A study is made of electron tunneling in semiconductor heterostructures having a complex dispersion law. A generalized Fabry–Perot approach is used to describe tunneling across the barrier. Mixing of electron states at the heterojunctions is responsible for the asymmetric resonant structure of the transmission which is characterized by a resonance–antiresonance pair. The resonance corresponds t...

We adapt Berlekamp’s light bulb switching game to finite projective plans and finite affine planes, then find the worst arrangement of lit bulbs for planes of even and odd orders. The results are then extended from the planes to spaces of higher dimension. 1 Berlekamp’s Switching Game The original game is simple enough. 100 light bulbs arranged in 10 rows and 10 columns. Each column and row has...

Let M be a matroid on E ∪ {`}, where ` 6∈ E is a distinguished element of M . The `-port of M is the set P = {P : P ⊆ E with P ∪ {`} a circuit of M}. Let A be the P-E incidence matrix. Let U2,4 be the uniform matroid on four elements of rank two, F7 be the Fano matroid, F ∗ 7 be the dual of F7, and F 7 be the unique series extension of F7. In this paper, we prove that the system Ax ≥ 1, x ≥ 0 i...

It is shown that every n vertex triple system with every pair of vertices lying in at least (1/2 + o(1))n triples contains a copy of the Fano plane. The constant 1/2 is sharp. This is the first triple system for which such a (nonzero) constant is determined. Let X be a finite set. An r-graph F with vertex set X is a family of r-element subsets of X. These subsets are called edges, and X is writ...

The automorphism groups of the 27 lines on the smooth cubic surface or the 28 bitangents to the general quartic plane curve are well-known to be closely related to the Weyl groups of E6 and E7. We show how classical subconfigurations of lines, such as double-sixes, triple systems or Steiner sets, are easily constructed from certain models of the exceptional Lie algebras. For e7 and e8 we are le...

For various k-uniform hypergraphs F , we give tight lower bounds on the number of copies of F in a k-uniform hypergraph with a prescribed number of vertices and edges. These are the first such results for hypergraphs, and extend earlier theorems of various authors who proved that there is one copy of F . A sample result is the following: Füredi-Simonovits [11] and independently KeevashSudakov [...

A binary clutter is cycling if its packing and covering linear program have integral optimal solutions for all Eulerian edge capacities. We prove that the clutter of odd stwalks of a signed graph is cycling if and only if it does not contain as a minor the clutter of odd circuits of K5 nor the clutter of lines of the Fano matroid. Corollaries of this result include, of many, the characterizatio...

We describe a method that we believe may be foundational for a comprehensive theory of generalised Turán problems. The cornerstone of our approach is a quasirandom counting lemma for quasirandom hypergraphs, which extends the standard counting lemma by not only counting copies of a particular configuration but also showing that these copies are evenly distributed. We demonstrate the power of th...

Query algorithms are used to compute Boolean functions. The definition of the function is known, but input is hidden in a black box. The aim is to compute the function value using as few queries to the black box as possible. As in other computational models, different kinds of query algorithms are possible: deterministic, probabilistic, as well as nondeterministic. In this paper, we present a n...

We review some recently established connections between the mathematics of black hole entropy in string theory and that of multipartite entanglement in quantum information theory. In the case of N = 2 black holes and the entanglement of three qubits, the quartic [SL(2)] 3 invariant, Cayley's hyperdeterminant, provides both the black hole entropy and the measure of tripartite entanglement. In th...