Abstract:
Gaussian elimination with partial pivoting (GEPP) has long been among the most widely used methods for computing the LU factorization of a given matrix. However, this method is also known to fail for matrices that induce large element growth during the factorization process. In this paper, we propose a new scheme, Gaussian elimination with randomized complete pivoting (GERCP) for the efficient and reliable LU factorization of a given matrix. GERCP satisfies GECP (Gaussian elimination with complete pivoting) style element growth bounds with high probability, yet costs only marginally higher than GEPP. Our numerical experimental results strongly suggest that GERCP is as reliable as GECP and as efficient as GEPP for computing the LU factorization.

Abstract:
Spectral graph sparsification has emerged as a powerful tool in the analysis of large-scale networks by reducing the overall number of edges, while maintaining a comparable graph Laplacian matrix. In this paper, we present an efficient algorithm for the construction of a new type of spectral sparsifier, the unweighted spectral sparsifier. Given a general undirected and unweighted graph $G = (V, E)$ and an integer $\ell < |E|$ (the number of edges in $E$), we compute an unweighted graph $H = (V, F)$ with $F \subset E$ and $|F| = \ell$ such that for every $x \in \mathbb{R}^{V}$ \[ {\displaystyle \frac{x^T L_G x}{\kappa} \leq x^T L_H x \leq x^T L_G x,} \] where $L_G$ and $L_H$ are the Laplacian matrices for $G$ and $H$, respectively, and $\kappa \geq 1$ is a slowly-varying function of $|V|, |E|$ and $\ell$. This work addresses the open question of the existence of unweighted graph sparsifiers for unweighted graphs. Additionally, our algorithm can efficiently compute unweighted graph sparsifiers for weighted graphs, leading to sparsified graphs that retain the weights of the original graphs.

Abstract:
In patients suffering from neurogenic detrusor overactivity, continence can be regained by conditional stimulation of the dorsal genital nerve (DGN); that is applying electrical stimulation to the DGN at the onset of an involuntary contraction. For this scheme to work, a sensor capable of reliably detecting the onset of bladder contractions is needed. This article reviews the methods proposed for or associated with detection of bladder contractions, and their applicability to onset detection is assessed. Fourteen methods are described in this review; bladder pressure, urethral sphincter EMG and anal sphincter EMG are the most promising options for onset detection. For all three modalities, however, further research is needed before clinical application becomes viable.

Abstract:
We establish existence of infinitely many distinct solutions to the semilinear elliptic Hartree-Fock equations for -electron Coulomb systems with quasirelativistic kinetic energy ？？2Δ

Abstract:
We study the standard and extended Kohn-Sham models for quasi-relativistic N-electron Coulomb systems; that is, systems where the kinetic energy of the electrons is given by the quasi-relativistic operator $$ sqrt{-alpha^{-2}Delta_{x_n}+alpha^{-4}}-alpha^{-2}. $$ For spin-unpolarized systems in the local density approximation, we prove existence of a ground state (or minimizer) provided that the total charge $Z_{ m tot}$ of K nuclei is greater than N-1 and that $Z_{ m tot}$ is smaller than a critical charge $Z_{ m c}=2 alpha^{-1} pi^{-1}$.

Abstract:
We consider self-adjoint Dirac operators $\ham{D}=\ham{D}_0 + V(x)$, where $\ham{D}_0$ is the free three-dimensional Dirac operator and $V(x)$ is a smooth compactly supported Hermitian matrix. We define resonances of $\ham{D}$ as poles of the meromorphic continuation of its cut-off resolvent. An upper bound on the number of resonances in disks, an estimate on the scattering determinant and the Lifshits-Krein trace formula then leads to a global Poisson wave trace formula for resonances of $\ham{D}$.

Abstract:
We study the existence of quantum resonances of the three-dimensional semiclassical Dirac operator perturbed by smooth, bounded and real-valued scalar potentials $V$ decaying like $\langle x \rangle ^{-\d}$ at infinity for some $\d >0$. By studying analytic singularities of a certain distribution related to $V$ and by combining two trace formulas, we prove that the perturbed Dirac operators possess resonances near $\sup V + 1$ and $\inf V -1$. We also provide a lower bound for the number of resonances near these points expressed in terms of the semiclassical parameter.

Abstract:
The Complex Absorbing Potential (CAP) method is widely used to compute resonances in Quantum Chemistry, both for nonrelativistic and relativistic Hamiltonians. In the semiclassical limit $\hbar \to 0$ we consider resonances near the real axis and we establish the CAP method rigorously for the perturbed Dirac operator by proving that individual resonances are perturbed eigenvalues of the nonselfadjoint CAP Hamiltonian, and vice versa. The proofs are based on pseudodifferential operator theory and microlocal analysis.

Abstract:
We establish geometric properties of Stiefel and Grassmann manifolds which arise in relation to Slater type variational spaces in many-particle Hartree-Fock theory and beyond. In particular, we prove that they are analytic homogeneous spaces and submanifolds of the space of bounded operators on the single-particle Hilbert space. As a by-product we obtain that they are complete Finsler manifolds. These geometric properties underpin state-of-the-art results on existence of solutions to Hartree-Fock type equations.

Abstract:
Contact theory has primarily been applied to the study of interactions between Blacks and Whites, with particular emphasis on changes in the attitudes of Whites towards Blacks. How individual contact with an out-group can influence not just attitudes, but also actual behavior, has not been thoroughly explored. Through an analysis of the 2006 Latino National Survey, using a measure that contrasts the intensity of individual social interaction with various ethnic and racial groups, the study shows that a high intensity of friendly social contact with African-Americans increases the likelihood Latino immigrants will establish a closer link to the social and political structures of the United States. Latino immigrants are potentially experiencing movement towards deprovincialization through high levels of friendly social interaction with African-Americans. The development of friendly personal interactions with an out-group stigmatized in the mother country can help Latino immigrants develop an optimistic view of life in the host country.