well as a variety of discrete developability energies are also introduced and applied to problems such as approximation of a surface by a piecewise developable surface, and the design and fabrication of piecewise developable surfaces. This will enable users to more easily take advantages of these cheap and quick fabrication methods. The novel methods, algorithms and the mathematical proof introduced in this thesis will be useful in many applications and fields, including numerical analysis of elliptic partial differential equations, geometry processing of triangle meshes, character animation, data denoising, data smoothing, scattered data interpolation, fabrication from simple materials, computer-controlled fabrication, and more.

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.

The aim of the thesis is to bring new insights into banks' internal credit risk estimates and their application in estimation of credit transition matrices, which are an important part of credit risk modelling with limited publicly available sources. The doctoral thesis consists of three essays that jointly analyse features of bank- sourced credit risk data and practicalities of transition matrices estimation. In the first essay, I empirically test two assumptions widely used for estimation of transition matrices: Markovian property and time homogeneity. The results indicate that internal credit risk estimates do not satisfy the two assumptions, showing evidence of both path-dependency and time heterogeneity even within a period of economic expansion. Contradicting previous findings based on data from credit rating agencies, banks tend to revert their past rating actions. The second essay analyses the extent to which transition matrices depend on the characteristics of the underlying overlapping bank-sourced credit risk datasets and the aggregation method. It outlines that the choice of aggregation approach has a substantial effect on credit risk model results. I also show that bank-sourced transition matrices are more dynamic than those produced by credit rating agencies and introduce industry-specific...

The study of matrices have been of interest to mathematicians for some time. Recently the use of matrices has assumed greater importance also in the fields of management, social science, and natural science because they are very useful in the organization and presentation of data and in the solution of linear equations. The theory of matrices is yet another type of mathematical model which we can use to solve many problems that arise in these fields. The aim of this paper is to show how we can use matrices and their mathematical model to solve some problems in the process of routing. First we will introduce the term of routing and the new approach in the process of selecting paths. We will show some simple examples. We will also pint out how we can learn about matrices in the classroom. At the end we will discuss about advantages and potential disadvantages that may occur in the described technique.

A k × n array with entries from a q-letter alphabet is called a t-covering array if each t × n submatrix contains amongst its columns each one of the gt different words of length t that can be produced by the q letters. In the present article we use a probabilistic approach based on an appropriate Markov chain embedding technique, to study a t-covering problem where, instead of looking at all possible t ×n submatrices, we consider only submatrices of dimension t ×n with its rows being consecutive rows of the original k × n array. Moreover, an exact formula is established for the probability distribution function of the random variable, which enumerates the number of deficient submatrices (i.e., submatrices with at least one missing word, amongst their columns), in the case of a k × n binary matrix (q = 2) obtained by realizing kn Bernoulli variables.

Consider an m x zero-one matrix A. An s x t submatrix of A is said to be even if the sum of its entries is even. In this paper, we focus on the case m = n and s = t = 2. The maximum number M(n) of even 2 x 2 submatrices of A is clearly ( 2n) 2, and corresponds to the matrix A having, e.g., all ones (or zeros). A more interesting question, motivated by Turán numbers and Hadamard matrices, is that of the minimum number m(n) of such matrices. It has recently been shown that m(n) ≥ 1/2 ( 2n) 2 - Bn 3 for some constant B. In this paper we show that if the matrix A = A n is considered to be induced by an infinite zero one matrix obtained at random, then P(E n ≤1/2( 2n) 2 - Cn 2 log n infinitely often) = 0, where E n denotes the number of even 2 x 2 submatrices of A n. Results such as these provide us with specific information about the tightness of the concentration of E n around its expected value of 1/2 ( 2n) 2.