Kinetics of the Near Infrared Electrochemiluminescence from Metal Nanoclusters on Surface and in Solution

The electronics structures of some metal nanoclusters enable strong photoluminescence in the near infrared spectrum range. Activation of the luminescence via electrode reactions, rather than light source, i.e., electrochemiluminescence (ECL), has received growing interests due to the various potential benefits, but has been mostly limited to steady-state behaviors such as overall emission intensity and materials optimizations. Here, the ECL kinetics in representative experiments where nanoclusters as luminophores are either immobilized on the surface or free diffusing in solution were investigated based on classic theory. An analytical equation derived under a sequential mass transport limit regime quantitates the experimental ECL kinetics features in a wide range of conditions. Deconvolution of non-faradic charging current from redox current provides the threshold in time ranges for the analysis of ECL kinetics. The ECL kinetics profiles suggest that bimolecular or pseudo first order reactions limit the ECL generation immediately following the establishment of the applied potentials, while later ECL generation is governed by diffusion or mass transport displaying a Cottrell type decay over inverse square root time. Physical meanings of key parameters as defined in classic theorem are discussed in representative experimental systems for appropriate quantitation and evaluation of ECLs properties from different materials systems.

Laue’s theorem revisited: Energy–momentum tensors, symmetries, and the habitat of globally conserved quantities

The energy–momentum tensor for a particular matter component summarises its local energy–momentum distribution in terms of densities and current densities. We re-investigate under what conditions these local distributions can be integrated to meaningful global quantities. This leads us directly to a classic theorem by Max von Laue concerning integrals of components of the energy–momentum tensor, whose statement and proof we recall. In the first half of this paper, we do this within the realm of Special Relativity (SR) and in the traditional mathematical language using components with respect to affine charts, thereby focusing on the intended physical content and interpretation. In the second half, we show how to do all this in a proper differential-geometric fashion and on arbitrary spacetime manifolds, this time focusing on the group-theoretic and geometric hypotheses underlying these results. Based on this we give a proper geometric statement and proof of Laue’s theorem, which is shown to generalise from Minkowski space (which has the maximal number of isometries) to spacetimes with significantly less symmetries. This result, which seems to be new, not only generalizes but also clarifies the geometric content and hypotheses of Laue’s theorem. A series of three appendices lists our conventions and notation and summarises some of the conceptual and mathematical background needed in the main text.

Extension of Kelvin’s minimum energy theorem to incompressible fluid domains with open regions

Kelvin’s minimum energy theorem predicts that the irrotational motion of a homogeneously incompressible fluid in a simply connected region will carry less kinetic energy than any other profile that shares the same normal velocity conditions on the domain’s boundary. In this work, Kelvin’s analysis is extended to regions with boundaries on which the normal velocity requirements are relaxed. Given the ubiquity of practical configurations in which such boundaries exist, the question of whether Kelvin’s theorem continues to hold is one of significant interest. In reconstructing Kelvin’s proof, we find it useful to define a net rotational velocity as the difference between the generally rotational flow and the corresponding potential motion. In Kelvin’s classic theorem, the normal component of the net rotational velocity at all domain boundaries is zero. In contrast, the present analysis derives a sufficient condition for ensuring the validity of Kelvin’s theorem in a domain where the normal component of net rotational velocity at some or all of the boundaries is not zero. The corresponding criterion requires the evaluation of a simple surface integral over the boundary.

On the Number of 4-Edge Paths in Graphs With Given Edge Density

We investigate the number of 4-edge paths in graphs with a given number of vertices and edges, proving an asymptotically sharp upper bound on this number. The extremal construction is the quasi-star or the quasi-clique graph, depending on the edge density. An easy lower bound is also proved. This answer resembles the classic theorem of Ahlswede and Katona about the maximal number of 2-edge paths, and a recent theorem of Kenyon, Radin, Ren and Sadun about k-edge stars.

Block imbedding and interlacing results for normal matrices

A pair of matrices is said to be imbeddable precisely when one is an isometric projection of the other on a suitable subspace. The concept of imbedding has been the subject of extensive study. Particular emphasis has been placed on relating the spectra of the matrices involved, especially when both matrices are Hermitian or normal. In this paper, the notion of block imbedding is introduced and shown to be intimately connected to an extension of interlacing for eigenvalues of normal matrices. Thus, a generalization of a classic Theorem of K. Fan and G. Pall is obtained, which is then applied to yield bounds on the number of eigenvalues of a block imbeddable pair in a closed, convex set. Moreover, a wide class of normal matrices, for which block imbedding applies, is indicated. Finally, comments and links on the necessary imbedding conditions of D. Carlson and E.M. de Sa, and J.P. Queiro and A.L. Duarte are provided.

Rational approximations and quantum algorithms with postselection

We study the close connection between rational functions that approximate a given Boolean function, and quantum algorithms that compute the same function using postselection. We show that the minimal degree of the former equals (up to a factor of 2) the minimal query complexity of the latter. We give optimal (up to constant factors) quantum algorithms with postselection for the Majority function, slightly improving upon an earlier algorithm of Aaronson. Finally we show how Newman's classic theorem about low-degree rational approximation of the absolute-value function follows from these algorithms.

Ore-degree threshold for the square of a Hamiltonian cycle

Graph Theory International audience A classic theorem of Dirac from 1952 states that every graph with minimum degree at least n=2 contains a Hamiltonian cycle. In 1963, P´osa conjectured that every graph with minimum degree at least 2n=3 contains the square of a Hamiltonian cycle. In 1960, Ore relaxed the degree condition in the Dirac’s theorem by proving that every graph with deg(u) + deg(v) ≥ n for every uv =2 E(G) contains a Hamiltonian cycle. Recently, Chˆau proved an Ore-type version of P´osa’s conjecture for graphs on n ≥ n0 vertices using the regularity–blow-up method; consequently the n0 is very large (involving a tower function). Here we present another proof that avoids the use of the regularity lemma. Aside from the fact that our proof holds for much smaller n0, we believe that our method of proof will be of independent interest.

A Multipartite Version of the Hajnal–Szemerédi Theorem for Graphs and Hypergraphs

A perfect Kt-matching in a graph G is a spanning subgraph consisting of vertex-disjoint copies of Kt. A classic theorem of Hajnal and Szemerédi states that if G is a graph of order n with minimum degree δ(G) ≥ (t − 1)n/t and t|n, then G contains a perfect Kt-matching. Let G be a t-partite graph with vertex classes V1, …, Vt each of size n. We show that, for any γ > 0, if every vertex x ∈ Vi is joined to at least $\bigl ((t-1)/t + \gamma \bigr )n$ vertices of Vj for each j ≠ i, then G contains a perfect Kt-matching, provided n is large enough. Thus, we verify a conjecture of Fischer [6] asymptotically. Furthermore, we consider a generalization to hypergraphs in terms of the codegree.

AMENABLE ACTIONS, INVARIANT MEANS AND BOUNDED COHOMOLOGY

We show that amenability of an action of a discrete group on a compact space X is equivalent to vanishing of bounded cohomology for a class of Banach G-modules associated to the action, that can be viewed as analogs of continuous bundles of dual modules over the G-space X. In the case when the compact space is a point, our result reduces to a classic theorem of Johnson, characterising amenability of groups. In the case when the compact space is the Stone–Čech compactification of the group, we obtain a cohomological characterisation of exactness, or equivalently, Yu's property A for the group, answering a question of Higson.