The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

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Electronic Transport In Mesoscopic Systems

Fragmentation Methods: A Route to Accurate Calculations on Large Systems Mark S. Gordon,* Dmitri G. Fedorov, Spencer R. Pruitt, and Lyudmila V. Slipchenko Department of Chemistry and Ames Laboratory, Iowa State University, Ames Iowa 50011, United States Nanosystem Research Institute, National Institute of Advanced Industrial Science and Technology (AIST), 1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan Department of Chemistry, Purdue University, West Lafayette, Indiana 47907, United States

Fragmentation methods: a route to accurate calculations on large systems.

Single-cell expression profiles of melanoma Tumors harbor multiple cell types that are thought to play a role in the development of resistance to drug treatments. Tirosh et al. used single-cell sequencing to investigate the distribution of these differing genetic profiles within melanomas. Many cells harbored heterogeneous genetic programs that reflected two different states of genetic expression, one of which was linked to resistance development. Following drug treatment, the resistance-linked expression state was found at a much higher level. Furthermore, the environment of the melanoma cells affected their gene expression programs. Science, this issue p. 189 Melanoma cells show transcriptional heterogeneity. To explore the distinct genotypic and phenotypic states of melanoma tumors, we applied single-cell RNA sequencing (RNA-seq) to 4645 single cells isolated from 19 patients, profiling malignant, immune, stromal, and endothelial cells. Malignant cells within the same tumor displayed transcriptional heterogeneity associated with the cell cycle, spatial context, and a drug-resistance program. In particular, all tumors harbored malignant cells from two distinct transcriptional cell states, such that tumors characterized by high levels of the MITF transcription factor also contained cells with low MITF and elevated levels of the AXL kinase. Single-cell analyses suggested distinct tumor microenvironmental patterns, including cell-to-cell interactions. Analysis of tumor-infiltrating T cells revealed exhaustion programs, their connection to T cell activation and clonal expansion, and their variability across patients. Overall, we begin to unravel the cellular ecosystem of tumors and how single-cell genomics offers insights with implications for both targeted and immune therapies.

Dissecting the multicellular ecosystem of metastatic melanoma by single-cell RNA-seq

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Methods of Numerical Integration

http://www.cs.tsukuba.ac.jp/~sakurai/Publications_files/ISE-TR-02-189.pdf

A projection method for generalized eigenvalue problems using numerical integration

A contour integral method is proposed to solve nonlinear eigenvalue problems numerically. The target equation is F (λ)x = 0, where the matrix F (λ) is an analytic matrix function of λ. The method can extract only the eigenvalues λ in a domain defined by the integral path, by reducing the original problem to a linear eigenvalue problem that has identical eigenvalues in the domain. Theoretical aspects of the method are discussed, and we illustrate how to apply of the method with some numerical examples.

A numerical method for nonlinear eigenvalue problems using contour integrals

http://www.cs.tsukuba.ac.jp/~sakurai/Publications_files/CS-TR-08-13.pdf

A filter diagonalization for generalized eigenvalue problems based on the Sakurai-Sugiura projection method

We consider a Rayleigh-Ritz type eigensolver for finding a limited set of eigenvalues and their corresponding eigenvectors in a certain region of generalized eigen- value problems. When the matrices are very large, iterative methods are used to generate an invariant subspace that contains the desired eigenvectors. Approximations are ex- tracted from the subspace through a Rayleigh-Ritz projection. In this paper, we present a Rayleigh-Ritz type method with a contour integral (CIRR method). In this method, numerical integration along a circle that contains relatively small number of eigenval- ues is used to construct a subspace. Since the process to derive the subspace can be performed in parallel, the presented method is suitable for master-worker programming models. Numerical experiments illustrate the property of the proposed method.

/pdf/cirr-a-rayleigh-ritz-type-method-with-contour-integral-for-4e847fm7xl.pdf

CIRR: a Rayleigh-Ritz type method with contour integral for generalized eigenvalue problems

The Rayleigh-Ritz-type approach of the contour integral (CIRR) eigensolver is extended to be generally applicable to non-Hermitian systems The CIRR method can extract only the eigenvalues in a given domain, which was previously formulated for non-degenerated Hermitian systems In this method, the Ritz space for the domain is constructed by numerical evaluation of a contour integral The effect of the numerical approximation is analyzed from the viewpoint of a filter operator, which supports the use of moderate approximations The numerical accuracy of the original moment-based approach is also assured A block version of the CIRR method is proposed with a detailed algorithm, which allows us to resolve degenerated systems

/pdf/contour-integral-eigensolver-for-non-hermitian-systems-a-5fbmrn8io9.pdf

Tetsuya Sakurai

Papers

A projection method for generalized eigenvalue problems using numerical integration

A numerical method for nonlinear eigenvalue problems using contour integrals

A filter diagonalization for generalized eigenvalue problems based on the Sakurai-Sugiura projection method

CIRR: a Rayleigh-Ritz type method with contour integral for generalized eigenvalue problems

Contour integral eigensolver for non-hermitian systems: a rayleigh-ritz-type approach