The maximum multiplicity of the largest k-th eigenvalue in a matrix whose graph is acyclic or unicyclic

Given a graph $G$ we are interested in studying the symmetric matrices associated to $G$ with a fixed number of negative eigenvalues. For this class of matrices we focus on the maximum possible nullity. For trees this parameter has already been studied and plenty of applications are known. In this work we derive a formula for the maximum nullity and completely describe its behavior as a function of the number of negative eigenvalues. In addition, we also carefully describe the matrices associated with trees that attain this maximum nullity. The analysis is then extended to the more general class of unicyclic graphs. Further our work is applied to re-describing all possible partial inertias associated with trees, and is employed to study an instance of the inverse eigenvalue problem for certain trees.     

A Buchwald–Hartwig Protocol to Enable Rapid Linker Exploration of Cereblon E3-Ligase PROTACs**

A palladium-catalysed Buchwald–Hartwig amination for lenalidomide-derived aryl bromides was optimised using high throughput experimentation (HTE). The substrate scope of the optimised conditions was evaluated for a range of alkyl- and aryl- amines and functionalised aryl bromides. The methodology allows access to new cereblon-based bifunctional proteolysis targeting chimeras with a reduced step count and improved yields.     

Force-Free States, Relative Strain, and Soft Elasticity in Nematic Elastomers

The remarkable ability of nematic elastomers to exhibit large deformations under small applied forces is known as soft elasticity. The recently proposed neo-classical free-energy density for nematic elastomers, derived by molecular-statistical arguments, has been used to model soft elasticity. In particular, the neo-classical free-energy density allows for a continuous spectrum of equilibria, which implies that deformations may occur in the complete absence of force and energy cost. Here we study the notion of force-free states in the context of a continuum theory of nematic elastomers that allows for isotropy, uniaxiality, and biaxiality of the polymer microstructure. Within that theory, the neo-classical free-energy density is an example of a free-energy density function that depends on the deformation gradient only through a nonlinear strain measure associated with the deformation of the polymer microstructure relative to the macroscopic continuum. Among the force-free states for a nematic elastomer described by the neo-classical free energy density, there is, in particular, a continuous spectrum of states parameterized by a pair of tensors that allows for soft deformations. In these force-free states the polymer microstructure is material in the sense that it stretches and rotates with the macroscopic continuum. Limitations of and possible improvements upon the neo-classical model are also discussed. This revised version was published online in July 2006 with corrections to the Cover Date.     

Exploring the Effect of Ligand Structural Isomerism in Langmuir–Blodgett Films of Chiral Luminescent EuIII Self‐Assemblies

Here we have investigated the influence of the antenna group position on both the formation of chiral amphiphilic Eu^III‐based self‐assemblies in CH₃CN solution and, on the ability to form monolayers on the surface of quartz substrates using the Langmuir–Blodgett technique, by changing from the 1‐naphthyl ( 2(R) , 2(S) ) to the 2‐naphthyl ( 1(R) , 1(S) ) position. The evaluation of binding constants of the self‐ assemblies in CH₃CN solution was achieved using conventional techniques such as UV/Visible and luminescence spectroscopies along with more specific circular dichroism (CD) spectroscopy. The binding constants obtained for EuL , EuL₂ and EuL₃ species in the case of 2‐naphthyl derivatives were comparable to those obtained for 1‐naphthyl derivatives. The analysis of the changes in the CD spectra of 1(R) and 1(S) upon addition of Eu^III not only allowed us to evaluate the values of the binding constants but the resulting recalculated spectra may also be used as fingerprints for assignment of the chiral self‐assembly species formed in solution. The obtained monolayers were predominantly formed from EuL₃ (≈85 %) with the minor species present in ≈15 % EuL₂ .     

A Multifunctional Surface That Simultaneously Balances Hydrophilic Enzyme Catalysis and Hydrophobic Water Repellency

The compatibility of multiple functions at a single interface is difficult to achieve, but is even more challenging when the functions directly counteract one another. This study provides insight into the creation of a simultaneously multifunctional surface formed by balancing two orthogonal functions; water repellency and enzyme catalysis. A partially fluorinated thiol is used to impart bulk hydrophobicity on the surface, and an N‐hydroxysuccinimide ester‐terminated thiol provides a specific anchoring sites for the covalent enzyme attachment. Different ratios of the two thiols are mixed together to form amphiphilic self‐assembled monolayers, which are characterized with polarization‐modulation infrared reflection–absorption spectroscopy and contact angle goniometry. The enzyme activity is measured by a fluorescence assay. With the results collected here, specific surface compositions are identified at which the orthogonal functions of water repellency and enzyme catalysis are balanced and exist simultaneously. An understanding of how to effectively balance orthogonal functions at surfaces can be extended to a number of higher‐scale applications.     

Rapid Aqueous Borohydride Reduction of Carbonyls Under Sealed-Tube Microwave Conditions

Ketones and aldehydes are conveniently and rapidly reduced to the corresponding alcohols in good yields using sodium borohydride under sealed-tube microwave conditions in either 95% ethanol or water. In purely aqueous systems, highly aliphatic substrates are sluggish, but this can be overcome by introducing sodium dodecyl sulfate (SDS) at the critical micelle concentration. With a 2:1 substrate/borohydride ratio and a reaction temperature of 100 °C, reduction is typically complete within 1 min in 95% ethanol and 5 min in water/SDS. The methodology is well suited for parallel and combinatorial synthetic approaches.     

Invariant circles for the piecewise linear standard map

We investigate invariant circles for a one-parameter family of piecewise linear twist homeomorphisms of the annulus. We show that invariant circles of all types and rotation numbers occur and we classify them into families. We compute parameter ranges in which there are no invariant circles.     

Fierz bilinear formulation of the Maxwell–Dirac equations and symmetry reductions

We study the Maxwell–Dirac equations in a manifestly gauge invariant presentation using only the spinor bilinear scalar and pseudoscalar densities, and the vector and pseudovector currents, together with their quadratic Fierz relations. The internally produced vector potential is expressed via algebraic manipulation of the Dirac equation, as a rational function of the Fierz bilinears and first derivatives (valid on the support of the scalar density), which allows a gauge invariant vector potential to be defined. This leads to a Fierz bilinear formulation of the Maxwell tensor and of the Maxwell–Dirac equations, without any reference to gauge dependent quantities. We show how demanding invariance of tensor fields under the action of a fixed (but arbitrary) Lie subgroup of the Poincaré group leads to symmetry reduced equations. The procedure is illustrated, and the reduced equations worked out explicitly for standard spherical and cylindrical cases, which are coupled third order nonlinear PDEs. Spherical symmetry necessitates the existence of magnetic monopoles, which do not affect the coupled Maxwell–Dirac system due to magnetic terms cancelling. In this paper we do not take up numerical computations. As a demonstration of the power of our approach, we also work out the symmetry reduced equations for two distinct classes of dimension 4 one-parameter families of Poincaré subgroups, one splitting and one non-splitting. The splitting class yields no solutions, whereas for the non-splitting class we find a family of formal exact solutions in closed form.     

Semisimple strata for p-adic classical groups

Let F₀ be a non-archimedean local field, of residual characteristic different from 2, and let G be a unitary, symplectic or orthogonal group defined over F₀. In this paper, we prove some fundamental results towards the classification of the representations of G via types [8]. In particular, we show that any positive level supercuspidal representation of G contains a semisimple skew stratum, that is, a special character of a certain compact open subgroup of G. The intertwining of such a stratum has been calculated in [19].