Analytical techniques for characterization of organic molecular assemblies in molecular electronics devices

The analytical techniques used for the physical characterization of organic molecular electronic-based devices are surveyed and discussed. These protocols include methods that are used to probe molecular assemblies such as single wavelength ellipsometry, water contact angle goniometry, cyclic voltammetry, infrared spectroscopy, and X-ray photoelectron spectroscopy, and methods used to measure charge transport properties of devices such as scanning tunneling microscopy, and inelastic electron tunneling spectroscopy. Examples from our laboratory and the literature are given for each of these analytical techniques.     

Filters in the partition lattice

Given a filter \(\Delta \) in the poset of compositions of n, we form the filter \(\Pi ^{*}_{\Delta }\) in the partition lattice. We determine all the reduced homology groups of the order complex of \(\Pi ^{*}_{\Delta }\) as \({\mathfrak S}_{n-1}\)-modules in terms of the reduced homology groups of the simplicial complex \(\Delta \) and in terms of Specht modules of border shapes. We also obtain the homotopy type of this order complex. These results generalize work of Calderbank–Hanlon–Robinson and Wachs on the d-divisible partition lattice. Our main theorem applies to a plethora of examples, including filters associated with integer knapsack partitions and filters generated by all partitions having block sizes a or b. We also obtain the reduced homology groups of the filter generated by all partitions having block sizes belonging to the arithmetic progression \(a, a + d, \ldots , a + (a-1) \cdot d\), extending work of Browdy.     

Ultrafine nanofibers fabricated from an arylene-ethynylene macrocyclic molecule using surface assisted self-assembly

Large area uniform nanofibers have been fabricated from a hexameric arylene-ethynylene macrocycle (1) through in situ self-assembly on a glass substrate during solvent evaporation. The fibril morphology is controlled by the solvophilic core of 1, in conjunction with the interfacial interactions between the side chains of 1 and the substrate.     

Full Spark Frames

Finite frame theory has a number of real-world applications. In applications like sparse signal processing, data transmission with robustness to erasures, and reconstruction without phase, there is a pressing need for deterministic constructions of frames with the following property: every size-M subcollection of the M-dimensional frame elements is a spanning set. Such frames are called full spark frames, and this paper provides new constructions using the discrete Fourier transform. Later, we prove that full spark Parseval frames are dense in the entire set of Parseval frames, meaning full spark frames are abundant, even if one imposes an additional tightness constraint. Finally, we prove that testing whether a given matrix is full spark is hard for NP under randomized polynomial-time reductions, indicating that deterministic full spark constructions are particularly significant because they guarantee a property which is otherwise difficult to check.     

Substituted potassium internal vinyltrifluoroborates: preparation and use in Suzuki–Miyaura cross-coupling reactions

A collection of substituted potassium internal vinyltrifluoroborates was prepared and utilized in palladium-catalyzed cross-coupling reactions with aryl and heteroaryl halides. The cross-coupling reactions proceeded readily with Pd(PPh₃)₄ as a catalyst in toluene/H₂O under reflux in the presence of Cs₂CO₃. The process is regiospecific with regard to the internal vinyltrifluoroborate starting material and tolerates a variety of functional groups. This unique collection of substituted internal vinyltrifluoroborates can be readily utilized in research pharmaceutical applications.     

Computing isogeny volcanoes of composite degree

Isogeny volcanoes are an interesting structure that have had several recent applications. An isogeny volcano is a connected component of a larger graph called a cordillera. In this paper, we further explore properties of how to compute volcanoes given that we have already computed one of a different degree. This allows us to compute volcanoes of composite degree more efficiently than a direct construction using modular polynomials.     

Solving equations with fractions: An analysis of prospective teachers’ solution pathways and errors

We analyzed the solution pathways and errors found in the written responses of 469 prospective teachers solving an equation containing fractions. The majority (332, or 70%) used an algebraic method; 141 of the 332 (42%) were correct, and 22% of the algebraic methods were abandoned before a solution was obtained. We identified the steps in the written solutions, determined which solution pathways led to the correct solution, and identified common errors in the solution pathways of respondents who incorrectly solved the equation. Respondents initially attempted different methods. The most common method was solving by using fractions, but the majority of respondents who solved by using mixed numbers were able to correctly solve the problem. Common errors related to fraction arithmetic and the distributive property. Nearly all of the abandoned pathways contained no errors, but ended with a step that likely would precede an operation with fractions. Our findings suggest that the ability to solve an arithmetic equation with no fractions was necessary, but not sufficient, to solve an arithmetic equation involving fractions, and that the problem of solving equations with fractions was more closely tied to one's difficulties with rational number arithmetic and less with one's understanding of algebra.     

Accuracy and Stability of Computing High-order Derivatives of Analytic Functions by Cauchy Integrals

High-order derivatives of analytic functions are expressible as Cauchy integrals over circular contours, which can very effectively be approximated, e.g., by trapezoidal sums. Whereas analytically each radius r up to the radius of convergence is equal, numerical stability strongly depends on r. We give a comprehensive study of this effect; in particular, we show that there is a unique radius that minimizes the loss of accuracy caused by round-off errors. For large classes of functions, though not for all, this radius actually gives about full accuracy; a remarkable fact that we explain by the theory of Hardy spaces, by the Wiman–Valiron and Levin–Pfluger theory of entire functions, and by the saddle-point method of asymptotic analysis. Many examples and nontrivial applications are discussed in detail.     

Derandomizing Restricted Isometries via the Legendre Symbol

The restricted isometry property (RIP) is an important matrix condition in compressed sensing, but the best matrix constructions to date use randomness. This paper leverages pseudorandom properties of the Legendre symbol to reduce the number of random bits in an RIP matrix with Bernoulli entries. In this regard, the Legendre symbol is not special—our main result naturally generalizes to any small-bias sample space. We also conjecture that no random bits are necessary for our Legendre symbol-based construction.