A non-commutative formula for the colored Jones function

The colored Jones function of a knot is a sequence of Laurent polynomials that encodes the Jones polynomial of a knot and its parallels. It has been understood in terms of representations of quantum groups and Witten gave an intrinsic quantum field theory interpretation of the colored Jones function as the expectation value of Wilson loops of a 3-dimensional gauge theory, the Chern–Simons theory. We present the colored Jones function as an evaluation of the inverse of a non-commutative fermionic partition function. This result is in the form familiar in quantum field theory, namely the inverse of a generalized determinant. Our formula also reveals a direct relation between the Alexander polynomial and the colored Jones function of a knot and immediately implies the extensively studied Melvin–Morton–Rozansky conjecture, first proved by Bar–Natan and the first author about 10 years ago. Our results complement recent work of Huynh and Le, who also give a non-commutative formulae for the colored Jones function of a knot, starting from a non-commutative formula for the R matrix of the quantum group ; see Huynh and Le (in math.GT/0503296).     

Grothendieck groups of unbounded complexes of finitely generated modules

Let Λ be a left Artinian ring, D⁺(mod Λ) (resp., D⁻(mod Λ), D(mod Λ)) the derived category of bounded below complexes (resp., bounded above complexes, unbounded complexes) of finitely generated left Λ-modules. We show that the Grothendieck groups K₀(D⁺(mod Λ)), K₀(D⁻(mod Λ)) and K₀(D(mod Λ)) are trivial. Received: 7 April 2005     

Well-posedness and sharp uniform decay rates at the L 2(Ω)-Level of the Schrödinger equation with nonlinear boundary dissipation

We prove that the Schr?dinger equation defined on a bounded open domain of and subject to a certain attractive, nonlinear, dissipative boundary feedback is (semigroup) well-posed on L₂(Ω) for any n = 1, 2, 3, ..., and, moreover, stable on L₂(Ω) for n = 2, 3, with sharp (optimal) uniform rates of decay. Uniformity is with respect to all initial conditions contained in a given L₂(Ω)-ball. This result generalizes the corresponding linear case which was proved recently in [L-T-Z.2]. Both results critically rely—at the outset—on a far general result of interest in its own right: an energy estimate at the L₂(Ω)-level for a fully general Schr?dinger equation with gradient and potential terms. The latter requires a heavy use of pseudo-differential/micro-local machinery [L-T-Z.2, Section 10], to shift down the more natural H¹(Ω)-level energy estimate to the L₂(Ω)-level. In the present nonlinear boundary dissipation case, the resulting energy estimate is then shown to fit into the general uniform stabilization strategy, first proposed in [La-Ta.1] in the case of wave equations with nonlinear (interior and) boundary dissipation.     

Symmetric Boundary Value Methods for Second Order Initial and Boundary Value Problems

We introduce symmetric Boundary Value Methods for the solution of second order initial and boundary value problems (in particular Hamiltonian problems). We study the conditioning of the methods and link it to the boundary loci of the roots of the associated characteristic polynomial. Some numerical tests are provided to assess their reliability. Dedicated to the memory of Professor Aldo Cossu     

Sobolev Spaces with only Trivial Isometries

We will give some conditions for Sobolev spaces on bounded Lipschitz domains to admit only trivial isometries.     

Maximal partial spreads in PG (3, q)

We transfer the whole geometry of PG(3, q) over a non-singular quadric Q_4,q of PG(4, q) mapping suitably PG(3, q) over Q_4,q. More precisely the points of PG(3, q) are the lines of Q_4,q; the lines of PG(3, q) are the tangent cones of Q_4,q and the reguli of the hyperbolic quadrics hyperplane section of Q_4,q. A plane of PG(3, q) is the set of lines of Q_4,q meeting a fixed line of Q_4,q. We remark that this representation is valid also for a projective space over any field K and we apply the above representation to construct maximal partial spreads in PG(3, q). For q even we get new cardinalities for For q odd the cardinalities are partially known.     

Eigenvalues of Two Parameter Polynomial Operator Pencils of Waveguide Type

The spectral structure of two parameter unbounded operator pencils of waveguide type is studied. Theorems on discreteness of the spectrum for a fixed parameter are proved. Variational principles for real eigenvalues in some parts of the root zones are established. In the case of n = 1 (quadratic pencils) domains containing the spectrum are described (see Fig. 1–3). Conditions in the definition of the pencils of waveguide type arise naturally from physical problems and each of them has a physical meaning. In particular a connection between the energetic stability condition and a perturbation problem for the coefficients is given.     

Hedging Uncertainty: Approximation Algorithms for Stochastic Optimization Problems

We study two-stage, finite-scenario stochastic versions of several combinatorial optimization problems, and provide nearly tight approximation algorithms for them. Our problems range from the graph-theoretic (shortest path, vertex cover, facility location) to set-theoretic (set cover, bin packing), and contain representatives with different approximation ratios. The approximation ratio of the stochastic variant of a typical problem is found to be of the same order of magnitude as its deterministic counterpart. Furthermore, we show that common techniques for designing approximation algorithms such as LP rounding, the primal-dual method, and the greedy algorithm, can be adapted to obtain these results.     

Optimal estimates on rotation number of almost periodic systems

In this paper, we will give some optimal estimates on the rotation number of the linear equation and that of the asymmetric equation: where p(t) and q(t) are almost periodic functions and These estimates are obtained by introducing some kind of new norms in the spaces of almost periodic functions. Supported by the National Natural Science Foundation of China (no. 10325102), TRAPOYT-M.O.E. of China (2001), and the National 973 Project of China (no. G1999075108). Received: April 6, 2004; revised: July 7, 2004     

Boundary Value Problems with Local Generalized Nevanlinna Functions in the Boundary Condition

For a class of abstract λ-dependent boundary value problems where a local variant of generalized Nevanlinna functions appears in the boundary condition, linearizations are constructed and their local spectral properties are investigated.