Dimensions of irreducible modules for partition algebras and tensor power multiplicities for symmetric and alternating groups

The partition algebra \(\mathsf {P}_k(n)\) and the symmetric group \(\mathsf {S}_n\) are in Schur–Weyl duality on the k-fold tensor power \(\mathsf {M}_n^{\otimes k}\) of the permutation module \(\mathsf {M}_n\) of \(\mathsf {S}_n\), so there is a surjection \(\mathsf {P}_k(n) \rightarrow \mathsf {Z}_k(n) := \mathsf {End}_{\mathsf {S}_n}(\mathsf {M}_n^{\otimes k})\), which is an isomorphism when \(n \ge 2k\). We prove a dimension formula for the irreducible modules of the centralizer algebra \(\mathsf {Z}_k(n)\) in terms of Stirling numbers of the second kind. Via Schur–Weyl duality, these dimensions equal the multiplicities of the irreducible \(\mathsf {S}_n\)-modules in \(\mathsf {M}_n^{\otimes k}\). Our dimension expressions hold for any \(n \ge 1\) and \(k\ge 0\). Our methods are based on an analog of Frobenius reciprocity that we show holds for the centralizer algebras of arbitrary finite groups and their subgroups acting on a finite-dimensional module. This enables us to generalize the above result to various analogs of the partition algebra including the centralizer algebra for the alternating group acting on \(\mathsf {M}_n^{\otimes k}\) and the quasi-partition algebra corresponding to tensor powers of the reflection representation of \(\mathsf {S}_n\).     

Quantum integer-valued polynomials

We define a q-deformation of the classical ring of integer-valued polynomials which we call the ring of quantum integer-valued polynomials. We show that this ring has a remarkable combinatorial structure and enjoys many positivity properties: For instance, the structure constants for this ring with respect to its basis of q-binomial coefficient polynomials belong to \(\mathbb {N}[q]\). We then classify all maps from this ring into a field, extending a known classification in the classical case where \(q=1\).     

Constructing all self-adjoint matrices with prescribed spectrum and diagonal

The Schur–Horn Theorem states that there exists a self-adjoint matrix with a given spectrum and diagonal if and only if the spectrum majorizes the diagonal. Though the original proof of this result was nonconstructive, several constructive proofs have subsequently been found. Most of these constructive proofs rely on Givens rotations, and none have been shown to be able to produce every example of such a matrix. We introduce a new construction method that is able to do so. This method is based on recent advances in finite frame theory which show how to construct frames whose frame operator has a given prescribed spectrum and whose vectors have given prescribed lengths. This frame construction requires one to find a sequence of eigensteps, that is, a sequence of interlacing spectra that satisfy certain trace considerations. In this paper, we show how to explicitly construct every such sequence of eigensteps. Here, the key idea is to visualize eigenstep construction as iteratively building a staircase. This visualization leads to an algorithm, dubbed Top Kill, which produces a valid sequence of eigensteps whenever it is possible to do so. We then build on Top Kill to explicitly parametrize the set of all valid eigensteps. This yields an explicit method for constructing all self-adjoint matrices with a given spectrum and diagonal, and moreover all frames whose frame operator has a given spectrum and whose elements have given lengths.     

Melt-mediated coalescence of solution-deposited ZnO nanoparticles by excimer laser annealing for thin-film transistor fabrication

Nanoparticle solutions are considered promising for realizing low cost printable high performance flexible electronics. In this letter, excimer laser annealing (ELA) was employed to induce melting of solution-deposited ZnO nanoparticles and form electrically conductive porous films. The properties of the films were characterized by scanning electron microscopy, high-resolution transmission electron microscopy, DC conductance, and photoluminescence measurements. Thin-film field-effect transistors have been fabricated by ELA without the use of conventional vacuum or any high temperature thermal annealing processes. The transistors show n-type accumulation mode behavior with mobility greater than 0.1 cm²/V s and current on/off ratios of more than 10⁴. Optimization and control of the laser processing parameters minimized thermal impact on the substrate. This technique can be beneficial in the fabrication of metal oxide based electronics on heat sensitive flexible plastic substrates using low-cost, large-area solution processing combined with direct printing techniques.     

Photolysis of polymeric organosilicon systems. III. Synthesis and photochemical behavior of polysiloxanes containing phenyldisilanyl units

The synthesis and photochemical behavior of three types of polysiloxanes containing phenyldisilanyl units as chromophore have been reported. Irradiation of thin liquid films of the polymer (2) prepared from copolymerization of octamethylcyclotetrasiloxane and 1,3,5-triphenyl-1,3,5-tris(trimethylsilyl)cyclostrisiloxane (1) with a 500-W Xe-Hg lamp afforded transparent solid films which are insoluble in common organic solvents. Similar irradiation of thin liquid films of polysiloxane (8) containing both phenyldisilanyl units and vinyl groups resulted in formation of crosslinked solid films. When thick films of the polymers 2 and 8 were irradiated with a Xe-Hg lamp, crosslinking occurred only on the surface of the films, but inside remained as liquid. Irradiation of the films prepared from the polysiloxane involving the phenyldisilanyl units and butenyl groups, however, gave crosslinked solid films, whatever thickness of the films. The mechanism of photochemically induced crosslinking of the copolymers is discussed on the basis of the results obtained from the photolysis of 1,1-bis(trimethylsiloxy)-1-phenyl(trimethyl)disilane as a model compound.     

Directed supramolecular assembly of infinite 1-D M(II)-containing chains (M = Cu, Co, Ni) using structurally bifunctional ligands

The directed assembly of six different M(II) complexes (M = Cu, Co, and Ni) into infinite chains has been achieved by combining anionic chelating ligands (for controlling the coordination geometry) with bifunctional ligands containing a metal-coordinating pyridyl moiety and a self-complementary hydrogen-bonding moiety. Six crystal structures are presented, and in each case, the chelating acac ligand occupies the four equatorial coordination sites leaving room for the bifunctional ligand to coordinate in the axial positions. The supramolecular chemistry, which organizes the coordination complexes into the desired infinite 1-D chains, is driven by a combination of N-H...N and N-H...O hydrogen bonds.     

Polymeric conjugates for drug delivery

The field of polymer therapeutics has evolved over the past decade and has resulted in the development of polymer-drug conjugates with a wide variety of architectures and chemical properties. Whereas traditional non-degradable polymeric carriers such as poly(ethylene glycol) (PEG) and N-(2-hydroxypropyl methacrylamide) (HPMA) copolymers have been translated to use in the clinic, functionalized polymer-drug conjugates are increasingly being utilized to obtain biodegradable, stimuli-sensitive, and targeted systems in an attempt to further enhance localized drug delivery and ease of elimination. In addition, the study of conjugates bearing both therapeutic and diagnostic agents has resulted in multifunctional carriers with the potential to both "see and treat" patients. In this paper, the rational design of polymer-drug conjugates will be discussed followed by a review of different classes of conjugates currently under investigation. The design and chemistry used for the synthesis of various conjugates will be presented with additional comments on their potential applications and current developmental status.