Independent Columns Research Articles

Uniform Uncertainty Principle for Bernoulli and Subgaussian Ensembles

The paper considers random matrices with independent subgaussian columns and provides a new elementary proof of the Uniform Uncertainty Principle for such matrices. The Principle was introduced by Candes, Romberg and Tao in 2004; for subgaussian random matrices it was carlier proved by the present authors, as a consequence of a general result based on a generic chaining method of Talagrand. The present proof combines a simple measure concentration and a covering argument, which are standard tools of high-dimensional convexity.

A harmonic approach to global ocean tide analysis based on TOPEX/Poseidon satellite

The constant and harmonic parts of the global ocean tide are modeled by up to nine major tidal constituents, namely, S2, M2, N2, K1, P1, O1, Mf, Mm, and Ssa. Our computations start with the Fourier sine and cosine series expansion for the tidal constituents, including the constant Mean Sea Level (MSL). Although the frequencies of the tidal constituents are considered known, the coefficients of the sine and cosine functions are assumed to be unknown. Subsequently, the coefficients of the sine and cosine functions, as well as the constant part of the Fourier expansion, were expanded into spherical harmonics up to degree and order n, where n corresponds to the number of linearly independent spherical harmonic base functions needed to model the tidal constituents, determined via independent columns of the Gram matrix. The unknown coefficients of the spherical harmonic expansions are computed using sea level observations within cycles #1–#350 of the TOPEX/Poseidon satellite altimetry over 11 years of its mission. A set of orthonormal base functions was generated for the marine areas covered by TOPEX/Poseidon observations from the spherical harmonics using a Gram-Schmidt orthogonalization process. These were used for modeling the dominant tidal constituents. The computed models based on orthonormal base functions for the nine tidal constituents and the constant part of the Fourier expansion, were tested numerically for their validity and accuracy, proving centimeter accuracy.

Fully Automated Flow-Through Synthesis of Secondary Sulfonamides in a Binary Reactor System

A fully automated flow-through process for the production of secondary sulfonamides is presented. Primary sulfonamides were monoalkylated using a two-step "catch and release" protocol to generate library products of high purity. The automated flow synthesis platform incorporates four independent reactor columns and is able to perform automated column regeneration. A 48-member sulfonamide library was prepared as two 24-member sublibraries, affording library compounds in good yields and high purities without the need for further column chromatographic purification.

Multivariate Analysis Without “vec” and “⊗”

This article proposes a simple notation for random matrices with independent normal columns that share a common covariance matrix. It presents a list of convenient properties regarding transformation and independence. The applications of the notation and its properties simplify tedious analysis for many multivariate problems.

Is There a Common Origin to Surround-Inhibition as Seen Through Electrical Activity Versus Hemodynamic Changes? Focus on “Duration-Dependent Response in SI to Vibrotactile Stimulation in Squirrel Monkey”

A common aspect of neocortical electrophysiology is that even a brief and spatially localized stimulus leads to activation that both outlives the stimulus and spreads well beyond the patch of cortex that receives direct afferent input. In this issue of the Journal of Neurophysiology , Simons et al

Complementary bases in symplectic matrices and a proof that their determinant is one

New results on the patterns of linearly independent rows and columns among the blocks of a symplectic matrix are presented. These results are combined with the block structure of the inverse of a symplectic matrix, together with some properties of Schur complements, to give a new and elementary proof that the determinant of any symplectic matrix is +1. The new proof is valid for any field. Information on the zero patterns compatible with the symplectic structure is also presented.

Investigation hydrologic scaling: Observed effects of heterogeneity and nonlocal processes across hillslope, watershed, and regional scales

The relationship between soil moisture and outflow is fundamental to land surface water and energy balance monitoring and modeling. However, characterizing it at levels above a point scale is complicated by factors including climate and surface heterogeneity, presence of lateral transports, and mismatches between spatial resolutions of measurements and models. We investigate the distinct roles of heterogeneity and nonlocal interactions in scaling this relationship from points to areas. Locations are modeled as independent columns, using measured soil moisture and precipitation data in a conditional averaging approach to estimate the local moisture outflow relationship. These estimates are aggregated using a distributional approach to account for the effect of heterogeneity. The locations are then treated as one large column; that is, the moisture outflow relationship is estimated directly from spatially aggregated data. We demonstrate that statistically significant differences between the two estimates indicate that the system is not well represented by the independence assumption; that is, local outflow is dependent on local moisture and is also independently influenced by large‐scale moisture. We applied these methods to data from a hillslope, a watershed, and the state of Illinois and found that heterogeneity and nonlocal processes significantly affected scaling in all three. The identified nonlocal effect is to decrease (increase) local outflow during large‐scale dry (wet) anomalies. A possible pathway is through decreased wind speed during dry anomalies. The combined effect increases the sensitivity of outflow to soil moisture as scale increases. These results could have implications for specifying parameters in large‐scale models, especially those calibrated with smaller‐scale field data.

Matrix analysis of 2-D microresonator lattice optical filters

We present a transfer matrix analysis of a two-dimensional (2-D) filter to study its frequency response functions. The (M/spl times/N) array consists of N independent columns of microring resonators side-coupled to two channel bus waveguides, with equal spacing between columns and each column consisting of M coupled resonators. We show that such a general 2-D lattice network of lossless and symmetric resonators can approximate an ideal bandpass filter characterized by a flat-top box-like amplitude response without out-of-band sidelobes, and a linear phase response. The bandwidth is determined by the coupling factor between resonators. The 2-D periodic structure exhibits nonoverlapping photonic bandgaps arising from the complementary transmission properties of the row and column arrays. The row array behaves as a distributed feedback grating giving rise to narrow bandgaps corresponding to the flat reflection passbands of the filter with out-of-band sidelobes. The column array, on the other hand, acts as a high-order coupled-cavities filter with broad bandgaps that overlap with the sidelobe regions, thereby effectively suppressing the sidelobes. The phase response is linear except near the band edges, where enhanced group delay limits the usable bandwidth of the filter to about 80%. The minimum size of the array required is about 3/spl times/10, but is ultimately limited by waveguide loss.

Novel design of disturbance decoupled reduced‐order observer

PurposeIn order to improve the practicability of the design in state estimation, the paper aims to present a novel disturbance decoupled reduced‐order observer (DDRO) design scheme.Design/methodology/approachThe paper first uses equivalence transformation to eliminate unknown input appearing in measurement. Then based on Luenberger observer and using two non‐singular coordinate transformation, the design observer can get no bias error in the state estimation.FindingsBy using this approach we find that the conditions of DDRO depend on the system itself that is weaker than other observers. It is a qualified and simple and straightforward approach to be applied in wide domains.Research limitations/implicationsWe should note that the number of independent rows of the matrix C must not be less than the number of the independent columns of the matrix E to satisfy condition rank(CE)=rank(E)=q. In other words, the maximum number of disturbances which can be decoupled cannot be larger than the number of independent measurements.Practical implicationsIt is a very useful approach to solve the problem that the measurement is contaminated by disturbances.Originality/valueThe paper proposed an equivalence transformation. It is used to eliminate unknown input appearing in measurement. At the same time the algebraic transformation guaranteed that it would lose no information of the unknown inputs. And compared with other known results, the design condition of the reduced‐order observer which proposed in this paper depends on system itself, especially, which is weaker than others.

Probability theory for number of mixture components resolved by n independent columns

A general theory is proposed for the probability of different outcomes of success and failure of component resolution, when complex mixtures are partially separated by n independent columns. Such a separation is called an n-column separation. An outcome of particular interest is component resolution by at least one column. Its probability is identified with the probability of component resolution by a single column, thereby defining the effective saturation of the n-column separation. Several trends are deduced from limiting expressions of the effective saturation. In particular, at low saturation the probability that components cluster together as unresolved peaks decreases exponentially with the number of columns, and the probability that components cluster together on addition of another column decreases by a factor equal to twice the column saturation. The probabilities of component resolution by n-column and two-dimensional separations also are compared. The theory is applied by interpreting three sets of previously reported retention indices of the 209 polychlorinated biphenyls (PCBs), as determined by GC. The origin of column independence is investigated from two perspectives. First, it is suggested that independence exists when the difference between indices of the same compound on two columns is much larger than the interval between indices required for separation. Second, it is suggested that independence exists when the smaller of the two intervals between a compound and its adjacent neighbors is not correlated with its counterpart on another column.

Computing expectation values for RNA motifs using discrete convolutions

BackgroundComputational biologists use Expectation values (E-values) to estimate the number of solutions that can be expected by chance during a database scan. Here we focus on computing Expectation values for RNA motifs defined by single-strand and helix lod-score profiles with variable helix spans. Such E-values cannot be computed assuming a normal score distribution and their estimation previously required lengthy simulations.ResultsWe introduce discrete convolutions as an accurate and fast mean to estimate score distributions of lod-score profiles. This method provides excellent score estimations for all single-strand or helical elements tested and also applies to the combination of elements into larger, complex, motifs. Further, the estimated distributions remain accurate even when pseudocounts are introduced into the lod-score profiles. Estimated score distributions are then easily converted into E-values.ConclusionA good agreement was observed between computed E-values and simulations for a number of complete RNA motifs. This method is now implemented into the ERPIN software, but it can be applied as well to any search procedure based on ungapped profiles with statistically independent columns.

Approximation theorems for random permanents and associated stochastic processes

The limit theorems for certain stochastic processes generated by permanents of random matrices of independent columns with exchangeable components are established. The results are based on the martingale decomposition of a random permanent function similar to the one known for U-statistics and on relating the components of this decomposition to some multiple stochastic integrals.

The dual electron emission spectromicroscope: a novel multi-technique instrument for surface analysis

The concept, design and performance of the novel dual emission electron spectromicroscope (DEEM) are presented. The DEEM consists of four fully electrostatic lens systems comprising two independent and parallel projective columns, objective lens system and deflector lens system. Switching on the deflector lens potentials causes a parallel shift of the electron beam from the axis of the of the first projective lens column to the axis of the second projective column. This original concept allows for the quasi-simultaneous observation of the complementary diffraction (angular distribution) and imaging planes at the two independent imaging units. The dispersive properties of the deflector lens system can also be used to select the energy range of the electrons emitted from the sample for analysis. Consequently, energetically and laterally resolved measurements are possible and a plurality of image contrast mechanisms are available.

Limit theorems for random permanents with exchangeable structure

Permanents of random matrices extend the concept of U-statistics with product kernels. In this paper, we study limiting behavior of permanents of random matrices with independent columns of exchangeable components. Our main results provide a general framework which unifies already existing asymptotic theory for projection matrices as well as matrices of all-iid entries. The method of the proofs is based on a Hoeffding-type orthogonal decomposition of a random permanent function. The decomposition allows us to relate asymptotic behavior of permanents to that of elementary symmetric polynomials based on triangular arrays of rowwise independent rv's.

On the hardness of efficiently approximating maximal non- L submatrices

The sign pattern of a real matrix A is the matrix obtained by replacing each entry of A by its sign. A real matrix A is an L- matrix if every real matrix with the same sign pattern as A has linearly independent columns. L-matrices arise naturally in and are essential to the study of sign-solvability and related notions. In special cases, the L-matrix property has connections to the even dicycle problem, Pfaffian orientations, and Pólya’s permanent problem. Unfortunately, the problem of recognizing L-matrices is known to be co- NP -complete in general. We elaborate in this vein by showing a polynomial-time inapproximability result for Max-not- L -rows, a particular optimization version of L-matrix recognition, by means of an approximation-preserving reduction from the previously studied problem Max-not-equal-2-Sat.

Obtaining the QR Decomposition by Pairs of Row and Column Operations

Let S = {vl, 2, ..., Vm } be a set of linearly independent vectors in Rn and let A be the matrix that has these vectors as its columns. The Gram-Schmidt process can be applied to the column vectors to produce a new matrix whose column vectors are orthonormal and whose column space is the same as A's. The process replaces each column of A by a linear combination of that column and its predecessors. If Q denotes the matrix with orthonormal columns, then A = QR, where R is an upper-triangular nonsingular matrix. This is the factorization or of A. In this note, we show how to obtain the QR decomposition by using pairs of row and column operations. Suppose the n by m matrix A = [aij] has linearly independent columns vj (alj, a2j, ..., anj)T for j = 1, 2,..., m. Then A = [aij] for i = 1, 2,..., n. Since ATA is symmetric, and its diagonal elements are positive, we can use n(n 1)/2 pairs of row and column operations on ATA to annihilate all of its off-diagonal entries. That is, we obtain BTATAB = diag[dl, d2,..., dm], where B and BT are products of respective lower-triangular and upper-triangular matrices. Since BTATAB = (AB)T(AB) = diag[dl, d2, ..., dm], the columns of AB are orthonormal. Letting C be a square root of this diagonal matrix, we have (C-1BTAT)(ABC -) = I. Thus, the matrix Q = ABC-~ has orthonormal columns. So A = QR, where R = CB-1 is an upper-triangular nonsingular matrix.

Determination of organochlorine pesticides in ground water using solid-phase microextraction followed by dual-column gas chromatography with electron-capture detection

A rapid, sensitive, convenient, and highly quality-assured method is presented for the determination of 19 organochlorine pesticides (OCPs) in small samples (10 ml) of ground water. Samples are initially fortified with 2,4,5,6-tetrachloro- m-xylene (surrogate) and decachlorobiphenyl (retention time marker), then extracted with a 30-μm thickness polydimethylsiloxane solid-phase microextraction fiber. The analytes collected are thermally desorbed in a heated gas chromatographic inlet, separated using independent fused-silica capillary columns (“primary” and “confirmatory”), and detected using electron-capture detection. Two independent statistical procedures were used to evaluate the detection limits, which typically range between 10 and 40 ng l −1, for these analytes. Method performance was also evaluated using two additional protocols employing “performance evaluation” samples, in which authentic ground water samples were fortified to ca. 100 ng l −1 in each of at least six OCPs. The method satisfies additional strict criteria based on uniformity of fiber performance and minimal degradation of the thermally-sensitive analytes endrin and DDT.

Functionally Independent Columns of Rat Somatosensory Barrel Cortex Revealed with Voltage-Sensitive Dye Imaging

Whisker movement is somatotopically represented in rodent neocortex by electrical activity in clearly defined barrels, which can be visualized in living brain slices. The functional architecture of this part of the cortex can thus be mapped in vitro with respect to its physiological input and compared with its anatomical architecture. The spatial extent of excitation was measured at high temporal resolution by imaging optical signals from voltage-sensitive dye evoked by stimulation of individual barrels in layer 4. The optical signals correlated closely with subthreshold EPSPs recorded simultaneously from excitatory neurons in layer 4 and layer 2/3, respectively. Excitation was initially (<2 msec) limited to the stimulated barrel and subsequently (>3 msec) spread in a columnar manner into layer 2/3 and then subsided in both layers after approximately 50 msec. The lateral extent of the response was limited to the cortical column defined structurally by the barrel in layer 4. Two experimental interventions increased the spread of excitation. First, blocking GABA(A) receptor-mediated synaptic inhibition caused excitation to spread laterally throughout wide regions of layer 2/3 and layer 5 but not into neighboring barrels, suggesting that the local excitatory connections within layer 4 are restricted to single barrels and that inhibitory neurons control spread in supragranular and infragranular layers. Second, NMDA receptor-dependent increase of the spread of excitation was induced by pairing repetitive stimulation of a barrel column with coincident stimulation of layer 2/3 in a neighboring column. Such plasticity in the spatial extent of excitation in a barrel column could underlie changes in cortical map structure induced by alterations of sensory experience.

Structural characterization of the columnar alkali metal cyclopentadienide [K{C5H2(SiMe3)3-1,2,4}]∞†

Unsolvated 1,2,4-tris(trimethylsilyl)cyclopentadienyl potassium (K[Cp3Si]) crystallizes as a base-free “super-sandwich” of alternating potassium ions and Cp′ rings. The polymeric chains in the structure are not connected as in other unsolvated KCp′ structures, but uniquely consist of independent columns. This type of structure has previously been observed in base-free lithium cyclopentadienyl complexes; however, K[Cp3Si] is the first base-free potassium cyclopentadienyl complex that crystallizes without interactions between neighboring chains.

Nearly L-matrices and generalized row sign balanced matrices

A real matrix A is called an L-matrix if every matrix with the same sign pattern as A has linearly independent columns. A nearly L-matrix is a matrix which is not an L-matrix, but each matrix obtained by deleting one of its columns is an L-matrix. A generalized row sign balanced (GRSB) matrix is a matrix which can be transformed to a matrix having both positive and negative entries in each row by multiplying some of its columns by −1. In this paper, we study the relations between L-matrices, nearly L-matrices and GRSB matrices. We obtain a complete characterization of nearly L-matrices in terms of GRSB matrices. By comparing this result with a well-known theorem about L-indecomposable, barely L-matrices, we find an interesting duality relation between L-matrices and GRSB matrices. We also use GRSB matrices to characterize the conditional S * -matrices (which are closely related to nearly L-matrices and the conditionally sign solvable linear systems). Finally, we propose some unsolved problems for further research.