When solving problems involving quantum mechanical systems, it is frequently desirable to find the matrix elements of a unitary representation $T$ of a real algebraic Lie group $G$. This requires defining an inner product on the Hilbert space $\mathbb{H}$ that carries the representation $T$. In the case where the representation is determined by a representation of a subgroup containing the lowest weight vector of $T$, this can be achieved through the coherent state construction. In both the scalar and vector coherent state methods, the process of finding the overlaps can be simplified by introducing the coherent state triplet ($\mathfrak{F}_{\mathbb{H}_D}$, $\mathbb{H}_D$, $\mathfrak{F}^{\mathfrak{H}_D}$) of Bargmann spaces. Coherent state wave functions -- the elements of $\mathfrak{F}_{\mathbb{H}_D}$ and of $\mathfrak{F}^{\mathbb{H}_D}$ -- are used to define the inner product on $\mathbb{H}_D$ in a way that simplifies the calculation of the overlaps. This inner product and the group action $\Gamma$ of $G$ on $\mathfrak{F}^{\mathbb{H}_D}$ are used to formulate expressions for the matrix elements of $T$ with coefficients from the given subrepresentation. The process of finding an explicit expression for $\Gamma$ relies on matrix factorizations in the complexification of $G$ even though the representation $T$ does not extend to the complexification. It will be shown that these factorizations are, in fact, justified, that the overlaps and $\Gamma$ action can be expressed in terms of the given subrepresentation, and that it is possible to find numerical values for the inner product in $\mathbb{H}$. The scalar and vector coherent state methods will both be applied to Sp($n$) and Sp($n,\mathbb{R}$).

When solving problems involving quantum mechanical systems, it is frequently desirable to find the matrix elements of a unitary representation $T$ of a real algebraic Lie group $G$. This requires defining an inner product on the Hilbert space $\mathbb{H}$ that carries the representation $T$. In the case where the representation is determined by a representation of a subgroup containing the lowest weight vector of $T$, this can be achieved through the coherent state construction. In both the scalar and vector coherent state methods, the process of finding the overlaps can be simplified by introducing the coherent state triplet ($\mathfrak{F}_{\mathbb{H}_D}$, $\mathbb{H}_D$, $\mathfrak{F}^{\mathfrak{H}_D}$) of Bargmann spaces. Coherent state wave functions -- the elements of $\mathfrak{F}_{\mathbb{H}_D}$ and of $\mathfrak{F}^{\mathbb{H}_D}$ -- are used to define the inner product on $\mathbb{H}_D$ in a way that simplifies the calculation of the overlaps. This inner product and the group action $\Gamma$ of $G$ on $\mathfrak{F}^{\mathbb{H}_D}$ are used to formulate expressions for the matrix elements of $T$ with coefficients from the given subrepresentation. The process of finding an explicit expression for $\Gamma$ relies on matrix factorizations in the complexification of $G$ even though the representation $T$ does not extend to the complexification. It will be shown that these factorizations are, in fact, justified, that the overlaps and $\Gamma$ action can be expressed in terms of the given subrepresentation, and that it is possible to find numerical values for the inner product in $\mathbb{H}$. The scalar and vector coherent state methods will both be applied to Sp($n$) and Sp($n,\mathbb{R}$).

In this thesis we generalize the problem of phase retrieval of vector to that of multi-vector. The identification of the multi-vector is done up to some special classes of isometries in the space. We give some upper and lower estimates on the minimal number of multi-linear operators needed for the retrieval. The results are preliminary and far from sharp.

Autonomous underwater vehicles (AUVs) have the potential to dramatically improve safety, quality of life and general scientific knowledge. Our coasts, lakes and rivers are filled with various forms of marine infrastructure including dams, bridges, ship hulls, communication lines, and oil rigs. Each of these structures requires regular inspection, and current methods utilize divers, which is dangerous, expensive, and time consuming. AUVs have the potential to alleviate these difficulties and enable more regular inspection of these structures. Furthermore, there are significant scientific discoveries in the fields of geology, marine biology and medicine that AUV exploration of our oceans will enable. Since field trials of AUVs can be both expensive and high-risk, making a simulation method to generate data for algorithm development is a necessity. For this purpose, we present HoloOcean, an open-source, fully-featured, underwater robotics simulator. Built upon Unreal Engine 4 (UE4), HoloOcean comes equipped with multi-agent communications, common underwater sensors, high-fidelity graphics, and a novel sonar simulation method. Our novel sonar simulation framework is built upon an octree structure, allowing for rapid data generation and flexible usage to simulate a variety of sonars. Further, we have augmented this simulation to incorporate various probabilistic models to account for the heavy noise found in sonar imagery. Simulation enables development of many algorithms such as mapping, localization, structure from motion, controls, and many others. Localization is one essential algorithm for AUV navigation. Recent developments in the utilization of Lie Groups for robotic localization have lead to dramatic performance improvements in convergence and uncertainty characterization. One such method, the Invariant Extended Kalman Filter (InEKF), leverages that invariant error dynamics defined on matrix Lie Groups satisfy a log-linear differential equation. We lay out the various practical decisions required for the InEKF, and show that the primary sensors used in underwater robotics with minor modifications can be used in the InEKF. We show the convergence improvements of the InEKF over the quaternion-based extended Kalman filter (QEKF) on HoloOcean data, both in low and high uncertainty scenarios.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第24165号 / 理博第4856号 / 新制||理||1695(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 加藤 毅, 教授 入谷 寛, 教授 藤原 耕二 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

No description available.

The modelling power of Itˆo integrals has a far reaching impact on a spectrum of diverse fields. For example, in mathematics of finance, its use has given insights into the relationship between call options and their non-deterministic underlying stock prices; in the study of blood clotting dynamics, its utility has helped provide an understanding of the behaviour of platelets in the blood stream; and in the investigation of experimental psychology, it has been used to build random fluctuations into deterministic models which model the dynamics of repetitive movements in humans. Finding the quadrature for these integrals using continuous groups or Lie groups has to take families of time indexed random variables, known as Wiener processes, into consideration. Adaptations of Sophus Lie’s work to stochastic ordinary differential equations (SODEs) have been done by Gaeta and Quintero [1], Wafo Soh and Mahomed [2], ¨Unal [3], Meleshko et al. [4], Fredericks and Mahomed [5], and Fredericks and Mahomed [6]. The seminal work [1] was extended in Gaeta [7]; the differential methodology of [2] and [3] were reconciled in [5]; and the integral methodology of [4] was corrected and reconciled in [5] via [6]. Symmetries of SODEs are analysed. This work focuses on maintaining the properties of the Weiner processes after the application of infinitesimal transformations. The determining equations for first-order SODEs are derived in an Itˆo calculus context. These determining equations are non-stochastic. Many methods of deriving Lie point-symmetries for Itˆo SODEs have surfaced. In the Itˆo calculus context both the formal and intuitive understanding of how to construct these symmetries has led to seemingly disparate results. The impact of Lie point-symmetries on the stock market, population growth and weather SODE models, for example, will not be understood until these different results are reconciled as has been attempted here. Extending the symmetry generator to include the infinitesimal transformation of the Wiener process for Itˆo stochastic differential equations (SDEs), has successfully been done in this thesis. The impact of this work leads to an intuitive understanding of the random time change formulae in the context of Lie point symmetries without having to consult much of the intense Itˆo calculus theory needed to derive it formerly (see Øksendal [8, 9]). Symmetries of nth-order SODEs are studied. The determining equations of these SODEs are derived in an Itˆo calculus context. These determining equations are not stochastic in nature. SODEs of this nature are normally used to model nature (e.g. earthquakes) or for testing the safety and reliability of models in construction engineering when looking at the impact of random perturbations. The symmetries of high-order multi-dimensional SODEs are found using form invariance arguments on both the instantaneous drift and diffusion properties of the SODEs. We then apply this to a generalised approximation analysis algorithm. The determining equations of SODEs are derived in an It¨o calculus context. A methodology for constructing conserved quantities with Lie symmetry infinitesimals in an Itˆo integral context is pursued as well. The basis of this construction relies on Lie bracket relations on both the instantaneous drift and diffusion operators.

In this thesis, we examine key geometric properties of a class of Carnot groups of Heisenberg type. After first computing the geodesics, we consider some partial differential equations in such groups and discuss viscosity solutions to these equations.

No description available.

No description available.