Indiana University-Purdue University Indianapolis (IUPUI) / We consider actions of the current Lie algebras $\gl_{n}[t]$ and $\gl_{k}[t]$ on the space $\mathfrak{P}_{kn}$ of polynomials in $kn$ anticommuting variables. The actions depend on parameters $\bar{z}=(z_{1},\dots ,z_{k})$ and $\bar{\alpha}=(\alpha_{1},\dots ,\alpha_{n})$, respectively. We show that the images of the Bethe algebras $\mathcal{B}_{\bar{\alpha}}^{\langle n \rangle}\subset U(\gl_{n}[t])$ and $\mathcal{B}_{\bar{z}}^{\langle k \rangle}\subset U(\gl_{k}[t])$ under these actions coincide. To prove the statement, we use the Bethe ansatz description of eigenvectors of the Bethe algebras via spaces of quasi-exponentials. We establish an explicit correspondence between the spaces of quasi-exponentials describing eigenvectors of $\mathcal{B}_{\bar{\alpha}}^{\langle n \rangle}$ and the spaces of quasi-exponentials describing eigenvectors of $\mathcal{B}_{\bar{z}}^{\langle k \rangle}$. One particular aspect of the duality of the Bethe algebras is that the Gaudin Hamiltonians exchange with the Dynamical Hamiltonians. We study a similar relation between the trigonometric Gaudin and Dynamical Hamiltonians. In trigonometric Gaudin model, spaces of quasi-exponentials are replaced by spaces of quasi-polynomials. We establish an explicit correspondence between the spaces of quasi-polynomials describing eigenvectors of the trigonometric Gaudin Hamiltonians and the spaces of quasi-exponentials describing eigenvectors of the trigonometric Dynamical Hamiltonians. We also establish the $(\gl_{k},\gl_{n})$-duality for the rational, trigonometric and difference versions of Knizhnik-Zamolodchikov and Dynamical equations.

We investigate a special case of the quantum integrable Heisenberg spin chain known as Gaudin model. The Gaudin model is an important example of quantum integrable systems. We study the Gaudin model for the Lie algebra s[z(<C). The key problem is to find the spectrum and the corresponding eigenvectors of the commuting Hamiltonians. The standard method to solve this type of classical problem was introduced by H. Bethe and is known as the Bethe-Ansatz. Bethe's technique has proven to be very powerful in various areas of modem many-body theory and statistical mechanics. [19], [14], [4] Following Sklyanin's ideas in [19], we derive the Bethe-Ansatz equations for sl2(<C). Solving the Bethe-Ansatz equations is equivalent to finding polynomial solutions of the Lame differential equation, which has a meaning in electrostatics. We derive this equation for sl2(<C), and investigate its special cases. We discuss classical and more recent results on the Gaudin spin chain for sl2(<C) and provide numerical evidence for new observations in the real case of the Lame equation. Using roots of classical polynomials known as Jacobi polynomials, which are solutions to a special case of the Lame equation, we numerically approximate solutions to the Lame equation in more complicated settings. We discuss the Gaudin model associated to the Lie algebra sl3(C). Using the Bethe-Ansatz equations for sl3(C), we provide solutions in special cases. / Thesis / Master of Science (MSc)

Ingeniero Civil de Minas / Actualmente se ha tomado consciencia lo fundamental que es medir y controlar la granulometría resultante de la etapa de tronadura, ya que siendo ésta la primera instancia de conminución de la roca in-situ, tendrá gran influencia en la eficiencia de los procesos que lo prosiguen, tales como el carguío, transporte y reducción secundaria entre otros. La presente investigación pretende elaborar un modelo que tenga como objetivo la predicción de la distribución granulométrica en túneles, en función de los parámetros de diseño de perforación y tronadura, y de las propiedades físicas de la roca. A pesar de que existen en la literatura modelos que cumplen con el objetivo como el modelo Kuz-Ram y Swebrec , éstos son complejos, es por eso que el modelo elaborado en este trabajo tendrá la ventaja de ser simple, sin que eso merme el acierto de la predicción. Para construir el modelo se han obtenido fotografías de la marina resultante de la tronadura de una faena en particular (en este caso de la Mina Esmeralda, de la división El Teniente), de las cuales se estimará una distribución granulométrica por medio del análisis de imágenes. Se utiliza la distribución de Gaudin-Schuhmann para la construcción del modelo, relacionando de manera lineal los parámetros de ésta con las variables litológicas y de diseño. El modelo construido presenta un error promedio de 7.8%, y es válido para el rango de parámetros bajo los cuáles fue ajustado. Una aplicación directa del mismo es la capacidad predictiva del modelo con el cual fue posible diseñar una malla de perforación en las galerías del nivel de acarreo, y que tenga por finalidad una redistribución de los tiros sin que esto implique un aumento significativo en el tamaño máximo de fragmento resultante. Como conclusiones principales de este trabajo, se observa que el tamaño máximo resultante está condicionado por los parámetros burden, espaciamiento y largo de avance de los tiros (quienes limitan el volumen del fragmento en sus tres dimensiones), es decir, al aumentar los parámetros geométricos de diseño del diagrama de perforación, se observa un aumento del tamaño máximo de partícula, lo cual concuerda con la teoría básica de perforación y tronadura. Por otro lado se tiene que al aumentar el burden, espaciamiento y el factor de carga, se observa un aumento en la heterogeneidad de la muestra. La aplicación de esta herramienta trae consigo grandes beneficios para el ciclo minero, ya que permite al ingeniero de perforación y tronadura rediseñar los diagramas de disparo en función de una distribución granulométrica deseada, permitiendo optimizar la cantidad de perforaciones realizadas por avance, lo cual a largo plazo se traduce, no tan sólo en un ahorro de tiempo, sino que además en un cuantioso ahorro de insumos para la operación. Cabe destacar, además, que esta metodología es replicable para cualquier otra condición de perforación y tronadura.

Indiana University-Purdue University Indianapolis (IUPUI) / We study the Gaudin model associated to Lie algebras of classical types. First, we derive explicit formulas for solutions of the Bethe ansatz equations of the Gaudin model associated to the tensor product of one arbitrary finite-dimensional irreducible module and one vector representation for all simple Lie algebras of classical type. We use this result to show that the Bethe Ansatz is complete in any tensor product where all but one factor are vector representations and the evaluation parameters are generic. We also show that except for the type D, the joint spectrum of Gaudin Hamiltonians in such tensor products is simple. Second, we define a new stratification of the Grassmannian of N planes. We introduce a new subvariety of Grassmannian, called self-dual Grassmannian, using the connections between self-dual spaces and Gaudin model associated to Lie algebras of types B and C. Then we obtain a stratification of self-dual Grassmannian.

This thesis explores connections between the Gaudin Hamiltonians in type A and the combinatorics of tableaux. The cactus group acts on standard tableaux via the Schützenberger involution. We show in this thesis that the action of the cactus group on standard tableaux can be recovered as a monodromy action of the cactus group on the simultaneous spectrum of the Gaudin Hamiltonians. More precisely, we consider the action of the Bethe algebra, which contains the Gaudin Hamiltonians, on the multiplicity space of a tensor product of irreducible glr-modules. The spectrum of this algebra forms a flat and finite family over M0,n+1(C). We use work of Mukhin, Tarasov and Varchenko, who link this spectrum to certain Schubert intersections, and work of Speyer, who extends these Schubert intersections to a flat and finite map over the entire moduli space of stable curves M0,n+1(C). We show the monodromy over the real points M0,n+1(R) can be identified with the action of the cactus group on a tensor product of irreducible glr-crystals. Furthermore we show this identification is canonical with respect to natural labelling sets on both sides.

Cette thèse a pour sujet une classe de théories des champs intégrables appelées modèles avec fonction twist. Les principaux exemples de tels modèles sont les modèles sigma non-linéaires intégrables, tel le Modèle Principal Chiral, et leurs déformations. Un premier résultat obtenu est la preuve que le modèle dit de Bi-Yang-Baxter, qui est une déformation à deux paramètres du Modèle Principal Chiral, est lui aussi un modèle avec fonction twist. Il est ensuite montré que les déformations de type Yang-Baxter modifient certaines symétries globales du modèle non déformé en symétries de Poisson-Lie. Un autre chapitre concerne la construction d'une infinité de charges locales en involution pour tous les modèles sigma intégrables et leurs déformations : ce résultat repose sur le formalisme général partagé par tous ces modèles en tant que théories des champs avec fonction twist.La seconde partie de la thèse a pour sujet les modèles de Gaudin. Ceux-ci sont des modèles intégrables associés à des algèbres de Lie. En particulier, les théories des champs avec fonction twist sont liées aux modèles de Gaudin associés à des algèbres de Lie affines. Une approche standard pour l'étude du spectre des modèles de Gaudin quantiques sur des algèbres finies est celle de Feigin-Frenkel-Reshetikhin. Dans cette thèse, des généralisations de cette approche sont conjecturées, motivées et testées. L'une d'elles concerne les modèles de Gaudin finis dits cyclotomiques. La seconde porte sur les modèles de Gaudin associés à des algèbres affines. / This thesis deals with a class of integrable field theories called models with twist function. The main examples of such models are integrable non-linear sigma models, such as the Principal Chiral Model, and their deformations. A first obtained result is the proof that the so-called Bi-Yang-Baxter model, which is a two-parameter deformation of the Principal Chiral Model, is also a model with twist function. It is then shown that Yang-Baxter type deformations modify certain global symmetries of the undeformed model into Poisson-Lie symmetries. Another chapter concerns the construction of an infinite number of local charges in involution for all integrable sigma models and their deformations: this result is based on the general formalism shared by all these models as field theories with twist function.The second part of the thesis concerns Gaudin models. These are integrable models associated with Lie algebras. In particular, field theories with twist function are related to Gaudin models associated with affine Lie algebras. A standard approach for studying the spectrum of quantum Gaudin models over finite algebras is the one of Feigin-Frenkel-Reshetikhin. In this thesis, generalisations of this approach are conjectured, motivated and tested. One of them deals with the so-called cyclotomic finite Gaudin models. The second one concerns the Gaudin models associated with affine Lie algebras.

<div>We study the Gaudin model associated to Lie algebras of classical types.</div><div><br></div><div>First, we derive explicit formulas for solutions of the Bethe ansatz equations of the Gaudin model associated to the tensor product of one arbitrary finite-dimensional irreducible module and one vector representation for all simple Lie algebras of classical type. We use this result to show that the Bethe Ansatz is complete in any tensor product where all but one factor are vector representations and the evaluation parameters are generic. We also show that except for the type D, the joint spectrum of Gaudin Hamiltonians in such tensor products is simple.</div><div><br></div><div>Second, we define a new stratification of the Grassmannian of N planes. We introduce a new subvariety of Grassmannian, called self-dual Grassmannian, using the connections between self-dual spaces and Gaudin model associated to Lie algebras of types B and C. Then we obtain a stratification of self-dual Grassmannian. </div>

Indiana University-Purdue University Indianapolis (IUPUI) / We describe a reproduction procedure which, given a solution of the gl(m|n) Gaudin Bethe ansatz equation associated to a tensor product of polynomial modules, produces a family P of other solutions called the population. To a population we associate a rational pseudodifferential operator R and a superspace W of rational functions. We show that if at least one module is typical then the population P is canonically identified with the set of minimal factorizations of R and with the space of full superflags in W. We conjecture that the singular eigenvectors (up to rescaling) of all gl(m|n) Gaudin Hamiltonians are in a bijective correspondence with certain superspaces of rational functions. We establish a duality of the non-periodic Gaudin model associated with superalgebra gl(m|n) and the non-periodic Gaudin model associated with algebra gl(k). The Hamiltonians of the Gaudin models are given by expansions of a Berezinian of an (m+n) by (m+n) matrix in the case of gl(m|n) and of a column determinant of a k by k matrix in the case of gl(k). We obtain our results by proving Capelli type identities for both cases and comparing the results. We study solutions of the Bethe ansatz equations of the non-homogeneous periodic XXX model associated to super Yangian Y(gl(m|n)). To a solution we associate a rational difference operator D and a superspace of rational functions W. We show that the set of complete factorizations of D is in canonical bijection with the variety of superflags in W and that each generic superflag defines a solution of the Bethe ansatz equation. We also give the analogous statements for the quasi-periodic supersymmetric spin chains.

The repulsive delta-function interaction model in one dimension is reviewed for spinless particles and for spin-1/2 fermions. The problem of solving the differential equation related to the Schrödinger equation is reduced by the Bethe ansatz to a system of algebraic equations. The delta-function interaction is shown to have no effect on spinless fermions which therefore behave like free fermions, in agreement with Pauli's exclusion principle. The ground-state problem of spinless bosons is reduced to an inhomogeneous Fredholm equation of the second kind. In the limit of impenetrable interactions, the spinless bosons are shown to have the energy spectrum of free fermions. The model for spin-1/2 fermions is reduced by the Bethe ansatz to an eigenvalue problem of matrices of the same sizes as the irreducible representations R of the permutation group of N elements. For some R's this eigenvalue problem itself is solved by a generalized Bethe ansatz. The ground-state problem of spin-1/2 fermions is reduced to a generalized Fredholm equation.

We describe a reproduction procedure which, given a solution of the gl(m|n) Gaudin Bethe ansatz equation associated to a tensor product of polynomial modules, produces a family P of other solutions called the population. <br>To a population we associate a rational pseudodifferential operator R and a superspace W of rational functions. <br><br>We show that if at least one module is typical then the population P is canonically identified with the set of minimal factorizations of R and with the space of full superflags in W. We conjecture that the singular eigenvectors (up to rescaling) of all gl(m|n) Gaudin Hamiltonians are in a bijective correspondence with certain superspaces of rational functions.<br><br>We establish a duality of the non-periodic Gaudin model associated with superalgebra gl(m|n) and the non-periodic Gaudin model associated with algebra gl(k).<br><br>The Hamiltonians of the Gaudin models are given by expansions of a Berezinian of an (m+n) by (m+n) matrix in the case of gl(m|n) <br>and of a column determinant of a k by k matrix in the case of gl(k). We obtain our results by proving Capelli type identities for both cases and comparing the results.<br><br>We study solutions of the Bethe ansatz equations of the non-homogeneous periodic XXX model associated to super Yangian Y(gl(m|n)).<br>To a solution we associate a rational difference operator D and a superspace of rational functions W. We show that the set of complete factorizations of D is in canonical bijection with the variety of superflags in W and that each generic superflag defines a solution of the Bethe ansatz equation. We also give the analogous statements for the quasi-periodic supersymmetric spin chains.<br>