General Polyhedral Finite Elements for Rapid Nonlinear Analysis

An analysis system for solid mechanics applications is described in which a new finite element method that can accommodate general polyhedral elements is exploited. The essence of the method is direct polynomial approximation of the shape functions on the physical element, without transformation to a canonical element. The main motive is elimination of the requirement that all elements be similar to a canonical element via the usual isoparametric mapping. It is this topological restriction that largely drives the design of mesh-generation algorithms, and ultimately leads to the considerable human effort required to perform complex analyses. An integrated analysis system is described in which the flexibility of the polyhedral element method is leveraged via a robust computational geometry processor. The role of the latter is to perform rapid Boolean intersection operations between hex meshes and surface representations of the body to be analyzed. A typical procedure is to create a space-filling structured hex mesh that contains the body, and then extract a polyhedral mesh of the body by intersecting the hex mesh and the body’s surface. The result is a mesh that is directly usable in the polyhedral finite element method. Some example applications are: 1) simulation on very complex geometries; 2) rapid geometry modification and re-analysis; and 3) analysis of material-removal process steps following deformation processing. This last class of problems is particularly challenging for the conventional FE methodology, because the element boundaries are, in general, not aligned with the cutting geometry following the deformation (e.g. forging) step.

Thom isomorphism and push-forward map in twisted K-theory

We establish the Thom isomorphism in twisted K-theory for any real vector bundle and develop the push-forward map in twisted K-theory for any differentiable map f : X → Y (not necessarily K-oriented). We also obtain the wrong way functoriality property for the push-forward map in twisted K-theory. For D-branes satisfying Freed-Witten's anomaly cancellation condition in a manifold with a non-trivial B-field, we associate a canonical element in the twisted K-group to get the so-called D-brane charges.

A Simple Mathematical Method to Predict the Course of a First Episode of Liver Viral Cytolysis

Early detection of the self-limiting or chronic tendency towards disease at the beginning of the hepatitis process is a canonical element of the analysis and synthesis to formulate the diagnosis and to decide the most useful and economic therapeutic strategy. Early treatment with antiviral drugs before liver functions undergo deterioration may reduce death and disability especially in young patients.A formula suggested by a Max Planck equation provides mathematical curves that can be obtained very early during the course of hepatitis using just a few initial serum measurements of the alanine-aminotransferase (ALT) enzyme. The clinical ALT data forming the unpredictable early natural curve of ALT can be transformed into mathematical values capable of providing a deterministic curve that intersects the normal ALT value at what is known as theGpoint, and allows some predictions to be made concerning the acute or chronic evolution of the disease.TheGpoint seems to be a good predictor of whether or not a patient with clinically evident first episode of viral hepatitis will develop chronic liver disease as it revealed the initial phase of the first cytolytic episode in 56 untreated patients of our 59 cases; the three wrong predictions involved patients being treated with interferon.

The groupoid interpretation of type theory

Many will agree that identity sets are the most intriguing concept of intensional Martin-Löf type theory. For instance, it may appear surprising that their axiomatisation as an inductive family allows one to deduce the usual properties of equality, notably the replacement rule (Leibniz’s principle) which gives P(a′) from P(a) and a proof that a equals a′. This holds for arbitrary families of sets P, not only those corresponding to a predicate. This is not in conflict with decidability of type checking since if a equals a′ and p : P(a) then one does not in general have p : P(a′), but only subst(s, p) : P(a′) where s is the proof that a equals a′ and subst is defined from the eliminator for identity sets. It is a natural question to ask whether these translation functions subst(s, _) actually depend upon the nature of the proof s or, more generally, the question whether any two elements of an identity set are equal. We will call UIP(A) (t/niqueness of Identity Proofs) the following property. If a1, a2 are objects of type A then for any two proofs p and q of the proposition “a1 equals a2” we can prove that p and q are equal. More generally, UIP will stand for UIP(A) for all types A. Note that in traditional logical formalism a principle like UIP cannot even be expressed sensibly as proofs cannot be referred to by terms of the object language and thus are not within the scope of prepositional equality. The question of whether UIP is valid in intensional Martin-Löf type theory was open for a while, though it was commonly believed that UIP is underivable as any attempt for constructing a proof has failed (Coquand 1992; Streicher 1993; Altenkirch 1992). On the other hand, the intuition that a type is determined by its canonical objects might be seen as evidence for the validity of UIP as the identity sets have at most one canonical element corresponding to an instance of reflexivity.