This thesis studies various aspects of differential algebra, from fundamental concepts to practical computations. A characteristic feature of this work is the use of combinatorial techniques, which offer a unique and original perspective on the subject matter. First, we establish the connection between the n-jet space of the fat point defined by xm and the stable set polytope of a perfect graph. We prove that the dimension of the coordinate ring of the scheme defined by polynomial arcs of degree less than or equal to n is a polynomial in m of degree n + 1. This is based on Zobnin’s result which states that the set {x^m} is a differential Gr ̈obner basis for its differential ideal. We generalize this statement to the case of two independent variables and link the dimensions in this case to some triangulations of the p × q rectangle, where the pair (p, q) now plays the role of n. Second, we study the arc space of the fat point x^m on a line from the point of view of filtration by finite-dimensional differential algebras. We prove that the generating series of the dimensions of these differential algebras is m/(1 -mt) . Based on this we propose a definition of the multiplicity of a solution of an algebraic differential equation as the growth of the dimensions of these differential algebras. This generalizes the concept of the multiplicity of an ideal in a polynomial ring. Furthermore, we determine a full description of the set of standard monomials of the differential ideal generated by x^m. This description proves a conjecture by Afsharijoo concerning a new version of the Roger-Ramanujan identities. Every homogeneous linear system of partial differential equations with constant coef- ficients can be encoded by a submodule of the ring of polynomials. We develop practical methods for computing the space of solutions to these PDEs. These spaces are typically infinite dimensional, and we use the Ehrenpreis–Palamodov Theorem for finite encoding. We apply this finite encoding to the solutions of the PDEs associated with the arc spaces of a double point. We prove that these vector spaces are spanned by determinants of some special Wronskians, and we relate them to differentially homogeneous polynomials. Finally, we introduce D-algebraic functions: they are solutions to algebraic differential equations. We study closure properties of these functions. We present practical algorithms and their implementations for carrying out arithmetic operations on D-algebraic functions. This amounts to solving elimination problems for differential ideals.

Determining the chromatic number of the partition graph P(33) poses a considerable challenge. We can bound it to 4 ≤ χ(P(33)) ≤ 6, with exhaustive search confirming χ(P(33)) = 6. A potential mathematical proof strategy for this equality involves identifying a Z2-invariant S4 with non-trivial homology in the box complex of the partition graph P(33), namely Bedge(︁P(33))︁, and applying the Borsuk-Ulam theorem to compute its Z2-index. This provides a robust topological lower bound for the chromatic number of P(33), termed the Lovász bound. We have verified the absence of such an S4 within certain sections of Bedge(︁P(33))︁. We also validated this approach through a case study on the Petersen graph. This thesis offers a thorough examination of various topological lower bounds for a graph’s chromatic number, complete with proofs and examples. We demonstrate instances where these lower bounds converge to a single value and others where they diverge significantly from a graph’s actual chromatic number. We also classify all vertex pairs, triples, and quadruples of P(33) into unique equivalence classes, facilitating the derivation of all maximal complete bipartite subgraphs. This classification informs the construction of all simplices of Bedge(︁P(33)). Following a detailed and technical exploration, we uncover both the maximal size of the pairwise intersections of its maximal simplices and their underlying structure. Our study proposes an algorithm for building the box complex of the partition graph P(33) using our method of identifying maximal complete bipartite subgraphs. This reduces time complexity to O(n3), marking a substantial enhancement over brute-force techniques. Lastly, we apply discrete Morse theory to construct a simplicial complex homotopy equivalent to the box complex of P(33), using two methods: elementary collapses and the determination of a discrete Morse function on the box complex. This process reduces the dimension of the box complex from 35 to 12, streamlining future calculations of the Z2-index and the Lovász bound.

<p>Despite the unprecedented success achieved by Kohn-Sham density functional theory (KS-DFT) in the past few decades, the standard approximations used for the KS exchange-correlation functional typically lead to unacceptably large errors when applied to strongly-correlated electronic systems. Partition-DFT (P-DFT) is a formally exact reformulation of KS-DFT in which the ground-state density and energy of a system are obtained through self-consistent calculations on isolated fragments, with a partition energy introduced to account for the inter-fragment interactions. The unique advantage of this partitioning scheme lies in the fact that it adopts the electron density of fragments as the main variable, in place of the density of the entire system in KS-DFT, so that novel approximations can be constructed in terms of fragment properties. With a simple overlap approximation (OA) of the partition energy proposed for binary-partitioned systems, P-DFT is able to rectify the static correlation error caused by standard density functional approximations for strongly-correlated diatomic molecules. In this work, we first implement P-DFT on a one-dimensional (1D) real-space grid and calculate the ground-state energy and density of a series of 1D hydrogen chains using the local density approximation (LDA) as the density functional approximation for fragments. We then propose the generalized overlap approximation (GOA) and the corrected generalized overlap approximation (cGOA), which extends the applicability of OA to systems partitioned into more than two fragments. Combining LDA with cGOA leads to quantitatively correct dissociation curves of hydrogen chains. The static correlation error of LDA is suppressed by cGOA in the strongly-correlation regime when the calculations are performed in a spin-restricted manner, i.e., without the spin symmetry breaking. Additionally, GOA induces an improvement of the ground-state density upon LDA results, and hence helps P-DFT provide a better description of the density dimerization in hydrogen chains.</p>

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[pt] A injeção química, principalmente de surfactantes, é um dos métodos mais utilizados na recuperação melhorada de petróleo. Uma das principais limitações deste método é a perda devido à partição do surfactante para o óleo presente no reservatório. O estudo do equilíbrio de partição água-óleo de formulações de surfactantes torna-se relevante devido a perda existente nos reservatórios, afetando a inviabilidade econômica e ambiental da aplicação. O objetivo deste trabalho foi determinar o coeficiente de partição de surfactantes entre as fases água e óleo através de diferentes métodos analíticos. Foram estudados os surfactantes dodecilbenzeno sulfonato de sódio (SDBS, de tipo aniônico) e o polioxietileno (9- 10) p-teroctil fenol (Triton X-100, de tipo não iônico), e foi utilizado hexadecano como óleo modelo. Com o objetivo de identificar qual seria o melhor procedimento para a quantificação dos surfactantes na fase aquosa e, por conseguinte determinar o coeficiente de partição, foram desenvolvidas três metodologias de quantificação utilizando diferentes técnicas (medidas de tensão interfacial, medidas de absorbância UV-Vis e HPLC com detecção UV). Testes iniciais de solubilidade mostraram que o SDBS é praticamente insolúvel em presença de sal, pelo qual só foram realizados experimentos com este surfactante, em ausência de sal. Para ambos os surfactantes os resultados mostraram uma baixa partição para a fase oleosa, tanto na ausência quanto na presença de sal. Em algumas soluções foi identificada a formação de emulsões devido à concentração do surfactante e a proporção água/óleo utilizada, o qual interferiu com o método espectrofotométrico. Adicionalmente, se estabeleceu uma comparação entre os resultados obtidos pelas metodologias desenvolvidas que permitiram identificar que o melhor dos métodos estudados para a avaliação do equilíbrio de partição foi a cromatografia líquida de alta eficiência (HPLC). A partir destes resultados, pode-se concluir que os surfactantes estudados possuem um baixo valor de coeficiente de partição para a fase óleo, tornando o método de injeção química favorável para a recuperação avançada de petróleo. / [en] Chemical injection, mainly of surfactants, is one of the most used methods for improved oil recovery. One of the main limitations of this method is the loss due to partitioning of the surfactant into the oil present in the reservoir. The study of the water-oil partition balance of surfactant formulations becomes relevant due to the existing loss in the reservoirs, affecting the economic and environmental unfeasibility of the application. The objective of this work was to determine the partition coefficient of surfactants between the water and oil phases using different analytical methods. The surfactants sodium dodecylbenzene sulfonate (SDBS, anionic type) and polyoxyethylene (9-10) p-teroctyl phenol (Triton X-100, nonionic type) were studied, and hexadecane was used as model oil. In order to identify which would be the best procedure for the quantification of surfactants in the aqueous phase and, therefore, to determine the partition coefficient, three quantification methodologies were developed using different techniques (interfacial tension measurements, UV-Vis absorbance measurements, and HPLC with UV detection). Initial solubility tests showed that SDBS is practically insoluble in the presence of salt, so experiments with this surfactant were only carried out in the absence of salt. For both surfactants, the results showed a low partition for the oil phase, both in the absence and in the presence of salt. In some solutions, the formation of emulsions was identified due to the surfactant concentration and the water/oil ratio used, which interfered with the spectrophotometric method. In addition, a comparison was established between the results obtained by the developed methodologies, which allowed to identify that the best method for the evaluation of the partition equilibrium was the high performance liquid chromatography (HPLC). From these results, it can be concluded that the studied surfactants have a low partition coefficient for the oil phase, making the chemical injection method favorable for advanced oil recovery.