Two-dimensional Banach spaces with polynomial numerical index zero

We study two-dimensional Banach spaces with polynomial numerical indices equal to zero.

Unconditional constants and polynomial inequalities

If P is a polynomial on Rm of degree at most n then we define the polynomial |P|. Now if B is a convex compact set in Rm, we define the norm ||P||B of P as the maximum of P on B, and then we investigate the inequality || |P| ||B

MODELED DIFFUSION-CONTROLLED RESPONSE AND RECOVERY BEHAVIOR OF A NAKED OPTICAL FILM SENSOR WITH A HYPERBOLIC-TYPE RESPONSE TO ANALYTE CONCENTRATION

The diffusion-controlled response and recovery behaviour of a naked optical film sensor (i.e., with no protective membrane) with a hyperbolic-type response [i.e., S0/S = (1 + Kc), where S is the measured value of the absorbance or luminescence intensity of one form of the sensor dye in the presence of the analyte, S0 is the observed value of S in the absence of analyte and K is a constant] to changes in analyte concentration, c, in a system under test is approximated using a simple model, and described more accurately using a numerical model; in both models it is assumed that the system under test represents an infinite reservoir. Each model predicts the variations in the response and recovery times of such an optical sensor, as a function of the final external analyte concentration, the film thickness (I) and the analyte diffusion coefficient (D). From an observed signal versus time profile for a naked optical film sensor it is shown how values for K and D/I2 can be extracted using the numerical model. Both models provide a qualitative description of the often cited asymmetric nature of the response and recovery for hyperbolic-type response naked optical film sensors. It is envisaged that the models will help in the interpretation of the response and recovery behaviour exhibited by many naked optical film sensors and might be especially apposite when the analyte is a gas.

Polynomial Functions and the Riesz Interpolation Property

Building on a proof by D. Handelman of a generalisation of an example due to L. Fuchs, we show that the space of real-valued polynomials on a non-empty set X of reals has the Riesz Interpolation Property if and only if X is bounded.

Finite domination and Novikov rings: Laurent polynomial rings in two variables

Let C be a bounded cochain complex of finitely generatedfree modules over the Laurent polynomial ring L = R[x, x−1, y, y−1].The complex C is called R-finitely dominated if it is homotopy equivalentover R to a bounded complex of finitely generated projective Rmodules.Our main result characterises R-finitely dominated complexesin terms of Novikov cohomology: C is R-finitely dominated if andonly if eight complexes derived from C are acyclic; these complexes areC ⊗L R[[x, y]][(xy)−1] and C ⊗L R[x, x−1][[y]][y−1], and their variants obtainedby swapping x and y, and replacing either indeterminate by its inverse.

Updating Credal Networks is Approximable in Polynomial Time

Credal networks relax the precise probability requirement of Bayesian networks, enabling a richer representation of uncertainty in the form of closed convex sets of probability measures. The increase in expressiveness comes at the expense of higher computational costs. In this paper, we present a new variable elimination algorithm for exactly computing posterior inferences in extensively specified credal networks, which is empirically shown to outperform a state-of-the-art algorithm. The algorithm is then turned into a provably good approximation scheme, that is, a procedure that for any input is guaranteed to return a solution not worse than the optimum by a given factor. Remarkably, we show that when the networks have bounded treewidth and bounded number of states per variable the approximation algorithm runs in time polynomial in the input size and in the inverse of the error factor, thus being the first known fully polynomial-time approximation scheme for inference in credal networks.

Hyperbolic Polaritonic Crystals Based on Nanostructured Nanorod Metamaterials

Surface plasmon polaritons usually exist on a few suitable plasmonic materials; however, nanostructured plasmonic metamaterials allow a much broader range of optical properties to be designed. Here, bottom-up and top-down nanostructuring are combined, creating hyperbolic metamaterial-based photonic crystals termed hyperbolic polaritonic crystals, allowing free-space access to the high spatial frequency modes supported by these metamaterials.

Finite domination and Novikov rings. Laurent polynomial rings in several variables

We present a homological characterisation of those chain complexes of modules over a Laurent polynomial ring in several indeterminates which are finitely dominated over the ground ring (that is, are a retract up to homotopy of a bounded complex of finitely generated free modules). The main tools, which we develop in the paper, are a non-standard totalisation construction for multi-complexes based on truncated products, and a high-dimensional mapping torus construction employing a theory of cubical diagrams that commute up to specified coherent homotopies.