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We define the Laplacian on an arbitrary set with a not necessarily symmetric weight function and discuss the Dirichlet problem and other classical topics in this setting.

This paper presents the foundational ideas for a new way of modeling social aggregation. Traditional approaches have been using network theory, and the theory of random networks. Under that paradigm, every social agent is represented by a node, and every social interaction is represented by a segment connecting two nodes. Early work in family interactions,… (More)

In this paper we offer a new definition of monogenicity for functions defined on R n+1 with values in the Clifford algebra R n following an idea inspired by the recent papers [6], [7]. This new class of monogenic functions contains the polynomials (and, more in general, power series) with coefficients in the Clifford algebra R n. We will prove a Cauchy… (More)

This paper presents some algorithmic techniques to compute explicitly the noetherian operators associated to a class of ideals and modules over a polynomial ring. The procedures we include in this work can be easily encoded in computer algebra packages such as CoCoA [5] and Singular [9].

In this paper we use the notion of slice monogenic functions [2] to define a new functional calculus for an n-tuple T of not necessarily commuting operators. This calculus is different from the one discussed in [5] and it allows the explicit construction of the eigenvalue equation for the n-tuple T based on a new notion of spectrum for T. Our functional… (More)

We employ a classical idea of Ehrenpreis, together with a new algebraic result, to give a new proof that regular functions of several quaternionic variables cannot have compact singularities. As a byproduct we characterize those inhomogeneous Cauchy{ Fueter systems which admit solutions on convex sets. Our method readily extends to the case of monogenic… (More)

In this paper we prove that the projective dimension of Mn = R 4 =hAni is 2n ? 1, where R is the ring of polynomials in 4n variables with complex coeecients, and hAni is the module generated by the columns of a 44n matrix which arises as the Fourier transform of the matrix of diierential operators associated with the regularity condition for a function of n… (More)

In this paper we develop a functional calculus for bounded operators defined on quater-nionic Banach spaces. This calculus is based on the notion of slice-regularity, see [4], and the key tools are a new resolvent operator and a new eigenvalue problem.

We prove that regular functions of one quaternionic variable which satisfy a large class of diierential equations cannot have compact singularities. This result is equivalent to the fact that a large family of 8 4 matrices has torsion-free cokernel. The result (obvious in the complex case) easily extends to Cliiord algebras. 1. Introduction The classical… (More)

The problem of reconstructing and identifying intracellular protein signaling and biochemical networks is of critical importance in biology. We propose a mathematical approach called augmented sparse reconstruction for the identification of links among nodes of ordinary differential equation (ODE) networks, given a small set of observed trajectories with… (More)