Fundamental approach to exchange rate modeling: Toward an augmented theory of purchasing power parity

In this dissertation, I test the homogeneity and symmetry conditions of PPP by applying the Johansen multivariate cointegration methodology to quarterly data for six countries. I perform the tests in the framework of both a traditional version and an augmented version of PPP. The results of tests on the traditional version of PPP reveal that in all cases the theoretical PPP-vector [1, 1, −1] is not contained in the cointegrating space. This finding is consistent with that of existing literature and indicates the empirical failure of the homogeneity and symmetry conditions of PPP. However, when the traditional PPP is augmented with several non-price variables (real interest rate differential, relative growth rate of real GDP, relative current account balance as a percentage of GDP, and relative terms of trade), the theoretical PPP-vector [1, 1, −1] exists in all the instances except Germany. The fact that the theoretical PPP-vector exists in the augmented model but not in the traditional model indicates that the empirical failure of the PPP is caused by misspecification as a result of missing variables. The true relationship between exchange rates and prices, i.e., the homogeneity and symmetry conditions of PPP, is revealed once those ‘missing variables’ are added to the model. One potential reason for the failure to find the theoretical PPP-vector [1, 1, −1] in the case of Germany is the structural break caused by monetary reunification between Eastern and Western Germany in 1992

Formalizing the SSA-based Compiler for Verified Advanced Program Transformations

Compilers are not always correct due to the complexity of language semantics and transformation algorithms, the trade-offs between compilation speed and verifiability,etc.The bugs of compilers can undermine the source-level verification efforts (such as type systems, static analysis, and formal proofs) and produce target programs with different meaning from source programs. Researchers have used mechanized proof tools to implement verified compilers that are guaranteed to preserve program semantics and proved to be more robust than ad-hoc non-verified compilers. The goal of the dissertation is to make a step towards verifying an industrial strength modern compiler--LLVM, which has a typed, SSA-based, and general-purpose intermediate representation, therefore allowing more advanced program transformations than existing approaches. The dissertation formally defines the sequential semantics of the LLVM intermediate representation with its type system, SSA properties, memory model, and operational semantics. To design and reason about program transformations in the LLVM IR, we provide tools for interacting with the LLVM infrastructure and metatheory for SSA properties, memory safety, dynamic semantics, and control-flow-graphs. Based on the tools and metatheory, the dissertation implements verified and extractable applications for LLVM that include an interpreter for the LLVM IR, a transformation for enforcing memory safety, translation validators for local optimizations, and verified SSA construction transformation. This dissertation shows that formal models of SSA-based compiler intermediate representations can be used to verify low-level program transformations, thereby enabling the construction of high-assurance compiler passes

Bidiagonal factorizations with some parameters equal to zero

AbstractMotivated by the results of Fiedler and Markham [2], we provide necessary and sufficient conditions for a matrix to have a bidiagonal factorization with some of the parameters of the bidiagonal factors equal to zero

Exact Weighted-FBP Algorithm for Three-Orthogonal-Circular Scanning Reconstruction

Recently, 3D image fusion reconstruction using a FDK algorithm along three-orthogonal circular isocentric orbits has been proposed. On the other hand, we know that 3D image reconstruction based on three-orthogonal circular isocentric orbits is sufficient in the sense of Tuy data sufficiency condition. Therefore the datum obtained from three-orthogonal circular isocentric orbits can derive an exact reconstruction algorithm. In this paper, an exact weighted-FBP algorithm with three-orthogonal circular isocentric orbits is derived by means of Katsevich's equations of filtering lines based on a circle trajectory and a modified weighted form of Tuy's reconstruction scheme

Disc Separation of the Schur Complement of Diagonally Dominant Matrices and Determinantal Bounds

We consider the Gersgorin disc separation from the origin for (doubly) diagonally dominant matrices and their Schur complements, showing that the separation of the Schur complement of a (doubly) diagonally dominant matrix is greater than that of the original grand matrix. As application we discuss the localization of eigenvalues and present some upper and lower bounds for the determinant of diagonally dominant matrices

Improved Complexity Analysis of the Sinkhorn and Greenkhorn Algorithms for Optimal Transport

The Sinkhorn algorithm is a widely used method for solving the optimal transport problem, and the Greenkhorn algorithm is one of its variants. While there are modified versions of these two algorithms whose computational complexities are $O({n^2\|C\|_\infty^2\log n}/{\varepsilon^2})$ to achieve an $\varepsilon$-accuracy, the best known complexities for the vanilla versions are $O({n^2\|C\|_\infty^3\log n}/{\varepsilon^3})$. In this paper we fill this gap and show that the complexities of the vanilla Sinkhorn and Greenkhorn algorithms are indeed $O({n^2\|C\|_\infty^2\log n}/{\varepsilon^2})$. The analysis relies on the equicontinuity of the dual variables of the entropic regularized optimal transport problem, which is of independent interest

Characterizations of inverse M-matrices with special zero patterns

AbstractIn this paper, we provide some characterizations of inverse M-matrices with special zero patterns. In particular, we give necessary and sufficient conditions for k-diagonal matrices and symmetric k-diagonal matrices to be inverse M-matrices. In addition, results for triadic matrices, tridiagonal matrices and symmetric 5-diagonal matrices are presented as corollaries