Spherical functions and invariant differential operators on complex Grassmann manifolds

Proofs are given of two theorems of Berezin and Karpelevič, which as far as we know never have been proved correctly. By using eigenfunctions of the Laplace-Beltrami operator it is shown that the spherical functions on a complex Grassmann manifold are given by a determinant of certain hypergeometric functions. By application of this result, it is proved that a certain system of operators, fow which explicit expressions are given, generates the algebra of radial parts of invariant differential operators.     

The Sasaki Hook Is Not a [Static] Implicative Connective but Induces a Backward [in Time] Dynamic One That Assigns Causes

The Sasaki adjunction, which formally encodes the logicality that different authors tried to attach to the Sasaki hook as a ‘quantum implicative connective,’ has a fundamental dynamic nature and encodes the so-called ‘causal duality’ (Coecke et al., 2001) for the particular case of a quantum measurement with a projector as corresponding self-adjoint operator. The action of the Sasaki hook ( $a\xrightarrow{S} - $ ) for fixed antecedent a assigns to some property “the weakest cause before the measurement of actuality of that property after the measurement,” i.e., ( $a\xrightarrow{S}b$ ) is the weakest property that guarantees actuality of b after performing the measurement represented by the projector that has the ‘subspace a’ as eigenstates for eigenvalue 1, say, the measurement that ‘tests’ a. The logicality attributable to quantum systems contains a fundamentally dynamic ingredient: Causal duality actually provides a new dynamic interpretation of orthomodularity. We also reconsider the status of the Sasaki hook within ‘dynamic (operational) quantum logic,’ what leads us to the claim made in the title of this paper. The Sasaki adjunction has a physical significance in terms of causal duality. The labeled dynamic hooks (forwardly and backwardly) that encode quantum measurements, act on properties as $(a_1 \xrightarrow{{\varphi _a }}a_2 ): = (a_1 \to _L (a\xrightarrow{S}a_2 ))$ and $(a_1 \xleftarrow{{\varphi _a }}a_2 ): = ((a\xrightarrow{S}a_2 ) \to _L a_1 )$ , taking values in the ‘disjunctive extension’ $DI(L)$ of the property lattice L, where $a \in L$ is the tested property and $( - \to _L - )$ is the Heyting implication that lives on DI(L). Since these hooks $( - \xrightarrow{{\varphi _a }} - )$ and $( - \xleftarrow{{\varphi _a }} - )$ extend to DI(L)×DI(L) they constitute internal operations. The transition from either classical or constructive/intuitionistic logic to quantum logic entails besides the introduction of an additional unary connective ‘operational resolution’ (Coecke, 2002a) the shift from a binary connective implication to a ternary connective where two of the arguments refer to qualities of the system and the third, the new one, to an obtained outcome (in a measurement)     

Quantaloids Describing Causation and Propagation of Physical Properties

A general principle of causal duality for physical systems, lying at the base of representation theorems for both compound and evolving systems, is proved; formally it is encoded in a quantaloidal setting. Other particular examples of quantaloids and quantaloidal morphisms appear naturally within this setting; as in the case of causal duality, they originate from primitive physical reasonings on the lattices of properties of physical systems. Furthermore, an essentially dynamical operational foundation for studying physical systems is outlined; complementary as it is to the existing static operational foundation, it leads to the natural axiomatization of causal duality in operational quantum logic.     

The smallest enclosing ball problem and the smallest intersecting ball problem: existence and uniqueness of solutions

In this paper we study the following problems and their further generalizations: given a finite number of nonempty closed subsets of a normed space, find a ball with the smallest radius that encloses all of the sets, and find a ball with the smallest radius that intersects all of the sets. These problems can be viewed as generalized versions of the smallest enclosing circle problem introduced in the nineteenth century by Sylvester which asks for the circle of smallest radius enclosing a given set of finite points in the plane. We will focus on sufficient conditions for the existence and uniqueness of an optimal solution for each problem, while the study of optimality conditions and numerical implementation will be addressed in our next projects.     

Following the cat: A preface for the next three papers

The three papers to follow, by Chris Rasmussen and three of his students, build from a task design the authors of this note developed in the 1990s. To introduce these papers, which extend our prior work in new directions, we sketch some background here.     

Modeling the Behavior of Heat-Shrinkable Thin Films

We describe an asymptotic model for the behavior of PET-like heat-shrinkable thin films that includes both membrane and bending energies when the thickness of the film is positive. We compare the model to Koiter’s shell model and to models in which a membrane energy or a bending energy are obtained by Γ-convergence techniques. We also provide computational results for various temperature distributions applied to the films.