On-line reaction monitoring by extractive electrospray ionisation

The design and development of a novel extractive electrospray ionisation (EESI) device for on-line reaction monitoring is described. The EESI apparatus uses a secondary, grounded nebuliser to produce an analyte aerosol and a Venturi pump is then used to transfer a sample of the aerosol to an electrospray source where it is ionised. The EESI apparatus was then tested with a variety of small, organic molecules to assess sensitivity, linearity and dynamic range. The performance of the technique will depend on the mass spectrometer used for the experiments; in the configurations used here it has a usable dynamic range of around 3.5 orders of magnitude with a linear range of around 2.5 orders of magnitude and is capable of analysing species present down to low μg/mL with signal-to-noise ratio greater than 2.5. The use of EESI for reaction monitoring was validated using a series of mock reaction mixtures and then used to monitor the base hydrolysis of ethyl salicylate to salicylic acid.     

Proper analytic free maps

This paper concerns analytic free maps. These maps are free analogs of classical analytic functions in several complex variables, and are defined in terms of non-commuting variables amongst which there are no relations - they are free variables. Analytic free maps include vector-valued polynomials in free (non-commuting) variables and form a canonical class of mappings from one non-commutative domain D in say g variables to another non-commutative domain in variables. As a natural extension of the usual notion, an analytic free map is proper if it maps the boundary of D into the boundary of . Assuming that both domains contain 0, we show that if is a proper analytic free map, and f(0)=0, then f is one-to-one. Moreover, if also , then f is invertible and f−1 is also an analytic free map. These conclusions on the map f are the strongest possible without additional assumptions on the domains D and .     

Stillman's question for exterior algebras and Herzog's conjecture on Betti numbers of syzygy modules

Let K be a field of characteristic 0 and consider exterior algebras of finite dimensional K-vector spaces. In this short paper we exhibit principal quadric ideals in a family whose Castelnuovo–Mumford regularity is unbounded. This negatively answers the analogue of Stillman's Question for exterior algebras posed by I. Peeva. We show that, via the Bernstein–Gel'fand–Gel'fand correspondence, these examples also yields counterexamples to a conjecture of J. Herzog on the Betti numbers in the linear strand of syzygy modules over polynomial rings.     

Free actions on handlebodies

The equivalence (or weak equivalence) classes of orientation-preserving free actions of a finite group G on an orientable three-dimensional handlebody of genus g?1 can be enumerated in terms of sets of generators of G. They correspond to the equivalence classes of generating n-vectors of elements of G, where n=1+(g−1)/|G|, under Nielsen equivalence (or weak Nielsen equivalence). For Abelian and dihedral G, this allows a complete determination of the equivalence and weak equivalence classes of actions for all genera. Additional information is obtained for other classes of groups. For all G, there is only one equivalence class of actions on the genus g handlebody if g is at least 1+?(G)|G|, where ?(G) is the maximal length of a chain of subgroups of G. There is a stabilization process that sends an equivalence class of actions to an equivalence class of actions on a higher genus, and some results about its effects are obtained.     

Analytic model for surface-induced molecular weight segregation in thermoplastic composites

An analytic model based on a random-walk formulation for absorbing barriers is presented. The range of applicability of the model is established from results derived from Monte Carlo simulations. The model takes into account the influences of neutral, repulsive, and attractive polymer-surface interactions on the spatial variation of species concentrations in the vicinity of a surface. The model predicts that the lower molecular weight species of a polydisperse mixture will accumulate in the vicinity of a surface to create an interphase region. The extent of this surface-perturbed interphase region is on the order of three times the radius of gyration of the longest species of the mixture. The degree of molecular weight segregation is dependent upon the breadth of the molecular weight distribution. © 1994 John Wiley & Sons. Inc.     

The influence of draw-induced structural changes on the activity of a semicrystalline polymer-supported catalyst

The activity of a solid, polymer-supported catalyst (a semicrystalline polyethylene film containing grafted, sulfonated styrene) was shown to be altered by cold drawing. The catalytic activity was measured by a test reaction, the dehydration of isopropyl alcohol to give propylene. Catalytic reaction rates were measured with variously drawn films clamped in a differential flow reactor operated at 100°C and 1 atm. The catalytic activity increased with the elongation of the polymer up to a draw ratio of 2.5; the activity decreased upon further drawing. The drawn films were characterized by x-ray diffraction, dynamic mechanical measurements, electron microscopy, birefringence, and density measurements. Since no evidence was found for the formation of surface sites by creation of microcracks, the changes in activity are attributed to modifications in the polymer structure induced by drawing. The kinetics of the catalytic reaction and the data giving percentage crystallinity and crystalline and amorphous orientation factors suggest that, in the undrawn polymer, the catalytically active ? SO₃H groups form a hydrogen-bonded network, which is excluded from the crystalline regions. Initial elongation partially breaks up the network, allowing more ? SO₃H groups to bond to alcohol and become catalytically engaged. Elongation beyond a draw ratio of 2.5 leads to a separation of ? SO₃H groups greater than that required for the formation of the reaction intermediate, which involves the alcohol hydrogen bonded to several ? SO₃H groups. The combined results of the catalytic kinetics experiments and structural characterizations imply that the preliminary deformation response of the semicrystalline polymer occurs predominantly within the noncrystalline regions and is accommodated by the rigid slip and tilting of crystal lamellae. The hydrogen bonding among the ? SO₃H groups in the noncrystalline regions hinders lamellar breakup and suppresses the formation of highly aligned fibrillar morphologies.     

Matrix Representations for Positive Noncommutative Polynomials

In real semialgebraic geometry it is common to represent a polynomial q which is positive on a region R as a weighted sum of squares. Serious obstructions arise when q is not strictly positive on the region R. Here we are concerned with noncommutative polynomials and obtaining a representation for them which is valid even when strict positivity fails. Specifically, we treat a ``symmetric' polynomial q(x, h) in noncommuting variables, {x<sub>1</sub>, . . . , } and {h<sub>1</sub>, . . . , } for which q(X,H) is positive semidefinite whenever are tuples of selfadjoint matrices with ||X<sub>j</sub>|| ≤ 1 but H<sub>j</sub> unconstrained. The representation we obtain is a Gram representation in the variables h where P<sub>q</sub> is a symmetric matrix whose entries are noncommutative polynomials only in x and V is a ``vector' whose entries are polynomials in both x and h. We show that one can choose P_q such that the matrix P_q(X) is positive semidefinite for all ||X_j|| ≤ 1. The representation covers sum of square results ([Am. Math. (to appear); Linear Algebra Appl. 326 (2001), 193–203; Non commutative Sums of Squares, preprint]) when g_x = 0. Also it allows for arbitrary degree in h, rather than degree two, in the main result of [Matrix Inequalities: A Symbolic Procedure to Determine Convexity Automatically to appear IOET July 2003] when restricted to x-domains of the type ||X_j|| ≤ 1. Partially supported by NSF, DARPA and Ford Motor Co. Partially supported by NSF grant DMS-0140112 Partially supported by NSF grant DMS-0100367     

An energetics approach to the analysis of molecular motions in polymeric solids

The feasibility of certain modes of intra-and intermolecular motion in polymeric solids are examined in terms of interaction energetics and geometrical constraints. Energy methods for determining equilibrium chain structures and stable deformation structures are reviewed. A systematic approach to the specification of explicit geometrical constraints in condensed phase polymeric systems is developed by matrix techniques. Combination rules are derived which provide straightforward means of constructing deformation structures for oriented polymers.

Applications of these general methods are illustrated for paraffinic systems. Structure-energy maps, based on empirically justified potential functions, are presented for assemblies of paraff inic chains. The potential energy surface in the vicinity of the minima associated with the orthorhombic crystal structure is described by a Taylor's series in the unit cell parameters. Applications of this approximate representation of the energy surface to the prediction of the moduli, thermal expansion coefficients, and bulk compressibility of crystalline paraffins and polyethylene are discussed. Detailed energy maps for coupled rotations of rigid chains and coupled translation-rotation motions are developed. Feasible modes of motion are identified as low-energy structural conversion paths; activation energies are determined from the saddle points in the structure-energy map. The temperature dependence of barriers to motion is estimated by introducing empirical functions which relate average chain spacings to temperature.     

The matricial relaxation of a linear matrix inequality

Given linear matrix inequalities (LMIs) L ₁ and L ₂ it is natural to ask:

(Q₁) when does one dominate the other, that is, does ${L_1(X) \succeq 0}$ imply ${L_2(X) \succeq 0}$ ?

(Q₂) when are they mutually dominant, that is, when do they have the same solution set?

The matrix cube problem of Ben-Tal and Nemirovski (SIAM J Optim 12:811–833, 2002) is an example of LMI domination. Hence such problems can be NP-hard. This paper describes a natural relaxation of an LMI, based on substituting matrices for the variables x _j . With this relaxation, the domination questions (Q₁) and (Q₂) have elegant answers, indeed reduce to constructible semidefinite programs. As an example, to test the strength of this relaxation we specialize it to the matrix cube problem and obtain essentially the relaxation given in Ben-Tal and Nemirovski (SIAM J Optim 12:811–833, 2002). Thus our relaxation could be viewed as generalizing it. Assume there is an X such that L ₁(X) and L ₂(X) are both positive definite, and suppose the positivity domain of L ₁ is bounded. For our “matrix variable” relaxation a positive answer to (Q₁) is equivalent to the existence of matrices V _j such that $$\begin{array}{ll}L_2(x) = V_1^{*} L_1(x) V_1 + \cdots + V_\mu^{*} L_1(x) V_{\mu}. \quad \quad \quad ({\rm A}_1)\end{array}$$ As for (Q₂) we show that L ₁ and L ₂ are mutually dominant if and only if, up to certain redundancies described in the paper, L ₁ and L ₂ are unitarily equivalent. Algebraic certificates for positivity, such as (A₁) for linear polynomials, are typically called Positivstellensätze. The paper goes on to derive a Putinar-type Positivstellensatz for polynomials with a cleaner and more powerful conclusion under the stronger hypothesis of positivity on an underlying bounded domain of the form ${ \{X \mid L(X)\succeq0\}. }$ An observation at the core of the paper is that the relaxed LMI domination problem is equivalent to a classical problem. Namely, the problem of determining if a linear map τ from a subspace of matrices to a matrix algebra is “completely positive”. Complete positivity is one of the main techniques of modern operator theory and the theory of operator algebras. On one hand it provides tools for studying LMIs and on the other hand, since completely positive maps are not so far from representations and generally are more tractable than their merely positive counterparts, the theory of completely positive maps provides perspective on the difficulties in solving LMI domination problems.