hamiltonian stationary cones and self similar solutions in higher dimension in lw we construct examples of two dimensional hamiltonian stationary self shrinkers and self expanders for lagrangian mean curvature flows which are asymptotic to the union of two schoen wolfson cones these self shrinkers and self expanders can be glued together to yield solutions of the brakke flow a weak formulation of the mean curvature flow moreover there is no mass loss along the brakke flow in this paper we generalize these results to higher dimension we construct new higher dimensional hamiltonian stationary cones of different topology as generalizations of the schoen wolfson cones hamiltonian stationary self shrinkers and self expanders that are asymptotic to these hamiltonian stationary cones are also constructed they can also be glued together to produce eternal solutions of the brakke flow without mass loss finally we show the same conclusion holds for those lagrangian self similar examples recently found by joyce tsui and the first author in jlt

totally real minimal tori in in this paper we show that all totally real minimal tori in open image in new window correspond to doubly periodic finite gap solutions of the tzitzeica equation open image in new window using the results on the tzitzeica equation in integrable systems theory we describe explicitly these tori by prym theta functions

a symmetric approach to cyclization for systems over c y abstract the cyclization problem from the theory of systems over rings is transformed to a symmetric equivalent problem which is remarkably easier and leads to a solution in the 4 dimensional case for c y the complex polynomial ring

local empirical likelihood inference for varying coefficient density ratio models based on case control data in this article we develop a varying coefficient density ratio model for case control studies the case and control samples come from two different distributions under the model assumption the ratio of the two densities is related to the linear combination of covariates with varying coefficients through a known function a special case is the exponential tilt model where the log ratio of the two densities is a linear function of covariates we propose a local empirical likelihood el approach to estimate the nonparametric coefficient functions under some regularity assumptions the proposed estimators are shown to be consistent and asymptotically normally distributed the sieve empirical likelihood ratio selr test statistic for detecting whether the varying coefficients are really constant and other related hypotheses is constructed and it follows approximately a chi squared distribution we introduce a modified bootstrap procedure to estimate the null distribution of the selr when sample size is s

hypothesis testing in a mixture case control model summary we consider a problem of testing mixture proportions using two sample data one from group one and the other from a mixture of groups one and two with unknown proportion for being in group two various statistical applications including microarray study infectious epidemiological studies case control studies with contaminated controls clinical trials allowing nonresponders genetic studies for gene mutation and fishery applications can be formulated in this setup under the assumption that the log ratio of probability density functions from the two groups is linear in the observations we propose a generalized score test statistic to test the mixture proportion under some regularity conditions it is shown that this statistic converges to a weighted chi squared random variable under the null hypothesis of 0 where the weight depends only on the sampling fraction of both groups the permutation method is used to provide more reliable finite sample approximation simulation results and two real data applications are presented

matrix cyclization over complex polynomials abstract let a be an n n b an n m complex polynomial matrix the following open problem is investigated does there exist a complex polynomial m n matrix f such that a bf has a cyclic vector in the image of b an explicit solution of the problem is given for the following generic situation n 4 b ab a n 1 b is rightinvertible necessary the entries of a specific normalized form of a b do not satisfy certain polynomial equations and one specific entry has at least one simple zero outside a certain finite set the exceptional equations are given explicitely and can easily be checked for

coefficient assignability and a block decomposition for systems over rings abstract we characterize the class of commutative rings r such that for any linear system a b with coefficients in r one can extract the reachable part of the system in a way similar to the classical kalman controllability decomposition for systems over fields the notion of strong ca ring is introduced as the class of commutative rings over which any system verifies a strong form of the coefficient assignability property it is shown that the class of strong ca rings lies strictly between the classes of rings with strong versions of the known pole assignability pa and feedback cyclization fc properties defined only for reachable systems we prove that for ucu rings for example polynomials with coefficients in a field the usual pa ca and fc properties are equivalent to the corresponding strong forms of these properties in particular c y is a strong ca ring

the angle criterion we complete the proof of the angle conjecture on when a pair of orientedm planes is area minimizing the nonzero sum oriented union p1 p2 is area minimizing if and only if the characterizing angles between them satisfy the inequality beta m leqq beta 1 beta m 1

pole assignability in polynomial rings power series rings and prufer domains the topics treated in this paper have their origins in the area of algebraic systems theory however the paper itself should be classified as pure commutative algebra and we shall present it as such in the body of the text still it is appropriate to give a brief paragraph of motivation if a physical system is governed by a pair f g of matrices then the stability of the system can be determined by examining the eigenvalues of the matrix f if the system is unstable a feedback matrix k can sometimes be employed in such a way that the eigenvalues of the matrix f gk measure the stability of the modified system f gk g in this manner an unstable system can be rendered stable the pole assignability problem over commutative rings is one method of attacking the problem of finding such matrices k the paper is divided into four sections section 1 is given over almost entirely to defining the properties in which we shall be interested it concludes with a theorem about residuating and lifting the properties section 2 is concerned with the preservation of the properties under polynomial ring and power series ring formation section 3 is concerned with feedback cyclization a strong form of pole assignability section 4 is concerned with pole assignability over priifer domains a complete elaboration of our results must await the introduction of the necessary terminology for now we mention the following results in somewhat vague language if r is a zero dimensional ring then the pole assignability problem is solvable in r x and in r xl from a ringtheoretic standpoint almost any class of commutative rings contains members for which the feedback cyclization problem is solvable on the other hand if r is a ring with 1 in its stable range then the feedback cyclization problem is solvable in r if and only if a certain nice matricial property

a note on feedback cyclicity of c y abstract let a b be an n dimensional linear system with 2 inputs over c y the ring of polynomials in one variable over the field of complex numbers we prove the feedback cyclicity of a b under certain conditions on their entries and deduce that a b is feedback cyclic in an exceptional case left open in w schmale linear algebra appl 275 276 1998 551 562

confluent cauchy and cauchy vandermonde matrices abstract the s c confluent cauchy and confluent cauchy vandermonde matrices are introduced corresponding to an interpolation problem with multiple interpolation nodes inversion formulas are derived and a factorization of confluent loewner matrices by means of confluent cauchy vandermonde matrices is described

remarks on the schur complement abstract first a generalization of the schur determinantal formula is given using properties of quasidirect sums of matrices a new characterization of the schur complement is proved

generalized invertibility of hankel and toeplitz matrices abstract a result for the computation of the moore penrose inverse of hankel matrices over the field of the complex numbers is given by heinig and rost in this note some methods of characterizing and computing generalized inverses of hankel and toeplitz matrices over fields and over rings with the extended rao condition are presented the results were obtained by combining classical theory on the special structure of those classes of matrices with recent theory on generalized invertibility

u 1 invariant special lagrangian 3 folds iii properties of singular solutions abstract this is the last of three papers studying special lagrangian 3 submanifolds slv 3 folds n in c 3 invariant under the u 1 action ei z1 z2 z3 ei z1 ei z2 z3 using analytic methods if n is such a 3 fold then z1 2 z2 2 2a on n for some a r locally n can be written as a kind of graph of functions u v r 2 r satisfying a nonlinear cauchy riemann equation depending on a so that u iv is like a holomorphic function of x iy the first paper studied the case a nonzero and proved existence and uniqueness for solutions of two dirichlet problems derived from the nonlinear cauchy riemann equation this yields existence and uniqueness of a large class of nonsingular u 1 invariant sl 3 folds in c 3 with boundary conditions the second paper extended these results to weak solutions of the dirichlet problems when a 0 giving existence and uniqueness of many singular u 1 invariant sl 3 folds in c 3 with boundary conditions this third paper studies the singularities of these sl 3 folds we show that under mild conditions the singularities are isolated and have a multiplicityn 0 and one of two types examples are constructed with every multiplicity and type we also prove the existence of large families of u 1 invariant special lagrangian fibrations of open sets in c 3 including singular fibres

the multipoint pade approximation problem and its hankel vector abstract the multipoint pade approximation problem and the structure of multipoint pade approximation table have been investigated thoroughly in the literature in this paper we show that each multipoint pade approximation problem can be reduced to what amounts to a certain pade approximation problem at infinity by using the so called hankel vector method that was introduced in chen et al 1996 for the general rational interpolation problem this connection offers a new way to study the properties of multipoint pade approximants and the structure and computation of entries of multipoint pade approximation table based on the characteristic polynomials of hankel matrices

schur complements and statistics abstract in this paper we discuss various properties of matrices of the type s h ge 1 f which we call the schur complement of e in a e f g h the matrix e is assumed to be nonsingular when e is singular or rectangular we consider the generalized schur complements s h ge f where e is a generalized inverse of e a comprehensive account of results pertaining to the determinant the rank the inverse and generalized inverses of partitioned matrices and the inertia of a matrix is given both for schur complements and for generalized schur complements we survey the known results in a historical perspective and obtain several extensions numerous applications in numerical analysis and statistics are included the paper ends with an exhaustive bibliography of books and articles related to schur complements

calculation of power and sample size with bounded outcome scores the two sample wilcoxon rank sum test is the most popular non parametric test for the comparison of two samples when the underlying distributions are not normal although the underlying distributions need not be known in detail to calculate the null distribution of the test statistic parametric assumptions are often made to determine the power of the test or the sample size we encountered difficulties with this approach in the planning of a recent clinical trial in stroke patients it is shown that for power and sample size estimation it can be dangerous to apply the classical formulae routinely especially with outcome scores having a u shaped or a j shaped distribution as an example we have taken the barthel index a quality of life outcome measure in stroke patients further we have investigated alternative methods by means of monte carlo simulation the distributional characteristics of the estimated powers were compared our findings suggest more appropriate computer software is necessary for the calculation of power and sample size when efficacy is measured by a non parametric method

a unified approach to loewner and hankel matrices abstract we introduce so called generalized loewner matrices the definition is almost the same as antoulas and anderson s our class of generalized loewner matrices contains both classical loewner and hankel matrices as special cases the properties of such matrices are studied together with the associated interpolation problem the detailed characterization of its nonsolvable cases is given by means of the s c minimal generalized solution by means of rank properties of the loewner matrix and by means of the s c l polynomial introduced by fiedler

the general rational interpolation problem and its connection with the nevanlinna pick interpolation and power moment problem abstract the hankel vector approach in a recent work of the author with zhao and zhang on the general rational interpolation problem grip allows the problem to be reduced to what amounts to a limiting case i e a high order rational interpolation at only a single node infinity in particular it gives rise to a new method to find the coefficient matrix of the linear fractional parametrization of interpolants to the grip this approach is extended in the present paper to the nevanlinna pick np interpolation problem with multiple nodes in the class of nevanlinna functions which can be in essence considered as a grip with certain symmetries the hankel vector suitably adapted to the present situation is in fact a finite nonnegative sequence relative to an axis if the problem is solvable in particular a finite positive sequence in the indeterminate case this development leads to an intrinsic connection between the np problem and problems of the truncated power moment and reconstructing generators of the hankel vector in question and therefore opens up certain ways for the discussion and solution of the np problem and its relatives on the basis of the theories of the rational interpolation and algebraic moment problems

on deriving the inverse of a sum of matrices available expressions are reviewed and new ones derived for the inverse of the sum of two matrices one of them being nonsingular particular attention is given to text bf a text bf ubv 1 where textbf a is nonsingular and textbf u b and textbf v may be rectangular generalized inverses of textbf a textbf ubv are also considered several statistical applications are discussed

quasidirect decompositions of hankel and toeplitz matrices decompositions over an algebraically closed field of a hankel matrix into a sum of hankel matrices the sum of the ranks of which is equal to the rank of the original matrix are completely described similar results hold for toeplitz matrices

inversion of a block lo wner matrix in this paper we give a fast algorithm to compute the parameters of an inversion formula for any nonsingular block lowner matrix the connection with matrix rational interpolation is given

note on products of bezoutians and hankel matrices abstract a quantitative theorem due to fiedler and ptak on products of bezoutians and their inverses is extended to a more general case using as the main tool an alternative to fiedler s assertion on the quasidirect decomposition of hankel matrices

polynomials and hankel matrices abstract compatibility of a hankel n n matrix h and a polynomial f of degree m m n is defined if m n compatibility means that hc t f c f h where c f is t companion matrix of f with a suitable generalization of c f this theorem is generalized to the case that m n

special lagrangian cones we study special lagrangian cones in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i with isolated singularities especially the case n 3 our main result constructs an infinite family of special lagrangian cones in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i each of which has a toroidal link we obtain a detailed geometric description of these tori we prove a regularity result for special lagrangian cones in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i with a spherical link any such cone must be a plane we also construct a one parameter family of asymptotically conical special lagrangian submanifolds from any special lagrangian cone

the use of bootstrap methods for estimating sample size and analysing health related quality of life outcomes health related quality of life hrqol measures are increasingly used in trials as primary outcome measures investigators are now asking statisticians for advice on how to plan and analyse studies using such outcomes hrqol outcomes like the sf 36 are usual measured on an ordinal scale although most investigators assume that there exists an underlying continuous latent variable and that the actual measured outcomes the ordered categories reflect contiguous intervals along this continuum the ordinal scaling of hrqol measures means they tend to generate data that have discrete bounded and skewed distributions thus standard methods of analysis that assume normality and constant variance may not be appropriate for this reason conventional statistical advice would suggest non parametric methods be used to analyse hrqol data the bootstrap is one such computer intensive non parametric method for estimating sample sizes and analysing data we describe three methods of estimating sample sizes for two group cross sectional comparisons of hrqol outcomes we then compared the power of the three methods for a two group cross sectional study design using bootstrap simulation the results showed that under the location shift alternative hypothesis conventional methods of sample size estimation performed well particularly whitehead s method whitehead s method is recommended if the hrqol outcome has a limited number of discrete values 7 and or the expected proportion of cases at either of the bounds is high if a pilot data set is readily available then bootstrap simulation will provide a more accurate and reliable estimate than conventional methods finally we used the bootstrap for hypothesis testing and the estimation of standard errors and confidence intervals for parameters in an example data set we then compared and contrasted the bootstrap with standard methods of analysing hrqol outcomes in the data set studied with the sf 36 outcome the use of the bootstrap for estimating sample sizes and analysing hrqol data produces results similar to conventional statistical methods these results suggest that bootstrap methods are not more appropriate for analysing hrqol outcome data than standard methods

choice of test for comparing two groups with particular application to skewed outcomes we consider a clinical trial where a skewed outcome variable is to be compared between two groups while comparison of sample means may lack power we show that power also depends on the nature of the anticipated treatment effect for any given distribution in the control arm there is a family of true distributions in the intervention arm for which the most powerful test is a comparison of arithmetic means similar results hold for a comparison of geometric means and approximately for the wilcoxon rank sum test and a comparison of medians we discuss how these methods could be used in planning the analysis of a clinical trial in which the intervention effect alters the shape of the distribution these ideas are illustrated by a trial in community psychiatry where the primary outcome days in hospital was highly skewed but the intervention was mainly expected to reduce the frequency of values in the tail we show that a comparison of sample means is a reasonable choice in this case despite the skewness

a modified wilcoxon test for non negative distributions with a clump of zeros comparing two samples with a continuous non negative score e g a utility score over 0 1 with a substantial proportion say 50 per cent scoring 0 presents distributional problems for most standard tests a wilcoxon rank test can be used but the large number of ties reduces power i propose a new test the wilcoxon rank sum test performed after removing an equal and maximal number of 0 s from each sample this test recovers much of the power compared with a directional modification of a two part test proposed by lachenbruch the truncated wilcoxon has similar power when the non zero scores are independent of the proportion of zeros but unlike the two part test the truncated wilcoxon is relatively unaffected when these processes are dependent

algebraic condition for the singularity of certain toeplitz pencils abstract an algebraic condition for the singularity of certain toeplitz matrix pencils is derived which involves only the principal minors of the constant parts of the pencils this leads to an algebraic conjecture which is equivalent to the so called toeplitz pencil conjecture

some power considerations when deciding to use transformations conventional wisdom suggests that for small data sets having substantial skew one should attempt to determine the correct distributional form if possible and apply statistical methods appropriate for that distribution transformations such as the log or square root are often used if an appropriate distributional form cannot be determined a distribution free procedure such as a rank transformation or a randomization test procedure can be used to better appreciate the effect of such alternatives on both the type i error and power of detecting differences between treatment groups simulation studies were conducted for responses having specific gamma g r theta and log normal ln m v distributions the gamma and log normal distributions were selected so that they had the same first two moments a simple two group design was assumed the reference group always had an average disease level mu 3 0 mu r theta for gamma mu m for log normal and the treatment group always had means whose reductions ranged from 0 per cent to 50 per cent the effect of distributional type and the degree of skewness was investigated by varying the population parameter values six statistical test procedures were compared for the gamma distributions all test procedures were robust relative to the type i error the ump test based on a ratio of sample means produced the greatest power for all combinations of n r and rt the power losses associated with the randomization test the t test on original scale and the t test on the square root scale were very small 3 per cent to 6 per cent in absolute value for n 10 and 15 and less than 2 per cent for group sizes of 25 or more the power loss associated with the t test on the log scale was much larger ranging from 5 per cent to 10 per cent smaller power than the t test on original scale the wilcoxon rank test produced similar results to that of the log t test for small samples the power for the shifted log x c test increased monotonically to the asymptotic value of the orig t test the same five test procedures based on differences in sample means were then compared for the corresponding log normal distributions the ump test that is log x produced the highest power there was very little power lost for the sqrt t test the loss in power varied between 2 per cent and 5 per cent for the rank test abstract truncated at 400 words

comparison of two treatments with skewed ordinal responses in clinical studies the proportional odds model is widely used to compare treatment efficacies when the responses are categorically ordered however this model has been shown to be inappropriate when the proportional odds assumption is invalid mainly because it is unable to control the type i error rate in such circumstances to remedy this problem the latent normal model was recently promoted and has been demonstrated to be superior to the proportional odds model however the application of the latent normal model is limited to compare treatments with similar underlying distributions except possibly their means and variances when the underlying distributions are very different in skewness both of the aforementioned procedures suffer from the undesirable inflation of the type i error rate to solve the problem for clinical studies with ordinal responses we provide a viable solution that relies on the use of the latent weibull distribution which is a member of the log location scale family the proposed model is able to control the type i error rate regardless of the degree of skewness of the treatment responses in addition the power of the test also outperforms that of the latent normal model the testing procedure draws on newly developed theoretical results related to latent distributions from the location scale family the testing procedure is illustrated with two clinical examples

a semiparametric two component compound mixture model and its application to estimating malaria attributable fractions summary malaria remains a major epidemiologic problem in many developing countries malaria is defined as the presence of parasites and symptoms usually fever due to the parasites in endemic areas an individual may have symptoms attributable either to malaria or to other causes from a clinical viewpoint it is important to correctly diagnose an individual who has developed symptoms so that the appropriate treatments can be given from an epidemiologic and economic viewpoint it is important to determine the proportion of malaria affected cases in individuals who have symptoms so that policies on intervention program can be developed once symptoms have developed in an individual the diagnosis of malaria can be based on the analysis of the parasite levels in blood samples however even a blood test is not conclusive as in endemic areas many healthy individuals can have parasites in their blood slides therefore data from this type of study can be viewed as coming from a mixture distribution with the components corresponding to malaria and nonmalaria cases a unique feature in this type of data however is the fact that a proportion of the nonmalaria cases have zero parasite levels therefore one of the component distributions is itself a mixture distribution in this article we propose a semiparametric likelihood approach for estimating the proportion of clinical malaria using parasite level data from a group of individuals with symptoms our approach assumes the density ratio for the parasite levels in clinical malaria and nonclinical malaria cases can be modeled using a logistic model we use empirical likelihood to combine the zero and nonzero data the maximum semiparametric likelihood estimate is more efficient than existing nonparametric estimates using only the frequencies of zero and nonzero data on the other hand it is more robust than a fully parametric maximum likelihood estimate that assumes a parametric model for the nonzero data simulation results show that the performance of the proposed method is satisfactory the proposed method is used to analyze data from a malaria survey carried out in tanzania

semiparametric estimation and inference for distributional and general treatment effects summary there is a large literature on methods of analysis for randomized trials with noncompliance which focuses on the effect of treatment on the average outcome the paper considers evaluating the effect of treatment on the entire distribution and general functions of this effect for distributional treatment effects fully non parametric and fully parametric approaches have been proposed the fully non parametric approach could be inefficient but the fully parametric approach is not robust to the violation of distribution assumptions we develop a semiparametric instrumental variable method based on the empirical likelihood approach our method can be applied to general outcomes and general functions of outcome distributions and allows us to predict a subject s latent compliance class on the basis of an observed outcome value in observed assignment and treatment received groups asymptotic results for the estimators and likelihood ratio statistic are derived a simulation study shows that our estimators of various treatment effects are substantially more efficient than the currently used fully non parametric estimators the method is illustrated by an analysis of data from a randomized trial of an encouragement intervention to improve adherence to prescribed depression treatments among depressed elderly patients in primary care practices

u 1 invariant special lagrangian 3 folds i nonsingular solutions abstract this is the first of three papers studying special lagrangian 3 submanifolds sl 3 folds n in c 3 invariant under the u 1 action ei z1 z2 z3 ei z1 e i z2 z3 using analytic methods let n be such a u 1 invariant sl 3 fold then z1 2 z2 2 2a on n for some a r locally n can be written as a kind of graph of functions u v r 2 r satisfying a nonlinear cauchy riemann equation depending on a so that u iv is like a holomorphic function of x iy when a is nonzero u v are always smooth and n is always nonsingular but if a 0 there may be points x 0 where u v are not differentiable which correspond to singular points of n this paper focusses on the nonsingular case when a is nonzero we prove analogues for our nonlinear cauchy riemann equation of well known results in complex analysis in particular we prove existence and uniqueness for solutions of two dirichlet problems derived from it this yields existence and uniqueness of a large class of nonsingular u 1 invariant sl 3 folds in c 3 with two kinds of boundary conditions the sequels extend these to the case a 0 study the singularities of the sl 3 folds that arise and construct special lagrangian fibrations of open sets in c 3

computer intensive methods for tests about the mean of an asymmetrical distribution abstract for one sided tests about the mean of a skewed distribution the t test is asymptotically robust for validity however it can be quite inaccurate and inefficient with small sample sizes results presented here confirm that a procedure due to johnson should be preferred to the t test when the parent distribution is asymmetrical because it reduces the probability of type i error in cases where the t test has an inflated type i error rate and it is more powerful in other situations but if the skewness is severe and the sample size is small then johnson s test can also be appreciably inaccurate for such situations computer intensive test procedures using bootstrap resampling are proposed and with an extensive monte carlo study it is shown that these procedures are remarkably robust and can result in reduced probabilities of type i and type ii errors compared to johnson s test

three dimensional feedback cyclization over c y abstract this paper answers in the affirmative the following open question for 3 dimensional linear systems over the ring of complex polynomials in one variable if the system is reachable does there exist a feedback that makes it reachable from a single input channel the result given remains valid for slightly more general coefficient domains

the hankel pencil conjecture abstract the toeplitz pencil conjecture stated in w schmale p k sharma problem 30 3 singularity of a toeplitz matrix image 30 2003 w schmale p k sharma cyclizable matrix pairs over c x and a conjecture on toeplitz pencils linear algebra appl 389 2004 33 42 is equivalent to a conjecture for n n hankel pencils of the form h n x c i j n 1 where c 0 x is an indeterminate c l 0 for l 0 and c l c c 0 for l 1 in this paper it is shown to be implied by another conjecture which we call the root conjecture the root conjecture asserts a strong relationship between the roots of certain submaximal minors of h n x specialized to have c 1 c 2 1 we give explicit formulae in the c i for these minors and prove the root conjecture for minors m nn m n 1 n of degree 6 this implies the hankel pencil conjecture for matrices up to size 8 8 the main tools involved are a partial parametrization of the set of solutions of systems of polynomial equations that are both homogeneous and index sum homogeneous and use of the sylvester identity for matrices

determinantal identities gauss schur cauchy sylvester kronecker jacobi binet laplace muir and cayley abstract we give a common concise derivation of some important determinantal identities attributed to the mathematicians in the title we also give a formal treatment of determinantal identities of the minors of a matrix

cyclizable matrix pairs over c x and a conjecture on toplitz pencils abstract the 22 year old conjecture is addressed which claims that any reachable matrix pair a b from c x n n m is cyclizable i e allows for f u over c x s t a bf k bu k 0 n 1 is a basis for c y n it is shown that for a whole class of pairs the correctness of the conjecture is a consequence of the correctness of a conjecture on certain toplitz pencils an algebraic computable test for the validity of the latter is given based on these results the validity of the conjecture on cyclizability can be extended up to dimension 5

on some properties of positive definite toeplitz matrices and their possible applications abstract various properties of a real symmetric toeplitz matrix m with elements jk a j k 1 j k m are reviewed here matrices of this kind often arise in applications in statistics econometrics psychometrics structural engineering multichannel filtering reflection seismology etc and it is desirable to have techniques which exploit their special structure possible applications of the results related to their inverse determinant and eigenvalue problem are suggested

multiplication of diagonal transforms of loewner matrices abstract a new formula for multiplication of matrices arising as appropriate diagonal transformations of loewner matrices is derived in the case of symmetric loewner matrices it is shown that special subclasses form multiplicative groups some extensions of known results about polynomials associated with hankel and loewner matrices are given also a previously found formula of the author describing the inverses of loewner matrices is shown to be a consequence of the present results

the log transformation is special the logarithmic log transformation is a simple yet controversial step in the analysis of positive continuous data measured on an interval scale situations where a log transformation is indicated will be reviewed this paper contends that the log transformation should not be classed with other transformations as it has particular advantages problems with using the data themselves to decide whether or not to transform will be discussed it is recommended that log transformed analyses should frequently be preferred to untransformed analyses and that careful consideration should be given to use of a log transformation at the protocol design stage