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Given an acyclic representation  of the fundamental group of a compact oriented odd-dimensional manifold, which is close enough to an acyclic unitary representation, we define a refinement T of the Ray-Singer torsion associated to , which can be viewed as the analytic counterpart of the refined combinatorial torsion introduced by Turaev. T is equal to the graded determinant of the odd signature operator up to a correction term, the metric anomaly, needed to make it independent of the choice of the Riemannian metric.T is a holomorphic function on the space of such representations of the fundamental group. When  is a unitary representation, the absolute value of T is equal to the Ray-Singer torsion and the phase of T is proportional to the -invariant of the odd signature operator. The fact that the Ray-Singer torsion and the-invariant can be combined into one holomorphic function allows one to use methods of complex analysis to study both invariants. In particular, using these methods we compute the quotient of the refined analytic torsion and Turaev’s refinement of the combinatorial torsion generalizing in this way the classical Cheeger-M¨uller theorem. As an application, we extend and improve a result of Farber about the relationship between the Farber-Turaev absolute torsion and the -invariant. As part of our construction of T we prove several new results about determinants and -invariants of non self-adjoint elliptic operators.

Let Ω⊂ℝ^{$n$}be an arbitrary open set. We characterize the space$W$^{1,1}_{loc}(Ω) using variants of the classical area and coarea formulas. We use these characterizations to obtain a norm approximation and trace theorems for functions in the space$W$^{1,1}(ℝ^{$n$}).

In the past decades, many methods for computing conformal mesh parameterizations have been developed in response to demand of numerous applications in the field of geometry processing. Spectral conformal parameterization (SCP) (Mullen et al. in Proceedings of the symposium on geometry processing, SGP ’08. Eurographics Association, Aire-la-Ville, Switzerland, pp 1487–1494, 2008) is one of these methods used to compute a quality conformal parameterization based on the spectral techniques. SCP focuses on a generalized eigenvalue problem (GEP) $L_{C} \bf{f} = \lambda B \bf{f}$ whose eigenvector(s) associated with the smallest positive eigenvalue(s) provide the conformal parameterization result. This paper is devoted to studying a novel eigensolver for this GEP. Based on structures of the matrix pair $(L_{C}, B)$, we show that this GEP can be transformed into a small-scale compressed and deflated standard eigenvalue problem with a symmetric positive definite skew-Hamiltonian operator. We then propose a symmetric skew-Hamiltonian isotropic Lanczos algorithm (SHILA) to solve the reduced problem. Numerical experiments show that our compressed deflating technique can exclude the impact of convergence from the kernel of $L_{C}$ and transform the original problem to a more robust system. The novel SHILA method can effectively avoid the disturbance of duplicate eigenvalues. As a result, based on the spectral model of SCP, our numerical eigensolver can compute the conformal parameterization accurately and efficiently.