Deletion Of Rows Research Articles

On the continuity of the maximum size of P-sets of acyclic matrices

For a given n×n real symmetric matrix A, if the nullity of the principal submatrix, obtained from the deletion of rows and columns of A of the same index set α, goes up by |α|, we call α a P-set. The maximum size of a P-set of A is commonly denoted by Ps(A). The study of Ps(A) seems in general a difficult problem to tackle. In this note, we show that if there is a matrix A whose graph is a tree T with Ps(A)=⌊n2⌋, then for any integer k∈0,1,…,n2, there is a matrix M, whose graph is T, such that Ps(M)=k. An illustrative example is provided as well as an open question.

Submatrices of Four Dimensional Summability Matrices

In this paper, we show that a matrix that maps l ′′ into l ′′ can be obtained from any RH-regular matrix by the deletion of rows. Also a four dimensional conservative matrix can be obtained by the deletion of rows from a matrix that preserves boundedness. We will use these techniques to derive a sufficient condition for a four dimensional matrix to sum an unbounded sequence. RESUMEN En este trabajo probamos que una matriz que lleva l ′′ en l ′′ se puede obtener a partir de

QM/MM kinetic isotope effects for chloromethane hydrolysis in water

Computational simulations for chloromethane hydrolysis have been performed using hybrid quantum‐mechanical/molecular‐mechanical methods with explicit solvation by large numbers of water molecules. In the first part of the paper, we present results for 2° 2H3, 1° 14C, and 1° 37Cl kinetic isotope effects (KIEs) at 298 K with both the AM1/TIP3P and B3LYP/6‐31G* QM methods for the nucleophile H2O and electrophile CH3Cl surrounded by 496 solvating TIP3P water molecules. An initial Hessian computed for a subset of this system including up to 104 MM water molecules was reduced in size by successive deletion of rows and columns, and KIEs were evaluated for each. We suggest that accurate calculations of KIEs in solvated systems should involve a subset Hessian including the substrate together with any solvent atoms making specific interactions with any isotopically substituted atom. In the second part of the paper, the ensemble‐averaged 2° α‐2H3 KIE calculated with the B3LYP/6‐31+G(d,p)/TIP3P method is shown to be in good agreement with experiment. This comparison is meaningful because it includes consideration of uncertainties owing to sampling of a range of representative thermally accessible solvent configurations. We also present ensemble‐averaged 14C and 37Cl KIEs which have not as yet been determined experimentally. Copyright © 2013 John Wiley & Sons, Ltd.

Kinetic Isotope Effects from QM/MM Subset Hessians: "Cut-Off" Analysis for SN2 Methyl Transfer in Solution.

Isotopic partition-function ratios and kinetic isotope effects for reaction of S-adenosylmethionine with catecholate in water are evaluated using a subset of 324 atoms within its surrounding aqueous environment at the AM1/TIP3P level. Two alternative methods for treating motion in the six librational degrees of freedom of the subset atoms relative to their environment are compared. A series of successively smaller subset Hessians are generated by cumulative deletion of rows and columns from the initial 972 × 972 Hessian. We find that it is better to treat these librations as vibrations than as translations and rotations and that there is no need to invoke the Teller-Redlich product rule. The validity of "cut-off" procedures for computation of isotope effects with truncated atomic subsets is assessed: to ensure errors in ln(KIE) < 1% (or 2% for the quantum-corrected KIE) for all isotopic substitutions considered, it is necessary to use a less-restrictive procedure than is suggested by the familiar two-bond cutoff rule.

PetroPlot: A plotting and data management tool set for Microsoft Excel

PetroPlot is a 4000‐line software code written in Visual Basic for the spreadsheet program Excel that automates plotting and data management tasks for large amount of data. The major plotting functions include: automation of large numbers of multiseries XY plots; normalized diagrams (e.g., spider diagrams); replotting of any complex formatted diagram with multiple series for any other axis parameters; addition of customized labels for individual data points; and labeling flexible log scale axes. Other functions include: assignment of groups for samples based on multiple customized criteria; removal of nonnumeric values; calculation of averages/standard deviations; calculation of correlation matrices; deletion of nonconsecutive rows; and compilation of multiple rows of data for a single sample to single rows appropriate for plotting. A cubic spline function permits curve fitting to complex time series, and comparison of data to the fits. For users of Excel, PetroPlot increases efficiency of data manipulation and visualization by orders of magnitude and allows exploration of large data sets that would not be possible making plots individually. The source codes are open to all users.

A modified Gram–Schmidt-based downdating technique for ULV decompositions with applications to recursive TLS problems

The ULV decomposition (ULVD) is an important member of a class of rank-revealing two-sided orthogonal decompositions used to approximate the singular value decomposition (SVD). The ULVD can be updated and downdated much faster than the SVD, hence its utility in the solution of recursive total least squares (TLS) problems. However, the robust implementation of ULVD after the addition and deletion of rows (called updating and downdating, respectively) is not altogether straightforward. When updating or downdating the ULVD, the accurate computation of the subspaces necessary to solve the TLS problem is of great importance. In this paper, algorithms are given to compute simple parameters that can often show when good subspaces have been computed.

A fault-counting algorithm for repairing redundant memories

This paper describes a new approach for repairing memories. Repair is implemented by deletion of either rows and/or columns on which faulty cells lie. These devices are commonly referred to as redundant memories, because redundant columns and rows are added. A new repair technique and an algorithm are proposed. The algorithm is based on a fault-counting technique and on a reduced-covering approach. The innovative feature is that reduced covering permits an heuristic, but efficient, criterion to be included in the selection of the rows and/or columns to be deleted. This retains independence of the repair process on the distribution of faulty cells in memory, while allowing a good repairability/ unrepairability detection. The main benefits that result by using the proposed repair algorithm are a reduction in execution time to determine the repair-solution for the device under test and its suitability for implementation in a defect analysis system. Illustrative examples and theoretical results are provided t...

Node-Deletion Problems on Bipartite Graphs

A set of problems which has attracted considerable interest recently is the set of node-deletion problems. The general node-deletion problem can be stated as follows: Given a graph, find the minimum number of nodes whose deletion results in a subgraph satisfying property $\pi $. In [LY] this problem was shown to be NP-complete for a large class of properties (the class of properties that are hereditary on induced subgraphs) using a small number of reduction schemes from the node cover problem. Since the node cover problem becomes polynomial on bipartite graphs, it might be hoped that this is the case with other node-deletion problems too. In this paper we characterize those properties for which the bipartite restriction of the node-deletion problem is polynomial and those for which it remains NP-complete. Similar results follow for analogous problems on other structures such as families of sets, hypergraphs and 0,1 matrices. For example, in the case of matrices, our result states that if M is a class of 0,1 matrices which is closed under permutation and deletion of rows and columns, then finding the largest submatrix in M of a matrix is polynomial if the matrices of M have bounded rank and NP-complete otherwise.

Submatrices of summability matrices

It is proved that a matrix that maps ℓ1 into ℓ1 can be obtained from any regular matrix by the deletion of rows. Similarly, a conservative matrix can be obtained by deletion of rows from a matrix that preserves boundedness. These techniques are also used to derive a simple sufficient condition for a matrix to sum an unbounded sequence.

Hex Must Have a Winner: An Inductive Proof

The game of Hex is an excellent example of a game for which a winning strategy is known to exist, even though it is not known what the strategy is. It is easy to show the existence of a strategy once it is known that either black or white must win, that is, that Hex cannot end in a draw. The available proofs of the latter fact are all rather difficult (see, for instance, [1, pp. 334-338]). In this note we give a simple proof. A Hex board is a parallelogram divided into m rows and n columns of hexagons. Players alternate turns placing black and white stones on the board with the objective of completing a chain from one side of the board to the opposite side, one player seeking a chain joining top to bottom, the other joining right to left. We will show that the game cannot end without a winner. Specifically, whenever a Hex board is completely filled with black and white stones, there must be either a black chain from right to left or a white chain from top to bottom. And, equivalently, there must be either a white chain from right to left or a black chain from top to bottom. Consequently, one of the players must have achieved his objective and won the game. We will prove our proposition by induction on m and n, the dimensions of the board. The proposition is clear for a 1 x n, m x 1, or 2 x 2 board. Now we assume it true for any board smaller than m x n. Consider the (m 1) x n board obtained by deleting row m. By the inductive hypothesis there is either a black chain from column 1 to column n or else a white chain W1 from row 1 to row m 1. In the former case we are done, so we assume the latter. It follows from an analogous argument involving deletion of row 1 that there must be a white chain W2 from row 2 to row m. We assume that W1 and W2 do not meet, or else we would be done. By deleting column n and then column 1 we can show in a similar manner that there are nonintersecting black chains B1 from column 1 to column n 1 and B2 from column 2 to column n. Since these horizontal black chains do not meet, the number of rows m must be greater than 2.

Analysis of linear active (n+l)-poles by simple transformations of the nodal admittance matrix

An analysis method is presented for linear networks containing arbitrary linear, passive or active, and ideal or nonideal two-port elements (three-poles and four-poles) with existing chain matrices. It is suitable, especially in the analysis of networks containing two-ports with nonexisting admittince matrices like ideal transformers, proportionally controlled sources, and nullors. After the determination of the nodal admittance matrix of the subnetwork consisting only of two-pole elements, the remaining two-ports are inserted by simple row and column transformations in the matrix. The order of the matrix is reduced by the successive deletion of rows and columns corresponding to inner nodes of the network.

Configurational Averages for Chain Molecules: Higher Moments and Related Quantitites

The higher even moments 〈r2p〉0 of the end-to-end vector r of a chain molecule are formulated according to a straightforward procedure which is free of complications for large values of p. It is generally applicable also to averages of other configuration-dependent quantities, and simplifies their treatment. The condensation of self-direct matrix products derived by Nagai for reducing the complexity of computation of higher moments is achieved directly through an appropriate transformation, which is defined for such a product of any degree. Major reductions of the orders of the statistical weight matrix and associated expressions permissible for symmetric chains are achieved succinctly by similar matrix transformations. It is pointed out that further condensations can be realized through deletion of superfluous rows and columns from the generator matrices (G, G⊗G, etc.) for the moments.

On the Realization of a Linear Graph Given Its Algebraic Specification

The following problem is discussed in this paper. Given an algebraic specification of an oriented linear graph in the form of a branch-mesh or branch-node matrix on any arbitrary basis, how does one proceed systematically to construct the graph? This rather basic problem has been considered by other authors and a few procedures have been described in the literature. On closer examination the problem is not simply one of finding any old procedure that will work but rather a question of finding an efficient procedure in terms of the time required by an analyst to carry through the computation. A new procedure is described which is believed to be quite efficient. This procedure is simple since matrix manipulations are limited primarily to the deletion of rows and columns and the other main manipulation is that of freehand drawing. The procedure is based upon the Decomposition Theorem published earlier and involves the notion of dividing a large graph into a sequence of small ones. A reassembly process then yields the desired result.