Tidal oscillations at the head of Monterey Submarine Canyon and their relation to oceanographic sampling and the circulation of water in Monterey Bay. Annual report, Part 6, September 1972

During a 25-hour hydrographic times series at two stations near the head of Monterey Submarine Canyon, an internal tide was observed with an amplitude of 80 to 115 m in water depths of 120 and 220 m respectively. These large oscillations produced daily variations in hydrographic and chemical parameters that were of the same magnitude as seasonal variations in Monterey Bay. Computed velocities associated with the internal tide were on the order of 10 em/sec, and this tidally induced circulation may have a significant role in the exchange of deep water between Monterey Submarine Canyon and the open ocean. (PDF contains 49 pages

Matrix Convex Hulls of Free Semialgebraic Sets

This article resides in the realm of the noncommutative (free) analog of real algebraic geometry - the study of polynomial inequalities and equations over the real numbers - with a focus on matrix convex sets $C$ and their projections $\hat C$. A free semialgebraic set which is convex as well as bounded and open can be represented as the solution set of a Linear Matrix Inequality (LMI), a result which suggests that convex free semialgebraic sets are rare. Further, Tarski's transfer principle fails in the free setting: The projection of a free convex semialgebraic set need not be free semialgebraic. Both of these results, and the importance of convex approximations in the optimization community, provide impetus and motivation for the study of the free (matrix) convex hull of free semialgebraic sets. This article presents the construction of a sequence $C^{(d)}$ of LMI domains in increasingly many variables whose projections $\hat C^{(d)}$ are successively finer outer approximations of the matrix convex hull of a free semialgebraic set $D_p=\{X: p(X)\succeq0\}$. It is based on free analogs of moments and Hankel matrices. Such an approximation scheme is possibly the best that can be done in general. Indeed, natural noncommutative transcriptions of formulas for certain well known classical (commutative) convex hulls does not produce the convex hulls in the free case. This failure is illustrated on one of the simplest free nonconvex $D_p$. A basic question is which free sets $\hat S$ are the projection of a free semialgebraic set $S$? Techniques and results of this paper bear upon this question which is open even for convex sets.Comment: 41 pages; includes table of contents; supplementary material (a Mathematica notebook) can be found at http://www.math.auckland.ac.nz/~igorklep/publ.htm

The Tracial Hahn-Banach Theorem, Polar Duals, Matrix Convex Sets, and Projections of Free Spectrahedra

This article investigates matrix convex sets and introduces their tracial analogs which we call contractively tracial convex sets. In both contexts completely positive (cp) maps play a central role: unital cp maps in the case of matrix convex sets and trace preserving cp (CPTP) maps in the case of contractively tracial convex sets. CPTP maps, also known as quantum channels, are fundamental objects in quantum information theory. Free convexity is intimately connected with Linear Matrix Inequalities (LMIs) L(x) = A_0 + A_1 x_1 + ... + A_g x_g > 0 and their matrix convex solution sets { X : L(X) is positive semidefinite }, called free spectrahedra. The Effros-Winkler Hahn-Banach Separation Theorem for matrix convex sets states that matrix convex sets are solution sets of LMIs with operator coefficients. Motivated in part by cp interpolation problems, we develop the foundations of convex analysis and duality in the tracial setting, including tracial analogs of the Effros-Winkler Theorem. The projection of a free spectrahedron in g+h variables to g variables is a matrix convex set called a free spectrahedrop. As a class, free spectrahedrops are more general than free spectrahedra, but at the same time more tractable than general matrix convex sets. Moreover, many matrix convex sets can be approximated from above by free spectrahedrops. Here a number of fundamental results for spectrahedrops and their polar duals are established. For example, the free polar dual of a free spectrahedrop is again a free spectrahedrop. We also give a Positivstellensatz for free polynomials that are positive on a free spectrahedrop.Comment: v2: 56 pages, reworked abstract and intro to emphasize the convex duality aspects; v1: 60 pages; includes an index and table of content

The use of Bioceramics as root-end filling materials in periradicular surgery: a literature review

Introduction: Periradicular surgery involves the placement of a root-end filling following root-end resection, to provide an apical seal to the root canal system. Historically several materials have been used in order to achieve this seal. Recently a class of materials known as Bioceramics have been adopted. The aim of this article is to provide a review of the outcomes of periradicular surgery when Bioceramic root-end filling materials are used on human permanent teeth in comparison to “traditional” materials. Methods & results: An electronic literature search was performed in the databases of Web of Science, PubMed and Google Scholar, between 2006 and 2017, to collect clinical studies where Bioceramic materials were utilised as retrograde filling materials, and to compare such materials with traditional materials. In this search, 1 systematic review and 14 clinical studies were identified. Of these, 8 reported the success rates of retrograde Bioceramics, and 6 compared treatment outcomes of mineral trioxide aggregate (MTA) and traditional cements when used as root-end filling materials. Conclusion: Bioceramic root-end filling materials are shown to have success rates of 86.4–95.6% (over 1–5 years). Bioceramics has significantly higher success rates than amalgam, but they were statistically similar to intermediate restorative material (IRM) and Super ethoxybenzoic acid (Super EBA) when used as retrograde filling materials in apical surgery. However, it seems that the high success rates were not solely attributable to the type of the root-end filling materials. The surgical/microsurgical techniques and tooth prognostic factors may significantly affect treatment outcome

Free bianalytic maps between spectrahedra and spectraballs in a generic setting

Given a tuple $E=(E_1,\dots,E_g)$ of $d\times d$ matrices, the collection of those tuples of matrices $X=(X_1,\dots,X_g)$ (of the same size) such that $\| \sum E_j\otimes X_j\|\le 1$ is called a spectraball $\mathcal B_E$. Likewise, given a tuple $B=(B_1,\dots,B_g)$ of $e\times e$ matrices the collection of tuples of matrices $X=(X_1,\dots,X_g)$ (of the same size) such that $I + \sum B_j\otimes X_j +\sum B_j^* \otimes X_j^*\succeq 0$ is a free spectrahedron $\mathcal D_B$. Assuming $E$ and $B$ are irreducible, plus an additional mild hypothesis, there is a free bianalytic map $p:\mathcal B_E\to \mathcal D_B$ normalized by $p(0)=0$ and $p'(0)=I$ if and only if $\mathcal B_E=\mathcal B_B$ and $B$ spans an algebra. Moreover $p$ is unique, rational and has an elegant algebraic representation.Comment: 19 page