We show that replicator equations follow naturally from the exponential affine structure of the simplex known from information geometry. It is then natural to call replicator equations linear if their fitness function is affine. For such linear replicator equations an explicit solution can be found. The approach is also demonstrated for the example of Eigen’s hypercycle, where some new analytic results are obtained using the explicit solution.

In this thesis we explore the geometric representation theory of shifted quantum affine algebras 𝒜^𝜇, using the critical K-theory of certain moduli spaces of infinite flags of quiver representations resembling the moduli of quasimaps to Nakajima quiver varieties. These critical K-theories become 𝒜^𝜇-modules via the so-called critical R-matrix 𝑅, which generalizes the geometric R-matrix of Maulik, Okounkov, and Smirnov. In the asymptotic limit corresponding to taking infinite instead of finite flags, singularities appear in 𝑅 and are responsible for the shift in 𝒜^𝜇. The result is a geometric construction of interesting infinite-dimensional modules in the category 𝒪 of 𝒜^𝜇, including e.g. the pre-fundamental modules previously introduced and studied algebraically by Hernandez and Jimbo. Following Nekrasov, we provide a very natural geometric definition of qq-characters for our asymptotic modules compatible with the pre-existing definition of q-characters. When 𝒜^𝜇 is the shifted quantum toroidal gl₁ algebra, we construct asymptotic modules DT_𝜇 and PT_𝜇 whose combinatorics match those of (1-legged) vertices in Donaldson--Thomas and Pandharipande--Thomas theories. Such vertices control enumerative invariants of curves in toric 3-folds, and finding relations between (equivariant, K-theoretic) DT and PT vertices with descendent insertions is a typical example of a wall-crossing problem. We prove a certain duality between our DT_𝜇 and PT_𝜇 modules which, upon taking q-/qq-characters, provides one such wall-crossing relation.

Le problème de discernabilité des comportements entrées-sorties de deux systèmes dynamiquesse pose dans de nombreuses applications telles que l’observation et la commande dessystèmes dynamiques hybrides. Dans cette thèse, nous nous intéressons à la caractérisation decette propriété de discernabilité des comportements entrées-sorties. Pour la classe des systèmesdynamiques linéaires et non-linéaires affines en la commande, nous établissons : des conditionsde discernabilité stricte qui garantissent la discernabilité des systèmes quelles que soient lescommandes qui leur sont conjointement appliquées ; des conditions de discernabilité contrôlablequi assurent l’existence d’au moins une commande qui rend discernable les sorties ; desconditions de résidu-discernabilité qui caractérisent la discernabilité à travers les résidus issusde la méthode de l’espace de parité. Outre ces différentes conditions, nous spécifions dans le caslinéaire, une forme de distance qui permet de quantifier pour une commande donnée, le degréde discernabilité des systèmes ainsi que la robustesse de la propriété de discernabilité. / The distinguishability of the input-output behavior of two dynamical systems plays a crucialrole in many applications such as control and observation of hybrid dynamical systems. Thisthesis aims to characterize this property of distinguishability. For linear systems and nonlinearcontrol-affine systems, we establish: conditions for strict distinguishability that ensure thedistinguishability of the systems for every control input jointly applied to them; conditions forcontrolled-distinguishability that guarantee the existence of a control input which makes distinguishable the outputs of the systems; conditions for residual-distinguishability that characterize the distinguishability of the modes through parity-space residuals. Moreover, in the linear case, a metric is specified in order to quantify for a given control input, the distinguishability degreeof the systems and the robustness of the property of distinguishability.

Affine term structure models (ATSMs) are one set of popular models for yield curve modeling. Given that the models forecast yields based on the speed of mean reversion, under what circumstances can we distinguish one ATSM from another? The objective of my dissertation is to quantify the benefit of knowing the “true” model as well as the cost of being wrong when choosing between ATSMs. In particular, I detail the power of out-of-sample forecasts to statistically distinguish one ATSM from another given that we only know the data are generated from an ATSM and are observed without errors. My study analyzes the power and size of affine term structure models (ATSMs) by evaluating their relative out-of-sample performance. Essay one focuses on the study of the oneactor ATSMs. I find that the model’s predictive ability is closely related to the bias of mean reversion estimates no matter what the true model is. The smaller the bias of the estimate of the mean reversion speed, the better the out-of-sample forecasts. In addition, my finding shows that the models' forecasting accuracy can be improved, in contrast, the power to distinguish between different ATSMs will be reduced if the data are simulated from a high mean reversion process with a large sample size and with a high sampling frequency. In the second essay, I extend the question of interest to the multiactor ATSMs. My finding shows that adding more factors in the ATSMs does not improve models' predictive ability. But it increases the models' power to distinguish between each other. The multiactor ATSMs with larger sample size and longer time span will have more predictive ability and stronger power to differentiate between models.

In this Thesis we study two types of mechanical nonholonomic systems, namely systems with linear constraints and lagrangian with a linear term in the velocities, and nonholonomic systems with affine constraints and lagrangian without a linear term in the velocities. For the former type of systems we construct an almost-Poisson bracket using elements related to a riemannian metric induced by the kinetic energy, and we show that under certain conditions gauge momenta exist. For the latter type of systems, we focus on the ones possessing a \emph{Noether symmetry}. To everyone of these systems we associate an equivalent system of the former type, and we exhibit the procedure to relate them and their gauge momentum. As a test case for the theory, we analyze the system of a heavy ball rolling without slipping on a rotating surface of revolution: we elucidate that also in this framework the so-called Routh integrals are related to symmetries, we give conditions for boundedness of the motions. In the particular case the surface of revolution is an inverted cone we characterize the qualitative behavior of the motions.

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Given a finite Weyl group W_fin with root system Phi_fin, one can create the affine Weyl group W_aff by taking the semidirect product of the translation group associated to the coroot lattice for Phi_fin, with W_fin. The double affine Weyl semigroup W can be created by using a similar semidirect product where one replaces W_fin with W_aff and the coroot lattice with the Tits cone of W_aff. We classify cocovers and covers of a given element of W with respect to the Bruhat order, specifically when W is associated to a finite root system that is irreducible and simply laced. We show two approaches: one extending the work of Lam and Shimozono, and its strengthening by Milicevic, where cocovers are characterized in the affine case using the quantum Bruhat graph of W_fin, and another, which takes a more geometrical approach by using the length difference set defined by Muthiah and Orr. / Doctor of Philosophy / The Bruhat order is a way of organizing elements of the double affine Weyl semigroup so that we have a better understanding of how the elements interact. In this dissertation, we study the Bruhat order, specifically looking for when two elements are separated by exactly one step in the order. We classify these elements and show that there are only finitely many of them.

We construct a strong filtered graph $\Gamma_s(\Lambda)$ dependent on the dominant weight $\Lambda$, and a weak filtered graph $\Gamma_w(\Kcen)$ dependent on the canonical central element $\Kcen$ for an arbitrary Kac-Moody algebra $g$. In our construction, both graphs $(\Gamma_s(\Lambda), \Gamma_w(\Kcen))$ have the vertex set as the Weyl group of $g$, with the grading given by the length function. The edges of the graph $\Gamma_s(\La)$ are labeled versions of the $\lambda$-chain model of K-Chevalley rules for Kac-Moody flag manifolds as developed by Lenart and Shimozono, originally defined by Lenart and Postnikov. Meanwhile, the labels on $\Gamma_w(\Kcen)$ come from the dual multiplication map of K-cohomology of affine Grassmannian $Gr_G$. We conjecture that the strong filtered graph and weak filtered graph are dual, which means we get an identity when we apply the up and down operators on the vertices. We proved this identity except one case that where we call the chain is $j$-present. Our identity is similar to the Möbius construction of the dual filtered graph, as previously studied by Patrias and Pylyavskyy, and in fact, in the limit $n\rightarrow \infty$ of the $A^{(1)}_{n-1}$, our construction recovers their identity. We also expect to recover their combinatorics of Möbius deformation of the shifted Young's lattice in type $C^{(1)}_n$ as $n$ approaches infinity. / Doctor of Philosophy / In this thesis, we introduce a pair of graphs $(\Gamma_s(\La),\Gamma_w(\Kcen))$ motivated by the study of affine Schubert calculus. Affine Schubert calculus emerges as an extension and generalization of classical Schubert calculus, which involves questions such as determining the number of lines intersecting four lines in three-dimensional space. This type of questions can often be translated into computations aimed at finding the structure constants for the Schubert basis in the K-(co)homology of the flag varieties such as affine Grassmannian. These structure constants represent the coefficients of the Schubert basis in the product of the other two Schubert bases, all indexed by the Weyl group of the affine Lie algebra $g$. We define up and down operators on the vertices of graphs $(\Gamma_s(\Lambda), \Gamma_w(\Kcen))$, which are elements in Weyl group of $g$, utilizing the structure constants as essential components. We conjecture that, in general, and prove in certain cases, this approach yields new identities for these operators, leading us to define this pair of graphs as a dual filtered graph.

Yes / This paper compares several widely-used and recently-developed methods to extract risk-neutral densities (RND) from option prices in terms of estimation accuracy. It shows that positive convolution approximation method consistently yields the most accurate RND estimates, and is insensitive to the discreteness of option prices. RND methods are less likely to produce accurate RND estimates when the underlying process incorporates jumps and when estimations are performed on sparse data, especially for short time-to-maturities, though sensitivity to the discreteness of the data differs across different methods.