Intersection theory and the Alesker product

Alesker has introduced the space $\mathcal V^\infty(M)$ of {\it smooth valuations} on a smooth manifold $M$, and shown that it admits a natural commutative multiplication. Although Alesker's original construction is highly technical, from a moral perspective this product is simply an artifact of the operation of intersection of two sets. Subsequently Alesker and Bernig gave an expression for the product in terms of differential forms. We show how the Alesker-Bernig formula arises naturally from the intersection interpretation, and apply this insight to give a new formula for the product of a general valuation with a valuation that is expressed in terms of intersections with a sufficiently rich family of smooth polyhedra.Comment: further revisons, now 23 page

Convolution of convex valuations

We show that the natural "convolution" on the space of smooth, even, translation-invariant convex valuations on a euclidean space $V$, obtained by intertwining the product and the duality transform of S. Alesker, may be expressed in terms of Minkowski sum. Furthermore the resulting product extends naturally to odd valuations as well. Based on this technical result we give an application to integral geometry, generalizing Hadwiger's additive kinematic formula for $SO(V)$ to general compact groups $G \subset O(V)$ acting transitively on the sphere: it turns out that these formulas are in a natural sense dual to the usual (intersection) kinematic formulas.Comment: 18 pages; Thm. 1.4. added; references updated; other minor changes; to appear in Geom. Dedicat

Riemannian curvature measures

A famous theorem of Weyl states that if $M$ is a compact submanifold of euclidean space, then the volumes of small tubes about $M$ are given by a polynomial in the radius $r$, with coefficients that are expressible as integrals of certain scalar invariants of the curvature tensor of $M$ with respect to the induced metric. It is natural to interpret this phenomenon in terms of curvature measures and smooth valuations, in the sense of Alesker, canonically associated to the Riemannian structure of $M$. This perspective yields a fundamental new structure in Riemannian geometry, in the form of a certain abstract module over the polynomial algebra $\mathbb R[t]$ that reflects the behavior of Alesker multiplication. This module encodes a key piece of the array of kinematic formulas of any Riemannian manifold on which a group of isometries acts transitively on the sphere bundle. We illustrate this principle in precise terms in the case where $M$ is a complex space form.Comment: Corrected version, to appear in GAF

The effect of pre-processing and grain structure on the bio-corrosion and fatigue resistance of magnesium alloy AZ31

Magnesium alloys are broadly used for structural applications in the aerospace and automotive industries as well as in consumer electronics. While a high specific strength is the forte of magnesium alloys, one serious limitation for Mg alloys is their corrosion performance. Unlike aluminium, it does not form a stable passive film to provide long-term protection from further corrosion. The poor corrosion resistance of magnesium and magnesium alloys is regarded as a major drawback, and significant effort has been focused on improving this.[1-3] However, the high reactivity of magnesium alloys in corrosive media can be used to advantage in biomedical applications, particularly in temporary implants where the capacity of a material for bio-degradation is one of the most sought after properties. Indeed, permanent implant materials, such as stainless steel, titanium alloys or Nitinol (55Ni-45Ti), are the only choices currently available for hard tissue implantation. They can cause permanent physical irritation, long-term endothelial dysfunction and chronic inflammatory local reaction. Sometimes a second operation is needed for the implant to be removed. Given the ability of the human body to gradually recover and regenerate damaged tissue, the ideal solution would thus be a degradable implant, which would offer a physiologically less invasive repair and temporary support during tissue recovery. After fulfilling its function, this implant would be obliterated, being absorbed by the body. This philosophy of implant surgery would also be of particular interest for endovascular stent

Symmetric Criticality for Tight Knots

We prove a version of symmetric criticality for ropelength-critical knots. Our theorem implies that a knot or link with a symmetric representative has a ropelength-critical configuration with the same symmetry. We use this to construct new examples of ropelength critical configurations for knots and links which are different from the ropelength minima for these knot and link types.Comment: This version adds references, and most importantly an acknowledgements section which should have been in the original postin

Phase properties of hypergeometric states and negative hypergeometric states

We show that the three quantum states (P$\acute{o}$lya states, the generalized non-classical states related to Hahn polynomials and negative hypergeometric states) introduced recently as intermediates states which interpolate between the binomial states and negative binomial states are essentially identical. By using the Hermitial-phase-operator formalism, the phase properties of the hypergeometric states and negative hypergeometric states are studied in detail. We find that the number of peaks of phase probability distribution is one for the hypergeometric states and $M$ for the negative hypergeometric states.Comment: 7 pages, 4 figure