This work studies the kernel of a linear operator associated with the generalized k-fold commutator. Given a set $\mathfrak{A}= \left\{ A_{1}, \ldots ,A_{k} \right\}$ of real $n \times n$ matrices, the commutator is denoted by$[A_{1}| \ldots |A_{k}]$. For a fixed set of matrices $\mathfrak{A}$ we introduce a multilinear skew-symmetric linear operator $T_{\mathfrak{A}}(X)=T(A_{1}, \ldots ,A_{k})[X]=[A_{1}| \ldots |A_{k} |X] $. For fixed $n$ and $k \ge 2n-1, \; T_{\mathfrak{A}} \equiv 0$ by the Amitsur--Levitski Theorem [2] , which motivated this work. The matrix representation $M$ of the linear transformation $T$ is called the k-commutator matrix. $M$ has interesting properties, e.g., it is a commutator; for $k$ odd, there is a permutation of the rows of $M$ that makes it skew-symmetric. For both $k$ and $n$ odd, a provocative matrix $\mathcal{S}$ appears in the kernel of $T$. By using the Moore--Penrose inverse and introducing a conjecture about the rank of $M$, the entries of $\mathcal{S}$ are shown to be quotients of polynomials in the entries of the matrices in $\mathfrak{A}$. One case of the conjecture has been recently proven by Brassil. The Moore--Penrose inverse provides a full rank decomposition of $M$.

Recent studies have reported the prevalence of pituitary tumors to be ~1/1000 population. Many are prolactin-producing tumors that are managed medically, however, the epidemiology of surgically resected pituitary adenohypophysial neuroendocrine tumors has not been reported in a large series with detailed characterization. We reviewed 1055 adenohypophysial tumors from 1169 transsphenoidal resections from the pathology files of University Health Network, Toronto, 2001-2016. Tumors were characterized by immunohistochemical localization of transcription factors (Pit-1, ERα, SF-1, Tpit), hormones (adrenocorticotropin, growth hormone, prolactin, β-thyrotropin, β-folliculotropin, β-luteotropin, α-subunit), and other biomarkers (keratins, Ki67, p27, FGFR4). Electron microscopy was used only for unusual lesions. In this cohort, 51.3% of patients were female; the average age was 51 years. Gonadotroph tumors represented 42.5%. Pit-1-lineage-tumors represented 29.9%; these were subclassified as growth-hormone-predominant (somatotroph/mammosomatotroph/mixed; 53%), prolactin-predominant (lactotroph/acidophil-stem-cell; 28%), thyrotrophs (2%), plurihormonal (14%), and not-otherwise-specified (3%). Corticotroph tumors represented 17.1%. Only 4.5% were null cell tumors and 0.5% were unusual plurihormonal tumors. In 5.5% the tumor was not characterized for technical reasons (sample size, fixation, necrosis or other artifact). All corticotroph and plurihormonal tumors were positive for keratins; others tumors showed variable negativity with highest rates in gonadotroph (37.1%) and null cell tumors (28.2%). Tumors with a Ki67 ≥ 3% comprised 60% of this cohort. Global loss of p27 was most frequent in corticotroph neoplasms, specifically those associated with elevated glucocorticoid levels. Corticotroph and lactotroph tumors were more common among females; gonadotroph tumors were more common among males. Younger patients had mainly corticotroph and Pit-1-lineage neoplasms, whereas older patients harbored mainly gonadotroph tumors. This represents one of the largest surgical series of morphologically characterized pituitary tumors reported to date and the first to include the routine use of transcription factors for tumor classification. The data provide the basis for clinicopathologic correlations that are helpful for prognostic and predictive patient management.

The generalized commutator [ A1 vertical bar center dot center dot center dot vertical bar A(k)] of a list A1,..., A(k) of k real n x n matrices is defined as a multilinear skew-symmetric function and the linear operator T = T(A1,..., A(k)) on the vector space Mn(R) is defined by TX := [ A1 vertical bar center dot center dot center dot vertical bar A(k) vertical bar X]. The Amitsur-Levitzki theorem shows that T = 0 when k = 2n-1. We investigate the kernel of T and prove that for all integers k and n such that 2 = k = 2n-2 we have dim ker T(A1,..., A(k)) =.0(n, k) where.0(n, k) := k if k is even; k + 1 if k is odd and n is even; and k + 2 if k and n are both odd. We conjecture that this result is best possible and that dim ker T(A1,..., A(k))=.0(n, k) for almost all A1,..., A(k) when k and n are in this range. This conjecture is supported by some computational evidence but so far remains open.

Comparison of the shapes of barefoot impressions from an individual with footprints or shoes linked to a crime may be useful as a means of including or excluding that individual as possibly being at the scene of a crime. The question of the distinguishability of a person's barefoot print arises frequently. This study indicates that measurements taken from the outlines of inked footprint impressions show a great degree of variability between donors and a great degree of similarity for multiple impressions taken from the same donor. The normality of the set of measurements on footprint outlines that we have selected for this study is confirmed. A statistical justification for the use of the product rule on individual statistical precisions is developed.

Given a data set arising from a series of observations, an outlier is a value that deviates substantially from the natural variability of the data set as to arouse suspicions that it was generated by a different mechanism. We call an observation an extreme outlier if it lies at an abnormal distance from the "center" of the data set. We introduce the Monte Carlo SCD algorithm for detecting extreme outliers. The algorithm finds extreme outliers in terms of a subset of the data set called the outer shell. Each iteration of the algorithm is polynomial. This could be reduced by preprocessing the data to reduce its size.This approach has an interesting new feature. It estimates a relative measure of the degree to which a data point on the outer shell is an outlier (its "outlierness"). This measure has potential for serendipitous discoveries in data mining where unusual or special behavior is of interest. Other applications include spatial filtering and smoothing in digital image processing. We apply this method to baseball data and identify the ten most exceptional pitchers of the 1998 American League. To illustrate another useful application, we also show that the SCD can be used to reduce the solution time of the D-optimal experimental design problem.

. We study the optimal placement of replicas of data objects in a connected network with the topology of a straight-line segment. This special case of a NP-complete location problem has a remarkably attractive algebraic solution. The minimum cost problem gives rise to tridiagonal matrices that are both persymmetric and symmetric and these are used to prove the symmetry of the optimal solution. The eigenvalues and eigenvectors of these matrices are completely described by Chebyshev polynomials of the second kind to give a complete solution to the replica location problem. We denote the k th Chebyshev polynomials of the second kind by U k . The Chebyshev identity arises naturally in examining the norms of the eigenvectors that occur.

. We consider the problem of locating replicas in a network to minimize communications costs. Under the assumption that the read-one-write-all policy is used to ensure data consistency, an optimization problem is formulated in which the cost function estimates the total communications costs. The paper concentrates on the study of the optimal communications cost as a function of the ratio between the frequency of the read and write operations. The problem is reformulated as a zero-one linear programming problem, and its connection to the p -median problem is explained. The general problem is proved to be NP-complete. For path graphs a dynamic programming algorithm for the problem is presented.

We reduce the size of large semidefinite programming problems by identifying necessary linear matrix inequalities (LMI's) using Monte Carlo techniques. We describe three algorithms for detecting necessary LMI constraints that extend algorithms used in linear programming to semidefinite programming. We demonstrate that they are beneficial and could serve as tools for a semidefinite programming preprocessor.A necessary LMI is one whose removal changes the feasible region defined by all the LMI constraints. The general problem of checking whether or not a particular LMI is necessary is NP-complete. However, the methods we describe are polynomial in each iteration, and the number of iterations can be limited by stopping rules. This provides a practical method for reducing the size of some large Semidefinite Programming problems before one attempts to solve them. We demonstrate the applicability of this approach to solving instances of the Lowner ellipsoid problem. We also consider the problem of classification of all the constraints of a semidefinite programming problem as redundant or necessary.

Let R be the convex subset of IR defined by q simultaneous linear matrix inequalities (LMI) A 0 + ∑n i=1 xiA (j) i 0, j = 1, 2, . . . , q. Given a strictly positive vector ω = (ω1, ω2, · · · , ωq), the weighted analytic center xac(ω) is the minimizer argmin (φω(x)) of the strictly convex function φω(x) = ∑q j=1 ωj log det[A (j)(x)]−1 over R. We give a necessary and sufficient condition for a point of R to be a weighted analytic center. We study the argmin function in this instance and show that it is a continuously differentiable open function. In the special case of linear constraints, all interior points are weighted analytic centers. We show that the region W = {xac(ω) | ω > 0} ⊆ R of weighted analytic centers for LMI’s is not convex and does not generally equal R. These results imply that the techniques in linear programming of following paths of analytic centers may require special consideration when extended to semidefinite programming. We show that the regionW and its boundary are described by real algebraic varieties, and provide slices of a non-trivial real algebraic variety to show that W isn’t convex. Stiemke’s Theorem of the alternative provides a practical test of whether a point is in W . Weighted analytic centers are used to improve the location of standing points for the Stand and Hit method of identifying necessary LMI constraints in semidefinite programming.

Twelve known symmetry patterns of matrices are combined with three modest patterns to form a steiner triple system. We investigate matrices satisfying more than one symmetry pattern. We show how a group of operators on GL(n, C) gives rise to distinct types of matrices which satisfy sets of patterns, and which give unique decompositions of matrices into components of each type. These give a new characterization of normal and unitary matrices. We extend symmetry patterns to vectors to study spectral properties of these matrices. When a (skew) symmetric basis of eigenvectors exist, we can infer symmetry properties of these matrices.

This paper describes the application of Reed-Solomon (RS) error correcting codes to the new Canada Post Corporation bar codes, which are used in the mechanised processing of mail. The use of RS codes provides a capability to detect and correct errors and/or erasures which is a substantial improvement on the single parity check code previously employed. This significantly reduces the incidence of costly machine rejects due to errors in the bar codes, and contributes to improved service.

This paper studies the problem of the optimal placement of replicas in a network so that communications costs are minimized when a strict consistency policy is in effect. A READ ANYWRITE ALL policy is introduced, and the appropriate cost function is developed to measure the relative effectiveness of each choice of replica placement. The general problem is proven to be NP complete. An efficient polynomial-time dynamic programming algorithm is introduced for linear graphs. A zero-one linear programming solution is provided for the general problem, using the cost function introduced here. A parametric approach is taken to the problem in the situation in which all activities of each node are identical. This leads to the introduction of piecewise linear solutions to particular problems and the existence of an underlying enveloping curve for the problem.The approach taken to the problem here is to examine simple models to develop some understanding of the issues. This has led to a number of interesting and challenging mathematical questions.

Given a long exact sequence of abelian groups L :. . ., L'-1 1 --+ L' -L' -}1 ~. . .a short exact sequence of complexes of free abelian groups is constructed whose cohomology long exact sequence is precisely L. In this sense, L is realized .Two techniques which are introduced to reduce or replace lengthy diagram chasing arguments may be of interest to some readers.One is an arithmetic of bicartesian squares; the other is the use of the fact that categories of morphisms of abelian categories are themselves abelian.

A shuffle is the horizontal interchange of a pair of blocks of the same size in a matrix. A general algorithm using row reduction and shuffles was first introduced by Luenberger, and then used by Anstreicher and Rothblum to give an algorithm to compute generalized nullspaces. We present a new, concise proof of this shuffle algorithm, and show how the shuffle algorithm can be used in deriving the Jordan blocks for a square matrix with known eigenvalues.

A commutative square (1) of morphisms is said to have a lifting if there is a morphism λ: B1 → A2 such that λϕ1 = α and ϕ2λ = β1Let us assume that we are working in a fixed abelian category . Therefore, ϕi will have a kernel “Ki” and a cokernel “Ci” for i = 1, 2. Let k : K1 → K2 and c: C1 → C2 denote the canonical morphisms induced by α and β.We shall construct a short exact sequence (s.e.s.)2using the data of (1). We shall prove that (1) has a lifting if and only if k = 0, c = 0, and (2) represents the zero class in Ext1(C1, K2). Furthermore, if (1) has one lifting, then the liftings will be in one-to-one correspondence with the elements of the set |Hom(G1, K2)|.