In this paper we address some of the most fundamental questions regarding the differentiability structure of locally Lipschitz functions defined on separable Banach spaces. For example, we examine the relationship between integrability, D-representability, and strict differentiability. In addition to this, we show that on any separable Banach space there is a significant family of locally Lipsc...

Consider a Hilbert space obtained as the completion of the polynomials C[z ] in m-variables for which the monomials are orthogonal. If the commuting weighted shifts defined by the coordinate functions are essentially normal, then the same is true for their restrictions to invariant subspaces spanned by monomials. This generalizes the result of Arveson [4] in which the Hilbert space is the m-shi...

In this paper I offer a defence of a Russellian analysis of the referential uses of incomplete (mis) descriptions, in a contextual setting. With regard to descriptions (introduction), I will support the former against the latter. In 1. I explain what I mean by “essentially” incomplete descriptions: incomplete descriptions are context dependent descriptions. In 2. I examine one of the best versi...

Although sensory perception and neurobiology are traditionally investigated one modality at a time, real world behaviour and perception are driven by the integration of information from multiple sensory sources. Mounting evidence suggests that the neural underpinnings of multisensory integration extend into early sensory processing. This article examines the notion that neocortical operations a...

An algebraic quantum group is a regular multiplier Hopf algebra with integrals. In this paper we will develop a theory of algebraic quantum hypergroups. It is very similar to the theory of algebraic quantum groups, except that the comultiplication is no longer assumed to be a homomorphism. We still require the existence of a left and of a right integral. There is also an antipode but it is char...

An algebraic quantum group is a multiplier Hopf algebra with integrals. In this paper we will develop a theory of algebraic quantum hypergroups. It is very similar to the theory of algebraic quantum groups, except that the comultiplication is no longer assumed to be a homomorphism. We still require the existence of a left and of a right integral. There is also an antipode but it is characterize...

Abstract Second-order (maximally) superintegrable systems in dimensions two and three are essentially classified. With increasing dimension, however, the non-linear partial differential equations employed current methods become unmanageable. Here we propose a new, algebraic-geometric approach to classification problem—based on proof that space for irreducible non-degenerate second-order is natu...

In this paper, we propose an abstract definition of dependent type theories as essentially algebraic theories. One of the main advantages of this definition is its composability: simple theories can be combined into more complex ones, and different properties of the resulting theory may be deduced from properties of the basic ones. We define a category of algebraic dependent type theories which...

We propose a new definition for abstract syntax (with binding constructions), and, accordingly, for initial semantics and algebraicity. Our definition is based on the notion of module over a monad and its companion notion of linearity. In our setting, we give a one-line definition of an untyped lambda-calculus. Among untyped lambda-calculi, the initial one, the pure untyped lambda-calculus, app...